Optimal Control for a Class of Chaotic Systems

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 [14–17]. 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 𝐴−𝐵>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: 𝐱̇=𝐴𝐱+𝐹(𝐱)+𝐵𝐮,(2.1) where 𝐴 and 𝐵 are constant matrices, 𝐱∈ℜ^𝑛 is the state vector, 𝐮∈ℜ^𝑚(𝑚≤𝑛) is the control input variable, and the nonlinear term 𝐹(𝐱)∈ℜ^𝑛 is assumed to satisfy Lipschitz continuity condition, uniform or local, and 𝐹(0)=0.

Associated with this system is the cost function: 𝐽=^∞0𝐱^𝑇(𝑡)𝑄𝐱(𝑡)+𝐮^𝑇(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)=𝐾_𝑖𝐱+𝑎_𝑖+Δ_𝑖(𝐱),𝐱∈𝑋_𝑖,𝑖∈𝕀,(2.3) where 𝐾_𝑖∈ℜ^𝑛×𝑛, 𝑎_𝑖∈ℜ^𝑛 are some given parameters, {𝑋_𝑖}_𝑖∈𝕀⊆ℜ^𝑛 denotes a partition of the state space of chaotic system, 𝕀 is the index set, and Δ_𝑖(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: 𝐱̇=𝐴+𝐾_𝑖𝐱+𝑎_𝑖+Δ_𝑖(𝐱)+𝐵𝐮,𝐱∈𝑋_𝑖,𝑖∈𝕀.(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: 𝐹(𝐱)=0,0,𝑓𝑥₁^𝑇,(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 𝑓𝑥₁=𝑘_𝑖𝑥₁+𝑙_𝑖+𝛿_𝑖𝑥₁,𝐱∈𝑋_𝑖,𝑖∈𝕀,(2.6)where 𝑘_𝑖,𝑙_𝑖∈ℜ are some given parameters, 𝛿_𝑖(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 𝐱̇=𝐴_𝑖𝐱+𝑎_𝑖+Δ_𝑖+𝐵𝐮,𝐱∈𝑋_𝑖,𝑖∈𝕀(2.7) with𝐴_𝑖=𝐴+⎡⎢⎢⎢⎢⎣𝑘000000_𝑖00⎤⎥⎥⎥⎥⎦,𝑎^𝑖=⎡⎢⎢⎢⎢⎣𝑙00_𝑖⎤⎥⎥⎥⎥⎦,Δ_𝑖=⎡⎢⎢⎢⎢⎣𝛿_𝑖00𝑥₁⎤⎥⎥⎥⎥⎦.(2.8)

3. State Feedback Optimal Control of Systems

Without loss of generality, consider the uncertain piecewise linear system of the form 𝐱̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖𝐱(𝑡)+𝐵_𝑖+Δ𝐵_𝑖𝐮(𝑡)+𝑎_𝑖+Δ𝑎_𝑖(3.1) for 𝐱(𝑡)∈𝑋_𝑖, where {𝑋_𝑖}_𝑖∈𝕀⊆ℜ^𝑛 denotes a partition of the state space into a number of polyhedral cells, 𝕀 is the index set of the cells, (𝐴_𝑖,𝐵_𝑖,𝑎_𝑖) is the 𝑖th nominal local model of the system, 𝑎_𝑖 is the offset term. Δ𝐴_𝑖, Δ𝐵_𝑖, and Δ𝑎_𝑖 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: Δ𝐴_𝑖,Δ𝐵_𝑖,Δ𝑎_𝑖=𝑀^𝑖𝐻𝑁_𝐴𝑖,𝑁_𝐵𝑖,𝑁_𝑎𝑖,(3.2) where 𝐻∈ℜ^𝑖×𝑗 is an uncertain matrix bounded by 𝐻^𝑇𝐻≤𝐼, and 𝑀_𝑖, 𝑁_𝐴𝑖, 𝑁_𝐵𝑖, 𝑁_𝑎𝑖 are known constant matrices of appropriate dimensions which specify how the elements of the nominal matrices 𝐴_𝑖,𝐵_𝑖, and 𝑎_𝑖 are affected by the uncertain parameters in 𝐻.

