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Journal of Control Science and Engineering
VolumeΒ 2011, Article IDΒ 859159, 6 pages
http://dx.doi.org/10.1155/2011/859159
Research Article

Non-Fragile Sliding Mode Control of Uncertain Chaotic Systems

1College of Electrical and Information Engineering, Henan University of Science and Technology, Luoyang 471003, China
2School of Electrical and Information Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Received 5 April 2011; Accepted 27 June 2011

Academic Editor: OnurΒ Toker

Copyright Β© 2011 Leipo Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with non-fragile sliding mode control of uncertain chaotic systems with external disturbance. Firstly, a new sliding surface is proposed, and sufficient conditions are derived to guarantee that sliding mode dynamics is asymptotically stable with a generalized 𝐻2 disturbance rejection level. Secondly, non-fragile sliding mode controller is established to make the state of system reach the sliding surface in a finite time. Finally, an example is given to illustrate the effectiveness of the proposed method.

1. Introduction

Chaotic behavior is a seemingly random phenomenon of a deterministic system that is characterized by sensitive deaspendence on initial conditions. Many electronic, mechanical, and chemical systems exhibit chaotic dynamics. Therefore, chaos is a very interesting nonlinear phenomenon, and control of chaotic systems has been paid much attention by researchers since the pioneering work of Ott et al. [1]. The main aim of chaos control is to suppress chaotic behavior and to stabilize the chaotic system, and various effective techniques have been presented and successfully applied to obtain chaos control such as adaptive control [2], backstepping control [3], fuzzy control [4], optimal control [5], and sliding mode control [6–9].

Among the above-mentioned methods, sliding mode control is a very effective approach to control chaos because of its attractive features such as fast response, good transient response, and insensitivity to variations in system parameters and external disturbances [6–11]. Furthermore, in some engineering practices, for example, in order to avoid actuator saturation, one often guarantees that whenever the exogenous input has bounded energy, the controlled output should have a specified maximum peak value. In this case, generalized 𝐻2 control is a good tool [12, 13].

Generally speaking, accurate controllers are required to stabilize control systems, so all of the controller coefficients are exact values in designing a desired controller. However, in practice, the uncertainty is not avoided, and it may be caused by many reasons, such as finite word length in digital systems, the imprecision inherent in analog systems, and the need for additional tuning of parameters in the final controller implementation [14]. The existence of uncertainty leads to the study of non-fragile control problem, that is, to design a controller such that the controller is insensitive to uncertainties. Many results about non-fragile control problems have been reported [15–17]. However, so far, non-fragile sliding mode control problem for uncertain chaotic systems has been paid little attention.

In this paper, the problem of non-fragile sliding mode control of uncertain chaotic systems with external disturbance is considered. Firstly, a new integral-type sliding surface is proposed, and the reaching phase is eliminated, and the proposed sliding surface function includes a norm-bounded parameter uncertainty; this term is added not only to make the design of sliding surface more relaxed, but also to guarantee that the sliding mode dynamics has better robust performance. For both the cases with additive and multiplicative uncertainties, sufficient conditions are derived to make sliding mode dynamics stable with a generalized 𝐻2 performance. Secondly, non-fragile sliding mode controller is established to make the state of system reach the sliding surface in a finite time. Finally, an example is given to illustrate the effectiveness of the proposed method.

Notations
The following notations will be used throughout this paper. 𝑅𝑛 and π‘…π‘›Γ—π‘š denote, respectively, the 𝑛-dimensional Euclidean space and the space of π‘›Γ—π‘š real matrices. For a real symmetric matrix 𝑀, 𝑀>0 means 𝑀 is positive definite. 𝐿2[0,∞) is the space of square-integrable vector functions over [0,∞), and 𝐿∞ is the space of essential bounded functions. βˆ— represents the symmetric form of matrix.

