research article passivity-sliding mode control of...

Research ArticlePassivity-Sliding Mode Control of Uncertain Chaotic Systemswith Stochastic Disturbances

Zhumu Fu12 Leipo Liu1 and Xiaohong Wang1

1College of Information Engineering Henan University of Science and Technology Luoyang 471023 China2School of Control Science and Engineering Shandong University Jinan 250061 China

Correspondence should be addressed to Zhumu Fu fzm1974163com

Received 22 September 2014 Accepted 9 December 2014 Published 23 December 2014

Academic Editor Wuneng Zhou

Copyright copy 2014 Zhumu Fu 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 is concerned with the stabilization problem of uncertain chaotic systems with stochastic disturbances A novel slidingfunction is designed and then a slidingmode controller is established such that the trajectory of the system converges to the slidingsurface in a finite time Using a virtual state feedback control technique sufficient condition for themean square asymptotic stabilityand passivity of sliding mode dynamics is derived via linear matrix inequality (LMI) Finally a simulation example is presented toshow the validity and advantage of the proposed method

1 Introduction

Chaos control has received a great deal of interest in thelast three decades Many techniques for chaos control areused such as adaptive control [1 2] backstepping control[3] fuzzy control [4] and sliding mode control [5ndash7] In theabove methods the chaotic system model has a determin-istic differential equation there is no random parameter orrandom excitation on the system governing equation Butthe chaotic systemmay be affected by stochastic disturbancesdue to environmental noise [8 9] Stochastic chaotic systemsappear inmany fields such as chemistry [10] physics and laserscience [11] and economics [12] So it is necessary to studysuch systems

There have been some meaningful results about chaossystem with stochastic disturbances [13ndash15] In [13] theergodic theory and stochastic noise are considered Time-delay feedback control of the Van der Pol oscillator underthe influence of white noise is investigated in [14] In [15] anadaptive control of of chaotic systems with white Gaussiannoise is considered On the other hand sliding mode controlis a very effective approach for the robust control systemsIt has many attractive features such as fast response good

transient response and insensitivity to variations Somerelated results have been presented [16 17] In [16] controlof stochastic chaos via slidingmode control is investigated In[17] chaos synchronization of nonlinear gyroswith stochasticexcitation is considered by using sliding mode control Sincethen from a practical point of view many systems need tobe passive in order to attenuate noises So it is necessaryto investigate the passivity of chaotic systems via slidingmode control However there have been few results in thisrespect

Motivated by the above reasons passivity-sliding modecontrol problem of uncertain chaotic systems with stochas-tic disturbances is considered Stochastic disturbance is astandard Wienner process the derivative of which gener-ates a continuous time Gaussian white noise The maincontributions of this paper lie in the following aspects(1) A new sliding function is designed compared withthe existing results [18 19] (2) By incorporating the vir-tual state feedback control into the performance anal-ysis of the sliding mode dynamics a sufficient condi-tion for the mean square asymptotic stability and pas-sivity of the sliding mode dynamics is easily derived viaLMI

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2014 Article ID 390151 5 pageshttpdxdoiorg1011552014390151

2 Journal of Control Science and Engineering

2 Problem Formulation

Consider the following chaotic system with stochastic distur-bances

119889119909 (119905) = [(119860 + Δ119860) 119909 (119905) + 119861 (119906 (119905) + 119891 (119909) + Δ119891 (119909))

+ 119867V (119905)] 119889119905 + 119863119909 (119905) 119889119908 (119905)

119910 (119905) = 119862119909 (119905)

(1)

where 119909(119905) isin 119877119899 is the state vector 119910(119905) isin 119877119897 is the systemoutput and 119906(119905) isin 119877119898times119899 is the control input 119891(119909) is anonlinear real-valued function vector Δ119891(119909) represents theuncertainty 119860 119861 119862 119863 and 119867 are matrices with compatibledimensions and (119860 119861) is controllable V(119905) is an externalbounded disturbance 119908(119905) is a one-dimensional Brownianmotion Δ119860(119905) is an uncertainty which is assumed to be inthe form of

Δ119860 (119905) = 1198631119865 (119905) 119864

1 (2)

where1198631and 119864

1are real matrices with appropriate sizes and

119865(119905) is an unknown time-varying matrix function satisfying

119865119879

(119905) 119865 (119905) le 119868 (3)

The following assumption is necessary for further study

Assumption 1 The uncertainty Δ119891(119909) satisfies

1003817100381710038171003817Δ119891 (119909)1003817100381710038171003817 le 120588 119909 (119905) (4)

where 120588 is a known positive constant

3 Main Results

For system (1) firstly a sliding function is designed and asliding mode controller is designed to drive the state onto thesliding surface in a finite time Secondly a sufficient conditionis given which guarantees system (1) on the sliding surface119904(119905) = 0 is mean square asymptotically stable and robustlypassive

31 Sliding Mode Controller Design For system (1) a slidingfunction is designed as

119904 (119905) = 119861119879

119875 [119909 (119905) minus 120578119890minus120582119905

119909 (0)] (5)

where 119875 gt 0 is a matrix to be chosen later and 120582 and 120578 arepositive constants

