2266 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 12, DECEMBER 2004 New Delay-Dependent Stability Criteria and Stabilizing Method for Neutral Systems Min Wu, Yong He, and Jin-Hua She Abstract—This note concerns delay-dependent robust stability criteria and a design method for stabilizing neutral systems with time-varying structured uncertainties. A new way of deriving such criteria is pre- sented that combines the parameterized model transformation method with a method that takes the relationships between the terms in the Leibniz–Newton formula into account. The relationships are expressed as free weighting matrices obtained by solving linear matrix inequalities. Moreover, the stability criteria are also used to design a stabilizing state-feedback controller. Numerical examples illustrate the effectiveness of the method and the improvement over some existing methods. Index Terms—Delay-dependent criterion, linear matrix inequality (LMI), neutral system, robust stability, state feedback stabilizing con- troller, time-varying structured uncertainties. I. INTRODUCTION Stability criteria for neutral systems can be classified into two types: delay-dependent, which include information on the size of delays, [1]–[10], and delay-independent, which are applicable to delays of arbitrary size [11]. Delay-independent stability criteria tend to be conservative, especially for small delays, while delay-dependent ones are usually less conservative. The Lyapunov functional method is the main method employed to derive delay-dependent criteria. The discretized-Lyapunov-func- tional method (e.g., [5], [12], and [13]) is one of the most efficient among them, but it is difficult to extend to the synthesis of a control system. Another method involves a fixed model transformation, which expresses the delay term in terms of an integral. Four basic model transformations have been proposed [9]. The descriptor model transformation method combined with Park’s or Moon et al.’s in- equalities [14], [15] is the most efficient [8], [9], [16]. However, there is room for further investigation. For example, in the derivative of the Lyapunov functional, the Leibniz–Newton formula was used, and the term was replaced by in some places but not in others. Moreover, the relationship between these two terms was not considered. Recently, He et al. [10] devised a new method that employs free weighting matrices to express the relationships between the terms in the Leibniz–Newton formula. This overcomes the conservativeness of methods involving a fixed model transformation. A different idea is the application of a parameterized model trans- formation with a parameter matrix. The delayed matrix (the coefficient matrix of the delayed term) is decomposed into two parts. One part is kept; and the other part is replaced either with , which is in the derivative of the Lyapunov functional [6], or with the neutral transformation [4]. However, in the former treatment [6], the weighting matrices are fixed, as in [8], [9], [14]–[16]; and in both treat- ments, the method of decomposing the parameter matrix [4], [6] needs Manuscript received July 24, 2003; revised February 10, 2004 and June 22, 2004. Recommended by Associate Editor Z. Lin. This work was supported by the National Science Foundation of China and by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institu- tions of MOE, P.R.C. M. Wu and Y. He are with the School of Information Science and Engineering, Central South University, Changsha 410083, China (e-mail: heyong08@yahoo.com.cn). J.-H. She is with the School of Bionics, Tokyo University of Technology, Tokyo 192-0982, Japan. Digital Object Identifier 10.1109/TAC.2004.838484 more investigation. Han presented a method of selecting the parameter matrix (Remark 7) in [6]. However, a severe restriction was imposed, namely, that three of the matrices must be chosen to be the same, which may lead to conservativeness. This note presents a new parameterized-matrix form expressed in terms of the solution of a linear matrix inequality (LMI) [17]. This is combined with the free-weighting-matrix method [10] to yield a new stability criterion for a neutral system with no uncertainties. The cri- terion is further extended to a system with time-varying structured un- certainties. Based on this criterion, a method of designing a stabilizing state feedback controller is derived. II. NOTATION AND PRELIMINARIES Consider the following neutral system, , with time-varying struc- tured uncertainties: (1) where is the state vector; is the control input; is a constant time delay; and , , , and are constant matrices with appropriate dimensions. The uncertainties are of the form (2) where , and are appropriately dimensioned constant ma- trices, and is an unknown real and possibly time-varying matrix with Lebesgue-measurable elements satisfying (3) where is the Euclidean norm. The problem is to find a state feedback gain, , in the control law (4) that stabilizes . First, the nominal system, , of is discussed. It is given by . (5) The following lemma is used to deal with a system with time-varying uncertainties [20]. Lemma 1: Given matrices , , ,and with appropriate dimensions for all satisfying , if and only if there exists a scalar such that The operator : is defined to be Its stability is defined as follows [18]. Definition 1: The operator is said to be stable if the zero solution of the homogeneous difference equation , , is uniformly asymptotically stable. 0018-9286/04$20.00 © 2004 IEEE