2. Problem Formulation

Consider the following chaotic system:𝑧̇π‘₯(𝑑)=𝐴π‘₯+𝐡(𝑓(π‘₯)+Δ𝑓(π‘₯)+𝑒(𝑑))+𝐷𝑀(𝑑),(𝑑)=𝐢π‘₯(𝑑),(1) where π‘₯(𝑑)βˆˆπ‘…π‘› is the state vector, 𝑒(𝑑)βˆˆπ‘…π‘š is the control input. 𝑧(𝑑)βˆˆπ‘…π‘ž is the system output. 𝑓(π‘₯) is a nonlinear real-valued function. 𝐴,𝐡,𝐢,and𝐷 are matrices with compatible dimensions and (𝐴,𝐡) is completely controllable. 𝐡 is of full column rank. 𝑀(𝑑) is a bounded disturbance. Δ𝑓(π‘₯) represents the uncertainty part, satisfying ‖Δ𝑓(π‘₯)β€–β‰€πœŒ with a positive constant 𝜌.

The block diagram of the system (1) is given in Figure 1.

859159.fig.001
Figure 1: The block diagram of the system (1).

The following lemmas are necessary for future discussion.

Lemma 1 (see [8]). For a given matrix 𝑆=𝑆11𝑆12𝑆𝑇12𝑆22ξ‚„ with 𝑆11=𝑆𝑇11, 𝑆22=𝑆𝑇22, then the following conditions are equivalent: (1)𝑆<0, (2)𝑆22<0, 𝑆11βˆ’π‘†12π‘†βˆ’122𝑆𝑇12<0,(3)𝑆11<0, 𝑆22βˆ’π‘†π‘‡12π‘†βˆ’111𝑆12<0.

Lemma 2 (see [17]). Let Ξ©,Ξ“ and Ξ be matrices of appropriate dimensions, and let Ξ© be symmetrical. Then Ξ©+Γ𝐹(𝑑)Ξ+(Γ𝐹(𝑑)Ξ)𝑇<0,(2) for an uncertain matrix 𝐹(𝑑) satisfying 𝐹𝑇(𝑑)𝐹(𝑑)≀𝐼 if and only if there exists a scalar πœ€>0 such that Ξ©+πœ€Ξ“Ξ“π‘‡+1πœ€ξ€·Ξžπ‘‡Ξžξ€Έ<0.(3)

3. Main Results

For system (1), firstly, a new sliding surface is designed, and sufficient conditions are given to guarantee that the sliding mode dynamics is asymptotically stable with a generalized 𝐻2-norm bound 𝛾. Secondly, non-fragile sliding mode controller is designed to drive the state arriving at the sliding surface in a finite time.

3.1. Sliding Surface Design

For system (1), we design the following integral-type sliding surface function:ξ‚Έξ€œπ‘ (𝑑)=𝐺π‘₯(𝑑)βˆ’π‘₯(0)βˆ’π‘‘0(ξ‚Ή,𝐴+𝐡(𝐾+Δ𝐾(𝑠)))π‘₯(𝑠)𝑑𝑠(4) where 𝐺 is a matrix such that 𝐺𝐡 is invertible. 𝐾>0 is a gain matrix to be chosen later, and the matrix Δ𝐾(𝑑) represents the norm-bounded parameter uncertainty.

In this paper, the following two classes of parameter uncertainty will be considered:(a)Δ𝐾(𝑑) is with the norm-bounded additive form Δ𝐾(𝑑)=Ξ”1=𝐻1𝐹1(𝑑)𝐸1,(5) where 𝐻1 and 𝐸1 are known matrices and 𝐹1(𝑑) is an unknown matrix satisfying 𝐹𝑇1(𝑑)𝐹1(𝑑)≀𝐼;(6)(b)Δ𝐾(𝑑) is with the norm-bounded multiplicative form Δ𝐾(𝑑)=Ξ”2=𝐻2𝐹2(𝑑)𝐸2𝐾,(7) where 𝐻2 and 𝐸2 are known matrices, and 𝐹2(𝑑) is an unknown matrix satisfying 𝐹𝑇2(𝑑)𝐹2(𝑑)≀𝐼.(8)

