Characterization of well-posedness of piecewise-linear systems

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. T...

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Bibliographic Details
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
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Summary: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.
ISSN:0018-9286
1558-2523
DOI:10.1109/9.880612