research article fuzzy finite-time stability of chaotic...

Research ArticleFuzzy Finite-Time Stability of Chaotic Systems withTime-Varying Delay and Parameter Uncertainties

Dong Li12 and Jinde Cao34

1School of Automation Southeast University Nanjing 210096 China2School of Mathematical Sciences Anhui University Hefei 230601 China3Department of Mathematics Southeast University Nanjing 210096 China4Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Jinde Cao jdcaoseueducn

Received 27 February 2015 Revised 12 May 2015 Accepted 13 May 2015

Academic Editor Mohammed Chadli

Copyright copy 2015 D Li and J Cao 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 discusses the finite-time stability of chaotic systems with time-varying delay and parameter uncertainties A newmodelbased on Takagi-Sugeno (T-S)model is proposed for representing chaotic systems By the newmodel finite-time stability of chaoticsystems can be converted into stabilization of fuzzy T-S systems with parameter uncertainties A sufficient condition is given interms of matrix inequalities which guarantees the finite-time stability for fuzzy systems can be achieved Numerical simulationson the chaotic systems are presented to demonstrate the effectiveness of the theoretical results

1 Introduction

In the last decades the study of controlling chaotic systemshas received considerable attention due to its broad appli-cations in biological systems information processing securecommunications and so forth In practical physical systemsthe parameters of chaotic systems may not be known exactlySo there are many papers concerning uncertain chaoticsystems [1 2] For the character of the uncertain chaoticsystems many significant results have been obtained by usingadaptive control technique for example output feedbackcontrol [3] fuzzy adaptive control [4] and optimal control[5]

In order to simplify the controller design of the chaoticsystems various schemes have been developed amongwhichfuzzy model is one of the successful approaches to obtainnonlinear chaotic systems Specifically a so-called Takagi-Sugeno (TS) fuzzymodel was proposed in [6] where complexnonlinear systems are represented by some local linearsubsystems Based on the T-S fuzzy model some chaoticcontrol design methods have been developed [7ndash12] In [7]a fuzzy model is presented to simulate two different chaoticsystems with different numbers of nonlinear terms and a

new adaptive approach is proposed to synchronize these twodifferent fuzzy chaotic systems Predictive control for chaoticsystems is based on a T-S fuzzy model in [8] Wang and Wu[9] and Zhang et al [10] investigate the problem of fuzzyimpulsive control to stabilize chaotic systems

Although there have been many works to discuss thestabilization of chaotic systems most of the existing stabi-lization algorithms are asymptotically convergent algorithms[13ndash17] which means that the convergence rate is at bestexponential with infinite settling time Compared to theasymptotically convergent algorithms the finite-time conver-gence algorithms demonstrate not only faster convergencerates but also better disturbance rejection properties androbustness against uncertainties [18ndash20] References [21ndash24]discuss finite-time control for chaotic systems But there arefew works considering finite-time control for chaotic systemsbased on the fuzzy models

Motivated by the aforementioned analysis the finite-timestabilization control for chaotic systems based on the T-Sfuzzy models is a significative topic Our main contributionin this paper is the introduction of the fuzzy model to discussfinite-time stabilization of chaotic systems in a generalmodelIn this model we not only consider time-varying delay but

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 393860 7 pageshttpdxdoiorg1011552015393860

2 Mathematical Problems in Engineering

also study the chaotic system with parameter uncertaintiesObviously on the one hand our results have optimality inthe convergence time of stabilization in [14 15] on the otherhand they are more practically applicable than those in[22ndash24] Based on the Lyapunov stability theory a feedbackcontroller is designed to guarantee the stabilization in finitetimeThe example is given to illustrate our theoretical results

The rest of the paper is organized as follows In Section 2some preliminaries are briefly given Section 3 presents themain results In Section 4 simulation results aiming at sub-stantiating the theoretical analysis are presented This paperis concluded in Section 5

2 Preliminaries and Problem Formulation

In this paper we want to develop a finite-time control law fora class of chaotic systems In system analysis and design it isimportant to select an appropriate model representing a realsystem The T-S fuzzy model can express a highly nonlinearfunctional relation with a small number of rules Manychaotic systems can be represented by fuzzy linear modelssuch as Lorenz Rossler Chua Chen and Lu systems Thisfuzzy modeling method is simple and natural So considera class of time-delayed chaotic systems represented by thefollowing time-delayed T-S fuzzy model

Plant Rule is as followsIf 1205791(119905) is1198721198941 and 120579119901(119905) is119872119894119901 then

(119905) = 119860 119894119909 (119905) + 119861119894119909 (119905 minus 120591 (119905)) 119894 = 1 2 119873 (1)

where 119909(119905) = (1199091(119905) 119909119899(119905))119879isin 119877119899 is the state variable the

premise variables 1205791(119905) 120579119901(119905) are proper state variables119873 is the number of the fuzzy rules 119872

119894119895(119895 = 1 2 119901)

are fuzzy sets 119860119894and 119861

119894are known constant matrices with

appropriate dimensions 120591(119905) is the transmission delayIn practice however some parameters of chaotic systems

cannot be exactly known a priori If the information ofuncertain matrices is considered the results will be less con-servative than those results that do not utilize the elementaluncertain informationThus it is very interesting to study thedesign of the fuzzy systems with parameter perturbations


(119905) = (119860 119894 +nabla119860 119894) 119909 (119905) + (119861119894 +nabla119861119894) 119909 (119905 minus 120591 (119905))

119894 = 1 2 119873(2)

The parameter uncertainties are classically written as [10]nabla119860119894= 119863119860

119894119865119860

119894119864119860

119894and nabla119861

119894= 119863119861

119894119865119861

119894119864119861

119894where 119863119860

119894 119863119861119894 119864119860119894 119864119861119894

are known real matrices of appropriate dimension and119865119860119894119865119861119894

are the unknownmatrix functionswith Lebesgue-measurableelements and satisfy the conditions 119865119860

119894

119879

119865119860

119894le 119868 119865119861

119894

119879

119865119861

119894le 119868

where 119868 is the identity matrix of appropriate dimension

The overall fuzzy system with control is inferred asfollows

(119905) =

119873

sum

119894=1ℎ119894 (120579 (119905)) [(119860 119894 +nabla119860 119894) 119909 (119905)

+ (119861119894+nabla119861119894) 119909 (119905 minus 120591 (119905)) + 119906 (119905)]

