Re: System Stability

Constantin von Altrock (cva@inform-ac.com)
Wed, 8 May 1996 13:19:49 +0200


Dear Anthony

You asked for a reference on the discussion about fuzzy stability:

> 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. 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?

Let me give you a citation from the book "Fuzzy Logic and NeuroFuzzy
Applications Explained" by Prentice Hall Publisher 1995 (ISBN
0-1336-8465-2), that discusses the subject in some detail:

Stability and Verification of Fuzzy Logic Systems
========================================

Many critics of fuzzy logic claim that there is no such thing as a
stability proof for fuzzy logic systems in closed-loop control. Some
of them even stem from a sound scientific background. In some
countries, the discussion about fuzzy logic and stability analysis has
almost gotten into a "religious war". Let me give you some facts on
the topic here. I will restrict it to what is of practical interest.
For an excellent treatment of this story in general, refer to [97, pp.
341].

Stability Proof for Fuzzy Logic Controller: The Good News

There is good news and there is bad news for those who ask whether a
stability proof for fuzzy logic controllers exists. First the good
news: of course, such a stability proof exist. Now the even better
news: stability theory of conventional control completely suffices for
this.

If you try to define mathematically what a fuzzy logic system is, you
could state that it is a mapping of an input space Rn into an output
space Qm with the following properties:

- deterministic
(the same input condition always resutls in the same output condition)

- time invariant
(the transfer function describing the mapping does not change over
time)

- non-linear
(the output variables are no linear combination of the input
variables)

Control theory classifies such a system as a " non-linear
multivariable controller" or "multiband controller". Hence, all
stability analysis methods applicable to these controller types are
applicable to fuzzy logic controller.

However, due to the complex non-linearities of a fuzzy logic system,
an analytical solution is not possible in most cases. This is no
limitation, as fuzzy logic development tools offer links to process
simulation software such as Mathlab/Simulink, VisSim and Matrixx that
support numerical stability analysis (refer to section 3.1. for more
details). On the other hand, analytical stability analysis is also
impossible for conventional controller with a similar degree of
complexity.

Also, there is work on a stability theory dedicated to fuzzy logic
systems. In this scientific discipline, researchers try to establish
general stability proofs for fuzzy logic systems. The methods
developed so far are only applicable to very simple fuzzy logic
controller architectures. Hence, this fuzzy stability theory has no
practical relevance yet. On the other hand, it is a hot topic among
the control theory reseach community. If you are interested in this
work, start with the papers [161, 92, 95, 162] and the literature
cited herein.

Now the Bad News

In fuzzy logic or conventional control engineering: stability analysis
plays a minor role in most engineering applications. Some
theoreticians claim that this is due to the fact that most
practioneers do not have a sound background in control theory to
properly conduct stability analysis. Even though, this may be a factor
in some applications, in most applications the reason is much simpler.
To do stability analysis, you need a mathematical model of the process
under control. For a reliable stability analysis, the model must be
very accurate. The problem is, only for a small fraction of complex
applications, building such a model is possible with reasonable
effort.

To illustrate the role of stability analysis in industry, let me cite
a friend of mine, who develops closed-loop control systems for fighter
airplanes:

"Every engineer who went from campus directly into management,
requires us to prove stability for al control systems we design. Also,
Government and the FAA require these analyses. However, the
mathematical models we developed for our systems under control do not
even suffice to implement model-based controllers just from them. We
develop most of our controller from experience and optimize them in
test flights. Likewise, the quality of our mathematical models are
also not sufficient to do a sound stability proof. However, we must do
a stability proof. So we do them and use our limited mathamatical
models and reference all premises and restrictions of the model. We
never had to discuss with our management or the FAA whether these
premises are reasonable or not. On the other hand, the control systems
we do are very stable. Our experience allows us to evaluate the
stability also by experiments and test flights."

Here is another potential benfit of fuzzy logic design. In a fuzzy
logic system, the individual rules represent local behavior. That is,
each fuzzy logic rule describes the reaction to a certain situation.
Such a situation could be "high temperature, low conversion, and
medium pressure". If you encounter stability problems in this
situation, you modify the respective rule. The advantage of this is,
that all the other rules remain unchanged. This enables a
goal-oriented stability optimization. With conventional control
techniques most of the fudge factors used to tweak the system, do have
global effects of the system's performance. Hence, a modification that
optimizes one situation can have bad effects on other situations.
Optimization can then become a trial-and-error task.

Practical Experience with Stability Analysis

- A stability theory dedicated for fuzzy logic systems exists.
However, it has no practical relevance, as it is only applicable for
very small systems or special cases.

- Conventional stability analysis techniques for multiband controller
is completely applicable for fuzzy logic controller.

- Stability analysis with fuzzy logic systems and conventional systems
suffer from the same deficiency: mathematical models of the process
under control often either are of prohibitive effort to derive or are
not sufficiently accurate.

Constantin von Altrock
e-mail: cva@inform-ac.com

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