Define 𝕀₀⊆𝕀 as the set of indices for cells that contain origin and 𝕀₁⊆𝕀 the set of indices for cells that do not contain the origin. It is assumed that 𝑎_𝑖=Δ𝑎_𝑖=0 for all 𝑖∈𝕀₀.

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 𝐴_𝑖𝐱across 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: 𝐱=⎡⎢⎢⎣𝐱1⎤⎥⎥⎦,𝐴_𝑖=⎡⎢⎢⎣𝐴^𝑖𝑎_𝑖00⎤⎥⎥⎦,𝐵_𝑖=⎡⎢⎢⎣𝐵^𝑖0⎤⎥⎥⎦,𝑀_𝑖=⎡⎢⎢⎣𝑀^𝑖0⎤⎥⎥⎦,Δ𝑁_𝐴𝑖=𝑁_𝐴𝑖,𝑁_𝑎𝑖,𝐴_𝑖=⎡⎢⎢⎣Δ𝐴^𝑖Δ𝑎_𝑖00⎤⎥⎥⎦=𝑀_𝑖𝐻𝑁_𝐴𝑖,Δ𝐵_𝑖=⎡⎢⎢⎣Δ𝐵^𝑖0⎤⎥⎥⎦=𝑀_𝑖𝐻𝑁_𝐵𝑖,(3.3) then system (3.1) can be expressed as 𝐱̇‌(𝑡)=𝐴_𝑖+Δ𝐴_𝑖𝐱(𝑡)+𝐵_𝑖+Δ𝐵_𝑖𝐮(𝑡),𝑖∈𝕀.(3.4)

Associated with this system is the following cost function: 𝐽=^∞0𝐱^𝑇(𝑡)𝑄_𝑖𝐱(𝑡)+𝐮^𝑇(𝑡)𝑅_𝑖𝐮(𝑡)d𝑡,(3.5) where 𝑖 is defined so that 𝐱(𝑡)∈𝑋_𝑖, and 𝑄_𝑖>0, 𝑅_𝑖>0 are given weighting matrices.

Note that if 𝑄_𝑖,𝑅_𝑖 in (3.5) are set to be the same, respectively, for every 𝑖∈𝕀, the cost function (3.5) will reduce to (2.2). In addition, the matrix 𝑄_𝑖=diag{𝑄_𝑖,0}∈ℜ^{(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 𝐹_𝑖=[𝐹_𝑖,𝑓_𝑖], 𝑖∈𝕀 with 𝑓_𝑖=0 for 𝑖∈𝕀₀ should be constructed, which are used to characterize the boundaries between the regions: 𝐹_𝑖𝐱=𝐹_𝑗𝐱,𝐱∈𝑋_𝑖∩𝑋_𝑗,𝑖,𝑗∈𝕀.(3.6) Then, the piecewise Lyapunov function candidates that are continuous across the region boundaries can be parameterized as 𝑉(𝐱)=𝐱^𝑇𝑃_𝑖𝐱,𝐱∈𝑋_𝑖,𝑖∈𝕀₀,𝐱^𝑇𝑃_𝑖𝐱,𝐱∈𝑋_𝑖,𝑖∈𝕀₁,(3.7) with 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖 and 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖, where 𝑆 is a symmetric matrix which characterizes the free parameters of the Lyapunov function candidates.

Note the form of 𝑃_𝑖 and the characteristics of the matrices 𝐹_𝑖. 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 𝐸_𝑖=[𝐸_𝑖,𝑒_𝑖], 𝑖∈𝕀 with 𝑒_𝑖=0 for 𝑖∈𝕀₀, such that 𝐸_𝑖𝐱≥0,𝐱∈𝑋_𝑖,𝑖∈𝕀,(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 𝐸_𝑖, 𝐹_𝑖 for a given piecewise linear system was suggested in [20].

Consider the following piecewise linear feedback control law: 𝐮=−𝐿_𝑖𝐱−𝑙_𝑖∶=−𝐿_𝑖𝐱,𝐱∈𝑋_𝑖,𝑖∈𝕀,(3.9) with 𝑙_𝑖=0 for 𝑖∈𝕀₀.

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 𝐿_𝑖 as follows 𝐿_𝑖=𝐿^𝑇𝐹_𝑖,𝑖∈𝕀,(3.10) where 𝐿 is a parameter matrix characterizing the free parameters of the state feedback controller, and 𝐹_𝑖 is the matrix defined in (3.6). It should be pointed out that the gain matrix 𝐿_𝑖 should take the form of 𝐿_𝑖=𝐿^𝑇𝐹_𝑖 for 𝑖∈𝕀₀.

