Journal of Applied MathematicsVolume 2012 (2012), Article ID 859542, 20 pagesdoi:10.1155/2012/859542

Research Article

Optimal Control for a Class of Chaotic Systems

Jianxiong Zhang and Wansheng Tang

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 [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    denotes being a positive definite (positive semidefinite, or negative semidefinite) matrix, and means . denotes an identity matrix of appropriate dimension. The superscript “” 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: where and are constant matrices, is the state vector, is the control input variable, and the nonlinear term is assumed to satisfy Lipschitz continuity condition, uniform or local, and .

Associated with this system is the cost function: where , are given weighting matrices.

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

It is known that the control law 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 in system (2.1) can be approximated by a piecewise linear function as follows: where , are some given parameters, denotes a partition of the state space of chaotic system, is the index set, and 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:

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 taking the following form: where is the nonlinear term in the 3rd dimension of the system and can be approximated by a piecewise linear function as where are some given parameters, 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 with

3. State Feedback Optimal Control of Systems

Without loss of generality, consider the uncertain piecewise linear system of the form for , where denotes a partition of the state space into a number of polyhedral cells, is the index set of the cells, is the th nominal local model of the system, is the offset term. , , and represent parametric perturbations in the system state matrix, input matrix, and offset term of the th nominal local model, respectively, and are assumed to be of the following form: where is an uncertain matrix bounded by , and , , , are known constant matrices of appropriate dimensions which specify how the elements of the nominal matrices , and are affected by the uncertain parameters in .

Define as the set of indices for cells that contain origin and the set of indices for cells that do not contain the origin. It is assumed that for all .

For any given initial condition , and input signals , it is assumed that system (3.1) has a unique solution, and there is no sliding mode. Note that with possible discontinuities in across the boundaries of the partitions, the solution of system (3.1) may be just continuous and piecewise . 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: then system (3.1) can be expressed as

Associated with this system is the following cost function: where is defined so that , and , are given weighting matrices.

Note that if in (3.5) are set to be the same, respectively, for every , the cost function (3.5) will reduce to (2.2). In addition, the matrix 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 , with for should be constructed, which are used to characterize the boundaries between the regions: Then, the piecewise Lyapunov function candidates that are continuous across the region boundaries can be parameterized as with and , where is a symmetric matrix which characterizes the free parameters of the Lyapunov function candidates.

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

The -procedure has been used in [20, 22] to reduce the conservatism of the stability result. Specifically, the matrices , with for , such that 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 , for a given piecewise linear system was suggested in [20].

Consider the following piecewise linear feedback control law: with for .

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 as follows where is a parameter matrix characterizing the free parameters of the state feedback controller, and is the matrix defined in (3.6). It should be pointed out that the gain matrix should take the form of for .

Substituting the control law (3.9) into system (3.4), we can get the following closed-loop system:

Our goal in this section is to find a parameter matrix 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 , , and such that and have nonnegative entries, while , and , , satisfy for , and for , then every continuous and piecewise trajectory of system (3.4) with , and for all tends to zero exponentially.

Lemma 3.2 (Xie [27]). Given matrices , , and of appropriate dimensions with symmetric, then for all matrices satisfying , if and only if there exists some such that

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 . If there exist a set of constants and symmetric matrices , , and such that and have nonnegative entries, while , , and , , satisfy for , for , where then the closed-loop system is globally exponentially stable, and the cost function (3.5) satisfies

Proof. By Schur complement [19], the first inequality of (3.15) is equivalent to Note the definitions of (3.3) and (3.17). By virtue of Lemma 3.2, inequality (3.19) is equivalent to Along a similar proof technique as used above, it can also be shown that the first inequality of (3.16) is equivalent to where . Note that and . 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 Multiplying from left and right by and , respectively, and removing the nonnegative term render 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 and when both the Lyapunov matrix and the feedback gain matrix become the variables to be determined. Our interest is to find a parameter matrix to minimize the upper bound on the cost function (3.5) for the state feedback closed-loop system (3.11). Then, the optimization problem can be formulated as where , and is the set of admissible values for the state feedback gain matrix , 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 and the corresponding objective . In general, one can set the parameter matrix to be the decision variables searched by GA. For a given chromosome corresponding to , 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 and symmetric matrices and such that have nonnegative entries, while , and , satisfy for , for , where then for every trajectory of the uncertain system (3.4) with , , the cost function (3.5) satisfies

