Research ArticleImpact-Time-Control Guidance Law forMissile with Time-Varying Velocity

Jun Zhou Yang Wang and Bin Zhao

Institute of Precision Guidance and Control Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Yang Wang 2013100280mailnwpueducn

Received 25 January 2016 Accepted 5 June 2016

Academic Editor Andrzej Swierniak

Copyright copy 2016 Jun Zhou et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problem of impact-time-control guidance (ITCG) for the homingmissile with time-varying velocity is addressed First a novelITCG law is proposed based on the integral sliding mode control (ISMC) methodThen a salvo attack algorithm is designed basedon the proposed guidance lawThe performances of the conventional ITCG laws strongly depend on the accuracy of the estimatedtime-to-go (TTG) However the accurate estimated TTG can be obtained only if the missile velocity is constant The conventionalITCG laws were designed under the assumption that the missile velocity is constantThe most attractive feature of this work is thatthe newly proposed ITCG law relaxes the constant velocity assumption which only needs the variation range of themissile velocityFinally the numerical simulation demonstrates the effectiveness of the proposed method

1 Introduction

Proportional navigation guidance law (PNGL) [1ndash4] has beenwidely used in the area of homing guidance which canachieve excellent performance in the presence of a nonma-neuvering target However in recent years with the devel-opment of defense systems such as space defense antimissilesystem [5] electronic countermeasure system (ECMS) [6]and close-in weapon system (CIWS) [7] the survivability ofattack missile with conventional guidance scheme has beenintimidated Fortunately most defense systems have ldquoone-to-onerdquo feature thus the salvo attack of multiple missiles canbe one of the most effective countermeasures for missilesagainst the threats of defense systems The salvo attack canbe realized if all missiles hit the target simultaneously this iscalled impact-time-control guidance (ITCG)

In 2006 Jeon et al [8] first proposed an ITCG law torealize the salvo attack The solution is a combination of thePNGL and the feedback of the impact time error In [9] acooperative-proportional navigation guidance (CPNG) lawwas designed to realize the salvo attack by decreasing thetime-to-go (TTG) variance of missiles In [10] a bias pro-portional navigation guidance (BPNG) law was developedin which the desired impact time and angle were achieved

simultaneously In [11] the authors proposed an optimalguidance law for controlling the impact time and angleIn [12] an ITCG law based on a two-step control strategywas proposed In [13] the authors proposed a polynomialguidance law to control the impact time and angle

Actually the above-mentioned ITCG laws in [8ndash13] weredesigned based on linearized-engagement-dynamics How-ever the linearized-engagement-dynamic-based method canachieve high precision only if the missilersquos flight-path angle issmall It is well known that the sliding mode control (SMC)method is powerful in controlling nonlinear system [14ndash17]So far some SMC-based ITCG laws have been developedin [18ndash20] And they were derived based on the nonlinearengagement dynamics In [18] a guidance law based on thesecond-order SMCmethod and the backstepping schemewasdeveloped In order to satisfy the impact time constraint onecoefficient of the guidance law was searched for by using theoff-line simplex algorithm In [19] a SMC-based guidance lawwas developed tomeet the requirement of ITCG inwhich thesliding mode was defined as the combination of the line-of-sight (LOS) angle rate and the impact time error Howeverif the initial LOS angle is zero the guidance law proposedin [19] will generate zero-acceleration command during thewhole guidance process To overcome this problem in 2015

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7951923 14 pageshttpdxdoiorg10115520167951923

2 Mathematical Problems in Engineering

Cho et al [20] proposed an ITCG law based on a nonsingularsliding mode Compared with the sliding mode in [19] thesliding mode in [20] was defined as only the impact timeerror

Note that the aforementioned ITCG laws in [8ndash13 18ndash20]all need the TTG However the TTG cannot be measureddirectly To overcome this problem [8ndash13 18ndash20] all designestimation algorithm to estimate the TTG These estimationresults in [8ndash13 18ndash20] are obtained under the assumptionthat themissile velocity is invariant However the assumptionis too restrictive for many cases In practical missile systemsthe missile velocity is affected inevitably by various forcessuch as the driving force of engine and the aerodynamicforces When the constant velocity assumption is invalid theestimation error of TTG may be large Thus these guidancelaws in [8ndash13 18ndash20]maynotworkwell in practical situations

In this paper a novel ITCG law is proposed for missilewith time-varying velocity based on integral sliding modecontrol (ISMC) The main contributions of this paper are thefollowing

(1) As far as we know the ITCG law for the missile withtime-varying velocity is achieved for the first timeCompared with the conventional ITCG laws in [8ndash13 18ndash20] the constant velocity assumption is relaxedin this paper In otherwords the proposed law ismorepractical than the conventional ITCG laws

(2) The proposed guidance law is derived based onthe nonlinear engagement dynamic rather than thelinearized dynamics used in [8ndash13]

The remaining parts of this paper are as follows InSection 2 the problems of existing ITCG laws are formulatedThe main results are presented in Section 3 In Section 3an ITCG law based on ISMC method is proposed and thepermissible set of the desired impact times is discussed Sec-tion 4 shows a salvo attack algorithm based on the proposedguidance law In Section 5 the numerical simulations verifythe effectiveness of the proposed method in comparison withthemethods presented in [8 20] In Section 6 the conclusionsof the whole paper are presented

Notations The following notations will be used in this paper119905 denotes the elapsed time after launching the missile 119909

0

denotes 119909 at initial time 1199050 and 119909(119905

119896) denotes 119909 at time 119905

119896

2 Problem Formulation

As shown in Figure 1119874119909 represents the horizontal direction119874119910 represents the longitudinal direction 119879 represents thestationary target 119872 represents the missile 120582 represents theLOS angle 119877 represents the relative distance between themissile and the target 120574 represents the heading angle 119886

119872

represents the missile acceleration and 119881 represents themissile velocity From Figure 1 the following equalities canbe established [20]

= 119881 cos 120574

= 119881 sin 120574

(1)

T

V

M

bull

x

O y

R

120574

120582

aM

Figure 1 Missile-target engagement geometry

where (119909 119910) are the coordinates of missile The equations forrelative motion can be expressed as [20]

= minus119881 cos (120582 minus 120574) = minus119881 cos120601

119877 = minus119881 sin (120582 minus 120574) = minus119881 sin120601

119886119872

= 119881

=1

21198621199091205881198812 119878119872

119898

(2)

where 119862119909denotes the aerodynamic coefficients 120588 denotes

the air density 119878119872denotes the reference area 119898 denotes the

missile mass and 119886119872

denotes the lateral acceleration It isassumed that air resistance plays a major role in the changeof the missile velocity thus we have

119862119909lt 0 (3)

Note that if condition (4) is satisfied then the objective ofITCG is realized [8ndash13 18ndash20]

119905119891= 119905119889

119891 (4)

where 119905119891denotes the impact time and 119905

119889

119891denotes the desired

impact time 119905119891can be rewritten as

119905119891= 119905 + 119905go (5)

where 119905 is the elapsed time after launching the missile and 119905gois the time-to-go (TTG)

Combining (4) with (5) condition (4) is equivalent to thefollowing condition

119905119889

119891= 119905 + 119905go (6)

Mathematical Problems in Engineering 3

In order to satisfy (6) the conventional ITCG laws in [8ndash1318ndash20] need 119905go However 119905go can not be measured directlyThese literatures design estimation algorithm to estimate theTTGThe performances of these ITCG laws strongly dependon the accuracy of the estimated TTG

The length of the trajectory of missile 119878 is given by

119878 = int

119905119891

1199050

119881119889119905 = int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 (7)

If 119881 is constant ( = 0) then we have

119905119891=

1

119881int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 + 1199050 (8)

Combining (5) with (8) we have

119905go =1

119881int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 + 1199050minus 119905 (9)

Based on (9) the estimated TTG go is obtained in [8ndash13 18ndash20] For example in [8] the estimated TTG is

go =119877

119881(1 +

(120574 minus 120582)2

10) (10)

In [9] the estimated TTG is

go =119877

119881(1 +

(120574 minus 120582)2

2 (2119873 minus 1)) (11)

where119873 is a constant parameterHowever if119881 is time-varying (8) cannot be derived from

(7) In other word the estimation error of TTG may be verylarge if we still use the estimation methods in [8ndash13 18ndash20] And the requirement of the ITCG cannot be satisfiedwith their guidance laws (in Section 5 of this paper thesimulation result also demonstrates that the performance ofexisting ITCG laws is poor when the missile velocity is time-varying) This motivates the research topic of this paper thatis designing a new ITCG law for missile with time-varyingvelocity to satisfy condition (4)

The following assumptions should be assumed to be validthroughout this paper

(A1) Missile velocity 119881 satisfies 119881min le 119881 le 119881max where119881min and 119881max are the known constants

(A2) is available (see [21ndash23])

Assumption (A1) implies that we only need the bound ofthe time-varying velocity Obviously assumption (A1) is arelaxed version of the constant velocity assumption used in[8ndash13 18ndash20] Because can be measured by accelerometerassumption (A2) is a commonly used assumption for thedesign of guidance law (see [21ndash23])

3 New Impact-Time-Control GuidanceLaw Based on ISMC

31 Design of ITCG Law Condition (4) is equivalent to thefollowing conditions

119877 (119905) gt 0 forall1199050le 119905 lt 119905

119889

119891

119877 (119905119889

119891) = 0

(12)

Conditions (12) mean that the missile can attack the targetonly if 119905 = 119905

119889

119891 In this section a novel ITCG law is developed

to satisfy (12)A state variable 119911

1is defined as

1199111= 119877 minus 119881min (119905

119889

119891minus 119905) (13)

Differentiating (13) gives

1= + 119881min (14)

Let 1= 1199112 and the guidance system (2) can be rewritten as

1= 1199112

2= 119891 (119911) + 119892 (119911) 119906

=1

21198621199091205881198812 119878119872

119898

(15)

where 119906 = 119886119872 119891(119911) = minus cos (120582 minus 120574) + 119881 sin (120582 minus 120574) and

119892(119911) = minus sin (120582 minus 120574)A novel integral sliding mode (ISM) is constructed as

119904ISM = 1199112minus 1199112(1199050) + int

119905

1199050

(

119887(119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2)119889120591 (16)

where 119886 119887 and 119888 are the guidance parameters which are allconstants Then using the following guidance law

119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

(17)

where 120573 is a small positive constant and sgn (sdot) denotes thesignum function we have the following results

4 Mathematical Problems in Engineering

Theorem 1 Consider system (15) with the guidance law (17)provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the guidance parametersin (17) If 119905119889

119891satisfies

1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

119894119891 1199022gt 1199021

1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

119894119891 1199021gt 1199022

(18)

1199021ge 2 119902

2ge 2 119902

1= 1199022 and |119881

0cos (120582

0minus 1205740)119881119898119894119899

| lt 1 thenthe following conditions will be satisfied

119877 (119905) gt 0 forall1199050le 119905 lt 119905

119889

119891

119877 (119905119889

119891) = 0

(19)

Proof Differentiating (16) with respect to time yields

119904ISM = 2+

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (20)

Combining (15) with (20) we have

119904ISM = 119891 (119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (21)

Construct a Lyapunov function 1198811as

1198811=

1

21199042

ISM (22)

Differentiating (22) and then using (21) and (17) yield

1= 119904ISM 119904ISM

= 119904ISM (119891(119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2)

= minus119904ISM (120573 sgn (119904ISM)) lt 0 when 119904ISM = 0

(23)

Note that 119904ISM(1199050) = 0 and one can imply that

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (24)

As 119904ISM(119905) remains as zero we get

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (25)

The following equation can be obtained by combining (20)with (25)

119886 (119905119889

119891minus 119905)2

1+ 119887 (119905119889

119891minus 119905)

1+ 1198881199111= 0

forall1199050le 119905 le 119905

119889

119891

(26)

Provided that 1199021and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 if 1199021

= 1199022 1199021ge 2 and 119902

2ge 2 the solutions

to the differential equation (26) can be easily obtained as

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (27)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(28)

where 1198881and 1198882are constant

Substituting (13) into (27) and substituting (14) into (28)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(29)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(30)

Substituting the initial values 1198770and

0into (29) and (30)

we get

1198770minus 119881min (119905

119889

119891minus 1199050) = 1198881(119905119889

119891minus 1199050)1199021

+ 1198882(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(0+ 119881min) (119905

119889

119891minus 1199050) = minus119888

11199021(119905119889

119891minus 1199050)1199021

minus 11988821199022(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(31)

From (31) we have

1198881=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199021

1198882=

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199022

(32)

According to (18) |1198810cos (120582

0minus 1205740)119881min| lt 1 119902

1ge 2 and

1199022ge 2 we have

1198881lt 0

1198882lt 0

(33)

Mathematical Problems in Engineering 5

From (32) (33) and 119905119889

119891minus 1199050gt 0 one can imply that

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

lt 0

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

lt 0

(34)

Substituting (32) into (29) we have

119877 minus 119881min (119905119889

119891minus 119905)

=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199021

(119905119889

119891minus 1199050)

1199021

+

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199022

(119905119889

119891minus 1199050)

1199022 forall1199050le 119905 le 119905

119889

119891

(35)

Considering (34) 0 le (119905119889

119891minus119905)(119905

119889

119891minus1199050) le 1 119902

1ge 2 and 119902

2ge 2

we get (36) from (35) Consider

119877 minus 119881min (119905119889

119891minus 119905)

ge

(0+ (1 minus 119902

2) 119881min) (119905

119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

+

(minus0minus (1 minus 119902

1) 119881min) (119905

119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

= 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

minus 119881min (119905119889

119891minus 119905)

forall1199050le 119905 le 119905

119889

119891

(36)