(3)

where 119906(119905) denotes the feedback control ℎ119894(120579(119905)) = 120596

119894(120579(119905))

sum119873

119894=1 120596119894(120579(119905)) 120596119894(120579(119905)) = prod119901

119895=1119872119894119895(120579119895(119905)) and 120596119894(120579(119905)) ge 0sum119873

119894=1 120596119894(120579(119905)) ge 0 thus ℎ119894(120579(119905)) ge 0 sum119873

119894=1 ℎ119894(120579(119905)) = 1Assume 119862([minus120591 0] 119877119899) is a Banach space of continuous

functions mapping the interval [minus120591 0] into 119877119899 with the

norm 120601 = supminus120591le120579le0120601(120579) For the functional differential

equation (3) its initial conditions are given by 119909119894(119905) =

120601119894(119905) isin 119862([minus120591 0] 119877119899)We always assume that (3) has a unique

solution with respect to initial conditionsFor starting simplification one has the following funda-

mental assumption

Assumption 1 Consider 0 le 120591(119905) le ℎ lt 1whereℎ is constant

To end this section we introduce the following lemmaswhich are useful in deriving sufficient conditions of finite-time stability

Lemma 2 (see [25]) Consider system = 119891(119909) 119891(0) = 0119909 isin 119877

119899 where 119891(sdot) 119877119899 rarr 119877119899 is a continuous vector function

Suppose there exist a 1198621 positive definite and proper function119881 119877119899rarr 119877 and real numbers 120583 gt 0 and 120578 isin (0 1) such that

+ 120583119881120578 is negative semidefinite Then the origin is a globally

finite-time stable equilibrium of system = 119891(119909) Moreoverthe settling time 119879 le 119881

1minus120578(0)(1 minus 120578)120583

Lemma 3 (see [26]) Given any real matrices 119860 119861 Σ ofappropriate dimensions and a scalar 119904 gt 0 such that 0 lt Σ =

Σ119879 then the following inequality holds

119860119879119861+119861119879119860 le 119904119860

119879Σ119860+ 119904

minus1119861119879Σminus1119861 (4)

Lemma 4 (see [27]) Let 119860 119863 119864 and 119865 be real matrices ofappropriate dimensions with 119865 satisfying 119865 le 1 Then onehas the following

(a) For any scalar 120582 gt 0

119863119865119864+119864119879119865119879119863119879le 120582minus1119863119863119879+120582119864119864

119879 (5)

(b) For any matrix 119875 gt 0 and scalar 120585 gt 0 such that 120585119868 minus119864119879119875119864 gt 0

(119860 +119863119865119864)119879119875 (119860+119863119865119864)

le 119860119879119875119860+119860

119879119875119864 (120585119868 minus119864

119879119875119864)minus1119864119879119875119860+ 120585

minus1119863119879119863

(6)

Lemma 5 ((Schur complement) [28]) For a given matrix

119878 = [11987811 11987812

119878119879

12 11987822] lt 0 (7)

where 11987811 = 119878119879

11 and 11987822 = 119878119879

22 are equivalent to any one of thefollowing conditions

Mathematical Problems in Engineering 3

(a) 11987822 lt 0 11987811 minus 11987812119878minus122 119878119879

12 lt 0(b) 11987811 lt 0 11987822 minus 11987811987912119878

minus111 11987812 lt 0

3 Finite-Time Stabilization

First we present our main result on the finite-time stabiliza-tion of system (3)

Theorem 6 Suppose the positive constants 120582119894 119894 = 1 2 3 1198961

1198962 and positive definite matrix 119875 such that(1)

(

119860119879

119894119875 + 119875119860

119894+ (1205822 + 1198962 minus 1198961) 119875 119864

119860

119894119875119863119860

119894

119864119860

119894

119879

minus1205821119868 0

119863119860

119894

119879

119875 0 minus120582minus11 119868

)

le 0 119894 = 1 119873

(8)

(2)

(

119861119879

119894119875119861119894minus 11989621205822 (1 minus ℎ) 119875 119863

119861

119894119864119861

119894

119879

119875119861119894

119863119861

119894

119879

minus120582minus13 119868 0

119861119879

119894119875119864119861

1198940 minus (1205823119868 minus 119864

119861

119894

119879

119875119864119861

119894)

)

le 0 119894 = 1 119873

(9)

Then stabilization of system (3) under Assumption 1 can beachieved in finite time if the control law 119906 is designed as

119906 = minus11989612119909 (119905) minus

120583

2120582(120578+1)2max (119875)

120582min (119875)sign (119909 (119905)) |119909 (119905)|120578

minus120583

2120582min (119875)(1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)

119909 (119905)2

(10)

where 120583 is an arbitrary positive constant

Proof Construct a Lyapunov function

119881 (119905) = 119909119879(119905) 119875119909 (119905) + 1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904 (11)

so we get

(119905)10038161003816100381610038161003816(3)

=

119873

sum

119894=1ℎ119894 (120579 (119905)) 119909

119879(119905)

sdot (119860119879

119894119875+119875119860

119894+119864119860

119894

119879

119865119860

119894

119879

119863119860

119894

119879

119875+119875119863119860

119894119865119860

119894119864119860

119894) 119909 (119905)

+ 119909119879(119905 minus 120591) (119861

119879

119894+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875119909 (119905) + 119909119879(119905)

sdot 119875 (119861119894+119863119860

119894119865119860

119894119864119860

119894) 119909 (119905 minus 120591) + 2119909119879 (119905) 119875119906 (119905)

+ 1198962119909119879(119905) 119875119909 (119905) minus 1198962119909

119879(119905 minus 120591) 119875119909 (119905 minus 120591) (1minus 120591 (119905))

(12)

Using Lemma 4(a) we have

119864119860

119894

119879

119865119860

119894

119879

119863119860

119894

119879

119875+119875119863119860

119894119865119860

119894119864119860

119894

le 120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

(13)

Applying Lemmas 3 and 4(b) we get

119909119879(119905 minus 120591) (119861

119879

119894+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875119909 (119905) + 119909119879(119905) 119875 (119861119894

+119863119861

119894119865119861

119894119864119861

119894) 119909 (119905 minus 120591) le 120582

minus12 119909119879(119905 minus 120591) (119861

119879

119894

+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875 (119861119894+119863119861

119894119865119861

119894119864119861

119894) 119909 (119905 minus 120591)

