research article predictive sliding mode control for...

Research ArticlePredictive Sliding Mode Control for Attitude Tracking ofHypersonic Vehicles Using Fuzzy Disturbance Observer

Xianlei Cheng Guojian Tang Peng Wang and Luhua Liu

College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Xianlei Cheng xianleicheng163com

Received 7 July 2014 Accepted 29 October 2014

Academic Editor Chunyu Yang

Copyright copy 2015 Xianlei Cheng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a predictive sliding mode control (PSMC) scheme for attitude control of hypersonic vehicle (HV) with systemuncertainties and external disturbances based on an improved fuzzy disturbance observer (IFDO) First for a class of uncertainaffine nonlinear systemswith systemuncertainties and external disturbances we propose a predictive slidingmode control based onfuzzy disturbance observer (FDO-PSMC) which is used to estimate the composite disturbances containing system uncertaintiesand external disturbances Afterward to enhance the composite disturbances rejection performance an improved FDO-PSMC(IFDO-PSMC) is proposed by incorporating a hyperbolic tangent function with FDO to compensate for the approximate error ofFDO Finally considering the actuator dynamics the proposed IFDO-PSMC is applied to attitude control system design for HVto track the guidance commands with high precision and strong robustness Simulation results demonstrate the effectiveness androbustness of the proposed attitude control scheme

1 Introduction

Near space is the airspace of Earth altitudes from 20 km to100 km which has shown strategy and spatial superioritywith the prosperous development of aerospace technology[1] The near space hypersonic vehicle (HV) which has manyoutstanding advantages such as large flight envelope highmaneuverability and fast response ability compared withthe ordinary aircraft has attracted a growing worldwideinterest [2 3] Due to the hypersonic velocity and changeableflight environment the HV possesses some distinct dynamiccharacters of highly coupled control channels serious non-linearity and strong uncertainty which all contribute tothe design difficulty of the attitude control system withremarkable precision and strong robustness Due to thepoor performance of the traditional control approaches inaddressing the nonlinear and uncertain problem advancedcontrol methods should be employed in the attitude controlsystem design for the HV which is still an open problem

The sliding mode control (SMC) is insensitive to systemuncertainties and disturbances It is one of the most impor-tant approaches to the control of systems with modelingimprecision and it has been widely applied to the flight

control system design [4ndash11] References [12 13] incorporatethe sliding mode surface into the quadratic performanceindex of the predictive control and then the predictivesliding mode control law which has an explicitly analyticalform is derived by minimizing the performance index Byusing this method the undesirable chattering phenomenonis attenuated and the large online computational issue ofthe predictive control is avoided The predictive slidingmode control (PSMC) takes merits of the strong robustnessof sliding model control and the outstanding optimizationperformance of predictive control which appears to be a verypromising control method in control engineering

For the design of control system for the HV manyadvanced control methods mainly focus on the stability orrobust stability rather than considering the system uncer-tainties and disturbances explicitly in the controller design[14ndash18] They may encounter some unexpected problemsand the flight control system may even become unstable inthe presence of strong disturbances Thus it is of profoundsignificance to adopt new strategies to eliminate the influenceof the composite disturbances and improve the precision androbustness of the flight control system Fuzzy disturbanceobserver (FDO) is a particular type of disturbance observer

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 727162 13 pageshttpdxdoiorg1011552015727162

2 Mathematical Problems in Engineering

which combines the distinct merits of fuzzy control withdisturbance observer technology These advantages make itan appropriate candidate for robust control of uncertainnonlinear systems [19ndash25] Nevertheless the robust flightcontrol scheme based on FDO needs to be further researchedfor the HV to improve the control ability which needs to duelwith the changeable flight environment and problems causedby system uncertainties

Motivated by the precise and robust attitude controldemand of the HV with uncertain model and external dis-turbances this paper considers the composite disturbancesin flight control system design of the HV to improve therobust control performance Three major contributions arepresented as follows

(1) We propose a predictive sliding mode control basedon fuzzy disturbance observer (FDO-PSMC)methodfor a class of uncertain affine nonlinear systems withsystem uncertainties and external disturbances Thecomposite disturbances are considered which resultin poor performance and instability of the controlsystem

(2) To address the problems which are brought by thecomposite disturbances an improved FDO (IFDO) isproposed through utilizing the special properties ofa hyperbolic tangent function to compensate for theapproximate error of FDO By using IFDO the com-posite disturbances can be approximated effectively

(3) Considering the actuator dynamics we apply theimproved fuzzy disturbance observer based predic-tive sliding mode control (IFDO-PSMC) scheme tothe attitude control system design for theHVNumer-ical simulation verifies the surpassing performance ofthe proposed control scheme

The rest of this paper is organized as follows In Section 2the control-oriented hypersonic flight model during cruisephase is presented In order to approximate the compos-ite disturbances Section 3 presents a FDO-PSMC methodand the FDO is improved for a better performance for aclass of uncertain affine nonlinear systems with externaldisturbances In Section 4 the proposed control scheme isapplied to the attitude control system design for the HV inconsideration of the actuator dynamics Simulation resultsare presented in Section 5 to validate the effectiveness of thedesigned flight control system Finally Section 6 providessome conclusions of this paper

2 HV Modeling

Suppose that the fuel slosh is not considered and the productsof inertia are negligible [26] Based on the hypothesis ofan inverse-square-law gravitational model the centripetal

acceleration for the nonrotating Earth the mathematicalmodel of HV during the cruise phase can be described as

V = minus

120583

119903

3119862

1minus

119863

119898

+

119879

119898

cos120572 cos120573

120579 =

1

119898V(minus119898

120583

119903

3119862

2+ 119871 cos 120574

119881minus 119873 sin 120574

119881+ 119879119862

4)

=

minus1

119898V cos 120579(minus119898

120583

119903

3119862

3+ 119871 sin 120574

119881+ 119873 cos 120574

119881+ 119879119862

5)

(1)

= 120596

119910sin 120574 + 120596

119911cos 120574

120595 = (120596

119910cos 120574 minus 120596

119911sin 120574) sec120593

120574 = 120596

119909minus (120596

119910cos 120574 minus 120596

119911sin 120574) tan120593

119909= 119869

minus1

119909119872

119909+ 119869

minus1

119909(119869

119910minus 119869

119911) 120596

119911120596

119910

119910= 119869

minus1

119910119872

119910+ 119869

minus1

119910(119869

119911minus 119869

119909) 120596

119909120596

119911

119911= 119869

minus1

119911119872

119911+ 119869

minus1

119911(119869

119909minus 119869

119910) 120596

119910120596

119909

(2)

sin120573

= (sin (120595 minus 120590) cos 120574 + sin120593 sin 120574 cos (120595 minus 120590)) cos 120579

minus cos120593 sin 120574 sin 120579

sin120572 cos120573 = cos (120595 minus 120590) sin120593 cos 120574 cos 120579

minus sin (120595 minus 120590) sin 120574 cos 120579

minus cos120593 cos 120574 sin 120579

sin 120574119881cos120573

= cos (120595 minus 120590) sin120593 sin 120574 sin 120579

+ sin (120595 minus 120590) cos 120574 sin 120579 + cos120593 sin 120574 cos 120579

(3)

119862

1= 119909 cos 120579 cos120590 + (119910 + 119877

119890) sin 120579 minus 119911 cos 120579 sin120590

119862

2= minus119909 sin 120579 cos120590 + (119910 + 119877

119890) cos 120579 + 119911 sin 120579 sin120590

119862

3= 119909 sin120590 + 119911 cos120590

119862

4= sin120572 cos 120574

119881+ cos120572 sin120573 sin 120574

119881

119862

5= sin120572 sin 120574

119881minus cos120572 sin120573 cos 120574

119881

(4)

where 120572 120573 and 120574119881are the angle-of-attack sideslip angle and

bank angle respectively 120593 120595 and 120574 are the pitch yaw androlling angles respectively V 120579 and 120590 are the flight velocityflight path angle and trajectory angle respectively 120596

119909 120596

119910

and 120596

119911are pitch yaw and roll rates respectively 119909 119910 119911

are the three position coordinates in the ground coordinateframe 119877

119890 119903 are radial distance from the Earthrsquos center and

the mean radius of the spherical Earth respectively119872119909119872

119910

and 119872

119911are control moments including roll yaw and pitch

control moments 119871119873 and 119879 are the lift side and thrustforces respectively 119869

119909 119869

119910 and 119869

119911are the main moments of

inertia of the three axes respectively

Mathematical Problems in Engineering 3

Remark 1 For the convenience of control system design[120593 120595 120574]

T is selected as the state variables In view of the aimof attitude control system design for HV which is to trackthe guidance commands [120572

119888 120573

119888 120574

119881119888

]

T precisely it is necessaryto transform [120572

119888 120573

119888 120574

119881119888

]

T into [120593

119888 120595

119888 120574

119888]

T by means of (3)which is obtained by the coordinate system transformation

3 Predictive Sliding Mode Control Based onFuzzy Disturbance Observer

31 Predictive Sliding Mode Control for Systems with Uncer-tainties and Disturbances To develop the predictive slidingmode control we consider the following MIMO uncertainaffine nonlinear system with external disturbances

x = f (x) + Δf (x) + (g (x) + Δg (x)) u + d

y = h (x) (5)

where x isin R119899 is the state vector of the uncertain nonlinearsystem u isin R119898 is the control input vector y isin R119898 is theoutput vector of the uncertain nonlinear system f(x) isin R119899and g(x) isin R119899times119898 are the given function vector and controlgain matrix respectively Δf(x) and Δg(x) are the internaluncertainty and modeling error and d is the external time-varying disturbance

Define

D (x u d) = Δf (x) +Δg (x)u + d (6)

where D(x u d) denotes the system uncertainties and exter-nal disturbances

Without loss of generality the equilibrium of uncertainnonlinear system (5) is supposed to be x

0= 0 In accordance

with the Lie derivative operation [27] in differential geometrytheory the vector relative degree of MIMO system can bedefined as follows

Definition 2 (see [27]) The vector relative degree of system(5) is 120588

1 120588

2 120588

119898 at x

0under the following conditions

where 1205881 120588

2 120588

119898denote the relative degree of each chan-

nel respectively

(1) For all x in a neighborhood of x0 119871g119895

119871

119896

fℎ119894(x) = 0 (1 le119894 119895 le 119898 0 lt 119896 lt 120588

119894)

(2) The following119898 times 119898matrix is nonsingular at x0= 0

A (x)

=

[

[

[

[

[

[

119871g1

119871

1205881minus1

f ℎ

1(x) 119871g

2

119871

1205881minus1

f ℎ

1(x) sdot sdot sdot 119871g

119898

119871

1205881minus1

f ℎ

1(x)

119871g1

119871

1205882minus1

f ℎ

2(x) 119871g

2

119871

1205882minus1

f ℎ


119898

119871

1205882minus1

f ℎ

2(x)

d

119871g1

119871

120588119898minus1

f ℎ

119898(x) 119871g

2

119871

120588119898minus1

f ℎ


119898

119871

120588119898minus1

f ℎ

119898(x)

]

]

]

]

]

]