Proof. We will first show the conditions for the cost function (3.5) satisfying the lower bound (3.28) can be guaranteed by for , and for .Actually, for , we can get from (3.29) and (3.30) that Multiplying from left and right by and , respectively, and removing the nonnegative term yield Integration from 0 to , and noticing , gives the result of (3.28).Next, we will show that inequality (3.29) is equivalent to (3.25). For simplifying the presentation, denote Note the uncertain form (3.2). Then, inequality (3.29) can be written as By Lemma 3.2, inequality (3.34) is equivalent to the existence of some such that that is, 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 , , and . 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 is given or known in advance. Note that the bounds in (3.18) and (3.28) depend on the initial state . 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 is a random variable subjected to uniform distribution on a certain bounded region . 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: Substituting the control law (3.37) into system (2.1), we can get the following closed-loop system:

Additionally, note the boundedness of the chaotic attractor and the Lipschitz continuity condition for the nonlinear term . There exist some matrix and a bounded set which bounds the chaotic attractor, such that

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 . If there exist positive constants , , positive definite matrix , and any matrix with appropriate dimensions such that then the closed-loop system (3.38) is globally exponentially stable, and the cost function (2.2) satisfies Furthermore, the corresponding control law can be obtained as .

Proof. Denote . Construct the Lyapunov function candidate as By virtue of the fact that , for all , and matrices and with appropriate dimensions, calculating the time derivative of along the trajectory of the closed-loop system (3.38) and noticing (3.39) yield On the other hand, by Schur complement, the first inequality of (3.40) is equivalent to Noticing , , pre- and postmultiplying both sides of (3.44) by implies Thus, it follows from (3.43) and (3.45) that Note that and . It is obvious that which guarantees the global stability of closed-loop system (3.38), that is, .Integration both sides of (3.46) from 0 to , and noticing , renders 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 , , , . 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: where , , .

Denote that and . Note the boundedness of the chaotic attractor shown in [23]. The state space can be confined to by simulation. The partition of state space is set to be Then, the nonlinear term can be described as where denotes the approximation error. Taking , , , , , , one can obtain that

Note the expressions (4.3) and (4.4). System (4.1) can be converted to the piecewise linear system (3.1) with where , and is an uncertain matrix bounded by .

It is worthwhile to mention that the nominal autonomous piecewise linear system (3.1) with parameters (4.5), that is, , , , 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 , , and the initial value of system (4.1). The matrices and can be constructed by virtue of the method proposed in [20]. Assume that the feedback gain matrix is bounded by , where denotes the largest absolute value among all the entries of vector . 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 , denoted as , and the corresponding optimal parameter matrix as follows: According to the expression of (3.10), we can get the following state feedback gain matrices: with which the optimal control 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 . 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 , denoted as , can be obtained by solving the corresponding SDP with the LMI toolbox in MATLAB 7.0 as follows:

On the other hand, note that . The matrix in (3.39) can be obtained as . According to Theorem 3.9, we solve the corresponding SDP, and obtain the optimal gain matrix in (3.37) and upper bound as follows: 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: where , , and the hyperbolic function . The strange attractor of the autonomous system (4.10) with 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 by simulation. The partition of state space is set to be Then, the nonlinear term can be described as where denotes the approximation error. Taking , , , , , , one can obtain that

Note the expressions (4.12) and (4.13). System (4.10) can be converted to the piecewise linear system (3.1) with Consider the cost function (2.2) with , and the system initial value . Assume that the feedback gain matrix is bounded by . Then, similarly to the above subsection, we get the maximal lower bound , the optimal upper bound , and the corresponding optimal parameter matrix as follows: According to the expression of (3.10), we can get the following state feedback gain matrices: with which the optimal control 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 . 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 . The matrix in (3.39) can be obtained as . According to Theorem 3.9, we solve the corresponding SDP, and obtain the optimal gain matrix in (3.37) and upper bound as follows: which is significantly greater than the optimal upper bound obtained from Theorem 3.3.

It is obviously shown from the above examples that the optimal upper bounds obtained above get close to the corresponding lower bounds , 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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