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Journal of Applied Mathematics
Volume 2012, Article ID 859542, 20 pages
http://dx.doi.org/10.1155/2012/859542
Research Article

Optimal Control for a Class of Chaotic Systems

Institute of Systems Engineering, Tianjin University, Tianjin 300072, China

Received 19 October 2011; Revised 17 January 2012; Accepted 10 February 2012

Academic Editor: Chuanhou Gao

Copyright © 2012 Jianxiong Zhang and Wansheng Tang. 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 proposes the optimal control methods for a class of chaotic systems via state feedback. By converting the chaotic systems to the form of uncertain piecewise linear systems, we can obtain the optimal controller minimizing the upper bound on cost function by virtue of the robust optimal control method of piecewise linear systems, which is cast as an optimization problem under constraints of bilinear matrix inequalities (BMIs). In addition, the lower bound on cost function can be achieved by solving a semidefinite programming (SDP). Finally, numerical examples are given to illustrate the results.

1. Introduction

As a very interesting nonlinear phenomenon, chaos has been widely applied in many areas, such as secure communication, signal generator design, biology, economics, and many other engineering systems, which has been researched thoroughly over the past two decades [1]. Recently, chaos control of chaotic systems has become an active research topic [2]. In general, there are several schemes to achieve the control of continuous time chaotic systems, such as OGY method [3], parametric resonance method [4], adaptive feedback method [5, 6], delay feedback method [7], backstepping design method [8], fractional controller design method [9], sliding mode control method [10, 11], internal model approach [12], impulsive control approach [13], as well as linear and nonlinear feedback control methods [1417]. However, most of the existing methods were used to achieve chaos control either by employing the linearization scheme in the neighborhood of the objective point which is difficult to accomplish the global analysis, or by applying the nonlinear feedback controller which often limits practical applications. Based on the fuzzy control theory, Tanaka et al. [18] studied the feedback control of chaotic systems. The result formulated in terms of linear matrix inequalities (LMIs, [19]) was convenient to solve, but the controller design for the associated fuzzy systems was fulfilled by virtue of global quadratic Lyapunov function which is conservative in the control synthesis.

As pointed out in [20], piecewise linear systems, which can approximate general nonlinear systems to any degree of accuracy, can be analyzed based on piecewise quadratic Lyapunov function technique that introduces more flexibility than the classical global quadratic Lyapunov function technique. Thus, the piecewise linear systems provide a powerful way of analysis and synthesis for nonlinear systems. Chaotic systems belong to complex nonlinear systems. In fact, it is significant to design a practicable piecewise linear feedback controller to stabilize globally a chaotic system with a performance measure for the control synthesis. We recently [21] proposed a new chaotic system and designed a piecewise linear feedback controller to stabilize globally the new system based on piecewise linear systems method. So far, there have been very few results dealing with the optimal control for chaotic systems. In this paper, we investigate the problem of designing piecewise linear feedback controller to stabilize a class of chaotic systems, and meanwhile minimize a quadratic cost function for the closed-loop systems. Particularly, in this paper, a class of chaotic systems are converted to uncertain piecewise linear systems. Then, based on piecewise quadratic Lyapunov function technique and Hamilton-Jacobi-Bellman (HJB) inequality method, the optimal chaos control via piecewise linear state feedback controller is studied. It is shown that the optimal controller minimizing the upper bound on cost function can be obtained by solving an optimization problem under constraints of bilinear matrix inequalities (BMIs). The lower bound on cost function can be attained by solving a semidefinite programming (SDP). If the upper and lower bounds obtained are sufficiently tight, it is concluded that the associated solutions achieve or get close to optimality.

This paper is organized as follows. In Section 2, the optimal control problem of chaotic systems is introduced. In Section 3, the optimal control for a class of chaotic systems via piecewise linear state feedback controller is proposed. The upper bound and lower bound on cost function are designed. Illustrative examples are given in Section 4, and the conclusion is drawn in Section 5.

Throughout this paper, a real symmetric matrix 𝑃>0(0,0) denotes 𝑃 being a positive definite (positive semidefinite, or negative semidefinite) matrix, and 𝐴>𝐵 means 𝐴𝐵>0. 𝐼 denotes an identity matrix of appropriate dimension. The superscript “𝑇” represents the transpose of a matrix. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Problem Formulation

Consider the chaotic system of the form: ̇𝐱=𝐴𝐱+𝐹(𝐱)+𝐵𝐮,(2.1) where 𝐴 and 𝐵 are constant matrices, 𝐱𝑛 is the state vector, 𝐮𝑚(𝑚𝑛) is the control input variable, and the nonlinear term 𝐹(𝐱)𝑛 is assumed to satisfy Lipschitz continuity condition, uniform or local, and 𝐹(0)=0.

Associated with this system is the cost function: 𝐽=0𝐱𝑇(𝑡)𝑄𝐱(𝑡)+𝐮𝑇(𝑡)𝑅𝐮(𝑡)d𝑡,(2.2) where 𝑄>0, 𝑅>0 are given weighting matrices.

The goal of this paper is to design a state feedback law 𝐮(𝑡) stabilizing the chaotic system (2.1) and meanwhile minimizing the cost function (2.2).

It is known that the control law 𝐮(𝑡) can be derived from the solution to the associated HJB equation. However, generally speaking, the HJB equation corresponding to a general nonlinear system is notoriously hard to solve. Many numerical methods have been devised for the solution of optimal control problems but tended to suffer from combinatorial explosion. Piecewise linear systems, which can approximate nonlinear systems to any degree of accuracy, provide a powerful means of analysis for nonlinear systems. By virtue of HJB inequalities rather than equations, the authors in [20, 22] have investigated the state feedback optimal control of piecewise linear systems. It was shown that the upper bound on piecewise quadratic cost function can be obtained by solving a nonconvex BMIs problem, and the lower bound on cost function can be obtained by solving an SDP. Motivated by this, we first convert the chaotic system (2.1) to the form of uncertain piecewise linear systems and then extend the corresponding results of optimal control for the ordinary piecewise linear systems in [20] to the case of uncertain piecewise linear systems. Thus, we can achieve the optimal control for the original chaotic system.

