Research ArticleGlobal Finite-Time Stabilization for a Class of UncertainHigh-Order Nonlinear Systems

Jian Wang1 Jing Xie2 and Fangzheng Gao1

1 School of Mathematics and Statistics Anyang Normal University Anyang 455002 China2 School of Mathematics and Physics Anyang Institute of Technology Anyang 455000 China

Correspondence should be addressed to Fangzheng Gao gaofz126com

Received 10 February 2014 Accepted 27 March 2014 Published 22 April 2014

Academic Editor Douglas R Anderson

Copyright copy 2014 Jian Wang 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

This paper addresses the problem of global finite-time stabilization by state feedback for a class of high-order nonlinear systemsunder weaker condition By using the methods of adding a power integrator a continuous state feedback controller is successfullyconstructed to guarantee the global finite-time stability of the resulting closed-loop system A simulation example is provided toillustrate the effectiveness of the proposed approach

1 Introduction

In this paper we consider the following high-order nonlinearsystems

119894= 119909

119901119894

119894+1+ 119891

119894 (119905 119909 119906) 119894 = 1 119899 minus 1

119899= 119906

119901119899+ 119891

119899 (119905 119909 119906)

(1)

where 119909 = (1199091 119909

119899)

119879isin 119877

119899 119906 isin 119877 are the system state andinput respectively 119901

119894isin 119877

ge1

odd=119901119902 | 119901 and 119902 are positiveodd integers and 119901 ge 119902 are said to be the high orders of thesystem 119891

119894 119877

+times 119877

119899times 119877 rarr 119877 119894 = 1 119899 are unknown

continuous functions of all the states and the control inputThe importance for studying such system is exemplified in

[1] where state feedback controller was used to stabilize theunderactuated weakly coupled unstable mechanical systemSince Jacobian linearization of system (1) at the origin isneither controllable nor feedback linearizable for the caseof 119901 gt 1 the traditional design tools including feedbacklinearization or backstepping are hardly applicable to thesystem (1) Mainly thanks to the adding a power integratormethod a series of stabilizing results have been achieved overthe last decades for example one can see [2ndash10] and thereferences therein However it should be mentioned that theaforementioned works only consider the feedback stabilizer

that makes the trajectories of the systems converge to theequilibrium as the time goes to infinity

Compared to the asymptotic stabilization the finite-timestabilization which renders the trajectories of the closed-loopsystems convergent to the origin in a finite time has manyadvantages such as fast response high tracking precisionand disturbance-rejection properties [11] Hence it is moremeaningful to investigate the finite-time stabilization prob-lem than the classical asymptotical stabilization In recentyears the finite-time stabilization of system (1) has beenstudied fairly extensively with various restrictions on theintegrator powers and the system nonlinearities [12ndash21] Inparticular [22] solved the finite-time stabilization problemunder the condition that 119891

119894satisfies

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 119872

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895 (2)

with constants 119872 gt 0 and 120596 isin (minus2(2119899 + 1)119901

1sdot sdot sdot 119901

119899minus1 0)

However from both practical and theoretical points of viewit is somewhat restrictive to require system (1) to satisfy suchrestriction Therefore the following interesting problem isproposed is it possible to relax the nonlinear growth conditionUnder the weaker condition can a finite-time stabilizingcontroller be designed

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 384713 7 pageshttpdxdoiorg1011552014384713

2 Abstract and Applied Analysis

In this paper by necessarily modifying the method ofadding a power integrator and by successfully overcomingsome essential difficulties such as the weaker assumption onthe system growth the appearance of the sign function andthe construction of a continuously differentiable positive-definite and proper Lyapunov function we will focus onsolving the above problem

NotationsThroughout this paper the following notations areadopted 119877+ denotes the set of all nonnegative real numbersand 119877119899 denotes the real 119899-dimensional space For any vector119909 = (119909

1 119909

119899)

119879isin 119877

119899 denote 119909119894= (119909

1 119909

119894)

119879isin 119877

119894119894 = 1 119899 119909 = (sum

119899

119894=1119909

2

119894)

12 119870 denotes the set ofall functions 119877+ rarr 119877

+ which are continuous strictlyincreasing and vanishing at zero 119870

infindenotes the set of

all functions which are of class 119870 and unbounded A signfunction sign(119909) is defined as follows sign(119909) = 1 if 119909 gt 0sign(119909) = 0 if 119909 = 0 and sign(119909) = minus1 if 119909 lt 0 Besidesthe arguments of the functions (or the functionals) will beomitted or simplified whenever no confusion can arise fromthe context For instance we sometimes denote a function119891(119909(119905)) by simply 119891(119909) 119891(sdot) or 119891

2 Problem Statement and Preliminaries

The objective of this paper is to develop a recursive designmethod for globally finite-time stabilizing system (1) via statefeedback under the following assumption

Assumption 1 For 119894 = 1 119899 there are smooth functions120593

119894(119909

119894) ge 0 and constant 120596 isin (minus1sum119899

119897=1119901

1sdot sdot sdot 119901

119897minus1 0) such that

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120593

119894(119909

119894)

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895 (3)

where 119903119894rsquos are defined as

119903

1= 1 119903

119894+1=

119903

119894+ 120596

119901

119894

119894 = 1 119899 (4)

Remark 2 It is worth pointing out that Assumption 1 whichgives the nonlinear growth condition on the system driftterms encompasses the assumptions in the closely relatedworks [14 21 22] To clearly show this we would like to makethe following comparisons to reveal the relationship betweenAssumption 1 and the counterparts in [14 21 22] that isAssumption 1 includes those as special cases

(i) In [14 21] the system nonlinearities 119891119894rsquos are required

to satisfy

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120574

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

) (5)

where 120574119894(119909

119894) ge 0 119894 = 1 119899 are 1198621 functions By (4) we get

that 119903119894= (1119901

1sdot sdot sdot 119901

119894minus1) + sum

119894minus1

119897=1(120596119901

119897sdot sdot sdot 119901

119894minus1) Furthermore

from 120596 isin (minus1sum

119899

119897=1119901

1sdot sdot sdot 119901

119897minus1 0) we have (119903

119894+ 120596)119903

119895lt 1

119894 = 1 119899 It means that1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120574

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

)

le 120574

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

(119903119894+120596)1199031 10038161003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

1minus(119903119894+120596)1199031

+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

(119903119894+120596)119903119894 10038161003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

1minus(119903119894+120596)119903119894)

(6)

Letting 120593119894(119909

119894) be smooth functions and satisfying 120593

119894(119909

119894) ge

max120574119894|119909

1|

1minus(119903119894+120596)1199031 120574

119894|119909

119894|

1minus(119903119894+120596)119903119894 we have

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120593

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

(119903119894+120596)1199031+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

(119903119894+120596)119903119894) (7)

from this we can see that the main assumption (4) in [14 21]is a special case of Assumption 1 above

(ii) In [22] the system nonlinearities 119891119894rsquos are required to

satisfy

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 119872

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895 (8)

with constants 119872 gt 0 and 120596 isin (minus2(2119899 + 1)119901

1sdot sdot sdot 119901

119899minus1 0)

Obviously when 120593119894(119909

119894) = 119872 inequality (4) degenerates to

inequality (8) Moreover the value range of 120596 in (4) is largerthan that in (8) Thus the main assumption (5) in [22] is aspecial case of Assumption 1 above

In the remainder of this section we present the followinglemmas which play an important role in the design process

Lemma 3 (see [11]) Consider the nonlinear system

= 119891 (119909)with 119891 (0) = 0 119909 isin 119877

119899 (9)

where 119891 1198800rarr 119877

119899 is continuous with respect to 119909 on an openneighborhood119880

0of the origin 119909 = 0 Suppose that there is a 1198621

function 119881(119909) defined in a neighborhood 119880 isin 119877119899 of the originreal numbers 119888 gt 0 and 0 lt 120572 lt 1 such that

(i) 119881(119909) is positive definite on 119880(ii)

119881(119909) + 119888119881

120572(119909) le 0 forall119909 isin 119880

Then the origin of (9) is finite-time stable with

119879 le

119881

1minus120572(119909 (0))

119888 (1 minus 120572)

(10)

for initial condition 119909(0) in some open neighborhood 119880 isin

119880

of the origin If 119880 = 119877

119899 and 119881(119909) is radially unbounded (ie119881(119909) rarr +infin as 119909 rarr +infin) the origin of system (9) is globallyfinite-time stable

Lemma 4 (see [6]) For 119909 119910 isin 119877 and 119901 ge 1 being a constantthe following inequalities hold

1003816

1003816

1003816

1003816

119909 + 119910

1003816

1003816

1003816

1003816

119901le 2

119901minus1 10038161003816

1003816

1003816

119909

119901+ 119910

11990110038161003816

1003816

1003816

(|119909| +

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

)

1119901le |119909|

1119901+

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

1119901le 2

(119901minus1)119901(|119909| +

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

)

1119901

(11)

Abstract and Applied Analysis 3

If 119901 ge 1 is odd then1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119901le 2

119901minus1 10038161003816

1003816

1003816

119909

119901minus 119910

11990110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

1119901minus 119910

111990110038161003816

1003816

1003816

1003816

le 2

(119901minus1)11990110038161003816

1003816

1003816

119909 minus y1003816100381610038161003816

1119901

(12)

Lemma 5 (see [10]) If 119901 = 119886119887 isin 119877ge1119900119889119889

with 119887 ge 1 then forany 119909 119910 isin 119877

1003816

1003816

1003816

1003816

119909

119901minus 119910

11990110038161003816

1003816

1003816

le 2

1minus111988710038161003816

1003816

1003816

1003816

sgn (119909) |119909|119886 minus sgn (119910) 1003816100381610038161003816

119910

1003816

1003816

1003816

1003816

11988610038161003816

1003816

1003816

1003816

1119887

(13)

Lemma 6 (see [23]) Let 119909 119910 be real variables then for anypositive real numbers 119886119898 and 119899 one has

119886|119909|

11989810038161003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

119899le 119887|119909|

119898+119899+

119899

119898 + 119899

(

119898 + 119899

119898

)

minus119898119899

times 119886

(119898+119899)119899119887

minus11989811989910038161003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

119898+119899

(14)

where 119887 gt 0 is any real number

3 Finite-Time Control Design

In this section we will construct a continuous state feedbackcontroller by applying the method of adding a power integra-tor For simplicity we denote sgn(119909)|119909|119886 ≜ [119909]119886 for any 119886 isin 119877+and 119909 isin 119877

Step 1 Let 1205851= [119909

1] and choose 119881

1= 119882

1= int

1199091

0119904

2minus11990321199011119889119904

Using (1) and Assumption 1 we have

119881

1= 119909

2minus11990321199011

1119909

1199011

2+ 119909

2minus11990321199011

1119891

1le 119909

2minus11990321199011

1119909

1199011

2+ 119909

2

1120593

1

(15)

Obviously the first virtual controller

119909

lowast

2= minus120573

1199032

1119909

1199032

1= minus120573

1199032

1[120585

1]

1199032 (16)

with 1205731ge (119899 + 120593

1)

111990321199011 being smooth results in

119881

1le minus119899

1003816

1003816

1003816

1003816

120585

1

1003816

1003816

1003816

1003816

2+ [120585

1]

2minus11990321199011(119909

1199011

2minus 119909

lowast1199011

2) (17)

Step k (119896 = 2 119899) Suppose at step 119896 minus 1 there are a 1198621proper and positive definite Lyapunov function 119881

119896minus1 and a

set of virtual controllers 119909lowast1 119909

lowast

119896defined by

119909

lowast

1= 0

119909

lowast

2= minus 120573

1199032

1[120585

1]

1199032

119909

lowast

119896= minus 120573

119903119896

119896minus1[120585

119896minus1]

119903119896

120585

1= [119909

1]

11199031minus [119909

lowast

1]

11199031

120585

2= [119909

2]

11199032minus [119909

lowast

2]

11199032

120585

119896= [119909

119896]

1119903119896minus [119909

lowast

119896]

1119903119896

(18)

with 1205731gt 0 120573

119896minus1gt 0 being smooth such that

119881

119896minus1le minus (119899 minus 119896 + 2)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896minus1]

2minus119903119896119901119896minus1

times (119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

(19)

To complete the induction at the 119896th step we choose thefollowing Lyapunov function

119881

119896(119909

119896) = 119881

119896minus1(119909

119896minus1) + 119882

119896(119909

119896) (20)

where

119882

119896= int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904 (21)

Noting that 2 minus 119903119896+1119901

119896ge 1 and using a similar method as

in [10]119881119896can be shown to be1198621 proper and positive definite

Moreover we can obtain

120597119882

119896

120597119909

119896

= [120585

119896]

2minus119903119896+1119901119896

120597119882

119896

120597119909

119895

= minus (2 minus 119903

119896+1119901

119896)

times int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(22)

where 119895 = 1 119896 minus 1Using (20)ndash(22) it follows that

119881

119896le minus (119899 minus 119896 + 2)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

+[120585

119896]

2minus119903119896+1119901119896119909

119901119896

119896+1+[120585

119896]

2minus119903119896+1119901119896119891

119896+

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

(23)

In order to proceed further an appropriate boundingestimate should be given for the last three terms on the right-hand side of inequality (23) This is accomplished in thefollowing propositions whose technical proofs are given inthe appendix

Proposition 7 There exists a positive constant 1198971198961such that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896) le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (24)

Proposition 8 There exists a nonnegative smooth function 1198971198962

such that

[120585

119896]

2minus119903119896+1119901119896119891

119896le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (25)

