# About passivity and stability in fuzzy control

Calcev George (calcev@auto.ucl.ac.be)
Mon, 13 May 1996 13:37:06 +0200

Jobst Schmalenbach wrote:
>
> Calcev George <Calcev@auto.ucl.ac.be> writes:
>
> >I agree with the opinion of Dr. Contantin von Altrock concerning
> > stability of fuzzy control systems.
> >Nevertheless, I would like to add that this problem (very often controversial
> >one) seems to be not very hard. In most of the applications of fuzzy
> >control systems, the stability can be obtained as a simple application of
> > Passivity Theorem, ( not only for simple fuzzy controllers).
> ^^^^^^^^^^^^^^^^^
> There maight be others who would be interrested to know....
>
> [snip]
>
> thanks
> jobst

If one consider a nonlinear dynamic system:

x'=f(x,u)
y= h(x,u)

this system it will be called passive if there exists a nonnegative
function V(x) with V(0)=0 called storage function s.t
V(x(t))- V(x(0)) \leq \int_0^{t} y^T(\tau) u(\tau) d \tau

i.e (in terms of an energy equivalent) if the stored energy is less than
supplied energy. A passive system is Lyapunov stable if the storage
function is positive definite.

Now, if two passive systems are in a feedback connection, the resulted
system it is still passive.

In the same manner one can define strictly passive systems, which are
asymptotic stable systems.

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.

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.

References (some of the best) for passivity theory:

V. M. Popov Hiperstabilitatea Sistemelor Automate, ed,Academiei, 1966
(in rumanian)
V. M. Popov Hyperstabilite des Systemes Automatiques, Dunod, 1973
(in french)

Sorry I have not the reference for the english version (but there
exists).

J.C. Willems 'Dissipative dynamical systems - Part I : General Theory'
Arch. Rational Mechanics and Analysis vol 45 pp 321-351, 1972

Hill D. and P Moylan 'Stability results for nonlinear feedback systems'
Automatica, vol 13, pp377-382, 1977

C. Byrnes, A. Isidori, J, Willems,'Passivity, Feedback Equivalence, and
the Global Stabilization of Minimum Phase Nonlinear Systems' ,
IEEE Tr. AC vol 36, pp. 1228-1240, 1991

....................

George Calcev