TY - JOUR

T1 - Topological entropy of switched linear systems

T2 - general matrices and matrices with commutation relations

AU - Yang, Guosong

AU - Schmidt, A. James

AU - Liberzon, Daniel

AU - Hespanha, João P.

N1 - Funding Information:
The work of G. Yang and J. P. Hespanha was supported by the Office of Naval Research under the MURI Grant N00014-16-1-2710, and by the National Science Foundation under the Grants CNS-1329650 and EPCN-1608880. The work of D. Liberzon was supported by the National Science Foundation under the Grant CMMI-1662708, and by the Air Force Office of Scientific Research under the Grant FA9550-17-1-0236. The authors thank Raphaël M. Jungers for his comments on a preliminary version of the paper.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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.

AB - 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.

KW - Commutation relations

KW - Exponential stability

KW - Switched linear systems

KW - Topological entropy

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U2 - 10.1007/s00498-020-00265-9

DO - 10.1007/s00498-020-00265-9

M3 - Article

AN - SCOPUS:85090864443

VL - 32

SP - 411

EP - 453

JO - Mathematics of Control, Signals, and Systems

JF - Mathematics of Control, Signals, and Systems

SN - 0932-4194

IS - 3

ER -