IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008 739 Filter Design for Nonlinear Systems With Time-Delay Through T–S Fuzzy Model Approach Chong Lin, Senior Member, IEEE, Qing-Guo Wang, Member, IEEE, Tong Heng Lee, Member, IEEE, and Bing Chen Abstract—This paper is concerned with the filter design for nonlinear systems with time-varying delay via Takagi–Sugeno fuzzy model approach. Delay-dependent design method is pro- posed in terms of linear matrix inequalities (LMIs), which forms the main contribution of this paper. The main technique used is the free-weighting matrix method combined with a matrix decoupling approach. The results for rate-independent case, delay-indepen- dent case, and delay-free case are also given as easy corollaries. An illustrative example is given to show the effectiveness of the present method. Index Terms—Filter, fuzzy systems, time delay. I. INTRODUCTION N ONLINEAR filtering is of both theoretical and practical importance in signal processing and this area keeps at- tracting researchers for decades. So far, various methodologies have been developed for the filter designs with/without perfor- mance ( and/or ) [1]–[4]. The filtering problem is concerned with the design of estimators guaranteeing that the gain from the disturbance (noise signals) to the estimation error is within a prescribed level. Such designs are also robust against unmodelled dynamics and system uncertainties. The fil- tering approach is a good complement to the limited applica- tions of the conventional Kalman filtering approach. For linear time-delay/delay-free systems, there have been fruitful results of filter designs based on the linear matrix inequality (LMI) ap- proach [5]–[9]. Nonlinear filtering design methods for specific class of nonlinear systems have also been presented in the liter- ature [1]–[3]. For complex nonlinear systems, it in general lacks of common techniques in filter designs. During the last two decades, the well-known Takagi–Sugeno (T–S) fuzzy model [10] has been recognized effective in approx- imating a complex nonlinear system. Consequently, much effort has been made in the control literature to investigate T–S fuzzy systems and various techniques have been developed for the analysis and synthesis [11]–[14]. A recent paper [15] provides Manuscript received October 6, 2006; revised February 6, 2007 and April 9, 2007. This work was supported in part by the National Natural Science Founda- tion of China (60674055, 60774047) and in part by the Taishan Scholar Program of Shandong province, China. C. Lin was with the Department of Electrical and Computer Engineering, Na- tional University of Singapore, 119260 Singapore. He is now with the Institute of Complexity Science, Qingdao University, Qingdao 266071, China (e-mail: linchong_2004@hotmail.com). Q.-G. Wang and T. H. Lee are with the Department of Electrical and Computer Engineering, National University of Singapore, 119260 Singapore (e-mail: elewqg@nus.edu.sg; eleleeth@nus.edu.sg). B. Chen is with the Institute of Complexity Science, Qingdao University, Qingdao 266071, China (e-mail: chenbing1958@yahoo.com.cn). Digital Object Identifier 10.1109/TFUZZ.2007.905915 a survey on this aspect. As the time delay is often inevitable in practical systems and is usually a source of instability, great ef- fort has been made to investigate time-delay systems from var- ious aspects [16]–[20]. Recently, the T–S fuzzy model approach has been extended to deal with nonlinear systems with time-de- lays; see [21]–[25] for delay-independent methods and [26] for delay-dependent methods. In [27], a quadratic method is used to study fuzzy estimation for discrete-time T–S fuzzy sys- tems. However, there are few works appeared to cope with the nonlinear filtering problem for T–S fuzzy systems with time-de- lays. In [28], a delay-independent LMI approach is proposed for exponential filter design for T–S fuzzy systems with time delay. In [29], mixed filtering design is presented in terms of delay-independent LMIs for discrete-time delay fuzzy systems. In [30], a delay-dependent design scheme is proposed for time-delay T–S fuzzy systems. The method therein is suit- able for the case that the filter form is of the extended Kalman filtering type. So far, to the best of our knowledge, there has been no delay-dependent method reported for the general dynamic nonlinear filter design via T–S fuzzy model approach. This motivates the research in this paper. To achieve our purpose, we will adopt the newly developed technique for the filter design. The main technique used is a combination of the free-weighting matrix method [31], [32] and the matrix decoupling procedure similar to that of [7]. As a result, a delay-dependent LMI scheme is presented for the filter design for time-delay T–S fuzzy systems. The present re- sult of general dynamic filter design is an important complement to the work in [30]. A numerical example is given to illustrate the effectiveness of the present design method. This paper is organized as follows. Section II gives the problem formulation and preliminaries. Section III presents the main result and various corollaries. Section IV gives an illustrative example to show the effectiveness of the results. This paper is concluded in Section V. Notation: denotes the n-dimensional real Euclidean space; is the identity matrix of appropriate dimensions; the superscripts “ ,” and “ ” stand for the matrix transpose and inverse, respectively; and mean that is, respectively, a real symmetric positive (nega- tive) definite and positive (negative) semi-definite matrix; is the spectral norm; denotes the space of square-in- tegrable vector functions over . II. PROBLEM FORMULATION Consider a nonlinear system with time-delay which could be approximated by a time-delay T–S fuzzy model with plant rules [21], [22], [25], [26], [28]. 1063-6706/$25.00 © 2007 IEEE