Substituting the control law (3.9) into system (3.4), we can get the following closed-loop system: 𝐱̇̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖𝐱(𝑡),for𝑖∈𝕀₀,𝐱(𝑡)=𝐴_𝑖+Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖𝐱(𝑡),for𝑖∈𝕀₁.(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, 𝑈_𝑖, and 𝑊_𝑖 such that 𝑈_𝑖 and 𝑊_𝑖 have nonnegative entries, while 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖, 𝑖∈𝕀₀ and 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖, 𝑖∈𝕀₁, satisfy 𝐴^𝑇𝑖𝑃_𝑖+𝐸𝑃_𝑖𝐴_𝑖+𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖<0,_𝑇𝑖𝑊_𝑖𝐸_𝑖<𝑃_𝑖,(3.12) for 𝑖∈𝕀₀, and 𝐴^𝑇𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖<0,𝐸^𝑇𝑖𝑊_𝑖𝐸_𝑖<𝑃_𝑖,(3.13) for 𝑖∈𝕀₁, then every continuous and piecewise 𝐶¹ trajectory 𝐱(𝑡) of system (3.4) with Δ𝐴_𝑖=0, Δ𝑎_𝑖=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+𝑁^𝑇𝐻^𝑇𝑀^𝑇<0 for all matrices 𝐻 satisfying 𝐻^𝑇𝐻≤𝐼, if and only if there exists some 𝜀>0 such that 𝐺+𝜀⁻¹𝑀𝑀^𝑇+𝜀𝑁^𝑇𝑁<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 𝐱₀∈𝑋_𝑖0. If there exist a set of constants 𝜀_𝑖>0 and symmetric matrices 𝑆, 𝑈_𝑖, and 𝑊_𝑖 such that 𝑈_𝑖 and 𝑊_𝑖 have nonnegative entries, while 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖, 𝑖∈𝕀₀, and 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖, 𝑖∈𝕀₁, satisfy ⎡⎢⎢⎢⎢⎢⎢⎣Φ_𝑖𝜀_𝑖𝑁_𝐴𝑖−𝑁_𝐵𝑖𝐿_𝑖^𝑇𝑃_𝑖𝑀_𝑖𝐿^𝑇𝑖𝜀_𝑖𝑁𝑀_𝐴𝑖−𝑁_𝐵𝑖𝐿_𝑖−𝜀_𝑖𝐼00_𝑇𝑖𝐿𝑃_𝑖0−𝜀_𝑖𝐼0_𝑖00−𝑅^𝑖−1𝐸⎤⎥⎥⎥⎥⎥⎥⎦<0,_𝑇𝑖𝑊_𝑖𝐸_𝑖<𝑃_𝑖(3.15) for 𝑖∈𝕀₀, ⎡⎢⎢⎢⎢⎢⎢⎢⎣Φ_𝑖𝜀_𝑖𝑁_𝐴𝑖−𝑁_𝐵𝑖𝐿_𝑖^𝑇𝑃_𝑖𝑀_𝑖𝐿^𝑇𝑖𝜀_𝑖𝑁_𝐴𝑖−𝑁_𝐵𝑖𝐿_𝑖−𝜀^𝑖𝐼00𝑀^𝑇𝑖𝑃_𝑖0−𝜀_𝑖𝐼0𝐿_𝑖00−𝑅^𝑖−1⎤⎥⎥⎥⎥⎥⎥⎥⎦<0𝐸^𝑇𝑖𝑊_𝑖𝐸_𝑖<𝑃_𝑖(3.16) for 𝑖∈𝕀₁, where Φ_𝑖∶=𝐴_𝑖−𝐵_𝑖𝐿_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖−𝐵_𝑖𝐿_𝑖+𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖+𝑄_𝑖,Φ_𝑖∶=𝐴_𝑖−𝐵_𝑖𝐿_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖−𝐵_𝑖𝐿_𝑖+𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖+𝑄_𝑖,(3.17) then the closed-loop system is globally exponentially stable, and the cost function (3.5) satisfies 𝐽≤inf_{𝑆,𝑈𝑖,𝑊𝑖,𝜀𝑖}𝐱^𝑇0𝑃_𝑖0𝐱₀.(3.18)