(7)

where g119895is the jth row vector of g(x) and ℎ

119894(x) is the ith output

of system (5) Similarly the disturbance relative degree at x0

can be defined as 1205911 120591

2 120591

119898

To facilitate the process of control system design the fol-lowing reasonable assumptions are required before develop-ing predictive sliding mode control of the uncertain MIMOnonlinear system (5)

Assumption 3 f(x) and h(x) are continuously differentiableand g(x) is continuous

Assumption 4 All states are available moreover the outputand reference signals are also continuously differentiable

Assumption 5 The zero dynamics are stable

Assumption 6 The vector relative degree is 1205881 120588

2 120588

119898

and the disturbance relative degree of composite disturbancesis 1205911 120591

2 120591

119898 where 120591

119894ge 120588

119894(119894 = 1 119898)

According to Assumption 6 we can obtain

119910

(120588119894)

119894= 119871

120588119894

119891ℎ

119894 (x) + (119871

119892119871

120588119894minus1

119891ℎ

119894 (x)) u

+ 119871

119863119871

120588119894minus1

119891ℎ

119894(x) (119894 = 1 119898)

(8)

Associate the 119898 equations the nonlinear system (5) canbe written as

y(120588) = 120572 (x) + 120573 (x)u + Δ (x uD) (9)

where

120572 (x) = [120572

1 120572

2sdot sdot sdot 120572

119898]

Tisin R119898

120573 (x) = [120573

1 120573


119898]

Tisin R119898times119898

Δ (x uD) = [120575

1 120575


119898]

Tisin R119898

(10)

where 120572119894= 119871

120588119894

119891ℎ

119894(x) 120573119894= 119871

119892119871

120588119894minus1

119891ℎ

119894(x) 120575119894= 119871

119863119871

120588119894minus1

119891ℎ

119894(x) (119894 =

1 119898) and Δ(x uD) is related to the composite distur-bances

Furthermore the sliding surface vector is defined as

s (119905) = [119904

1 119904


119898]

Tisin R119898 (11)

where each sliding surface is the combination of the trackingerror and its higher order derivatives which can be depictedas

119904

119894= 119890

[120588119894minus1]

119894+ 119896

119894120588119894minus1119890

[120588119894minus2]

119894+ sdot sdot sdot + 119896

1198940119890

119894 (12)

where 119890

119894= 119910

119894minus 119910

119888119894(119894 = 1 119898) and 119896

119894119895(119895 =

0 1 120588

119894minus1)mustmake the polynomial (12)Hurwitz stable

Differentiating (12) we have

119904

119894= 119890

[120588119894]

119894+ 119896

119894120588119894minus1119890

[120588119894minus1]

119894+ sdot sdot sdot + 119896

1198940119890

119894

= 119910

[120588119894]

119894minus 119910

[120588119894]

119888119894+ 119896

119894120588119894minus1119890

[120588119894minus1]

119894

+ sdot sdot sdot + 119896

1198940119890

119894 (119894 = 1 119898)

(13)

For the sake of simplified notation define 119911119894as

119911

119894= 119896

119894120588119894minus1119890

[120588119894minus1]

119894+ sdot sdot sdot + 119896

1198940119890

119894(119894 = 1 119898)

(14)


Then the vector z can be written as

z = [119911

1 119911


119898]

Tisin R119898 (15)

Consequently differentiating the sliding surface vectorwe obtain

s (119905) = y(120588) minus y(120588)c + z (16)

where y(120588)c = [119910

(1205881)

1198881 119910

(1205882)


(120588119898)

119888119898]

Tisin R119898

Within themoving time frame the sliding surface s(119905+119879)at the time 119879 is approximately predicted by

s (119905 + 119879) = s (119905) + 119879 s (119905) (17)

For the derivation of the control law the receding-horizon performance index at the time 119879 is given by

119869 (x u 119905) = 1

2

sT (119905 + 119879) s (119905 + 119879) (18)

The necessary condition for the optimal control to mini-mize (18) with respect to u is given by

120597119869

120597u= 0

(19)

According to (19) substitute (17) into (18) then thepredictive sliding mode control law can be proposed as

u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) + Δ (x uD) minus y(120588)c + z)) (20)

Remark 7 If D = 0 the control law given by (20) is thenominal predictive sliding mode control law If D = 0Δ(x uD) cannot be directly obtained due to the immeasur-able unknown composite disturbancesDTheperformance offlight control systemmay becomeworse even unstable whenD is big enough To handle this problem a fuzzy observerwould be designed to estimate the composite disturbances

Remark 8 For the convenience of notation Δ(x uD)wouldbe denoted by Δ(x) in the rest of this paper

32 Fuzzy Disturbance Observer

321 Fuzzy Logic System The fuzzy inference engine adoptsthe fuzzy if-then rules to perform a mapping from an inputlinguistic vector X = [119883

1 119883

2 119883

119899]

Tisin R119899 to an output

variable 119884 isin R The ith fuzzy rule can be expressed as [19]

119877

(119894) If 1198831is 1198601198941 119883

119899is 119860119894119899 then 119884 is 119884119894 (21)

where 119860

119894

1 119860

119894

119899are fuzzy sets characterized by fuzzy

membership functions and 119884119894 is a singleton numberBy adopting a center-average and singleton fuzzifier and

the product inference the output of the fuzzy system can bewritten as

119884 (X) =sum

119897

119894=1119884

119894(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

=

120579

T120599 (X) (22)

where 119897 is the number of fuzzy rules 120583119860119894

119895

(119883

119895) is the mem-

bership function value of fuzzy variable 119883119895(119895 = 1 2 119899)

120579 = [119884

1 119884

2 119884

119897]

T is an adjustable parameter vector and120599(x) = [120599

1 120599

2 120599

119897]

T is a fuzzy basis function vector 120599119894 canbe expressed as

120599

119894=

prod

119899

119895=1120583

119860119894

119895

(119883

119895)

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

(23)

Lemma 9 (see [28]) For any given real continuous functiong(x) on the compact set U isin R119899 and arbitrary 120576 gt 0 thereexists a fuzzy system 119901(X) presented as (22) such that

sup119909isinU

1003816

1003816

1003816

1003816

119901 (X) minus 119884 (X)1003816100381610038161003816

lt 120576 (24)

According to Lemma 9 a fuzzy system can be designedto approximate the composite disturbances Δ(x) by adjustingthe parameter vector 120579 onlineThe designed fuzzy system canbe written as

Δ (x | 120579TΔ) = [

120575

1

120575

2sdot sdot sdot

120575

119898]

T=

120579

TΔ120599Δ(x) (25)

where

120579

TΔ= diag 120579

T1

120579

T2

120579

T119898

120579

T119894= (119884

1

119894 119884

2

119894 119884

119897

119894)

T

120599Δ (

x) = [1205991 (x)T 1205992 (x)

T 120599

119898 (x)T]

T

120599 (x) = [120599

1

119894 120599

2

119894 120599

119897

119894]

T

(26)

Let x belong to a compact set Ωx the optimal parametervector 120579lowastT

Δcan be defined as

120579lowastTΔ

= argminTΔ

supxisinΩx

1003817

1003817

1003817

1003817

1003817

1003817

1003817

Δ (x) minus

Δ (x | 120579TΔ)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

(27)

where the optimal parameter matrix 120579lowastTΔ

lies in a convexregion given by

M120579 =

120579Δ|

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

le 119898

120579 (28)

where sdot is the Frobenius norm and119898120579is a design parameter

with119898120579gt 0

Consequently the composite disturbances Δ(x) can bewritten as

Δ (x) = 120579lowastTΔ120599Δ(x) + 120576 (29)

where 120576 is the smallest approximation error of the fuzzysystem Apparently the norm of 120579lowastT

Δis bounded with

120576 le 120576 (30)

where 120576 is the upper bound of the approximation error 120576


Through adjusting the parameter vector

120579 online thecomposite disturbances Δ(x) can be approximated by thefuzzy system hence the performance of the control sys-tem with composite disturbances can be improved As theperformance of the control system is closely related to theselected fuzzy system the control precision will be reducedwhen the approximation ability of the selected fuzzy system isdissatisfactory In order to improve the approximation abilitya hyperbolic tangent function is integrated with the fuzzydisturbance observer to compensate for the approximateerror

322 Fuzzy Disturbance Observer Consider the followingdynamic system

= minus120594120583 + 119901 (x u 120579Δ) (31)

where 119901(x u 120579Δ) = 120594y(120588minus1) + 120572(x) + 120573(x)u +

Δ(x | 120579TΔ) 120594 gt 0

is a design parameter and

Δ(x |

120579

TΔ) =

120579

TΔ120599Δ(x) is utilized to

compensate for the composite disturbancesDefine the disturbance observation error 120577 = y(120588minus1) minus 120583

invoking (9) and (31) yields

120577 = 120572 (x) + 120573 (x)u + Δ (x) + 120590120583 minus 120590y(120588minus1) minus 120572 (x)

minus 120573 (x) u minus

Δ (x | 120579TΔ) = minus120590120577 +

120579

TΔ120599Δ(x) + 120576

(32)

where 120579Δ= 120579lowast

Δminus

120579Δis the adjustable parameter error vector

Then the FDO-PSMC is proposed as

u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ(x) minus y(120588)c + z))

(33)

Invoking (16) and (33) we have

s = y(120588) minus y(120588)c + z = 120572 (x) + 120573 (x) u + Δ (x) minus y(120588)c + z

= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576

(34)

Invoking (32) and (34) the augmented system is obtainedas

= A120589 + B (

120579

TΔ120599Δ(x) + 120576) (35)

where = (120577T sT)TA = diag(minus120594I

119898 minus119879

minus1I119898) and B =

(I119898 I119898)

T

Theorem 10 Assume that the disturbance of (9) is monitoredby the system (32) and the system (9) is controlled by (33) Ifthe adjustable parameter vector of FDO is tuned by (36) thenthe augmented error 120589 is uniformly ultimately bounded withinan arbitrarily small region

120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(36)

where119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(37)

Proof Consider the Lyapunov function candidate

119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ (38)

Invoking (35) and (36) the time derivative of 119881 is

119881 = 120589T +

1

120582

120579

TΔ

120579Δ= 120577

T

120577 + sT s + 1

120582

120579

TΔ

120579Δ

= 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576) + sT (minus119879minus1s +

120579

TΔ120599Δ (

x) + 120576)

minus

1

120582

120579

TΔProj [120582120593

Δ(x) 120589TB] + 1

120578

120599

120599

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576) + sT (minus119879minus1s +

120579

TΔ120599Δ(x) + 120576)

minus

120579

TΔ120599Δ (

x) 120589T +

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(39)

Considering the fact

120579

TΔ

120579Δ= 120579lowastTΔ

120579Δminus

120579

TΔ

120579Δ

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2+

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

) minus

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2minus 119898

120579

2)

le 0 (when 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579)

(40)

and invoking (37) we have

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ= 0

or

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δlt 0

(41)

Consequently we obtain

119881 le 120577T(minus120594120577 +

120579

TΔ120593Δ(x) + 120576)

+ sT (minus119879minus1s +

120579

TΔ120593Δ(x) + 120576) minus

120579

TΔ120593Δ(x) 120589T

le 120577T(minus120594120577 +

120579


120579

TΔ120593Δ(x) + 120576)

minus

120579

TΔ120593Δ(x) 120577 minus

120579

TΔ120593Δ(x) s

le 120577T(minus120594120577 + 120576) + sT (minus119879minus1s + 120576)


le minus120594 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576

le minus

120594

2

1205772+

1

2120594

1205762minus

1

2119879

s2 + 119879

2

1205762

(42)