Usually the system (1) on the sliding surface 119904(119905) = 0 iscalled the sliding mode dynamics of the system (1)

Remark 2 The sliding function satisfies 119904(0) = 0 so thereaching interval is eliminated Furthermore if the parameter120578 = 0 then 119904(119905) = 119861

119879119875119909(119905) which is widely used such as in[18 19]

Definition 3 The sliding mode dynamics of system (1) is saidto be robustly passive if there exists a scalar 120574 gt 0 such that

E2int119905lowast

0

V119879 (119904) 119910 (119904) 119889119904 ge minus120574int119905lowast

0

V119879 (119904) V (119904) 119889119904

for any V (119905) isin 1198712[0infin)

(6)

for all 119905lowast ge 0 under zero initial conditions and for all admis-sible uncertainties

To achieve the control objective the input 119906(119905) is designedas follows

119906 (119905) =

minus(119861119879119875119861)minus1

times[119861119879119875119860119909 (119905) + 119891 (119909)

+ 120582120578119890minus120582119905

119861119879

119875119909 (0)

+ (120573 +10038171003817100381710038171003817119861119879119875119863

1

10038171003817100381710038171003817sdot10038171003817100381710038171198641119909 (119905)

1003817100381710038171003817

+ 120588 119909 (119905) sdot10038171003817100381710038171003817119861119879119875119861

10038171003817100381710038171003817

+ V (119905) sdot10038171003817100381710038171003817119861119879119875119867

10038171003817100381710038171003817) sgn 119904 (119905)

+

10038171003817100381710038171003817119861119879119875119863119909 (119905)

10038171003817100381710038171003817119904 (119905)

119904(119905)2

] E 119904 (119905) = 0

0 E 119904 (119905)= 0

(7)

Theorem 4 If the sliding mode controller 119906(119905) is taken as (7)then the trajectory of the system (1) converges to the slidingsurface 119904(119905) = 0 in a finite time

Proof Consider a Lyapunov function candidate as follows

119881 (119905) = 119904119879

(119905) 119904 (119905) (8)

In fact we have

119904 (119905) = 119861119879

119875 minus 120578119890minus120582119905

119861119879

119875119909 (0)

= 119861119879

119875 (119860 + Δ119860) 119909 (119905) + 119861119879

119875119861 (119906 (119905) + 119891 (119909) + Δ119891 (119909))

+ 119861119879

119875119867V (119905) + 119861119879

119875119863119909 (119905) (119905) + 120582120578119890minus120582119905

119861119879

119875119909 (0)

(9)

If E119904(119905) = 0 calculating the time derivative of 119881(119905) alongthe trajectory of system (1) then we have

= 2119904119879

(119905) 119904 (119905) = 2119904119879

(119905)

times (119861119879

119875 (119860 + Δ119860) 119909 (119905) + 119861119879

119875119861 (119906 (119905) + 119891 (119909) + Δ119891 (119909))

+119861119879

119875119867V (119905) + 119861119879

119875119863119909 (119905) (119905) + 120582120578119890minus120582119905

119861119879

119875119909 (0))

(10)

Substitute (7) into (10) by which the weak infinitesimaloperatorL119881(119905) can be given by

L119881 (119905) le minus2120573 119904 (119905) (11)

Journal of Control Science and Engineering 3

That is

L119881 (119905) le minus2120573radic119881 (119905) (12)

By Itorsquos formula this yields

L 119904 (119905) = Lradic119881 (119905) le minus120573 (13)

and hence

E 119904 (119905) le E 119904 (0) minus 120573119905 (14)

which impliesE119904(119905) converges to zero in a finite timeTheproof is completed

32 Stability Analysis of the Sliding Mode Dynamics Thefollowing theorem gives a sufficient condition to guaranteethe sliding mode dynamics is mean square asymptoticallystable and robustly passive

Theorem 5 Consider system (1) and a given constant 120590 gt 0 Ifthere exist matrices 119883 gt 0 and 119884 and a scalar 120603 gt 0 such thatthe following linear matrix inequality (LMI) is true

[[[

[

Δ 119867 + 119883119862119879 119883119863119879 1198831198641198791

⋆ minus120574119868 0 0

⋆ ⋆ minus119883 0

⋆ ⋆ ⋆ minus120603119868

]]]

]

lt 0 (15)

where Δ = 119860119883 + 119861119884 + 119883119879119860119879 + 119884119879119861119879 + 12060311986311198631198791

+ 120590119883then the sliding mode dynamics of system (1) is mean squareasymptotically stable and robustly passive Furthermore 119875 =

119883minus1 and 119870 = 119884119883minus1

Proof Consider the following Lyapunov functional candi-date

119881 (119905) = 119909119879

(119905) 119875119909 (119905) (16)

The time derivative of119881(119905) along the trajectories of system (1)is given by

= 2119909119879

119875 [(119860 + Δ119860) 119909 (119905) + 119861 (119906 (119905) + 119891 (119909)) + 119867V (119905)]

