Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays
Introduction
During the last decade, a great deal effort has been done to the stability analysis of time-delay systems since the occurrence of time delays may cause poor performance or instability. This time delay occurs in many industrial systems such as chemical processes, biological systems, population dynamics, neural networks, large-scale systems, network control systems, and so on. Therefore, many researchers have focused on the stability analysis of the systems with time delays. For more details, see [1], [2], [3] and references therein.
In general, delay-dependent stability criteria, which include information on the size of delays, are less conservative than delay-independent ones. Thus, much attention has been paid to the delay-dependent stability criteria [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. An important issue in this field is to enlarge the feasibility region of the stability criteria. To do this, a new bounding technique was studied in Park [4] to reduce the conservatism of the stability criteria by introducing variables in cross-terms. A descriptor form approach had been proposed by Fridman and Shaked [6]. Model transformations such as neutral model transformation [16] and parameterized neutral one [8] were utilized to derive the stability criteria of time-delay systems. Recently, methods introducing free-weighting matrices were used to give large delay bounds [10], [11], [12], [13], [14], [15]. In [11], an augmented Lyapunov function was proposed to treat the cross-terms of variables. He et al. [15] improved the stability region of uncertain systems with time-varying by including the ignored integral term obtained by calculating the time-derivative of Lyapunov–Krasovskii functional and utilizing free-weighting matrices in zero equations.
On the other hand, the stability analysis of neutral differential systems, which have delays in both its state and the derivatives of the state, has been widely investigated by many researchers [16], [17], [18], [19], [20], [21], [22], [23], [24]. Yue and Han [17] studied the delay-dependent stability problem of partial element equivalent circuit model which can be represented as a differential equation of neutral-type. Park and Kwon [18] investigated delay-dependent stability criteria for the systems of neutral-type by utilizing the parameterized model transformation and free-weighting matrices. Lien [19] solved the stability and stabilization problem for uncertain neutral systems by using a genetic algorithm.
In this paper, we propose an improved delay-dependent stability criterion for neutral systems with time-varying delays and norm-bounded parameter uncertainties. By constructing a suitable augmented Lyapunov functional, a delay-dependent criterion is derived in terms of LMIs which can be solved efficiently by using the interior-point algorithms [25]. Six numerical examples are included to show the effectiveness of the proposed method.
In the sequel, the following notation will be used. is the n-dimensional Euclidean space. denotes the set of real matrix. denotes the symmetric part. means that X is a real symmetric positive definitive matrix (positive semi-definite). I denotes the identity matrix with appropriate dimensions. refers to the induced matrix 2-norm. denotes the block diagonal matrix. denotes the Banach space of continuous functions mapping the interval into , with the topology of uniform convergence.
Section snippets
Problem statements
Consider the following uncertain neutral systems with time-varying delays:where is the state vector, A and are known constant matrices with appropriate dimensions, and , and are the uncertainties of system matrices of the formin which the time-varying nonlinear function satisfiesThe delays, , are time-varying continuous functions that satisfies
Main results
In this section, we propose a new stability criterion for uncertain neutral systems (1) with time-varying delays. For simplicity, the notations of several matrices are defined in Appendix.
Now, we have the following theorem. Theorem 1 For given positive scalars and , system (6) is globally asymptotically stable if and there exist positive definite matrices , , Q, H, , , , , and any matrices , , ,
Numerical examples
Example 1 Consider the following neutral systems (1) with parameters [17]:where and and denote the parameters uncertainties which satisfyFor , our results are compared with Yue and Han ones [17] and listed in Table 1. For , the comparison of our results with Yue and Han ones [17] is listed in Table 2. From these two tables, one can see that Theorem 1 provides larger delay bounds for
Conclusions
In this paper, a new delay-dependent stability criterion for uncertain neutral systems with time-varying delays is proposed. To obtain a less conservative result, the suitable augmented Lyapunov–Krasovskii functional and free-weighting matrices are utilized by combining with the LMI framework to obtain the stability criterion of the system. The effectiveness of the proposed stability criterion is successfully verified by six numerical examples.
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