Note that the nonlinear term 𝐹(𝐱) in system (2.1) can be approximated by a piecewise linear function as follows: 𝐹(𝐱)=𝐾𝑖𝐱+𝑎𝑖+Δ𝑖(𝐱),𝐱𝑋𝑖,𝑖𝕀,(2.3) where 𝐾𝑖𝑛×𝑛, 𝑎𝑖𝑛 are some given parameters, {𝑋𝑖}𝑖𝕀𝑛 denotes a partition of the state space of chaotic system, 𝕀 is the index set, and Δ𝑖(𝐱) is the approximation error, which can be regarded as uncertainties in the system. Then, it is obvious that system (2.1) can be converted to the uncertain piecewise linear system: ̇𝐱=𝐴+𝐾𝑖𝐱+𝑎𝑖+Δ𝑖(𝐱)+𝐵𝐮,𝐱𝑋𝑖,𝑖𝕀.(2.4)

It is worth mentioning that system (2.1) can represent a large class of chaotic systems such as Genesio-Tesi chaotic system [23], Coullet chaotic system [24], Chua’s Circuit system [25], and the new chaotic systems presented in [21, 26]. A simple but typical case is the three-dimensional chaotic system with the nonlinear term 𝐹(𝐱) taking the following form: 𝑥𝐹(𝐱)=0,0,𝑓1𝑇,(2.5) where 𝑓(𝑥1) is the nonlinear term in the 3rd dimension of the system and can be approximated by a piecewise linear function as 𝑓𝑥1=𝑘𝑖𝑥1+𝑙𝑖+𝛿𝑖𝑥1,𝐱𝑋𝑖,𝑖𝕀,(2.6)where 𝑘𝑖,𝑙𝑖 are some given parameters, 𝛿𝑖(𝑥1) is the approximation error. Then, system (2.1) with the nonlinear term (2.5) can be converted to the form of the uncertain piecewise linear system (2.4) as ̇𝐱=𝐴𝑖𝐱+𝑎𝑖+Δ𝑖+𝐵𝐮,𝐱𝑋𝑖,𝑖𝕀(2.7) with𝐴𝑖𝑘=𝐴+000000𝑖00,𝑎𝑖=00𝑙𝑖,Δ𝑖=00𝛿𝑖𝑥1.(2.8)

3. State Feedback Optimal Control of Systems

Without loss of generality, consider the uncertain piecewise linear system of the form ̇𝐱𝐴(𝑡)=𝑖+Δ𝐴𝑖𝐱𝐵(𝑡)+𝑖+Δ𝐵𝑖𝐮(𝑡)+𝑎𝑖+Δ𝑎𝑖(3.1) for 𝐱(𝑡)𝑋𝑖, where {𝑋𝑖}𝑖𝕀𝑛 denotes a partition of the state space into a number of polyhedral cells, 𝕀 is the index set of the cells, (𝐴𝑖,𝐵𝑖,𝑎𝑖) is the 𝑖th nominal local model of the system, 𝑎𝑖 is the offset term. Δ𝐴𝑖, Δ𝐵𝑖, and Δ𝑎𝑖 represent parametric perturbations in the system state matrix, input matrix, and offset term of the 𝑖th nominal local model, respectively, and are assumed to be of the following form: Δ𝐴𝑖,Δ𝐵𝑖,Δ𝑎𝑖=𝑀𝑖𝐻𝑁𝐴𝑖,𝑁𝐵𝑖,𝑁𝑎𝑖,(3.2) where 𝐻𝑖×𝑗 is an uncertain matrix bounded by 𝐻𝑇𝐻𝐼, and 𝑀𝑖, 𝑁𝐴𝑖, 𝑁𝐵𝑖, 𝑁𝑎𝑖 are known constant matrices of appropriate dimensions which specify how the elements of the nominal matrices 𝐴𝑖,𝐵𝑖, and 𝑎𝑖 are affected by the uncertain parameters in 𝐻.

Define 𝕀0𝕀 as the set of indices for cells that contain origin and 𝕀1𝕀 the set of indices for cells that do not contain the origin. It is assumed that 𝑎𝑖=Δ𝑎𝑖=0 for all 𝑖𝕀0.

For any given initial condition 𝐱(0)=𝐱0, and input signals 𝐮, it is assumed that system (3.1) has a unique solution, and there is no sliding mode. Note that with possible discontinuities in 𝐴𝑖𝐱across the boundaries of the partitions, the solution of system (3.1) may be just continuous and piecewise 𝐶1. For a definition of the state trajectory of the system in (3.1) refer to [20] for details.

For convenience, the following notations are introduced: 𝐱1,𝐱=𝐴𝑖=𝐴𝑖𝑎𝑖,00𝐵𝑖=𝐵𝑖0,𝑀𝑖=𝑀𝑖0,𝑁𝐴𝑖=𝑁𝐴𝑖,𝑁𝑎𝑖,Δ𝐴𝑖=Δ𝐴𝑖Δ𝑎𝑖=00𝑀𝑖𝐻𝑁𝐴𝑖,Δ𝐵𝑖=Δ𝐵𝑖0=𝑀𝑖𝐻𝑁𝐵𝑖,(3.3) then system (3.1) can be expressed as ̇𝐱(𝑡)=𝐴𝑖+Δ𝐴𝑖𝐱(𝑡)+𝐵𝑖+Δ𝐵𝑖𝐮(𝑡),𝑖𝕀.(3.4)

Associated with this system is the following cost function: 𝐽=0𝐱𝑇(𝑡)𝑄𝑖𝐱(𝑡)+𝐮𝑇(𝑡)𝑅𝑖𝐮(𝑡)d𝑡,(3.5) where 𝑖 is defined so that 𝐱(𝑡)𝑋𝑖, and 𝑄𝑖>0, 𝑅𝑖>0 are given weighting matrices.

