Characterization of well-posedness of piecewise-linear systems

• Published in: IEEE transactions on automatic control 2000-09, Vol.45 (9), p.1600-1619
• Main Authors: Imura, J., van der Schaft, A.
• Format: Article
• Language: English
• Subjects: Automata; Automatic control; Computer science; Control system synthesis; Control systems; Controllability; Dynamical systems; Dynamics; Gain; Inequalities; Piecewise linear techniques; Stability; State feedback; Studies; Sufficient conditions; Switching; Well posed problems
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/9.880612

One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. The paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Caratheodory. The concepts of jump solutions or of sliding modes are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multimodal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-posed.