Re: System Stability

Ralf Mikut (
Wed, 8 May 1996 14:15:38 +0200

Dear Anthony,

> Recently at a technical conference I again heard someone questioning the
> stability and robustness of fuzzy systems, and in a separate incident, I
> heard someone refer to fuzzy systems as inherently stable.

Fuzzy controllers use human expert knowledge to control systems.
However, most of the implementations obtain the routine behaviour
of the human operator only. More sophisticated tasks like supervision
and adaptation are not implemented although this supervision
guarantees the ability of human operators to control complex systems in a
robust and stable way. Typical risks for the stability behaviour are:
1. uncertainties in the knowledge base (nobody is able to formulate it's
complete knowledge including the unconscious parts)
2. the definition of the membership functions of the fuzzy controller
3. inference problems in complex fuzzy rulebases using some today's fuzzy
operators (some strange results between different rules).

> These remarks were made to separate audiences, and in neither case were
> the statements challenged!
> Does anyone know of a good refererence for a discussion of this topic?

>From a control point of view, a fuzzy system is nothing else than a
nonlinear control system which can be described e.g by nonlinear
algebraic equations and nonlinear differential equations.

Many mathematical methods have been developed to solve the stability problem of
fuzzy systems, but they have a very sophisticated theoretical level and are only
available for simple-structured systems. An overview is given for instance in

BRETTHAUER, G.; OPITZ, H.-P.: Stability of Fuzzy Systems - A Survey.
Proc., EUFIT'94,Aachen, pp. 283 - 290; 1994

TITLI, A.; MARIN, P.: Comparative analysis of stability methods for fuzzy
controllers. Proc.,EUFIT'94, Aachen, pp. 1183 - 1187; 1994

The base of these techniques is the nonlinear control theory with the steps
1. building a mathematical model (e.g. using fuzzy modelling or ANNs)
2. interpreting the fuzzy controller as a static nonlinearity
3. using the test methods for nonlinear systems.

Some methods can be found e.g. in

KIENDL, H.; RUEGER, J. J.: Stability analysis of fuzzy control systems using
facet functions. Fuzzy Sets and Systems, 70, pp 275 - 285; 1995

TANAKA, K.; SUGENO, M.: Stability analysis and design of fuzzy control
systems. Fuzzy Sets and Systems, 45, pp. 135 - 156; 1992

WANG, L.: Supervisory controller for fuzzy control systems that
guarantees stability. Proc., IEEE Conf. on Fuzzy Systems, Orlando,
pp. 1035 - 1039; 1994

WANG, L. X.: Stable adaptive fuzzy control of nonlinear systems.
IEEE Trans. on Fuzzy Systems, 1, pp. 146 - 155; 1993

Today, all these methods are not in use for practical processes.

We try to develop a supervision component using fuzzy approaches
and some ideas of the nonlinear control theory to identify critical
situations and to start, in these cases, a strategy to save the stability
(e.g. by switching to a more conservativ strategy), in the same way as
human being realizes it.

The concept developed for it has been described in

On-line stability supervision with a hierarchical fuzzy
concept and Fuzzy-Lyapunov functions.
Proc., EUFIT'95, Aachen, pp. 775 - 780; 1995

I hope it could be helpful for you.

Best regards


Ralf Mikut
University of Mining and Technology Freiberg
Institute of Automatic Control
D-09596 Freiberg, Lessingstr.45
phone: ++49-(0)3731-393247
fax: ++49-(0)3731-392925