rojas@aic.dpl.scg.hac.com (07/21/90)
Joseph Maurice Rojas
Hughes Research Laboratories
Malibu, CA 90265
(213) 317-5765
rojas@math.berkeley.edu
rojas@aic.hrl.hac.com
July 18, 1990
Dear colleagues,
I was reading the paper "Stability Regions of Nonlinear Autonomous
Dynamical Systems" by Chiang, Hirsch, and Wu (IEEE Transactions on Automatic
Control, vol. 33, No. 1, Jan. 1988) when I noticed that "autonomous" in the
dynamical systems context meant non time-varying. My summer work deals with
the control of an autonomous submarine. In the context of AI, "autonomous"
means intelligent in some task-oriented sense. My project is to define and
characterize "instability" in the autonomous control logic of this submarine,
and I would like to know if the ideas of the above paper have been generalized
enough to include my project.
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? I apologize in advance for any
abuse of terminology but control theory is not my strong point. I'm actually
a math graduate student specializing in modular functions. Any ideas or
leads regarding stability regions of time-varying systems would be greatly
appreciated. Thank you in advance for your time.
Sincerely,
Maurice
P.S.: I realize this bulletin board may not be the best place for my topic,
but is there a sci.control-theory or something like that? If so, please
let me know! nadel@aerospace.aero.org (Miriam H. Nadel) (08/04/90)
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