If 120577 gt 120576120594 or s gt 119879120576

119881 lt 0Therefore the augmentederror 120589 is uniformly ultimately bounded

Remark 11 In order to make sure that the FDO

Δ(x |

120579

TΔ)

outputs zero signal in case of the composite disturbancesD =

0 the consequent parameters should be set to be 0

120579Δ(0) = 0

(43)

Remark 12 A projection operator (36) is used in the tuningmethod of the adjustable parameter vector then 120579

Δ can be

guaranteed to be bounded [29]

Remark 13 The adjustable parameter vector of the FDO (36)includes the observation error 120577 and the sliding surface sIn view of the fact that the sliding surface s which is thelinearization combination of the tracking error e is Hurwitzstable the observation error 120577 and tracking error e are bothuniformly ultimately bounded which ensures stability of thecontrol system

33 Improved Fuzzy Disturbance Observer

Lemma 14 (see [30]) For all 119888 gt 0 and w isin 119877119898 the followinginequality holds

0 lt w minus wT tanh(w119888

) le 119898120581119888 (44)

where 120581 is a constant such that 120581 = eminus(120581+1) that is 120581 = 02785

The hyperbolic tangent function is incorporated to com-pensate for the approximate error and improve the precisionof FDO Consider the following dynamic system

= minus120594120583 + 119901 (x u 120579Δ) (45)


Δ(x |

120579

TΔ) minus 120592 tanh(120577119888) 120594 gt 0 is a design parameter and

Δ(x |

120579

TΔ) =

120579

TΔ120599Δ(x) is utilized to compensate for the composite

disturbancesDefine the disturbance observation error 120577 = y(120588minus1) minus 120583


120577 = 120572 (x) + 120573 (x) u + Δ (x) + 120590120583 minus 120590y(120588minus1) minus 120572 (x)


Δ (x | 120579TΔ)

= minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

)

(46)


Δminus

120579Δis an adjustable parameter vector

The IFDO-PSMC is proposed as

u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ (

x)

minus y(120588)c + z + 120592 tanh( s (119905)119888

)))

(47)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

)

(48)


= A120589 + B(

120579

TΔ120599Δ(x) + 120576120592 tanh(120589

TB119888

)) (49)

where 120589 = (120577T sT)T A = diag(minus120594I

119898 minus119879


(I119898 I119898)

T

Theorem 15 Assume that the disturbance of the system (9) ismonitored by the system (49) and the system (9) is controlledby (47) If the adjustable parameter vector of IFDO is tunedby (50) and the approximate error compensation parameter istuned by (51) the augmented error 120589 is uniformly ultimatelybounded within an arbitrarily small region

120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(50)

= 120578

1003817

1003817

1003817

1003817

1003817

120589TB100381710038171003817

1003817

1003817

(51)

where

119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(52)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ+

1

2120578

120592

2 (53)


where 120592 = 120592 minus 120576 Invoking (49) (50) and (51) the timederivative of 119881 is

119881 = 120589T +

1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T

120577 + sT s + 1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576)

minus 120592 tanh( s119888

) minus

1

120582

120579

TΔProj [120582120599

Δ(x) 120589TB] + 1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ (

x) + 120576)

minus 120592 tanh( s119888

) minus

120579

TΔ120599Δ(x) 120589T

+

120579

TΔ119868120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ+

1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

))


120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh( s119888

))

minus

120579

TΔ120599Δ(x) 120589T + 1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

))

minus

120579

TΔ120599Δ (

x) 120577 minus

120579

TΔ120599Δ (

x) s + 1

120578

120592

le 120577T(minus120594120577 + 120576) + sT (minus119879minus1s + 120576) minus 120577T120592 tanh(120577

119888

)

minus sT120592 tanh( s119888

) + 120592 120577 + 120592 s

(54)

According to Lemma 14 we have

minus 120577T120592 tanh(120577

119888

) le minus |120592| 120577 + 119898120581119888


) le minus |120592| s + 119898120581119888

(55)


119881 le minus120590 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576 minus |120592| 120577

+ 119898120581119888 minus |120592| s + 119898120581119888 + 120592 120577 + 120592 s

le minus120590 1205772+ 120577 120576 minus 119879

minus1s2 + s 120576 minus 120592 120577

+ 120592 120577 minus 120592 120577 + 120592 s + 2119898120581119888

le minus120590 1205772minus 119879

minus1s2 + 2119898120581119888

(56)

If 120577 gt radic2119898120581119888120594 or s gt radic

2119898120581119888119879

119881 lt 0 Thereforethe augmented error 120589 is uniformly ultimately bounded

Remark 16 Thecomposite disturbances can be approximatedeffectively by adjusting the parameter law (50) and (51) Ifthe fuzzy system approximates the composite disturbancesaccurately the approximate error compensation will havesmall effect on compensating the approximate error and viceversa Based on this the approximate precision of FDO canbe improved through overall consideration

4 Design of Attitude Control ofHypersonic Vehicle

41 Control Strategy The structure of the IFDO-PSMC flightcontrol system is shown in Figure 1 Considering the strongcoupled nonlinear and uncertain features of the HV predic-tive sliding mode control approach which has distinct meritsis applied to the design of the nominal flight control systemTo further improve the performance of flight control systeman improved fuzzy disturbance observer is proposed to esti-mate the composite disturbances The adjustable parameterlaw of IFDO can be designed based on Lyapunov theorem toapproximate the composite disturbances online Accordingto the estimated composite disturbances the compensationcontroller can be designed and integrated with the nominalcontroller to track the guidance commands precisely In orderto improve the quality of flight control system three samecommand filters are designed to smooth the changing ofthe guidance commands in three channels Their transferfunctions have the same form as

119909

119903

119909

119888

=

2

119909 + 2

(57)

where 119909

119903and 119909

119888are the reference model states and the

external input guidance commands respectively

42 Attitude Controller Design The states of the attitudemodel (2) are pitch angle 120593 yaw angle120595 rolling angle 120574 pitchrate 120596

119909 yaw rate 120596

119910 and roll rate 120596

119911 and the outputs are

selected as pitch angle 120593 yaw angle 120595 and rolling angle 120574which can be written asΘ = [120593 120595 120574]

T and120596 = [120596

119909 120596

119910 120596

119911]

Tand the output states are selected as y = [120593 120595 120574]

T then theattitude model (2) can be rewritten as

Θ = F120579120596

= Jminus1F120596+ Jminus1u

(58)


Predictivesliding mode

controller

Attitudemodel

Fuzzydisturbance

observerCompensating

controller

Commandfilter

Commandtransformation

Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590

Figure 1 IFDO-PSMC based flight control system

where

F120579=

[

[

0 sin 120574 cos 1205740 cos 120574 sec120593 minus sin 120574 sec1205931 minus tan120593 cos 120574 tan120593 sin 120574

]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)

According to the PSMC approach (58) can be derived as

y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)

Then the sliding surface can be selected as

s (119905) = e + 119870e (62)

where e = y minus y119888is the attitude tracking error vector y

119888

is attitude command vector [120593119888 120595

119888 120574

119888]

T transformed fromthe input attitude command vector [120572

119888 120573

119888 120574

119881119888

]

T and 119870 isthe parameter matrix which must make the polynomial (62)Hurwitz stable Define z as

z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)

Substituting (61) (62) (63) and (64) into (20) thenominal predictive slidingmodeflight controllerwhich omitsthe composite disturbances Δ(x) is proposed as

u0= minus119879

minus1120573 (x)minus1 (s (119905) minus 119879 (120572 (x) minus Yc120588 + z)) (65)

In order to improve performance of the flight controlsystem the adaptive compensation controller based on IFDOis designed as

u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where

Δ(x) is the approximate composite disturbances forΔ(x)


Magnitude limiter Rate limiter

+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc

Figure 2 Actuator model

Accordingly a fuzzy system can be designed to approx-imate the composite disturbances Δ(x) First each state isdivided into five fuzzy sets namely 119860119894

1(negative big) 119860119894

2

(negative small) 1198601198943(zero) 119860119894

4(positive small) and 119860

119894

5

(positive big) where 119894 = 1 2 3 represent120593120595 and 120574 Based onthe Gaussianmembership function 25 if-then fuzzy rules areused to approximate the composite disturbances 120575

1 1205752 and

120575

3 respectivelyConsidering the adaptive parameter law (50) and (51) the

approximate composite disturbances can be expressed as

Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579

T3 is the adjustable parameter

vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599

3(x)T]T is a fuzzy basis

function vector related to the membership functions hencethe HV attitude control based on the IFDO can be written as

u = u0+ u1 (68)

Theorem 17 Assume that the HV attitude model satisfiesAssumptions 3ndash6 and the sliding surface is selected as (62)which is Hurwitz stable Moreover assume that the compositedisturbances of the attitude model are monitored by the system(49) and the system is controlled by (68) If the adjustableparameter vector of IFDO is tuned by (50) and the approximateerror compensation parameter is tuned by (51) then theattitude tracking error and the disturbance observation errorare uniformly ultimately bounded within an arbitrarily smallregion

Remark 18 Theorem 17 can be proved similarly asTheorem 15 Herein it is omitted

Remark 19 By means of the designed IFDO-PSMC system(68) the proposed HV attitude control system can notonly track the guidance commands precisely with strongrobustness but also guarantee the stability of the system

43 Actuator Dynamics The HV attitude system tracks theguidance commands by controlling actuator to generatedeflection moments then the HV is driven to change theattitude angles However there exist actuator dynamics togenerate the input moments119872

119909119872119910 and119872

119911 which are not

considered inmost of the existing literature In order tomake

the study in accordance with the actual case consider theactuator model expressed as

119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596

119899are the damping and bandwidth respectively

119878

119860(119904) is the deflection angle limiter and 119878V(119904) is the deflection

rate limiter Figure 2 shows the actuator model where 1199091198880 119909119888

and

119888are the deflection angle command virtual deflection

angle and deflection rate respectively

5 Simulation Results Analysis

To validate the designed HV attitude control system simula-tion studies are conducted to track the guidance commands[120572

119888 120573

119888 120574

119881119888

]

T Assume that the HV flight lies in the cruisephase with the flight altitude 335 km and the flight machnumber 15 The initial attitude and attitude angular velocityconditions are arbitrarily chosen as 120572

0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110

= 0 the initial flight path angle andtrajectory angle are 120579

0= 120590

0= 0 the guidance commands are

chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5

∘ respectivelyThe damping 120577

119899and bandwidth 120596

119899of the actuator are

0707 and 100 rads respectively the deflection angle limit is[minus20 20]∘ the deflection rate limit is [minus50 50]∘s

To exhibit the superiority of the proposed schemethe variation of the aerodynamic coefficients aerodynamicmoment coefficients atmosphere density thrust coefficientsand moments of inertia is assumed to be minus50 minus50 +50minus30 and minus10 which are much more rigorous than thoseconsidered in [18] On the other hand unknown externaldisturbance moments imposed on the HV are expressed as

119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)