Then we get

119877 ge 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

gt 0 forall1199050le 119905 lt 119905

119889

119891 (37)

Moreover according to (29) we get

119877 (119905119889

119891) = 0 (38)

The proof is finished

Remark 2 From Theorem 1 it is clear that the proposedguidance law (17) can guarantee eachmissile attacks the targetonly when 119905 = 119905

119889

119891

Remark 3 Equation (17) shows that the control law contains(119905119889

119891minus 119905) in the denominator which may bring the singularity

when 119905 = 119905119889

119891 Fortunately the singularity can be eliminated

combining (27) with (28) yields

119887 (119905119889

119891minus 119905)

1+ 1198881199111

119886 (119905119889

119891minus 119905)

2= (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

forall1199050le 119905 le 119905

119889

119891

(39)

From (39) we can know that the singularity brought by (119905119889119891minus119905)

is eliminated as long as 1199021ge 2 and 119902

2ge 2

32 Nonsingular Impact-Time-Control Guidance Law If119892(119911) = 0 (17) shows that the guidance command 119906 rarr infinwhich means it is impossible to realize the guidance lawIn what follows Theorem 4 shows the condition which canensure 119892(119911) = 0 during the guidance process

Theorem 4 Consider system (15) with the guidance law (17)Provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the parameters in (17)if 119905119889119891satisfies (18) 119902

1ge 2 119902

2ge 2 119902

1= 1199022 sin (120582

0minus 1205740) = 0

and |1198810cos (120582

0minus 1205740)119881119898119894119899

| lt 1 minus 120595 lt 1 then 119892(119911) satisfies thefollowing condition

1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (40)

where 120576 is a sufficiently small time and

120579 = min radic1 minus 1205951003816100381610038161003816sin (120582

0minus 1205740)1003816100381610038161003816 (41)

120595 = max(119881119898119894119899

119881

)

2

(minus1 + 120595)2 (42)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Proof From the demonstration ofTheorem 4 we known thatthe guidance law (17) can ensure the state variables of system(15) satisfy the following equations

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (43)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(44)

where 1198881and 1198882are constants and 119902

1and 1199022are the solutions

to the equation 1198861199022minus (119886 + 119887)119902 + 119888 = 0

6 Mathematical Problems in Engineering

Substituting (13) into (43) and substituting (14) into (44)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(45)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(46)

According to (18) 1199021ge 2 and 119902

2ge 2 we can get the following

results from the demonstration of Theorem 4

1198881lt 0

1198882lt 0

(47)

Combining (47) with (46) yields

+ 119881min gt 0 forall1199050le 119905 lt 119905

119889

119891 (48)

Substituting (2) into (48) yields

cos (120582 minus 120574) lt119881min119881

forall1199050le 119905 lt 119905

119889

119891 (49)

Differentiating (44) gives

1= 11988811199021(1199021minus 1) (119905

119889

119891minus 119905)(1199021minus2)

+ 11988821199022(1199022minus 1) (119905

119889

119891minus 119905)(1199022minus2)

forall1199050le 119905 lt 119905

119889

119891

(50)

Then considering 1199021ge 2 119902

2ge 2 1198881lt 0 and 119888

2lt 0 we have

1lt 0 forall119905

0le 119905 lt 119905

119889

119891 (51)

Equation (51) means that 1reduces monotonically with

increasing 119905 and we get

1lt 1(1199050) forall119905

0lt 119905 lt 119905

119889

119891 (52)

From (52) we get

cos (120582 minus 120574) gt1198810cos (120582

0minus 1205740)

119881 forall1199050lt 119905 lt 119905

119889

119891 (53)

Combining (49) and (53) yields

1198810cos (120582

0minus 1205740)

119881lt cos (120582 minus 120574) lt

119881min119881

forall1199050lt 119905 lt 119905

119889

119891

(54)

From (3) it can be known that lt 0 Then we have

119881min119881

le119881min

119881

lt 1 forall1199050lt 119905 le 119905

119889

119891minus 120576 (55)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Since 119881min119881 le 119881min119881 lt 1 and 1198810cos (120582

0minus 1205740)119881 ge

minus|1198810cos (120582

0minus 1205740)119881min| gt minus1 + 120595 gt minus1 we have

minus1 + 120595 lt cos (120582 minus 120574) le119881min

119881

forall1199050lt 119905 le 119905

119889

119891minus 120576 (56)

Let 120595 = max (119881min119881)2 (minus1 + 120595)

2 then we have

cos2 (120582 minus 120574) le 120595 forall1199050lt 119905 le 119905

119889

119891minus 120576 (57)

It is clear that 0 lt 120595 lt 1 Since | sin (120582 minus 120574)| =

radic1 minus cos2 (120582 minus 120574) we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge radic1 minus 120595 forall119905

0lt 119905 le 119905

119889

119891minus 120576 (58)

Let 120579 = minradic1 minus 120595 | sin (1205820minus 1205740)| we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 forall119905

0le 119905 le 119905

119889

119891minus 120576 (59)

Since | sin(1205820minus 1205740)| gt 0 andradic1 minus 120595 gt 0 we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (60)

Then we have1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (61)

where 120576 is a sufficiently small time The proof is finished

Remark 5 In the practical engineering the available missileacceleration is bounded

|119906| le 119906max (62)

where 119906max is the maximum available acceleration of missileand is determined by the performance of missile Let 119906 =

minus119892minus1(119911)(119891(119911) + (119887(119905

119889

119891minus 119905)1199112+ 1198881199111)119886(119905119889

119891minus 119905)2+ 120573 sgn (119904ISM))

To satisfy (62) in the practical engineering the guidance law(17) must be modified as

119906 =

119906max if 119906 gt 119906max

119906 if |119906| le 119906max

minus119906max if 119906 lt minus119906max

(63)

In the most ideal case (119906max = infin) we can expect the relativedistance119877(119905119889

119891) = 0 However 119906max is bounded in engineering

Fortunately from [8] it can be known that the objective ofITCG can be realized if the impact time error is smaller than01 sThus if the following conditions (63) (64) and (65) canbe satisfied simultaneously the objective of ITCG also can beobtained

119877 gt 0 forall1199050le 119905 le 119905

119889

119891minus 120576 (64)

10038161003816100381610038161003816119877 (119905119889

119891minus 120576)

10038161003816100381610038161003816le 119877set (65)

|119906| le 119906max forall1199050le 119905 le 119905

119889

119891minus 120576 (66)

Mathematical Problems in Engineering 7

where 120576 is a sufficiently small time and 120576 lt 01 119877set is a smalldistance that can guarantee the precision of attack (eg ifthe target is ship 119877set = 1m can guarantee the precision ofattack)

From (2) we have

=minus119881 sin (120582 minus 120574)

119877=

119881

119877119892 (119911) (67)

Substituting (39) into guidance law (17) and considering (67)yield

119906 = minus119892minus1

(119911) (119891 (119911) + (1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = minus119892minus1

(119911)

sdot (minus cos (120582 minus 120574) +1198812 sin (120582 minus 120574) 119892 (119911)

119877+ (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = cos (120582 minus 120574)

119892 (119911)minus 1198812sin (120582 minus 120574)

119877

minus

((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

119892 (119911) forall1199050le 119905 le 119905

119889

119891minus 120576

(68)

From (68) we have

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

119892 (119911)

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

1003816100381610038161003816100381610038161003816100381610038161003816119892 (119911)

1003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

1198812sin (120582 minus 120574)

119877

100381610038161003816100381610038161003816100381610038161003816

forall1199050le 119905 le 119905

119889

119891minus 120576

(69)

Combining (37) (61) and (69) yields

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

120579

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)

1003816100381610038161003816100381610038161003816100381610038161003816100381612057910038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050)

1198770sdot (119905119889

119891minus 119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

(10038161003816100381610038161003816 cos (120582 minus 120574)

10038161003816100381610038161003816+1003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886)

1003816100381610038161003816(119905119889

119891minus 119905)1199021minus2

+10038161003816100381610038161198882119888119886 minus 119888

21199022119887119886

1003816100381610038161003816(119905119889

119891minus 1199050)1199022minus2

+ 120573)

1003816100381610038161003816100381612057910038161003816100381610038161003816

+

100381610038161003816100381610038161198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050) 1198770

10038161003816100381610038161003816

|120576| forall1199050le 119905 le 119905

119889

119891minus 120576

(70)

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Cho et al [20] proposed an ITCG law based on a nonsingularsliding mode Compared with the sliding mode in [19] thesliding mode in [20] was defined as only the impact timeerror

Note that the aforementioned ITCG laws in [8ndash13 18ndash20]all need the TTG However the TTG cannot be measureddirectly To overcome this problem [8ndash13 18ndash20] all designestimation algorithm to estimate the TTG These estimationresults in [8ndash13 18ndash20] are obtained under the assumptionthat themissile velocity is invariant However the assumptionis too restrictive for many cases In practical missile systemsthe missile velocity is affected inevitably by various forcessuch as the driving force of engine and the aerodynamicforces When the constant velocity assumption is invalid theestimation error of TTG may be large Thus these guidancelaws in [8ndash13 18ndash20]maynotworkwell in practical situations

In this paper a novel ITCG law is proposed for missilewith time-varying velocity based on integral sliding modecontrol (ISMC) The main contributions of this paper are thefollowing

(1) As far as we know the ITCG law for the missile withtime-varying velocity is achieved for the first timeCompared with the conventional ITCG laws in [8ndash13 18ndash20] the constant velocity assumption is relaxedin this paper In otherwords the proposed law ismorepractical than the conventional ITCG laws

(2) The proposed guidance law is derived based onthe nonlinear engagement dynamic rather than thelinearized dynamics used in [8ndash13]

The remaining parts of this paper are as follows InSection 2 the problems of existing ITCG laws are formulatedThe main results are presented in Section 3 In Section 3an ITCG law based on ISMC method is proposed and thepermissible set of the desired impact times is discussed Sec-tion 4 shows a salvo attack algorithm based on the proposedguidance law In Section 5 the numerical simulations verifythe effectiveness of the proposed method in comparison withthemethods presented in [8 20] In Section 6 the conclusionsof the whole paper are presented

Notations The following notations will be used in this paper119905 denotes the elapsed time after launching the missile 119909

0

denotes 119909 at initial time 1199050 and 119909(119905

119896) denotes 119909 at time 119905

119896

2 Problem Formulation

As shown in Figure 1119874119909 represents the horizontal direction119874119910 represents the longitudinal direction 119879 represents thestationary target 119872 represents the missile 120582 represents theLOS angle 119877 represents the relative distance between themissile and the target 120574 represents the heading angle 119886

119872

represents the missile acceleration and 119881 represents themissile velocity From Figure 1 the following equalities canbe established [20]

= 119881 cos 120574

= 119881 sin 120574

(1)

T

V

M

bull

x

O y

R

120574

120582

aM

Figure 1 Missile-target engagement geometry

where (119909 119910) are the coordinates of missile The equations forrelative motion can be expressed as [20]

= minus119881 cos (120582 minus 120574) = minus119881 cos120601

119877 = minus119881 sin (120582 minus 120574) = minus119881 sin120601

119886119872

= 119881

=1

21198621199091205881198812 119878119872

119898

(2)

where 119862119909denotes the aerodynamic coefficients 120588 denotes

the air density 119878119872denotes the reference area 119898 denotes the

missile mass and 119886119872

denotes the lateral acceleration It isassumed that air resistance plays a major role in the changeof the missile velocity thus we have

119862119909lt 0 (3)

Note that if condition (4) is satisfied then the objective ofITCG is realized [8ndash13 18ndash20]

119905119891= 119905119889

119891 (4)

where 119905119891denotes the impact time and 119905

119889

119891denotes the desired

impact time 119905119891can be rewritten as

119905119891= 119905 + 119905go (5)

where 119905 is the elapsed time after launching the missile and 119905gois the time-to-go (TTG)

Combining (4) with (5) condition (4) is equivalent to thefollowing condition

119905119889

119891= 119905 + 119905go (6)

Mathematical Problems in Engineering 3

In order to satisfy (6) the conventional ITCG laws in [8ndash1318ndash20] need 119905go However 119905go can not be measured directlyThese literatures design estimation algorithm to estimate theTTGThe performances of these ITCG laws strongly dependon the accuracy of the estimated TTG

The length of the trajectory of missile 119878 is given by

119878 = int

119905119891

1199050

119881119889119905 = int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 (7)

If 119881 is constant ( = 0) then we have

119905119891=

1

119881int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 + 1199050 (8)

Combining (5) with (8) we have

119905go =1

119881int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 + 1199050minus 119905 (9)

Based on (9) the estimated TTG go is obtained in [8ndash13 18ndash20] For example in [8] the estimated TTG is

go =119877

119881(1 +

(120574 minus 120582)2

10) (10)

In [9] the estimated TTG is

go =119877

119881(1 +

(120574 minus 120582)2

2 (2119873 minus 1)) (11)

where119873 is a constant parameterHowever if119881 is time-varying (8) cannot be derived from

(7) In other word the estimation error of TTG may be verylarge if we still use the estimation methods in [8ndash13 18ndash20] And the requirement of the ITCG cannot be satisfiedwith their guidance laws (in Section 5 of this paper thesimulation result also demonstrates that the performance ofexisting ITCG laws is poor when the missile velocity is time-varying) This motivates the research topic of this paper thatis designing a new ITCG law for missile with time-varyingvelocity to satisfy condition (4)

The following assumptions should be assumed to be validthroughout this paper

(A1) Missile velocity 119881 satisfies 119881min le 119881 le 119881max where119881min and 119881max are the known constants