+ 2119909119879

119875119863119909 (119905) (119905) + 119909119879

(119905) 119863119879

119875119863119909 (119905)

(17)

where119870 is an arbitrary matrixThe weak infinitesimal operatorL119881(119905) can be given by

L119881 (119905) = 119909119879

(119905) [119875 (119860 + 119861119870) + (119860 + 119861119870)119879

119875] 119909 (119905)

+ 2119909119879

(119905) 119875Δ119860119909 (119905) + 2119909119879

(119905) 119875119867V (119905)

+ 2119909119879

(119905) 119875119861 [119906 (119905) + 119891 (119909 119905) minus 119870119909 (119905)]

+ 119909119879

(119905) 119863119879

119875119863119909 (119905)

(18)

Note that

2119909119879

(119905) 119875Δ119860119909 (119905) le 120603119909119879

(119905) 1198751198631119863119879

1119875119909 (119905)

+1

120603119909119879

(119905) 119864119879

11198641119909 (119905)

(19)

On the sliding surface 119904(119905) = 0 from (5) we have

2119909119879

(119905) 119875119861 = 2120578119890minus120582119905

119909119879

(0) 119875119861 (20)

Under the zero-initial conditions we have

2119909119879

(119905) 119875119861 = 2120578119890minus120582119905

119909119879

(0) 119875119861 = 0 (21)

From (18) and (19) we have

L119881 (119905) minus 2V119879 (119905) 119910 (119905) minus 120574V119879 (119905) V (119905)

le [119909 (119905)

V (119905)]119879

[Δ 119875119867 + 119862119879

lowast minus120574119868] [

119909 (119905)

V (119905)]

(22)

whereΔ = 119875(119860+119861119870)+(119860+119861119870)119879

119875+12060311987511986311198631198791119875+(1120603)119864119879

11198641+

119863119879119875119863Before and aftermultiplyingmatrix (15) by diag119875 119868 119875 119868

and using 119875 = 119883minus1 and119884 = 119870119883 by Shur complement we get

[Δ + 120590119875 119875119867 + 119862119879

lowast minus120574119868] lt 0 (23)


L119881 (119905) minus 2V119879 (119905) 119910 (119905) minus 120574V119879 (119905) V (119905) lt minus120590119881 (119905) lt 0 (24)

Integrating (24) from 0 to 119905 and noting that 119909(0) = 0 weobtain

0 le E 119881 (119905) lt Eint119905

0

[minus120574V119879 (119905) V (119905) minus 2119910119879

(119905) 119910 (119905)] 119889119905

(25)

We easily get

E2int119905

0

V119879 (119905) 119910 (119905) 119889119905 lt 120574Eint119905

0

V119879 (119905) V (119905) 119889119905 (26)

Thus system (1) on the sliding surface 119904(119905) = 0 is robustlypassive

In fact when V(119905) = 0 on the sliding surface 119904(119905) = 0 wecan have

L119881 (119905) = 119909119879

(119905) [119875 (119860 + 119861119870) + (119860 + 119861119870)119879

119875] 119909 (119905)

+ 2119909119879

(119905) 119875Δ119860119909 (119905) + 2120578119890minus120582119905

119909119879

(0) 119875119861

times [119906 (119905) + 119891 (119909 119905) minus 119870119909 (119905)] + 119909119879

(119905) 119863119879

119875119863119909 (119905)

(27)

Since E119909(119905) is bounded then E119906(119905) is bounded from (7)So the termE2119909119879(0)119875119861[119906(119905)minus119870119909+119891(119909)] is bounded whereEsdot is an expect operator Assume that its boundedness is 120575

Noting that (15) implies Δ lt minus120590119875 from (27) we have

E L119881 (119905) le minus120590E 119881 (119905) + 120575119890minus120582119905

(28)

Solving (28) we obtain

E 119881 (119905) lt 119890minus120590119905

E 119881 (0) + 120575119890minus120590119905

int119905

0

119890(120590minus120582)120591

119889120591

=

119890minus120590119905E 119881 (0) + 120575119890minus120590119905119905 120590 = 120582

119890minus120590119905E 119881 (0) + 120575119890minus120582119905 minus 119890minus120590119905

120590 minus 120582 120590 = 120582

(29)


We easily get

lim119905rarrinfin

E 119881 (119905) = 0 (30)

From the above we get

lim119905rarrinfin

E 119909 (119905) = 0 (31)

This implies that system (1) with V(119905) = 0 is mean squareasymptotically stable on the sliding surface 119904(119905) = 0Theproofis completed

Remark 6 From (28) the boundedness of the chaotic systemstate is used because the state of chaotic system withstochastic disturbances is indeterminate the expectation isused to define the boundedness of the state

4 Example

In this section we use Genesiorsquos chaotic system to show theeffectiveness of the method Genesiorsquos system with stochasticdisturbance is as follows

1(119905) = 119909

2(119905) + 01119909

1(119905) (119905)

2(119905) = 119909

3(119905) + 01119909

3(119905) (119905)

3(119905) = minus 6119909

1(119905) minus 292119909

2(119905) minus 12119909

3(119905) + 119909

2

1(119905)