The simulation time is set to be 20 s and the updatestep of the controller is 001 s The predictive sliding modecontroller design parameters are chosen as 119879 = 01 and119870 = diag(2 50 4) the IFDO design parameters are chosenas 120582 = 275 120594 = 15 120578 = 95 and 119888 = 025 In additionthe membership functions of the fuzzy system are chosen as


0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)

Figure 3 Comparison of tracking curves and control inputs under SMC and PSMC in nominal condition

Gaussian functions expressed as (71) The simulation resultsare shown in Figures 3 4 and 5 Consider

120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909

119895) = exp[minus(

119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)

Figure 3 presents the simulation results under PSMC andSMC without considering system uncertainties and externaldisturbance Figures 3(a)sim3(c) show the results of trackingthe guidance commands under PSMC and SMC As shownthe guidance commands of PSMC and SMC can be trackedboth quickly andprecisely Figures 3(d)sim3(f) show the controlinputs under PSMC and SMC It can be observed thatthere exists distinct chattering of SMC while the results ofPSMC are smooth owing to the outstanding optimizationperformance of the predictive control The simulation resultsin Figure 3 indicate that the PSMC is more effective thanSMC for system without considering system uncertaintiesand external disturbances

Figure 4 shows the simulation results under PSMC andSMCwith considering systemuncertainties where the resultsare denoted as those in Figure 4 From Figures 4(a)sim4(c)it can be seen that the flight control system adopted SMCand PSMC can track the commands well in spite of systemuncertainties which proves strong robustness of SMCMean-while it is obvious that SMC takes more time to achieveconvergence and the control inputs become larger than thatin Figure 4 However PSMC can still perform as well as thatin Figure 3 with the settling time increasing slightly and the


0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)

Figure 4 Comparison of tracking curves and control inputs under SMC and PSMC in the presence of system uncertainties

control inputs almost remain the same In accordance withFigures 4(d)sim4(f) we can observe that distinct chattering isalso caused by SMC while the results of PSMC are smoothIt can be summarized that PSMC is more robust and moreaccurate than SMC for system with uncertain model

Figure 5 gives the simulation results under PSMC FDO-PSMC and IFDO-PSMC in consideration of system uncer-tainties and external disturbances Figures 5(a)sim5(c) showthe results of tracking the guidance commands It can beobserved that PSMC cannot track the guidance commands inthe presence of big external disturbances while FDO-PSMCand IFDO-PSMC can both achieve satisfactory performancewhich verifies the effectiveness of FDO in suppressing theinfluence of system uncertainties and external disturbancesFigures 5(d)sim5(f) are the partial enlarged detail correspond-ing to Figures 5(a)sim5(c) As shown the IFDO-PSMC cantrack the guidance commands more precisely which declaresstronger disturbance approximate ability of IFDO whencompared with FDO To sum up the results of Figure 5indicate that IFDO is effective in dealing with system uncer-tainties and external disturbances In addition the proposed

IFDO-PSMC is applicative for the HV which has rigoroussystem uncertainties and strong external disturbances

It can be concluded from the above-mentioned simula-tion analysis that the proposed IFDO-PSMC flight controlscheme is valid

6 Conclusions

An effective control scheme based on PSMC and IFDO isproposed for the HV with high coupling serious nonlinear-ity strong uncertainty unknown disturbance and actuatordynamics PSMC takes merits of the strong robustness ofsliding model control and the outstanding optimizationperformance of predictive control which seems to be a verypromising candidate for HV control system design PSMCand FDO are combined to solve the system uncertaintiesand external disturbance problems Furthermorewe improvethe FDO by incorporating a hyperbolic tangent functionwith FDO Simulation results show that IFDO can betterapproximate the disturbance than FDO and can assure the


0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)

Figure 5 Comparison of tracking curves under PSMC FDO-PSMC and IFDO-PSMC in the presence of system uncertainties and externaldisturbances

control performance of HV attitude control system in thepresence of rigorous system uncertainties and strong externaldisturbances

Conflict of Interests

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

References

[1] E Tomme and S Dahl ldquoBalloons in todayrsquos military Anintroduction to the near-space conceptrdquo Air and Space PowerJournal vol 19 no 4 pp 39ndash49 2005

[2] M J Marcel and J Baker ldquoIntegration of ultra-capacitors intothe energy management system of a near space vehiclerdquo in Pro-ceedings of the 5th International Energy Conversion EngineeringConference June 2007

[3] M A Bolender and D B Doman ldquoNonlinear longitudinaldynamical model of an air-breathing hypersonic vehiclerdquo Jour-nal of Spacecraft and Rockets vol 44 no 2 pp 374ndash387 2007

[4] Y Shtessel C Tournes and D Krupp ldquoReusable launch vehiclecontrol in sliding modesrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 335ndash345 1997

[5] D Zhou CMu andWXu ldquoAdaptive sliding-mode guidance ofa homing missilerdquo Journal of Guidance Control and Dynamicsvol 22 no 4 pp 589ndash594 1999

[6] F-K Yeh H-H Chien and L-C Fu ldquoDesign of optimalmidcourse guidance sliding-mode control for missiles withTVCrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 39 no 3 pp 824ndash837 2003

[7] H Xu M D Mirmirani and P A Ioannou ldquoAdaptive slidingmode control design for a hypersonic flight vehiclerdquo Journal ofGuidance Control and Dynamics vol 27 no 5 pp 829ndash8382004

[8] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012

[9] Y N Yang J Wu and W Zheng ldquoTrajectory tracking for anautonomous airship using fuzzy adaptive slidingmode controlrdquoJournal of Zhejiang University Science C vol 13 no 7 pp 534ndash543 2012

[10] Y Yang J Wu and W Zheng ldquoConcept design modelingand station-keeping attitude control of an earth observationplatformrdquo Chinese Journal of Mechanical Engineering vol 25no 6 pp 1245ndash1254 2012

[11] Y Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and external


disturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012

[12] K-B Park and T Tsuji ldquoTerminal sliding mode control ofsecond-order nonlinear uncertain systemsrdquo International Jour-nal of Robust and Nonlinear Control vol 9 no 11 pp 769ndash7801999

[13] S N Singh M Steinberg and R D DiGirolamo ldquoNonlinearpredictive control of feedback linearizable systems and flightcontrol system designrdquo Journal of Guidance Control andDynamics vol 18 no 5 pp 1023ndash1028 1995

[14] L Fiorentini A Serrani M A Bolender and D B DomanldquoNonlinear robust adaptive control of flexible air-breathinghypersonic vehiclesrdquo Journal of Guidance Control and Dynam-ics vol 32 no 2 pp 401ndash416 2009

[15] B Xu D Gao and S Wang ldquoAdaptive neural control based onHGO for hypersonic flight vehiclesrdquo Science China InformationSciences vol 54 no 3 pp 511ndash520 2011

[16] D O Sigthorsson P Jankovsky A Serrani S YurkovichM A Bolender and D B Doman ldquoRobust linear outputfeedback control of an airbreathing hypersonic vehiclerdquo Journalof Guidance Control and Dynamics vol 31 no 4 pp 1052ndash1066 2008

[17] B Xu F Sun C Yang D Gao and J Ren ldquoAdaptive discrete-time controller designwith neural network for hypersonic flightvehicle via back-steppingrdquo International Journal of Control vol84 no 9 pp 1543ndash1552 2011

[18] P Wang G Tang L Liu and J Wu ldquoNonlinear hierarchy-structured predictive control design for a generic hypersonicvehiclerdquo Science China Technological Sciences vol 56 no 8 pp2025ndash2036 2013

[19] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002

[20] E Kim ldquoA discrete-time fuzzy disturbance observer and itsapplication to controlrdquo IEEE Transactions on Fuzzy Systems vol11 no 3 pp 399ndash410 2003

[21] E Kim and S Lee ldquoOutput feedback tracking control ofMIMO systems using a fuzzy disturbance observer and itsapplication to the speed control of a PM synchronous motorrdquoIEEE Transactions on Fuzzy Systems vol 13 no 6 pp 725ndash7412005

[22] Z Lei M Wang and J Yang ldquoNonlinear robust control ofa hypersonic flight vehicle using fuzzy disturbance observerrdquoMathematical Problems in Engineering vol 2013 Article ID369092 10 pages 2013

[23] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its application on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013

[24] C-S Tseng and C-K Hwang ldquoFuzzy observer-based fuzzycontrol design for nonlinear systems with persistent boundeddisturbancesrdquo Fuzzy Sets and Systems vol 158 no 2 pp 164ndash179 2007

[25] C-C Chen C-H Hsu Y-J Chen and Y-F Lin ldquoDisturbanceattenuation of nonlinear control systems using an observer-based fuzzy feedback linearization controlrdquo Chaos Solitons ampFractals vol 33 no 3 pp 885ndash900 2007

[26] S Keshmiri M Mirmirani and R Colgren ldquoSix-DOF mod-eling and simulation of a generic hypersonic vehicle for con-ceptual design studiesrdquo in Proceedings of AIAA Modeling andSimulation Technologies Conference and Exhibit pp 2004ndash48052004

[27] A Isidori Nonlinear Control Systems An Introduction IIISpringer New York NY USA 1995

[28] L X Wang ldquoFuzzy systems are universal approximatorsrdquo inProceedings of the IEEE International Conference on FuzzySystems pp 1163ndash1170 San Diego Calif USA March 1992

[29] S Huang K K Tan and T H Lee ldquoAn improvement onstable adaptive control for a class of nonlinear systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1398ndash14032004

[30] C Y Zhang Research on Modeling and Adaptive Trajectory Lin-earization Control of an Aerospace Vehicle Nanjing Universityof Aeronautics and Astronautics 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


which combines the distinct merits of fuzzy control withdisturbance observer technology These advantages make itan appropriate candidate for robust control of uncertainnonlinear systems [19ndash25] Nevertheless the robust flightcontrol scheme based on FDO needs to be further researchedfor the HV to improve the control ability which needs to duelwith the changeable flight environment and problems causedby system uncertainties

Motivated by the precise and robust attitude controldemand of the HV with uncertain model and external dis-turbances this paper considers the composite disturbancesin flight control system design of the HV to improve therobust control performance Three major contributions arepresented as follows

(1) We propose a predictive sliding mode control basedon fuzzy disturbance observer (FDO-PSMC)methodfor a class of uncertain affine nonlinear systems withsystem uncertainties and external disturbances Thecomposite disturbances are considered which resultin poor performance and instability of the controlsystem

(2) To address the problems which are brought by thecomposite disturbances an improved FDO (IFDO) isproposed through utilizing the special properties ofa hyperbolic tangent function to compensate for theapproximate error of FDO By using IFDO the com-posite disturbances can be approximated effectively

(3) Considering the actuator dynamics we apply theimproved fuzzy disturbance observer based predic-tive sliding mode control (IFDO-PSMC) scheme tothe attitude control system design for theHVNumer-ical simulation verifies the surpassing performance ofthe proposed control scheme

The rest of this paper is organized as follows In Section 2the control-oriented hypersonic flight model during cruisephase is presented In order to approximate the compos-ite disturbances Section 3 presents a FDO-PSMC methodand the FDO is improved for a better performance for aclass of uncertain affine nonlinear systems with externaldisturbances In Section 4 the proposed control scheme isapplied to the attitude control system design for the HV inconsideration of the actuator dynamics Simulation resultsare presented in Section 5 to validate the effectiveness of thedesigned flight control system Finally Section 6 providessome conclusions of this paper