Proof. By Schur complement [19], the first inequality of (3.15) is equivalent to Φ_𝑖+𝐿^𝑇𝑖𝑅_𝑖𝐿_𝑖+𝜀^𝑖−1𝑃_𝑖𝑀_𝑖𝑀^𝑇𝑖𝑃_𝑖+𝜀_𝑖𝑁_𝐴𝑖−𝑁_𝐵𝑖𝐿_𝑖^𝑇𝑁_𝐴𝑖−𝑁_𝐵𝑖𝐿_𝑖<0.(3.19) Note the definitions of (3.3) and (3.17). By virtue of Lemma 3.2, inequality (3.19) is equivalent to 𝐴_𝑖+Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖+𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖+𝑄_𝑖+𝐿^𝑇𝑖𝑅_𝑖𝐿_𝑖<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 𝐴_𝑖+Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖+𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖+𝑄_𝑖+𝐿^𝑇𝑖𝑅_𝑖𝐿_𝑖<0(,3.21) where 𝑄_𝑖=diag{𝑄_𝑖,0}. Note that 𝑄_𝑖>0 and 𝑅_𝑖>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) that𝐴_𝑖++Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖−𝐵_𝑖+Δ𝐵_𝑖𝐿_𝑖𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖+𝑄_𝑖+𝐿^𝑇𝑖𝑅_𝑖𝐿_𝑖≤0,𝑖∈𝕀.(3.22) Multiplying from left and right by 𝐱^𝑇and 𝐱, respectively, and removing the nonnegative term 𝐱^𝑇𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖𝐱render dd𝑡𝐱^𝑇𝑃_𝑖𝐱+𝐱^𝑇𝑄_𝑖𝐱+𝐮^𝑇𝑅_𝑖𝐮≥0.(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 𝑃_𝑖𝐵_𝑖𝐿_𝑖 and 𝜀_𝑖𝐿_𝑖 when both the Lyapunov matrix 𝑃_𝑖 and the feedback gain matrix 𝐿_𝑖 become the variables to be determined. Our interest is to find a parameter matrix 𝐿 to minimize the upper bound 𝐱^𝑇0𝑃_𝑖0𝐱₀ on the cost function (3.5) for the state feedback closed-loop system (3.11). Then, the optimization problem can be formulated as min_{𝐿,𝑆,𝑈𝑖,𝑊𝑖,𝜀𝑖}𝐱^𝑇𝟎𝑃_𝑖0𝐱₀s.t.𝐿(3_𝑖.1∈5)𝕃-(3.16),(3.24) where 𝑖∈𝕀, and 𝕃 is the set of admissible values for the state feedback gain matrix 𝐿_𝑖, 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 𝐱^𝑇0𝑃_𝑖0𝐱₀. 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 𝜀_𝑖>0 and symmetric matrices 𝑆 and 𝑈_𝑖 such that 𝑈_𝑖 have nonnegative entries, while 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖, 𝑖∈𝕀₀ and 𝑃_𝑖=𝐹^𝑇𝑖𝑆𝐹_𝑖, 𝑖∈𝕀₁ satisfy ⎡⎢⎢⎢⎢⎢⎣Ψ_𝑖𝑃_𝑖𝐵_𝑖−𝜀_𝑖𝑁^𝑇𝐴_𝑖𝑁_𝐵𝑖𝑃_𝑖𝑀_𝑖𝐵^𝑇𝑖𝑃_𝑖𝑀−𝜀_𝑖𝑇𝑖𝑁^𝑇𝐵_𝑖𝑁_𝐴𝑖𝑅_𝑖−𝜀_𝑖𝑁^𝑇𝐵_𝑖𝑁_𝐵𝑖0𝑃_𝑖0𝜀_𝑖𝐼⎤⎥⎥⎥⎥⎥⎦>0,(3.25) for 𝑖∈𝕀₀, ⎡⎢⎢⎢⎢⎣Ψ_𝑖𝑃_𝑖𝐵_𝑖−𝜀_𝑖𝑁^𝑇𝐴_𝑖𝑁_𝐵𝑖𝑃_𝑖𝑀_𝑖𝐵^𝑇𝑖𝑃_𝑖−𝜀_𝑖𝑁^𝑇𝐵_𝑖𝑁_𝐴𝑖𝑅_𝑖−𝜀_𝑖𝑁^𝑇𝐵_𝑖𝑁_𝐵𝑖0𝑀^𝑇𝑖𝑃_𝑖0𝜀_𝑖𝐼⎤⎥⎥⎥⎥⎦>0,(3.26) for 𝑖∈𝕀₁, where Ψ_𝑖∶=𝐴^𝑇𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖−𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖−𝜀_𝑖𝑁^𝑇𝐴_𝑖𝑁_𝐴𝑖,Ψ_𝑖∶=𝐴^𝑇𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖−𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖−𝜀_𝑖𝑁^𝑇𝐴_𝑖𝑁_𝐴𝑖,(3.27) then for every trajectory 𝐱(𝑡) of the uncertain system (3.4) with 𝐱(∞)=0, 𝐱(0)=𝐱₀∈𝑋_𝑖0, the cost function (3.5) satisfies 𝐽≥sup_{𝑆,𝑈𝑖,𝜀𝑖}𝐱^𝑇0𝑃_𝑖0𝐱₀.(3.28)