+ 01 sin (1199091(119905)) + 0001 sin (119905) + 119906 (119905)

119910 (119905) = 1199093(119905)

(32)

where

119860 = [

[

0 1 0

0 0 1

minus6 minus292 minus12

]

]

119861 = [

[

0

0

1

]

]

119867 = [

[

0

0

0001

]

]

119862 = [0 0 1] 119863 = [

[

01 0 0

0 0 01

0 0 0

]

]

1198631= [

[

01 0 0

0 0 0

0 0 0

]

]

1198641= 01 119865 (119905) = sin 119905 119891 (119909) = 119909

2

1(119905)

Δ119891 (119909) = 01 cos (1199091(119905)) V (119905) = sin (119905)

(33)

Let 120590 = 05 and 120574 = 02 solving LMI (15) yields

119883 = [

[

10290 minus03534 00052

minus03534 02286 minus01346

00052 minus01346 04802

]

]

(34)

Then we have

119875 = 119883minus1

= [

[

26316 48524 13317

48524 141869 39244

13317 39244 31684

]

]

(35)

The initial value is 119909(0) = [minus1 0 1]119879

0 2 4 6 8 10 12 14 16 18 20minus15

minus1

minus05

0

05

1

t (s)

x1

x2

x3

Figure 1 State 119909(119905) of the closed-loop system

0 5 10 15 20minus7minus6minus5minus4minus3minus2minus10123

t (s)

Figure 2 The control input 119906(119905)

From Assumption 1 we have 120588 = 01 The parameters ofcontroller (7) are 120578 = 1 and120573 = 00001The simulation resultsare as shown in Figures 1 2 3 and 4

Figures 1 2 and 3 show the time responses of thestate control input and sliding function respectively It isconcluded that the proposed method is effective In view ofFigure 3 and Figure 4 it is obvious that the sliding functiondesigned in this paper eliminates the reaching interval andreduces the chattering

5 Conclusion

In this paper the stabilization problem of uncertain chaoticsystems with stochastic disturbances is investigated A newsliding function is proposed which not only makes thereaching interval eliminated but also reduces the difficultyof systems analysis and design A sliding mode controller isdesigned to make the state of system reach the sliding surfacein a finite time Finally the simulation shows the effectivenessof the proposed method

Conflict of Interests

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


0 5 10 15 20

000501015020250303504

minus005

t (s)

Figure 3 The sliding function 119904(119905)

0 5 10 15 20minus01

001020304050607

s

t (s)

Figure 4 The sliding function 119904(119905) in [18 19]

Acknowledgments

The authors would like to thank the anonymous reviewersfor their constructive and insightful comments for furtherimproving the quality of this paper This work was par-tially supported by National Natural Science Foundation ofChina under Grant nos 61473115 51277116 and 51375145China Postdoctoral Science Foundation under Grant no2013T60670 Science and Technology Innovative Foundationfor Distinguished Young Scholar of Henan Province underGrant no 144100510004 the Science and Technology Pro-gramme Foundation for the Innovative Talents of HenanProvince University Grant no 13HASTIT038 and the KeyScientific andTechnological Project ofHenan ProvinceGrantno 132102210247

References

[1] C Hua and X Guan ldquoAdaptive control for chaotic systemsrdquoChaos Solitons amp Fractals vol 22 no 1 pp 55ndash60 2004

[2] S Kuntanapreeda and T Sangpet ldquoSynchronization of chaoticsystems with unknown parameters using adaptive passivity-based controlrdquo Journal of the Franklin Institute Engineering andApplied Mathematics vol 349 no 8 pp 2547ndash2569 2012

[3] G-H Li ldquoProjective synchronization of chaotic system usingbackstepping controlrdquo Chaos Solitons and Fractals vol 29 no2 pp 490ndash494 2006

[4] JWang GQiao and BDeng ldquoObserver-based robust adaptivevariable universe fuzzy control for chaotic systemrdquo ChaosSolitons and Fractals vol 23 no 3 pp 1013ndash1032 2005

[5] H Wang X-L Zhang X-H Wang and X-J Zhu ldquoFinitetime chaos control for a class of chaotic systems with inputnonlinearities via TSM schemerdquo Nonlinear Dynamics vol 69no 4 pp 1941ndash1947 2012

[6] J Li W Li and Q Li ldquoSliding mode control for uncertainchaotic systems with input nonlinearityrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 1 pp341ndash348 2012

[7] M P Aghababa S Khanmohammadi andG Alizadeh ldquoFinite-time synchronization of two different chaotic systems withunknown parameters via sliding mode techniquerdquo AppliedMathematical Modelling vol 35 no 6 pp 3080ndash3091 2011

[8] W Zhou D Tong Y Gao C Ji and H Su ldquoMode anddelay-dependent adaptive exponential synchronization in pthmoment for stochastic delayed neural networkswithmarkovianswitchingrdquo IEEE Transactions on Neural Networks and LearningSystems vol 23 no 4 pp 662ndash668 2012

