Topological entropy of switched linear systems: general matrices and matrices with commutation relations
Topological entropy of switched linear systems: general matrices and matrices with commutation relations
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DOI：
10.1007/s00498-020-00265-9
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发表时间：
2020-09
期刊：
影响因子：
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通讯作者：
Guosong Yang;A. J. Schmidt;D. Liberzon;J. Hespanha
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文献类型：
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作者：
Guosong Yang;A. J. Schmidt;D. Liberzon;J. Hespanha
本文研究了切换系统的拓扑熵的概念，它是用有限精度逼近所有轨迹所需的最小轨迹数来表示的。对于一般切换线性系统，我们证明了拓扑熵与初始状态集无关。我们构造了一个拓扑熵的上界，这个上界是用各个模的系统矩阵的度量的平均值来表示的，它是由它们相应的活动时间加权得到的，它的下界是用它们的迹的活动时间加权平均值来表示的。对于具有标量状态的切换线性系统和具有成对交换矩阵的切换线性系统，我们建立了用单个模式的系统矩阵的特征值的主动时间加权平均来表示拓扑熵的公式。对于更一般的可同时三角化矩阵的情形，我们构造了拓扑熵的上界，它仅依赖于特征值、它们在同时三角化中的顺序和活动时间。在上述每种情况下，我们还建立了更保守但需要更少关于系统矩阵或关于切换的信息的上界，并用数值例子说明了它们之间的关系。给出了由拓扑熵的上界引起的稳定性条件。
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.