(A2) is available (see [21ndash23])

Assumption (A1) implies that we only need the bound ofthe time-varying velocity Obviously assumption (A1) is arelaxed version of the constant velocity assumption used in[8ndash13 18ndash20] Because can be measured by accelerometerassumption (A2) is a commonly used assumption for thedesign of guidance law (see [21ndash23])

3 New Impact-Time-Control GuidanceLaw Based on ISMC

31 Design of ITCG Law Condition (4) is equivalent to thefollowing conditions

119877 (119905) gt 0 forall1199050le 119905 lt 119905

119889

119891

119877 (119905119889

119891) = 0

(12)

Conditions (12) mean that the missile can attack the targetonly if 119905 = 119905

119889

119891 In this section a novel ITCG law is developed

to satisfy (12)A state variable 119911

1is defined as

1199111= 119877 minus 119881min (119905

119889

119891minus 119905) (13)

Differentiating (13) gives

1= + 119881min (14)

Let 1= 1199112 and the guidance system (2) can be rewritten as

1= 1199112

2= 119891 (119911) + 119892 (119911) 119906

=1

21198621199091205881198812 119878119872

119898

(15)

where 119906 = 119886119872 119891(119911) = minus cos (120582 minus 120574) + 119881 sin (120582 minus 120574) and

119892(119911) = minus sin (120582 minus 120574)A novel integral sliding mode (ISM) is constructed as

119904ISM = 1199112minus 1199112(1199050) + int

119905

1199050

(

119887(119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2)119889120591 (16)

where 119886 119887 and 119888 are the guidance parameters which are allconstants Then using the following guidance law

119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

(17)

where 120573 is a small positive constant and sgn (sdot) denotes thesignum function we have the following results

4 Mathematical Problems in Engineering

Theorem 1 Consider system (15) with the guidance law (17)provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the guidance parametersin (17) If 119905119889

119891satisfies

1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

119894119891 1199022gt 1199021

1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

119894119891 1199021gt 1199022

(18)

1199021ge 2 119902

2ge 2 119902

1= 1199022 and |119881

0cos (120582

0minus 1205740)119881119898119894119899

| lt 1 thenthe following conditions will be satisfied

119877 (119905) gt 0 forall1199050le 119905 lt 119905

119889

119891

119877 (119905119889

119891) = 0

(19)

Proof Differentiating (16) with respect to time yields

119904ISM = 2+

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (20)

Combining (15) with (20) we have

119904ISM = 119891 (119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (21)

Construct a Lyapunov function 1198811as

1198811=

1

21199042

ISM (22)

Differentiating (22) and then using (21) and (17) yield

1= 119904ISM 119904ISM

= 119904ISM (119891(119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2)

= minus119904ISM (120573 sgn (119904ISM)) lt 0 when 119904ISM = 0

(23)

Note that 119904ISM(1199050) = 0 and one can imply that

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (24)

As 119904ISM(119905) remains as zero we get

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (25)

The following equation can be obtained by combining (20)with (25)

119886 (119905119889

119891minus 119905)2

1+ 119887 (119905119889

119891minus 119905)

1+ 1198881199111= 0

forall1199050le 119905 le 119905

119889

119891

(26)

Provided that 1199021and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 if 1199021

= 1199022 1199021ge 2 and 119902

2ge 2 the solutions

to the differential equation (26) can be easily obtained as

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (27)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(28)

where 1198881and 1198882are constant

Substituting (13) into (27) and substituting (14) into (28)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(29)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(30)

Substituting the initial values 1198770and

0into (29) and (30)

we get

1198770minus 119881min (119905

119889

119891minus 1199050) = 1198881(119905119889

119891minus 1199050)1199021

+ 1198882(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(0+ 119881min) (119905

119889

119891minus 1199050) = minus119888

11199021(119905119889

119891minus 1199050)1199021

minus 11988821199022(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(31)

From (31) we have

1198881=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199021

1198882=

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199022

(32)

According to (18) |1198810cos (120582

0minus 1205740)119881min| lt 1 119902

1ge 2 and

1199022ge 2 we have

1198881lt 0

1198882lt 0

(33)

Mathematical Problems in Engineering 5

From (32) (33) and 119905119889

119891minus 1199050gt 0 one can imply that

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

lt 0

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

lt 0

(34)

Substituting (32) into (29) we have

119877 minus 119881min (119905119889

119891minus 119905)

=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199021

(119905119889

119891minus 1199050)

1199021

+

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199022

(119905119889

119891minus 1199050)

1199022 forall1199050le 119905 le 119905

119889

119891

(35)

Considering (34) 0 le (119905119889

119891minus119905)(119905

119889

119891minus1199050) le 1 119902

1ge 2 and 119902

2ge 2

we get (36) from (35) Consider

119877 minus 119881min (119905119889

119891minus 119905)

ge

(0+ (1 minus 119902

2) 119881min) (119905

119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

+

(minus0minus (1 minus 119902

1) 119881min) (119905

119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

= 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

minus 119881min (119905119889

119891minus 119905)

forall1199050le 119905 le 119905

119889

119891

(36)

Then we get

119877 ge 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

gt 0 forall1199050le 119905 lt 119905

119889

119891 (37)

Moreover according to (29) we get

119877 (119905119889

119891) = 0 (38)

The proof is finished

Remark 2 From Theorem 1 it is clear that the proposedguidance law (17) can guarantee eachmissile attacks the targetonly when 119905 = 119905

119889

119891

Remark 3 Equation (17) shows that the control law contains(119905119889

119891minus 119905) in the denominator which may bring the singularity

when 119905 = 119905119889

119891 Fortunately the singularity can be eliminated

combining (27) with (28) yields

119887 (119905119889

119891minus 119905)

1+ 1198881199111

119886 (119905119889

119891minus 119905)

2= (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

forall1199050le 119905 le 119905

119889

119891

(39)

From (39) we can know that the singularity brought by (119905119889119891minus119905)

is eliminated as long as 1199021ge 2 and 119902

2ge 2

32 Nonsingular Impact-Time-Control Guidance Law If119892(119911) = 0 (17) shows that the guidance command 119906 rarr infinwhich means it is impossible to realize the guidance lawIn what follows Theorem 4 shows the condition which canensure 119892(119911) = 0 during the guidance process

Theorem 4 Consider system (15) with the guidance law (17)Provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the parameters in (17)if 119905119889119891satisfies (18) 119902

1ge 2 119902

2ge 2 119902

1= 1199022 sin (120582

0minus 1205740) = 0

and |1198810cos (120582

0minus 1205740)119881119898119894119899

| lt 1 minus 120595 lt 1 then 119892(119911) satisfies thefollowing condition

1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (40)

where 120576 is a sufficiently small time and

120579 = min radic1 minus 1205951003816100381610038161003816sin (120582

0minus 1205740)1003816100381610038161003816 (41)

120595 = max(119881119898119894119899

119881

)

2

(minus1 + 120595)2 (42)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Proof From the demonstration ofTheorem 4 we known thatthe guidance law (17) can ensure the state variables of system(15) satisfy the following equations

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (43)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(44)

where 1198881and 1198882are constants and 119902

1and 1199022are the solutions

to the equation 1198861199022minus (119886 + 119887)119902 + 119888 = 0

6 Mathematical Problems in Engineering

Substituting (13) into (43) and substituting (14) into (44)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(45)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(46)

According to (18) 1199021ge 2 and 119902

2ge 2 we can get the following

results from the demonstration of Theorem 4

1198881lt 0

1198882lt 0

(47)

Combining (47) with (46) yields

+ 119881min gt 0 forall1199050le 119905 lt 119905

119889

119891 (48)

Substituting (2) into (48) yields

cos (120582 minus 120574) lt119881min119881

forall1199050le 119905 lt 119905

119889

119891 (49)

Differentiating (44) gives

1= 11988811199021(1199021minus 1) (119905

119889

119891minus 119905)(1199021minus2)

+ 11988821199022(1199022minus 1) (119905

119889

119891minus 119905)(1199022minus2)

forall1199050le 119905 lt 119905

119889

119891

(50)

Then considering 1199021ge 2 119902

2ge 2 1198881lt 0 and 119888

2lt 0 we have

1lt 0 forall119905

0le 119905 lt 119905

119889

119891 (51)

Equation (51) means that 1reduces monotonically with

increasing 119905 and we get

1lt 1(1199050) forall119905

0lt 119905 lt 119905

119889

119891 (52)

From (52) we get

cos (120582 minus 120574) gt1198810cos (120582

0minus 1205740)

119881 forall1199050lt 119905 lt 119905

119889

119891 (53)

Combining (49) and (53) yields

1198810cos (120582

0minus 1205740)

119881lt cos (120582 minus 120574) lt

119881min119881

forall1199050lt 119905 lt 119905

119889

119891

(54)

From (3) it can be known that lt 0 Then we have

119881min119881

le119881min

119881

lt 1 forall1199050lt 119905 le 119905

119889

119891minus 120576 (55)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Since 119881min119881 le 119881min119881 lt 1 and 1198810cos (120582

0minus 1205740)119881 ge

minus|1198810cos (120582

0minus 1205740)119881min| gt minus1 + 120595 gt minus1 we have

minus1 + 120595 lt cos (120582 minus 120574) le119881min

119881

forall1199050lt 119905 le 119905

119889

119891minus 120576 (56)

Let 120595 = max (119881min119881)2 (minus1 + 120595)

2 then we have

cos2 (120582 minus 120574) le 120595 forall1199050lt 119905 le 119905

119889

119891minus 120576 (57)

It is clear that 0 lt 120595 lt 1 Since | sin (120582 minus 120574)| =

radic1 minus cos2 (120582 minus 120574) we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge radic1 minus 120595 forall119905

0lt 119905 le 119905

119889

119891minus 120576 (58)

Let 120579 = minradic1 minus 120595 | sin (1205820minus 1205740)| we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 forall119905

0le 119905 le 119905

119889

119891minus 120576 (59)

Since | sin(1205820minus 1205740)| gt 0 andradic1 minus 120595 gt 0 we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (60)

Then we have1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (61)

where 120576 is a sufficiently small time The proof is finished

Remark 5 In the practical engineering the available missileacceleration is bounded

|119906| le 119906max (62)

where 119906max is the maximum available acceleration of missileand is determined by the performance of missile Let 119906 =

minus119892minus1(119911)(119891(119911) + (119887(119905

119889

119891minus 119905)1199112+ 1198881199111)119886(119905119889

119891minus 119905)2+ 120573 sgn (119904ISM))

To satisfy (62) in the practical engineering the guidance law(17) must be modified as

119906 =

119906max if 119906 gt 119906max

119906 if |119906| le 119906max

minus119906max if 119906 lt minus119906max

(63)

In the most ideal case (119906max = infin) we can expect the relativedistance119877(119905119889

119891) = 0 However 119906max is bounded in engineering

Fortunately from [8] it can be known that the objective ofITCG can be realized if the impact time error is smaller than01 sThus if the following conditions (63) (64) and (65) canbe satisfied simultaneously the objective of ITCG also can beobtained

119877 gt 0 forall1199050le 119905 le 119905

119889

119891minus 120576 (64)

10038161003816100381610038161003816119877 (119905119889

119891minus 120576)

10038161003816100381610038161003816le 119877set (65)

|119906| le 119906max forall1199050le 119905 le 119905

119889

119891minus 120576 (66)

Mathematical Problems in Engineering 7

where 120576 is a sufficiently small time and 120576 lt 01 119877set is a smalldistance that can guarantee the precision of attack (eg ifthe target is ship 119877set = 1m can guarantee the precision ofattack)

From (2) we have

=minus119881 sin (120582 minus 120574)

119877=

119881

119877119892 (119911) (67)

Substituting (39) into guidance law (17) and considering (67)yield

119906 = minus119892minus1

(119911) (119891 (119911) + (1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = minus119892minus1

(119911)

sdot (minus cos (120582 minus 120574) +1198812 sin (120582 minus 120574) 119892 (119911)

119877+ (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = cos (120582 minus 120574)

119892 (119911)minus 1198812sin (120582 minus 120574)

119877

minus

((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

119892 (119911) forall1199050le 119905 le 119905

119889

119891minus 120576

(68)

From (68) we have

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

119892 (119911)

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

1003816100381610038161003816100381610038161003816100381610038161003816119892 (119911)

1003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

1198812sin (120582 minus 120574)

119877

100381610038161003816100381610038161003816100381610038161003816

forall1199050le 119905 le 119905

119889

119891minus 120576

(69)

Combining (37) (61) and (69) yields

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

120579

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)

1003816100381610038161003816100381610038161003816100381610038161003816100381612057910038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050)

1198770sdot (119905119889

119891minus 119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

(10038161003816100381610038161003816 cos (120582 minus 120574)

10038161003816100381610038161003816+1003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886)

1003816100381610038161003816(119905119889

119891minus 119905)1199021minus2

+10038161003816100381610038161198882119888119886 minus 119888

21199022119887119886

1003816100381610038161003816(119905119889

119891minus 1199050)1199022minus2

+ 120573)

1003816100381610038161003816100381612057910038161003816100381610038161003816

+

100381610038161003816100381610038161198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050) 1198770

10038161003816100381610038161003816

|120576| forall1199050le 119905 le 119905

119889

119891minus 120576

(70)

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

In order to satisfy (6) the conventional ITCG laws in [8ndash1318ndash20] need 119905go However 119905go can not be measured directlyThese literatures design estimation algorithm to estimate theTTGThe performances of these ITCG laws strongly dependon the accuracy of the estimated TTG