[9] W Zhou Q Zhu P Shi H Su J Fang and L Zhou ldquoAdaptivesynchronization for neutraltype neural networks with stochas-tic perturbation and Markovian switching parametersrdquo IEEETransactions onCybernetics vol 44 no 12 pp 2848ndash2860 2014

[10] M Pascual and P Mazzega ldquoQuasicycles revisited apparentsensitivity to initial conditionsrdquoTheoretical Population Biologyvol 64 no 3 pp 385ndash395 2003

[11] P R Patnaik ldquoThe extended Kalman filter as a noise modulatorfor continuous yeast cultures under monotonic oscillating andchaotic conditionsrdquo Chemical Engineering Journal vol 108 no1-2 pp 91ndash99 2005

[12] N Kikuchi Y Liu and J Ohtsubo ldquoChaos control and noisesuppression in external-cavity semiconductor lasersrdquo IEEEJournal of Quantum Electronics vol 33 no 1 pp 56ndash65 1997

[13] C Kyrtsou and M Terraza ldquoStochastic chaos or ARCH effectsin stock series A comparative studyrdquo International Review ofFinancial Analysis vol 11 no 4 pp 407ndash431 2002

[14] E Scholl J Pomplun A Amann A G Balanov and N BJanson ldquoTime-delay feedback control of nonlinear stochasticoscillationsrdquo in Proceedings of the 5th EUROMECH NonlinearDynamics Conference (ENOC rsquo05) Eindhoven The Nether-lands August 2005

[15] H Salarieh and A Alasty ldquoAdaptive control of chaotic systemswith stochastic time varying unknown parametersrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 168ndash177 2008

[16] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[17] H Salarieh and A Alasty ldquoChaos synchronization of nonlineargyros in presence of stochastic excitation via sliding modecontrolrdquo Journal of Sound and Vibration vol 313 no 3ndash5 pp760ndash771 2008

[18] S Qu and Y Wang ldquoRobust control of uncertain time delaysystem a novel sliding mode control design via LMIrdquo Journal ofSystems Engineering and Electronics vol 17 no 3 pp 624ndash6282006

[19] T-Z Wu and Y-T Juang ldquoDesign of variable structure controlfor fuzzy nonlinear systemsrdquo Expert Systems with Applicationsvol 35 no 3 pp 1496ndash1503 2008

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2 Problem Formulation

Consider the following chaotic system with stochastic distur-bances

119889119909 (119905) = [(119860 + Δ119860) 119909 (119905) + 119861 (119906 (119905) + 119891 (119909) + Δ119891 (119909))

+ 119867V (119905)] 119889119905 + 119863119909 (119905) 119889119908 (119905)

119910 (119905) = 119862119909 (119905)

(1)

where 119909(119905) isin 119877119899 is the state vector 119910(119905) isin 119877119897 is the systemoutput and 119906(119905) isin 119877119898times119899 is the control input 119891(119909) is anonlinear real-valued function vector Δ119891(119909) represents theuncertainty 119860 119861 119862 119863 and 119867 are matrices with compatibledimensions and (119860 119861) is controllable V(119905) is an externalbounded disturbance 119908(119905) is a one-dimensional Brownianmotion Δ119860(119905) is an uncertainty which is assumed to be inthe form of

Δ119860 (119905) = 1198631119865 (119905) 119864

1 (2)

where1198631and 119864

1are real matrices with appropriate sizes and

119865(119905) is an unknown time-varying matrix function satisfying

119865119879

(119905) 119865 (119905) le 119868 (3)

The following assumption is necessary for further study

Assumption 1 The uncertainty Δ119891(119909) satisfies

1003817100381710038171003817Δ119891 (119909)1003817100381710038171003817 le 120588 119909 (119905) (4)

where 120588 is a known positive constant

3 Main Results

For system (1) firstly a sliding function is designed and asliding mode controller is designed to drive the state onto thesliding surface in a finite time Secondly a sufficient conditionis given which guarantees system (1) on the sliding surface119904(119905) = 0 is mean square asymptotically stable and robustlypassive

31 Sliding Mode Controller Design For system (1) a slidingfunction is designed as

119904 (119905) = 119861119879

119875 [119909 (119905) minus 120578119890minus120582119905

119909 (0)] (5)

where 119875 gt 0 is a matrix to be chosen later and 120582 and 120578 arepositive constants

Usually the system (1) on the sliding surface 119904(119905) = 0 iscalled the sliding mode dynamics of the system (1)

Remark 2 The sliding function satisfies 119904(0) = 0 so thereaching interval is eliminated Furthermore if the parameter120578 = 0 then 119904(119905) = 119861

119879119875119909(119905) which is widely used such as in[18 19]

Definition 3 The sliding mode dynamics of system (1) is saidto be robustly passive if there exists a scalar 120574 gt 0 such that

E2int119905lowast

0

V119879 (119904) 119910 (119904) 119889119904 ge minus120574int119905lowast

0

V119879 (119904) V (119904) 119889119904

for any V (119905) isin 1198712[0infin)

(6)

for all 119905lowast ge 0 under zero initial conditions and for all admis-sible uncertainties