2 HV Modeling

Suppose that the fuel slosh is not considered and the productsof inertia are negligible [26] Based on the hypothesis ofan inverse-square-law gravitational model the centripetal

acceleration for the nonrotating Earth the mathematicalmodel of HV during the cruise phase can be described as

V = minus

120583

119903

3119862

1minus

119863

119898

+

119879

119898

cos120572 cos120573

120579 =

1

119898V(minus119898

120583

119903

3119862

2+ 119871 cos 120574

119881minus 119873 sin 120574

119881+ 119879119862

4)

=

minus1

119898V cos 120579(minus119898

120583

119903

3119862

3+ 119871 sin 120574

119881+ 119873 cos 120574

119881+ 119879119862

5)

(1)

= 120596

119910sin 120574 + 120596

119911cos 120574

120595 = (120596

119910cos 120574 minus 120596

119911sin 120574) sec120593

120574 = 120596

119909minus (120596

119910cos 120574 minus 120596

119911sin 120574) tan120593

119909= 119869

minus1

119909119872

119909+ 119869

minus1

119909(119869

119910minus 119869

119911) 120596

119911120596

119910

119910= 119869

minus1

119910119872

119910+ 119869

minus1

119910(119869

119911minus 119869

119909) 120596

119909120596

119911

119911= 119869

minus1

119911119872

119911+ 119869

minus1

119911(119869

119909minus 119869

119910) 120596

119910120596

119909

(2)

sin120573

= (sin (120595 minus 120590) cos 120574 + sin120593 sin 120574 cos (120595 minus 120590)) cos 120579

minus cos120593 sin 120574 sin 120579

sin120572 cos120573 = cos (120595 minus 120590) sin120593 cos 120574 cos 120579

minus sin (120595 minus 120590) sin 120574 cos 120579

minus cos120593 cos 120574 sin 120579

sin 120574119881cos120573

= cos (120595 minus 120590) sin120593 sin 120574 sin 120579

+ sin (120595 minus 120590) cos 120574 sin 120579 + cos120593 sin 120574 cos 120579

(3)

119862

1= 119909 cos 120579 cos120590 + (119910 + 119877

119890) sin 120579 minus 119911 cos 120579 sin120590

119862

2= minus119909 sin 120579 cos120590 + (119910 + 119877

119890) cos 120579 + 119911 sin 120579 sin120590

119862

3= 119909 sin120590 + 119911 cos120590

119862

4= sin120572 cos 120574

119881+ cos120572 sin120573 sin 120574

119881

119862

5= sin120572 sin 120574

119881minus cos120572 sin120573 cos 120574

119881

(4)

where 120572 120573 and 120574119881are the angle-of-attack sideslip angle and

bank angle respectively 120593 120595 and 120574 are the pitch yaw androlling angles respectively V 120579 and 120590 are the flight velocityflight path angle and trajectory angle respectively 120596

119909 120596

119910

and 120596

119911are pitch yaw and roll rates respectively 119909 119910 119911

are the three position coordinates in the ground coordinateframe 119877

119890 119903 are radial distance from the Earthrsquos center and

the mean radius of the spherical Earth respectively119872119909119872

119910

and 119872

119911are control moments including roll yaw and pitch

control moments 119871119873 and 119879 are the lift side and thrustforces respectively 119869

119909 119869

119910 and 119869

119911are the main moments of

inertia of the three axes respectively




119888 120573

119888 120574

119881119888

]


119888 120573

119888 120574

119881119888

]

T into [120593

119888 120595

119888 120574

119888]




x = f (x) + Δf (x) + (g (x) + Δg (x)) u + d

y = h (x) (5)


Define

D (x u d) = Δf (x) +Δg (x)u + d (6)



0= 0 In accordance



1 120588

2 120588

119898 at x


where 1205881 120588

2 120588


nel respectively


119871

119896

fℎ119894(x) = 0 (1 le119894 119895 le 119898 0 lt 119896 lt 120588

119894)


A (x)

=

[

[

[

[

[

[

119871g1

119871

1205881minus1

f ℎ

1(x) 119871g

2

119871

1205881minus1

f ℎ


119898

119871

1205881minus1

f ℎ

1(x)

119871g1

119871

1205882minus1

f ℎ

2(x) 119871g

2

119871

1205882minus1

f ℎ


119898

119871

1205882minus1

f ℎ

2(x)

d

119871g1

119871

120588119898minus1

f ℎ

119898(x) 119871g

2

119871

120588119898minus1

f ℎ


119898

119871

120588119898minus1

f ℎ

119898(x)

]

]

]

]

]

]

(7)





2 120591

119898






2 120588

119898


2 120591

119898 where 120591

119894ge 120588

119894(119894 = 1 119898)


119910

(120588119894)

119894= 119871

120588119894

119891ℎ

119894 (x) + (119871

119892119871

120588119894minus1

119891ℎ

119894 (x)) u

+ 119871

119863119871

120588119894minus1

119891ℎ

119894(x) (119894 = 1 119898)

(8)


y(120588) = 120572 (x) + 120573 (x)u + Δ (x uD) (9)

where

120572 (x) = [120572

1 120572


119898]

Tisin R119898

120573 (x) = [120573

1 120573


119898]


Δ (x uD) = [120575

1 120575


119898]

Tisin R119898

(10)

where 120572119894= 119871

120588119894

119891ℎ

119894(x) 120573119894= 119871

119892119871

120588119894minus1

119891ℎ

119894(x) 120575119894= 119871

119863119871

120588119894minus1

119891ℎ

119894(x) (119894 =



s (119905) = [119904

1 119904


119898]

Tisin R119898 (11)


119904

119894= 119890

[120588119894minus1]

119894+ 119896

119894120588119894minus1119890

[120588119894minus2]

119894+ sdot sdot sdot + 119896

1198940119890

119894 (12)

where 119890

119894= 119910

119894minus 119910

119888119894(119894 = 1 119898) and 119896

119894119895(119895 =

0 1 120588



119904

119894= 119890

[120588119894]

119894+ 119896

119894120588119894minus1119890

[120588119894minus1]

119894+ sdot sdot sdot + 119896

1198940119890

119894

= 119910

[120588119894]

119894minus 119910

[120588119894]

119888119894+ 119896

119894120588119894minus1119890

[120588119894minus1]

119894


1198940119890

119894 (119894 = 1 119898)

(13)


119911

119894= 119896

119894120588119894minus1119890

[120588119894minus1]

119894+ sdot sdot sdot + 119896

1198940119890

119894(119894 = 1 119898)

(14)



z = [119911

1 119911


119898]

Tisin R119898 (15)


s (119905) = y(120588) minus y(120588)c + z (16)

where y(120588)c = [119910

(1205881)

1198881 119910

(1205882)


(120588119898)

119888119898]

Tisin R119898


s (119905 + 119879) = s (119905) + 119879 s (119905) (17)


119869 (x u 119905) = 1

2

sT (119905 + 119879) s (119905 + 119879) (18)


120597119869

120597u= 0

(19)


u = minus119879

minus1120573 (119909)minus1






1 119883

2 119883

119899]



119877

(119894) If 1198831is 1198601198941 119883

119899is 119860119894119899 then 119884 is 119884119894 (21)

where 119860

119894

1 119860

119894




119884 (X) =sum

119897

119894=1119884

119894(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

=

120579

T120599 (X) (22)


119895

(119883

119895) is the mem-


120579 = [119884

1 119884

2 119884

119897]


1 120599

2 120599

119897]


120599

119894=

prod

119899

119895=1120583

119860119894

119895

(119883

119895)

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

(23)


sup119909isinU

1003816

1003816

1003816

1003816

119901 (X) minus 119884 (X)1003816100381610038161003816

lt 120576 (24)


Δ (x | 120579TΔ) = [

120575

1

120575

2sdot sdot sdot

120575

119898]

T=

120579

TΔ120599Δ(x) (25)

where

120579

TΔ= diag 120579

T1

120579

T2

120579

T119898

120579

T119894= (119884

1

119894 119884

2

119894 119884

119897

119894)

T

120599Δ (

x) = [1205991 (x)T 1205992 (x)

T 120599

119898 (x)T]

T

120599 (x) = [120599

1

119894 120599

2

119894 120599

119897

119894]

T

(26)


Δcan be defined as

120579lowastTΔ

= argminTΔ

supxisinΩx

1003817

1003817

1003817

1003817

1003817

1003817

1003817

Δ (x) minus

Δ (x | 120579TΔ)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

(27)



M120579 =

120579Δ|

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

le 119898

120579 (28)


with119898120579gt 0


Δ (x) = 120579lowastTΔ120599Δ(x) + 120576 (29)


Δis bounded with

120576 le 120576 (30)






= minus120594120583 + 119901 (x u 120579Δ) (31)


Δ(x | 120579TΔ) 120594 gt 0


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ) = minus120590120577 +

120579

TΔ120599Δ(x) + 120576

(32)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ(x) minus y(120588)c + z))

(33)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576

(34)


= A120589 + B (

120579

TΔ120599Δ(x) + 120576) (35)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(36)

where119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(37)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ (38)


119881 = 120589T +

1

120582

120579

TΔ

120579Δ= 120577

T

120577 + sT s + 1

120582

120579

TΔ

120579Δ

= 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576) + sT (minus119879minus1s +

120579

TΔ120599Δ (

x) + 120576)

minus

1

120582

120579

TΔProj [120582120593

Δ(x) 120589TB] + 1

120578

120599

120599

= 120577T(minus120594120577 +

120579


120579

TΔ120599Δ(x) + 120576)

minus

120579

TΔ120599Δ (

x) 120589T +

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(39)


120579

TΔ

120579Δ= 120579lowastTΔ

120579Δminus

120579

TΔ

120579Δ

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2+

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

) minus

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2minus 119898

120579

2)

le 0 (when 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579)

(40)


120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ= 0

or

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δlt 0

(41)


119881 le 120577T(minus120594120577 +

120579

TΔ120593Δ(x) + 120576)


120579

TΔ120593Δ(x) + 120576) minus

120579

TΔ120593Δ(x) 120589T

le 120577T(minus120594120577 +

120579


120579

TΔ120593Δ(x) + 120576)

minus

120579

TΔ120593Δ(x) 120577 minus

120579

TΔ120593Δ(x) s



le minus120594 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576

le minus

120594

2

1205772+

1

2120594

1205762minus

1

2119879

s2 + 119879

2

1205762

(42)

If 120577 gt 120576120594 or s gt 119879120576



Δ(x |

120579

TΔ)



120579Δ(0) = 0

(43)


Δ can be






) le 119898120581119888 (44)



= minus120594120583 + 119901 (x u 120579Δ) (45)


Δ(x |

120579


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ)

= minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

)

(46)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ (

x)

minus y(120588)c + z + 120592 tanh( s (119905)119888

)))

(47)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

)

(48)


= A120589 + B(

120579

TΔ120599Δ(x) + 120576120592 tanh(120589

TB119888

)) (49)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(50)

= 120578

1003817

1003817

1003817

1003817

1003817

120589TB100381710038171003817

1003817

1003817

(51)

where

119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(52)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ+

1

2120578

120592

2 (53)