+ 1205822119909119879(119905) 119875119909 (119905) le 120582

minus12 119909119879(119905 minus 120591) (119861

119879

119894119875119861119894

+119861119879

119894119875119864119861

119894(1205823119868 minus119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)119909 (119905 minus 120591) + 1205822119909

119879(119905) 119875119909 (119905)

(14)

where 120582119894 119894 = 1 2 3 are positive constants

Substituting (13)-(14) into (12) and using Assumption 1we have

(119905) le

119873

sum

119894=1ℎ119894 (120579 (119905)) 119909

119879(119905) (119860

119879

119894119875+119875119860

119894

+120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

+1205822119875+ 1198962119875)119909 (119905)

+ 2119909119879 (119905) 119875119906 (119905) + 119909119879 (119905 minus 120591) [120582minus12 (119861119879

119894119875119861119894

+119861119879

119894119875119864119861

119894(1205823119868 minus 119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)minus 1198962119875 (1minus ℎ)] 119909 (119905 minus 120591)

(15)

Applying Schur criterion on conditions (1) and (2) yields

119860119879

119894119875+119875119860

119894+120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

+1205822119875

+ 1198962119875 le 1198961119875

119861119879

119894119875119861119894+119861119879

119894119875119864119861

119894(1205823119868 minus119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894le 11989621205822 (1minus ℎ) 119875

(16)


Substituting (16) into (15) and using control inputs in (10)one obtains

(119905) le minus 120583120582(1+120578)2max (119875)

120582min (119875)119909119879(119905) 119875 sign (119909 (119905)) |119909 (119905)|120578

minus120583

120582min (119875)(1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)119879119875119909 (119905)

119909 (119905)2

le minus120583[120582(1+120578)2max (119875) (119909

119879(119905) 119909 (119905))

(1+120578)2

+(1198962 int119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2]

le minus120583 (119881 (119905))(1+120578)2

(17)

Therefore by Lemma 2 the synchronization error system(3) is globally finite-time stable and the finite time is estimatedby

119879 = 1199050 +119881

1minus(1+120578)2(1199050)

120583 (1 minus (1 + 120578) 2) (18)

This completes the proof of the theorem

Remark 7 In [10 14 15] the authors investigated the sta-bilization of chaotic systems via fuzzy models But theseresults were all based on the stability time in large enoughIn Theorem 6 we introduce the feedback control method toguarantee finite-time stability

Remark 8 Theorem 6 provides a sufficient condition offinite-time stabilization of chaotic systems In [22ndash24] theauthors also study the finite-time control for chaotic systemsbut the delay is a constant or there is no delay moreover theparameters of chaotic systems are known exactly Obviouslythe results of this section extend and improve existing results

Remark 9 The magnitude of 119909(119905)119909(119905)2 in the controller119906(119905) will turn to infinity as 119909(119905) rarr 0 in order to avoid theoccurrence of this phenomenon we can add a sufficient smallpositive constant 120576 to its denominator in practice [2 29]

Similar to Huang et al [30] and Khoo et al [31] thecandidate Lyapunov function 119881(119905) can be chosen as 119881(119905) =119909119879(119905)119909(119905) A summary is shown in the next corollary

Corollary 10 Suppose the positive constants 120582119894 119894 = 1 2 3

1198961 1198962 and positive constant 119901 such that

(1)

119860119879

119894119901+119901119860

119894+120582minus11 119901

2119863119860

119894119863119860

119894

119879

+1205821119864119860

119894119864119860

119894

119879

+ (1205822 + 1198962 minus 1198961) 119901119868 le 0 119894 = 1 119873(19)

(2)

120582minus12 (119901119861

119879

119894119861119894+119901

2119861119879

119894119864119861

119894(1205823119868 minus 119901119864

119861

119894

119879

119864119861

119894)

minus1119864119861

119894

119879

119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)minus 1198962 (1minus ℎ) 119901119868 le 0 119894 = 1 119873

(20)


119906 = minus11989612119909 (119905) minus

120583

2119901(120578minus1)2 sign (119909 (119905)) |119909 (119905)|120578 minus

120583

2

sdot 119901(120578minus1)2

(1198962 int119905

119905minus120591

119909119879(119904) 119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)

119909 (119905)2

(21)


Proof Letting 119875 = 119901119868 we can obtain Corollary 10 directlyfromTheorem 6

4 Numerical Example

In this section simulation example is presented to illustratethe utility of theoretical analysis in this paper

Example 1 We consider the following time-delayed Lorenzsystem [32]

1 (119905) = minus 101199091 (119905) + 101199092 (119905 minus 120591)

2 (119905) = 281199091 (119905) minus 1199092 (119905) minus 1199091 (119905) 1199093 (119905)

3 (119905) = 1199091 (119905) 1199092 (119905) minus8

31199093 (119905 minus 120591)

(22)

where 120591 = 16 Figure 1 is the state trajectory of system (22)

Then we have the following fuzzy control model withparameter uncertainties

Plant Rule is as follows If 1199091(119905) is119872119894 then


+ 119906 (119905) 119894 = 1 2(23)


minus50

0

50

minus100minus50

050

100minus40

minus20

0

20

40

60

80

x2

x3

x1

Figure 1 The state trajectory of system (22)

where 119909(119905) = (1199091(119905) 1199092(119905) 1199093(119905))1198791198721 = (12)(1 + 1199091(119905)30)

1198722 = (12)(1 minus 1199091(119905)30)

1198601 = (

minus10 0 028 minus1 minus300 30 0

)

1198602 = (

minus10 0 028 minus1 300 minus30 0

)

1198611 = 1198612 = (

0 10 00 0 0

0 0 minus83

)

(24)

The elements ofnabla119860119894andnabla119861

119894are randomly chosen within

20 of their nominal values corresponding to 119860119894and 119861

119894

respectively Based on assumption of uncertainty we define119865119860

119894and 119865119861

119894to be random matrices and satisfy the conditions

119865119860

119894

119879

119865119860

119894le 119868 119865119861

119894

119879

119865119861

119894le 119868

119863119860

119894= 119863119861

119894= diag (02 02 02)

119864119860

119894= (

minus10 0 028 minus1 00 0 0

)

119864119861

119894= 119861119894

119894 = 1 2

(25)

Select 1205821 = 1205822 = 1 1205823 = 15 1198961 = 100 1198962 = 20 and120578 = 025 By the simulation of the Matlab LMI Toolbox 119875 =

