Topological entropy of switched linear systems: general matrices and matrices with commutation relations
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Topological entropy of switched linear systems: general matrices and matrices with commutation relations Guosong Yang1 · A. James Schmidt2 · Daniel Liberzon2 · João P. Hespanha1 Received: 15 July 2019 / Accepted: 24 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract This paper studies a notion of topological entropy for switched systems, formulated in terms of the minimal number of trajectories needed to approximate all trajectories with a finite precision. For general switched linear systems, we prove that the topological entropy is independent of the set of initial states. We construct an upper bound for the topological entropy in terms of an average of the measures of system matrices of individual modes, weighted by their corresponding active times, and a lower bound in terms of an active-time-weighted average of their traces. For switched linear systems with scalar-valued state and those with pairwise commuting matrices, we establish formulae for the topological entropy in terms of active-time-weighted averages of the eigenvalues of system matrices of individual modes. For the more general case with simultaneously triangularizable matrices, we construct upper bounds for the topological entropy that only depend on the eigenvalues, their order in a simultaneous triangularization, and the active times. In each case above, we also establish upper bounds that are more conservative but require less information on the system matrices or on the switching, with their relations illustrated by numerical examples. Stability conditions inspired by the upper bounds for the topological entropy are presented as well.
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Guosong Yang [email protected] A. James Schmidt [email protected] Daniel Liberzon [email protected] João P. Hespanha [email protected]
1
Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara, CA 93106, USA
2
Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
123
Mathematics of Control, Signals, and Systems
Keywords Topological entropy · Switched linear systems · Exponential stability · Commutation relations
1 Introduction Since its introduction for dynamical systems by Kolmogorov [21], entropy has been an invaluable tool for understanding system behaviors. The Ornstein isomorphism theorem [34], which characterizes Bernoulli shifts entirely according to their entropy, further solidified its importance. Broadly, the entropy of a dynamical system captures the rate at which uncertainty about the state grows as time evolves, which intuitively corresponds to entropy notions in other disciplines such as thermodynamics and information theory [10]. In systems theory, topological entropy describes the information generation rate in terms of the number of distinguishable trajectories with a finite precision, or the complexity growth rate of a system acting on a set with finite measure. The latter idea corresponds to Kolmogorov’s original definition in [21], and shares a striking rese
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