119881 = 120589T +

1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T

120577 + sT s + 1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576)

minus 120592 tanh( s119888

) minus

1

120582

120579

TΔProj [120582120599

Δ(x) 120589TB] + 1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ (

x) + 120576)

minus 120592 tanh( s119888

) minus

120579

TΔ120599Δ(x) 120589T

+

120579

TΔ119868120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ+

1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

))


120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh( s119888

))

minus

120579

TΔ120599Δ(x) 120589T + 1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

))

minus

120579

TΔ120599Δ (

x) 120577 minus

120579

TΔ120599Δ (

x) s + 1

120578

120592


119888

)


) + 120592 120577 + 120592 s

(54)


minus 120577T120592 tanh(120577

119888

) le minus |120592| 120577 + 119898120581119888


) le minus |120592| s + 119898120581119888

(55)


119881 le minus120590 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576 minus |120592| 120577

+ 119898120581119888 minus |120592| s + 119898120581119888 + 120592 120577 + 120592 s

le minus120590 1205772+ 120577 120576 minus 119879

minus1s2 + s 120576 minus 120592 120577

+ 120592 120577 minus 120592 120577 + 120592 s + 2119898120581119888

le minus120590 1205772minus 119879

minus1s2 + 2119898120581119888

(56)


2119898120581119888119879





119909

119903

119909

119888

=

2

119909 + 2

(57)

where 119909

119903and 119909




119909 yaw rate 120596




T and120596 = [120596

119909 120596

119910 120596

119911]



Θ = F120579120596


(58)



controller

Attitudemodel

Fuzzydisturbance


controller

Commandfilter


Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590


where

F120579=

[

[


]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)


y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)


s (119905) = e + 119870e (62)


119888


119888 120574

119888]


119888 120573

119888 120574

119881119888

]


z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)


u0= minus119879



u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where




+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119888 120573

119888 120574

119881119888

]


119888 120573

119888 120574

119881119888

]

T into [120593

119888 120595

119888 120574

119888]




x = f (x) + Δf (x) + (g (x) + Δg (x)) u + d

y = h (x) (5)


Define

D (x u d) = Δf (x) +Δg (x)u + d (6)



0= 0 In accordance



1 120588

2 120588

119898 at x


where 1205881 120588

2 120588


nel respectively


119871

119896

fℎ119894(x) = 0 (1 le119894 119895 le 119898 0 lt 119896 lt 120588

119894)


A (x)

=

[

[

[

[

[

[

119871g1

119871

1205881minus1

f ℎ

1(x) 119871g

2

119871

1205881minus1

f ℎ


119898

119871

1205881minus1

f ℎ

1(x)

119871g1

119871

1205882minus1

f ℎ

2(x) 119871g

2

119871

1205882minus1

f ℎ


119898

119871

1205882minus1

f ℎ

2(x)

d

119871g1

119871

120588119898minus1

f ℎ

119898(x) 119871g

2

119871

120588119898minus1

f ℎ


119898

119871

120588119898minus1

f ℎ

119898(x)

]

]

]

]

]

]

(7)





2 120591

119898






2 120588

119898


2 120591

119898 where 120591

119894ge 120588

119894(119894 = 1 119898)


119910

(120588119894)

119894= 119871

120588119894

119891ℎ

119894 (x) + (119871

119892119871

120588119894minus1

119891ℎ

119894 (x)) u

+ 119871

119863119871

120588119894minus1

119891ℎ

119894(x) (119894 = 1 119898)

(8)


y(120588) = 120572 (x) + 120573 (x)u + Δ (x uD) (9)

where

120572 (x) = [120572

1 120572


119898]

Tisin R119898

120573 (x) = [120573

1 120573


119898]


Δ (x uD) = [120575

1 120575


119898]

Tisin R119898

(10)

where 120572119894= 119871

120588119894

119891ℎ

119894(x) 120573119894= 119871

119892119871

120588119894minus1

119891ℎ

119894(x) 120575119894= 119871

119863119871

120588119894minus1

119891ℎ

119894(x) (119894 =



s (119905) = [119904

1 119904


119898]

Tisin R119898 (11)


119904

119894= 119890

[120588119894minus1]

119894+ 119896

119894120588119894minus1119890

[120588119894minus2]

119894+ sdot sdot sdot + 119896

1198940119890

119894 (12)

where 119890

119894= 119910

119894minus 119910

119888119894(119894 = 1 119898) and 119896

119894119895(119895 =

0 1 120588



119904

119894= 119890

[120588119894]

119894+ 119896

119894120588119894minus1119890

[120588119894minus1]

119894+ sdot sdot sdot + 119896

1198940119890

119894

= 119910

[120588119894]

119894minus 119910

[120588119894]

119888119894+ 119896

119894120588119894minus1119890

[120588119894minus1]

119894


1198940119890

119894 (119894 = 1 119898)

(13)


119911

119894= 119896

119894120588119894minus1119890

[120588119894minus1]

119894+ sdot sdot sdot + 119896

1198940119890

119894(119894 = 1 119898)

(14)



z = [119911

1 119911


119898]

Tisin R119898 (15)


s (119905) = y(120588) minus y(120588)c + z (16)

where y(120588)c = [119910

(1205881)

1198881 119910

(1205882)


(120588119898)

119888119898]

Tisin R119898


s (119905 + 119879) = s (119905) + 119879 s (119905) (17)


119869 (x u 119905) = 1

2

sT (119905 + 119879) s (119905 + 119879) (18)


120597119869

120597u= 0

(19)


u = minus119879

minus1120573 (119909)minus1






1 119883

2 119883

119899]



119877

(119894) If 1198831is 1198601198941 119883

119899is 119860119894119899 then 119884 is 119884119894 (21)

where 119860

119894

1 119860

119894




119884 (X) =sum

119897

119894=1119884

119894(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

=

120579

T120599 (X) (22)


119895

(119883

119895) is the mem-


120579 = [119884

1 119884

2 119884

119897]


1 120599

2 120599

119897]


120599

119894=

prod

119899

119895=1120583

119860119894

119895

(119883

119895)

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

(23)


sup119909isinU

1003816

1003816

1003816

1003816

119901 (X) minus 119884 (X)1003816100381610038161003816

lt 120576 (24)


Δ (x | 120579TΔ) = [

120575

1

120575

2sdot sdot sdot

120575

119898]

T=

120579

TΔ120599Δ(x) (25)

where

120579

TΔ= diag 120579

T1

120579

T2

120579

T119898

120579

T119894= (119884

1

119894 119884

2

119894 119884

119897

119894)

T

120599Δ (

x) = [1205991 (x)T 1205992 (x)

T 120599

119898 (x)T]

T

120599 (x) = [120599

1

119894 120599

2

119894 120599

119897

119894]

T

(26)


Δcan be defined as

120579lowastTΔ

= argminTΔ

supxisinΩx

1003817

1003817

1003817

1003817

1003817

1003817

1003817

Δ (x) minus

Δ (x | 120579TΔ)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

(27)



M120579 =

120579Δ|

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

le 119898

120579 (28)


with119898120579gt 0


Δ (x) = 120579lowastTΔ120599Δ(x) + 120576 (29)


Δis bounded with

120576 le 120576 (30)






= minus120594120583 + 119901 (x u 120579Δ) (31)


Δ(x | 120579TΔ) 120594 gt 0


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ) = minus120590120577 +

120579

TΔ120599Δ(x) + 120576

(32)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ(x) minus y(120588)c + z))

(33)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576

(34)


= A120589 + B (

120579

TΔ120599Δ(x) + 120576) (35)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(36)

where119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(37)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ (38)


119881 = 120589T +

1

120582

120579

TΔ

120579Δ= 120577

T

120577 + sT s + 1

120582

120579

TΔ

120579Δ

= 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576) + sT (minus119879minus1s +

120579

TΔ120599Δ (

x) + 120576)

minus

1

120582

120579

TΔProj [120582120593

Δ(x) 120589TB] + 1

120578

120599

120599

= 120577T(minus120594120577 +

120579


120579

TΔ120599Δ(x) + 120576)

minus

120579

TΔ120599Δ (

x) 120589T +

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(39)


120579

TΔ

120579Δ= 120579lowastTΔ

120579Δminus

120579

TΔ

120579Δ

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2+

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

) minus

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2minus 119898

120579

2)

le 0 (when 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579)

(40)


120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ= 0

or

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δlt 0

(41)


119881 le 120577T(minus120594120577 +

120579

TΔ120593Δ(x) + 120576)


120579

TΔ120593Δ(x) + 120576) minus

120579

TΔ120593Δ(x) 120589T

le 120577T(minus120594120577 +

120579


120579

TΔ120593Δ(x) + 120576)

minus

120579

TΔ120593Δ(x) 120577 minus

120579

TΔ120593Δ(x) s



le minus120594 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576

le minus

120594

2

1205772+

1

2120594

1205762minus

1

2119879

s2 + 119879

2

1205762

(42)

If 120577 gt 120576120594 or s gt 119879120576



Δ(x |

120579

TΔ)



120579Δ(0) = 0

(43)


Δ can be






) le 119898120581119888 (44)



= minus120594120583 + 119901 (x u 120579Δ) (45)


Δ(x |

120579


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ)

= minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

)

(46)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ (

x)

minus y(120588)c + z + 120592 tanh( s (119905)119888

)))

(47)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

)

(48)


= A120589 + B(

120579

TΔ120599Δ(x) + 120576120592 tanh(120589

TB119888

)) (49)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(50)

= 120578

1003817

1003817

1003817

1003817

1003817

120589TB100381710038171003817

1003817

1003817

(51)

where

119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(52)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ+

1

2120578

120592

2 (53)



119881 = 120589T +

1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T

120577 + sT s + 1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576)

minus 120592 tanh( s119888

) minus

1

120582

120579

TΔProj [120582120599

Δ(x) 120589TB] + 1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ (

x) + 120576)

minus 120592 tanh( s119888

) minus

120579

TΔ120599Δ(x) 120589T

+

120579

TΔ119868120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ+

1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

))


120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh( s119888

))

minus

120579

TΔ120599Δ(x) 120589T + 1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

))

minus

120579

TΔ120599Δ (

x) 120577 minus

120579

TΔ120599Δ (

x) s + 1

120578

120592


119888

)


) + 120592 120577 + 120592 s

(54)


minus 120577T120592 tanh(120577

119888

) le minus |120592| 120577 + 119898120581119888


) le minus |120592| s + 119898120581119888

(55)


119881 le minus120590 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576 minus |120592| 120577

+ 119898120581119888 minus |120592| s + 119898120581119888 + 120592 120577 + 120592 s

le minus120590 1205772+ 120577 120576 minus 119879

minus1s2 + s 120576 minus 120592 120577

+ 120592 120577 minus 120592 120577 + 120592 s + 2119898120581119888

le minus120590 1205772minus 119879

minus1s2 + 2119898120581119888

(56)


2119898120581119888119879





119909

119903

119909

119888

=

2

119909 + 2

(57)

where 119909

119903and 119909




119909 yaw rate 120596




T and120596 = [120596

119909 120596

119910 120596

119911]



Θ = F120579120596


(58)



controller

Attitudemodel

Fuzzydisturbance


controller

Commandfilter


Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590


where

F120579=

[

[


]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)


y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)


s (119905) = e + 119870e (62)