[01345 minus02493 00000 minus02493 102021 minus00000 00000minus00000 13286] The trajectory of system states 119909

119894(119905) under

finite-time control is shown in Figure 2 Then the simulationresults show the correctness of Theorem 6

0 05 1 15 2

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)

Figure 2 The trajectory of system states 119909119894(119905) with finite-time

control

0 2 4 6 8 10

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)

Figure 3 The trajectory of system (22) with 120591(119905) = 119890119905(1 + 119890119905) via

finite-time control

More generally we can choose a time-varying delay toverify the correctness of the theory We choose the delayas 120591(119905) = 119890

119905(1 + 119890

119905) in system (22) Obviously 0 lt

120591(119905) lt 12 lt 1 By the simulation of the Matlab LMIToolbox 119875 = [01439 minus03021 00004 minus03021 7822900856 00004 00856 17595]The trajectory of system states119909119894(119905) under finite-time control is shown in Figure 3

Remark 11 When 120591 is a time-varying delay system (22) maynot be a chaotic system In [33] when the systems have no


chaotic phenomenon the method is not feasible Howeverour results do not need this restriction

5 Conclusions

This paper presents finite-time stability of chaotic systemswith time-varying delay by fuzzy-model-based controllersIn order to obtain better robustness results we discuss thefuzzy system with parameter uncertainties Various sufficientconditions for finite-time control are derived by Lyapunovmethods and linear matrix inequalities (LMI) techniquesThese results are novel some numerical examples are given toverify our theoretical results Next wewill continue the finite-time control problem for coupled chaotic systems based onthe T-S fuzzy models

Conflict of Interests

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

References

[1] J Mei M Jiang and J Wang ldquoFinite-time structure identi-fication and synchronization of drive-response systems withuncertain parameterrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 18 no 4 pp 999ndash1015 2013

[2] M P Aghababa and H P Aghababa ldquoA general nonlinearadaptive control scheme for finite-time synchronization ofchaotic systems with uncertain parameters and nonlinearinputsrdquo Nonlinear Dynamics vol 69 no 4 pp 1903ndash1914 2012

[3] J Zhou and M J Er ldquoAdaptive output control of a class ofuncertain chaotic systemsrdquo Systems and Control Letters vol 56no 6 pp 452ndash460 2007

[4] A Boulkroune A Bouzeriba S Hamel and T Bouden ldquoAprojective synchronization scheme based on fuzzy adaptivecontrol for unknown multivariable chaotic systemsrdquo NonlinearDynamics vol 78 no 1 pp 433ndash447 2014

[5] Y Miladi M Feki and N Derbel ldquoStabilizing the unstableperiodic orbits of a hybrid chaotic system using optimalcontrolrdquo Communications in Nonlinear Science and NumericalSimulation vol 20 no 3 pp 1043ndash1056 2015

[6] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[7] S-Y Li H-K Chen L-M Tam S-C Huang and Z-M GeldquoPragmatical adaptive synchronizationmdashnew fuzzy model oftwo different and complex chaotic systems by new adaptivecontrolrdquo Information Sciences vol 277 pp 458ndash480 2014

[8] A Senouci and A Boukabou ldquoPredictive control and synchro-nization of chaotic and hyperchaotic systems based on a T-Sfuzzy modelrdquo Mathematics and Computers in Simulation vol105 pp 62ndash78 2014

[9] Z-P Wang and H-N Wu ldquoSynchronization of chaotic systemsusing fuzzy impulsive controlrdquoNonlinear Dynamics vol 78 no1 pp 729ndash742 2014

[10] X Zhang A Khadra D Yang and D Li ldquoUnified impulsivefuzzy-model-based controllers for chaotic systems with param-eter uncertainties via LMIrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 1 pp 105ndash1142010

[11] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010

[12] K Yuan H-X Li and J Cao ldquoRobust stabilization of thedistributed parameter systemwith time delay via fuzzy controlrdquoIEEE Transactions on Fuzzy Systems vol 16 no 3 pp 567ndash5842008

[13] WHe and J Cao ldquoAdaptive synchronization of a class of chaoticneural networks with known or unknown parametersrdquo PhysicsLetters A vol 372 no 4 pp 408ndash416 2008

[14] M H Asemani and V J Majd ldquoStability of output-feedbackDPDC-based fuzzy synchronization of chaotic systems viaLMIrdquo Chaos Solitons and Fractals vol 42 no 2 pp 1126ndash11352009

[15] M S Ali and P Balasubramaniam ldquoGlobal exponential stabilityof uncertain fuzzy BAM neural networks with time-varyingdelaysrdquo Chaos Solitons amp Fractals vol 42 no 4 pp 2191ndash21992009

[16] M Chadli I Zelinka and T Youssef ldquoUnknown inputsobserver design for fuzzy systems with application to chaoticsystem reconstructionrdquo Computers amp Mathematics with Appli-cations vol 66 no 2 pp 147ndash154 2013

[17] M Chadli and I Zelinka ldquoChaos synchronization of unknowninputs Takagi-Sugeno fuzzy application to secure communica-tionsrdquo Computers and Mathematics with Applications vol 68no 12 pp 2142ndash2147 2014

[18] H Du S Li and C Qian ldquoFinite-time attitude tracking controlof spacecraftwith application to attitude synchronizationrdquo IEEETransactions on Automatic Control vol 56 no 11 pp 2711ndash27172011

[19] S Ding S Li and Q Li ldquoDisturbance analysis for continuousfinite-time control systemsrdquo Journal of Control Theory andApplications vol 7 pp 271ndash276 2009

[20] S Li H Du and X Lin ldquoFinite-time consensus algorithm formulti-agent systems with double-integrator dynamicsrdquo Auto-matica vol 47 no 8 pp 1706ndash1712 2011

[21] E Moulay M Dambrine N Yeganefar and W PerruquettildquoFinite-time stability and stabilization of time-delay systemsrdquoSystems amp Control Letters vol 57 no 7 pp 561ndash566 2008

[22] J Mei M Jiang W Xu and B Wang ldquoFinite-time synchro-nization control of complex dynamical networks with timedelayrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 9 pp 2462ndash2478 2013

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] Q Wei X Wang and X Hu ldquoFinite-time hybrid projectivesynchronization of unified chaotic systemrdquo Transactions of theInstitute of Measurement and Control vol 36 no 8 pp 1069ndash1073 2014

