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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 714525, 10 pages
Passivity and Passification for Delay Fuzzy System Based on Delay Partitioning Approach
1The State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China
2Department of Automation, North China Electric Power University, Baoding 071003, China
Received 17 February 2013; Accepted 28 March 2013
Academic Editor: Constantinos Siettos
Copyright © 2013 Xiangjie Liu 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.
A delay partitioning approach is introduced to solve problems of passivity and passification for continuous T-S fuzzy system with time delay. Our aim is to design a state feedback controller such that the resulting closed system is passive. By constructing a Lyapunov-Krasovskii functional, delay-dependent passivity/passification performance conditions are formulated in terms of linear matrix inequalities (LMIs). Finally, numerical examples are used to illustrate the effectiveness of the proposed approaches which can further reduce conservatism and become more obvious with partitioning getting thinner.
The passivity concept was introduced by Willems  and developed further by Hill and Moylan . Passivity of nonlinear systems has attracted great interest in the control area mainly because of the link between stability and passivity. The passivity theory provides a nice tool for analyzing the stability of systems and has found applications in diverse areas such as stability, signal processing, chaos control, and synchronization.
Since most physical systems in real world are nonlinear, researchers have been devoting their efforts to the control of nonlinear systems. Among the methods, the fuzzy control has been proven to be effective in dealing with the analysis of nonlinear systems. especially the T-S fuzzy control [3, 4]. It is denoted by a group of IF-THEN rules that the conventional linear system theory can be applied to the analysis of the class of nonlinear systems, and numerous nonlinear analysis problems have been studied based on this T-S fuzzy model, such as [5–7] reported the problem of stability analysis, and [8–10] investigated the control designs. References [11–13] mentioned the fault detection of the T-S fuzzy systems. model reduction is addressed in . The fuzzy controller was carried out via Parallel Distributed Compensation (PDC) technique . Based on the PDC technique, the fuzzy controller can also be designed to guarantee the passivity of T-S fuzzy systems.
On the other hand, the time delay exists naturally in various control systems. Time delays often degrade the system’s performance and even cause instability. Therefore, time delays have received great attention in recent years and many researchers have studied various analytical techniques and developed many synthesis methods for time-delay systems. For instance, model reduction is addressed in  and filtering problems are investigated in [17, 18]. So, the passivity and passification analysis of nonlinear systems with time delays is worth to be discussed and researched. To date, researches have gained many results in passivity control of T-S fuzzy systems; the passivity of delayed neural networks is considered in , passivity of fuzzy time-delay systems is investigated in  which adopt delay moom’s inequality, and passive controller design for T-S fuzzy systems is addressed in . The passivity of uncertain fuzzy systems is considered in . However, the above methods still have strong conservation, and it is necessary for us to further study.
In this paper, we adopt a delay partitioning approach to study the passivity and passification of T-S fuzzy systems with time delay. Based on this idea, we can further reduce the conservatism, and it becomes even less conservative when the partitioning goes finer. The results of this paper are given in terms of LMIs. The rest of the paper is organized as follows. In Section 2, the problem to be studied is stated and some preliminaries are presented. Passivity analysis results are presented in Section 3. Based on the results obtained in Section 3, we design the controller in Section 4. In Section 5, numerical examples are given to demonstrate the effectiveness of the theoretical results. Finally, conclusions are drawn in Section 6.
Notations. Throughout the paper, and denote the inverse and transpose of a square matrix . denotes the -dimensional Euclidean apace and refers to the Euclidean vector norm. The notation is used to define a symmetric positive definite matrix and sym () is defined as . Matrices are assumed to have compatible dimensions.
2. Problem Statement and Preliminaries
Consider the T-S fuzzy system with time delay has the following form.
Plant Rule :
IF is and … and is THEN where is the state vector; is the control input vector; is a time delay; is the initial condition.
Controller Rule :
IF is and… and is THEN where is the fuzzy set; is the number of IF-THEN rules; is the premise variables vector; , are constant matrices representing state-feedback control gains. Let be the normalized membership function of the fuzzy set where is the grade of membership function of in . It is assumed that , , and for all . Therefore, and for all . By applying (3) into (1), the fuzzy system can be expressed as with By applying (2) into (4), we can get the following closed-loop system: Before formulating the main problem, we first give the following definition.