Note that if 𝑄𝑖,𝑅𝑖 in (3.5) are set to be the same, respectively, for every 𝑖𝕀, the cost function (3.5) will reduce to (2.2). In addition, the matrix 𝑄𝑖=diag{𝑄𝑖,0}(𝑛+1)×(𝑛+1) is introduced, which will be used in the sequel.

As noted in [20], to find a piecewise Lyapunov function that is continuous across region boundaries, the matrices 𝐹𝑖=[𝐹𝑖,𝑓𝑖], 𝑖𝕀 with 𝑓𝑖=0 for 𝑖𝕀0 should be constructed, which are used to characterize the boundaries between the regions: 𝐹𝑖𝐱=𝐹𝑗𝐱,𝐱𝑋𝑖𝑋𝑗,𝑖,𝑗𝕀.(3.6) Then, the piecewise Lyapunov function candidates that are continuous across the region boundaries can be parameterized as 𝐱𝑉(𝐱)=𝑇𝑃𝑖𝐱,𝐱𝑋𝑖,𝑖𝕀0,𝐱𝑇𝑃𝑖𝐱,𝐱𝑋𝑖,𝑖𝕀1,(3.7) with 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖 and 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖, where 𝑆 is a symmetric matrix which characterizes the free parameters of the Lyapunov function candidates.

Note the form of 𝑃𝑖 and the characteristics of the matrices 𝐹𝑖. The continuity of the Lyapunov function 𝑉(𝐱) across the partition boundaries is ensured from (3.6) and (3.7).

The 𝑆-procedure has been used in [20, 22] to reduce the conservatism of the stability result. Specifically, the matrices 𝐸𝑖=[𝐸𝑖,𝑒𝑖], 𝑖𝕀 with 𝑒𝑖=0 for 𝑖𝕀0, such that 𝐸𝑖𝐱0,𝐱𝑋𝑖,𝑖𝕀,(3.8) should be constructed to verify the positivity of a piecewise quadratic function of the form (3.7) on a polyhedral partition. It should be noted that the above vector inequalities imply that each entry of the vector is nonnegative.

A systematic procedure for constructing the matrices 𝐸𝑖, 𝐹𝑖 for a given piecewise linear system was suggested in [20].

Consider the following piecewise linear feedback control law: 𝐮=𝐿𝑖𝐱𝑙𝑖=𝐿𝑖𝐱,𝐱𝑋𝑖,𝑖𝕀,(3.9) with 𝑙𝑖=0 for 𝑖𝕀0.

In general, the control law of form (3.9) will bring more flexibility in stability analysis than that of the ordinary linear feedback form. However, this control law may be discontinuous and give rise to sliding modes [20]. To avoid this case, we should construct the control law continuously across subspace boundaries and take the feedback gain matrix 𝐿𝑖 as follows 𝐿𝑖=𝐿𝑇𝐹𝑖,𝑖𝕀,(3.10) where 𝐿 is a parameter matrix characterizing the free parameters of the state feedback controller, and 𝐹𝑖 is the matrix defined in (3.6). It should be pointed out that the gain matrix 𝐿𝑖 should take the form of 𝐿𝑖=𝐿𝑇𝐹𝑖 for 𝑖𝕀0.

Substituting the control law (3.9) into system (3.4), we can get the following closed-loop system: ̇𝐴𝐱(𝑡)=𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖𝐱(𝑡),for𝑖𝕀0,̇𝐱(𝑡)=𝐴𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖𝐱(𝑡),for𝑖𝕀1.(3.11)

Our goal in this section is to find a parameter matrix 𝐿 to stabilize system (3.11) and meanwhile minimize the cost function (3.5). Before presenting the main results of this paper, we introduce the following lemmas.

Lemma 3.1 (Johansson and Rantzer [22]). Consider symmetric matrices 𝑆, 𝑈𝑖, and 𝑊𝑖 such that 𝑈𝑖 and 𝑊𝑖 have nonnegative entries, while 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖, 𝑖𝕀0 and 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖, 𝑖𝕀1, satisfy 𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝐸<0,𝑇𝑖𝑊𝑖𝐸𝑖<𝑃𝑖,(3.12) for 𝑖𝕀0, and 𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+𝐸𝑇𝑖𝑈𝑖𝐸𝑖<0,𝐸𝑇𝑖𝑊𝑖𝐸𝑖<𝑃𝑖,(3.13) for 𝑖𝕀1, then every continuous and piecewise 𝐶1 trajectory 𝐱(𝑡) of system (3.4) with Δ𝐴𝑖=0, Δ𝑎𝑖=0 and 𝑢=0 for all 𝑡>0 tends to zero exponentially.

Lemma 3.2 (Xie [27]). Given matrices 𝐺, 𝑀, and 𝑁 of appropriate dimensions with 𝐺 symmetric, then 𝐺+𝑀𝐻𝑁+𝑁𝑇𝐻𝑇𝑀𝑇<0 for all matrices 𝐻 satisfying 𝐻𝑇𝐻𝐼, if and only if there exists some 𝜀>0 such that 𝐺+𝜀1𝑀𝑀𝑇+𝜀𝑁𝑇𝑁<0.(3.14)

Motivated by the result in [20], we can get the upper bound on the cost function (3.5) for uncertain piecewise linear systems based on the HJB inequality method. The result is presented as follows.

