Elsevier

Automatica

Volume 39, Issue 9, September 2003, Pages 1571-1582
Automatica

New approaches to H controller designs based on fuzzy observers for T-S fuzzy systems via LMI

https://doi.org/10.1016/S0005-1098(03)00172-9Get rights and content

Abstract

The problems of relaxed quadratic stability conditions, fuzzy observer designs and H controller designs for T-S fuzzy systems have been studied. First new stability conditions are obtained by relaxing the stability conditions derived in previous papers. Secondly, new fuzzy observers based on the relaxed stability conditions for the T-S fuzzy systems have been proposed. Thirdly two sufficient LMI conditions, which guarantee the existence of the H controllers based on fuzzy observers for the T-S fuzzy systems have been proposed. The conditions are not only simple, but also consider the interactions among the fuzzy subsystems. Finally by some examples, using the LMI technique, we show that the regulators, the fuzzy observers and the H controller designs based on new observers for the T-S fuzzy systems are very practical and efficient.

Introduction

T-S fuzzy systems (Takagi & Sugeno, 1985) are nonlinear systems described by a set of If–Then rules. The authors of Feng, Cao, Rees, and Chak (1997), Cao, Rees, & Feng 1996a, Cao, Rees, & Feng 1996b, Cao, Rees, & Feng 1997 have proved that the T-S fuzzy systems can approximate to any continuous functions in a compact set of Rn at any preciseness. This allows the designers to take advantage of conventional linear systems to analyze and design the nonlinear systems. So that T-S fuzzy control has become one of the most popular and promising research platform in the model-based fuzzy control and the theoretic researches on the issue have been conducted actively by many fuzzy control theorists. Originally, Tanaka and his colleagues have provided a sufficient condition for the quadratic stability of the T-S fuzzy systems in the sense of Lyapunov in a series of papers (Takagi & Sugeno, 1993; Tanaka & Sano, 1994; Wang, Tanaka, & Griffin, 1996; Tanaka & Kosaki, 1997; Tanaka, Ikede, & Wang 1998a, Tanaka, Ikeda, & Wang 1998b) by considering a Lyapunov function of the sub-fuzzy systems of the T-S fuzzy systems.

Recently and notably, in Tanaka, Ikede, & Wang 1998a, Tanaka, Ikeda, & Wang 1998b and Tanaka, Ikede, and Wang (1997), the relaxed stabilizability conditions for fuzzy control systems were reported to relax the conservatism of the conventional conditions. In Kim and Lee (2000), new approaches to relaxed quadratic stability conditions of fuzzy control systems were suggested and relaxed the conservatism of the previous works such as Tanaka, Ikede, & Wang 1998a, Tanaka, Ikeda, & Wang 1998b, Tanaka et al. (1997). In 2 The relaxed quadratic stability conditions, 3 Observer design for T-S fuzzy systems of this paper, new stability conditions are obtained by relaxing the stability conditions derived in previous papers such as Kim and Lee (2000), Tanaka, Ikede, & Wang 1998a, Tanaka, Ikeda, & Wang 1998b, Tanaka, Ikede, & Wang 1997, Tanaka, Ikede and Wang (1996) and Wang, et al. (1996). Theorem 2 is obtained by allowing the symmetrical off-diagonal matrix blocks in Kim and Lee (2000) to be non-symmetrical, hence there are more variables in the matrices than that in Kim and Lee (2000) and admit more great freedom (or dimension) in guaranteeing the stability of the fuzzy control systems than previous papers. It is useful in designing the fuzzy controllers, especially when the design problem involves not only stability, but also the other performance requirements such as the speed of response, constrains on control input and output, robust stabilization and so on. By the same example as Kim and Lee (2000), we show that the new conditions greatly relax the quadratic stability conditions in Kim and Lee (2000). Since the observer designs in this paper are based on the new stability conditions proposed by this paper, hence the new approaches of observers designs relax the conservatism of the conditions derived by the previous works such as Tanaka, Ikede, & Wang 1998a, Tanaka, Ikeda, & Wang 1998b, Yoneyama, Masahiro, Hitoshi, and Akria (2000) and Ma, Sun, and He (1998). By the examples, we show the new fuzzy observer design methods are very practical and efficient.

