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Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach Kazuo Tanaka, Hua O. Wang
Copyright Q 2001 John Wiley & Sons, Inc. CHAPTER 3 ISBNs: 0-471-32324-1 ŽHardback.; 0-471-22459-6 ŽElectronic.
LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
The preceding chapter introduced the concept and basic procedure of parallel distributed compensation and LMI-based designs. The goal of this chapter is to present the details of analysis and design via LMIs. This chapter forms a basic and important component of this book. To this end, it will be shown that various kinds of control performance specifications can be repre-sented in terms of LMIs. The control performance specifications include stability conditions, relaxed stability conditions, decay rate conditions, con-strains on control input and output, and disturbance rejection for both continuous and discrete fuzzy control systems w1]3x. Other more advanced control performance considerations utilizing LMI conditions will be pre-sented in later chapters.
3.1 STABILITY CONDITIONS
In the 1990’s, the issue of stability of fuzzy control systems has been investigated extensively in the framework of nonlinear system stability w1]18x. Today, there exist a large number of papers on stability analysis of fuzzy control in the literature. This section discusses some basic results on the stability of fuzzy control systems.
In the following, Theorems 5 and 6 deal with stability conditions for the open-loop systems. Theorem 5 can be readily obtained via Lyapunov stability theory. The proof of Theorem 6 is given in w4,7x.
49
50 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
THEOREM 5 wCFSx The equilibrium of the continuous fuzzy system Ž2.3. with uŽt.s 0 is globally asymptotically stable if there exists a common positi®edefinite matrix P such that
Ai P q PAi -0, is 1, 2, . . . , r, Ž3.1.
that is, a common P has to exist for all subsystems.
THEOREM 6 wDFSx The equilibrium of the discrete fuzzy system Ž2.5. with uŽt.s 0 is globally asymptotically stable if there exists a common positi®edefinite matrix P such that
Ai PAi y P -0, is 1, 2, . . . , r, Ž3.2.
that is, a common P has to exist for all subsystems.
Next, let us consider the stability of the closed-loop system. By substituting Ž2.23. into Ž2.3. and Ž2.5., we obtain Ž3.3. and Ž3.4., respectively.
CFS
r r
xŽt. s Ý Ý hiŽzŽt..hjŽzŽt..Ai y Bi Fj4xŽt.. Ž3.3.
is1 js1
DFS
r r
xŽtq 1. s Ý Ý hiŽzŽt..hjŽzŽt..Ai y Bi Fj4xŽt.. Ž3.4.
is1 js1
Denote
Gij s Ai y Bi Fj.
Equations Ž3.3. and Ž3.4. can be rewritten as Ž3.5. and Ž3.6., respectively.
CFS
r
xŽt. s ÝhiŽzŽt..hiŽzŽt..Gii xŽt.
is1
q 2 Ý ÝhiŽzŽt..hjŽzŽt..½Gij q Gji 5xŽt.. Ž3.5. is1 i-j
DFS
r
xŽtq 1. s ÝhiŽzŽt..hiŽzŽt..Gii xŽt.
is1
q 2 Ý ÝhiŽzŽt..hjŽzŽt..½Gij q Gji 5xŽt.. Ž3.6. is1 i-j
STABILITY CONDITIONS 51
By applying the stability conditions for the open-loop system ŽTheorems 5 and 6. to Ž3.5. and Ž3.6., we can derive stability conditions for the CFS and the DFS, respectively.
THEOREM 7 wCFSx The equilibrium of the continuous fuzzy control system described by Ž3.5. is globally asymptotically stable if there exists a common positi®e definite matrix P such that
Gii P q PGii -0, Ž3.7.
žGij q Gji /T P q P žGij q Gji /F 0,
i-j s.t. hi l hj /f. Ž3.8.
Proof. It follows directly from Theorem 5.
For the explanation of the notation i-j s.t. hi l hj /f, refer to Chapter 1.
THEOREM 8 wDFSx The equilibrium of the discrete fuzzy control system described by Ž3.6. is globally asymptotically stable if there exists a common positi®e definite matrix P such that
Gii PGii y P -0, Ž3.9.