Theorem 3.3. Consider the closed-loop uncertain system (3.11) with 𝐱0𝑋𝑖0. If there exist a set of constants 𝜀𝑖>0 and symmetric matrices 𝑆, 𝑈𝑖, and 𝑊𝑖 such that 𝑈𝑖 and 𝑊𝑖 have nonnegative entries, while 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖, 𝑖𝕀0, and 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖, 𝑖𝕀1, satisfy Φ𝑖𝜀𝑖𝑁𝐴𝑖𝑁𝐵𝑖𝐿𝑖𝑇𝑃𝑖𝑀𝑖𝐿𝑇𝑖𝜀𝑖𝑁𝐴𝑖𝑁𝐵𝑖𝐿𝑖𝜀𝑖𝑀𝐼00𝑇𝑖𝑃𝑖0𝜀𝑖𝐿𝐼0𝑖00𝑅𝑖1𝐸<0,𝑇𝑖𝑊𝑖𝐸𝑖<𝑃𝑖(3.15) for 𝑖𝕀0, Φ𝑖𝜀𝑖𝑁𝐴𝑖𝑁𝐵𝑖𝐿𝑖𝑇𝑃𝑖𝑀𝑖𝐿𝑇𝑖𝜀𝑖𝑁𝐴𝑖𝑁𝐵𝑖𝐿𝑖𝜀𝑖𝐼00𝑀𝑇𝑖𝑃𝑖0𝜀𝑖𝐼0𝐿𝑖00𝑅𝑖1<0𝐸𝑇𝑖𝑊𝑖𝐸𝑖<𝑃𝑖(3.16) for 𝑖𝕀1, where Φ𝑖𝐴=𝑖𝐵𝑖𝐿𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖𝐵𝑖𝐿𝑖+𝐸𝑇𝑖𝑈𝑖𝐸𝑖+𝑄𝑖,Φ𝑖=𝐴𝑖𝐵𝑖𝐿𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖𝐵𝑖𝐿𝑖+𝐸𝑇𝑖𝑈𝑖𝐸𝑖+𝑄𝑖,(3.17) then the closed-loop system is globally exponentially stable, and the cost function (3.5) satisfies 𝐽inf𝑆,𝑈𝑖,𝑊𝑖,𝜀𝑖𝐱𝑇0𝑃𝑖0𝐱0.(3.18)

Proof. By Schur complement [19], the first inequality of (3.15) is equivalent to Φ𝑖+𝐿𝑇𝑖𝑅𝑖𝐿𝑖+𝜀𝑖1𝑃𝑖𝑀𝑖𝑀𝑇𝑖𝑃𝑖+𝜀𝑖𝑁𝐴𝑖𝑁𝐵𝑖𝐿𝑖𝑇𝑁𝐴𝑖𝑁𝐵𝑖𝐿𝑖<0.(3.19) Note the definitions of (3.3) and (3.17). By virtue of Lemma 3.2, inequality (3.19) is equivalent to 𝐴𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖+𝐸𝑇𝑖𝑈𝑖𝐸𝑖+𝑄𝑖+𝐿𝑇𝑖𝑅𝑖𝐿𝑖<0.(3.20) Along a similar proof technique as used above, it can also be shown that the first inequality of (3.16) is equivalent to 𝐴𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖+𝐸𝑇𝑖𝑈𝑖𝐸𝑖+𝑄𝑖+𝐿𝑇𝑖𝑅𝑖𝐿𝑖<0,(3.21) where 𝑄𝑖=diag{𝑄𝑖,0}. Note that 𝑄𝑖>0 and 𝑅𝑖>0. By Lemma 3.1, it is obviously shown from inequalities (3.20), (3.21), and the second inequalities of (3.15) and (3.16) that the closed-loop system (3.11) is stable.
In addition, it can be seen from inequalities (3.20) and (3.21) that𝐴𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖+Δ𝐴𝑖𝐵𝑖+Δ𝐵𝑖𝐿𝑖+𝐸𝑇𝑖𝑈𝑖𝐸𝑖+𝑄𝑖+𝐿𝑇𝑖𝑅𝑖𝐿𝑖0,𝑖𝕀.(3.22) Multiplying from left and right by 𝐱𝑇and 𝐱, respectively, and removing the nonnegative term 𝐱𝑇𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝐱render dd𝑡𝐱𝑇𝑃𝑖𝐱+𝐱𝑇𝑄𝑖𝐱+𝐮𝑇𝑅𝑖𝐮0.(3.23) Integration from 0 to , and noticing the global stability of closed-loop system (3.11), gives the result of (3.18). The proof is thus completed.

It is shown that the matrix inequalities (3.15) and (3.16) are BMIs due to the bilinear forms of 𝑃𝑖𝐵𝑖𝐿𝑖 and 𝜀𝑖𝐿𝑖 when both the Lyapunov matrix 𝑃𝑖 and the feedback gain matrix 𝐿𝑖 become the variables to be determined. Our interest is to find a parameter matrix 𝐿 to minimize the upper bound 𝐱𝑇0𝑃𝑖0𝐱0 on the cost function (3.5) for the state feedback closed-loop system (3.11). Then, the optimization problem can be formulated as min𝐿,𝑆,𝑈𝑖,𝑊𝑖,𝜀𝑖𝐱𝑇𝟎𝑃𝑖0𝐱0s.t.𝐿𝑖𝕃(3.15)-(3.16),(3.24) where 𝑖𝕀, and 𝕃 is the set of admissible values for the state feedback gain matrix 𝐿𝑖, bounded by practical design constraints.

Remark 3.4. It should be noted that the optimization problem (3.24) is a nonconvex optimization problem with the BMIs constraints of (3.15) and (3.16). For BMIs problem, we [28] recently have already designed a mixed algorithm combining genetic algorithm (GA) and interior point method to solve it. Here, we can use the mixed algorithm proposed in [28] to obtain the optimal controller parameter matrix L and the corresponding objective 𝐱𝑇0𝑃𝑖0𝐱0. In general, one can set the parameter matrix L to be the decision variables searched by GA. For a given chromosome corresponding to L, the nonconvex problem (3.24) reduces to an SDP involving LMIs which can be solved efficiently by Matlab LMI toolbox.

Remark 3.5. It should be pointed out that when solving the BMIs problem which is an NP hard problem in essence, the mixed algorithm combining GA with the interior point method may suffer from long computational time, especially for high-dimensional systems. Therefore, the optimal control problem can only be solved offline. In addition, the approximation error introduced by the linearization procedure for the chaotic system in Section 2 may adversely impact the stability analysis of the closed-loop system. To overcome this negative impact, one can divide the state space into a more sophisticated partition, but this will also increase the computational burden. Thus, one should seek a balance between the solution accuracy and the computational burden. On the other hand, for the chaotic systems there exists at least a bounded attractor. Due to the boundedness of the chaotic attractor, a relatively fine partition can be achieved to reduce the approximation error in the piecewise linearization procedure, which leads to a controller with a good performance.