H control has been an attractive research topic during the last decade (Doyle, Glover, Khargonekar, & Francis, 1989; Isidori & Astofi, 1992; Khargoneker, Petersen, & Zhou, 1990). So far some papers (Lee, Jeung, & Park, 2001; Chen, Tseng, & Uang, 1999; Cao, Rees, & Feng, 2000) have discussed the H feedback control for fuzzy systems. They dealt with a state feedback control design that requires all states of systems to be measured. In many cases, this requirement is too restrictive. Very recently, some papers (Lee, Kim, Jeung, & Park, 2000; Tseng, Chen, & Uang, 2001; Chen, Tseng, & Uang, 2000) have discussed the H control based on fuzzy observers for fuzzy systems. In Section 4 of this paper, two new designs of the H control based on the new fuzzy observers for T-S fuzzy systems have been proposed. Comparing to Lee et al. (2000), Tseng et al. (2001) and Chen et al. (2000), the new approaches are different in the following ways: First, the T-S fuzzy system model presented in this paper is different from that in them. In Tseng et al. (2001) and Chen et al. (2000), they consider the H control attenuating x̃, which is the observer of x, from the disturbance w̃, i.e. 0tfx̃T(t)Q̃x̃(t)dt⩽x̃T(0)P̃x̃(0)+ρ20tfw̃T(t)w̃(t)dt. But this paper considers the H controller u(t)=∑i=1rλi(ζ)Kix̃(t) to attenuate z(t)=∑i=1rλi(ζ)(C1ix(t)+D12iu(t)) from the disturbance w(t) i.e. ||z||2γ||w||2. In many paper such as Cao, Rees, & Feng 1996a, Cao, Rees, & Feng 1996b; Feng, Cao, and Rees (1996), Tseng et al. (2001) and Chen et al. (2000)D12i=0,i=1,2,…,r, this may results in that ||Ki||,i=1,2,…,r, can be very large such as examples in Chen et al. (2000) and fail to satisfy ||z||2γ||w||2 when D12i≠0. One advantage of designs in this paper is that the H controller can achieve ||z||2γ||w||2 by the relative small feedback gains Ki, i=1,2,…,r. Secondly, the conditions for the existence of the H controllers in this paper relax the restrictions given in Lee et al. (2000), Tseng et al. (2001), Zhang and Feng (2001) and Chen et al. (2000), in which the left parts of the matrix inequalities ij in the conditions must be negative definite. By the analysis of some groups of often used condition forms for H control designs, we show that the conditions in Lemma 1 admit more freedom in guaranteeing the existence of the H control based on fuzzy observers for T-S fuzzy systems. Thirdly, notice that each Hk<0,k=1,2,…,r, whereHk=Z11Z1rV1kTZr1ZrrVrkTV1kVrk−Iand Zij is determined by the subfuzzy systems i,j, is a single matrix built up by the coefficient matrices of all subfuzzy systems in Theorem 4, i.e. the interactions among the subfuzzy systems are aggregated in a single matrix Hk and solved in a numerical manner. Hence the designs for the H controller based on fuzzy observers are the results of the interactions among the subfuzzy systems. As opposed to the previous works about H controller designs such as Lee et al. (2000), Tseng et al. (2001), Zhang and Feng (2001) and Chen et al. (2000), in each matrix inequality, just interactions between two subfuzzy systems were considered. We notice often more than two subfuzzy systems are fired at each time in fuzzy systems specially for high-dimension systems.

The paper is organized as follows. Section 2 presents the relaxed quadratic stability conditions. In Section 3, the fuzzy observers based on the relaxed stability conditions for the T-S fuzzy systems are presented. In Section 4, two new H control designs based on the new fuzzy observers are considered. In Section 5, simulation examples are provided to demonstrate the designs effectiveness. Finally, concluding remarks are made in Section 6. The proof of Lemma 1 is given in Appendix A.

Section snippets

The relaxed quadratic stability conditions

Consider the T-S fuzzy dynamic model described by the following fuzzy IF–Then rules:

IF ξ1(t) is M1i and …and ξp(t) is Mpi, THENẋ(t)=Aix(t)+B1iw(t)+B2iu(t),z(t)=C1ix(t)+D12iu(t),y(t)=C2ix(t)+D21iw(t),i=1,2,…,r,x(t)∈Rn is the state variable, z(t)∈Rq is the controlled output variable, w(t)∈Rl is the disturbance variable, u(t)∈Rm is the input variable, y(t)∈Rh is the output variable,AiRn×n, B1iRn×l, B2iRn×m, C1iRq×n, D12iRq×m, C2iRh×n, D21iRh×l, ξ1,…,ξp are premise variables and measurable.

Observer design for T-S fuzzy systems

In this section, we propose new fuzzy observers based on Theorem 2 to estimate states of the T-S fuzzy model (1). Fuzzy observers are required to satisfy x(t)−η(t)→0 exponentially as t→∞, where η denotes the state vector estimated by a fuzzy observer. The new fuzzy observers we proposed for T-S fuzzy system (1) are the following (9):η̇(t)=i=1rλiAiη+B1iw+B2iu−Liy−j=1rλj(C2jη+D21jw).