Remark 3. From (4), we have 𝑠(0)=0. It is worth to note that the integral-type sliding surface function (4) is different from the existing methods, for example, those presented in [10, 11], since it takes the term Δ𝐾(𝑑) into account. Moreover, these existing results can be considered as a special form of the integral-type sliding surface designed in this paper, when Δ𝐾(𝑑) is zero. In fact, uncertainty Δ𝐾(𝑑) is added not only to make the design of sliding surface more relaxed, but also to guarantee that the sliding mode dynamics has better robust performance.
Let ̇𝑠(𝑑)=0. Then, the equivalent control law can be obtained as follows: 𝑒eq(𝑑)=(𝐾+Δ𝐾(𝑑))π‘₯(𝑑)βˆ’(𝐺𝐡)βˆ’1𝐺𝐷𝑀(𝑑)βˆ’π‘“(π‘₯)βˆ’Ξ”π‘“(π‘₯).(9) Substituting (9) into the system (1) and denoting 𝐷=π·βˆ’π΅(𝐺𝐡)βˆ’1𝐺𝐷, sliding mode dynamics and the output equation can be formulated as Μ‡π‘₯(𝑑)=(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))π‘₯(𝑑)+𝐷𝑀(𝑑),𝑧(𝑑)=𝐢π‘₯(𝑑).(10)

Definition 4. Sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻2-norm bound 𝛾 if the following conditions hold.(i)When 𝑀(𝑑)=0, sliding mode dynamics (10) is asymptotically stable.(ii)Sliding mode dynamics (10) has a given generalized 𝐻2 (or 𝐿2βˆ’πΏβˆž) disturbance rejection level; that is, the 𝐿2βˆ’πΏβˆž gain from disturbance input 𝑀(𝑑) to control output 𝑧(𝑑) of the system (10) is less than a given value. It is to make, under the zero-valued initial condition, the following inequality hold: supπ‘€βˆˆπΏ2βˆ’0‖𝑧‖2βˆžβ€–π‘€β€–22<𝛾2,(11) where ‖𝑀‖22=ξ€œβˆž0𝑀𝑇(𝑑)𝑀(𝑑)𝑑𝑑,‖𝑧‖2∞=sup𝑑𝑧𝑇.(𝑑)𝑧(𝑑)(12)

Theorem 5. Consider the system (10) and a given constant 𝛾>0. If there exist matrices 𝑃>0 and 𝐾 such that the following matrix inequalities are true ⎑⎒⎒⎣Θ=𝑃(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))+(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))π‘‡π‘ƒπ‘ƒπ·βŽ€βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£βˆ—βˆ’πΌ<0,(13)π‘ƒπΆπ‘‡βˆ—π›Ύ2𝐼⎀βŽ₯βŽ₯⎦>0,(14) then the sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻2-norm bound 𝛾.