The length of the trajectory of missile 119878 is given by

119878 = int

119905119891

1199050

119881119889119905 = int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 (7)

If 119881 is constant ( = 0) then we have

119905119891=

1

119881int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 + 1199050 (8)

Combining (5) with (8) we have

119905go =1

119881int

119909(119905119891)

1199090

radic1 + (119889119910

119889119909)

2

119889119909 + 1199050minus 119905 (9)

Based on (9) the estimated TTG go is obtained in [8ndash13 18ndash20] For example in [8] the estimated TTG is

go =119877

119881(1 +

(120574 minus 120582)2

10) (10)

In [9] the estimated TTG is

go =119877

119881(1 +

(120574 minus 120582)2

2 (2119873 minus 1)) (11)

where119873 is a constant parameterHowever if119881 is time-varying (8) cannot be derived from

(7) In other word the estimation error of TTG may be verylarge if we still use the estimation methods in [8ndash13 18ndash20] And the requirement of the ITCG cannot be satisfiedwith their guidance laws (in Section 5 of this paper thesimulation result also demonstrates that the performance ofexisting ITCG laws is poor when the missile velocity is time-varying) This motivates the research topic of this paper thatis designing a new ITCG law for missile with time-varyingvelocity to satisfy condition (4)

The following assumptions should be assumed to be validthroughout this paper

(A1) Missile velocity 119881 satisfies 119881min le 119881 le 119881max where119881min and 119881max are the known constants

(A2) is available (see [21ndash23])

Assumption (A1) implies that we only need the bound ofthe time-varying velocity Obviously assumption (A1) is arelaxed version of the constant velocity assumption used in[8ndash13 18ndash20] Because can be measured by accelerometerassumption (A2) is a commonly used assumption for thedesign of guidance law (see [21ndash23])

3 New Impact-Time-Control GuidanceLaw Based on ISMC

31 Design of ITCG Law Condition (4) is equivalent to thefollowing conditions

119877 (119905) gt 0 forall1199050le 119905 lt 119905

119889

119891

119877 (119905119889

119891) = 0

(12)

Conditions (12) mean that the missile can attack the targetonly if 119905 = 119905

119889

119891 In this section a novel ITCG law is developed

to satisfy (12)A state variable 119911

1is defined as

1199111= 119877 minus 119881min (119905

119889

119891minus 119905) (13)

Differentiating (13) gives

1= + 119881min (14)

Let 1= 1199112 and the guidance system (2) can be rewritten as

1= 1199112

2= 119891 (119911) + 119892 (119911) 119906

=1

21198621199091205881198812 119878119872

119898

(15)

where 119906 = 119886119872 119891(119911) = minus cos (120582 minus 120574) + 119881 sin (120582 minus 120574) and

119892(119911) = minus sin (120582 minus 120574)A novel integral sliding mode (ISM) is constructed as

119904ISM = 1199112minus 1199112(1199050) + int

119905

1199050

(

119887(119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2)119889120591 (16)

where 119886 119887 and 119888 are the guidance parameters which are allconstants Then using the following guidance law

119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

(17)

where 120573 is a small positive constant and sgn (sdot) denotes thesignum function we have the following results

4 Mathematical Problems in Engineering

Theorem 1 Consider system (15) with the guidance law (17)provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the guidance parametersin (17) If 119905119889

119891satisfies

1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

119894119891 1199022gt 1199021

1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

119894119891 1199021gt 1199022

(18)

1199021ge 2 119902

2ge 2 119902

1= 1199022 and |119881

0cos (120582

0minus 1205740)119881119898119894119899

| lt 1 thenthe following conditions will be satisfied

119877 (119905) gt 0 forall1199050le 119905 lt 119905

119889

119891

119877 (119905119889

119891) = 0

(19)

Proof Differentiating (16) with respect to time yields

119904ISM = 2+

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (20)

Combining (15) with (20) we have

119904ISM = 119891 (119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (21)

Construct a Lyapunov function 1198811as

1198811=

1

21199042

ISM (22)

Differentiating (22) and then using (21) and (17) yield

1= 119904ISM 119904ISM

= 119904ISM (119891(119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2)

= minus119904ISM (120573 sgn (119904ISM)) lt 0 when 119904ISM = 0

(23)

Note that 119904ISM(1199050) = 0 and one can imply that

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (24)

As 119904ISM(119905) remains as zero we get

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (25)

The following equation can be obtained by combining (20)with (25)

119886 (119905119889

119891minus 119905)2

1+ 119887 (119905119889

119891minus 119905)

1+ 1198881199111= 0

forall1199050le 119905 le 119905

119889

119891

(26)

Provided that 1199021and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 if 1199021

= 1199022 1199021ge 2 and 119902

2ge 2 the solutions

to the differential equation (26) can be easily obtained as

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (27)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(28)

where 1198881and 1198882are constant

Substituting (13) into (27) and substituting (14) into (28)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(29)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(30)

Substituting the initial values 1198770and

0into (29) and (30)

we get

1198770minus 119881min (119905

119889

119891minus 1199050) = 1198881(119905119889

119891minus 1199050)1199021

+ 1198882(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(0+ 119881min) (119905

119889

119891minus 1199050) = minus119888

11199021(119905119889

119891minus 1199050)1199021

minus 11988821199022(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(31)

From (31) we have

1198881=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199021

1198882=

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199022

(32)

According to (18) |1198810cos (120582

0minus 1205740)119881min| lt 1 119902

1ge 2 and

1199022ge 2 we have

1198881lt 0

1198882lt 0

(33)

Mathematical Problems in Engineering 5

From (32) (33) and 119905119889

119891minus 1199050gt 0 one can imply that

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

lt 0

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

lt 0

(34)

Substituting (32) into (29) we have

119877 minus 119881min (119905119889

119891minus 119905)

=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199021

(119905119889

119891minus 1199050)

1199021

+

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199022

(119905119889

119891minus 1199050)

1199022 forall1199050le 119905 le 119905

119889

119891

(35)

Considering (34) 0 le (119905119889

119891minus119905)(119905

119889

119891minus1199050) le 1 119902

1ge 2 and 119902

2ge 2

we get (36) from (35) Consider

119877 minus 119881min (119905119889

119891minus 119905)

ge

(0+ (1 minus 119902

2) 119881min) (119905

119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

+

(minus0minus (1 minus 119902

1) 119881min) (119905

119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

= 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

minus 119881min (119905119889

119891minus 119905)

forall1199050le 119905 le 119905

119889

119891

(36)

Then we get

119877 ge 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

gt 0 forall1199050le 119905 lt 119905

119889

119891 (37)

Moreover according to (29) we get

119877 (119905119889

119891) = 0 (38)

The proof is finished

Remark 2 From Theorem 1 it is clear that the proposedguidance law (17) can guarantee eachmissile attacks the targetonly when 119905 = 119905

119889

119891

Remark 3 Equation (17) shows that the control law contains(119905119889

119891minus 119905) in the denominator which may bring the singularity

when 119905 = 119905119889

119891 Fortunately the singularity can be eliminated

combining (27) with (28) yields

119887 (119905119889

119891minus 119905)

1+ 1198881199111

119886 (119905119889

119891minus 119905)

2= (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

forall1199050le 119905 le 119905

119889

119891

(39)

From (39) we can know that the singularity brought by (119905119889119891minus119905)

is eliminated as long as 1199021ge 2 and 119902

2ge 2

32 Nonsingular Impact-Time-Control Guidance Law If119892(119911) = 0 (17) shows that the guidance command 119906 rarr infinwhich means it is impossible to realize the guidance lawIn what follows Theorem 4 shows the condition which canensure 119892(119911) = 0 during the guidance process

Theorem 4 Consider system (15) with the guidance law (17)Provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the parameters in (17)if 119905119889119891satisfies (18) 119902

1ge 2 119902

2ge 2 119902

1= 1199022 sin (120582

0minus 1205740) = 0

and |1198810cos (120582

0minus 1205740)119881119898119894119899

| lt 1 minus 120595 lt 1 then 119892(119911) satisfies thefollowing condition

1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (40)

where 120576 is a sufficiently small time and

120579 = min radic1 minus 1205951003816100381610038161003816sin (120582

0minus 1205740)1003816100381610038161003816 (41)

120595 = max(119881119898119894119899

119881

)

2

(minus1 + 120595)2 (42)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Proof From the demonstration ofTheorem 4 we known thatthe guidance law (17) can ensure the state variables of system(15) satisfy the following equations

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (43)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(44)

where 1198881and 1198882are constants and 119902

1and 1199022are the solutions

to the equation 1198861199022minus (119886 + 119887)119902 + 119888 = 0

6 Mathematical Problems in Engineering

Substituting (13) into (43) and substituting (14) into (44)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(45)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(46)

According to (18) 1199021ge 2 and 119902

2ge 2 we can get the following

results from the demonstration of Theorem 4

1198881lt 0

1198882lt 0

(47)

Combining (47) with (46) yields

+ 119881min gt 0 forall1199050le 119905 lt 119905

119889

119891 (48)

Substituting (2) into (48) yields

cos (120582 minus 120574) lt119881min119881

forall1199050le 119905 lt 119905

119889

119891 (49)

Differentiating (44) gives

1= 11988811199021(1199021minus 1) (119905

119889

119891minus 119905)(1199021minus2)

+ 11988821199022(1199022minus 1) (119905

119889

119891minus 119905)(1199022minus2)

forall1199050le 119905 lt 119905

119889

119891

(50)

Then considering 1199021ge 2 119902

2ge 2 1198881lt 0 and 119888

2lt 0 we have

1lt 0 forall119905

0le 119905 lt 119905

119889

119891 (51)

Equation (51) means that 1reduces monotonically with

increasing 119905 and we get

1lt 1(1199050) forall119905

0lt 119905 lt 119905

119889

119891 (52)

From (52) we get

cos (120582 minus 120574) gt1198810cos (120582

0minus 1205740)

119881 forall1199050lt 119905 lt 119905

119889

119891 (53)

Combining (49) and (53) yields

1198810cos (120582

0minus 1205740)

119881lt cos (120582 minus 120574) lt

119881min119881

forall1199050lt 119905 lt 119905

119889

119891

(54)

From (3) it can be known that lt 0 Then we have

119881min119881

le119881min

119881

lt 1 forall1199050lt 119905 le 119905

119889

119891minus 120576 (55)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Since 119881min119881 le 119881min119881 lt 1 and 1198810cos (120582

0minus 1205740)119881 ge

minus|1198810cos (120582

0minus 1205740)119881min| gt minus1 + 120595 gt minus1 we have

minus1 + 120595 lt cos (120582 minus 120574) le119881min

119881

forall1199050lt 119905 le 119905

119889

119891minus 120576 (56)

Let 120595 = max (119881min119881)2 (minus1 + 120595)

2 then we have

cos2 (120582 minus 120574) le 120595 forall1199050lt 119905 le 119905

119889

119891minus 120576 (57)

It is clear that 0 lt 120595 lt 1 Since | sin (120582 minus 120574)| =

radic1 minus cos2 (120582 minus 120574) we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge radic1 minus 120595 forall119905

0lt 119905 le 119905

119889

119891minus 120576 (58)

Let 120579 = minradic1 minus 120595 | sin (1205820minus 1205740)| we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 forall119905

0le 119905 le 119905

119889

119891minus 120576 (59)

Since | sin(1205820minus 1205740)| gt 0 andradic1 minus 120595 gt 0 we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (60)

Then we have1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (61)

where 120576 is a sufficiently small time The proof is finished

Remark 5 In the practical engineering the available missileacceleration is bounded

|119906| le 119906max (62)

where 119906max is the maximum available acceleration of missileand is determined by the performance of missile Let 119906 =

minus119892minus1(119911)(119891(119911) + (119887(119905

119889

119891minus 119905)1199112+ 1198881199111)119886(119905119889

119891minus 119905)2+ 120573 sgn (119904ISM))

To satisfy (62) in the practical engineering the guidance law(17) must be modified as

119906 =

119906max if 119906 gt 119906max

119906 if |119906| le 119906max

minus119906max if 119906 lt minus119906max

(63)

In the most ideal case (119906max = infin) we can expect the relativedistance119877(119905119889

119891) = 0 However 119906max is bounded in engineering

Fortunately from [8] it can be known that the objective ofITCG can be realized if the impact time error is smaller than01 sThus if the following conditions (63) (64) and (65) canbe satisfied simultaneously the objective of ITCG also can beobtained

119877 gt 0 forall1199050le 119905 le 119905

119889

119891minus 120576 (64)

10038161003816100381610038161003816119877 (119905119889

119891minus 120576)

10038161003816100381610038161003816le 119877set (65)

|119906| le 119906max forall1199050le 119905 le 119905

119889

119891minus 120576 (66)

Mathematical Problems in Engineering 7

where 120576 is a sufficiently small time and 120576 lt 01 119877set is a smalldistance that can guarantee the precision of attack (eg ifthe target is ship 119877set = 1m can guarantee the precision ofattack)

From (2) we have

=minus119881 sin (120582 minus 120574)

119877=

119881

119877119892 (119911) (67)

Substituting (39) into guidance law (17) and considering (67)yield

119906 = minus119892minus1

(119911) (119891 (119911) + (1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = minus119892minus1

(119911)

sdot (minus cos (120582 minus 120574) +1198812 sin (120582 minus 120574) 119892 (119911)

119877+ (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = cos (120582 minus 120574)

119892 (119911)minus 1198812sin (120582 minus 120574)

119877

minus

((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

119892 (119911) forall1199050le 119905 le 119905

119889

119891minus 120576

(68)

From (68) we have

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

119892 (119911)

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

1003816100381610038161003816100381610038161003816100381610038161003816119892 (119911)