Theorem 3

If there exist matricesNi,X,Xij, whereXis symmetrical positive definite, Xiiare symmetrical, Xji=XijT, ij, i,j

H control based on fuzzy observers for the T-S fuzzy systems

In the following, we study H control based on observer (9) for the T-S fuzzy system (1). The H performance is defined as follows.

Definition 2

For T-S fuzzy system (1), when u(t)≡0. If for any w(t)∈L2(0,∞;Rl) (the space of square integrable functions), z(t)∈L2(0,∞;Rq), then (1) is called input–output stable. For a given positive number γ>0, if (1) is quadratically stable and input–output stable with ||z||2γ||w||2, under zero initial condition, where ||x(t)||2=(0xT(t)x(t)dt)1/2 is the L2-norm, then we

Numerical examples

To illustrate the H controller designs based on the fuzzy observers, we consider two examples. First, we apply Theorem 4, Theorem 5 to study the same H controller design problems in Example 2. Then consider the real world problem of balancing an inverted pendulum on a cart in Example 3.

Example 2

Let us consider the following T-S fuzzy systemẋ(t)=i=12λi(x(t))(Aix(t)+B1iw(t)+B2iu(t)),z(t)=i=12λi(x(t))(C1ix(t)+D12iu(t)),y(t)=i=12λi(x(t))(C2ix(t)+D21iu(t)),

where λ1=(2−0.5|x1|)/2, when x1⩽2; λ1=0, when x

Conclusion

By the examples and the theorems, we can draw the following conclusions: The new quadratic stability conditions and the existence conditions of the observers are much simple and relaxed than those in previous papers . Two new sufficient conditions in the terms of LMIs, which admit more freedom in guaranteeing the H control performances, have been proposed. In general, the new relaxed conditions of fuzzy regulators, fuzzy observers and H controller based on fuzzy observers are not only simple

Acknowledgements

This work is supported by the Natural Science Funds of China (60174014).

Xiaodong Liu received the B.S. and the M.S. degrees in mathematics from Northeastern Normal University in 1986 and Jilin University, Jilin, in 1989, P. R. China respectively, and the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, P. R. China in 2003. He is currently a Professor and Academic Director in Department of Mathematics and Physics, Dalian Maritime University, Dalian, P. R. China. He was a Senior Visiting Scientist in Department of

References (28)

  • A. Isidori et al.

    Disturbance attenuation and H control via measurement feedback in nonlinear systems

    IEEE Transactions on Automatica Control

    (1992)
  • E. Kim et al.

    New approaches to relaxed quadratic stability condition of fuzzy control systems

    IEEE Transactions on Fuzzy Systems

    (2000)
  • P.P. Khargoneker et al.

    Robust stabilization of uncertain linear systemsQuadratic stability and H control theory

    IEEE Transactions on Automatic Control

    (1990)
  • K.R. Lee et al.

    Robust fuzzy H control for uncertain nonlinear systems via state feedbackAn LMI approach

    Fuzzy Sets and Systems

    (2001)
  • Cited by (733)

    View all citing articles on Scopus

    Xiaodong Liu received the B.S. and the M.S. degrees in mathematics from Northeastern Normal University in 1986 and Jilin University, Jilin, in 1989, P. R. China respectively, and the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, P. R. China in 2003. He is currently a Professor and Academic Director in Department of Mathematics and Physics, Dalian Maritime University, Dalian, P. R. China. He was a Senior Visiting Scientist in Department of Electrical and Computer Engineering, University of Alberta, Edmonton Canada in 2003. He has been a Reviewer of American Mathematical Reviewer since 1993. He is a coauthor of three books. His research interests include algebra rings, combinatorics, topology molecular lattices, AFS (axiomatics fuzzy sets) theory and its applications, knowledge discovery and representations, data mining, pattern recognition and hitch diagnoses, analysis and design of intelligent control systems. Dr. Liu is a recipient of the 2002 Wufu-Zhenhua Best Teacher Award of the Ministry of Communications of People's Republic of China.

    Qingling Zhang received his Bachelor and Master degrees from Mathematics Department and Ph D degree from Automatic Control Department of Northeastern University in 1982, 1986 and 1995 respectively. He finished his two-year postdoctoral work in Automatic Control Department of Northwestern Polytechnical University in 1997. Since then he has been a professor and dean of college of science in Northeastern University. He is also a member of university teaching advisory committee of National Ministry of Education. He published 2 books and more than 170 papers about control theory and applications. He received 12 prizes from central and local governments for his researches respectively. He also received Gelden Scholarship from Australia in 2000. During these periods he visited Hong Kong University, Sydney University, Western Australia University and Niigata University as research associate, research fellow and senior research fellow respectively.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Gary G. Yen under the direction of Editor Robert R. Bitmead.

    View full text