žGij q Gji /T P žGij q Gji /y P F 0,
i-j s.t. hi l hj /f. Ž3.10.
Proof. It follows directly from Theorem 6.
The fuzzy control design problem is to determine Fj’s Ž js 1,2,. . . , r. which satisfy the conditions of Theorem 7 or 8 with a common positive definite matrix P.
Consider the common B matrix case, that is, B1 s B2 s ??? s Br. In this case, the stability conditions of Theorems 7 and 8 can be simplified as
follows.
COROLLARY 1 Assume that B1 s B2 s ??? s Br. The equilibrium of the fuzzy control system 3.5 is globally asymptotically stable if there exists a common positi®e definite matrix P satisfying Ž3.7..
COROLLARY 2 Assume that B1 s B2 s ??? s Br. The equilibrium of the fuzzy control system 3.6 is globally asymptotically stable if there exists a common positi®e definite matrix P satisfying Ž3.9..
52 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS
In other words, the corollaries state that in the common B case, Gii P q PGii -0 implies
žGij q Gji /T P q P žGij q Gji /F 0 and Gii PGii y P -0 implies
žGij q Gji /T P žGij q Gji /y P F 0
To check stability of the fuzzy control system, it has long been considered
difficult to find a common positive definite matrix P satisfying the conditions of Theorems 5]8. A trial-and-error type of procedure was first used w4,7,9x. In w19x, a procedure to construct a common P is given for second-order fuzzy systems, that is, the dimension of the state is 2. It was first stated in w11,12,17x that the common P problem for fuzzy controller design can be solved numerically, that is, the stability conditions of Theorems 5]8 can be expressed in LMIs. For example, to check the stability conditions of Theorem 7, we need to find P satisfying the LMIs
P )0, Gii P q PGii -0,
žGij q Gji /P q PžGij q Gji /F 0, i-j s.t. hi l hj /f,
or determine that no such P exists. This is a convex feasibility problem. As shown in Chapter 2, this feasibility problem can be numerically solved very efficiently by means of the most powerful tools available to date in the mathematical programming literature.
3.2 RELAXED STABILITY CONDITIONS
We have shown that the stability analysis of the fuzzy control system is reduced to a problem of finding a common P. If r, that is the number of IF-THEN rules, is large, it might be difficult to find a common P satisfying the conditions of Theorem 7 Žor Theorem 8.. This section presents new stability conditions by relaxing the conditions of Theorems 7 and 8. Theorems 9 and 10 provide relaxed stability conditions w1]3x. First, we need the following corollaries to prove Theorems 9 and 10.
COROLLARY 3
r r
Ýhi ŽzŽt.. y Ý Ý2hiŽzŽt..hjŽzŽt.. G 0,
is1 is1 i-j
RELAXED STABILITY CONDITIONS 53
where
for all i.
r
ÝhiŽzŽt.. s 1, hiŽzŽt.. G 0
is1
Proof. It holds since
r r
Ýhi ŽzŽt.. y Ý Ý2hiŽzŽt..hjŽzŽt..
is1 is1 i-j
s 1 Ý ÝhiŽzŽt.. y hjŽzŽt..42 G 0. Q.E.D. is1 i-j
COROLLARY 4 If the number of rules that fire for all t is less than or equal to s, where 1 -sF r, then
r r
Ýhi ŽzŽt.. y Ý Ý2hiŽzŽt..hjŽzŽt.. G 0,
is1 is1 i-j
where
for all i.
r
ÝhiŽzŽt.. s 1, hiŽzŽt.. G 0
is1
Proof. It follows directly from Corollary 3.
THEOREM 9 wCFSx Assume that the number of rules that fire for all t is less than or equal to s, where 1 -sF r. The equilibrium of the continuous fuzzy control system described by Ž3.5. is globally asymptotically stable if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q such that
Gii P q PGii q Žsy 1.Q -0 Ž3.11.
žGij q Gji /T P q P žGij q Gji /y Q F 0,
i-j s.t. hi l hj /f Ž3.12.
where s)1.
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