4 Abstract and Applied Analysis

Proposition 9 There exists a nonnegative smooth function 1198971198963

such that

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (26)

Substituting (24)ndash(26) into (23) yields

119881

119896le minus (119899 minus 119896 + 1)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896119909

119901119896

119896+1

+ (119897

1198961+ 119897

1198962+ 119897

1198963)

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(27)

Now it easy to see that the virtual controller

119909

lowast

119896+1= minus120573

119903119896+1

119896[120585

119896]

119903119896+1 (28)

where 120573119896ge (119899 minus 119896 + 1 + 119897

1198961+ 119897

1198962+ 119897

1198963)

1(119903119896+1119901119896) is a smoothfunction renders

119881

119896le minus (119899 minus 119896 + 1)

119896

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896(119909

119901119896

119896+1minus 119909

lowast119901119896

119896+1) (29)

This completes the inductive step

Using the inductive argument above we can concludethat at the 119899th step there exists a continuous state feedbackcontroller of the form

119906 = 119909

lowast

119899+1= minus120573

119903119899+1

119899[120585

119899]

119903119899+1 (30)

such that

119881

119899le minus

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

(31)

where

119881

119899=

119899

sum

119896=1

119882

119896=

119899

sum

119896=1

int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904 (32)

4 Stability Analysis

We state the main result in this paper

Theorem 10 If Assumption 1 holds for system (1) under thecontinuous state feedback controller (30) then the followingholds

(i) all the solutions of the closed-loop system are welldefined on [0 +infin)

(ii) the equilibrium 119909 = 0 of the closed-loop system isglobally finite-time stable

Proof (i) Considering (18) system (1) can be equivalentlytransformed into a 120585-system

120585

119894 (119905) = 120593119894 (

119905 120585 119906) 119894 = 1 119899 (33)

By the existence and continuation of the solutions states120585(119905) are defined on [0 119905

119872] where the number 119905

119872gt 0 may

be infinite or not The following analysis focuses on [0 119905119872]

Noting that 119881119899is positive definite and radially unbounded

by (31) (32) and Lemma 43 in [24] it is not hard to obtainthat there exist 119870

infinfunctions 120587

1 1205872 and 120587

3such that

120587

1(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

) le 119881

119899 (120585 (119905)) le 1205872

(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

119881

119899le minus120587

3(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

(34)

Since1205871is a class119870

infinfunction then for any 120576 gt 0 one can

always find a 120573 = 120573(120576)with 120573 gt 120576 gt 0 such that1205872(120576) le 120587

1(120573)

forall119905 isin [0 119905

119872] which means that 120585(119905) le 120573 forall119905 isin [0 119905

119872]

Suppose that 119905119872

is finite then lim119905rarr 119905119872

120585(119905) = +infin whichcontradicts 120585(119905) le 120573 forall119905 isin [0 119905

119872] Hence 120585(119905) is well

defined on [0 +infin) so is 119909(119905)

(ii) Noticing that 119905119872= +infin by using (34) and Lyapunov

stability theorem [24] we know that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) isglobally asymptotically stable According to the defi-nition of finite-time stability [11] if the global finite-time attractivity of the closed-loop system can beguaranteed then the global finite-time stabilizationresult will be obtained To this end let us prove theglobal finite-time attractivity First of all by usingLemma 5 it is easy to see that

119882

119896= int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904

le

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+1119901119896 10038161003816

1003816

1003816

119909

119896minus 119909

lowast

119896

1003816

1003816

1003816

1003816

le 2

1minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596

(35)

So we have the following estimate

119881

119899=

119899

sum

119896=1

119882

119896le 2

119899

sum

119896=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596 (36)

Let 120572 = 2(2 minus 120596) With (36) and (31) in mind byLemma 4 it is not difficult to obtain that

119881

119899le minus

1

2

119881

120572

119899 (37)

Therefore by Lemma 3 we obtain that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) is globallyfinite-time stable This together with the definitions of 119909lowast

119894rsquos

directly concludes that the globally finite-time stability of theclosed-loop systems (1) (18) and (30) at the equilibrium 119909 =

0

5 Simulation Example

To illustrate the effectiveness of the proposed approach weconsider the following low-dimensional system

1= 119909

53

2+ 119909

1011

1

2= 119906 + 119909

2

1sin1199092

(38)

Abstract and Applied Analysis 5

0 2 4 6 8 10Time (s)

minus2

minus15

minus1

minus05

0

05

1

x1

x2

(a) System states

0 2 4 6 8 10Time (s)

minus180

minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

uu

(b) Control input

Figure 1 The responses of the closed-loop system (38) and (39)

where 1199011= 53 and 119901

2= 1 It is worth pointing

out that although system (38) is simple it cannot beglobally finite-time stabilized using the design methodpresented in [14 21 22] because of the presence of bothlow-order term 1199091011

1and high-order term 1199092

1sin1199092 Choose

120596 = minus111 isin (minus38 0] then 1199031= 1 1199032= (119903

1+ 120596)119901

1= 611

and 1199033= (119903

2+ 120596)119901

2= 511 By Lemma 6 it is easy to

get |1198911| le |119909

1|

1011 and |1198912| le (1 + 119909

2

1)|119909

1|

511 ClearlyAssumption 1 holds with 120593

1= 1 and 120593

2= 1 + 119909

2

1 according

to the design procedure proposed in Section 3 we can get

119909

lowast

2= minus120573

611

1[119909

1]

611 120573

1= 3

1110

119906 = minus (1 + 119897

21+ 119897

22+ 119897

23) [120585

2]

511

(39)

where 1205852= [119909

2]

116minus [119909

lowast

2]

116 11989721= (511) sdot (1811)

65sdot 2

15119897

22= (1722) sdot (1522)

517sdot (1 + 119909

2

1)

2217 and 11989723

= (611) sdot (3011)

56sdot (1711)

116sdot (3

12160+ 3

12130)

In the simulation by choosing the initial values as119909

1(0) = 1 and 119909

2(0) = minus1 Figure 1 is obtained to demonstrate

the effectiveness of the control scheme

6 Conclusion

In this paper a continuous state feedback stabilizing con-troller is presented for a class of high-order nonlinearsystems under weaker condition The controller designedpreserves the equilibrium at the origin and guarantees theglobal finite-time stability of the systems It should be notedthat the proposed controller can only work well when thewhole state vector is measurable Therefore a natural andmore interesting problem is how to design output feedbackstabilizing controller for the systems studied in the paper

if only partial state vector being measurable which is nowunder our further investigation

Appendix

Proof of Proposition 7 Noting that minus1sum119899119897=1119901

1sdot sdot sdot 119901

119897minus1lt 120596 lt

0 and 119901119896minus1119903

119896= 119903

119896minus1+ 120596 we have 0 lt 119901

119896minus1119903

119896lt 1 Using (18)

it follows from Lemma 5 that1003816

1003816

1003816

1003816

1003816

119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

(119909

1119903119896

119896)

119903119896119901119896minus1

minus (119909

lowast(1119903119896)

119896)

119903119896119901119896minus1 10038161003816

1003816

1003816

1003816

1003816

le 2

1minus119903119896119901119896minus11003816

1003816

1003816

1003816

1003816

1003816

[119909

119896]

1119903119896minus [119909

lowast

119896]

11199031198961003816

1003816

1003816

1003816

1003816

1003816

119903119896119901119896minus1

= 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

(A1)

By (A1) and Lemma 6 it can be obtained that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

le 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A2)

where 1198971198961gt 0 is a constant

Proof of Proposition 8 According to (18) Assumption 1 andLemma 4 it follows that

1003816

1003816

1003816

1003816

119891

119894

1003816

1003816

1003816

1003816

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895

le 120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

+ 120573

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

)

119903119894+1119901119894

6 Abstract and Applied Analysis

le 2

1minus119903119894+1119901119894120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894+ 120573

119903119894+1119901119894

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894)

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894

(A3)

where 1205730= 120585

0= 0 and 120593

119894= 2

1minus119903119894+1119901119894sum

119894

119895=1(1 + 120573

119903119894+1119901119894

119895minus1)120593

119894ge 0 is

a smooth functionUsing (A3) and Lemmas 5 and 6 we obtain

[120585

119896]

2minus119903119896+1119901119896119891

119896le 120593

119896

119896

sum

119895=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+11199011198961003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119896+1119901119896

le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A4)

where 1198971198962ge 0 is a smooth function

Proof of Proposition 9 Note that

[119909

lowast

119895+1]

1(119903119895+1)

= minus120573

119895120585

119895= minus

119896minus1

sum

119895=1

119861

119895[119909

119895]

1119903119895 (A5)

where 119861119895= 120573

119896minus1sdot sdot sdot 120573

119895and 119895 = 1 119896 minus 1

Using (A5) after simple calculations it is not hard toobtain that for 119895 = 1 119896 minus 1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

= minus

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

[119909

119895]

1119903119895minus

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(A6)

By (22) (A3) (A6) and Lemmas 4 and 5 we get

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

= minus (2 minus 119903

119896+1119901

119896) int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

times

119896minus1

sum

119895=1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597[120572

119896minus1]

1119903119896

120597119909

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

+

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

times (

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(

1003816

1003816

1003816

1003816

1003816

119909j+11003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

120585

119895+ 120573

119895minus1120585

119895minus1

1003816

1003816

1003816

1003816

1003816

1minus119903119895

times (

1003816

1003816

1003816

1003816

1003816

120585

119895+1+ 120573

119895120585

119895

1003816

1003816

1003816

1003816

1003816

119903119895+1119901119895+ 120593

119894

119895

sum

119897=1

1003816

1003816

1003816

1003816

120585

119897

1003816

1003816

1003816

1003816

119903119895+1119901119895)

(A7)

where 119861119895ge 0 is a smooth function

Noting that 119903119895+1119901

119895= 119903

119895+ 120596 by using Lemma 6 we have

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (A8)

where 1198971198963ge 0 is a smooth function

Conflict of Interests

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

Acknowledgments

The authors thank the editor and the anonymous reviewersfor their constructive comments and suggestions for improv-ing the quality of the paper This work has been supportedin part by the National Nature Science Foundation of Chinaunder Grant 61073065 and the Key Program of ScienceTechnology Research of Education Department of HenanProvince under Grants 13A120016 and 14A520003

References

[1] C Rui M Reyhanoglu I Kolmanovsky S Cho and NHMcClamroch ldquoNonsmooth stabilization of an underactuatedunstable two degrees of freedom mechanical systemrdquo in Pro-ceedings of the 36th IEEE Conference Decision Control vol 4pp 3998ndash4003 1997

[2] W Lin and C Qian ldquoAdding one power integrator a toolfor global stabilization of high-order lower-triangular systemsrdquoSystems amp Control Letters vol 39 no 5 pp 339ndash351 2000

[3] C Qian andW Lin ldquoA continuous feedback approach to globalstrong stabilization of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 7 pp 1061ndash1079 2001

[4] W Lin and C Qian ldquoAdaptive control of nonlinearly param-eterized systems a nonsmooth feedback frameworkrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 757ndash7742002

[5] W Lin and C Qian ldquoAdaptive control of nonlinearly parame-terized systems the smooth feedback caserdquo IEEE Transactionson Automatic Control vol 47 no 8 pp 1249ndash1266 2002

[6] J Polendo and C Qian ldquoA generalized homogeneous domina-tion approach for global stabilization of inherently nonlinearsystems via output feedbackrdquo International Journal of Robustand Nonlinear Control vol 17 no 7 pp 605ndash629 2007

[7] J Polendo and C Qian ldquoAn expanded method to robustlystabilize uncertain nonlinear systemsrdquo Communications inInformation and Systems vol 8 no 1 pp 55ndash70 2008

Abstract and Applied Analysis 7

[8] Z Sun and Y Liu ldquoAdaptive stabilisation for a large class ofhigh-order uncertain non-linear systemsrdquo International Journalof Control vol 82 no 7 pp 1275ndash1287 2009

[9] J Zhang and Y Liu ldquoA new approach to adaptive control designwithout overparametrization for a class of uncertain nonlinearsystemsrdquo Science China Information Sciences vol 54 no 7 pp1419ndash1429 2011

[10] X-H Zhang and X-J Xie ldquoGlobal state feedback stabilisationof nonlinear systems with high-order and low-order nonlinear-itiesrdquo International Journal of Control vol 87 no 3 pp 642ndash6522014

[11] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[12] Y Hong J Huang and Y Xu ldquoOn an output feedback finite-time stabilization problemrdquo IEEE Transactions on AutomaticControl vol 46 no 2 pp 305ndash309 2001

[13] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[14] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[15] Y Hong J Wang and D Cheng ldquoAdaptive finite-time controlof nonlinear systems with parametric uncertaintyrdquo IEEE Trans-actions on Automatic Control vol 51 no 5 pp 858ndash862 2006

[16] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006

[17] S Li and Y-P Tian ldquoFinite-time stability of cascaded time-varying systemsrdquo International Journal of Control vol 80 no4 pp 646ndash657 2007

[18] S G Nersesov W M Haddad and Q Hui ldquoFinite-timestabilization of nonlinear dynamical systems via control vectorLyapunov functionsrdquo Journal of the Franklin Institute Engineer-ing and Applied Mathematics vol 345 no 7 pp 819ndash837 2008

[19] H Du C Qian M T Frye and S Li ldquoGlobal finite-timestabilisation using bounded feedback for a class of non-linearsystemsrdquo IET Control Theory amp Applications vol 6 no 14 pp2326ndash2336 2012

[20] S Ding S Li and W X Zheng ldquoNonsmooth stabilization of aclass of nonlinear cascaded systemsrdquoAutomatica vol 48 no 10pp 2597ndash2606 2012

