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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 587426, 16 pages
Delay-Dependent Guaranteed Cost Controller Design for Uncertain Neural Networks with Interval Time-Varying Delay
1Major of Mathematics and Statistics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand
2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50000, Thailand
Received 3 August 2012; Revised 24 September 2012; Accepted 25 September 2012
Academic Editor: Xiaodi Li
Copyright © 2012 M. Rajchakit et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper studies the problem of guaranteed cost control for a class of uncertain delayed neural networks. The time delay is a continuous function belonging to a given interval but not necessary to be differentiable. A cost function is considered as a nonlinear performance measure for the closed-loop system. The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles. By constructing a set of augmented Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, a guaranteed cost controller is designed via memoryless state feedback control, and new sufficient conditions for the existence of the guaranteed cost state feedback for the system are given in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the obtained result.
The last few decades have witnessed the use of artificial neural networks (ANNs) in many real-world applications and have offered an attractive paradigm for a broad range of adaptive complex systems. In recent years, ANNs have enjoyed a great deal of success and have proven useful in wide variety pattern recognition feature-extraction tasks. Examples include optical character recognition, speech recognition, and adaptive control, to name a few. To keep the pace with the huge demand in diversified application areas, many different kinds of ANN architecture and learning types have been proposed to meet varying needs as robustness and stability. Stability and control of neural networks with time delay have attracted considerable attention in recent years [1–8]. In many practical systems, it is desirable to design neural networks which are not only asymptotically or exponentially stable but can also guarantee an adequate level of system performance. In the area of control, signal processing, pattern recognition, and image processing, delayed neural networks have many useful applications. Some of these applications require that the equilibrium points of the designed network be stable. In both biological and artificial neural systems, time delays due to integration and communication are ubiquitous and often become a source of instability. The time delays in electronic neural networks are usually time varying, and sometimes vary violently with respect to time due to the finite switching speed of amplifiers and faults in the electrical circuitry. Guaranteed cost control problem [9–12] has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound. The Lyapunov-Krasovskii functional technique has been among the popular and effective tool in the design of guaranteed cost controls for neural networks with time delay. Nevertheless, despite such diversity of results available, most existing work either assumed that the time delays are constant or differentiable [13–16]. Although, in some cases, delay-dependent guaranteed cost control for systems with time-varying delays was considered in [12, 13, 15], the approach used there can not be applied to systems with interval, nondifferentiable time-varying delays. To the best of our knowledge, the guaranteed cost control and state feedback stabilization for uncertain neural networks with interval, non-differentiable time-varying delays have not been fully studied yet (see, e.g., [4–26] and the references therein), which are important in both theories and applications. This motivates our research.
In this paper, we investigate the guaranteed cost control for uncertain delayed neural networks problem. The novel features here are that the delayed neural network under consideration is with various globally Lipschitz continuous activation functions, and the time-varying delay function is interval, non-differentiable. A nonlinear cost function is considered as a performance measure for the closed-loop system. The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles. Based on constructing a set of augmented Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, new delay-dependent criteria for guaranteed cost control via memoryless feedback control are established in terms of LMIs, which allow simultaneous computation of two bounds that characterize the exponential stability rate of the solution and can be easily determined by utilizing Matlabs LMI control toolbox.
The outline of the paper is as follows. Section 2 presents definitions and some well-known technical propositions needed for the proof of the main result. LMI delay-dependent criteria for guaraneed cost control and a numerical examples showing the effectiveness of the result are presented in Section 3. The paper ends with conclusions and cited references.
The following notation will be used in this paper. denotes the set of all real nonnegative numbers; denotes the -dimensional space with the scalar product or of two vectors , and the vector norm ; denotes the space of all matrices of -dimensions. denotes the transpose of matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; ., ; denotes the set of all -valued continuously differentiable functions on ; denotes the set of all the -valued square integrable functions on .
Matrix is called semipositive definite if , for all is positive definite if for all means . The notation stands for a block-diagonal matrix. The symmetric term in a matrix is denoted by .