1003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

1198812sin (120582 minus 120574)

119877

100381610038161003816100381610038161003816100381610038161003816

forall1199050le 119905 le 119905

119889

119891minus 120576

(69)

Combining (37) (61) and (69) yields

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

120579

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)

1003816100381610038161003816100381610038161003816100381610038161003816100381612057910038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050)

1198770sdot (119905119889

119891minus 119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

(10038161003816100381610038161003816 cos (120582 minus 120574)

10038161003816100381610038161003816+1003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886)

1003816100381610038161003816(119905119889

119891minus 119905)1199021minus2

+10038161003816100381610038161198882119888119886 minus 119888

21199022119887119886

1003816100381610038161003816(119905119889

119891minus 1199050)1199022minus2

+ 120573)

1003816100381610038161003816100381612057910038161003816100381610038161003816

+

100381610038161003816100381610038161198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050) 1198770

10038161003816100381610038161003816

|120576| forall1199050le 119905 le 119905

119889

119891minus 120576

(70)

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Theorem 1 Consider system (15) with the guidance law (17)provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the guidance parametersin (17) If 119905119889

119891satisfies

1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

119894119891 1199022gt 1199021

1199050+

11987701199021

minus0minus (1 minus 119902

1) 119881119898119894119899

lt 119905119889

119891

lt 1199050+

11987701199022

minus0minus (1 minus 119902

2) 119881119898119894119899

119894119891 1199021gt 1199022

(18)

1199021ge 2 119902

2ge 2 119902

1= 1199022 and |119881

0cos (120582

0minus 1205740)119881119898119894119899

| lt 1 thenthe following conditions will be satisfied

119877 (119905) gt 0 forall1199050le 119905 lt 119905

119889

119891

119877 (119905119889

119891) = 0

(19)

Proof Differentiating (16) with respect to time yields

119904ISM = 2+

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (20)

Combining (15) with (20) we have

119904ISM = 119891 (119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2 (21)

Construct a Lyapunov function 1198811as

1198811=

1

21199042

ISM (22)

Differentiating (22) and then using (21) and (17) yield

1= 119904ISM 119904ISM

= 119904ISM (119891(119911) + 119892 (119911) 119906 +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2)

= minus119904ISM (120573 sgn (119904ISM)) lt 0 when 119904ISM = 0

(23)

Note that 119904ISM(1199050) = 0 and one can imply that

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (24)

As 119904ISM(119905) remains as zero we get

119904ISM (119905) = 0 forall1199050le 119905 le 119905

119889

119891 (25)

The following equation can be obtained by combining (20)with (25)

119886 (119905119889

119891minus 119905)2

1+ 119887 (119905119889

119891minus 119905)

1+ 1198881199111= 0

forall1199050le 119905 le 119905

119889

119891

(26)

Provided that 1199021and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 if 1199021

= 1199022 1199021ge 2 and 119902

2ge 2 the solutions

to the differential equation (26) can be easily obtained as

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (27)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(28)

where 1198881and 1198882are constant

Substituting (13) into (27) and substituting (14) into (28)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(29)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(30)

Substituting the initial values 1198770and

0into (29) and (30)

we get

1198770minus 119881min (119905

119889

119891minus 1199050) = 1198881(119905119889

119891minus 1199050)1199021

+ 1198882(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(0+ 119881min) (119905

119889

119891minus 1199050) = minus119888

11199021(119905119889

119891minus 1199050)1199021

minus 11988821199022(119905119889

119891minus 1199050)1199022

forall1199050le 119905 le 119905

119889

119891

(31)

From (31) we have

1198881=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199021

1198882=

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021) (119905119889

119891minus 1199050)

1199022

(32)

According to (18) |1198810cos (120582

0minus 1205740)119881min| lt 1 119902

1ge 2 and

1199022ge 2 we have

1198881lt 0

1198882lt 0

(33)

Mathematical Problems in Engineering 5

From (32) (33) and 119905119889

119891minus 1199050gt 0 one can imply that

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

lt 0

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

lt 0

(34)

Substituting (32) into (29) we have

119877 minus 119881min (119905119889

119891minus 119905)

=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199021

(119905119889

119891minus 1199050)

1199021

+

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199022

(119905119889

119891minus 1199050)

1199022 forall1199050le 119905 le 119905

119889

119891

(35)

Considering (34) 0 le (119905119889

119891minus119905)(119905

119889

119891minus1199050) le 1 119902

1ge 2 and 119902

2ge 2

we get (36) from (35) Consider

119877 minus 119881min (119905119889

119891minus 119905)

ge

(0+ (1 minus 119902

2) 119881min) (119905

119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

+

(minus0minus (1 minus 119902

1) 119881min) (119905

119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

= 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

minus 119881min (119905119889

119891minus 119905)

forall1199050le 119905 le 119905

119889

119891

(36)

Then we get

119877 ge 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

gt 0 forall1199050le 119905 lt 119905

119889

119891 (37)

Moreover according to (29) we get

119877 (119905119889

119891) = 0 (38)

The proof is finished

Remark 2 From Theorem 1 it is clear that the proposedguidance law (17) can guarantee eachmissile attacks the targetonly when 119905 = 119905

119889

119891

Remark 3 Equation (17) shows that the control law contains(119905119889

119891minus 119905) in the denominator which may bring the singularity

when 119905 = 119905119889

119891 Fortunately the singularity can be eliminated

combining (27) with (28) yields

119887 (119905119889

119891minus 119905)

1+ 1198881199111

119886 (119905119889

119891minus 119905)

2= (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

forall1199050le 119905 le 119905

119889

119891

(39)

From (39) we can know that the singularity brought by (119905119889119891minus119905)

is eliminated as long as 1199021ge 2 and 119902

2ge 2

32 Nonsingular Impact-Time-Control Guidance Law If119892(119911) = 0 (17) shows that the guidance command 119906 rarr infinwhich means it is impossible to realize the guidance lawIn what follows Theorem 4 shows the condition which canensure 119892(119911) = 0 during the guidance process

Theorem 4 Consider system (15) with the guidance law (17)Provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the parameters in (17)if 119905119889119891satisfies (18) 119902

1ge 2 119902

2ge 2 119902

1= 1199022 sin (120582

0minus 1205740) = 0

and |1198810cos (120582

0minus 1205740)119881119898119894119899

| lt 1 minus 120595 lt 1 then 119892(119911) satisfies thefollowing condition

1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (40)

where 120576 is a sufficiently small time and

120579 = min radic1 minus 1205951003816100381610038161003816sin (120582

0minus 1205740)1003816100381610038161003816 (41)

120595 = max(119881119898119894119899

119881

)

2

(minus1 + 120595)2 (42)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Proof From the demonstration ofTheorem 4 we known thatthe guidance law (17) can ensure the state variables of system(15) satisfy the following equations

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (43)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(44)

where 1198881and 1198882are constants and 119902

1and 1199022are the solutions

to the equation 1198861199022minus (119886 + 119887)119902 + 119888 = 0

6 Mathematical Problems in Engineering

Substituting (13) into (43) and substituting (14) into (44)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(45)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(46)

According to (18) 1199021ge 2 and 119902

2ge 2 we can get the following

results from the demonstration of Theorem 4

1198881lt 0

1198882lt 0

(47)

Combining (47) with (46) yields

+ 119881min gt 0 forall1199050le 119905 lt 119905

119889

119891 (48)

Substituting (2) into (48) yields

cos (120582 minus 120574) lt119881min119881

forall1199050le 119905 lt 119905

119889

119891 (49)

Differentiating (44) gives

1= 11988811199021(1199021minus 1) (119905

119889

119891minus 119905)(1199021minus2)

+ 11988821199022(1199022minus 1) (119905

119889

119891minus 119905)(1199022minus2)

forall1199050le 119905 lt 119905

119889

119891

(50)

Then considering 1199021ge 2 119902

2ge 2 1198881lt 0 and 119888

2lt 0 we have

1lt 0 forall119905

0le 119905 lt 119905

119889

119891 (51)

Equation (51) means that 1reduces monotonically with

increasing 119905 and we get

1lt 1(1199050) forall119905

0lt 119905 lt 119905

119889

119891 (52)

From (52) we get

cos (120582 minus 120574) gt1198810cos (120582

0minus 1205740)

119881 forall1199050lt 119905 lt 119905

119889

119891 (53)

Combining (49) and (53) yields

1198810cos (120582

0minus 1205740)

119881lt cos (120582 minus 120574) lt

119881min119881

forall1199050lt 119905 lt 119905

119889

119891

(54)

From (3) it can be known that lt 0 Then we have

119881min119881

le119881min

119881

lt 1 forall1199050lt 119905 le 119905

119889

119891minus 120576 (55)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Since 119881min119881 le 119881min119881 lt 1 and 1198810cos (120582

0minus 1205740)119881 ge

minus|1198810cos (120582

0minus 1205740)119881min| gt minus1 + 120595 gt minus1 we have

minus1 + 120595 lt cos (120582 minus 120574) le119881min

119881

forall1199050lt 119905 le 119905

119889

119891minus 120576 (56)

Let 120595 = max (119881min119881)2 (minus1 + 120595)

2 then we have

cos2 (120582 minus 120574) le 120595 forall1199050lt 119905 le 119905

119889

119891minus 120576 (57)

It is clear that 0 lt 120595 lt 1 Since | sin (120582 minus 120574)| =

radic1 minus cos2 (120582 minus 120574) we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge radic1 minus 120595 forall119905

0lt 119905 le 119905

119889

119891minus 120576 (58)

Let 120579 = minradic1 minus 120595 | sin (1205820minus 1205740)| we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 forall119905

0le 119905 le 119905

119889

119891minus 120576 (59)

Since | sin(1205820minus 1205740)| gt 0 andradic1 minus 120595 gt 0 we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (60)

Then we have1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (61)

where 120576 is a sufficiently small time The proof is finished

Remark 5 In the practical engineering the available missileacceleration is bounded

|119906| le 119906max (62)

where 119906max is the maximum available acceleration of missileand is determined by the performance of missile Let 119906 =

minus119892minus1(119911)(119891(119911) + (119887(119905

119889

119891minus 119905)1199112+ 1198881199111)119886(119905119889

119891minus 119905)2+ 120573 sgn (119904ISM))

To satisfy (62) in the practical engineering the guidance law(17) must be modified as

119906 =

119906max if 119906 gt 119906max

119906 if |119906| le 119906max

minus119906max if 119906 lt minus119906max

(63)

In the most ideal case (119906max = infin) we can expect the relativedistance119877(119905119889

119891) = 0 However 119906max is bounded in engineering

Fortunately from [8] it can be known that the objective ofITCG can be realized if the impact time error is smaller than01 sThus if the following conditions (63) (64) and (65) canbe satisfied simultaneously the objective of ITCG also can beobtained

119877 gt 0 forall1199050le 119905 le 119905

119889

119891minus 120576 (64)

10038161003816100381610038161003816119877 (119905119889

119891minus 120576)

10038161003816100381610038161003816le 119877set (65)

|119906| le 119906max forall1199050le 119905 le 119905

119889

119891minus 120576 (66)

Mathematical Problems in Engineering 7

where 120576 is a sufficiently small time and 120576 lt 01 119877set is a smalldistance that can guarantee the precision of attack (eg ifthe target is ship 119877set = 1m can guarantee the precision ofattack)

From (2) we have

=minus119881 sin (120582 minus 120574)

119877=

119881

119877119892 (119911) (67)

Substituting (39) into guidance law (17) and considering (67)yield

119906 = minus119892minus1

(119911) (119891 (119911) + (1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = minus119892minus1

(119911)

sdot (minus cos (120582 minus 120574) +1198812 sin (120582 minus 120574) 119892 (119911)

119877+ (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = cos (120582 minus 120574)

119892 (119911)minus 1198812sin (120582 minus 120574)

119877

minus

((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

119892 (119911) forall1199050le 119905 le 119905

119889

119891minus 120576

(68)

From (68) we have

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

119892 (119911)

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

1003816100381610038161003816100381610038161003816100381610038161003816119892 (119911)

1003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

1198812sin (120582 minus 120574)

119877

100381610038161003816100381610038161003816100381610038161003816

forall1199050le 119905 le 119905

119889

119891minus 120576

(69)

Combining (37) (61) and (69) yields

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

120579

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)

1003816100381610038161003816100381610038161003816100381610038161003816100381612057910038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050)

1198770sdot (119905119889

119891minus 119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

(10038161003816100381610038161003816 cos (120582 minus 120574)

10038161003816100381610038161003816+1003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886)

1003816100381610038161003816(119905119889

119891minus 119905)1199021minus2

+10038161003816100381610038161198882119888119886 minus 119888

21199022119887119886

1003816100381610038161003816(119905119889

119891minus 1199050)1199022minus2

+ 120573)

1003816100381610038161003816100381612057910038161003816100381610038161003816

+

100381610038161003816100381610038161198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050) 1198770

10038161003816100381610038161003816

|120576| forall1199050le 119905 le 119905

119889

119891minus 120576

(70)

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

From (32) (33) and 119905119889

119891minus 1199050gt 0 one can imply that

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

lt 0

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

lt 0

(34)

Substituting (32) into (29) we have

119877 minus 119881min (119905119889

119891minus 119905)

=

(0+ 119881min (1 minus 119902

2)) (119905119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199021

(119905119889

119891minus 1199050)

1199021

+

(minus0minus 119881min (1 minus 119902

1)) (119905119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)1199022

(119905119889

119891minus 1199050)

1199022 forall1199050le 119905 le 119905

119889

119891

(35)