[21] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 43no 1 pp 73ndash78 2012

[22] J Li C Qian and S Ding ldquoGlobal finite-time stabilisation byoutput feedback for a class of uncertain nonlinear systemsrdquoInternational Journal of Control vol 83 no 11 pp 2241ndash22522010

[23] B Yang and W Lin ldquoNonsmooth output feedback design witha dynamics gain for uncertain systems with strong nonlin-earitiesrdquo in Proceedings of the 46th IEEE Conference DecisionControl pp 3495ndash3500 New Orieans La USA 2007

[24] H K Khalil Nonlinear Systems Prentice-Hall New Jersey NJUSA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

In this paper by necessarily modifying the method ofadding a power integrator and by successfully overcomingsome essential difficulties such as the weaker assumption onthe system growth the appearance of the sign function andthe construction of a continuously differentiable positive-definite and proper Lyapunov function we will focus onsolving the above problem

NotationsThroughout this paper the following notations areadopted 119877+ denotes the set of all nonnegative real numbersand 119877119899 denotes the real 119899-dimensional space For any vector119909 = (119909

1 119909

119899)

119879isin 119877

119899 denote 119909119894= (119909

1 119909

119894)

119879isin 119877

119894119894 = 1 119899 119909 = (sum

119899

119894=1119909

2

119894)

12 119870 denotes the set ofall functions 119877+ rarr 119877

+ which are continuous strictlyincreasing and vanishing at zero 119870

infindenotes the set of

all functions which are of class 119870 and unbounded A signfunction sign(119909) is defined as follows sign(119909) = 1 if 119909 gt 0sign(119909) = 0 if 119909 = 0 and sign(119909) = minus1 if 119909 lt 0 Besidesthe arguments of the functions (or the functionals) will beomitted or simplified whenever no confusion can arise fromthe context For instance we sometimes denote a function119891(119909(119905)) by simply 119891(119909) 119891(sdot) or 119891

2 Problem Statement and Preliminaries

The objective of this paper is to develop a recursive designmethod for globally finite-time stabilizing system (1) via statefeedback under the following assumption

Assumption 1 For 119894 = 1 119899 there are smooth functions120593

119894(119909

119894) ge 0 and constant 120596 isin (minus1sum119899

119897=1119901

1sdot sdot sdot 119901

119897minus1 0) such that

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120593

119894(119909

119894)

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895 (3)

where 119903119894rsquos are defined as

119903

1= 1 119903

119894+1=

119903

119894+ 120596

119901

119894

119894 = 1 119899 (4)

Remark 2 It is worth pointing out that Assumption 1 whichgives the nonlinear growth condition on the system driftterms encompasses the assumptions in the closely relatedworks [14 21 22] To clearly show this we would like to makethe following comparisons to reveal the relationship betweenAssumption 1 and the counterparts in [14 21 22] that isAssumption 1 includes those as special cases

(i) In [14 21] the system nonlinearities 119891119894rsquos are required

to satisfy

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120574

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

) (5)

where 120574119894(119909

119894) ge 0 119894 = 1 119899 are 1198621 functions By (4) we get

that 119903119894= (1119901

1sdot sdot sdot 119901

119894minus1) + sum

119894minus1

119897=1(120596119901

119897sdot sdot sdot 119901

119894minus1) Furthermore

from 120596 isin (minus1sum

119899

119897=1119901

1sdot sdot sdot 119901

119897minus1 0) we have (119903

119894+ 120596)119903

119895lt 1

119894 = 1 119899 It means that1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120574

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

)

le 120574

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

(119903119894+120596)1199031 10038161003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

1minus(119903119894+120596)1199031

+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

(119903119894+120596)119903119894 10038161003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

1minus(119903119894+120596)119903119894)

(6)

Letting 120593119894(119909

119894) be smooth functions and satisfying 120593

119894(119909

119894) ge

max120574119894|119909

1|

1minus(119903119894+120596)1199031 120574

119894|119909

119894|

1minus(119903119894+120596)119903119894 we have

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 120593

119894(119909

119894) (

1003816

1003816

1003816

1003816

119909

1

1003816

1003816

1003816

1003816

(119903119894+120596)1199031+ sdot sdot sdot +

1003816

1003816

1003816

1003816

119909

119894

1003816

1003816

1003816

1003816

(119903119894+120596)119903119894) (7)

from this we can see that the main assumption (4) in [14 21]is a special case of Assumption 1 above

(ii) In [22] the system nonlinearities 119891119894rsquos are required to

satisfy

1003816

1003816

1003816

1003816

119891

119894 (119905 119909 119906)

1003816

1003816

1003816

1003816

le 119872

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895 (8)

with constants 119872 gt 0 and 120596 isin (minus2(2119899 + 1)119901

1sdot sdot sdot 119901

119899minus1 0)

Obviously when 120593119894(119909

119894) = 119872 inequality (4) degenerates to

inequality (8) Moreover the value range of 120596 in (4) is largerthan that in (8) Thus the main assumption (5) in [22] is aspecial case of Assumption 1 above

In the remainder of this section we present the followinglemmas which play an important role in the design process

Lemma 3 (see [11]) Consider the nonlinear system

= 119891 (119909)with 119891 (0) = 0 119909 isin 119877

119899 (9)

where 119891 1198800rarr 119877

119899 is continuous with respect to 119909 on an openneighborhood119880

0of the origin 119909 = 0 Suppose that there is a 1198621

function 119881(119909) defined in a neighborhood 119880 isin 119877119899 of the originreal numbers 119888 gt 0 and 0 lt 120572 lt 1 such that

(i) 119881(119909) is positive definite on 119880(ii)

119881(119909) + 119888119881

120572(119909) le 0 forall119909 isin 119880

Then the origin of (9) is finite-time stable with

119879 le

119881

1minus120572(119909 (0))

119888 (1 minus 120572)

(10)

for initial condition 119909(0) in some open neighborhood 119880 isin

119880

of the origin If 119880 = 119877

119899 and 119881(119909) is radially unbounded (ie119881(119909) rarr +infin as 119909 rarr +infin) the origin of system (9) is globallyfinite-time stable

Lemma 4 (see [6]) For 119909 119910 isin 119877 and 119901 ge 1 being a constantthe following inequalities hold

1003816

1003816

1003816

1003816

119909 + 119910

1003816

1003816

1003816

1003816

119901le 2

119901minus1 10038161003816

1003816

1003816

119909

119901+ 119910

11990110038161003816

1003816

1003816

(|119909| +

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

)

1119901le |119909|

1119901+

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

1119901le 2

(119901minus1)119901(|119909| +

1003816

1003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

)

1119901

(11)

Abstract and Applied Analysis 3

If 119901 ge 1 is odd then1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119901le 2

119901minus1 10038161003816

1003816

1003816

119909

119901minus 119910

11990110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

1119901minus 119910

111990110038161003816

1003816

1003816

1003816

le 2

(119901minus1)11990110038161003816

1003816

1003816

119909 minus y1003816100381610038161003816

1119901

(12)

Lemma 5 (see [10]) If 119901 = 119886119887 isin 119877ge1119900119889119889

with 119887 ge 1 then forany 119909 119910 isin 119877

1003816

1003816

1003816

1003816

119909

119901minus 119910

11990110038161003816

1003816

1003816

le 2

1minus111988710038161003816

1003816

1003816

1003816

sgn (119909) |119909|119886 minus sgn (119910) 1003816100381610038161003816

119910

1003816

1003816

1003816

1003816

11988610038161003816

1003816

1003816

1003816

1119887

(13)

Lemma 6 (see [23]) Let 119909 119910 be real variables then for anypositive real numbers 119886119898 and 119899 one has

119886|119909|

11989810038161003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

119899le 119887|119909|

119898+119899+

119899

119898 + 119899

(

119898 + 119899

119898

)

minus119898119899

times 119886

(119898+119899)119899119887

minus11989811989910038161003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

119898+119899

(14)

where 119887 gt 0 is any real number

3 Finite-Time Control Design

In this section we will construct a continuous state feedbackcontroller by applying the method of adding a power integra-tor For simplicity we denote sgn(119909)|119909|119886 ≜ [119909]119886 for any 119886 isin 119877+and 119909 isin 119877

Step 1 Let 1205851= [119909

1] and choose 119881

1= 119882

1= int

1199091

0119904

2minus11990321199011119889119904

Using (1) and Assumption 1 we have

119881

1= 119909

2minus11990321199011

1119909

1199011

2+ 119909

2minus11990321199011

1119891

1le 119909

2minus11990321199011

1119909

1199011

2+ 119909

2

1120593

1

(15)

Obviously the first virtual controller

119909

lowast

2= minus120573

1199032

1119909

1199032

1= minus120573

1199032

1[120585

1]

1199032 (16)

with 1205731ge (119899 + 120593

1)

111990321199011 being smooth results in

119881

1le minus119899

1003816

1003816

1003816

1003816

120585

1

1003816

1003816

1003816

1003816

2+ [120585

1]

2minus11990321199011(119909

1199011

2minus 119909

lowast1199011

2) (17)

Step k (119896 = 2 119899) Suppose at step 119896 minus 1 there are a 1198621proper and positive definite Lyapunov function 119881

119896minus1 and a

set of virtual controllers 119909lowast1 119909

lowast

119896defined by

119909

lowast

1= 0

119909

lowast

2= minus 120573

1199032

1[120585

1]

1199032

119909

lowast

119896= minus 120573

119903119896

119896minus1[120585

119896minus1]

119903119896

120585

1= [119909

1]

11199031minus [119909

lowast

1]

11199031

120585

2= [119909

2]

11199032minus [119909

lowast

2]

11199032

120585

119896= [119909

119896]

1119903119896minus [119909

lowast

119896]

1119903119896

(18)

with 1205731gt 0 120573

119896minus1gt 0 being smooth such that

119881

119896minus1le minus (119899 minus 119896 + 2)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896minus1]

2minus119903119896119901119896minus1

times (119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

(19)

To complete the induction at the 119896th step we choose thefollowing Lyapunov function

119881

119896(119909

119896) = 119881

119896minus1(119909

119896minus1) + 119882

119896(119909

119896) (20)

where

119882

119896= int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904 (21)

Noting that 2 minus 119903119896+1119901

119896ge 1 and using a similar method as

in [10]119881119896can be shown to be1198621 proper and positive definite

Moreover we can obtain

120597119882

119896

120597119909

119896

= [120585

119896]

2minus119903119896+1119901119896

120597119882

119896

120597119909

119895

= minus (2 minus 119903

119896+1119901

119896)

times int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(22)

where 119895 = 1 119896 minus 1Using (20)ndash(22) it follows that

119881

119896le minus (119899 minus 119896 + 2)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

+[120585

119896]

2minus119903119896+1119901119896119909

119901119896

119896+1+[120585

119896]

2minus119903119896+1119901119896119891

119896+

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

(23)

In order to proceed further an appropriate boundingestimate should be given for the last three terms on the right-hand side of inequality (23) This is accomplished in thefollowing propositions whose technical proofs are given inthe appendix

Proposition 7 There exists a positive constant 1198971198961such that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896) le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (24)

Proposition 8 There exists a nonnegative smooth function 1198971198962

such that

[120585

119896]

2minus119903119896+1119901119896119891

119896le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (25)

4 Abstract and Applied Analysis

Proposition 9 There exists a nonnegative smooth function 1198971198963

such that

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (26)

Substituting (24)ndash(26) into (23) yields

119881

119896le minus (119899 minus 119896 + 1)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896119909

119901119896

119896+1

+ (119897

1198961+ 119897

1198962+ 119897

1198963)

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(27)

Now it easy to see that the virtual controller

119909

lowast

119896+1= minus120573

119903119896+1

119896[120585

119896]

119903119896+1 (28)

where 120573119896ge (119899 minus 119896 + 1 + 119897

1198961+ 119897

1198962+ 119897

1198963)

1(119903119896+1119901119896) is a smoothfunction renders

119881

119896le minus (119899 minus 119896 + 1)

119896

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896(119909

119901119896

119896+1minus 119909

lowast119901119896

119896+1) (29)

This completes the inductive step

Using the inductive argument above we can concludethat at the 119899th step there exists a continuous state feedbackcontroller of the form

119906 = 119909

lowast

119899+1= minus120573

119903119899+1

119899[120585

119899]

119903119899+1 (30)

such that

119881

119899le minus

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

(31)

where

119881

119899=

119899

sum

119896=1

119882

119896=

119899

sum

119896=1

int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904 (32)

4 Stability Analysis

We state the main result in this paper

Theorem 10 If Assumption 1 holds for system (1) under thecontinuous state feedback controller (30) then the followingholds

(i) all the solutions of the closed-loop system are welldefined on [0 +infin)

(ii) the equilibrium 119909 = 0 of the closed-loop system isglobally finite-time stable

Proof (i) Considering (18) system (1) can be equivalentlytransformed into a 120585-system

120585

119894 (119905) = 120593119894 (

119905 120585 119906) 119894 = 1 119899 (33)

By the existence and continuation of the solutions states120585(119905) are defined on [0 119905

119872] where the number 119905

119872gt 0 may

be infinite or not The following analysis focuses on [0 119905119872]

Noting that 119881119899is positive definite and radially unbounded

by (31) (32) and Lemma 43 in [24] it is not hard to obtainthat there exist 119870

infinfunctions 120587

1 1205872 and 120587

3such that

120587

1(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

) le 119881

119899 (120585 (119905)) le 1205872

(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

119881

119899le minus120587

3(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

(34)