Proof. Consider the following Lyapunov functional candidate: 𝑉1(𝑑)=π‘₯𝑇(𝑑)𝑃π‘₯(𝑑).(15) The time derivative of 𝑉1(𝑑) along the trajectories of system (10) with 𝑀(𝑑)=0 is given by ̇𝑉1(𝑑)=π‘₯𝑇(𝑑)𝑃(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))+(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))𝑇𝑃π‘₯(𝑑).(16)
From (13), Θ<0 implies 𝑃(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))+(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))𝑇𝑃<0, then we have ̇𝑉1(𝑑)<0,(17) which implies sliding mode dynamics (10) is asymptotically stable, when 𝑀(𝑑)=0.
In order to construct the generalized 𝐻2 (𝐿2βˆ’πΏβˆž) performance index, we assume the zero-valued initial condition, then 𝑉1(𝑑)|𝑑=0=0. Now, we consider the following cost performance: 𝐽=𝑉1(ξ€œπ‘‘)βˆ’π‘‘0𝑀𝑇(𝑠)𝑀(𝑠)𝑑𝑠.(18) For any nonzero 𝑀(𝑑)∈𝐿2[0,∞), we have ξ€œπ½=𝑑0̇𝑉1(𝑠)βˆ’π‘€π‘‡(ξ€»=ξ€œπ‘ )𝑀(𝑠)𝑑𝑠𝑑0πœ‰π‘‡Ξ˜πœ‰π‘‘π‘ ,(19) where πœ‰=[π‘₯𝑇(𝑠)𝑀𝑇(𝑠)]𝑇.
From (13), we obtain 𝐽=𝑉1(ξ€œπ‘‘)βˆ’π‘‘0𝑀𝑇(𝑠)𝑀(𝑠)𝑑𝑠<0.(20) Equation (20) leads to π‘₯𝑇(𝑑)𝑃π‘₯(𝑑)=𝑉1(ξ€œπ‘‘)<𝑑0𝑀𝑇(𝑠)𝑀(𝑠)𝑑𝑠.(21) For (14), by Lemma 1, we have 𝐢𝑇𝐢<𝛾2𝑃(22) which leads to 𝑧𝑇(𝑑)𝑧(𝑑)=π‘₯𝑇(𝑑)𝐢𝑇𝐢π‘₯(𝑑)<𝛾2π‘₯𝑇(𝑑)𝑃π‘₯(𝑑)<𝛾2ξ€œπ‘‘0𝑀𝑇(𝑠)𝑀(𝑠)𝑑𝑠<𝛾2ξ€œβˆž0𝑀𝑇(𝑠)𝑀(𝑠)𝑑𝑠.(23) For any time 𝑑β‰₯0, taking the maximum value of 𝑧𝑇(𝑑)𝑧(𝑑), we have ‖𝑧‖2∞<𝛾2‖‖𝑀(𝑑)22.(24)
The proof is completed.

For sliding mode dynamics (10) with additive and multiplicative uncertainties, the following theorems give some sufficient conditions which guarantee the sliding mode dynamics is asymptotically stable with a generalized 𝐻2-norm bound 𝛾.

Theorem 6. Consider the system (10), uncertainty Δ𝐾(𝑑) in (5), and a given constant 𝛾>0. If there exist matrices 𝑋>0, π‘Œ and a scalar πœ€1>0 such that the following linear matrix inequalities (LMIs) are true βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄π‘‹+𝑋𝐴𝑇+π΅π‘Œ+π‘Œπ‘‡π΅π‘‡+πœ€1𝐡𝐻1𝐻𝑇1𝐡𝑇𝐷𝑋𝐸𝑇1βˆ—βˆ’πΌ0βˆ—βˆ—βˆ’πœ€1𝐼⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎣<0,(25)π‘‹π‘‹πΆπ‘‡βˆ—π›Ύ2𝐼⎀βŽ₯βŽ₯⎦>0,(26) then the sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻2-norm bound 𝛾. Furthermore, the gain matrix 𝐾 is chosen as 𝐾=π‘Œπ‘‹βˆ’1.

Proof. Pre- and postmultiplying (13) and (14) by diag{𝑋,𝐼} with 𝑋=π‘ƒβˆ’1, respectively, we have ⎑⎒⎒⎣(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))𝑋+𝑋(𝐴+𝐡𝐾+𝐡Δ𝐾(𝑑))π‘‡π·βŽ€βŽ₯βŽ₯βŽ¦βˆ—βˆ’πΌ<0,(27) and (26).
From (27), we can obtain ⎑⎒⎒⎣(𝐴+𝐡𝐾)𝑋+𝑋(𝐴+𝐡𝐾)π‘‡π·βŽ€βŽ₯βŽ₯⎦+βŽ‘βŽ’βŽ’βŽ£βˆ—βˆ’πΌπ΅π»10⎀βŽ₯βŽ₯⎦𝐹1𝐸(𝑑)1ξ‚„+βŽ›βŽœβŽœβŽβŽ‘βŽ’βŽ’βŽ£π‘‹0𝐡𝐻10⎀βŽ₯βŽ₯⎦𝐹1𝐸(𝑑)1ξ‚„βŽžβŽŸβŽŸβŽ π‘‹0𝑇<0.(28) From Lemma 2, (28) holds if and only if there exists a scalar πœ€1>0 such that the following inequality holds: ⎑⎒⎒⎣(𝐴+𝐡𝐾)𝑋+𝑋(𝐴+𝐡𝐾)π‘‡π·βŽ€βŽ₯βŽ₯βŽ¦βˆ—βˆ’πΌ+πœ€1⎑⎒⎒⎣𝐡𝐻10⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣𝐡𝐻10⎀βŽ₯βŽ₯βŽ¦π‘‡+1πœ€1βŽ‘βŽ’βŽ’βŽ£π‘‹πΈπ‘‡10⎀βŽ₯βŽ₯βŽ¦ξ‚ƒπΈ1𝑋0<0,(29) which, according to Lemma 1, is equivalent to (25).
The proof is completed.

