- Open Access
Guaranteed cost control for switched recurrent neural networks with interval time-varying delay
© Niamsup et al.; licensee Springer 2013
- Received: 14 July 2012
- Accepted: 11 May 2013
- Published: 10 June 2013
This paper studies the problem of guaranteed cost control for a class of switched recurrent neural networks with interval time-varying delay. The time delay is a continuous function belonging to a given interval, but not necessary 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, a guaranteed cost controller is designed via memoryless state feedback control, a switching rule for the exponential stabilization for the system is designed via linear matrix inequalities 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). A numerical example is given to illustrate the effectiveness of the obtained result.
- neural networks
- guaranteed cost control
- switching design
- interval time-varying delays
- Lyapunov function
- linear matrix inequalities
Stability and control of recurrent 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 tools in the design of guaranteed cost controls for neural networks with time delay. Nevertheless, despite such a diversity of results available, the most existing works 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 were considered in [12, 13, 15], the approach used there cannot be applied to systems with interval, non-differentiable time-varying delays. To the best of our knowledge, the guaranteed cost control and state feedback stabilization for switched recurrent neural networks with interval time-varying delay, non-differentiable time-varying delays have not been fully studied yet (see, e.g., [9–12, 15–25] 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 switched recurrent 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. Specifically, our goal is to develop a constructive way to design a switching rule to exponentially stabilize the system. 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 the 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 guaranteed cost control and a numerical example 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 non-negative numbers; denotes the n-dimensional space with the scalar product or of two vectors x, y, and the vector norm ; denotes the space of all matrices of dimensions. denotes the transpose of matrix A; A is symmetric if ; I denotes the identity matrix; denotes the set of all eigenvalues of A; . , ; denotes the set of all -valued continuously differentiable functions on ; denotes the set of all the -valued square integrable functions on .
Matrix A is called semi-positive definite () if for all ; A is positive definite () if for all ; means . The notation stands for a block-diagonal matrix. The symmetric term in a matrix is denoted by ∗.
are the activation functions; is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, implies that the system realization is chosen as the j th system, . It is seen that system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state hits predefined boundaries.
is exponentially stable and the closed-loop value of the cost function (2.3) is minimized.
Remark 2.1 It is worth noting that the time delay is a time-varying function belonging to a given interval, in which the lower bound of delay is not restricted to zero; therefore, the stability criteria proposed in [4–7, 9–13, 15–18, 21–24] are not applicable to this system.
Remark 2.2 It is worth noting that the time delay is a time-varying function belonging to a given interval, in which the delay function is non-differentiable; therefore, the stability criteria proposed in [5, 6, 8, 10–12, 14–19, 22–25] are not applicable to this system.
Definition 2.2 Consider control system (2.1). If there exist a memoryless state feedback control law and a positive number such that the zero solution of closed-loop system (2.5) is exponentially stable and the cost function (2.3) 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 2.1 (Schur complement lemma )
Proposition 2.2 (Integral matrix inequality )
This completes the proof of the theorem. □
Note that is non-differentiable, therefore, the stability criteria proposed in [4–7, 9–13, 15–18, 21–24] are not applicable to this system. Given , , , by using the Matlab LMI toolbox, we can solve for P, U, , , , , , and which satisfy the conditions (3.1)-(3.3) in Theorem 3.1.
In this paper, the problem of guaranteed cost control for Hopfield neural networks with interval non-differentiable 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, a switching rule for the exponential stabilization for the system is designed via linear matrix inequalities. 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 Office of Agricultural Research and Extension Maejo University, the Thailand Research Fund Grant, the Higher Education Commission and Faculty of Science, Maejo University, Thailand. The first 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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