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On well-posedness of relay systems

The talk is focused on the well-posedness of relay systems, namely, given a single-input single-output (non)-linear system S closed by a negative relay (sign) feedback; the problem addressed in this talk is to find conditions under which there exists a (locally) unique solution starting from any given initial condition. In general, a solution to this problem depends on a concept of the solutions for systems with discontinuous right-hand side. If the system S is linear, the closed loop system can be described by a Linear Complementarity Systems (LCS) framework. Using basic results from LCS theory it is known that a relay system is well posed (has a unique "forward" solution) provided the first nonzero Markov parameter of the linear counterpart is positive. This result is based on a solution concept employed in LCS theory. Particularly, a forward solution to a relay system is a continuous piece-wise real-analytic function of time. At the same time for the relay systems one can use Filippov's concept of solutions which extends the class of the solutions to a broader class of absolutely continuous functions of time. Thus one can pose the following problem: given a relay system, is it possible to derive the well-posedness conditions (in the Filippov sense) in terms of the Markov parameters of its linear counterpart? For relative degrees one and two, the answer is positive. However, for the relative degree three it is possible to design a counterexample, i.e. there is a relay system which is well-posed in the forward sense, but not well-posed in the Filippov sense. This counterexample is closely related with possible Zeno behavior in relay systems. Additionally we derive sufficient conditions which exclude a possibility of the Zeno behavior in relay systems. The results are obtained in cooperation with M.Heemels and H.Nijmeijer (TUE).

Type of Seminar:
Minisymposium on Analysis and Control of Dynamical Systems
Dr. Alexander Pogromsky
Electrical Engineering Department, Eindhoven University of Technology, The Netherlands
Dec 19, 2000   08:15

ETH Zentrum, VAW building room B1, Gloriastr. 37-39, 8006 Zurich
Contact Person:

Prof. M.Morari
No downloadable files available.
Biographical Sketch:
Alexander Pogromsky was born in Saint-Petersburg, Russia, on March 1, 1970. He received the M.Sc degree from the Baltic State Technical University, Russia in 1991 and Ph.D degree from the St.Petersburg Electrotechnical University in 1994. From 1995 till 1997 he was with the Laboratory ``Control of Complex Systems'' (IPME, St. Petersburg, Russia). From 1997 to 1998 he was a postdoc with the Department of Electrical Engineering, of Linkoping University, Sweden. Currently he is with the Electrical Engineering Department of Eindhoven University of Technology as a postdoc. In 1997 he was awarded by the Russian Presidential Fellowship for young scientists. His research interests include theory of nonlinear control, nonlinear oscillations.