Since1205871is a class119870

infinfunction then for any 120576 gt 0 one can

always find a 120573 = 120573(120576)with 120573 gt 120576 gt 0 such that1205872(120576) le 120587

1(120573)

forall119905 isin [0 119905

119872] which means that 120585(119905) le 120573 forall119905 isin [0 119905

119872]

Suppose that 119905119872

is finite then lim119905rarr 119905119872

120585(119905) = +infin whichcontradicts 120585(119905) le 120573 forall119905 isin [0 119905

119872] Hence 120585(119905) is well

defined on [0 +infin) so is 119909(119905)

(ii) Noticing that 119905119872= +infin by using (34) and Lyapunov

stability theorem [24] we know that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) isglobally asymptotically stable According to the defi-nition of finite-time stability [11] if the global finite-time attractivity of the closed-loop system can beguaranteed then the global finite-time stabilizationresult will be obtained To this end let us prove theglobal finite-time attractivity First of all by usingLemma 5 it is easy to see that

119882

119896= int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904

le

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+1119901119896 10038161003816

1003816

1003816

119909

119896minus 119909

lowast

119896

1003816

1003816

1003816

1003816

le 2

1minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596

(35)

So we have the following estimate

119881

119899=

119899

sum

119896=1

119882

119896le 2

119899

sum

119896=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596 (36)

Let 120572 = 2(2 minus 120596) With (36) and (31) in mind byLemma 4 it is not difficult to obtain that

119881

119899le minus

1

2

119881

120572

119899 (37)

Therefore by Lemma 3 we obtain that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) is globallyfinite-time stable This together with the definitions of 119909lowast

119894rsquos

directly concludes that the globally finite-time stability of theclosed-loop systems (1) (18) and (30) at the equilibrium 119909 =

0

5 Simulation Example

To illustrate the effectiveness of the proposed approach weconsider the following low-dimensional system

1= 119909

53

2+ 119909

1011

1

2= 119906 + 119909

2

1sin1199092

(38)

Abstract and Applied Analysis 5

0 2 4 6 8 10Time (s)

minus2

minus15

minus1

minus05

0

05

1

x1

x2

(a) System states

0 2 4 6 8 10Time (s)

minus180

minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

uu

(b) Control input

Figure 1 The responses of the closed-loop system (38) and (39)

where 1199011= 53 and 119901

2= 1 It is worth pointing

out that although system (38) is simple it cannot beglobally finite-time stabilized using the design methodpresented in [14 21 22] because of the presence of bothlow-order term 1199091011

1and high-order term 1199092

1sin1199092 Choose

120596 = minus111 isin (minus38 0] then 1199031= 1 1199032= (119903

1+ 120596)119901

1= 611

and 1199033= (119903

2+ 120596)119901

2= 511 By Lemma 6 it is easy to

get |1198911| le |119909

1|

1011 and |1198912| le (1 + 119909

2

1)|119909

1|

511 ClearlyAssumption 1 holds with 120593

1= 1 and 120593

2= 1 + 119909

2

1 according

to the design procedure proposed in Section 3 we can get

119909

lowast

2= minus120573

611

1[119909

1]

611 120573

1= 3

1110

119906 = minus (1 + 119897

21+ 119897

22+ 119897

23) [120585

2]

511

(39)

where 1205852= [119909

2]

116minus [119909

lowast

2]

116 11989721= (511) sdot (1811)

65sdot 2

15119897

22= (1722) sdot (1522)

517sdot (1 + 119909

2

1)

2217 and 11989723

= (611) sdot (3011)

56sdot (1711)

116sdot (3

12160+ 3

12130)

In the simulation by choosing the initial values as119909

1(0) = 1 and 119909

2(0) = minus1 Figure 1 is obtained to demonstrate

the effectiveness of the control scheme

6 Conclusion

In this paper a continuous state feedback stabilizing con-troller is presented for a class of high-order nonlinearsystems under weaker condition The controller designedpreserves the equilibrium at the origin and guarantees theglobal finite-time stability of the systems It should be notedthat the proposed controller can only work well when thewhole state vector is measurable Therefore a natural andmore interesting problem is how to design output feedbackstabilizing controller for the systems studied in the paper

if only partial state vector being measurable which is nowunder our further investigation

Appendix

Proof of Proposition 7 Noting that minus1sum119899119897=1119901

1sdot sdot sdot 119901

119897minus1lt 120596 lt

0 and 119901119896minus1119903

119896= 119903

119896minus1+ 120596 we have 0 lt 119901

119896minus1119903

119896lt 1 Using (18)

it follows from Lemma 5 that1003816

1003816

1003816

1003816

1003816

119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

(119909

1119903119896

119896)

119903119896119901119896minus1

minus (119909

lowast(1119903119896)

119896)

119903119896119901119896minus1 10038161003816

1003816

1003816

1003816

1003816

le 2

1minus119903119896119901119896minus11003816

1003816

1003816

1003816

1003816

1003816

[119909

119896]

1119903119896minus [119909

lowast

119896]

11199031198961003816

1003816

1003816

1003816

1003816

1003816

119903119896119901119896minus1

= 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

(A1)

By (A1) and Lemma 6 it can be obtained that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

le 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A2)

where 1198971198961gt 0 is a constant

Proof of Proposition 8 According to (18) Assumption 1 andLemma 4 it follows that

1003816

1003816

1003816

1003816

119891

119894

1003816

1003816

1003816

1003816

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895

le 120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

+ 120573

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

)

119903119894+1119901119894

6 Abstract and Applied Analysis

le 2

1minus119903119894+1119901119894120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894+ 120573

119903119894+1119901119894

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894)

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894

(A3)

where 1205730= 120585

0= 0 and 120593

119894= 2

1minus119903119894+1119901119894sum

119894

119895=1(1 + 120573

119903119894+1119901119894

119895minus1)120593

119894ge 0 is

a smooth functionUsing (A3) and Lemmas 5 and 6 we obtain

[120585

119896]

2minus119903119896+1119901119896119891

119896le 120593

119896

119896

sum

119895=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+11199011198961003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119896+1119901119896

le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A4)

where 1198971198962ge 0 is a smooth function

Proof of Proposition 9 Note that

[119909

lowast

119895+1]

1(119903119895+1)

= minus120573

119895120585

119895= minus

119896minus1

sum

119895=1

119861

119895[119909

119895]

1119903119895 (A5)

where 119861119895= 120573

119896minus1sdot sdot sdot 120573

119895and 119895 = 1 119896 minus 1

Using (A5) after simple calculations it is not hard toobtain that for 119895 = 1 119896 minus 1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

= minus

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

[119909

119895]

1119903119895minus

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(A6)

By (22) (A3) (A6) and Lemmas 4 and 5 we get

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

= minus (2 minus 119903

119896+1119901

119896) int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

times

119896minus1

sum

119895=1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597[120572

119896minus1]

1119903119896

120597119909

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

+

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

times (

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(

1003816

1003816

1003816

1003816

1003816

119909j+11003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

120585

119895+ 120573

119895minus1120585

119895minus1

1003816

1003816

1003816

1003816

1003816

1minus119903119895

times (

1003816

1003816

1003816

1003816

1003816

120585

119895+1+ 120573

119895120585

119895

1003816

1003816

1003816

1003816

1003816

119903119895+1119901119895+ 120593

119894

119895

sum

119897=1

1003816

1003816

1003816

1003816

120585

119897

1003816

1003816

1003816

1003816

119903119895+1119901119895)

(A7)

where 119861119895ge 0 is a smooth function

Noting that 119903119895+1119901

119895= 119903

119895+ 120596 by using Lemma 6 we have

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (A8)

where 1198971198963ge 0 is a smooth function

Conflict of Interests

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

Acknowledgments

The authors thank the editor and the anonymous reviewersfor their constructive comments and suggestions for improv-ing the quality of the paper This work has been supportedin part by the National Nature Science Foundation of Chinaunder Grant 61073065 and the Key Program of ScienceTechnology Research of Education Department of HenanProvince under Grants 13A120016 and 14A520003

References

[1] C Rui M Reyhanoglu I Kolmanovsky S Cho and NHMcClamroch ldquoNonsmooth stabilization of an underactuatedunstable two degrees of freedom mechanical systemrdquo in Pro-ceedings of the 36th IEEE Conference Decision Control vol 4pp 3998ndash4003 1997

[2] W Lin and C Qian ldquoAdding one power integrator a toolfor global stabilization of high-order lower-triangular systemsrdquoSystems amp Control Letters vol 39 no 5 pp 339ndash351 2000

[3] C Qian andW Lin ldquoA continuous feedback approach to globalstrong stabilization of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 7 pp 1061ndash1079 2001

[4] W Lin and C Qian ldquoAdaptive control of nonlinearly param-eterized systems a nonsmooth feedback frameworkrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 757ndash7742002

[5] W Lin and C Qian ldquoAdaptive control of nonlinearly parame-terized systems the smooth feedback caserdquo IEEE Transactionson Automatic Control vol 47 no 8 pp 1249ndash1266 2002

[6] J Polendo and C Qian ldquoA generalized homogeneous domina-tion approach for global stabilization of inherently nonlinearsystems via output feedbackrdquo International Journal of Robustand Nonlinear Control vol 17 no 7 pp 605ndash629 2007

[7] J Polendo and C Qian ldquoAn expanded method to robustlystabilize uncertain nonlinear systemsrdquo Communications inInformation and Systems vol 8 no 1 pp 55ndash70 2008

Abstract and Applied Analysis 7

[8] Z Sun and Y Liu ldquoAdaptive stabilisation for a large class ofhigh-order uncertain non-linear systemsrdquo International Journalof Control vol 82 no 7 pp 1275ndash1287 2009

[9] J Zhang and Y Liu ldquoA new approach to adaptive control designwithout overparametrization for a class of uncertain nonlinearsystemsrdquo Science China Information Sciences vol 54 no 7 pp1419ndash1429 2011

[10] X-H Zhang and X-J Xie ldquoGlobal state feedback stabilisationof nonlinear systems with high-order and low-order nonlinear-itiesrdquo International Journal of Control vol 87 no 3 pp 642ndash6522014

[11] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[12] Y Hong J Huang and Y Xu ldquoOn an output feedback finite-time stabilization problemrdquo IEEE Transactions on AutomaticControl vol 46 no 2 pp 305ndash309 2001

[13] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[14] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[15] Y Hong J Wang and D Cheng ldquoAdaptive finite-time controlof nonlinear systems with parametric uncertaintyrdquo IEEE Trans-actions on Automatic Control vol 51 no 5 pp 858ndash862 2006

[16] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006

[17] S Li and Y-P Tian ldquoFinite-time stability of cascaded time-varying systemsrdquo International Journal of Control vol 80 no4 pp 646ndash657 2007

[18] S G Nersesov W M Haddad and Q Hui ldquoFinite-timestabilization of nonlinear dynamical systems via control vectorLyapunov functionsrdquo Journal of the Franklin Institute Engineer-ing and Applied Mathematics vol 345 no 7 pp 819ndash837 2008

[19] H Du C Qian M T Frye and S Li ldquoGlobal finite-timestabilisation using bounded feedback for a class of non-linearsystemsrdquo IET Control Theory amp Applications vol 6 no 14 pp2326ndash2336 2012

[20] S Ding S Li and W X Zheng ldquoNonsmooth stabilization of aclass of nonlinear cascaded systemsrdquoAutomatica vol 48 no 10pp 2597ndash2606 2012

[21] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 43no 1 pp 73ndash78 2012

[22] J Li C Qian and S Ding ldquoGlobal finite-time stabilisation byoutput feedback for a class of uncertain nonlinear systemsrdquoInternational Journal of Control vol 83 no 11 pp 2241ndash22522010

[23] B Yang and W Lin ldquoNonsmooth output feedback design witha dynamics gain for uncertain systems with strong nonlin-earitiesrdquo in Proceedings of the 46th IEEE Conference DecisionControl pp 3495ndash3500 New Orieans La USA 2007

[24] H K Khalil Nonlinear Systems Prentice-Hall New Jersey NJUSA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

If 119901 ge 1 is odd then1003816

1003816

1003816

1003816

119909 minus 119910

1003816

1003816

1003816

1003816

119901le 2

119901minus1 10038161003816

1003816

1003816

119909

119901minus 119910

11990110038161003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

1119901minus 119910

111990110038161003816

1003816

1003816

1003816

le 2

(119901minus1)11990110038161003816

1003816

1003816

119909 minus y1003816100381610038161003816

1119901

(12)

Lemma 5 (see [10]) If 119901 = 119886119887 isin 119877ge1119900119889119889

with 119887 ge 1 then forany 119909 119910 isin 119877

1003816

1003816

1003816

1003816

119909

119901minus 119910

11990110038161003816

1003816

1003816

le 2

1minus111988710038161003816

1003816

1003816

1003816

sgn (119909) |119909|119886 minus sgn (119910) 1003816100381610038161003816

119910

1003816

1003816

1003816

1003816

11988610038161003816

1003816

1003816

1003816

1119887

(13)

Lemma 6 (see [23]) Let 119909 119910 be real variables then for anypositive real numbers 119886119898 and 119899 one has

119886|119909|

11989810038161003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

119899le 119887|119909|

119898+119899+

119899

119898 + 119899

(

119898 + 119899

119898

)

minus119898119899

times 119886

(119898+119899)119899119887

minus11989811989910038161003816

1003816

1003816

119910

1003816

1003816

1003816

1003816

119898+119899

(14)

where 119887 gt 0 is any real number

3 Finite-Time Control Design

In this section we will construct a continuous state feedbackcontroller by applying the method of adding a power integra-tor For simplicity we denote sgn(119909)|119909|119886 ≜ [119909]119886 for any 119886 isin 119877+and 119909 isin 119877