To achieve the control objective the input 119906(119905) is designedas follows

119906 (119905) =

minus(119861119879119875119861)minus1

times[119861119879119875119860119909 (119905) + 119891 (119909)

+ 120582120578119890minus120582119905

119861119879

119875119909 (0)

+ (120573 +10038171003817100381710038171003817119861119879119875119863

1

10038171003817100381710038171003817sdot10038171003817100381710038171198641119909 (119905)

1003817100381710038171003817

+ 120588 119909 (119905) sdot10038171003817100381710038171003817119861119879119875119861

10038171003817100381710038171003817

+ V (119905) sdot10038171003817100381710038171003817119861119879119875119867

10038171003817100381710038171003817) sgn 119904 (119905)

+

10038171003817100381710038171003817119861119879119875119863119909 (119905)

10038171003817100381710038171003817119904 (119905)

119904(119905)2

] E 119904 (119905) = 0

0 E 119904 (119905)= 0

(7)

Theorem 4 If the sliding mode controller 119906(119905) is taken as (7)then the trajectory of the system (1) converges to the slidingsurface 119904(119905) = 0 in a finite time

Proof Consider a Lyapunov function candidate as follows

119881 (119905) = 119904119879

(119905) 119904 (119905) (8)

In fact we have

119904 (119905) = 119861119879

119875 minus 120578119890minus120582119905

119861119879

119875119909 (0)

= 119861119879

119875 (119860 + Δ119860) 119909 (119905) + 119861119879

119875119861 (119906 (119905) + 119891 (119909) + Δ119891 (119909))

+ 119861119879

119875119867V (119905) + 119861119879

119875119863119909 (119905) (119905) + 120582120578119890minus120582119905

119861119879

119875119909 (0)

(9)

If E119904(119905) = 0 calculating the time derivative of 119881(119905) alongthe trajectory of system (1) then we have

= 2119904119879

(119905) 119904 (119905) = 2119904119879

(119905)

times (119861119879

119875 (119860 + Δ119860) 119909 (119905) + 119861119879

119875119861 (119906 (119905) + 119891 (119909) + Δ119891 (119909))

+119861119879

119875119867V (119905) + 119861119879

119875119863119909 (119905) (119905) + 120582120578119890minus120582119905

119861119879

119875119909 (0))

(10)

Substitute (7) into (10) by which the weak infinitesimaloperatorL119881(119905) can be given by

L119881 (119905) le minus2120573 119904 (119905) (11)


That is

L119881 (119905) le minus2120573radic119881 (119905) (12)


L 119904 (119905) = Lradic119881 (119905) le minus120573 (13)

and hence

E 119904 (119905) le E 119904 (0) minus 120573119905 (14)




[[[

[

Δ 119867 + 119883119862119879 119883119863119879 1198831198641198791

⋆ minus120574119868 0 0

⋆ ⋆ minus119883 0

⋆ ⋆ ⋆ minus120603119868

]]]

]

lt 0 (15)

where Δ = 119860119883 + 119861119884 + 119883119879119860119879 + 119884119879119861119879 + 12060311986311198631198791


119883minus1 and 119870 = 119884119883minus1


119881 (119905) = 119909119879

(119905) 119875119909 (119905) (16)


= 2119909119879

119875 [(119860 + Δ119860) 119909 (119905) + 119861 (119906 (119905) + 119891 (119909)) + 119867V (119905)]

+ 2119909119879

119875119863119909 (119905) (119905) + 119909119879

(119905) 119863119879

119875119863119909 (119905)

(17)


L119881 (119905) = 119909119879

(119905) [119875 (119860 + 119861119870) + (119860 + 119861119870)119879

119875] 119909 (119905)

+ 2119909119879

(119905) 119875Δ119860119909 (119905) + 2119909119879

(119905) 119875119867V (119905)

+ 2119909119879

(119905) 119875119861 [119906 (119905) + 119891 (119909 119905) minus 119870119909 (119905)]

+ 119909119879

(119905) 119863119879

119875119863119909 (119905)

(18)

Note that

2119909119879

(119905) 119875Δ119860119909 (119905) le 120603119909119879

(119905) 1198751198631119863119879

1119875119909 (119905)

+1

120603119909119879

(119905) 119864119879

11198641119909 (119905)

(19)


2119909119879

(119905) 119875119861 = 2120578119890minus120582119905

119909119879

(0) 119875119861 (20)


2119909119879

(119905) 119875119861 = 2120578119890minus120582119905

119909119879

(0) 119875119861 = 0 (21)


L119881 (119905) minus 2V119879 (119905) 119910 (119905) minus 120574V119879 (119905) V (119905)

le [119909 (119905)

V (119905)]119879

[Δ 119875119867 + 119862119879


119909 (119905)

V (119905)]

(22)

whereΔ = 119875(119860+119861119870)+(119860+119861119870)119879

119875+12060311987511986311198631198791119875+(1120603)119864119879

11198641+



[Δ + 120590119875 119875119867 + 119862119879

lowast minus120574119868] lt 0 (23)