Proof. We will first show the conditions for the cost function (3.5) satisfying the lower bound (3.28) can be guaranteed by⎡⎢⎢⎣𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+𝑃𝐵_𝑖_𝑖𝐴_𝑖+Δ𝐴_𝑖+𝑄_𝑖−𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃_𝑖𝑅_𝑖⎤⎥⎥⎦>0,(3.29) for 𝑖∈𝕀₀, and ⎡⎢⎢⎣𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖+𝑄_𝑖−𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃_𝑖𝑅_𝑖⎤⎥⎥⎦>0,(3.30) for 𝑖∈𝕀₁.
Actually, for 𝑖∈𝕀, we can get from (3.29) and (3.30) that⎡⎢⎢⎣𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖+𝑄_𝑖−𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃_𝑖𝑅_𝑖⎤⎥⎥⎦≥0.(3.31) Multiplying from left and right by [𝐱^𝑇,𝐮^𝑇] and [𝐱^𝑇,𝐮^𝑇]^𝑇, respectively, and removing the nonnegative term 𝐱^𝑇𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖𝐱yield 0≤=2d𝐱^𝑇𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖𝐱+𝐵_𝑖+Δ𝐵_𝑖𝐮+𝐱^𝑇𝑄_𝑖𝐱+𝐮^𝑇𝑅_𝑖𝐮d𝑡𝐱^𝑇𝑃_𝑖𝐱+𝐱^𝑇𝑄_𝑖𝐱+𝐮^𝑇𝑅_𝑖𝐮.(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, denote𝐺∶=⎡⎢⎢⎣𝐴^𝑇𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖−𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖𝑃_𝑖𝐵_𝑖𝐵^𝑇𝑖𝑃_𝑖𝑅_𝑖⎤⎥⎥⎦.(3.33) Note the uncertain form (3.2). Then, inequality (3.29) can be written as 𝐺+⎡⎢⎢⎣𝑃^𝑖𝑀_𝑖0⎤⎥⎥⎦𝐻𝑁^𝐴𝑖,𝑁_𝐵𝑖+𝑁_𝐴𝑖,𝑁_𝐵𝑖^𝑇𝐻^𝑇⎡⎢⎢⎣𝑃_𝑖𝑀_𝑖0⎤⎥⎥⎦^𝑇>0.(3.34) By Lemma 3.2, inequality (3.34) is equivalent to the existence of some 𝜀_𝑖>0 such that 𝐺−𝜀^𝑖−1⎡⎢⎢⎣𝑃_𝑖𝑀_𝑖0⎤⎥⎥⎦⎡⎢⎢⎣𝑃^𝑖𝑀_𝑖0⎤⎥⎥⎦^𝑇−𝜀_𝑖𝑁_𝐴𝑖,𝑁_𝐵𝑖^𝑇𝑁_𝐴𝑖,𝑁_𝐵𝑖>0,(3.35) that is, ⎡⎢⎢⎣𝐴^𝑇𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖−𝐸^𝑇𝑖𝑈_𝑖𝐸_𝑖−𝜀_𝑖𝑁^𝑇𝐴_𝑖𝑁_𝐴𝑖−𝜀^𝑖−1𝑃_𝑖𝑀_𝑖𝑀^𝑇𝑖𝑃_𝑖𝑃_𝑖𝐵_𝑖−𝜀_𝑖𝑁^𝑇𝐴_𝑖𝑁_𝐵𝑖𝐵^𝑇𝑖𝑃_𝑖−𝜀_𝑖𝑁^𝑇𝐵_𝑖𝑁_𝐴𝑖𝑅_𝑖−𝜀_𝑖𝑁^𝑇𝐵_𝑖𝑁_𝐵𝑖⎤⎥⎥⎦>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 𝑃_𝑖, 𝑃_𝑖, and 𝜀_𝑖. 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=−𝐿𝐱.(3.37) Substituting the control law (3.37) into system (2.1), we can get the following closed-loop system: 𝐱̇(𝑡)=𝐴−𝐵𝐿𝐱(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 Γ≥0 and a bounded set Ω which bounds the chaotic attractor, such that𝐹^𝑇(𝐱)𝐹(𝐱)≤𝐱^𝑇Γ²𝐱,∀𝐱∈Ω.(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 𝐱₀∈Ω. If there exist positive constants 𝛼, 𝛽, positive definite matrix 𝑌, and any matrix 𝑍 with appropriate dimensions such that ⎡⎢⎢⎢⎢⎢⎢⎣𝐴𝑌+𝑌𝐴^𝑇−𝐵𝑍−𝑍^𝑇𝐵^𝑇+𝛼𝐼𝑌Γ𝑌𝑍^𝑇Γ𝑌𝑌−0𝛼𝐼−𝑄0₋₁0𝑍00−𝑅0₋₁⎡⎢⎢⎣−𝛽𝐱_𝑇0⎤⎥⎥⎥⎥⎥⎥⎦<0,𝐱₀−𝑌⎤⎥⎥⎦<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=−𝑍𝑌⁻¹𝐱.

