General Solution of Stability Problem for Plane Linear Switched Systems and Differential Inclusions

Characterization and control of stability of switched dynamical systems and differential inclusions have attracted significant attention in the recent past. The most of the current results for this problem are obtained by application of the Lyapunov function method which provides sufficient but freq...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2008-10, Vol.53 (9), p.2149-2153
Main Authors: Zevin, A.A., Pinsky, M.A.
Format: Article
Language:English
Subjects:
Applied sciences
Asymptotic properties
Asymptotic stability
Breaking
Computer science
control theory
systems
Control system analysis
Control systems
Control theory. Systems
Differential equations
differential inclusion
Dynamical systems
Dynamics
Exact sciences and technology
Inclusions
Lyapunov method
Mathematics
necessary and sufficient conditions
Stability
Stability analysis
Sufficient conditions
switched plane system
Switched systems
Switching
Transmission line matrix methods
Upper bound
Vibration
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Summary:Characterization and control of stability of switched dynamical systems and differential inclusions have attracted significant attention in the recent past. The most of the current results for this problem are obtained by application of the Lyapunov function method which provides sufficient but frequently over conservative stability conditions. For planar systems, practically verifiable necessary and sufficient conditions are found only for switched systems with two subsystems. This paper provides explicit necessary and sufficient conditions for asymptotic stability of switched systems and differential inclusions with arbitrary number of subsystems; these conditions turned out to be identical for the both classes of systems. A precise upper bound for the number of switching points in a periodic solution, corresponding to the break of stability, is found. It is shown that, for a switched system, the break of stability may also occur on a solution with infinitely fast switching (chattering) between some two subsystems.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2008.930191