Brief On solution concepts and well-posedness of linear relay systems

Authors:
A. Yu. Pogromsky;W. P. M. H. Heemels;H. Nijmeijer
Affiliations:
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, Netherlands;Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, Netherlands;Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, Netherlands
Venue:
Automatica (Journal of IFAC)
Year:
2003

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Abstract

In this paper we study the well-posedness (existence and uniqueness of solutions) of linear relay systems with respect to two different solution concepts, Filippov solutions and forward solutions. We derive necessary and sufficient conditions for well-posedness in the sense of Filippov of linear systems of relative degree one and two in closed loop with relay feedback. To be precise, uniqueness of Filippov (and also forward) solutions follows in this case if the first non-zero Markov parameter is positive. By means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two. The influence of the Zeno phenomenon (an infinite number of relay switching times in a finite length time interval) on well-posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems, the consequences of the presence of the Zeno behaviour is typical for many other classes of non-smooth and hybrid systems.