To tell if the solutions obtained above are close to optimality or not, we must set up a lower bound on cost function (3.5). The result is presented as follows.

Theorem 3.6. If there exist a set of constants 𝜀𝑖>0 and symmetric matrices 𝑆 and 𝑈𝑖 such that 𝑈𝑖 have nonnegative entries, while 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖, 𝑖𝕀0 and 𝑃𝑖=𝐹𝑇𝑖𝑆𝐹𝑖, 𝑖𝕀1 satisfy Ψ𝑖𝑃𝑖𝐵𝑖𝜀𝑖𝑁𝑇𝐴𝑖𝑁𝐵𝑖𝑃𝑖𝑀𝑖𝐵𝑇𝑖𝑃𝑖𝜀𝑖𝑁𝑇𝐵𝑖𝑁𝐴𝑖𝑅𝑖𝜀𝑖𝑁𝑇𝐵𝑖𝑁𝐵𝑖0𝑀𝑇𝑖𝑃𝑖0𝜀𝑖𝐼>0,(3.25) for 𝑖𝕀0, Ψ𝑖𝑃𝑖𝐵𝑖𝜀𝑖𝑁𝑇𝐴𝑖𝑁𝐵𝑖𝑃𝑖𝑀𝑖𝐵𝑇𝑖𝑃𝑖𝜀𝑖𝑁𝑇𝐵𝑖𝑁𝐴𝑖𝑅𝑖𝜀𝑖𝑁𝑇𝐵𝑖𝑁𝐵𝑖0𝑀𝑇𝑖𝑃𝑖0𝜀𝑖𝐼>0,(3.26) for 𝑖𝕀1, where Ψ𝑖=𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+𝑄𝑖𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝜀𝑖𝑁𝑇𝐴𝑖𝑁𝐴𝑖,Ψ𝑖=𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+𝑄𝑖𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝜀𝑖𝑁𝑇𝐴𝑖𝑁𝐴𝑖,(3.27) then for every trajectory 𝐱(𝑡) of the uncertain system (3.4) with 𝐱()=0, 𝐱(0)=𝐱0𝑋𝑖0, the cost function (3.5) satisfies 𝐽sup𝑆,𝑈𝑖,𝜀𝑖𝐱𝑇0𝑃𝑖0𝐱0.(3.28)

Proof. We will first show the conditions for the cost function (3.5) satisfying the lower bound (3.28) can be guaranteed by𝐴𝑖+Δ𝐴𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖+Δ𝐴𝑖+𝑄𝑖𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝑃𝑖𝐵𝑖+Δ𝐵𝑖𝐵𝑖+Δ𝐵𝑖𝑇𝑃𝑖𝑅𝑖>0,(3.29) for 𝑖𝕀0, and 𝐴𝑖+Δ𝐴𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖+Δ𝐴𝑖+𝑄𝑖𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝑃𝑖𝐵𝑖+Δ𝐵𝑖𝐵𝑖+Δ𝐵𝑖𝑇𝑃𝑖𝑅𝑖>0,(3.30) for 𝑖𝕀1.
Actually, for 𝑖𝕀, we can get from (3.29) and (3.30) that𝐴𝑖+Δ𝐴𝑖𝑇𝑃𝑖+𝑃𝑖𝐴𝑖+Δ𝐴𝑖+𝑄𝑖𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝑃𝑖𝐵𝑖+Δ𝐵𝑖𝐵𝑖+Δ𝐵𝑖𝑇𝑃𝑖𝑅𝑖0.(3.31) Multiplying from left and right by [𝐱𝑇,𝐮𝑇] and [𝐱𝑇,𝐮𝑇]𝑇, respectively, and removing the nonnegative term 𝐱𝑇𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝐱yield 02𝐱𝑇𝑃𝑖𝐴𝑖+Δ𝐴𝑖𝐱+𝐵𝑖+Δ𝐵𝑖𝐮+𝐱𝑇𝑄𝑖𝐱+𝐮𝑇𝑅𝑖𝐮=dd𝑡𝐱𝑇𝑃𝑖𝐱+𝐱𝑇𝑄𝑖𝐱+𝐮𝑇𝑅𝑖𝐮.(3.32) Integration from 0 to , and noticing 𝐱()=0, gives the result of (3.28).
Next, we will show that inequality (3.29) is equivalent to (3.25). For simplifying the presentation, denote𝐴𝐺=𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+𝑄𝑖𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝑃𝑖𝐵𝑖𝐵𝑇𝑖𝑃𝑖𝑅𝑖.(3.33) Note the uncertain form (3.2). Then, inequality (3.29) can be written as 𝑃𝐺+𝑖𝑀𝑖0𝐻𝑁𝐴𝑖,𝑁𝐵𝑖+𝑁𝐴𝑖,𝑁𝐵𝑖𝑇𝐻𝑇𝑃𝑖𝑀𝑖0𝑇>0.(3.34) By Lemma 3.2, inequality (3.34) is equivalent to the existence of some 𝜀𝑖>0 such that 𝐺𝜀𝑖1𝑃𝑖𝑀𝑖0𝑃𝑖𝑀𝑖0𝑇𝜀𝑖𝑁𝐴𝑖,𝑁𝐵𝑖𝑇𝑁𝐴𝑖,𝑁𝐵𝑖>0,(3.35) that is, 𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐴𝑖+𝑄𝑖𝐸𝑇𝑖𝑈𝑖𝐸𝑖𝜀𝑖𝑁𝑇𝐴𝑖𝑁𝐴𝑖𝜀𝑖1𝑃𝑖𝑀𝑖𝑀𝑇𝑖𝑃𝑖𝑃𝑖𝐵𝑖𝜀𝑖𝑁𝑇𝐴𝑖𝑁𝐵𝑖𝐵𝑇𝑖𝑃𝑖𝜀𝑖𝑁𝑇𝐵𝑖𝑁𝐴𝑖𝑅𝑖𝜀𝑖𝑁𝑇𝐵𝑖𝑁𝐵𝑖>0,(3.36) which, by Schur complement, is equivalent to inequality (3.25). By similar techniques, it can also be shown that inequality (3.30) is equivalent to inequality (3.26). The proof is complete.