119888


119888 120574

119888]


119888 120573

119888 120574

119881119888

]


z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)


u0= minus119879



u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where




+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











z = [119911

1 119911


119898]

Tisin R119898 (15)


s (119905) = y(120588) minus y(120588)c + z (16)

where y(120588)c = [119910

(1205881)

1198881 119910

(1205882)


(120588119898)

119888119898]

Tisin R119898


s (119905 + 119879) = s (119905) + 119879 s (119905) (17)


119869 (x u 119905) = 1

2

sT (119905 + 119879) s (119905 + 119879) (18)


120597119869

120597u= 0

(19)


u = minus119879

minus1120573 (119909)minus1






1 119883

2 119883

119899]



119877

(119894) If 1198831is 1198601198941 119883

119899is 119860119894119899 then 119884 is 119884119894 (21)

where 119860

119894

1 119860

119894




119884 (X) =sum

119897

119894=1119884

119894(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

=

120579

T120599 (X) (22)


119895

(119883

119895) is the mem-


120579 = [119884

1 119884

2 119884

119897]


1 120599

2 120599

119897]


120599

119894=

prod

119899

119895=1120583

119860119894

119895

(119883

119895)

sum

119897

119894=1(prod

119899

119895=1120583

119860119894

119895

(119883

119895))

(23)


sup119909isinU

1003816

1003816

1003816

1003816

119901 (X) minus 119884 (X)1003816100381610038161003816

lt 120576 (24)


Δ (x | 120579TΔ) = [

120575

1

120575

2sdot sdot sdot

120575

119898]

T=

120579

TΔ120599Δ(x) (25)

where

120579

TΔ= diag 120579

T1

120579

T2

120579

T119898

120579

T119894= (119884

1

119894 119884

2

119894 119884

119897

119894)

T

120599Δ (

x) = [1205991 (x)T 1205992 (x)

T 120599

119898 (x)T]

T

120599 (x) = [120599

1

119894 120599

2

119894 120599

119897

119894]

T

(26)


Δcan be defined as

120579lowastTΔ

= argminTΔ

supxisinΩx

1003817

1003817

1003817

1003817

1003817

1003817

1003817

Δ (x) minus

Δ (x | 120579TΔ)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

(27)



M120579 =

120579Δ|

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

le 119898

120579 (28)


with119898120579gt 0


Δ (x) = 120579lowastTΔ120599Δ(x) + 120576 (29)


Δis bounded with

120576 le 120576 (30)






= minus120594120583 + 119901 (x u 120579Δ) (31)


Δ(x | 120579TΔ) 120594 gt 0


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ) = minus120590120577 +

120579

TΔ120599Δ(x) + 120576

(32)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ(x) minus y(120588)c + z))

(33)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576

(34)


= A120589 + B (

120579

TΔ120599Δ(x) + 120576) (35)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(36)

where119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(37)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ (38)


119881 = 120589T +

1

120582

120579

TΔ

120579Δ= 120577

T

120577 + sT s + 1

120582

120579

TΔ

120579Δ

= 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576) + sT (minus119879minus1s +

120579

TΔ120599Δ (

x) + 120576)

minus

1

120582

120579

TΔProj [120582120593

Δ(x) 120589TB] + 1

120578

120599

120599

= 120577T(minus120594120577 +

120579


120579

TΔ120599Δ(x) + 120576)

minus

120579

TΔ120599Δ (

x) 120589T +

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(39)


120579

TΔ

120579Δ= 120579lowastTΔ

120579Δminus

120579

TΔ

120579Δ

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2+

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

) minus

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2minus 119898

120579

2)

le 0 (when 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579)

(40)


120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ= 0

or

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δlt 0

(41)


119881 le 120577T(minus120594120577 +

120579

TΔ120593Δ(x) + 120576)


120579

TΔ120593Δ(x) + 120576) minus

120579

TΔ120593Δ(x) 120589T

le 120577T(minus120594120577 +

120579


120579

TΔ120593Δ(x) + 120576)

minus

120579

TΔ120593Δ(x) 120577 minus

120579

TΔ120593Δ(x) s



le minus120594 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576

le minus

120594

2

1205772+

1

2120594

1205762minus

1

2119879

s2 + 119879

2

1205762

(42)

If 120577 gt 120576120594 or s gt 119879120576



Δ(x |

120579

TΔ)



120579Δ(0) = 0

(43)


Δ can be






) le 119898120581119888 (44)



= minus120594120583 + 119901 (x u 120579Δ) (45)


Δ(x |

120579


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ)

= minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

)

(46)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ (

x)

minus y(120588)c + z + 120592 tanh( s (119905)119888

)))

(47)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

)

(48)


= A120589 + B(

120579

TΔ120599Δ(x) + 120576120592 tanh(120589

TB119888

)) (49)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(50)

= 120578

1003817

1003817

1003817

1003817

1003817

120589TB100381710038171003817

1003817

1003817

(51)

where

119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(52)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ+

1

2120578

120592

2 (53)



119881 = 120589T +

1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T

120577 + sT s + 1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576)

minus 120592 tanh( s119888

) minus

1

120582

120579

TΔProj [120582120599

Δ(x) 120589TB] + 1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ (

x) + 120576)

minus 120592 tanh( s119888

) minus

120579

TΔ120599Δ(x) 120589T

+

120579

TΔ119868120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ+

1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

))


120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh( s119888

))

minus

120579

TΔ120599Δ(x) 120589T + 1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

))

minus

120579

TΔ120599Δ (

x) 120577 minus

120579

TΔ120599Δ (

x) s + 1

120578

120592


119888

)


) + 120592 120577 + 120592 s

(54)


minus 120577T120592 tanh(120577

119888

) le minus |120592| 120577 + 119898120581119888


) le minus |120592| s + 119898120581119888

(55)


119881 le minus120590 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576 minus |120592| 120577

+ 119898120581119888 minus |120592| s + 119898120581119888 + 120592 120577 + 120592 s

le minus120590 1205772+ 120577 120576 minus 119879

minus1s2 + s 120576 minus 120592 120577

+ 120592 120577 minus 120592 120577 + 120592 s + 2119898120581119888

le minus120590 1205772minus 119879

minus1s2 + 2119898120581119888

(56)


2119898120581119888119879





119909

119903

119909

119888

=

2

119909 + 2

(57)

where 119909

119903and 119909




119909 yaw rate 120596




T and120596 = [120596

119909 120596

119910 120596

119911]



Θ = F120579120596


(58)



controller

Attitudemodel

Fuzzydisturbance


controller

Commandfilter


Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590


where

F120579=

[

[


]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)


y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)


s (119905) = e + 119870e (62)


119888


119888 120574

119888]


119888 120573

119888 120574

119881119888

]


z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)


u0= minus119879



u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where




+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













= minus120594120583 + 119901 (x u 120579Δ) (31)


Δ(x | 120579TΔ) 120594 gt 0


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ) = minus120590120577 +

120579

TΔ120599Δ(x) + 120576

(32)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ(x) minus y(120588)c + z))

(33)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576

(34)


= A120589 + B (

120579

TΔ120599Δ(x) + 120576) (35)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(36)

where119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(37)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ (38)


119881 = 120589T +

1

120582

120579

TΔ

120579Δ= 120577

T

120577 + sT s + 1

120582

120579

TΔ

120579Δ

= 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576) + sT (minus119879minus1s +

120579

TΔ120599Δ (

x) + 120576)

minus

1

120582

120579

TΔProj [120582120593

Δ(x) 120589TB] + 1

120578

120599

120599

= 120577T(minus120594120577 +

120579


120579

TΔ120599Δ(x) + 120576)

minus

120579

TΔ120599Δ (

x) 120589T +

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(39)


120579

TΔ

120579Δ= 120579lowastTΔ

120579Δminus

120579

TΔ

120579Δ

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2+

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

) minus

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

le

1

2

(

1003817

1003817

1003817

1003817

120579lowast

Δ

1003817

1003817

1003817

1003817

2minus 119898

120579

2)

le 0 (when 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579)

(40)


120579

TΔ119868

120579

120589TB120579

TΔ120599Δ (

x)1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ= 0

or

120579

TΔ119868

120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δlt 0

(41)


119881 le 120577T(minus120594120577 +

120579

TΔ120593Δ(x) + 120576)


120579

TΔ120593Δ(x) + 120576) minus

120579

TΔ120593Δ(x) 120589T

le 120577T(minus120594120577 +

120579


120579

TΔ120593Δ(x) + 120576)

minus

120579

TΔ120593Δ(x) 120577 minus

120579

TΔ120593Δ(x) s



le minus120594 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576

le minus

120594

2

1205772+

1

2120594

1205762minus

1

2119879

s2 + 119879

2

1205762

(42)

If 120577 gt 120576120594 or s gt 119879120576



Δ(x |

120579

TΔ)



120579Δ(0) = 0

(43)


Δ can be






) le 119898120581119888 (44)



= minus120594120583 + 119901 (x u 120579Δ) (45)


Δ(x |

120579


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ)

= minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

)

(46)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ (

x)

minus y(120588)c + z + 120592 tanh( s (119905)119888

)))

(47)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

)

(48)


= A120589 + B(

120579

TΔ120599Δ(x) + 120576120592 tanh(120589

TB119888

)) (49)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(50)

= 120578

1003817

1003817

1003817

1003817

1003817

120589TB100381710038171003817

1003817

1003817

(51)

where

119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(52)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ+

1

2120578

120592

2 (53)



119881 = 120589T +

1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T

120577 + sT s + 1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576)

minus 120592 tanh( s119888

) minus

1

120582

120579

TΔProj [120582120599

Δ(x) 120589TB] + 1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ (

x) + 120576)

minus 120592 tanh( s119888

) minus

120579

TΔ120599Δ(x) 120589T

+

120579

TΔ119868120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ+

1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

))


120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh( s119888

))

minus

120579

TΔ120599Δ(x) 120589T + 1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

))

minus

120579

TΔ120599Δ (

x) 120577 minus

120579

TΔ120599Δ (

x) s + 1

120578

120592


119888

)


) + 120592 120577 + 120592 s

(54)


minus 120577T120592 tanh(120577

119888

) le minus |120592| 120577 + 119898120581119888


) le minus |120592| s + 119898120581119888

(55)


119881 le minus120590 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576 minus |120592| 120577

+ 119898120581119888 minus |120592| s + 119898120581119888 + 120592 120577 + 120592 s

le minus120590 1205772+ 120577 120576 minus 119879

minus1s2 + s 120576 minus 120592 120577

+ 120592 120577 minus 120592 120577 + 120592 s + 2119898120581119888

le minus120590 1205772minus 119879

minus1s2 + 2119898120581119888

(56)


2119898120581119888119879





119909

119903

119909

119888

=

2

119909 + 2

(57)

where 119909

119903and 119909




119909 yaw rate 120596




T and120596 = [120596

119909 120596

119910 120596

119911]



Θ = F120579120596


(58)



controller

Attitudemodel

Fuzzydisturbance


controller

Commandfilter


Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590


where

F120579=

[

[


]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)


y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)


s (119905) = e + 119870e (62)


119888


119888 120574

119888]


119888 120573

119888 120574

119881119888

]


z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)


u0= minus119879



u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where




+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le minus120594 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576

le minus

120594

2

1205772+

1

2120594

1205762minus

1

2119879

s2 + 119879

2

1205762

(42)