[25] S P Bhat and D S Bernstein ldquoGeometric homogeneity withapplications to finite-time stabilityrdquo Mathematics of ControlSignals and Systems vol 17 no 2 pp 101ndash127 2005

[26] S Bovd L Ghaoui E E I Feron and V BalakrishnanLinear Matrix Inequalities in System and Control Theory SIAMPhiladephia Pa USA 1994

[27] C E de Souza and X Li ldquoDelay-dependent robust119867infincontrol

of uncertain linear state-delayed systemsrdquo Automatica vol 35no 7 pp 1313ndash1321 1999


[28] S Boyd L El Ghaoui E Feron and V Balakrishnan LinearMatrix Inequalities in SystemandControlTheory vol 15 of SIAMStudies in Applied Mathematics SIAM Philadelphia Pa USA1994

[29] L Zhou C Wang H He and Y Lin ldquoTime-controllable com-binatorial inner synchronization and outer synchronizationof anti-star networks and its application in secure communi-cationrdquo Communications in Nonlinear Science and NumericalSimulation vol 22 no 1ndash3 pp 623ndash640 2015

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

[31] S Khoo L Xie and Z Man ldquoRobust finite-time consensustracking algorithm for multirobot systemsrdquo IEEEASME Trans-actions on Mechatronics vol 14 no 2 pp 219ndash228 2009

[32] L Li H Peng Y Yang and X Wang ldquoOn the chaoticsynchronization of Lorenz systems with time-varying lagsrdquoChaos Solitons and Fractals vol 41 no 2 pp 783ndash794 2009

[33] U E Vincent and R Guo ldquoFinite-time synchronization for aclass of chaotic and hyperchaotic systems via adaptive feedbackcontrollerrdquo Physics Letters A vol 375 no 24 pp 2322ndash23262011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


also study the chaotic system with parameter uncertaintiesObviously on the one hand our results have optimality inthe convergence time of stabilization in [14 15] on the otherhand they are more practically applicable than those in[22ndash24] Based on the Lyapunov stability theory a feedbackcontroller is designed to guarantee the stabilization in finitetimeThe example is given to illustrate our theoretical results

The rest of the paper is organized as follows In Section 2some preliminaries are briefly given Section 3 presents themain results In Section 4 simulation results aiming at sub-stantiating the theoretical analysis are presented This paperis concluded in Section 5

2 Preliminaries and Problem Formulation

In this paper we want to develop a finite-time control law fora class of chaotic systems In system analysis and design it isimportant to select an appropriate model representing a realsystem The T-S fuzzy model can express a highly nonlinearfunctional relation with a small number of rules Manychaotic systems can be represented by fuzzy linear modelssuch as Lorenz Rossler Chua Chen and Lu systems Thisfuzzy modeling method is simple and natural So considera class of time-delayed chaotic systems represented by thefollowing time-delayed T-S fuzzy model


(119905) = 119860 119894119909 (119905) + 119861119894119909 (119905 minus 120591 (119905)) 119894 = 1 2 119873 (1)

where 119909(119905) = (1199091(119905) 119909119899(119905))119879isin 119877119899 is the state variable the

premise variables 1205791(119905) 120579119901(119905) are proper state variables119873 is the number of the fuzzy rules 119872

119894119895(119895 = 1 2 119901)

are fuzzy sets 119860119894and 119861

119894are known constant matrices with

appropriate dimensions 120591(119905) is the transmission delayIn practice however some parameters of chaotic systems

cannot be exactly known a priori If the information ofuncertain matrices is considered the results will be less con-servative than those results that do not utilize the elementaluncertain informationThus it is very interesting to study thedesign of the fuzzy systems with parameter perturbations



119894 = 1 2 119873(2)

The parameter uncertainties are classically written as [10]nabla119860119894= 119863119860

119894119865119860

119894119864119860

119894and nabla119861

119894= 119863119861

119894119865119861

119894119864119861

119894where 119863119860

119894 119863119861119894 119864119860119894 119864119861119894

are known real matrices of appropriate dimension and119865119860119894119865119861119894

are the unknownmatrix functionswith Lebesgue-measurableelements and satisfy the conditions 119865119860

119894

119879

119865119860

119894le 119868 119865119861

119894

119879

119865119861

119894le 119868

where 119868 is the identity matrix of appropriate dimension

The overall fuzzy system with control is inferred asfollows

(119905) =

119873

sum

119894=1ℎ119894 (120579 (119905)) [(119860 119894 +nabla119860 119894) 119909 (119905)

+ (119861119894+nabla119861119894) 119909 (119905 minus 120591 (119905)) + 119906 (119905)]

(3)

where 119906(119905) denotes the feedback control ℎ119894(120579(119905)) = 120596

119894(120579(119905))

sum119873

119894=1 120596119894(120579(119905)) 120596119894(120579(119905)) = prod119901

119895=1119872119894119895(120579119895(119905)) and 120596119894(120579(119905)) ge 0sum119873

119894=1 120596119894(120579(119905)) ge 0 thus ℎ119894(120579(119905)) ge 0 sum119873

119894=1 ℎ119894(120579(119905)) = 1Assume 119862([minus120591 0] 119877119899) is a Banach space of continuous

functions mapping the interval [minus120591 0] into 119877119899 with the

norm 120601 = supminus120591le120579le0120601(120579) For the functional differential

equation (3) its initial conditions are given by 119909119894(119905) =

120601119894(119905) isin 119862([minus120591 0] 119877119899)We always assume that (3) has a unique

solution with respect to initial conditionsFor starting simplification one has the following funda-

mental assumption

Assumption 1 Consider 0 le 120591(119905) le ℎ lt 1whereℎ is constant

To end this section we introduce the following lemmaswhich are useful in deriving sufficient conditions of finite-time stability

Lemma 2 (see [25]) Consider system = 119891(119909) 119891(0) = 0119909 isin 119877

119899 where 119891(sdot) 119877119899 rarr 119877119899 is a continuous vector function

Suppose there exist a 1198621 positive definite and proper function119881 119877119899rarr 119877 and real numbers 120583 gt 0 and 120578 isin (0 1) such that

+ 120583119881120578 is negative semidefinite Then the origin is a globally

finite-time stable equilibrium of system = 119891(119909) Moreoverthe settling time 119879 le 119881