Remark 3.7. It is shown that inequalities (3.25) and (3.26) are LMIs about the variables 𝑃𝑖, 𝑃𝑖, and 𝜀𝑖. So the problem of maximizing the lower bound (3.28) can be cast as an SDP with LMIs constraints of (3.25) and (3.26), and solved numerically effectively.

Remark 3.8. In the above analysis, it is assumed that the initial condition 𝐱0 is given or known in advance. Note that the bounds in (3.18) and (3.28) depend on the initial state 𝐱0. To remove this dependence on the initial state, we can use the techniques developed in [28] and extend the corresponding results to the case where the initial condition 𝐱0 is a random variable subjected to uniform distribution on a certain bounded region 𝑋0. For further details, please refer to [28].

The global quadratic Lyapunov function technique is often applied in the control synthesis of dynamical systems [26]. In the following, by virtue of the global quadratic Lyapunov function technique and linear feedback control law, we present an optimal guaranteed cost control method for the chaotic system (2.1) associated with the cost function (2.2), which with the comparisons in the simulation results will show advantages of the obtained results in Theorems 3.3 and 3.6.

Consider the following linear feedback control law: 𝐮=𝐿𝐱.(3.37) Substituting the control law (3.37) into system (2.1), we can get the following closed-loop system: ̇𝐿𝐱(𝑡)=𝐴𝐵𝐱(𝑡)+𝐹(𝐱).(3.38)

Additionally, note the boundedness of the chaotic attractor and the Lipschitz continuity condition for the nonlinear term 𝐹(𝐱). There exist some matrix Γ0 and a bounded set Ω which bounds the chaotic attractor, such that𝐹𝑇(𝐱)𝐹(𝐱)𝐱𝑇Γ2𝐱,𝐱Ω.(3.39)

The upper bound on the cost function (2.2) for the chaotic system (2.1) by applying linear feedback control law (3.37) is presented as follows.

Theorem 3.9. Consider system (2.1) with the initial condition 𝐱0Ω. If there exist positive constants 𝛼, 𝛽, positive definite matrix 𝑌, and any matrix 𝑍 with appropriate dimensions such that 𝐴𝑌+𝑌𝐴𝑇𝐵𝑍𝑍𝑇𝐵𝑇+𝛼𝐼𝑌Γ𝑌𝑍𝑇Γ𝑌𝛼𝐼00𝑌0𝑄10𝑍00𝑅1<0,𝛽𝐱𝑇0𝐱0𝑌<0,(3.40) then the closed-loop system (3.38) is globally exponentially stable, and the cost function (2.2) satisfies 𝐽<𝛽.(3.41) Furthermore, the corresponding control law can be obtained as 𝐮=𝑍𝑌1𝐱.

Proof. Denote 𝑃=𝑌1>0. Construct the Lyapunov function candidate as 𝑉(𝐱)=𝐱𝑇𝑃𝐱.(3.42)
By virtue of the fact that 𝑀𝑇𝑁+𝑁𝑇𝑀𝛼1𝑀𝑇𝑀+𝛼𝑁𝑇𝑁, for all 𝛼>0, and matrices 𝑀 and 𝑁 with appropriate dimensions, calculating the time derivative of 𝑉(𝐱) along the trajectory of the closed-loop system (3.38) and noticing (3.39) yieldd𝑉(𝐱)d𝑡=𝐱𝑇𝐿𝐴𝐵𝑇𝐿𝑃+𝑃𝐴𝐵𝐱+𝐹𝑇(𝐱)𝑃𝐱+𝐱𝑇𝑃𝐹(𝐱)𝐱𝑇𝐿𝐴𝐵𝑇𝐿𝑃+𝑃𝐴𝐵+𝛼𝑃2𝐱+𝛼1𝐹𝑇(𝐱)𝐹(𝐱)𝐱𝑇𝐿𝐴𝐵𝑇𝐿𝑃+𝑃𝐴𝐵+𝛼𝑃2+𝛼1Γ2𝐱.(3.43) On the other hand, by Schur complement, the first inequality of (3.40) is equivalent to 𝐴𝑌+𝑌𝐴𝑇𝐵𝑍𝑍𝑇𝐵𝑇+𝛼𝐼+𝛼1𝑌Γ2𝑌+𝑌𝑄𝑌+𝑍𝑇𝑅𝑍<0.(3.44) Noticing 𝑌𝑃=𝐼, 𝐿=𝑍𝑃, pre- and postmultiplying both sides of (3.44) by 𝑃 implies 𝐿𝐴𝐵𝑇𝐿𝑃+𝑃𝐴𝐵+𝛼𝑃2+𝛼1Γ2𝐿+𝑄+𝑇𝑅𝐿<0.(3.45) Thus, it follows from (3.43) and (3.45) that d𝑉(𝐱)d𝑡+𝐱𝑇𝑄𝐱+𝐱𝑇𝐿𝑇𝑅𝐿𝐱<0.(3.46) Note that 𝑄>0 and 𝑅>0. It is obvious that d𝑉(𝐱)/d𝑡<0 which guarantees the global stability of closed-loop system (3.38), that is, 𝐱()=0.
Integration both sides of (3.46) from 0 to , and noticing 𝑉(𝐱())=0, renders𝐱𝐽<𝑉0=𝐱0𝑌1𝐱0,(3.47) with which combining the second inequality of (3.40) shows the result of (3.41). The proof is complete.

Remark 3.10. It is shown that the inequalities in (3.40) are LMIs in the variables 𝑌, 𝑍, 𝛼, 𝛽. So the problem of minimizing the upper bound (3.41) can be cast as an SDP with LMIs constraints of (3.40) and can be solved numerically effectively. On the other hand, it should be pointed out that the control synthesis methods based on the global quadratic Lyapunov function (3.42) and linear feedback control law (3.37) are conservative in practice compared with those in Theorems 3.3 and 3.6, which will be shown in illustrative examples.