Proof. Denote 𝑃=𝑌⁻¹>0. Construct the Lyapunov function candidate as 𝑉(𝐱)=𝐱^𝑇𝑃𝐱.(3.42)
By virtue of the fact that 𝑀^𝑇𝑁+𝑁^𝑇𝑀≤𝛼⁻¹𝑀^𝑇𝑀+𝛼𝑁^𝑇𝑁, 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=𝐱^𝑇𝐴−𝐵𝐿^𝑇≤𝐱_𝑇𝑃+𝑃𝐴−𝐵𝐿𝐱+𝐹^𝑇(𝐱)𝑃𝐱+𝐱^𝑇𝑃𝐹(𝐱)𝐴−𝐵𝐿^𝑇𝑃+𝑃𝐴−𝐵𝐿+𝛼𝑃²𝐱+𝛼⁻¹≤𝐱_𝑇𝐹^𝑇(𝐱)𝐹(𝐱)𝐴−𝐵𝐿^𝑇𝑃+𝑃𝐴−𝐵𝐿+𝛼𝑃²+𝛼⁻¹Γ²𝐱.(3.43) On the other hand, by Schur complement, the first inequality of (3.40) is equivalent to 𝐴𝑌+𝑌𝐴^𝑇−𝐵𝑍−𝑍^𝑇𝐵^𝑇+𝛼𝐼+𝛼⁻¹𝑌Γ²𝑌+𝑌𝑄𝑌+𝑍^𝑇𝑅𝑍<0.(3.44) Noticing 𝑌𝑃=𝐼, 𝐿=𝑍𝑃, pre- and postmultiplying both sides of (3.44) by 𝑃 implies 𝐴−𝐵𝐿^𝑇𝑃+𝑃𝐴−𝐵𝐿+𝛼𝑃²+𝛼⁻¹Γ²+𝑄+𝐿^𝑇𝑅𝐿<0.(3.45) Thus, it follows from (3.43) and (3.45) that d𝑉(𝐱)d𝑡+𝐱^𝑇𝑄𝐱+𝐱^𝑇𝐿^𝑇𝑅𝐿𝐱<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, renders𝐽<𝑉𝐱₀=𝐱₀𝑌⁻¹𝐱₀,(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: ⎡⎢⎢⎢⎢⎣𝑥̇₁𝑥̇₂𝑥̇₃⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣−00𝑝₁1001−𝑝₂−𝑝₃⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑥₃⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣𝑥00₂₁⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦𝑢,(4.1) where 𝑝₁=6, 𝑝₂=2.92, 𝑝₃=1.2.

