144 CHAPTER 4. FUZZY CONTROL
4.3 Stability of fuzzy control systems..................
Fuzzy control designs seem appropriate for nonlinear systems whose mathemati-
cal models are not available. While the design of fuzzy controllers, based mostly
on linguistic rules, is relatively simple and leads to successful applications; it re-
mains to be seen how the most important problem in analysis of control systems,
namely the problem of stability, is addressed.
In standard control theory, one needs to study the stability of the control
system under consideration, either for its open-loop dynamics or its closed-loop
dynamics (assuming a control law has been designed). This is based on the
availability of the mathematical model for the system. Although in nonlinear
systems where solutions of nonlinear differential equations are difficult to obtain,
the knowledge of these equations is essential for stability analysis, such as in
the method of Lyapunov.
From a practical viewpoint, one can always use simulations to test stability
of the system when a fuzzy controller has been designed. But it is desirable to
have a theoretical analysis of stability rather than just ìblindî simulations. The
problem of stability analysis of fuzzy control systems is still an active research
area in the theory of fuzzy control. Now, as stated above, it is difficult to see
how stability analysis can be carried out without mathematical models of the
systems, but fuzzy control is intended primarily for systems whose mathematical
models are not available in thefirst place. Thus, to have a stability analysis
for fuzzy control, it seems plausible that some form of models should be known.
In other words, stability analysis of fuzzycontrol systems is only possible if the
problem is model-based.
A model-based approach to fuzzy control does not necessarily mean that
mathematical models are available. The models we refer to are fuzzy models
as opposed to precise mathematical models used in standard control. Mathe-
matically speaking, a fuzzy model is a mathematical model that involves fuzzy
terms. Thus, fuzzy models are generalizations of ìcrispî models.
The stability analysis of model-based fuzzy control systems was carried out
by Tanaka and Sugeno [73] in 1992 using the Takagi-Sugeno model [72]. Here
is a summary of their approach to stability analysis, using Lyapunovís direct
method. In the discrete form, the Takagi-Sugeno fuzzy dynamical model de-
scribes locally a linear relationship between inputs and outputs of the system.
Let the state-variable and control input be
x(k)=°
x 1 (k) x 2 (k) ∑∑∑ xn(k)¢T
u(k)=°
u 1 (k) u 2 (k) ∑∑∑ un(k)¢T
The fuzzy model is a collection of fuzzy rulesRj,j=1, 2 ,...,r,oftheform
Rj:Ifx 1 (k)isAj 1 andx 2 (k)isAj 2 and ... andxn(k)isAjn,
thenx(k+1)=Ajx(k)+Bju(k)where theAjiís,j=1,...,r;i=1,...,n, are fuzzy subsets of the state space.