4. Illustrative Examples

In this section, we will give two examples to illustrate the effectiveness of the proposed methods.

4.1. Genesio—Tesi Chaotic System

Consider the Genesio—Tesi chaotic system presented in [23], and the controlled system is described as follows: ̇𝑥1̇𝑥2̇𝑥3=010001𝑝1𝑝2𝑝3𝑥1𝑥2𝑥3+00𝑥21+111𝑢,(4.1) where 𝑝1=6, 𝑝2=2.92, 𝑝3=1.2.

Denote that 𝑐𝑇=[1,0,0] and 𝐱𝑇=[𝑥1,𝑥2,𝑥3]. Note the boundedness of the chaotic attractor shown in [23]. The state space can be confined to 𝑋={𝐱6𝑐𝑇𝐱6} by simulation. The partition of state space is set to be 𝑋1=𝐱𝑐𝑇[𝐱1,1),𝑋2=𝐱𝑐𝑇[𝐱1,3),𝑋3=𝐱𝑐𝑇[],𝑋𝐱3,64=𝐱𝑐𝑇[𝐱3,1),𝑋5=𝐱𝑐𝑇[.𝐱6,3)(4.2) Then, the nonlinear term 𝑥21 can be described as𝑥21=𝑘𝑖𝑥1+𝑙𝑖+𝛿𝑖𝑥1,𝐱𝑋𝑖,𝑖=1,2,3,4,5,(4.3) where 𝛿𝑖 denotes the approximation error. Taking 𝑘1=0, 𝑘2=𝑘4=4.5, 𝑘3=𝑘5=9, 𝑙1=0, 𝑙2=𝑙4=4.5, 𝑙3=𝑙5=18, one can obtain that ||𝛿1𝑥1||||𝑥1||,||𝛿2𝑥1||||𝛿1,3𝑥1||||𝛿2.25,4𝑥1||||𝛿1,5𝑥1||2.25.(4.4)

Note the expressions (4.3) and (4.4). System (4.1) can be converted to the piecewise linear system (3.1) with𝐴1=01000162.921.2,𝐴2=0100011.52.921.2,𝐴3=,𝐴01000132.921.24=01000110.52.921.2,𝐴5=010001152.921.2,𝐵𝑖=111,𝑎1=0,𝑎2=𝑎4=004.5𝑇,𝑎3=𝑎5=0018𝑇,Δ𝐴1=𝑀1𝐻𝑁𝐴1,Δ𝐴2=Δ𝐴3=Δ𝐴4=Δ𝐴5=0,Δ𝐵𝑖=0,Δ𝑎1=0,Δ𝑎2=𝑀2𝐻𝑁𝑎2,Δ𝑎3=𝑀3𝐻𝑁𝑎3,Δ𝑎4=𝑀4𝐻𝑁𝑎4,Δ𝑎5=𝑀5𝐻𝑁𝑎5,𝑀𝑖=002,𝑁𝐴1=,𝑁0000000.500𝑎2=𝑁𝑎4=000.5,𝑁𝑎3=𝑁𝑎5=00,1.125(4.5) where 𝑖=1,,5, and 𝐻 is an uncertain matrix bounded by 𝐻𝑇𝐻𝐼.

It is worthwhile to mention that the nominal autonomous piecewise linear system (3.1) with parameters (4.5), that is, 𝑢0, Δ𝐴𝑖=0, Δ𝑎𝑖=0, can exhibit chaotic dynamics, and the strange attractor is depicted in Figure 1. It is shown from Figure 1 that the system (3.1) with parameters (4.5) evolves to a single-scroll chaotic attractor, which is similar to the Genesio-Tesi chaotic attractor. Thus, it is indicated that the piecewise linear system approximating a chaotic system can preserve the complex dynamic behaviors of the original system.

859542.fig.001
Figure 1: Phase portraits of the nominal autonomous system (3.1) with parameters (4.5).

Consider the cost function (2.2) with 𝑄=diag{1,1,1}, 𝑅=1, and the initial value 𝐱0=[1.8,1,1]𝑇 of system (4.1). The matrices 𝐸𝑖 and 𝐹𝑖 can be constructed by virtue of the method proposed in [20]. Assume that the feedback gain matrix 𝐿𝑖 is bounded by 𝐿𝑖<12, where 𝐿𝑖 denotes the largest absolute value among all the entries of vector 𝐿𝑖. Then, applying the mixed algorithm provided in [28], we solve the BMIs problem (3.24) based on Theorem 3.3 with the code written in MATLAB 7.0 and get the optimal upper bound on 𝐽, denoted as 𝐽, and the corresponding optimal parameter matrix 𝐿 as follows: 𝐽𝐿=17.7528,=3.3088,0.3345,2.7111,1.2363,0.0548𝑇.(4.6) According to the expression of (3.10), we can get the following state feedback gain matrices: 𝐿1=,2.7111,1.2363,0.0548𝐿2=,2.3765,1.2363,0.0548,0.3345𝐿3=,6.0198,1.2363,0.0548,10.5954𝐿4=,6.0198,1.2363,0.0548,3.3088𝐿5=,2.3765,1.2363,0.0548,7.6212(4.7) with which the optimal control 𝑢 taking the form of piecewise linear feedback control law (3.9) can be obtained.

Actually, the cost function (2.2) for the closed-loop system (4.1) with above controller gain matrices is computed as 𝐽=13.1623. The numerical simulation of system (4.1) with the piecewise linear state feedback control is shown in Figure 2.

859542.fig.002
Figure 2: Time response of system (4.1) with the piecewise linear state feedback.

In addition, according to Theorem 3.6, the maximal lower bound on 𝐽, denoted as 𝐽, can be obtained by solving the corresponding SDP with the LMI toolbox in MATLAB 7.0 as follows:𝐽=10.2047.(4.8)

On the other hand, note that 6𝑐𝑇𝐱6. The matrix Γ in (3.39) can be obtained as Γ=diag{6,0,0}. According to Theorem 3.9, we solve the corresponding SDP, and obtain the optimal gain matrix 𝐿 in (3.37) and upper bound 𝛽 as follows: 𝐿=𝑍𝑌1=14.1843,1.1514,0.3980,𝛽=53.0699,(4.9) which shows a fact that the optimal control methods based on the global quadratic Lyapunov function are conservative compared with those in Theorem 3.3.