Step 1 Let 1205851= [119909

1] and choose 119881

1= 119882

1= int

1199091

0119904

2minus11990321199011119889119904

Using (1) and Assumption 1 we have

119881

1= 119909

2minus11990321199011

1119909

1199011

2+ 119909

2minus11990321199011

1119891

1le 119909

2minus11990321199011

1119909

1199011

2+ 119909

2

1120593

1

(15)

Obviously the first virtual controller

119909

lowast

2= minus120573

1199032

1119909

1199032

1= minus120573

1199032

1[120585

1]

1199032 (16)

with 1205731ge (119899 + 120593

1)

111990321199011 being smooth results in

119881

1le minus119899

1003816

1003816

1003816

1003816

120585

1

1003816

1003816

1003816

1003816

2+ [120585

1]

2minus11990321199011(119909

1199011

2minus 119909

lowast1199011

2) (17)

Step k (119896 = 2 119899) Suppose at step 119896 minus 1 there are a 1198621proper and positive definite Lyapunov function 119881

119896minus1 and a

set of virtual controllers 119909lowast1 119909

lowast

119896defined by

119909

lowast

1= 0

119909

lowast

2= minus 120573

1199032

1[120585

1]

1199032

119909

lowast

119896= minus 120573

119903119896

119896minus1[120585

119896minus1]

119903119896

120585

1= [119909

1]

11199031minus [119909

lowast

1]

11199031

120585

2= [119909

2]

11199032minus [119909

lowast

2]

11199032

120585

119896= [119909

119896]

1119903119896minus [119909

lowast

119896]

1119903119896

(18)

with 1205731gt 0 120573

119896minus1gt 0 being smooth such that

119881

119896minus1le minus (119899 minus 119896 + 2)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896minus1]

2minus119903119896119901119896minus1

times (119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

(19)

To complete the induction at the 119896th step we choose thefollowing Lyapunov function

119881

119896(119909

119896) = 119881

119896minus1(119909

119896minus1) + 119882

119896(119909

119896) (20)

where

119882

119896= int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904 (21)

Noting that 2 minus 119903119896+1119901

119896ge 1 and using a similar method as

in [10]119881119896can be shown to be1198621 proper and positive definite

Moreover we can obtain

120597119882

119896

120597119909

119896

= [120585

119896]

2minus119903119896+1119901119896

120597119882

119896

120597119909

119895

= minus (2 minus 119903

119896+1119901

119896)

times int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(22)

where 119895 = 1 119896 minus 1Using (20)ndash(22) it follows that

119881

119896le minus (119899 minus 119896 + 2)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

+[120585

119896]

2minus119903119896+1119901119896119909

119901119896

119896+1+[120585

119896]

2minus119903119896+1119901119896119891

119896+

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

(23)

In order to proceed further an appropriate boundingestimate should be given for the last three terms on the right-hand side of inequality (23) This is accomplished in thefollowing propositions whose technical proofs are given inthe appendix

Proposition 7 There exists a positive constant 1198971198961such that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896) le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (24)

Proposition 8 There exists a nonnegative smooth function 1198971198962

such that

[120585

119896]

2minus119903119896+1119901119896119891

119896le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (25)

4 Abstract and Applied Analysis

Proposition 9 There exists a nonnegative smooth function 1198971198963

such that

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (26)

Substituting (24)ndash(26) into (23) yields

119881

119896le minus (119899 minus 119896 + 1)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896119909

119901119896

119896+1

+ (119897

1198961+ 119897

1198962+ 119897

1198963)

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(27)

Now it easy to see that the virtual controller

119909

lowast

119896+1= minus120573

119903119896+1

119896[120585

119896]

119903119896+1 (28)

where 120573119896ge (119899 minus 119896 + 1 + 119897

1198961+ 119897

1198962+ 119897

1198963)

1(119903119896+1119901119896) is a smoothfunction renders

119881

119896le minus (119899 minus 119896 + 1)

119896

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896(119909

119901119896

119896+1minus 119909

lowast119901119896

119896+1) (29)

This completes the inductive step

Using the inductive argument above we can concludethat at the 119899th step there exists a continuous state feedbackcontroller of the form

119906 = 119909

lowast

119899+1= minus120573

119903119899+1

119899[120585

119899]

119903119899+1 (30)

such that

119881

119899le minus

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

(31)

where

119881

119899=

119899

sum

119896=1

119882

119896=

119899

sum

119896=1

int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904 (32)

4 Stability Analysis

We state the main result in this paper

Theorem 10 If Assumption 1 holds for system (1) under thecontinuous state feedback controller (30) then the followingholds

(i) all the solutions of the closed-loop system are welldefined on [0 +infin)

(ii) the equilibrium 119909 = 0 of the closed-loop system isglobally finite-time stable

Proof (i) Considering (18) system (1) can be equivalentlytransformed into a 120585-system

120585

119894 (119905) = 120593119894 (

119905 120585 119906) 119894 = 1 119899 (33)

By the existence and continuation of the solutions states120585(119905) are defined on [0 119905

119872] where the number 119905

119872gt 0 may

be infinite or not The following analysis focuses on [0 119905119872]

Noting that 119881119899is positive definite and radially unbounded

by (31) (32) and Lemma 43 in [24] it is not hard to obtainthat there exist 119870

infinfunctions 120587

1 1205872 and 120587

3such that

120587

1(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

) le 119881

119899 (120585 (119905)) le 1205872

(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

119881

119899le minus120587

3(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

(34)

Since1205871is a class119870

infinfunction then for any 120576 gt 0 one can

always find a 120573 = 120573(120576)with 120573 gt 120576 gt 0 such that1205872(120576) le 120587

1(120573)

forall119905 isin [0 119905

119872] which means that 120585(119905) le 120573 forall119905 isin [0 119905

119872]

Suppose that 119905119872

is finite then lim119905rarr 119905119872

120585(119905) = +infin whichcontradicts 120585(119905) le 120573 forall119905 isin [0 119905

119872] Hence 120585(119905) is well

defined on [0 +infin) so is 119909(119905)

(ii) Noticing that 119905119872= +infin by using (34) and Lyapunov

stability theorem [24] we know that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) isglobally asymptotically stable According to the defi-nition of finite-time stability [11] if the global finite-time attractivity of the closed-loop system can beguaranteed then the global finite-time stabilizationresult will be obtained To this end let us prove theglobal finite-time attractivity First of all by usingLemma 5 it is easy to see that

119882

119896= int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904

le

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+1119901119896 10038161003816

1003816

1003816

119909

119896minus 119909

lowast

119896

1003816

1003816

1003816

1003816

le 2

1minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596

(35)

So we have the following estimate

119881

119899=

119899

sum

119896=1

119882

119896le 2

119899

sum

119896=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596 (36)

Let 120572 = 2(2 minus 120596) With (36) and (31) in mind byLemma 4 it is not difficult to obtain that

119881

119899le minus

1

2

119881

120572

119899 (37)

Therefore by Lemma 3 we obtain that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) is globallyfinite-time stable This together with the definitions of 119909lowast

119894rsquos

directly concludes that the globally finite-time stability of theclosed-loop systems (1) (18) and (30) at the equilibrium 119909 =

0

5 Simulation Example

To illustrate the effectiveness of the proposed approach weconsider the following low-dimensional system

1= 119909

53

2+ 119909

1011

1

2= 119906 + 119909

2

1sin1199092

(38)

Abstract and Applied Analysis 5

0 2 4 6 8 10Time (s)

minus2

minus15

minus1

minus05

0

05

1

x1

x2

(a) System states

0 2 4 6 8 10Time (s)

minus180

minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

uu

(b) Control input

Figure 1 The responses of the closed-loop system (38) and (39)

where 1199011= 53 and 119901

2= 1 It is worth pointing

out that although system (38) is simple it cannot beglobally finite-time stabilized using the design methodpresented in [14 21 22] because of the presence of bothlow-order term 1199091011

1and high-order term 1199092

1sin1199092 Choose

120596 = minus111 isin (minus38 0] then 1199031= 1 1199032= (119903

1+ 120596)119901

1= 611

and 1199033= (119903

2+ 120596)119901

2= 511 By Lemma 6 it is easy to

get |1198911| le |119909

1|

1011 and |1198912| le (1 + 119909

2

1)|119909

1|

511 ClearlyAssumption 1 holds with 120593

1= 1 and 120593

2= 1 + 119909

2

1 according

to the design procedure proposed in Section 3 we can get

119909

lowast

2= minus120573

611

1[119909

1]

611 120573

1= 3

1110

119906 = minus (1 + 119897

21+ 119897

22+ 119897

23) [120585

2]

511

(39)

where 1205852= [119909

2]

116minus [119909

lowast

2]

116 11989721= (511) sdot (1811)

65sdot 2

15119897

22= (1722) sdot (1522)

517sdot (1 + 119909

2

1)

2217 and 11989723

= (611) sdot (3011)

56sdot (1711)

116sdot (3

12160+ 3

12130)

In the simulation by choosing the initial values as119909

1(0) = 1 and 119909

2(0) = minus1 Figure 1 is obtained to demonstrate

the effectiveness of the control scheme

6 Conclusion

In this paper a continuous state feedback stabilizing con-troller is presented for a class of high-order nonlinearsystems under weaker condition The controller designedpreserves the equilibrium at the origin and guarantees theglobal finite-time stability of the systems It should be notedthat the proposed controller can only work well when thewhole state vector is measurable Therefore a natural andmore interesting problem is how to design output feedbackstabilizing controller for the systems studied in the paper

if only partial state vector being measurable which is nowunder our further investigation

Appendix

Proof of Proposition 7 Noting that minus1sum119899119897=1119901

1sdot sdot sdot 119901

119897minus1lt 120596 lt

0 and 119901119896minus1119903

119896= 119903

119896minus1+ 120596 we have 0 lt 119901

119896minus1119903

119896lt 1 Using (18)

it follows from Lemma 5 that1003816

1003816

1003816

1003816

1003816

119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

(119909

1119903119896

119896)

119903119896119901119896minus1

minus (119909

lowast(1119903119896)

119896)

119903119896119901119896minus1 10038161003816

1003816

1003816

1003816

1003816

le 2

1minus119903119896119901119896minus11003816

1003816

1003816

1003816

1003816

1003816

[119909

119896]

1119903119896minus [119909

lowast

119896]

11199031198961003816

1003816

1003816

1003816

1003816

1003816

119903119896119901119896minus1

= 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

(A1)

By (A1) and Lemma 6 it can be obtained that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

le 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A2)

where 1198971198961gt 0 is a constant

Proof of Proposition 8 According to (18) Assumption 1 andLemma 4 it follows that

1003816

1003816

1003816

1003816

119891

119894

1003816

1003816

1003816

1003816

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895

le 120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

+ 120573

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

)

119903119894+1119901119894

6 Abstract and Applied Analysis

le 2

1minus119903119894+1119901119894120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894+ 120573

119903119894+1119901119894

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894)

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894

(A3)

where 1205730= 120585

0= 0 and 120593

119894= 2

1minus119903119894+1119901119894sum

119894

119895=1(1 + 120573

119903119894+1119901119894

119895minus1)120593

119894ge 0 is

a smooth functionUsing (A3) and Lemmas 5 and 6 we obtain

[120585

119896]

2minus119903119896+1119901119896119891

119896le 120593

119896

119896

sum

119895=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+11199011198961003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119896+1119901119896

le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A4)

where 1198971198962ge 0 is a smooth function

Proof of Proposition 9 Note that

[119909

lowast

119895+1]

1(119903119895+1)

= minus120573

119895120585

119895= minus

119896minus1

sum

119895=1

119861

119895[119909

119895]

1119903119895 (A5)

where 119861119895= 120573

119896minus1sdot sdot sdot 120573

119895and 119895 = 1 119896 minus 1

Using (A5) after simple calculations it is not hard toobtain that for 119895 = 1 119896 minus 1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

= minus

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

[119909

119895]

1119903119895minus

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(A6)

By (22) (A3) (A6) and Lemmas 4 and 5 we get

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

= minus (2 minus 119903

119896+1119901

119896) int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

times

119896minus1

sum

119895=1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597[120572

119896minus1]

1119903119896

120597119909

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

+

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

times (

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(

1003816

1003816

1003816

1003816

1003816

119909j+11003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

120585

119895+ 120573

119895minus1120585

119895minus1

1003816

1003816

1003816

1003816

1003816

1minus119903119895

times (

1003816

1003816

1003816

1003816

1003816

120585

119895+1+ 120573

119895120585

119895

1003816

1003816

1003816

1003816

1003816

119903119895+1119901119895+ 120593

119894

119895

sum

119897=1

1003816

1003816

1003816

1003816

120585

119897

1003816

1003816

1003816

1003816

119903119895+1119901119895)

(A7)

where 119861119895ge 0 is a smooth function

Noting that 119903119895+1119901

119895= 119903

119895+ 120596 by using Lemma 6 we have

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (A8)

where 1198971198963ge 0 is a smooth function

Conflict of Interests

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

Acknowledgments

The authors thank the editor and the anonymous reviewersfor their constructive comments and suggestions for improv-ing the quality of the paper This work has been supportedin part by the National Nature Science Foundation of Chinaunder Grant 61073065 and the Key Program of ScienceTechnology Research of Education Department of HenanProvince under Grants 13A120016 and 14A520003