3. Passivity Analysis
The problems to be addressed in this paper can be expressed as follows.
In this section, we will present a sufficient condition in terms of LMIs, under which the closed-loop system (6) is passive.
Theorem 2. Given matrices , an integer and a scalar , if there exist symmetric positive definite matrices , , , and matrices , and scalar , satisfying where Then the fuzzy system (1) is passive in the sense of Definition 1 for the time delay .
Proof. Choose a Lyapunov-Krasovskii functional as with where The time-derivative of along the trajectory of the system in (6) is given by Define and according to the Newton-Leibniz formula and the system in (6), we have Therefore If Then So, we can obtain where Besides Then Besides where So, we have with If then We can obtain It follows by integrating (31) with respect to over the time period that (7) holds, and hence the delayed fuzzy system (1) is passive in the sense of Definition 1. Our next objective is to convert the inequalities in (18) and (29) to some finite LMIs, then (29) can be rewritten as Equation (18) can be rewritten as From the Schur complement, we can get the inequalities (8), (9), and (10), and the proof is completed.
4. Controller Design
In this section, fuzzy state feedback controllers will be designed based on the result developed in the previous section.
Theorem 3. Given an integer and scalars , , there exists a fuzzy state feedback controller such that the closed-loop system (4) is passive if there exist symmetric positive definite matrices , , , , and matrices , , and scalars , , satisfying where If the above conditions are feasible, the gains of the controller are given by
Proof. Assume that is invertible, define , and Premultiplying and postmultiplying (34) with and , then we obtainwhere We defining Then then Thus, by the Schur complement, we can obtain (40) is equivalent to (8). Pre- and postmultiplying (35) with and , we obtain (9), pre- and postmultiplying (36) with and , we obtain (10). The proof is completed.
5. Numerical Example
Without delay and uncertainty, Example 1 designs different passive controllers by applying the theorem of our paper and the literature , respectively. We can compare the region of feasible solution.
Example 1. Consider a fuzzy system of the following form.
Rule 1: IF is , THEN
Rule 2: IF is , THEN with We can obtain Figure 1 by applying our method and obtain Figure 2 by applying method in . Symbol “*” shows that feasible solution exists at that point; symbol “o” shows that feasible solution does not exist at that point.
Example 2 (Li et al. ). Consider a delayed fuzzy system of the following form.
Rule 1: IF is , THEN Rule 2: IF is , THEN with we let , , , , , , and we can obtain By using the delay-dependent criterion Theorem 3 of , we can obtain when . And applying our result, is obtained. We can see that our results is not conservative since is much smaller than the result of .
Example 3. Consider a delayed fuzzy system of the following form.
Rule 1: IF is , THEN Rule 2: IF is , THEN with The membership function is The initial state is First, we should find the allowable maximum time delay which let the fuzzy system passive. Table 1 shows the maximum time delay obtained by  and our method.
It clearly shows that the method in our paper can get larger upper bound than before. It also shows that conservatism is further reduced when increases.
Then we let , , and we can obtain The fuzzy controller gains by our method are given by Figure 3 shows the state response of the closed-loop system with the controller gains in (58), and Figure 4 shows the state response of the closed-loop system.
This paper has adopted the delay partitioning approach to analyse the passivity and passification of delay fuzzy system based on T-S model. The theorems given in this paper are all in terms of LMIs. Examples have illustrated the effectiveness of our results. The method in our paper has further reduced the conservatism and the effect has been more apparent when increases. In addition, the results obtained in this paper also can be extended to the fuzzy system with time-varying delay and uncertainties.
This work was supported in part by the National Natural Science Foundation of China under Grant 61273144 and 61203041, the Natural Science Foundation of Beijing under Grant 4122071, the Chinese National Post-doctor Science Foundation under Grants 2011M500217 and 2012T50036, and the Doctoral Fund of Ministry of Education of China under Grant 20120036120013.
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