1minus120578(0)(1 minus 120578)120583

Lemma 3 (see [26]) Given any real matrices 119860 119861 Σ ofappropriate dimensions and a scalar 119904 gt 0 such that 0 lt Σ =

Σ119879 then the following inequality holds

119860119879119861+119861119879119860 le 119904119860

119879Σ119860+ 119904

minus1119861119879Σminus1119861 (4)

Lemma 4 (see [27]) Let 119860 119863 119864 and 119865 be real matrices ofappropriate dimensions with 119865 satisfying 119865 le 1 Then onehas the following

(a) For any scalar 120582 gt 0

119863119865119864+119864119879119865119879119863119879le 120582minus1119863119863119879+120582119864119864

119879 (5)

(b) For any matrix 119875 gt 0 and scalar 120585 gt 0 such that 120585119868 minus119864119879119875119864 gt 0

(119860 +119863119865119864)119879119875 (119860+119863119865119864)

le 119860119879119875119860+119860

119879119875119864 (120585119868 minus119864

119879119875119864)minus1119864119879119875119860+ 120585

minus1119863119879119863

(6)

Lemma 5 ((Schur complement) [28]) For a given matrix

119878 = [11987811 11987812

119878119879

12 11987822] lt 0 (7)

where 11987811 = 119878119879

11 and 11987822 = 119878119879

22 are equivalent to any one of thefollowing conditions


(a) 11987822 lt 0 11987811 minus 11987812119878minus122 119878119879

12 lt 0(b) 11987811 lt 0 11987822 minus 11987811987912119878

minus111 11987812 lt 0





(

119860119879

119894119875 + 119875119860

119894+ (1205822 + 1198962 minus 1198961) 119875 119864

119860

119894119875119863119860

119894

119864119860

119894

119879

minus1205821119868 0

119863119860

119894

119879

119875 0 minus120582minus11 119868

)

le 0 119894 = 1 119873

(8)

(2)

(

119861119879

119894119875119861119894minus 11989621205822 (1 minus ℎ) 119875 119863

119861

119894119864119861

119894

119879

119875119861119894

119863119861

119894

119879

minus120582minus13 119868 0

119861119879

119894119875119864119861

1198940 minus (1205823119868 minus 119864

119861

119894

119879

119875119864119861

119894)

)

le 0 119894 = 1 119873

(9)


119906 = minus11989612119909 (119905) minus

120583

2120582(120578+1)2max (119875)

120582min (119875)sign (119909 (119905)) |119909 (119905)|120578

minus120583

2120582min (119875)(1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)

119909 (119905)2

(10)



119881 (119905) = 119909119879(119905) 119875119909 (119905) + 1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904 (11)

so we get

(119905)10038161003816100381610038161003816(3)

=

119873

sum

119894=1ℎ119894 (120579 (119905)) 119909

119879(119905)

sdot (119860119879

119894119875+119875119860

119894+119864119860

119894

119879

119865119860

119894

119879

119863119860

119894

119879

119875+119875119863119860

119894119865119860

119894119864119860

119894) 119909 (119905)

+ 119909119879(119905 minus 120591) (119861

119879

119894+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875119909 (119905) + 119909119879(119905)

sdot 119875 (119861119894+119863119860

119894119865119860

119894119864119860

119894) 119909 (119905 minus 120591) + 2119909119879 (119905) 119875119906 (119905)

+ 1198962119909119879(119905) 119875119909 (119905) minus 1198962119909


(12)


119864119860

119894

119879

119865119860

119894

119879

119863119860

119894

119879

119875+119875119863119860

119894119865119860

119894119864119860

119894

le 120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

(13)


119909119879(119905 minus 120591) (119861

119879

119894+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875119909 (119905) + 119909119879(119905) 119875 (119861119894

+119863119861

119894119865119861

119894119864119861

119894) 119909 (119905 minus 120591) le 120582

minus12 119909119879(119905 minus 120591) (119861

119879

119894

+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875 (119861119894+119863119861

119894119865119861

119894119864119861

119894) 119909 (119905 minus 120591)

+ 1205822119909119879(119905) 119875119909 (119905) le 120582

minus12 119909119879(119905 minus 120591) (119861

119879

119894119875119861119894

+119861119879

119894119875119864119861

119894(1205823119868 minus119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)119909 (119905 minus 120591) + 1205822119909

119879(119905) 119875119909 (119905)

(14)



(119905) le

119873

sum

119894=1ℎ119894 (120579 (119905)) 119909

119879(119905) (119860

119879

119894119875+119875119860

119894

+120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

+1205822119875+ 1198962119875)119909 (119905)

+ 2119909119879 (119905) 119875119906 (119905) + 119909119879 (119905 minus 120591) [120582minus12 (119861119879

119894119875119861119894

+119861119879

119894119875119864119861

119894(1205823119868 minus 119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861


(15)


119860119879

119894119875+119875119860

119894+120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

+1205822119875

+ 1198962119875 le 1198961119875

119861119879

119894119875119861119894+119861119879

119894119875119864119861

119894(1205823119868 minus119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894le 11989621205822 (1minus ℎ) 119875

(16)



(119905) le minus 120583120582(1+120578)2max (119875)

120582min (119875)119909119879(119905) 119875 sign (119909 (119905)) |119909 (119905)|120578

minus120583

120582min (119875)(1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)119879119875119909 (119905)

119909 (119905)2

le minus120583[120582(1+120578)2max (119875) (119909

119879(119905) 119909 (119905))

(1+120578)2

+(1198962 int119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2]

le minus120583 (119881 (119905))(1+120578)2

(17)


119879 = 1199050 +119881

1minus(1+120578)2(1199050)

120583 (1 minus (1 + 120578) 2) (18)








(1)

119860119879

119894119901+119901119860

119894+120582minus11 119901

2119863119860

119894119863119860

119894

119879

+1205821119864119860

119894119864119860

119894

119879

+ (1205822 + 1198962 minus 1198961) 119901119868 le 0 119894 = 1 119873(19)

(2)

120582minus12 (119901119861

119879

119894119861119894+119901

2119861119879

119894119864119861

119894(1205823119868 minus 119901119864

119861

119894

119879

119864119861

119894)

minus1119864119861

119894

119879

119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)minus 1198962 (1minus ℎ) 119901119868 le 0 119894 = 1 119873

(20)