Consider the following uncertain neural networks with interval time-varying delay: where is the state of the neural; is the control; is the number of neurals, and are the activation functions; represents the self-feedback term; is control input matrix; denote the connection weights, the discretely delayed connection weights and the distributively delayed connection weight, respectively; the time-varying uncertain matrices , and are defined by where and are known constant real matrices with appropriate dimensions. and are unknown uncertain matrices satisfying The time-varying delay function satisfies the condition The initial functions , with the norm In this paper we consider various activation functions and assume that the activation functions are Lipschitzian with the Lipschitz constants : The performance index associated with the system (2.1) is the following function: where is a nonlinear cost function that satisfies for all and are given symmetric positive definite matrices. The objective of this paper is to design a memoryless state feedback controller for system (2.1) and the cost function (2.8) such that the resulting closed-loop system is exponentially stable and the closed-loop value of the cost function (2.10) is minimized.
Definition 2.1. Given . The zero solution of closed-loop system (2.8) is -exponentially stabilizable if there exists a positive number such that every solution satisfies the following condition:
Definition 2.2. Consider the control system (2.1). If there exists a memoryless state feedback control law and a positive number such that the zero solution of the closed-loop system (2.10) is exponentially stable and the cost function (2.8) satisfies , then the value is a guaranteed constant and is a guaranteed cost control law of the system and its corresponding cost function.
We introduce the following technical well-known propositions, which will be used in the proof of our results.
Proposition 2.3 (Schur complement lemma ). Given constant matrices , and with appropriate dimensions satisfying , then if and only if
Proposition 2.4 (integral matrix inequality ). For any symmetric positive definite matrix , scalar and vector function such that the integrations concerned are well defined, the following inequality holds
3. Design of Guaranteed Cost Controller
In this section, we give a design of memoryless guaranteed feedback cost control for uncertain neural networks (2.1). Let us set
Theorem 3.1. Consider control system (2.1) and the cost function (2.8). If there exist symmetric positive definite matrices and , and diagonal positive definite matrices , and satisfying the following LMIs then is a guaranteed cost control and the guaranteed cost value is given by Moreover, the solution of the system satisfies
Proof. Let . Using the feedback control (2.8) we consider the following Lyapunov-Krasovskii functional: It is easy to check that Taking the derivative of we have Applying Proposition 2.4 and the Leibniz-Newton formula We have for Note that Applying Proposition 2.4 gives Since , we have then Similarly, we have Then, we have Using (2.8) and multiplying both sides with , we have Adding all the zero items of (3.20) and , respectively, into (3.18) and using the condition (2.7) for the following estimations: we obtain where , and Note that by the Schur complement lemma, Proposition 2.3, the conditions and are equivalent to the conditions (3.3) and (3.4), respectively. Therefore, by condition (3.2), (3.3), and (3.4), we obtain from (3.22) that Integrating both sides of (3.24) from to , we obtain Furthermore, taking condition (3.9) into account, we have then which concludes the exponential stability of the closed-loop system (2.8). To prove the optimal level of the cost function (2.4), we derive from (3.22) and (3.2)–(3.4) that Integrating both sides of (3.28) from to leads to dute to . Hence, letting , we have This completes the proof of the theorem.
Example 3.3. Consider the uncertain neural networks with interval time-varying delays (2.1), where Note that is non-differentiable; therefore, the stability criteria proposed in [4–8, 12, 15–26] are not applicable to this system. Given and, by using the Matlab LMI toolbox, we can solve for , and which satisfy the conditions (3.2)–(3.4) in Theorem 3.1. A set of solutions are , Then is a guaranteed cost control law and the cost given by Moreover, the solution of the system satisfies
Example 3.4. Consider the uncertain neural networks with interval time-varying delays (2.1), where Note that is non-differentiable; therefore, the stability criteria proposed in [5–8, 12, 15–26] are not applicable to this system. Given , by using the Matlab LMI toolbox, we can solve for , and which satisfy the conditions (3.2)–(3.4) in Theorem 3.1. A set of solutions are , Then is a guaranteed cost control law and the cost given by Moreover, the solution of the system satisfies
In this paper, the problem of guaranteed cost control for uncertain neural networks with interval nondifferentiable time-varying delay has been studied. A nonlinear quadratic cost function is considered as a performance measure for the closed-loop system. The stabilizing controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles. By constructing a set of time-varying Lyapunov-Krasovskii functionals combined with Newton-Leibniz formula, a memoryless state feedback guaranteed cost controller design has been presented, and sufficient conditions for the existence of the guaranteed cost state-feedback for the system have been derived in terms of LMIs.
This work was supported by the Thai Research Fund Grant, the Higher Education Commission, and Faculty of Science, Maejo University, Thailand. The second author is supported by the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.
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