Considering (34) 0 le (119905119889

119891minus119905)(119905

119889

119891minus1199050) le 1 119902

1ge 2 and 119902

2ge 2

we get (36) from (35) Consider

119877 minus 119881min (119905119889

119891minus 119905)

ge

(0+ (1 minus 119902

2) 119881min) (119905

119889

119891minus 1199050) + 11990221198770

(1199022minus 1199021)

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

+

(minus0minus (1 minus 119902

1) 119881min) (119905

119889

119891minus 1199050) minus 11990211198770

(1199022minus 1199021)

sdot

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

= 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

minus 119881min (119905119889

119891minus 119905)

forall1199050le 119905 le 119905

119889

119891

(36)

Then we get

119877 ge 1198770

(119905119889

119891minus 119905)

(119905119889

119891minus 1199050)

gt 0 forall1199050le 119905 lt 119905

119889

119891 (37)

Moreover according to (29) we get

119877 (119905119889

119891) = 0 (38)

The proof is finished

Remark 2 From Theorem 1 it is clear that the proposedguidance law (17) can guarantee eachmissile attacks the targetonly when 119905 = 119905

119889

119891

Remark 3 Equation (17) shows that the control law contains(119905119889

119891minus 119905) in the denominator which may bring the singularity

when 119905 = 119905119889

119891 Fortunately the singularity can be eliminated

combining (27) with (28) yields

119887 (119905119889

119891minus 119905)

1+ 1198881199111

119886 (119905119889

119891minus 119905)

2= (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

forall1199050le 119905 le 119905

119889

119891

(39)

From (39) we can know that the singularity brought by (119905119889119891minus119905)

is eliminated as long as 1199021ge 2 and 119902

2ge 2

32 Nonsingular Impact-Time-Control Guidance Law If119892(119911) = 0 (17) shows that the guidance command 119906 rarr infinwhich means it is impossible to realize the guidance lawIn what follows Theorem 4 shows the condition which canensure 119892(119911) = 0 during the guidance process

Theorem 4 Consider system (15) with the guidance law (17)Provided that 119902

1and 1199022are the solutions to the equation 119886119902

2minus

(119886 + 119887)119902 + 119888 = 0 where 119886 119887 and 119888 are the parameters in (17)if 119905119889119891satisfies (18) 119902

1ge 2 119902

2ge 2 119902

1= 1199022 sin (120582

0minus 1205740) = 0

and |1198810cos (120582

0minus 1205740)119881119898119894119899

| lt 1 minus 120595 lt 1 then 119892(119911) satisfies thefollowing condition

1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (40)

where 120576 is a sufficiently small time and

120579 = min radic1 minus 1205951003816100381610038161003816sin (120582

0minus 1205740)1003816100381610038161003816 (41)

120595 = max(119881119898119894119899

119881

)

2

(minus1 + 120595)2 (42)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Proof From the demonstration ofTheorem 4 we known thatthe guidance law (17) can ensure the state variables of system(15) satisfy the following equations

1199111= 1198881(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022 forall1199050le 119905 le 119905

119889

119891 (43)

1= minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(44)

where 1198881and 1198882are constants and 119902

1and 1199022are the solutions

to the equation 1198861199022minus (119886 + 119887)119902 + 119888 = 0

6 Mathematical Problems in Engineering

Substituting (13) into (43) and substituting (14) into (44)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(45)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(46)

According to (18) 1199021ge 2 and 119902

2ge 2 we can get the following

results from the demonstration of Theorem 4

1198881lt 0

1198882lt 0

(47)

Combining (47) with (46) yields

+ 119881min gt 0 forall1199050le 119905 lt 119905

119889

119891 (48)

Substituting (2) into (48) yields

cos (120582 minus 120574) lt119881min119881

forall1199050le 119905 lt 119905

119889

119891 (49)

Differentiating (44) gives

1= 11988811199021(1199021minus 1) (119905

119889

119891minus 119905)(1199021minus2)

+ 11988821199022(1199022minus 1) (119905

119889

119891minus 119905)(1199022minus2)

forall1199050le 119905 lt 119905

119889

119891

(50)

Then considering 1199021ge 2 119902

2ge 2 1198881lt 0 and 119888

2lt 0 we have

1lt 0 forall119905

0le 119905 lt 119905

119889

119891 (51)

Equation (51) means that 1reduces monotonically with

increasing 119905 and we get

1lt 1(1199050) forall119905

0lt 119905 lt 119905

119889

119891 (52)

From (52) we get

cos (120582 minus 120574) gt1198810cos (120582

0minus 1205740)

119881 forall1199050lt 119905 lt 119905

119889

119891 (53)

Combining (49) and (53) yields

1198810cos (120582

0minus 1205740)

119881lt cos (120582 minus 120574) lt

119881min119881

forall1199050lt 119905 lt 119905

119889

119891

(54)

From (3) it can be known that lt 0 Then we have

119881min119881

le119881min

119881

lt 1 forall1199050lt 119905 le 119905

119889

119891minus 120576 (55)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Since 119881min119881 le 119881min119881 lt 1 and 1198810cos (120582

0minus 1205740)119881 ge

minus|1198810cos (120582

0minus 1205740)119881min| gt minus1 + 120595 gt minus1 we have

minus1 + 120595 lt cos (120582 minus 120574) le119881min

119881

forall1199050lt 119905 le 119905

119889

119891minus 120576 (56)

Let 120595 = max (119881min119881)2 (minus1 + 120595)

2 then we have

cos2 (120582 minus 120574) le 120595 forall1199050lt 119905 le 119905

119889

119891minus 120576 (57)

It is clear that 0 lt 120595 lt 1 Since | sin (120582 minus 120574)| =

radic1 minus cos2 (120582 minus 120574) we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge radic1 minus 120595 forall119905

0lt 119905 le 119905

119889

119891minus 120576 (58)

Let 120579 = minradic1 minus 120595 | sin (1205820minus 1205740)| we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 forall119905

0le 119905 le 119905

119889

119891minus 120576 (59)

Since | sin(1205820minus 1205740)| gt 0 andradic1 minus 120595 gt 0 we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (60)

Then we have1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (61)

where 120576 is a sufficiently small time The proof is finished

Remark 5 In the practical engineering the available missileacceleration is bounded

|119906| le 119906max (62)

where 119906max is the maximum available acceleration of missileand is determined by the performance of missile Let 119906 =

minus119892minus1(119911)(119891(119911) + (119887(119905

119889

119891minus 119905)1199112+ 1198881199111)119886(119905119889

119891minus 119905)2+ 120573 sgn (119904ISM))

To satisfy (62) in the practical engineering the guidance law(17) must be modified as

119906 =

119906max if 119906 gt 119906max

119906 if |119906| le 119906max

minus119906max if 119906 lt minus119906max

(63)

In the most ideal case (119906max = infin) we can expect the relativedistance119877(119905119889

119891) = 0 However 119906max is bounded in engineering

Fortunately from [8] it can be known that the objective ofITCG can be realized if the impact time error is smaller than01 sThus if the following conditions (63) (64) and (65) canbe satisfied simultaneously the objective of ITCG also can beobtained

119877 gt 0 forall1199050le 119905 le 119905

119889

119891minus 120576 (64)

10038161003816100381610038161003816119877 (119905119889

119891minus 120576)

10038161003816100381610038161003816le 119877set (65)

|119906| le 119906max forall1199050le 119905 le 119905

119889

119891minus 120576 (66)

Mathematical Problems in Engineering 7

where 120576 is a sufficiently small time and 120576 lt 01 119877set is a smalldistance that can guarantee the precision of attack (eg ifthe target is ship 119877set = 1m can guarantee the precision ofattack)

From (2) we have

=minus119881 sin (120582 minus 120574)

119877=

119881

119877119892 (119911) (67)

Substituting (39) into guidance law (17) and considering (67)yield

119906 = minus119892minus1

(119911) (119891 (119911) + (1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = minus119892minus1

(119911)

sdot (minus cos (120582 minus 120574) +1198812 sin (120582 minus 120574) 119892 (119911)

119877+ (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = cos (120582 minus 120574)

119892 (119911)minus 1198812sin (120582 minus 120574)

119877

minus

((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

119892 (119911) forall1199050le 119905 le 119905

119889

119891minus 120576

(68)

From (68) we have

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

119892 (119911)

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

1003816100381610038161003816100381610038161003816100381610038161003816119892 (119911)

1003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

1198812sin (120582 minus 120574)

119877

100381610038161003816100381610038161003816100381610038161003816

forall1199050le 119905 le 119905

119889

119891minus 120576

(69)

Combining (37) (61) and (69) yields

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

120579

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)

1003816100381610038161003816100381610038161003816100381610038161003816100381612057910038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050)

1198770sdot (119905119889

119891minus 119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

(10038161003816100381610038161003816 cos (120582 minus 120574)

10038161003816100381610038161003816+1003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886)

1003816100381610038161003816(119905119889

119891minus 119905)1199021minus2

+10038161003816100381610038161198882119888119886 minus 119888

21199022119887119886

1003816100381610038161003816(119905119889

119891minus 1199050)1199022minus2

+ 120573)

1003816100381610038161003816100381612057910038161003816100381610038161003816

+

100381610038161003816100381610038161198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050) 1198770

10038161003816100381610038161003816

|120576| forall1199050le 119905 le 119905

119889

119891minus 120576

(70)

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

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6 Mathematical Problems in Engineering

Substituting (13) into (43) and substituting (14) into (44)yield

119877 minus 119881min (119905119889

119891minus 119905) = 119888

1(119905119889

119891minus 119905)1199021

+ 1198882(119905119889

119891minus 119905)1199022

forall1199050le 119905 le 119905

119889

119891

(45)

+ 119881min = minus11988811199021(119905119889

119891minus 119905)1199021minus1

minus 11988821199022(119905119889

119891minus 119905)1199022minus1

forall1199050le 119905 le 119905

119889

119891

(46)

According to (18) 1199021ge 2 and 119902

2ge 2 we can get the following

results from the demonstration of Theorem 4

1198881lt 0

1198882lt 0

(47)

Combining (47) with (46) yields

+ 119881min gt 0 forall1199050le 119905 lt 119905

119889

119891 (48)

Substituting (2) into (48) yields

cos (120582 minus 120574) lt119881min119881

forall1199050le 119905 lt 119905

119889

119891 (49)

Differentiating (44) gives

1= 11988811199021(1199021minus 1) (119905

119889

119891minus 119905)(1199021minus2)

+ 11988821199022(1199022minus 1) (119905

119889

119891minus 119905)(1199022minus2)

forall1199050le 119905 lt 119905

119889

119891

(50)

Then considering 1199021ge 2 119902

2ge 2 1198881lt 0 and 119888

2lt 0 we have

1lt 0 forall119905

0le 119905 lt 119905

119889

119891 (51)

Equation (51) means that 1reduces monotonically with

increasing 119905 and we get

1lt 1(1199050) forall119905

0lt 119905 lt 119905

119889

119891 (52)

From (52) we get

cos (120582 minus 120574) gt1198810cos (120582

0minus 1205740)

119881 forall1199050lt 119905 lt 119905

119889

119891 (53)

Combining (49) and (53) yields

1198810cos (120582

0minus 1205740)

119881lt cos (120582 minus 120574) lt

119881min119881

forall1199050lt 119905 lt 119905

119889

119891

(54)

From (3) it can be known that lt 0 Then we have

119881min119881

le119881min

119881

lt 1 forall1199050lt 119905 le 119905

119889

119891minus 120576 (55)

where 119881 is the value of 119881 at time 119905 = 119905119889

119891minus 120576

Since 119881min119881 le 119881min119881 lt 1 and 1198810cos (120582

0minus 1205740)119881 ge

minus|1198810cos (120582

0minus 1205740)119881min| gt minus1 + 120595 gt minus1 we have

minus1 + 120595 lt cos (120582 minus 120574) le119881min

119881

forall1199050lt 119905 le 119905

119889

119891minus 120576 (56)

Let 120595 = max (119881min119881)2 (minus1 + 120595)

2 then we have

cos2 (120582 minus 120574) le 120595 forall1199050lt 119905 le 119905

119889

119891minus 120576 (57)

It is clear that 0 lt 120595 lt 1 Since | sin (120582 minus 120574)| =

radic1 minus cos2 (120582 minus 120574) we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge radic1 minus 120595 forall119905

0lt 119905 le 119905

119889

119891minus 120576 (58)

Let 120579 = minradic1 minus 120595 | sin (1205820minus 1205740)| we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 forall119905

0le 119905 le 119905

119889

119891minus 120576 (59)

Since | sin(1205820minus 1205740)| gt 0 andradic1 minus 120595 gt 0 we have

1003816100381610038161003816sin (120582 minus 120574)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (60)

Then we have1003816100381610038161003816119892 (119911)

1003816100381610038161003816ge 120579 gt 0 forall119905

0le 119905 le 119905

119889

119891minus 120576 (61)

where 120576 is a sufficiently small time The proof is finished

Remark 5 In the practical engineering the available missileacceleration is bounded

|119906| le 119906max (62)

where 119906max is the maximum available acceleration of missileand is determined by the performance of missile Let 119906 =

minus119892minus1(119911)(119891(119911) + (119887(119905

119889

119891minus 119905)1199112+ 1198881199111)119886(119905119889

119891minus 119905)2+ 120573 sgn (119904ISM))

To satisfy (62) in the practical engineering the guidance law(17) must be modified as

119906 =

119906max if 119906 gt 119906max

119906 if |119906| le 119906max

minus119906max if 119906 lt minus119906max

(63)

In the most ideal case (119906max = infin) we can expect the relativedistance119877(119905119889

