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From: nadel@aerospace.aero.org (Miriam H. Nadel)
Newsgroups: comp.theory.dynamic-sys
Subject: Re: Finding stability regions in time-varying dynamical systems.
Message-ID: <80040@aerospace.AERO.ORG>
Date: 3 Aug 90 20:54:15 GMT
References: <9636@hacgate.UUCP>
Reply-To: nadel@aero.UUCP (Miriam H. Nadel)
Organization: The Aerospace Corporation, El Segundo, CA
Lines: 39

In article <9636@hacgate.UUCP> rojas@aic.dpl.scg.hac.com () writes:
>
>	In particular, what is known about finding stability regions of 
>time-varying systems?  What about systems with unknown time-varying parameters,
>such as a submarine in unknown currents?

First off, there's some obvious things.  Lyapunov's stability theorem, for
example, is not restricted to time-invariant systems but it can be pretty
hard to come up with a suitable Lyapunov function and the local stability
definitions used in methods derived from it may not be appropriate for
your application (e.g. do you want to consider a limit cycle stable)

I can't think of any good reason why cell mapping techniques (I can't
come up with exact references offhand but there have been a number of papers
by Hsu and Guttalu) wouldn't work for time-varying systems but one might
need to generate different maps depending on initial conditions.

But the issue of unknown time-varying parameters raises an interesting
point about stability definitions.  It would seem to me that one designing
a controller for a submarine in unknown currents would care about something
analogous to "hyperstability."  I don't want to use the word "robustness" 
because it's defined too many different ways by different authors.  But if
you think of the Popov stability criterion, it specifies the conditions
under which a system with a linear part and a nonlinear part is stable for
a range of nonlinearities.  (This may be too vague a description for those
not familiar with Popov's hyperstability theorem, but detailed explanations
are in pretty much every book on advanced control techniques.  I suggest
Hsu and Meyer, _Modern Control Principals and Applications_ as a good general 
reference.)  At any rate, it seems like you need a criterion to impose on
the submarine which would be valid over the whole range of currents you
might expect, so stability in the conventional sense isn't enough.

Miriam Nadel

-- 
One of the 84% of Americans who would not oppose a marriage between a family
member and a person with a severe physical handicap.

nadel@aerospace.aero.org