References

[1] C Rui M Reyhanoglu I Kolmanovsky S Cho and NHMcClamroch ldquoNonsmooth stabilization of an underactuatedunstable two degrees of freedom mechanical systemrdquo in Pro-ceedings of the 36th IEEE Conference Decision Control vol 4pp 3998ndash4003 1997

[2] W Lin and C Qian ldquoAdding one power integrator a toolfor global stabilization of high-order lower-triangular systemsrdquoSystems amp Control Letters vol 39 no 5 pp 339ndash351 2000

[3] C Qian andW Lin ldquoA continuous feedback approach to globalstrong stabilization of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 7 pp 1061ndash1079 2001

[4] W Lin and C Qian ldquoAdaptive control of nonlinearly param-eterized systems a nonsmooth feedback frameworkrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 757ndash7742002

[5] W Lin and C Qian ldquoAdaptive control of nonlinearly parame-terized systems the smooth feedback caserdquo IEEE Transactionson Automatic Control vol 47 no 8 pp 1249ndash1266 2002

[6] J Polendo and C Qian ldquoA generalized homogeneous domina-tion approach for global stabilization of inherently nonlinearsystems via output feedbackrdquo International Journal of Robustand Nonlinear Control vol 17 no 7 pp 605ndash629 2007

[7] J Polendo and C Qian ldquoAn expanded method to robustlystabilize uncertain nonlinear systemsrdquo Communications inInformation and Systems vol 8 no 1 pp 55ndash70 2008

Abstract and Applied Analysis 7

[8] Z Sun and Y Liu ldquoAdaptive stabilisation for a large class ofhigh-order uncertain non-linear systemsrdquo International Journalof Control vol 82 no 7 pp 1275ndash1287 2009

[9] J Zhang and Y Liu ldquoA new approach to adaptive control designwithout overparametrization for a class of uncertain nonlinearsystemsrdquo Science China Information Sciences vol 54 no 7 pp1419ndash1429 2011

[10] X-H Zhang and X-J Xie ldquoGlobal state feedback stabilisationof nonlinear systems with high-order and low-order nonlinear-itiesrdquo International Journal of Control vol 87 no 3 pp 642ndash6522014

[11] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[12] Y Hong J Huang and Y Xu ldquoOn an output feedback finite-time stabilization problemrdquo IEEE Transactions on AutomaticControl vol 46 no 2 pp 305ndash309 2001

[13] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[14] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[15] Y Hong J Wang and D Cheng ldquoAdaptive finite-time controlof nonlinear systems with parametric uncertaintyrdquo IEEE Trans-actions on Automatic Control vol 51 no 5 pp 858ndash862 2006

[16] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006

[17] S Li and Y-P Tian ldquoFinite-time stability of cascaded time-varying systemsrdquo International Journal of Control vol 80 no4 pp 646ndash657 2007

[18] S G Nersesov W M Haddad and Q Hui ldquoFinite-timestabilization of nonlinear dynamical systems via control vectorLyapunov functionsrdquo Journal of the Franklin Institute Engineer-ing and Applied Mathematics vol 345 no 7 pp 819ndash837 2008

[19] H Du C Qian M T Frye and S Li ldquoGlobal finite-timestabilisation using bounded feedback for a class of non-linearsystemsrdquo IET Control Theory amp Applications vol 6 no 14 pp2326ndash2336 2012

[20] S Ding S Li and W X Zheng ldquoNonsmooth stabilization of aclass of nonlinear cascaded systemsrdquoAutomatica vol 48 no 10pp 2597ndash2606 2012

[21] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 43no 1 pp 73ndash78 2012

[22] J Li C Qian and S Ding ldquoGlobal finite-time stabilisation byoutput feedback for a class of uncertain nonlinear systemsrdquoInternational Journal of Control vol 83 no 11 pp 2241ndash22522010

[23] B Yang and W Lin ldquoNonsmooth output feedback design witha dynamics gain for uncertain systems with strong nonlin-earitiesrdquo in Proceedings of the 46th IEEE Conference DecisionControl pp 3495ndash3500 New Orieans La USA 2007

[24] H K Khalil Nonlinear Systems Prentice-Hall New Jersey NJUSA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Proposition 9 There exists a nonnegative smooth function 1198971198963

such that

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (26)

Substituting (24)ndash(26) into (23) yields

119881

119896le minus (119899 minus 119896 + 1)

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896119909

119901119896

119896+1

+ (119897

1198961+ 119897

1198962+ 119897

1198963)

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(27)

Now it easy to see that the virtual controller

119909

lowast

119896+1= minus120573

119903119896+1

119896[120585

119896]

119903119896+1 (28)

where 120573119896ge (119899 minus 119896 + 1 + 119897

1198961+ 119897

1198962+ 119897

1198963)

1(119903119896+1119901119896) is a smoothfunction renders

119881

119896le minus (119899 minus 119896 + 1)

119896

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ [120585

119896]

2minus119903119896+1119901119896(119909

119901119896

119896+1minus 119909

lowast119901119896

119896+1) (29)

This completes the inductive step

Using the inductive argument above we can concludethat at the 119899th step there exists a continuous state feedbackcontroller of the form

119906 = 119909

lowast

119899+1= minus120573

119903119899+1

119899[120585

119899]

119903119899+1 (30)

such that

119881

119899le minus

119899

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

(31)

where

119881

119899=

119899

sum

119896=1

119882

119896=

119899

sum

119896=1

int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904 (32)

4 Stability Analysis

We state the main result in this paper

Theorem 10 If Assumption 1 holds for system (1) under thecontinuous state feedback controller (30) then the followingholds

(i) all the solutions of the closed-loop system are welldefined on [0 +infin)

(ii) the equilibrium 119909 = 0 of the closed-loop system isglobally finite-time stable

Proof (i) Considering (18) system (1) can be equivalentlytransformed into a 120585-system

120585

119894 (119905) = 120593119894 (

119905 120585 119906) 119894 = 1 119899 (33)

By the existence and continuation of the solutions states120585(119905) are defined on [0 119905

119872] where the number 119905

119872gt 0 may

be infinite or not The following analysis focuses on [0 119905119872]

Noting that 119881119899is positive definite and radially unbounded

by (31) (32) and Lemma 43 in [24] it is not hard to obtainthat there exist 119870

infinfunctions 120587

1 1205872 and 120587

3such that

120587

1(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

) le 119881

119899 (120585 (119905)) le 1205872

(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

119881

119899le minus120587

3(

1003817

1003817

1003817

1003817

120585 (119905)

1003817

1003817

1003817

1003817

)

(34)

Since1205871is a class119870

infinfunction then for any 120576 gt 0 one can

always find a 120573 = 120573(120576)with 120573 gt 120576 gt 0 such that1205872(120576) le 120587

1(120573)

forall119905 isin [0 119905

119872] which means that 120585(119905) le 120573 forall119905 isin [0 119905

119872]

Suppose that 119905119872

is finite then lim119905rarr 119905119872

120585(119905) = +infin whichcontradicts 120585(119905) le 120573 forall119905 isin [0 119905

119872] Hence 120585(119905) is well

defined on [0 +infin) so is 119909(119905)

(ii) Noticing that 119905119872= +infin by using (34) and Lyapunov

stability theorem [24] we know that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) isglobally asymptotically stable According to the defi-nition of finite-time stability [11] if the global finite-time attractivity of the closed-loop system can beguaranteed then the global finite-time stabilizationresult will be obtained To this end let us prove theglobal finite-time attractivity First of all by usingLemma 5 it is easy to see that

119882

119896= int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

2minus119903119896+1119901119896

119889119904

le

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+1119901119896 10038161003816

1003816

1003816

119909

119896minus 119909

lowast

119896

1003816

1003816

1003816

1003816

le 2

1minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596

(35)

So we have the following estimate

119881

119899=

119899

sum

119896=1

119882

119896le 2

119899

sum

119896=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus120596 (36)

Let 120572 = 2(2 minus 120596) With (36) and (31) in mind byLemma 4 it is not difficult to obtain that

119881

119899le minus

1

2

119881

120572

119899 (37)

Therefore by Lemma 3 we obtain that the equilibrium120585 = 0 of the closed-loop 120585-systems (30) and (33) is globallyfinite-time stable This together with the definitions of 119909lowast

119894rsquos

directly concludes that the globally finite-time stability of theclosed-loop systems (1) (18) and (30) at the equilibrium 119909 =

0

5 Simulation Example

To illustrate the effectiveness of the proposed approach weconsider the following low-dimensional system

1= 119909

53

2+ 119909

1011

1

2= 119906 + 119909

2

1sin1199092

(38)

Abstract and Applied Analysis 5

0 2 4 6 8 10Time (s)

minus2

minus15

minus1

minus05

0

05

1

x1

x2

(a) System states

0 2 4 6 8 10Time (s)

minus180

minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

uu

(b) Control input

Figure 1 The responses of the closed-loop system (38) and (39)

where 1199011= 53 and 119901

2= 1 It is worth pointing

out that although system (38) is simple it cannot beglobally finite-time stabilized using the design methodpresented in [14 21 22] because of the presence of bothlow-order term 1199091011

1and high-order term 1199092

1sin1199092 Choose

120596 = minus111 isin (minus38 0] then 1199031= 1 1199032= (119903

1+ 120596)119901

1= 611

and 1199033= (119903

2+ 120596)119901

2= 511 By Lemma 6 it is easy to

get |1198911| le |119909

1|

1011 and |1198912| le (1 + 119909

2

1)|119909

1|

511 ClearlyAssumption 1 holds with 120593

1= 1 and 120593

2= 1 + 119909

2

1 according

to the design procedure proposed in Section 3 we can get

119909

lowast

2= minus120573

611

1[119909

1]

611 120573

1= 3

1110

119906 = minus (1 + 119897

21+ 119897

22+ 119897

23) [120585

2]

511

(39)

where 1205852= [119909

2]

116minus [119909

lowast

2]

116 11989721= (511) sdot (1811)

65sdot 2

15119897

22= (1722) sdot (1522)

517sdot (1 + 119909

2

1)

2217 and 11989723

= (611) sdot (3011)

56sdot (1711)

116sdot (3

12160+ 3

12130)

In the simulation by choosing the initial values as119909

1(0) = 1 and 119909

2(0) = minus1 Figure 1 is obtained to demonstrate

the effectiveness of the control scheme

6 Conclusion

In this paper a continuous state feedback stabilizing con-troller is presented for a class of high-order nonlinearsystems under weaker condition The controller designedpreserves the equilibrium at the origin and guarantees theglobal finite-time stability of the systems It should be notedthat the proposed controller can only work well when thewhole state vector is measurable Therefore a natural andmore interesting problem is how to design output feedbackstabilizing controller for the systems studied in the paper

if only partial state vector being measurable which is nowunder our further investigation

Appendix

Proof of Proposition 7 Noting that minus1sum119899119897=1119901

1sdot sdot sdot 119901

119897minus1lt 120596 lt

0 and 119901119896minus1119903

119896= 119903

119896minus1+ 120596 we have 0 lt 119901

119896minus1119903

119896lt 1 Using (18)

it follows from Lemma 5 that1003816

1003816

1003816

1003816

1003816

119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

(119909

1119903119896

119896)

119903119896119901119896minus1

minus (119909

lowast(1119903119896)

119896)

119903119896119901119896minus1 10038161003816

1003816

1003816

1003816

1003816

le 2

1minus119903119896119901119896minus11003816

1003816

1003816

1003816

1003816

1003816

[119909

119896]

1119903119896minus [119909

lowast

119896]

11199031198961003816

1003816

1003816

1003816

1003816

1003816

119903119896119901119896minus1

= 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

(A1)

By (A1) and Lemma 6 it can be obtained that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

le 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A2)

where 1198971198961gt 0 is a constant

Proof of Proposition 8 According to (18) Assumption 1 andLemma 4 it follows that

1003816

1003816

1003816

1003816

119891

119894

1003816

1003816

1003816

1003816

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895

le 120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

+ 120573

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

)

119903119894+1119901119894

6 Abstract and Applied Analysis

le 2

1minus119903119894+1119901119894120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894+ 120573

119903119894+1119901119894

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894)

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894

(A3)

where 1205730= 120585

0= 0 and 120593

119894= 2

1minus119903119894+1119901119894sum

119894

119895=1(1 + 120573

119903119894+1119901119894

119895minus1)120593

119894ge 0 is

a smooth functionUsing (A3) and Lemmas 5 and 6 we obtain

[120585

119896]

2minus119903119896+1119901119896119891

119896le 120593

119896

119896

sum

119895=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+11199011198961003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119896+1119901119896

le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A4)

where 1198971198962ge 0 is a smooth function

Proof of Proposition 9 Note that

[119909

lowast

119895+1]

1(119903119895+1)

= minus120573

119895120585

119895= minus

119896minus1

sum

119895=1

119861

119895[119909

119895]

1119903119895 (A5)

where 119861119895= 120573

119896minus1sdot sdot sdot 120573

119895and 119895 = 1 119896 minus 1

Using (A5) after simple calculations it is not hard toobtain that for 119895 = 1 119896 minus 1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

= minus

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

[119909

119895]

1119903119895minus

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(A6)

By (22) (A3) (A6) and Lemmas 4 and 5 we get

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

= minus (2 minus 119903

119896+1119901

119896) int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

times

119896minus1

sum

119895=1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597[120572

119896minus1]