119891) = 0 However 119906max is bounded in engineering

Fortunately from [8] it can be known that the objective ofITCG can be realized if the impact time error is smaller than01 sThus if the following conditions (63) (64) and (65) canbe satisfied simultaneously the objective of ITCG also can beobtained

119877 gt 0 forall1199050le 119905 le 119905

119889

119891minus 120576 (64)

10038161003816100381610038161003816119877 (119905119889

119891minus 120576)

10038161003816100381610038161003816le 119877set (65)

|119906| le 119906max forall1199050le 119905 le 119905

119889

119891minus 120576 (66)

Mathematical Problems in Engineering 7

where 120576 is a sufficiently small time and 120576 lt 01 119877set is a smalldistance that can guarantee the precision of attack (eg ifthe target is ship 119877set = 1m can guarantee the precision ofattack)

From (2) we have

=minus119881 sin (120582 minus 120574)

119877=

119881

119877119892 (119911) (67)

Substituting (39) into guidance law (17) and considering (67)yield

119906 = minus119892minus1

(119911) (119891 (119911) + (1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = minus119892minus1

(119911)

sdot (minus cos (120582 minus 120574) +1198812 sin (120582 minus 120574) 119892 (119911)

119877+ (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = cos (120582 minus 120574)

119892 (119911)minus 1198812sin (120582 minus 120574)

119877

minus

((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

119892 (119911) forall1199050le 119905 le 119905

119889

119891minus 120576

(68)

From (68) we have

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

119892 (119911)

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

1003816100381610038161003816100381610038161003816100381610038161003816119892 (119911)

1003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

1198812sin (120582 minus 120574)

119877

100381610038161003816100381610038161003816100381610038161003816

forall1199050le 119905 le 119905

119889

119891minus 120576

(69)

Combining (37) (61) and (69) yields

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

120579

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)

1003816100381610038161003816100381610038161003816100381610038161003816100381612057910038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050)

1198770sdot (119905119889

119891minus 119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

(10038161003816100381610038161003816 cos (120582 minus 120574)

10038161003816100381610038161003816+1003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886)

1003816100381610038161003816(119905119889

119891minus 119905)1199021minus2

+10038161003816100381610038161198882119888119886 minus 119888

21199022119887119886

1003816100381610038161003816(119905119889

119891minus 1199050)1199022minus2

+ 120573)

1003816100381610038161003816100381612057910038161003816100381610038161003816

+

100381610038161003816100381610038161198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050) 1198770

10038161003816100381610038161003816

|120576| forall1199050le 119905 le 119905

119889

119891minus 120576

(70)

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

where 120576 is a sufficiently small time and 120576 lt 01 119877set is a smalldistance that can guarantee the precision of attack (eg ifthe target is ship 119877set = 1m can guarantee the precision ofattack)

From (2) we have

=minus119881 sin (120582 minus 120574)

119877=

119881

119877119892 (119911) (67)

Substituting (39) into guidance law (17) and considering (67)yield

119906 = minus119892minus1

(119911) (119891 (119911) + (1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = minus119892minus1

(119911)

sdot (minus cos (120582 minus 120574) +1198812 sin (120582 minus 120574) 119892 (119911)

119877+ (

1198881119888

119886minus

11988811199021119887

119886) (119905119889

119891minus 119905)1199021minus2

+ (1198882119888

119886minus

11988821199022119887

119886) (119905119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)) = cos (120582 minus 120574)

119892 (119911)minus 1198812sin (120582 minus 120574)

119877

minus

((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

119892 (119911) forall1199050le 119905 le 119905

119889

119891minus 120576

(68)

From (68) we have

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

119892 (119911)

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816((1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM))

1003816100381610038161003816100381610038161003816100381610038161003816119892 (119911)

1003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

1198812sin (120582 minus 120574)

119877

100381610038161003816100381610038161003816100381610038161003816

forall1199050le 119905 le 119905

119889

119891minus 120576

(69)

Combining (37) (61) and (69) yields

|119906| le

100381610038161003816100381610038161003816100381610038161003816

cos (120582 minus 120574)

120579

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886) (119905

119889

119891minus 119905)1199021minus2

+ (1198882119888119886 minus 119888

21199022119887119886) (119905

119889

119891minus 119905)1199022minus2

+ 120573 sgn (119904ISM)

1003816100381610038161003816100381610038161003816100381610038161003816100381612057910038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050)

1198770sdot (119905119889

119891minus 119905)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

(10038161003816100381610038161003816 cos (120582 minus 120574)

10038161003816100381610038161003816+1003816100381610038161003816(1198881119888119886 minus 119888

11199021119887119886)

1003816100381610038161003816(119905119889

119891minus 119905)1199021minus2

+10038161003816100381610038161198882119888119886 minus 119888

21199022119887119886

1003816100381610038161003816(119905119889

119891minus 1199050)1199022minus2

+ 120573)

1003816100381610038161003816100381612057910038161003816100381610038161003816

+

100381610038161003816100381610038161198812 sin (120582 minus 120574) (119905

119889

119891minus 1199050) 1198770

10038161003816100381610038161003816

|120576| forall1199050le 119905 le 119905

119889

119891minus 120576

(70)

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Since 120579 gt 0 120576 gt 0 1199021ge 2 and 119902

2ge 2 it is clear that 119906 is

boundedwhen 1199050le 119905 le 119905

119889

119891minus120576Thus if themaximum available

acceleration 119906max is large enough then condition (66) can besatisfied even if 120576 is very small Since condition (66) can besatisfied we can know that

119906 = 119906 = minus119892minus1

(119911)

sdot (119891 (119911) +

119887 (119905119889

119891minus 119905) 1199112+ 1198881199111

119886 (119905119889

119891minus 119905)

2+ 120573 sgn (119904ISM))

forall1199050le 119905 le 119905

119889

119891minus 120576

(71)

That is the guidance law (17) is valid when 1199050le 119905 le 119905

119889

119891minus 120576

Thus the mathematical derivations from (12) to (61) are stillvalid when 119905

0le 119905 le 119905

119889

119891minus 120576 From (37) it is clear that (64) is

satisfied Substituting 119905 = 119905119889

119891minus 120576 into (45) yields

119877 (119905119889

119891minus 120576) = 119881min120576 + 119888

11205761199021

+ 11988821205761199022 (72)

From (72) it can be known that condition (65) can be satisfiedif 120576 is small enough

In short for the given small 120576 and119877set (64) (65) and (66)can be satisfied simultaneously if 119906max is large enough andbounded Actually from the simulation result in Section 5of this paper it can be known that the modified guidancelaw (63) can guarantee (64) (65) and (66) are satisfiedsimultaneously in the case that 119877set = 01m 120576 = 10

minus3 s and119906max = 100ms2 (from the results in [8ndash13 18ndash20] it canbe known that 100ms2 is a reasonable maximum availableacceleration for missile and the objective of ITCG can beaccomplished when 119877set = 01m and 120576 = 10

minus3 s)

33 The Permissible Set of the Desired Impact Times In thissection the permissible set of the desired impact times isdiscussed From the above Theorems 1 and 4 the desiredimpact time and the guidance parameters must satisfy

1199050+

1198770119902max

minus0minus (1 minus 119902max) 119881min

lt 119905119889

119891

lt 1199050+

1198770119902min

minus0minus (1 minus 119902min) 119881min

119902max gt 119902min ge 2

(73)

where 119902max is the maximum value of 1199021and 1199022and 119902min is the

minimum value of 1199021and 1199022

From (73) the bound of 119905119889119891can be defined as

120585 = 1199050+

1198770119902

minus0minus (1 minus 119902)119881min

119902 ge 2 (74)

where 119902 is variable If 119902 = 119902max 120585 is the lower bound If119902 = 119902min 120585 is the upper bound taking the following partialdifferential operation as

120597 (120585)

120597119902=

minus1198770(0+ 119881min)

(minus0minus (1 minus 119902)119881min)

2 119902 ge 2 (75)

Theorems 1 and 4 need the following condition

100381610038161003816100381610038161003816100381610038161003816

0

119881min

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

1198810cos (120582

0minus 1205740)

119881min

100381610038161003816100381610038161003816100381610038161003816

lt 1 (76)

Combining (75) and (76) yields

120597 (120585)

120597119902lt 0 119902 ge 2 (77)

From (75) we know that 120585 reduces monotonically withincreasing 119902 Then we get the following

(a) The lower bound of the permissible set of the desiredimpact times 119905119889

119891is 119905119889119891min which is described as

119905119889

119891min = 1199050+

1198770

119881min (78)

If 119902max rarr infin the lower bound can be achieved(b) The upper bound of the permissible set of the desired

impact times 119905119889119891is 119905119889119891max which is described as

119905119889

119891max = 1199050+

21198770

(119881min minus 0)

(79)

If 119902min = 2 we can get the upper boundFrom (a) and (b) the permissible set of the desired impact

times 119905119889119891can be given as

119865 = (119905119889

119891min 119905119889

119891max] (80)

If we select 119905119889119891isin 119865 and assume that (119905119889

119891minus 1199050)119881min minus119877

0gt 0

to meet condition (73) the permissible sets of 119902max and 119902mincan be described as

119902max isin (

(119905119889

119891minus 1199050) (119881min +

0)

((119905119889

119891minus 1199050)119881min minus 119877

0)

infin) cap (119902mininfin)

119902min isin [2infin)

(81)

Because 119902max and 119902min are the solutions to the equation 1198861199022minus

(119886 + 119887)119902 + 119888 = 0 the guidance parameters can be obtained as

119886 = 1

119887 = 119902max + 119902min minus 1

119888 = 119902max119902min

(82)

4 Salvo Attack

In this section an algorithm will be designed based on theproposed guidance law to realize the salvo attack Consider119899 missiles 119872

119894(119894 = 1 119899) engaged in a salvo attack against

a stationary target as shown in Figure 2 120582119894denotes the LOS

angle 119877119894denotes the relative distance between the missile119872

119894

and the target 120574119894denotes the heading angle 119886

119872119894denotes the

missile acceleration and 119881119894denotes the missile velocity

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

for each missile119872119894 119894 = 1 2 119899 do

Compute the permissible set 119865119894= (119905119889

119891119894min 119905119889

119891119894max] from (83)end forFind the intersection set119867 = cap

119899

119894=1119865119894

Select a 119905119889119891isin 119867

for each missile119872119894 119894 = 1 2 119899 do

Select 119902119894max and 119902

119894min from (84) and obtain theparameters 119886

119894 119887119894and 119888119894from (85)

end forfor each time step dofor each missile119872

119894 119894 = 1 2 119899 do

Use 119886119894 119887119894and 119888119894to generate the guidance input 119906

119894of (18)

end forend for

Algorithm 1 Salvo attack algorithm

O

y

x

Target

R1

V1

aM1M1

12057411205821

aMi

aMn

Mi

Mn120574i

120582i120582n

middot middot middot

Vimiddot middot middot

Ri Rn

120574n

Vn

Figure 2 Missile-target relative motion relationship

The permissible set of the desired impact times 119905119889

119891for

missile 119872119894is 119865119894

= (119905119889

119891119894min 119905119889

119891119894max] and the intersection setcan be described as 119867 = cap

119899

119894=1119865119894 It is assumed that 119867 is not

a null set Each missile can utilize the guidance law (17) toachieve the desired impact time at each time step Inspiredby the salvo attack strategy introduced in [24] the detailedalgorithm is given in Algorithm 1

5 Simulation

In this section to illustrate the effectiveness of the proposedguidance law the mathematical simulation is presented Toremove the chattering sgn (119909) is replaced by the sigmoidfunction given as [20]

sgmf (119909) = 2 (1

1 + expminus120572119909minus 1) (83)

where 120572 is selected as 30 In addition the maximum limit ofthe missile acceleration command is selected as 100ms2

51 Performance of the Proposed Guidance Law This sub-section shows the performance of the proposed guidancelaw The simulation result for the missile with constantvelocity is shown in Case 1 And the simulation result for

the missile with time-varying velocity is provided in Case 2For the comparison the traditional proportional navigationguidance law (PNGL) the impact-time-control guidance law(ITCGL) in [8] and the nonsingular sliding mode guidancelaw (NSMGL) in [20] are also considered The PNGL [8] isdefined as

119886PNGL = 119882119881 (84)

The parameter is chosen as119882 = 3 The ITCGL [8] is definedas

119886ITCGL = 119867119881(3

2minus

1

2

sdot radic1 +2401198815

(119867119881)2

1198773

(119905119889

119891minus 119905 minus

(1 + (120574 minus 120582)210) 119877

119881))

(85)

where the parameter is chosen as119867 = 3 The NSMGL [20] isdefined as

119886NSMGL = 119886eq119872

+ 119886119872con119872

+ 119886sw119872 (86)

where

119886eq119872

=1198812(2119873 minus 1)

119877

(cos 120574 minus 1)

120574+

051198812

119877120574 cos 120574

+ 119881

ℎ (120574) =

sgn (120574) 120574 if 10038161003816100381610038161205741003816100381610038161003816lt 1205761

1 minus 1205761

1205761minus 1205762

10038161003816100381610038161205741003816100381610038161003816+

1205761(1205762minus 1)

(1205762minus 1205761)

if 1205761lt10038161003816100381610038161205741003816100381610038161003816lt 1205762

1 otherwise

119886119872con119872

= minusℎ (120574)

120574119896119904119873

119886sw119872

= minus119872(119901 sgn (120574) + 1) sgn (119904119873)

119904119873

=119877

119881(1 +

05

2119873 minus 11205742)

(87)

where the parameters are chosen as 119896 = 50 119872 = 10 119901 = 21205761= 0001 120576

