Almost Global Stability of Nonlinear Switched Systems with Time-Dependent Switching

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Abstract

For a dynamical system, it is known that the
existence of a Lyapunov density implies almost global stability
of an equilibrium. It is then natural to ask whether the existence
of multiple Lyapunov densities for a nonlinear switched system
implies almost global stability, in the same way as the existence
of multiple Lyapunov functions implies global stability for
nonlinear switched systems. In this work, the answer to this
question is shown to be affirmative as long as switchings satisfy
a dwell time constraint with an arbitrarily small dwell time.
Specifically, as the main result, we show that a nonlinear
switched system with a minimum dwell time is almost globally
stable if there exist multiple Lyapunov densities which satisfy
some compatibility conditions depending on the value of the
minimum dwell time. This result can also be used to obtain a
minimum dwell time estimate to ensure almost global stability
of a nonlinear switched systems. In particular, the existence of
a common Lyapunov density implies almost global stability for
any arbitrary small minimum dwell time.
The results obtained for continuous-time switched systems
are based on some sufficient conditions for the almost global
stability of discrete-time non-autonomous systems. These conditions
are obtained using the duality between Frobenius-
Perron operator and Koopman operator.
Original languageEnglish
JournalI E E E Transactions on Automatic Control
ISSN0018-9286
DOIs
Publication statusE-pub ahead of print - 2020

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