119906 = minus11989612119909 (119905) minus

120583

2119901(120578minus1)2 sign (119909 (119905)) |119909 (119905)|120578 minus

120583

2

sdot 119901(120578minus1)2

(1198962 int119905

119905minus120591

119909119879(119904) 119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)

119909 (119905)2

(21)



4 Numerical Example



1 (119905) = minus 101199091 (119905) + 101199092 (119905 minus 120591)

2 (119905) = 281199091 (119905) minus 1199092 (119905) minus 1199091 (119905) 1199093 (119905)

3 (119905) = 1199091 (119905) 1199092 (119905) minus8

31199093 (119905 minus 120591)

(22)





+ 119906 (119905) 119894 = 1 2(23)


minus50

0

50

minus100minus50

050

100minus40

minus20

0

20

40

60

80

x2

x3

x1


where 119909(119905) = (1199091(119905) 1199092(119905) 1199093(119905))1198791198721 = (12)(1 + 1199091(119905)30)

1198722 = (12)(1 minus 1199091(119905)30)

1198601 = (


)

1198602 = (


)

1198611 = 1198612 = (

0 10 00 0 0

0 0 minus83

)

(24)




119894


119894and 119865119861


119865119860

119894

119879

119865119860

119894le 119868 119865119861

119894

119879

119865119861

119894le 119868

119863119860

119894= 119863119861

119894= diag (02 02 02)

119864119860

119894= (

minus10 0 028 minus1 00 0 0

)

119864119861

119894= 119861119894

119894 = 1 2

(25)



119894(119905) under


0 05 1 15 2

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


control

0 2 4 6 8 10

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


finite-time control


119905(1 + 119890






5 Conclusions




References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) 11987822 lt 0 11987811 minus 11987812119878minus122 119878119879

12 lt 0(b) 11987811 lt 0 11987822 minus 11987811987912119878

minus111 11987812 lt 0





(

119860119879

119894119875 + 119875119860

119894+ (1205822 + 1198962 minus 1198961) 119875 119864

119860

119894119875119863119860

119894

119864119860

119894

119879

minus1205821119868 0

119863119860

119894

119879

119875 0 minus120582minus11 119868

)

le 0 119894 = 1 119873

(8)

(2)

(

119861119879

119894119875119861119894minus 11989621205822 (1 minus ℎ) 119875 119863

119861

119894119864119861

119894

119879

119875119861119894

119863119861

119894

119879

minus120582minus13 119868 0

119861119879

119894119875119864119861

1198940 minus (1205823119868 minus 119864

119861

119894

119879

119875119864119861

119894)

)

le 0 119894 = 1 119873

(9)


119906 = minus11989612119909 (119905) minus

120583

2120582(120578+1)2max (119875)

120582min (119875)sign (119909 (119905)) |119909 (119905)|120578

minus120583

2120582min (119875)(1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)

119909 (119905)2

(10)



119881 (119905) = 119909119879(119905) 119875119909 (119905) + 1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904 (11)

so we get

(119905)10038161003816100381610038161003816(3)

=

119873

sum

119894=1ℎ119894 (120579 (119905)) 119909

119879(119905)

sdot (119860119879

119894119875+119875119860

119894+119864119860

119894

119879

119865119860

119894

119879

119863119860

119894

119879

119875+119875119863119860

119894119865119860

119894119864119860

119894) 119909 (119905)

+ 119909119879(119905 minus 120591) (119861

119879

119894+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875119909 (119905) + 119909119879(119905)

sdot 119875 (119861119894+119863119860

119894119865119860

119894119864119860

119894) 119909 (119905 minus 120591) + 2119909119879 (119905) 119875119906 (119905)

+ 1198962119909119879(119905) 119875119909 (119905) minus 1198962119909


(12)


119864119860

119894

119879

119865119860

119894

119879

119863119860

119894

119879

119875+119875119863119860

119894119865119860

119894119864119860

119894

le 120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

(13)


119909119879(119905 minus 120591) (119861

119879

119894+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875119909 (119905) + 119909119879(119905) 119875 (119861119894

+119863119861

119894119865119861

119894119864119861

119894) 119909 (119905 minus 120591) le 120582

minus12 119909119879(119905 minus 120591) (119861

119879

119894

+119864119861

119894

119879

119865119861

119894

119879

119863119861

119894

119879

)119875 (119861119894+119863119861

119894119865119861

119894119864119861

119894) 119909 (119905 minus 120591)

+ 1205822119909119879(119905) 119875119909 (119905) le 120582

minus12 119909119879(119905 minus 120591) (119861

119879

119894119875119861119894

+119861119879

119894119875119864119861

119894(1205823119868 minus119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)119909 (119905 minus 120591) + 1205822119909

119879(119905) 119875119909 (119905)

(14)



(119905) le

119873

sum

119894=1ℎ119894 (120579 (119905)) 119909

119879(119905) (119860

119879

119894119875+119875119860

119894

+120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

+1205822119875+ 1198962119875)119909 (119905)

+ 2119909119879 (119905) 119875119906 (119905) + 119909119879 (119905 minus 120591) [120582minus12 (119861119879

119894119875119861119894

+119861119879

119894119875119864119861

119894(1205823119868 minus 119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861


(15)


119860119879

119894119875+119875119860

119894+120582minus11 119875119863119860

119894119863119860

119894

119879

119875+1205821119864119860

119894119864119860

119894

119879

+1205822119875

+ 1198962119875 le 1198961119875

119861119879

119894119875119861119894+119861119879

119894119875119864119861

119894(1205823119868 minus119864

119861

119894

119879

119875119864119861

119894)

minus1119864119861

119894

119879

119875119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894le 11989621205822 (1minus ℎ) 119875

(16)



(119905) le minus 120583120582(1+120578)2max (119875)

120582min (119875)119909119879(119905) 119875 sign (119909 (119905)) |119909 (119905)|120578

minus120583

120582min (119875)(1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)119879119875119909 (119905)

119909 (119905)2

le minus120583[120582(1+120578)2max (119875) (119909

119879(119905) 119909 (119905))

(1+120578)2

+(1198962 int119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2]

le minus120583 (119881 (119905))(1+120578)2

(17)


119879 = 1199050 +119881

1minus(1+120578)2(1199050)

120583 (1 minus (1 + 120578) 2) (18)








(1)

119860119879

119894119901+119901119860

119894+120582minus11 119901

2119863119860

119894119863119860

119894

119879

+1205821119864119860

119894119864119860

119894

119879

+ (1205822 + 1198962 minus 1198961) 119901119868 le 0 119894 = 1 119873(19)