2= 0015 and 119873 = 3 In this paper the values

of119882119867 119896119872 119901 1205761 1205762 and119873 are the same as that in [8 20]

and used here to ensure the fairness of comparison

Case 1 (missile with constant velocity) In this case weconsider that the missile velocity is constant that is = 0The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (261008 424381] by using (80)Choose the desired impact time as 119905

119889

119891= 37 s isin 119865 The

permissible set of 119902max and 119902min can be calculated from (81) as119902max isin (26137infin) cap (119902mininfin) and 119902min isin [2infin) From thepermissible sets 119902max and 119902min are selected as 119902max = 4 and119902min = 25 Using 119902max and 119902min the guidance parameters 119886 119887and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and 119888 = 10

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 1 Initial condition for Cases 1 and 2

Parameters Case 1 (constantvelocity)

Case 2 (varyingvelocity)

Initial missile position (minus10000 minus3000) mInitial missile velocity 119881

0400ms

Initial missile headingangle 120574

0

minus60 deg

Target position (0 0)mDesired impact time 119905119889

11989137 s

Missile mass119898 400 kgReference area 119878

119872mdash 0039m2

Air density 120588 mdash 1293 kgm3

Minimum speed 119881min 400ms 350ms

Table 2 Performance of guidance laws in Section 51

Case Guidance law 119905119891minus 119905119889

119891(s) Miss distance (m)

1

Proposed law 124 times 10minus4 007

NSMGL 145 times 10minus4 005

ITCGL 312 times 10minus2 033

PNGL minus639 003

2

Proposed law 515 times 10minus4 015

NSMGL 057 006ITCGL 049 047PNGL minus3095 009

The simulation results of Case 1 are shown in Figures 3(a)3(b) 3(c) and 3(d) and Table 2 respectively From Figures3(a) and 3(b) and Table 2 it can be seen that the proposedlaw NSMGL and ITCGL all can guarantee that the impacttime errors are less than 4 times 10

minus2 s and the miss distances areless than 1m which means that the impact time constraintcan be satisfied by using these laws when the missile velocityis constant The PNGL makes the missile have a largeimpact time error minus639 s which means that the missionof impact time constraint cannot be accomplished underPNGL Moreover in order to control the impact time it canbe seen from Figure 3(c) that the proposed law NSMGLand ITCGL employ more control energy compared toPNGL

Case 2 (missile with time-varying velocity) In this case weconsider that themissile velocity is varyingThe aerodynamiccoefficients are given by

119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

(88)

The initial conditions used in Case 1 are listed in Table 1From Table 1 the permissible set of the desired impact timescan be calculated as 119865 = (298294 472385] by using (80)Choose the desired impact time as 119905119889

119891= 37 s isin 119865 Then the

permissible set of 119902max and 119902min can be calculated from (81)as 119902max isin (38033infin) cap (119902mininfin) and 119902max isin [2infin) Fromthe permissible sets 119902max and 119902min are selected as 119902max = 4

and 119902min = 25 Using 119902max and 119902min the guidance parameters

119886 119887 and 119888 can be obtained from (82) as 119886 = 1 119887 = 55 and119888 = 10

The simulation results of Case 2 are shown in Figures 4(a)4(b) 4(c) and 4(d) and Table 2 respectively From Figures4(a) and 4(b) and Table 2 it can be seen that the proposed lawcan guarantee that the impact time error is only 6times10

minus4 s andthe miss distance is 02m which means that the impact timeconstraint (37 s) can be satisfied by using the proposed lawHowever for NSMGL and ITCGL the impact time errors aregreater than 048 s As mentioned before this is because theNSMGL and ITCGL are designed under the assumption thatthe missile velocity is constant Compared with the impact-time-control guidance laws based on the assumption that themissile velocity is constant the proposed law can achievesmaller impact time errors when the velocity is varying

52 Application of Proposed Guidance Law to Salvo AttackScenario This subsection is performed with the proposedguidance law for a salvo attackThe air density is 1293 kgm3The aerodynamic coefficients for each missile are given by

Missile 1 119862119909=

minus037 119905 lt 6

minus0112 119905 ge 6

Missile 2 119862119909=

minus047 119905 lt 10

minus0325 119905 ge 10

Missile 3 119862119909=

minus057 119905 lt 10

minus05 119905 ge 10

(89)

The initial conditions used in this subsection are listed inTable 3

From Table 3 the permissible sets of the desired impacttimes for each missile can be calculated as

1198651= (298294 472385]

1198652= (249885 573847]

1198653= (206552 729195]

(90)

Choose the desired impact time as 119905119889119891= 37 s isin 119865

1cap 1198652cap 1198653

Then the permissible set of 119902max and 119902min can be calculatedfrom (81) as

1199021max isin (38033infin) cap (119902

1mininfin) 1199021min isin [2infin)

1199022max isin (34780infin) cap (119902

2mininfin) 1199022min isin [2infin)

1199023max isin (32450infin) cap (119902

3mininfin) 1199023min isin [2infin)

(91)

From the permissible sets we select that 1199021max = 119902

2max =

1199023max = 4 and 119902

1min = 1199022min = 119902

3min = 25 Then theguidance parameters can be obtained from (82) as

1198861= 1198862= 1198863= 1

1198871= 1198872= 1198873= 55

1198881= 1198882= 1198883= 10

(92)

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 3 Initial conditions for salvo attack

Object 1199090m 119910

0m 120574

0(deg) 119881

0(ms) 119881min(ms) Mass(kg) 119878

119872(m2)

Target 0 0 0 0 0 0 0Missile 1 minus10000 minus3000 minus60 400 350 400 0039

Missile 2 minus2000 minus8000 minus20 410 330 395 0040

Missile 3 minus5000 4000 70 420 310 410 0041

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus7000

minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

1000

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

399

3995

400

4005

401

Miss

ile v

eloci

ty (m

s)

(d) Missile velocity

Figure 3 Responses in Case 1 (constant velocity)

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Proposed law NSMGLITCGL PNGL

36 37 380

100

200

tdf = 37 s

(a) Miss distance 119877

minus10000 minus8000 minus6000 minus4000 minus2000 0 2000minus6000

minus5000

minus4000

minus3000

minus2000

minus1000

0

x (m)

y (m

)

Proposed law NSMGLITCGL PNGLTarget

(b) Flight trajectory

0 10 20 30 40minus20

0

20

40

60

80

100

Time (s)

Proposed law NSMGLITCGL PNGL

Acce

lera

tion

com

man

d (m

s2)

(c) Acceleration command

0 10 20 30 40Time (s)

350

360

370

380

390

400M

issile

velo

city

(ms

)

(d) Missile velocity

Figure 4 Responses in Case 2 (time-varying velocity)

Table 4 Performance of guidance law in Section 52

Object 119905119891minus 119905119889

119891(s) Miss distance (m)

Missile 1 313 times 10minus4 015

Missile 2 115 times 10minus4 045

Missile 3 089 times 10minus4 031

The simulation results are shown in Figures 4(a) 4(b)4(c) and 4(d) and Table 4 respectively From Figures 5(a)5(b) 5(c) and 5(d) and Table 4 it is clear that the proposedlaw can guarantee the miss distances are less than 05m

and the impact time errors are less than 4 times 10minus4 s which

means that the missiles can accomplish salvo attack even ifthe missile velocities are time-varying

6 Conclusions

In this paper a novel ITCG law has been proposed forthe homing missile with time-varying velocity The maincontribution here is that the proposed ITCG law relaxed theconstant velocity assumption which is needed for the existingITCG laws Since most missiles have time-varying velocitythe proposedmethod is more feasible than the existing ITCG

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

minus10000 minus5000 0 5000minus1

minus05

0

05

1

x (m)

y (m

)

Missile 1 Missile 2Missile 3 Target

times104

(a) Flight trajectory

0 10 20 30 40minus2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Miss

dist

ance

(m)

Missile 1 Missile 2Missile 3

tdf = 37 s

(b) Miss distance 119877

0 10 20 30 40300

320

340

360

380

400

420

Time (s)

Miss

ile v

eloci

ty (m

s)

Missile 1 Missile 2Missile 3

(c) Missile velocity

0 5 10 15 20 25 30 35 40minus100

minus50

0

50

100

Time (s)

Missile 1 Missile 2Missile 3

Acce

lera

tion

com

man

d (m

s2)

(d) Acceleration command

Figure 5 Responses in the salvo attack scenario

laws in engineering In addition the proposed law also exhib-ited one attractive feature The proposed guidance law wasderived based on the nonlinear engagement dynamic ratherthan the linearized-engagement-dynamics used in manytraditional ITCG laws A salvo attack algorithm has beendeveloped based on the proposed ITCG law The theoreticalderivations and simulation results all demonstrated that theproposed guidance law achieved a better performance thanthe existing ITCG laws when the missile velocity is time-varying and that the proposed salvo attack algorithm workedwell

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61304238) the National High-TechRampD Program of China (2012AA120602 2015AAxxx7033and 2015AAxxx1008) and the Chinese Aerospace SupportedFund (2015-HT-XGD)

References

[1] S N Ghawghawe and D Ghose ldquoPure proportional navigationagainst time-varying target maneuversrdquo IEEE Transaction onControl Systems Technology vol 22 no 4 pp 589ndash594 1999

[2] M Guelman ldquoThe closed-form solution of true proportionalnavigationrdquo IEEE Transaction on Aerospace and ElectronicSystems vol AES-12 no 4 pp 472ndash482 1976

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

[3] P-J Yuan and J-S Chern ldquoSolutions of true proportional nav-igation for maneuvering and nonmaneuvering targetsrdquo Journalof Guidance Control and Dynamics vol 15 no 1 pp 268ndash2711992

[4] C-D Yang F-B Hsiao and F-B Yeh ldquoGeneralized guidancelaw for homing missilesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 2 pp 197ndash212 1989

[5] R H Chen J L Speyer and D Lianos ldquoTerminal and boostphase intercept of ballistic missile defenserdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit Honolulu Hawaii USA August 2008

[6] S Fucci and P Zuzolo ldquoAvionics and ecm system integration onmilitary aircraftrdquo Journal of Aircraft vol 8 no 5 pp 388ndash3911971

[7] C-K Ryoo I Whang and M-J Tahk ldquo3-D evasive maneuverpolicy for anti-Ship missiles against close-in weapon systemsrdquoin Proceedings of the AIAA Guidance Navigation and controlconference and Exhibit Austin Tex USA 2003

[8] I-S Jeon J-I Lee and M-J Tahk ldquoImpact-time-control guid-ance law for anti-ship missilesrdquo IEEE Transactions on ControlSystems Technology vol 14 no 2 pp 260ndash266 2006

[9] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[10] Y Zhang G Ma and A Liu ldquoGuidance law with impact timeand impact angle constraintsrdquo Chinese Journal of Aeronauticsvol 26 no 4 pp 960ndash966 2013

[11] J-I Lee I-S Jeon and M-J Tahk ldquoGuidance law to controlimpact time and anglerdquo IEEE Transactions on Aerospace andElectronic Systems vol 43 no 1 pp 301ndash310 2007

[12] Y Zhang X Wang and H Wu ldquoImpact time control guidancewith field-of-view constraint accounting for uncertain systemlagrdquo Proceedings of the Institution of Mechanical Engineers PartG Journal of Aerospace Engineering vol 230 no 3 pp 515ndash5292016

[13] T-H Kim C-H Lee I-S Jeon and M-J Tahk ldquoAugmentedpolynomial guidance with impact time and angle constraintsrdquoIEEE Transactions on Aerospace and Electronic Systems vol 49no 4 pp 2806ndash2817 2013

[14] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006

[15] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993

[16] K-S Kim Y Park and S-H Oh ldquoDesigning robust slid-ing hyperplanes for parametric uncertain systems a Riccatiapproachrdquo Automatica vol 36 no 7 pp 1041ndash1048 2000

[17] J Wang S Li J Yang B Wu and Q Li ldquoExtended stateobserver-based sliding mode control for PWM-based DC-DCbuck power converter systems with mismatched disturbancesrdquoIET Control Theory and Applications vol 9 no 4 pp 579ndash5862015

[18] N Harl and S N Balakrishnan ldquoImpact time and angleguidance with sliding mode controlrdquo IEEE Transactions onControl Systems Technology vol 20 no 6 pp 1436ndash1449 2012

[19] S R Kumar and D Ghose ldquoSliding mode control basedguidance law with impact time constraintsrdquo in Proceedings ofthe 1st American Control Conference (ACC rsquo13) pp 5760ndash5765IEEE Washington DC USA June 2013

[20] D Cho H J Kim and M-J Tahk ldquoNonsingular sliding modeguidance for impact time controlrdquo Journal of Guidance Controland Dynamics vol 39 no 1 pp 61ndash68 2016

[21] H Cho C K Ryoo and M-J Tahk ldquoClosed-form optimalguidance law for missiles of time-varying velocityrdquo Journal ofGuidance Control and Dynamics vol 19 no 5 pp 1017ndash10221996

[22] Z Zhang S Li and S Luo ldquoComposite guidance laws based onslidingmode control with impact angle constraint and autopilotlagrdquo Transactions of the Institute of Measurement and Controlvol 35 no 6 pp 764ndash776 2013

[23] Z Zhang S Li and S Luo ldquoTerminal guidance laws of missilebased on ISMC and NDOB with impact angle constraintrdquoAerospace Science and Technology vol 31 no 1 pp 30ndash41 2013

[24] A Saleem and A Ratnoo ldquoLyapunov-based guidance lawfor impact time control and simultaneous arrivalrdquo Journal ofGuidance Control and Dynamics vol 39 no 1 pp 164ndash1722016

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of