Theorem 7. Consider the system (10), uncertainty Δ𝐾(𝑑) in (7), and a given constant 𝛾>0. If there exist matrices 𝑋>0, π‘Œ and the scalar πœ€2>0 such that the following linear matrix inequalities (LMIs) are true βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄π‘‹+𝑋𝐴𝑇+π΅π‘Œ+π‘Œπ‘‡π΅π‘‡+πœ€2𝐡𝐻2𝐻𝑇2π΅π‘‡π·π‘ŒπΈπ‘‡2βˆ—βˆ’πΌ0βˆ—βˆ—βˆ’πœ€2𝐼⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎣<0,(30)π‘‹π‘‹πΆπ‘‡βˆ—π›Ύ2𝐼⎀βŽ₯βŽ₯⎦>0,(31) then the sliding mode dynamics (10) is asymptotically stable with a generalized 𝐻2-norm bound 𝛾. Furthermore, the gain matrix 𝐾 is chosen as 𝐾=π‘Œπ‘‹βˆ’1.

Proof. The proof is similar to Theorem 6, and omitted.

3.2. Sliding Mode Controller Design

The input 𝑒(𝑑) in system (1) should be appropriately designed such that the state π‘₯(𝑑) in system (1) reaches the sliding surface 𝑠(𝑑)=0 within a finite time. The non-fragile sliding mode controller 𝑒(𝑑) is as follows:𝑒(𝑑)=(𝐾+Δ𝐾(𝑑))π‘₯(𝑑)βˆ’π‘“(π‘₯)βˆ’(𝐺𝐡)βˆ’1Γ—ξ€Ίξ€»,𝛼𝑠(𝑑)+(β€–G𝐷‖⋅‖𝑀(𝑑)β€–+β€–πΊπ΅β€–β‹…πœŒ+𝛽)sgn(𝑠(𝑑))(32) where sgn(𝑠(𝑑))=[sign(𝑠1),sign(𝑠2),…,sign(π‘ π‘š)]𝑇, 𝛼>0 and 𝛽>0.

Theorem 8. Consider the system (1). If the input 𝑒(𝑑) is taken as (32), then the trajectory of the system (1) converges to the sliding surface 𝑠(𝑑)=0 in a finite time.