0 le E 119881 (119905) lt Eint119905

0

[minus120574V119879 (119905) V (119905) minus 2119910119879

(119905) 119910 (119905)] 119889119905

(25)

We easily get

E2int119905

0

V119879 (119905) 119910 (119905) 119889119905 lt 120574Eint119905

0

V119879 (119905) V (119905) 119889119905 (26)



L119881 (119905) = 119909119879

(119905) [119875 (119860 + 119861119870) + (119860 + 119861119870)119879

119875] 119909 (119905)

+ 2119909119879

(119905) 119875Δ119860119909 (119905) + 2120578119890minus120582119905

119909119879

(0) 119875119861

times [119906 (119905) + 119891 (119909 119905) minus 119870119909 (119905)] + 119909119879

(119905) 119863119879

119875119863119909 (119905)

(27)




(28)


E 119881 (119905) lt 119890minus120590119905

E 119881 (0) + 120575119890minus120590119905

int119905

0

119890(120590minus120582)120591

119889120591

=

119890minus120590119905E 119881 (0) + 120575119890minus120590119905119905 120590 = 120582


120590 minus 120582 120590 = 120582

(29)


We easily get

lim119905rarrinfin

E 119881 (119905) = 0 (30)


lim119905rarrinfin

E 119909 (119905) = 0 (31)



4 Example


1(119905) = 119909

2(119905) + 01119909

1(119905) (119905)

2(119905) = 119909

3(119905) + 01119909

3(119905) (119905)

3(119905) = minus 6119909

1(119905) minus 292119909

2(119905) minus 12119909

3(119905) + 119909

2

1(119905)

+ 01 sin (1199091(119905)) + 0001 sin (119905) + 119906 (119905)

119910 (119905) = 1199093(119905)

(32)

where

119860 = [

[

0 1 0

0 0 1


]

]

119861 = [

[

0

0

1

]

]

119867 = [

[

0

0

0001

]

]

119862 = [0 0 1] 119863 = [

[

01 0 0

0 0 01

0 0 0

]

]

1198631= [

[

01 0 0

0 0 0

0 0 0

]

]

1198641= 01 119865 (119905) = sin 119905 119891 (119909) = 119909

2

1(119905)

Δ119891 (119909) = 01 cos (1199091(119905)) V (119905) = sin (119905)

(33)


119883 = [

[

10290 minus03534 00052

minus03534 02286 minus01346

00052 minus01346 04802

]

]

(34)

Then we have

119875 = 119883minus1

= [

[

26316 48524 13317

48524 141869 39244

13317 39244 31684

]

]

(35)


0 2 4 6 8 10 12 14 16 18 20minus15

minus1

minus05

0

05

1

t (s)

x1

x2

x3



t (s)




5 Conclusion





0 5 10 15 20

000501015020250303504

minus005

t (s)


0 5 10 15 20minus01

001020304050607

s

t (s)


Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










That is

L119881 (119905) le minus2120573radic119881 (119905) (12)


L 119904 (119905) = Lradic119881 (119905) le minus120573 (13)

and hence

E 119904 (119905) le E 119904 (0) minus 120573119905 (14)




[[[

[

Δ 119867 + 119883119862119879 119883119863119879 1198831198641198791

⋆ minus120574119868 0 0

⋆ ⋆ minus119883 0

⋆ ⋆ ⋆ minus120603119868

]]]

]

lt 0 (15)

where Δ = 119860119883 + 119861119884 + 119883119879119860119879 + 119884119879119861119879 + 12060311986311198631198791


119883minus1 and 119870 = 119884119883minus1


119881 (119905) = 119909119879

(119905) 119875119909 (119905) (16)


= 2119909119879

119875 [(119860 + Δ119860) 119909 (119905) + 119861 (119906 (119905) + 119891 (119909)) + 119867V (119905)]

+ 2119909119879

119875119863119909 (119905) (119905) + 119909119879

(119905) 119863119879

119875119863119909 (119905)

(17)


L119881 (119905) = 119909119879

(119905) [119875 (119860 + 119861119870) + (119860 + 119861119870)119879

119875] 119909 (119905)

+ 2119909119879

(119905) 119875Δ119860119909 (119905) + 2119909119879

(119905) 119875119867V (119905)

+ 2119909119879

(119905) 119875119861 [119906 (119905) + 119891 (119909 119905) minus 119870119909 (119905)]

+ 119909119879

(119905) 119863119879

119875119863119909 (119905)

(18)

Note that

2119909119879

(119905) 119875Δ119860119909 (119905) le 120603119909119879

(119905) 1198751198631119863119879

1119875119909 (119905)

+1

120603119909119879

(119905) 119864119879

11198641119909 (119905)

(19)


2119909119879

(119905) 119875119861 = 2120578119890minus120582119905

119909119879

(0) 119875119861 (20)


2119909119879

(119905) 119875119861 = 2120578119890minus120582119905

119909119879

(0) 119875119861 = 0 (21)


L119881 (119905) minus 2V119879 (119905) 119910 (119905) minus 120574V119879 (119905) V (119905)

le [119909 (119905)