4.2. A New Chaotic System

Consider the new chaotic system presented in [26], and the controlled system is described as follows: ̇𝑥1̇𝑥2̇𝑥3=010001𝑝1𝑝1𝑝1𝑥1𝑥2𝑥3+00𝑝2𝑥tanh1+110.5𝑢,(4.10) where 𝑝1=0.5, 𝑝2=5, and the hyperbolic function tanh(𝑥)=(exp(𝑥)exp(𝑥))/(exp(𝑥)+exp(𝑥)). The strange attractor of the autonomous system (4.10) with 𝑢0 is shown in Figure 3, which is a double-scroll chaotic attractor.

859542.fig.003
Figure 3: Phase portraits of the autonomous system (4.10).

Note the boundedness of the chaotic attractor shown in Figure 3. The state space can be confined to 𝑋={𝐱23.3𝑐𝑇𝐱23.3} by simulation. The partition of state space is set to be 𝑋1=𝐱𝑐𝑇[𝐱23.3,1.18),𝑋2=𝐱𝑐𝑇[,𝑋𝐱1.18,1.18)3=𝐱𝑐𝑇[].𝐱1.18,23.3(4.11) Then, the nonlinear term tanh(𝑥1) can be described as𝑥tanh1=𝑘1𝑥1+𝑙1+𝛿1𝑥1,𝐱𝑋1,𝑘2𝑥1+𝑙2+𝛿2𝑥1,𝐱𝑋2,𝑘3𝑥1+𝑙3+𝛿3𝑥1,𝐱𝑋3,(4.12) where 𝛿𝑖 denotes the approximation error. Taking 𝑘1=0, 𝑘2=0.85, 𝑘3=0, 𝑙1=1, 𝑙2=0, 𝑙3=1, one can obtain that||𝛿1𝑥1||||𝛿0.17,2𝑥1||||𝑥0.151||,||𝛿3𝑥1||0.17.(4.13)

Note the expressions (4.12) and (4.13). System (4.10) can be converted to the piecewise linear system (3.1) with 𝐴1=𝐴3=010001𝑝1𝑝1𝑝1,𝐴2=010001𝑝1+0.85𝑝2𝑝1𝑝1,𝐵1=𝐵2=𝐵3=110.5,𝑎1=𝑎3=00𝑝2,𝑎2=0,Δ𝐴1=Δ𝐴3=0,Δ𝐴2=𝑀2𝐻𝑁𝐴2,Δ𝐵1=Δ𝐵2=Δ𝐵3=0,Δ𝑎1=𝑀1𝐻𝑁𝑎1,Δ𝑎2=0,Δ𝑎3=𝑀3𝐻𝑁𝑎3,𝑀1=𝑀2=𝑀3=001,𝑁𝐴2=0000000.15𝑝200,𝑁𝑎1=𝑁𝑎3=000.17𝑝2.(4.14) Consider the cost function (2.2) with 𝑄=diag{0.8,0.8,0.8},𝑅=1.2, and the system initial value 𝐱0=[1.4,1,0.6]𝑇. Assume that the feedback gain matrix 𝐿𝑖 is bounded by 𝐿𝑖<8. Then, similarly to the above subsection, we get the maximal lower bound 𝐽, the optimal upper bound 𝐽, and the corresponding optimal parameter matrix 𝐿 as follows: 𝐽=5.5117,𝐽𝐿=9.7024,=4.3655,1.2292,2.3555,1.3460,1.4357𝑇.(4.15) According to the expression of (3.10), we can get the following state feedback gain matrices: 𝐿1=6.7210,1.3460,1.4357,5.1513,𝐿2=,2.3555,1.3460,1.4357𝐿3=,3.5847,1.3460,1.4357,1.4505(4.16) with which the optimal control 𝑢 taking the form of (3.9) is obtained.

Additionally, the cost function (2.2) for the closed-loop system (4.10) with above controller gain matrices is computed as 𝐽=7.8725. The numerical simulation of system (4.10) with piecewise linear state feedback control is shown in Figure 4.

859542.fig.004
Figure 4: The control law and time response of the controlled system (4.10).

Furthermore, note that tanh2(𝑥1)𝑥21. The matrix Γ in (3.39) can be obtained as Γ=diag{𝑝2,0,0}. According to Theorem 3.9, we solve the corresponding SDP, and obtain the optimal gain matrix 𝐿 in (3.37) and upper bound 𝛽 as follows: 𝐿=𝑍𝑌1=19.2415,2.5432,0.2071,𝛽=98.965,(4.17) which is significantly greater than the optimal upper bound 𝐽 obtained from Theorem 3.3.

It is obviously shown from the above examples that the optimal upper bounds 𝐽 obtained above get close to the corresponding lower bounds 𝐽, respectively. This implies that we have achieved or got close to the optimal control for the chaotic systems. Additionally, it should be pointed out that the newly reported chaotic system (4.10) is topologically not equivalent to the Genesio-Tesi chaotic system (4.1). However, by virtue of the optimal control methods proposed in this paper, both the different chaotic systems (4.1) and (4.10) can be optimally stabilized. The examples show the effectiveness of the proposed results.

5. Conclusion

In this paper, we first convert a class of chaotic systems to the form of uncertain piecewise linear systems then investigate the optimal control for the chaotic systems where the piecewise linear state feedback optimal controller can be obtained by solving an optimization problem with BMIs constraints. The performance of the controller can be evaluated by the upper and lower bounds on the cost function. The optimal chaos synchronization for this class of chaotic systems will be studied in the near future.

Acknowledgment

The authors thank the anonymous referees and editor for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (no. 61004015), the Research Fund for the Doctoral Programme of Higher Education of China (no. 20090032120034), the Program for New Century Excellent Talents in Universities of China, and the Program for Changjiang Scholars and Innovative Research Team in University of China (no. IRT1028).

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