Denote that 𝑐^𝑇=[1,0,0] and 𝐱^𝑇=[𝑥₁,𝑥₂,𝑥₃]. Note the boundedness of the chaotic attractor shown in [23]. The state space can be confined to 𝑋∶={𝐱∣−6≤𝑐^𝑇𝐱≤6} by simulation. The partition of state space is set to be 𝑋₁=𝐱∣𝑐^𝑇𝑋𝐱∈[−1,1),𝑋₂=𝐱∣𝑐^𝑇𝐱∈[1,3),𝑋₃=𝐱∣𝑐^𝑇𝐱∈[3,6₄],=𝐱∣𝑐^𝑇𝐱∈[−3,−1),𝑋₅=𝐱∣𝑐^𝑇𝐱∈[−6,−3).(4.2) Then, the nonlinear term 𝑥²¹ can be described as𝑥²¹=𝑘_𝑖𝑥₁+𝑙_𝑖+𝛿_𝑖𝑥₁,𝐱∈𝑋_𝑖,𝑖=1,2,3,4,5,(4.3) where 𝛿_𝑖 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 ||𝛿₁𝑥₁||≤||𝛿||𝑥₁||,||𝛿₂𝑥₁||≤1,||𝛿₃𝑥₁||≤2.25,₄𝑥₁||≤1,||𝛿₅𝑥₁||≤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=𝐴⎡⎢⎢⎢⎢⎣−006−210.92−101.2⎤⎥⎥⎥⎥⎦,𝐴₂=⎡⎢⎢⎢⎢⎣−100.5−210.92−101.2⎤⎥⎥⎥⎥⎦,𝐴₃=⎡⎢⎢⎢⎢⎣003−210.92−101.2₄⎤⎥⎥⎥⎥⎦,=𝑎⎡⎢⎢⎢⎢⎣₁−1000.5−210.92−101.2⎤⎥⎥⎥⎥⎦,𝐴₅=⎡⎢⎢⎢⎢⎣−0015−210.92−101.2⎤⎥⎥⎥⎥⎦,𝐵_𝑖=⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦,Δ𝐴₁=0,𝑎₂=𝑎₄=00−4.5^𝑇,𝑎₃=𝑎₅=00−18^𝑇,=Δ𝑀𝑎₁₁𝐻𝑁_𝐴1,Δ𝐴₂=Δ𝐴₃=Δ𝐴₄=Δ𝐴₅=0,Δ𝐵_𝑖=0,=0,Δ𝑎₄Δ𝑎₂=𝑀₂𝐻𝑁_𝑎2,Δ𝑎₃=𝑀₃𝐻𝑁_𝑎3,𝑀_𝑖=𝑀₄𝐻𝑁_𝑎4,Δ𝑎₅=𝑀₅𝐻𝑁_𝑎5,𝑁=⎡⎢⎢⎢⎢⎣002⎤⎥⎥⎥⎥⎦,𝑁_𝐴1=⎡⎢⎢⎢⎢⎣000.50000_𝑎200⎤⎥⎥⎥⎥⎦,=𝑁_𝑎4=⎡⎢⎢⎢⎢⎣000.5⎤⎥⎥⎥⎥⎦,𝑁_𝑎3=𝑁_𝑎5=⎡⎢⎢⎢⎢⎣1.10025⎤⎥⎥⎥⎥⎦,(4.5) where 𝑖=1,…,5, and 𝐻 is an uncertain matrix bounded by 𝐻^𝑇𝐻≤𝐼.

It is worthwhile to mention that the nominal autonomous piecewise linear system (3.1) with parameters (4.5), that is, 𝑢≡0, Δ𝐴_𝑖=0, Δ𝑎_𝑖=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.