Proof. Consider a Lyapunov function candidate as follows: 𝑉21(𝑑)=2𝑠𝑇(𝑑)𝑠(𝑑).(33) Calculating the time derivative of 𝑉2(𝑑) along the trajectory of system (1), we have ̇𝑉2(𝑑)=𝑠𝑇[].(𝑑)βˆ’πΊπ΅(𝐾+Δ𝐾(𝑑))π‘₯(𝑑)+𝐺𝐷𝑀(𝑑)+𝐺𝐡(𝑒(𝑑)+𝑓(π‘₯)+Δ𝑓(π‘₯))(34)
Substituting (32) into (34) yields ̇𝑉2(𝑑)=𝑠𝑇[βˆ’(ξ€»(𝑑)βˆ’πΊπ΅(𝐾+Δ𝐾(𝑑))π‘₯(𝑑)+𝐺𝐷𝑀(𝑑)+𝐺𝐡Δ𝑓(π‘₯)+𝐺𝐡𝑓(π‘₯)+𝐺𝐡(𝐾+Δ𝐾(𝑑))π‘₯(𝑑)βˆ’πΊπ΅π‘“(π‘₯)βˆ’π›Όπ‘ (𝑑)‖𝐺𝐷‖⋅‖𝑀(𝑑)β€–+β€–πΊπ΅β€–β‹…πœŒ+𝛽)sgn(𝑠(𝑑))≀‖𝑠(𝑑)β€–β‹…(‖𝐺𝐷‖⋅‖𝑀(𝑑)β€–+β€–πΊπ΅β€–β‹…πœŒ)βˆ’π›Όβ€–π‘ (𝑑)β€–2)βˆ’(‖𝐺𝐷‖⋅‖𝑀(𝑑)β€–+β€–πΊπ΅β€–β‹…πœŒ+𝛽‖𝑠(𝑑)β€–.(35) Then, we have ̇𝑉2β€–(𝑑)β‰€βˆ’π›Όβ€–π‘ (𝑑)2βˆ’π›½β€–π‘ (𝑑)β€–.(36) According to sliding mode theory, we conclude that the trajectory of the system (1) converges to the sliding surface 𝑠(𝑑)=0 in a finite time and remains in it thereafter. Thus, the proof is completed.

4. Simulation

In this section, we use Genesio's chaotic system to verify the effectiveness of the method. Genesio's system with additional input is as follows:𝑧̇π‘₯(𝑑)=𝐴π‘₯+𝐡(𝑓(π‘₯)+Δ𝑓(π‘₯)+𝑒(𝑑))+𝐷𝑀(𝑑),(𝑑)=𝐢π‘₯(𝑑),(37) where⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣001⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎑⎒⎒⎒⎒⎣111⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,π‘₯𝐴=010001βˆ’6βˆ’2.92βˆ’1.2,𝐡=,𝐷=𝐢=00βˆ’0.1,Δ𝑓(π‘₯)=sin3(𝑒𝑑),𝑀(𝑑)=βˆ’π‘‘.10+𝑑(38)

Without loss of generality, we consider Δ𝐾(𝑑) is with the norm-bounded additive form, where 𝐻1=[010], 𝐸1=βˆ’0.1 and 𝐹1(𝑑)=0.1sin𝑑, 𝐹1(𝑑) satisfies 𝐹𝑇1(𝑑)𝐹1(𝑑)=0.01sin2𝑑<1. Furthermore, we choose 𝐺=[001] and 𝛾=0.3162.

Then, solving LMIs (25) and (26) yields 𝐾=4.6545βˆ’0.8134βˆ’0.8913.(39)

The initial value is π‘₯(0)=[30βˆ’1]𝑇. The simulation results are shown in Figures 2, 3, 4, and 5.

859159.fig.002
Figure 2: The phase curves of uncontrolled Genesio's system.
859159.fig.003
Figure 3: State π‘₯(𝑑) of the controlled Genesio's system.
859159.fig.004
Figure 4: The control input 𝑒(𝑑).
859159.fig.005
Figure 5: Sliding mode 𝑠(𝑑).

Figure 2 shows the phase curves of uncontrolled Genesio's system. Figure 3 shows the time response of the state π‘₯(𝑑) of the controlled Genesio's system. Figure 4 shows the control input response curve. Sliding mode response curve is shown in Figure 5. From the simulation results, it is concluded that the proposed method is effective.

5. Conclusion

In this paper, the problem of non-fragile sliding mode control of uncertain chaotic systems with external disturbance is investigated. A new sliding surface is proposed, and sufficient conditions are derived for asymptotic stability with a generalized 𝐻2 disturbance rejection level of sliding mode dynamics. Non-fragile sliding mode controller is established to make the state of system reach the sliding surface in a finite time. Finally, the simulation shows the effectiveness of the proposed method.

Acknowledgment

The authors are grateful for the support of the National Natural Science Foundation of China under Grants nos. 61074003 and 60904023.

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