1119903119896

120597119909

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

+

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

times (

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(

1003816

1003816

1003816

1003816

1003816

119909j+11003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

120585

119895+ 120573

119895minus1120585

119895minus1

1003816

1003816

1003816

1003816

1003816

1minus119903119895

times (

1003816

1003816

1003816

1003816

1003816

120585

119895+1+ 120573

119895120585

119895

1003816

1003816

1003816

1003816

1003816

119903119895+1119901119895+ 120593

119894

119895

sum

119897=1

1003816

1003816

1003816

1003816

120585

119897

1003816

1003816

1003816

1003816

119903119895+1119901119895)

(A7)

where 119861119895ge 0 is a smooth function

Noting that 119903119895+1119901

119895= 119903

119895+ 120596 by using Lemma 6 we have

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (A8)

where 1198971198963ge 0 is a smooth function

Conflict of Interests

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

Acknowledgments

The authors thank the editor and the anonymous reviewersfor their constructive comments and suggestions for improv-ing the quality of the paper This work has been supportedin part by the National Nature Science Foundation of Chinaunder Grant 61073065 and the Key Program of ScienceTechnology Research of Education Department of HenanProvince under Grants 13A120016 and 14A520003

References

[1] C Rui M Reyhanoglu I Kolmanovsky S Cho and NHMcClamroch ldquoNonsmooth stabilization of an underactuatedunstable two degrees of freedom mechanical systemrdquo in Pro-ceedings of the 36th IEEE Conference Decision Control vol 4pp 3998ndash4003 1997

[2] W Lin and C Qian ldquoAdding one power integrator a toolfor global stabilization of high-order lower-triangular systemsrdquoSystems amp Control Letters vol 39 no 5 pp 339ndash351 2000

[3] C Qian andW Lin ldquoA continuous feedback approach to globalstrong stabilization of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 7 pp 1061ndash1079 2001

[4] W Lin and C Qian ldquoAdaptive control of nonlinearly param-eterized systems a nonsmooth feedback frameworkrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 757ndash7742002

[5] W Lin and C Qian ldquoAdaptive control of nonlinearly parame-terized systems the smooth feedback caserdquo IEEE Transactionson Automatic Control vol 47 no 8 pp 1249ndash1266 2002

[6] J Polendo and C Qian ldquoA generalized homogeneous domina-tion approach for global stabilization of inherently nonlinearsystems via output feedbackrdquo International Journal of Robustand Nonlinear Control vol 17 no 7 pp 605ndash629 2007

[7] J Polendo and C Qian ldquoAn expanded method to robustlystabilize uncertain nonlinear systemsrdquo Communications inInformation and Systems vol 8 no 1 pp 55ndash70 2008

Abstract and Applied Analysis 7

[8] Z Sun and Y Liu ldquoAdaptive stabilisation for a large class ofhigh-order uncertain non-linear systemsrdquo International Journalof Control vol 82 no 7 pp 1275ndash1287 2009

[9] J Zhang and Y Liu ldquoA new approach to adaptive control designwithout overparametrization for a class of uncertain nonlinearsystemsrdquo Science China Information Sciences vol 54 no 7 pp1419ndash1429 2011

[10] X-H Zhang and X-J Xie ldquoGlobal state feedback stabilisationof nonlinear systems with high-order and low-order nonlinear-itiesrdquo International Journal of Control vol 87 no 3 pp 642ndash6522014

[11] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[12] Y Hong J Huang and Y Xu ldquoOn an output feedback finite-time stabilization problemrdquo IEEE Transactions on AutomaticControl vol 46 no 2 pp 305ndash309 2001

[13] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[14] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[15] Y Hong J Wang and D Cheng ldquoAdaptive finite-time controlof nonlinear systems with parametric uncertaintyrdquo IEEE Trans-actions on Automatic Control vol 51 no 5 pp 858ndash862 2006

[16] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006

[17] S Li and Y-P Tian ldquoFinite-time stability of cascaded time-varying systemsrdquo International Journal of Control vol 80 no4 pp 646ndash657 2007

[18] S G Nersesov W M Haddad and Q Hui ldquoFinite-timestabilization of nonlinear dynamical systems via control vectorLyapunov functionsrdquo Journal of the Franklin Institute Engineer-ing and Applied Mathematics vol 345 no 7 pp 819ndash837 2008

[19] H Du C Qian M T Frye and S Li ldquoGlobal finite-timestabilisation using bounded feedback for a class of non-linearsystemsrdquo IET Control Theory amp Applications vol 6 no 14 pp2326ndash2336 2012

[20] S Ding S Li and W X Zheng ldquoNonsmooth stabilization of aclass of nonlinear cascaded systemsrdquoAutomatica vol 48 no 10pp 2597ndash2606 2012

[21] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 43no 1 pp 73ndash78 2012

[22] J Li C Qian and S Ding ldquoGlobal finite-time stabilisation byoutput feedback for a class of uncertain nonlinear systemsrdquoInternational Journal of Control vol 83 no 11 pp 2241ndash22522010

[23] B Yang and W Lin ldquoNonsmooth output feedback design witha dynamics gain for uncertain systems with strong nonlin-earitiesrdquo in Proceedings of the 46th IEEE Conference DecisionControl pp 3495ndash3500 New Orieans La USA 2007

[24] H K Khalil Nonlinear Systems Prentice-Hall New Jersey NJUSA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

0 2 4 6 8 10Time (s)

minus2

minus15

minus1

minus05

0

05

1

x1

x2

(a) System states

0 2 4 6 8 10Time (s)

minus180

minus160

minus140

minus120

minus100

minus80

minus60

minus40

minus20

0

20

uu

(b) Control input

Figure 1 The responses of the closed-loop system (38) and (39)

where 1199011= 53 and 119901

2= 1 It is worth pointing

out that although system (38) is simple it cannot beglobally finite-time stabilized using the design methodpresented in [14 21 22] because of the presence of bothlow-order term 1199091011

1and high-order term 1199092

1sin1199092 Choose

120596 = minus111 isin (minus38 0] then 1199031= 1 1199032= (119903

1+ 120596)119901

1= 611

and 1199033= (119903

2+ 120596)119901

2= 511 By Lemma 6 it is easy to

get |1198911| le |119909

1|

1011 and |1198912| le (1 + 119909

2

1)|119909

1|

511 ClearlyAssumption 1 holds with 120593

1= 1 and 120593

2= 1 + 119909

2

1 according

to the design procedure proposed in Section 3 we can get

119909

lowast

2= minus120573

611

1[119909

1]

611 120573

1= 3

1110

119906 = minus (1 + 119897

21+ 119897

22+ 119897

23) [120585

2]

511

(39)

where 1205852= [119909

2]

116minus [119909

lowast

2]

116 11989721= (511) sdot (1811)

65sdot 2

15119897

22= (1722) sdot (1522)

517sdot (1 + 119909

2

1)

2217 and 11989723

= (611) sdot (3011)

56sdot (1711)

116sdot (3

12160+ 3

12130)

In the simulation by choosing the initial values as119909

1(0) = 1 and 119909

2(0) = minus1 Figure 1 is obtained to demonstrate

the effectiveness of the control scheme

6 Conclusion

In this paper a continuous state feedback stabilizing con-troller is presented for a class of high-order nonlinearsystems under weaker condition The controller designedpreserves the equilibrium at the origin and guarantees theglobal finite-time stability of the systems It should be notedthat the proposed controller can only work well when thewhole state vector is measurable Therefore a natural andmore interesting problem is how to design output feedbackstabilizing controller for the systems studied in the paper

if only partial state vector being measurable which is nowunder our further investigation

Appendix

Proof of Proposition 7 Noting that minus1sum119899119897=1119901

1sdot sdot sdot 119901

119897minus1lt 120596 lt

0 and 119901119896minus1119903

119896= 119903

119896minus1+ 120596 we have 0 lt 119901

119896minus1119903

119896lt 1 Using (18)

it follows from Lemma 5 that1003816

1003816

1003816

1003816

1003816

119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

(119909

1119903119896

119896)

119903119896119901119896minus1

minus (119909

lowast(1119903119896)

119896)

119903119896119901119896minus1 10038161003816

1003816

1003816

1003816

1003816

le 2

1minus119903119896119901119896minus11003816

1003816

1003816

1003816

1003816

1003816

[119909

119896]

1119903119896minus [119909

lowast

119896]

11199031198961003816

1003816

1003816

1003816

1003816

1003816

119903119896119901119896minus1

= 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

(A1)

By (A1) and Lemma 6 it can be obtained that

[120585

119896minus1]

2minus119903119896119901119896minus1(119909

119901119896minus1

119896minus 119909

lowast119901119896minus1

119896)

le 2

1minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2minus119903119896119901119896minus1 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

119903119896119901119896minus1

le

1

3

1003816

1003816

1003816

1003816

120585

119896minus1

1003816

1003816

1003816

1003816

2+ 119897

1198961

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A2)

where 1198971198961gt 0 is a constant

Proof of Proposition 8 According to (18) Assumption 1 andLemma 4 it follows that

1003816

1003816

1003816

1003816

119891

119894

1003816

1003816

1003816

1003816

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

119909

119895 (119905)

1003816

1003816

1003816

1003816

1003816

(119903119894+120596)119903119895

le 120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

+ 120573

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

)

119903119894+1119901119894

6 Abstract and Applied Analysis

le 2

1minus119903119894+1119901119894120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894+ 120573

119903119894+1119901119894

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894)

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894

(A3)

where 1205730= 120585

0= 0 and 120593

119894= 2

1minus119903119894+1119901119894sum

119894

119895=1(1 + 120573

119903119894+1119901119894

119895minus1)120593

119894ge 0 is

a smooth functionUsing (A3) and Lemmas 5 and 6 we obtain

[120585

119896]

2minus119903119896+1119901119896119891

119896le 120593

119896

119896

sum

119895=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+11199011198961003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119896+1119901119896

le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A4)

where 1198971198962ge 0 is a smooth function

Proof of Proposition 9 Note that

[119909

lowast

119895+1]

1(119903119895+1)

= minus120573

119895120585

119895= minus

119896minus1

sum

119895=1

119861

119895[119909

119895]

1119903119895 (A5)

where 119861119895= 120573

119896minus1sdot sdot sdot 120573

119895and 119895 = 1 119896 minus 1

Using (A5) after simple calculations it is not hard toobtain that for 119895 = 1 119896 minus 1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

= minus

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

[119909

119895]

1119903119895minus

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(A6)

By (22) (A3) (A6) and Lemmas 4 and 5 we get

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

= minus (2 minus 119903

119896+1119901

119896) int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

times

119896minus1

sum

119895=1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597[120572

119896minus1]

1119903119896

120597119909

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

+

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

times (

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(

1003816

1003816

1003816

1003816

1003816

119909j+11003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

120585

119895+ 120573

119895minus1120585

119895minus1

1003816

1003816

1003816

1003816

1003816

1minus119903119895

times (

1003816

1003816

1003816

1003816

1003816

120585

119895+1+ 120573

119895120585

119895

1003816

1003816

1003816

1003816

1003816

119903119895+1119901119895+ 120593

119894

119895

sum

119897=1

1003816

1003816

1003816

1003816

120585

119897

1003816

1003816

1003816

1003816

119903119895+1119901119895)

(A7)

where 119861119895ge 0 is a smooth function

Noting that 119903119895+1119901

119895= 119903

119895+ 120596 by using Lemma 6 we have

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (A8)

where 1198971198963ge 0 is a smooth function

Conflict of Interests

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

Acknowledgments

The authors thank the editor and the anonymous reviewersfor their constructive comments and suggestions for improv-ing the quality of the paper This work has been supportedin part by the National Nature Science Foundation of Chinaunder Grant 61073065 and the Key Program of ScienceTechnology Research of Education Department of HenanProvince under Grants 13A120016 and 14A520003

References

[1] C Rui M Reyhanoglu I Kolmanovsky S Cho and NHMcClamroch ldquoNonsmooth stabilization of an underactuatedunstable two degrees of freedom mechanical systemrdquo in Pro-ceedings of the 36th IEEE Conference Decision Control vol 4pp 3998ndash4003 1997

[2] W Lin and C Qian ldquoAdding one power integrator a toolfor global stabilization of high-order lower-triangular systemsrdquoSystems amp Control Letters vol 39 no 5 pp 339ndash351 2000

[3] C Qian andW Lin ldquoA continuous feedback approach to globalstrong stabilization of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 7 pp 1061ndash1079 2001

[4] W Lin and C Qian ldquoAdaptive control of nonlinearly param-eterized systems a nonsmooth feedback frameworkrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 757ndash7742002

[5] W Lin and C Qian ldquoAdaptive control of nonlinearly parame-terized systems the smooth feedback caserdquo IEEE Transactionson Automatic Control vol 47 no 8 pp 1249ndash1266 2002

[6] J Polendo and C Qian ldquoA generalized homogeneous domina-tion approach for global stabilization of inherently nonlinearsystems via output feedbackrdquo International Journal of Robustand Nonlinear Control vol 17 no 7 pp 605ndash629 2007

[7] J Polendo and C Qian ldquoAn expanded method to robustlystabilize uncertain nonlinear systemsrdquo Communications inInformation and Systems vol 8 no 1 pp 55ndash70 2008

Abstract and Applied Analysis 7

[8] Z Sun and Y Liu ldquoAdaptive stabilisation for a large class ofhigh-order uncertain non-linear systemsrdquo International Journalof Control vol 82 no 7 pp 1275ndash1287 2009

[9] J Zhang and Y Liu ldquoA new approach to adaptive control designwithout overparametrization for a class of uncertain nonlinearsystemsrdquo Science China Information Sciences vol 54 no 7 pp1419ndash1429 2011