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 𝐸_𝑖 and 𝐹_𝑖 can be constructed by virtue of the method proposed in [20]. Assume that the feedback gain matrix 𝐿_𝑖 is bounded by ‖𝐿_𝑖‖_∞<12, where ‖𝐿_𝑖‖_∞ denotes the largest absolute value among all the entries of vector 𝐿_𝑖. 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: 𝐿𝐽^∗=17.7528,_∗=−3.3088,−0.3345,2.7111,1.2363,−0.0548^𝑇.(4.6) According to the expression of (3.10), we can get the following state feedback gain matrices: 𝐿₁=2.7111,1.2363,−0.0548,𝐿₂=2.3765,1.2363,−0.0548,0.3345,𝐿₃=6.0198,1.2363,−0.0548,−10.5954,𝐿₄=6.0198,1.2363,−0.0548,3.3088,𝐿₅=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.

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 −6≤𝑐^𝑇𝐱≤6. 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^∗=𝑍𝑌⁻¹=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: ⎡⎢⎢⎢⎢⎣𝑥̇₁𝑥̇₂𝑥̇₃⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣−00𝑝₁1001−𝑝₁−𝑝₁⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑥₁𝑥₂𝑥₃⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣𝑝₂00tanh𝑥₁⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣011.5⎤⎥⎥⎥⎥⎦𝑢,(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 𝑢≡0 is shown in Figure 3, which is a double-scroll chaotic attractor.

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∶={𝐱∣−23.3≤𝑐^𝑇𝐱≤23.3} by simulation. The partition of state space is set to be 𝑋₁=𝐱∣𝑐^𝑇𝐱∈[−2𝑋3.3,−1.18),𝑋₂=𝐱∣𝑐^𝑇𝐱∈[−1.18,1.18)₃,=𝐱∣𝑐^𝑇𝐱∈[1.18,23.3].(4.11) Then, the nonlinear term tanh(𝑥₁) can be described astanh𝑥₁=⎧⎪⎨⎪⎩𝑘¹𝑘₂𝑥₁+𝑙₁+𝛿₁𝑥₁,𝐱∈𝑋₁,𝑘₃𝑥₁+𝑙₂+𝛿₂𝑥₁,𝐱∈𝑋₂,𝑥₁+𝑙₃+𝛿₃𝑥₁,𝐱∈𝑋₃,(4.12) where 𝛿_𝑖 denotes the approximation error. Taking 𝑘₁=0, 𝑘₂=0.85, 𝑘₃=0, 𝑙₁=−1, 𝑙₂=0, 𝑙₃=1, one can obtain that||𝛿₁𝑥₁||≤0.17,||𝛿₂𝑥₁||≤0.15||𝑥₁||,||𝛿₃𝑥₁||≤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=⎡⎢⎢⎢⎢⎣−00𝑝₁1001−𝑝₁−𝑝₁⎤⎥⎥⎥⎥⎦,𝐴₂=⎡⎢⎢⎢⎢⎣−𝑝₁001001+0.85𝑝₂−𝑝₁−𝑝₁𝐵₁⎤⎥⎥⎥⎥⎦,=𝐵₂=𝐵₃=⎡⎢⎢⎢⎢⎣011.5⎤⎥⎥⎥⎥⎦,𝑎₁=−𝑎₃=⎡⎢⎢⎢⎢⎣−00𝑝₂⎤⎥⎥⎥⎥⎦,𝑎₂Δ𝐴₁=0,=ΔΔ𝐴𝑎₃₁=0,Δ𝐴₂=𝑀₂𝐻𝑁_𝐴2,Δ𝐵₁=Δ𝐵₂=Δ𝐵₃=0,𝑀₁=𝑀₁𝐻𝑁_𝑎1,Δ𝑎₂=0,Δ𝑎₃=𝑀₃𝐻𝑁_𝑎3,=𝑀₂=𝑀₃=⎡⎢⎢⎢⎢⎣001⎤⎥⎥⎥⎥⎦,𝑁_𝐴2=⎡⎢⎢⎢⎢⎣0.1005𝑝₂000000⎤⎥⎥⎥⎥⎦,𝑁^𝑎1=𝑁_𝑎3=⎡⎢⎢⎢⎢⎣0.1007𝑝₂⎤⎥⎥⎥⎥⎦.(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 𝐿_𝑖 is bounded by ‖𝐿_𝑖‖_∞<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: 𝐿𝐽^∗=5.5117,𝐽^∗=9.7024,_∗=−4.3655,1.2292,2.3555,1.3460,1.4357^𝑇.(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=2.3555,1.3460,1.4357,𝐿₃=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.

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^∗=𝑍𝑌⁻¹=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).