V (119905)]119879

[Δ 119875119867 + 119862119879


119909 (119905)

V (119905)]

(22)

whereΔ = 119875(119860+119861119870)+(119860+119861119870)119879

119875+12060311987511986311198631198791119875+(1120603)119864119879

11198641+



[Δ + 120590119875 119875119867 + 119862119879

lowast minus120574119868] lt 0 (23)




0 le E 119881 (119905) lt Eint119905

0

[minus120574V119879 (119905) V (119905) minus 2119910119879

(119905) 119910 (119905)] 119889119905

(25)

We easily get

E2int119905

0

V119879 (119905) 119910 (119905) 119889119905 lt 120574Eint119905

0

V119879 (119905) V (119905) 119889119905 (26)



L119881 (119905) = 119909119879

(119905) [119875 (119860 + 119861119870) + (119860 + 119861119870)119879

119875] 119909 (119905)

+ 2119909119879

(119905) 119875Δ119860119909 (119905) + 2120578119890minus120582119905

119909119879

(0) 119875119861

times [119906 (119905) + 119891 (119909 119905) minus 119870119909 (119905)] + 119909119879

(119905) 119863119879

119875119863119909 (119905)

(27)




(28)


E 119881 (119905) lt 119890minus120590119905

E 119881 (0) + 120575119890minus120590119905

int119905

0

119890(120590minus120582)120591

119889120591

=

119890minus120590119905E 119881 (0) + 120575119890minus120590119905119905 120590 = 120582


120590 minus 120582 120590 = 120582

(29)


We easily get

lim119905rarrinfin

E 119881 (119905) = 0 (30)


lim119905rarrinfin

E 119909 (119905) = 0 (31)



4 Example


1(119905) = 119909

2(119905) + 01119909

1(119905) (119905)

2(119905) = 119909

3(119905) + 01119909

3(119905) (119905)

3(119905) = minus 6119909

1(119905) minus 292119909

2(119905) minus 12119909

3(119905) + 119909

2

1(119905)

+ 01 sin (1199091(119905)) + 0001 sin (119905) + 119906 (119905)

119910 (119905) = 1199093(119905)

(32)

where

119860 = [

[

0 1 0

0 0 1


]

]

119861 = [

[

0

0

1

]

]

119867 = [

[

0

0

0001

]

]

119862 = [0 0 1] 119863 = [

[

01 0 0

0 0 01

0 0 0

]

]

1198631= [

[

01 0 0

0 0 0

0 0 0

]

]

1198641= 01 119865 (119905) = sin 119905 119891 (119909) = 119909

2

1(119905)

Δ119891 (119909) = 01 cos (1199091(119905)) V (119905) = sin (119905)

(33)


119883 = [

[

10290 minus03534 00052

minus03534 02286 minus01346

00052 minus01346 04802

]

]

(34)

Then we have

119875 = 119883minus1

= [

[

26316 48524 13317

48524 141869 39244

13317 39244 31684

]

]

(35)


0 2 4 6 8 10 12 14 16 18 20minus15

minus1

minus05

0

05

1

t (s)

x1

x2

x3



t (s)




5 Conclusion





0 5 10 15 20

000501015020250303504

minus005

t (s)


0 5 10 15 20minus01

001020304050607

s

t (s)


Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










We easily get

lim119905rarrinfin

E 119881 (119905) = 0 (30)


lim119905rarrinfin

E 119909 (119905) = 0 (31)



4 Example


1(119905) = 119909

2(119905) + 01119909

1(119905) (119905)

2(119905) = 119909

3(119905) + 01119909

3(119905) (119905)

3(119905) = minus 6119909

1(119905) minus 292119909

2(119905) minus 12119909

3(119905) + 119909

2

1(119905)

+ 01 sin (1199091(119905)) + 0001 sin (119905) + 119906 (119905)

119910 (119905) = 1199093(119905)

(32)

where

119860 = [

[

0 1 0

0 0 1


]

]

119861 = [

[

0

0

1

]

]

119867 = [

[

0

0

0001

]

]

119862 = [0 0 1] 119863 = [

[

01 0 0

0 0 01

0 0 0

]

]

1198631= [

[

01 0 0

0 0 0

0 0 0

]

]

1198641= 01 119865 (119905) = sin 119905 119891 (119909) = 119909

2

1(119905)

Δ119891 (119909) = 01 cos (1199091(119905)) V (119905) = sin (119905)

(33)


119883 = [

[

10290 minus03534 00052

minus03534 02286 minus01346

00052 minus01346 04802

]

]

(34)

Then we have

119875 = 119883minus1

= [

[

26316 48524 13317

48524 141869 39244

13317 39244 31684

]

]

(35)


0 2 4 6 8 10 12 14 16 18 20minus15

minus1

minus05

0

05

1

t (s)

x1

x2

x3



t (s)




5 Conclusion





0 5 10 15 20

000501015020250303504

minus005

t (s)


0 5 10 15 20minus01

001020304050607

s

t (s)


Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20

000501015020250303504

minus005

t (s)


0 5 10 15 20minus01

001020304050607

s

t (s)


Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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