[10] X-H Zhang and X-J Xie ldquoGlobal state feedback stabilisationof nonlinear systems with high-order and low-order nonlinear-itiesrdquo International Journal of Control vol 87 no 3 pp 642ndash6522014

[11] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[12] Y Hong J Huang and Y Xu ldquoOn an output feedback finite-time stabilization problemrdquo IEEE Transactions on AutomaticControl vol 46 no 2 pp 305ndash309 2001

[13] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[14] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[15] Y Hong J Wang and D Cheng ldquoAdaptive finite-time controlof nonlinear systems with parametric uncertaintyrdquo IEEE Trans-actions on Automatic Control vol 51 no 5 pp 858ndash862 2006

[16] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006

[17] S Li and Y-P Tian ldquoFinite-time stability of cascaded time-varying systemsrdquo International Journal of Control vol 80 no4 pp 646ndash657 2007

[18] S G Nersesov W M Haddad and Q Hui ldquoFinite-timestabilization of nonlinear dynamical systems via control vectorLyapunov functionsrdquo Journal of the Franklin Institute Engineer-ing and Applied Mathematics vol 345 no 7 pp 819ndash837 2008

[19] H Du C Qian M T Frye and S Li ldquoGlobal finite-timestabilisation using bounded feedback for a class of non-linearsystemsrdquo IET Control Theory amp Applications vol 6 no 14 pp2326ndash2336 2012

[20] S Ding S Li and W X Zheng ldquoNonsmooth stabilization of aclass of nonlinear cascaded systemsrdquoAutomatica vol 48 no 10pp 2597ndash2606 2012

[21] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 43no 1 pp 73ndash78 2012

[22] J Li C Qian and S Ding ldquoGlobal finite-time stabilisation byoutput feedback for a class of uncertain nonlinear systemsrdquoInternational Journal of Control vol 83 no 11 pp 2241ndash22522010

[23] B Yang and W Lin ldquoNonsmooth output feedback design witha dynamics gain for uncertain systems with strong nonlin-earitiesrdquo in Proceedings of the 46th IEEE Conference DecisionControl pp 3495ndash3500 New Orieans La USA 2007

[24] H K Khalil Nonlinear Systems Prentice-Hall New Jersey NJUSA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

le 2

1minus119903119894+1119901119894120593

119894

119894

sum

119895=1

(

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894+ 120573

119903119894+1119901119894

119895minus1

1003816

1003816

1003816

1003816

1003816

120585

119895minus1

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894)

le 120593

119894

119894

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119894+1119901119894

(A3)

where 1205730= 120585

0= 0 and 120593

119894= 2

1minus119903119894+1119901119894sum

119894

119895=1(1 + 120573

119903119894+1119901119894

119895minus1)120593

119894ge 0 is

a smooth functionUsing (A3) and Lemmas 5 and 6 we obtain

[120585

119896]

2minus119903119896+1119901119896119891

119896le 120593

119896

119896

sum

119895=1

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2minus119903119896+11199011198961003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

119903119896+1119901119896

le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198962

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2

(A4)

where 1198971198962ge 0 is a smooth function

Proof of Proposition 9 Note that

[119909

lowast

119895+1]

1(119903119895+1)

= minus120573

119895120585

119895= minus

119896minus1

sum

119895=1

119861

119895[119909

119895]

1119903119895 (A5)

where 119861119895= 120573

119896minus1sdot sdot sdot 120573

119895and 119895 = 1 119896 minus 1

Using (A5) after simple calculations it is not hard toobtain that for 119895 = 1 119896 minus 1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

= minus

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

[119909

119895]

1119903119895minus

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(A6)

By (22) (A3) (A6) and Lemmas 4 and 5 we get

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

= minus (2 minus 119903

119896+1119901

119896) int

119909119896

119909lowast

119896

[[119904]

1119903119896minus [119909

lowast

119896]

1119903119896]

1minus119903119896+1119901119896

119889119904

times

119896minus1

sum

119895=1

120597[119909

lowast

119896]

1119903119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597[120572

119896minus1]

1119903119896

120597119909

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le 2

2minus119903119896 10038161003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus120596119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119896minus1

sum

119897=1

120597119861

119897

120597119909

119895

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

+

1

119903

119895

119861

119895

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

times (

1003816

1003816

1003816

1003816

1003816

119909

119895+1

1003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

119909

119895

1003816

1003816

1003816

1003816

1003816

1119903119895minus1

(

1003816

1003816

1003816

1003816

1003816

119909j+11003816

1003816

1003816

1003816

1003816

119901119895+

1003816

1003816

1003816

1003816

1003816

119891

119895

1003816

1003816

1003816

1003816

1003816

)

le

119896minus1

sum

119895=1

119861

119895

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

1minus12059610038161003816

1003816

1003816

1003816

120585

119895+ 120573

119895minus1120585

119895minus1

1003816

1003816

1003816

1003816

1003816

1minus119903119895

times (

1003816

1003816

1003816

1003816

1003816

120585

119895+1+ 120573

119895120585

119895

1003816

1003816

1003816

1003816

1003816

119903119895+1119901119895+ 120593

119894

119895

sum

119897=1

1003816

1003816

1003816

1003816

120585

119897

1003816

1003816

1003816

1003816

119903119895+1119901119895)

(A7)

where 119861119895ge 0 is a smooth function

Noting that 119903119895+1119901

119895= 119903

119895+ 120596 by using Lemma 6 we have

119896minus1

sum

119895=1

120597119882

119896

120597119909

119895

(119909

119901119895

119895+1+ 119891

119895) le

1

3

119896minus1

sum

119895=1

1003816

1003816

1003816

1003816

1003816

120585

119895

1003816

1003816

1003816

1003816

1003816

2

+ 119897

1198963

1003816

1003816

1003816

1003816

120585

119896

1003816

1003816

1003816

1003816

2 (A8)

where 1198971198963ge 0 is a smooth function

Conflict of Interests

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

Acknowledgments

The authors thank the editor and the anonymous reviewersfor their constructive comments and suggestions for improv-ing the quality of the paper This work has been supportedin part by the National Nature Science Foundation of Chinaunder Grant 61073065 and the Key Program of ScienceTechnology Research of Education Department of HenanProvince under Grants 13A120016 and 14A520003

References

[1] C Rui M Reyhanoglu I Kolmanovsky S Cho and NHMcClamroch ldquoNonsmooth stabilization of an underactuatedunstable two degrees of freedom mechanical systemrdquo in Pro-ceedings of the 36th IEEE Conference Decision Control vol 4pp 3998ndash4003 1997

[2] W Lin and C Qian ldquoAdding one power integrator a toolfor global stabilization of high-order lower-triangular systemsrdquoSystems amp Control Letters vol 39 no 5 pp 339ndash351 2000

[3] C Qian andW Lin ldquoA continuous feedback approach to globalstrong stabilization of nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 46 no 7 pp 1061ndash1079 2001

[4] W Lin and C Qian ldquoAdaptive control of nonlinearly param-eterized systems a nonsmooth feedback frameworkrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 757ndash7742002

[5] W Lin and C Qian ldquoAdaptive control of nonlinearly parame-terized systems the smooth feedback caserdquo IEEE Transactionson Automatic Control vol 47 no 8 pp 1249ndash1266 2002

[6] J Polendo and C Qian ldquoA generalized homogeneous domina-tion approach for global stabilization of inherently nonlinearsystems via output feedbackrdquo International Journal of Robustand Nonlinear Control vol 17 no 7 pp 605ndash629 2007

[7] J Polendo and C Qian ldquoAn expanded method to robustlystabilize uncertain nonlinear systemsrdquo Communications inInformation and Systems vol 8 no 1 pp 55ndash70 2008

Abstract and Applied Analysis 7

[8] Z Sun and Y Liu ldquoAdaptive stabilisation for a large class ofhigh-order uncertain non-linear systemsrdquo International Journalof Control vol 82 no 7 pp 1275ndash1287 2009

[9] J Zhang and Y Liu ldquoA new approach to adaptive control designwithout overparametrization for a class of uncertain nonlinearsystemsrdquo Science China Information Sciences vol 54 no 7 pp1419ndash1429 2011

[10] X-H Zhang and X-J Xie ldquoGlobal state feedback stabilisationof nonlinear systems with high-order and low-order nonlinear-itiesrdquo International Journal of Control vol 87 no 3 pp 642ndash6522014

[11] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[12] Y Hong J Huang and Y Xu ldquoOn an output feedback finite-time stabilization problemrdquo IEEE Transactions on AutomaticControl vol 46 no 2 pp 305ndash309 2001

[13] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[14] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[15] Y Hong J Wang and D Cheng ldquoAdaptive finite-time controlof nonlinear systems with parametric uncertaintyrdquo IEEE Trans-actions on Automatic Control vol 51 no 5 pp 858ndash862 2006

[16] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006

[17] S Li and Y-P Tian ldquoFinite-time stability of cascaded time-varying systemsrdquo International Journal of Control vol 80 no4 pp 646ndash657 2007

[18] S G Nersesov W M Haddad and Q Hui ldquoFinite-timestabilization of nonlinear dynamical systems via control vectorLyapunov functionsrdquo Journal of the Franklin Institute Engineer-ing and Applied Mathematics vol 345 no 7 pp 819ndash837 2008

[19] H Du C Qian M T Frye and S Li ldquoGlobal finite-timestabilisation using bounded feedback for a class of non-linearsystemsrdquo IET Control Theory amp Applications vol 6 no 14 pp2326ndash2336 2012

[20] S Ding S Li and W X Zheng ldquoNonsmooth stabilization of aclass of nonlinear cascaded systemsrdquoAutomatica vol 48 no 10pp 2597ndash2606 2012

[21] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 43no 1 pp 73ndash78 2012

[22] J Li C Qian and S Ding ldquoGlobal finite-time stabilisation byoutput feedback for a class of uncertain nonlinear systemsrdquoInternational Journal of Control vol 83 no 11 pp 2241ndash22522010

[23] B Yang and W Lin ldquoNonsmooth output feedback design witha dynamics gain for uncertain systems with strong nonlin-earitiesrdquo in Proceedings of the 46th IEEE Conference DecisionControl pp 3495ndash3500 New Orieans La USA 2007

[24] H K Khalil Nonlinear Systems Prentice-Hall New Jersey NJUSA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

[8] Z Sun and Y Liu ldquoAdaptive stabilisation for a large class ofhigh-order uncertain non-linear systemsrdquo International Journalof Control vol 82 no 7 pp 1275ndash1287 2009

[9] J Zhang and Y Liu ldquoA new approach to adaptive control designwithout overparametrization for a class of uncertain nonlinearsystemsrdquo Science China Information Sciences vol 54 no 7 pp1419ndash1429 2011

[10] X-H Zhang and X-J Xie ldquoGlobal state feedback stabilisationof nonlinear systems with high-order and low-order nonlinear-itiesrdquo International Journal of Control vol 87 no 3 pp 642ndash6522014

[11] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[12] Y Hong J Huang and Y Xu ldquoOn an output feedback finite-time stabilization problemrdquo IEEE Transactions on AutomaticControl vol 46 no 2 pp 305ndash309 2001

[13] Y Hong ldquoFinite-time stabilization and stabilizability of a classof controllable systemsrdquo Systems amp Control Letters vol 46 no4 pp 231ndash236 2002

[14] X Huang W Lin and B Yang ldquoGlobal finite-time stabilizationof a class of uncertain nonlinear systemsrdquo Automatica vol 41no 5 pp 881ndash888 2005

[15] Y Hong J Wang and D Cheng ldquoAdaptive finite-time controlof nonlinear systems with parametric uncertaintyrdquo IEEE Trans-actions on Automatic Control vol 51 no 5 pp 858ndash862 2006

[16] Y Hong and Z-P Jiang ldquoFinite-time stabilization of nonlinearsystems with parametric and dynamic uncertaintiesrdquo IEEETransactions onAutomatic Control vol 51 no 12 pp 1950ndash19562006

[17] S Li and Y-P Tian ldquoFinite-time stability of cascaded time-varying systemsrdquo International Journal of Control vol 80 no4 pp 646ndash657 2007

[18] S G Nersesov W M Haddad and Q Hui ldquoFinite-timestabilization of nonlinear dynamical systems via control vectorLyapunov functionsrdquo Journal of the Franklin Institute Engineer-ing and Applied Mathematics vol 345 no 7 pp 819ndash837 2008

[19] H Du C Qian M T Frye and S Li ldquoGlobal finite-timestabilisation using bounded feedback for a class of non-linearsystemsrdquo IET Control Theory amp Applications vol 6 no 14 pp2326ndash2336 2012

[20] S Ding S Li and W X Zheng ldquoNonsmooth stabilization of aclass of nonlinear cascaded systemsrdquoAutomatica vol 48 no 10pp 2597ndash2606 2012

[21] Y Shen andYHuang ldquoGlobal finite-time stabilisation for a classof nonlinear systemsrdquo International Journal of Systems SciencePrinciples and Applications of Systems and Integration vol 43no 1 pp 73ndash78 2012

[22] J Li C Qian and S Ding ldquoGlobal finite-time stabilisation byoutput feedback for a class of uncertain nonlinear systemsrdquoInternational Journal of Control vol 83 no 11 pp 2241ndash22522010

[23] B Yang and W Lin ldquoNonsmooth output feedback design witha dynamics gain for uncertain systems with strong nonlin-earitiesrdquo in Proceedings of the 46th IEEE Conference DecisionControl pp 3495ndash3500 New Orieans La USA 2007

[24] H K Khalil Nonlinear Systems Prentice-Hall New Jersey NJUSA 3rd edition 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of