Re: About passivity and stability in fuzzy control

Ralf Mikut (
Tue, 14 May 1996 15:33:28 +0200

> Calcev George <> wrote

> Obvious this definition is for the systems with the same number of
> inputs and outputs. Therefore if one considers a fuzzy controller as a dynamic
> system with one input and one output by taking in consideration the
> dynamic relation between the inputs (error and derivative of error ) the
> stability of fuzzy control loops appears as trivial if the plant has
> some passivity properties (which is the case almost every time) and the
> fuzzy decision mapping some rough sectorial properties.

see e.g.
OPITZ, H.-P.: Fuzzy Control and Stability Criteria. Proc., EUFIT'93,
Aachen, pp. 130 - 136; 1993

Here, these problems have been discussed especially for the stability proof
of fuzzy systems.

> This result it's in the same class as Popov type results , because
> the hyperstability term introduced by rumanian scientist Popov it is
> nothing else than passivity concept from works of Willems, Hill and
> Moylan, Narendra.... Therefore proves the intrinsic robustness
> of such a systems, robustness specific to the absolute stability
> results.

The problem is that this passitivity proof in more complex loops can
cause a lot of problems. For linear subsystems, the so-called
Kalman-Yakubovich Lemma has to be fulfilled (e.g. a PT1 system
fulfills it but a PT2 system don't fulfill it).
Otherwise, some tricks has to be searched to transform the system in
such a manner that passive subsystems can be separated.
Two years ago, we tested the stability of a simple flow control
system with a static nonlinearity and a fuzzy controller which tunes
the PID gain. The way to testing the hyperstability required the moving of
some blocks through the system to handle a dead time of the process
and to assemble all subsystems to one linear and one nonlinear block.

Nonlinear subsystems of the process has to be handled searching a
form to proof an inequality describing the sector bounds...
The problems increases if the nonlinearity is not only a static one.

>From my point of view, a really sufficient way to proof the
hyperstability of complex technical system has not yet been developed.


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