If 120577 gt 120576120594 or s gt 119879120576



Δ(x |

120579

TΔ)



120579Δ(0) = 0

(43)


Δ can be






) le 119898120581119888 (44)



= minus120594120583 + 119901 (x u 120579Δ) (45)


Δ(x |

120579


Δ(x |

120579

TΔ) =

120579






Δ (x | 120579TΔ)

= minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

)

(46)


Δminus



u = minus119879

minus1120573 (119909)minus1

sdot (s (119905) + 119879 (120572 (119909) +

120579

TΔ120599Δ (

x)

minus y(120588)c + z + 120592 tanh( s (119905)119888

)))

(47)



= minus119879

minus1s +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

)

(48)


= A120589 + B(

120579

TΔ120599Δ(x) + 120576120592 tanh(120589

TB119888

)) (49)


119898 minus119879


(I119898 I119898)

T


120579Δ= Proj [120582120599

Δ(x) 120589TB]

= 120582120599Δ(x) 120589T minus 119868

120579120582

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ

(50)

= 120578

1003817

1003817

1003817

1003817

1003817

120589TB100381710038171003817

1003817

1003817

(51)

where

119868

120579

=

0 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

lt 119898

120579 or 1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) le 0

1 if 1003817100381710038171003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

= 119898

120579 120589

TB120579TΔ120599Δ(x) gt 0

(52)


119881 =

1

2

120589T120589 +

1

2120582

120579

TΔ

120579Δ+

1

2120578

120592

2 (53)



119881 = 120589T +

1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T

120577 + sT s + 1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576)

minus 120592 tanh( s119888

) minus

1

120582

120579

TΔProj [120582120599

Δ(x) 120589TB] + 1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ (

x) + 120576)

minus 120592 tanh( s119888

) minus

120579

TΔ120599Δ(x) 120589T

+

120579

TΔ119868120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ+

1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

))


120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh( s119888

))

minus

120579

TΔ120599Δ(x) 120589T + 1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

))

minus

120579

TΔ120599Δ (

x) 120577 minus

120579

TΔ120599Δ (

x) s + 1

120578

120592


119888

)


) + 120592 120577 + 120592 s

(54)


minus 120577T120592 tanh(120577

119888

) le minus |120592| 120577 + 119898120581119888


) le minus |120592| s + 119898120581119888

(55)


119881 le minus120590 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576 minus |120592| 120577

+ 119898120581119888 minus |120592| s + 119898120581119888 + 120592 120577 + 120592 s

le minus120590 1205772+ 120577 120576 minus 119879

minus1s2 + s 120576 minus 120592 120577

+ 120592 120577 minus 120592 120577 + 120592 s + 2119898120581119888

le minus120590 1205772minus 119879

minus1s2 + 2119898120581119888

(56)


2119898120581119888119879





119909

119903

119909

119888

=

2

119909 + 2

(57)

where 119909

119903and 119909




119909 yaw rate 120596




T and120596 = [120596

119909 120596

119910 120596

119911]



Θ = F120579120596


(58)



controller

Attitudemodel

Fuzzydisturbance


controller

Commandfilter


Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590


where

F120579=

[

[


]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)


y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)


s (119905) = e + 119870e (62)


119888


119888 120574

119888]


119888 120573

119888 120574

119881119888

]


z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)


u0= minus119879



u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where




+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881 = 120589T +

1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T

120577 + sT s + 1

120582

120579

TΔ

120579Δ+

1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576)

minus 120592 tanh( s119888

) minus

1

120582

120579

TΔProj [120582120599

Δ(x) 120589TB] + 1

120578

120592

= 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ (

x) + 120576)

minus 120592 tanh( s119888

) minus

120579

TΔ120599Δ(x) 120589T

+

120579

TΔ119868120579

120589TB120579

TΔ120599Δ(x)

1003817

1003817

1003817

1003817

1003817

120579Δ

1003817

1003817

1003817

1003817

1003817

2

120579Δ+

1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh(120577119888

))


120579

TΔ120599Δ (

x) + 120576 minus 120592 tanh( s119888

))

minus

120579

TΔ120599Δ(x) 120589T + 1

120578

120592

le 120577T(minus120594120577 +

120579

TΔ120599Δ(x) + 120576 minus 120592 tanh(120577

119888

))


120579

TΔ120599Δ(x) + 120576 minus 120592 tanh( s

119888

))

minus

120579

TΔ120599Δ (

x) 120577 minus

120579

TΔ120599Δ (

x) s + 1

120578

120592


119888

)


) + 120592 120577 + 120592 s

(54)


minus 120577T120592 tanh(120577

119888

) le minus |120592| 120577 + 119898120581119888


) le minus |120592| s + 119898120581119888

(55)


119881 le minus120590 1205772+ 120577

T120576 minus 119879

minus1s2 + sT120576 minus |120592| 120577

+ 119898120581119888 minus |120592| s + 119898120581119888 + 120592 120577 + 120592 s

le minus120590 1205772+ 120577 120576 minus 119879

minus1s2 + s 120576 minus 120592 120577

+ 120592 120577 minus 120592 120577 + 120592 s + 2119898120581119888

le minus120590 1205772minus 119879

minus1s2 + 2119898120581119888

(56)


2119898120581119888119879





119909

119903

119909

119888

=

2

119909 + 2

(57)

where 119909

119903and 119909




119909 yaw rate 120596




T and120596 = [120596

119909 120596

119910 120596

119911]



Θ = F120579120596


(58)



controller

Attitudemodel

Fuzzydisturbance


controller

Commandfilter


Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590


where

F120579=

[

[


]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)


y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)


s (119905) = e + 119870e (62)


119888


119888 120574

119888]


119888 120573

119888 120574

119881119888

]


z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)


u0= minus119879



u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where




+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











controller

Attitudemodel

Fuzzydisturbance


controller

Commandfilter


Centroidmodel

+

minus

u0

u1

ux

yd(t)

120572c120573c120574V119888 120593c

120595c120574c

120593

120595

120574

Δ(x)

120572

120573

120574V

120579120590


where

F120579=

[

[


]

]

u =

[

[

119872

119909

119872

119910

119872

119911

]

]

F120596=

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

J = [

[

119869

1199090 0

0 119869

1199100

0 0 119869

119911

]

]

(59)


y = 120572 (x) + 120573 (x) u (60)

where

120572 (x)

=

[

[


]

]

sdot

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

sdot

[

[

[

(119869

119910minus 119869

119911) 120596

119911120596

119910minus

119869

119909120596

119909

(119869

119911minus 119869

119909) 120596

119909120596

119911minus

119869

119910120596

119910

(119869

119909minus 119869

119910) 120596

119909120596

119910minus

119869

119911120596

119911

]

]

]

+

[

[

[

[

[

[

[

[

120574 (120596

119910cos 120574 minus 120596

119911sin 120574)

(120596

119910cos 120574 minus 120596

119911sin 120574) tan120593 sec120593

minus 120574 (120596

119910sin 120574 + 120596

119911cos 120574) sec120593

120574 (120596

119910sin 120574 + 120596

119911cos 120574) tan120593

minus (120596

119910cos 120574 minus 120596

119911sin 120574) sec2120593

]

]

]

]

]

]

]

]

120573 (x) = [

[


]

]

[

[

[

119869

minus1

1199090 0

0 119869

minus1

1199100

0 0 119869

minus1

119911

]

]

]

(61)


s (119905) = e + 119870e (62)


119888


119888 120574

119888]


119888 120573

119888 120574

119881119888

]


z = 119870e (63)

In addition

Yc120588 = [

119888

120595

119888 120574

119888]

T

(64)


u0= minus119879



u1= minus119879

minus1120573 (x)minus1 Δ (x) (66)

where




+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











+

minus

1205962n2120577n120596n

+

minus

2120577n120596n1

s

1

sxc

xc



1(negative big) 119860119894

2



119894

5


1 1205752 and

120575



Δ (x) = [

120575

1

120575

2

120575

3]

T=

120579

TΔ120599Δ(x) (67)

where

120579

TΔ

= diag120579T1

120579

T2

120579


vector and 120599Δ(x) = [120599

1(x)T 120599

2(x)T 120599



u = u0+ u1 (68)





119909119872119910 and119872




119909

119888(119904)

119909

119888119889(119904)

=

120596

2

119899

119904

2+ 2120577

119899120596

119899+ 120596

2

119899

119878

119860(119904) 119878V (119904) (69)

where 120577119899and120596


119878



and





119888 120573

119888 120574

119881119888

]


0= 120573

0= 120574

1198810

= 0

and 120596

1199090

= 120596

1199100

= 120596

1199110


0= 120590


chosen as 120572119888= 5

∘ 120573119888= 0

∘ and 120574119881119888

= 5






119889

1 (119905) = 10

5 sin (2119905) N sdotm

119889

2(119905) = 2 times 10

5 sin (2119905) N sdotm

119889

3 (119905) = 10

6 sin (2119905) N sdotm

(70)



0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 5 10 15 20minus015

minus01

minus005

0

005

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus10

0

10

20

Time (s)

120575120601

(deg

)

SMCPSMC

(d)

0 5 10 15 20minus5

0

5

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus4

minus2

0

2

Time (s)

120575120574

(deg

)SMCPSMC

(f)



120583

1198601

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 1205876)

(12058724)

)

2

]

]

120583

1198602

119895

(119909

119895) = exp[

[

minus(

(119909

119895+ 12058712)

(12058724)

)

2

]

]

120583

1198603

119895

(119909


119909

119895

(12058724)

)

2

]

120583

1198604

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 12058712)

(12058724)

)

2

]

]

120583

1198605

119895

(119909

119895) = exp[

[

minus(

(119909

119895minus 1205876)

(12058724)

)

2

]

]

(71)




0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

2

4

6

Time (s)

CommandSMCPSMC

120572(d

eg)

(a)

0 10 20minus04

minus02

0

02

04

Time (s)

CommandSMCPSMC

120573(d

eg)

(b)

0 5 10 15 20minus5

0

5

10

Time (s)

CommandSMCPSMC

120574V

(deg

)

(c)

0 5 10 15 20minus20

minus10

0

10

20

Time (s)

SMCPSMC

120575120601

(deg

)

(d)

0 5 10 15 20minus10

minus5

0

5

10

Time (s)

SMCPSMC

120575120595

(deg

)

(e)

0 5 10 15 20minus10

minus5

0

5

Time (s)120575120574

(deg

)

SMCPSMC

(f)






6 Conclusions



0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

2

4

6

CommandPSMC

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(a)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 20minus02

minus01

0

01

02

Time (s)

120573(d

eg)

(b)

CommandPSMC

FDO-PSMCIFDO-PSMC

0 5 10 15 200

2

4

6

Time (s)

120574V

(deg

)

(c)

10 15 20

498

499

5

FDO-PSMCIFDO-PSMC

Time (s)

120572(d

eg)

(d)

FDO-PSMCIFDO-PSMC

10 15 20

minus2

0

2

4times10minus3

Time (s)

120573(d

eg)

(e)

FDO-PSMCIFDO-PSMC

10 15 20499

4995

5

5005

501

Time (s)120574V

(deg

)

(f)





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article predictive sliding mode control for...

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