(2)

120582minus12 (119901119861

119879

119894119861119894+119901

2119861119879

119894119864119861

119894(1205823119868 minus 119901119864

119861

119894

119879

119864119861

119894)

minus1119864119861

119894

119879

119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)minus 1198962 (1minus ℎ) 119901119868 le 0 119894 = 1 119873

(20)


119906 = minus11989612119909 (119905) minus

120583

2119901(120578minus1)2 sign (119909 (119905)) |119909 (119905)|120578 minus

120583

2

sdot 119901(120578minus1)2

(1198962 int119905

119905minus120591

119909119879(119904) 119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)

119909 (119905)2

(21)



4 Numerical Example



1 (119905) = minus 101199091 (119905) + 101199092 (119905 minus 120591)

2 (119905) = 281199091 (119905) minus 1199092 (119905) minus 1199091 (119905) 1199093 (119905)

3 (119905) = 1199091 (119905) 1199092 (119905) minus8

31199093 (119905 minus 120591)

(22)





+ 119906 (119905) 119894 = 1 2(23)


minus50

0

50

minus100minus50

050

100minus40

minus20

0

20

40

60

80

x2

x3

x1


where 119909(119905) = (1199091(119905) 1199092(119905) 1199093(119905))1198791198721 = (12)(1 + 1199091(119905)30)

1198722 = (12)(1 minus 1199091(119905)30)

1198601 = (


)

1198602 = (


)

1198611 = 1198612 = (

0 10 00 0 0

0 0 minus83

)

(24)




119894


119894and 119865119861


119865119860

119894

119879

119865119860

119894le 119868 119865119861

119894

119879

119865119861

119894le 119868

119863119860

119894= 119863119861

119894= diag (02 02 02)

119864119860

119894= (

minus10 0 028 minus1 00 0 0

)

119864119861

119894= 119861119894

119894 = 1 2

(25)



119894(119905) under


0 05 1 15 2

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


control

0 2 4 6 8 10

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


finite-time control


119905(1 + 119890






5 Conclusions




References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905) le minus 120583120582(1+120578)2max (119875)

120582min (119875)119909119879(119905) 119875 sign (119909 (119905)) |119909 (119905)|120578

minus120583

120582min (119875)(1198962 int

119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)119879119875119909 (119905)

119909 (119905)2

le minus120583[120582(1+120578)2max (119875) (119909

119879(119905) 119909 (119905))

(1+120578)2

+(1198962 int119905

119905minus120591

119909119879(119904) 119875119909 (119904) d119904)

(1+120578)2]

le minus120583 (119881 (119905))(1+120578)2

(17)


119879 = 1199050 +119881

1minus(1+120578)2(1199050)

120583 (1 minus (1 + 120578) 2) (18)








(1)

119860119879

119894119901+119901119860

119894+120582minus11 119901

2119863119860

119894119863119860

119894

119879

+1205821119864119860

119894119864119860

119894

119879

+ (1205822 + 1198962 minus 1198961) 119901119868 le 0 119894 = 1 119873(19)

(2)

120582minus12 (119901119861

119879

119894119861119894+119901

2119861119879

119894119864119861

119894(1205823119868 minus 119901119864

119861

119894

119879

119864119861

119894)

minus1119864119861

119894

119879

119861119894

+120582minus13 119863119861

119894

119879

119863119861

119894)minus 1198962 (1minus ℎ) 119901119868 le 0 119894 = 1 119873

(20)


119906 = minus11989612119909 (119905) minus

120583

2119901(120578minus1)2 sign (119909 (119905)) |119909 (119905)|120578 minus

120583

2

sdot 119901(120578minus1)2

(1198962 int119905

119905minus120591

119909119879(119904) 119909 (119904) d119904)

(1+120578)2

sdot119909 (119905)

119909 (119905)2

(21)



4 Numerical Example



1 (119905) = minus 101199091 (119905) + 101199092 (119905 minus 120591)

2 (119905) = 281199091 (119905) minus 1199092 (119905) minus 1199091 (119905) 1199093 (119905)

3 (119905) = 1199091 (119905) 1199092 (119905) minus8

31199093 (119905 minus 120591)

(22)





+ 119906 (119905) 119894 = 1 2(23)


minus50

0

50

minus100minus50

050

100minus40

minus20

0

20

40

60

80

x2

x3

x1


where 119909(119905) = (1199091(119905) 1199092(119905) 1199093(119905))1198791198721 = (12)(1 + 1199091(119905)30)

1198722 = (12)(1 minus 1199091(119905)30)

1198601 = (


)

1198602 = (


)

1198611 = 1198612 = (

0 10 00 0 0

0 0 minus83

)

(24)




119894


119894and 119865119861


119865119860

119894

119879

119865119860

119894le 119868 119865119861

119894

119879

119865119861

119894le 119868

119863119860

119894= 119863119861

119894= diag (02 02 02)

119864119860

119894= (

minus10 0 028 minus1 00 0 0

)

119864119861

119894= 119861119894

119894 = 1 2

(25)



119894(119905) under


0 05 1 15 2

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


control

0 2 4 6 8 10

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


finite-time control


119905(1 + 119890






5 Conclusions




References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus50

0

50

minus100minus50

050

100minus40

minus20

0

20

40

60

80

x2

x3

x1


where 119909(119905) = (1199091(119905) 1199092(119905) 1199093(119905))1198791198721 = (12)(1 + 1199091(119905)30)

1198722 = (12)(1 minus 1199091(119905)30)

1198601 = (


)

1198602 = (


)

1198611 = 1198612 = (

0 10 00 0 0

0 0 minus83

)

(24)




119894


119894and 119865119861


119865119860

119894

119879

119865119860

119894le 119868 119865119861

119894

119879

119865119861

119894le 119868

119863119860

119894= 119863119861

119894= diag (02 02 02)

119864119860

119894= (

minus10 0 028 minus1 00 0 0

)

119864119861

119894= 119861119894

119894 = 1 2

(25)



119894(119905) under


0 05 1 15 2

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


control

0 2 4 6 8 10

0

1

2

3

4

t

x(t)

minus4

minus3

minus2

minus1

x1(t)

x2(t)

x3(t)


finite-time control


119905(1 + 119890






5 Conclusions




References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusions




References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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