On Topological Entropy of Switched Linear Systems with Diagonal, Triangular, and General Matrices
On Topological Entropy of Switched Linear Systems with Diagonal, Triangular, and General Matrices
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DOI：
10.1109/cdc.2018.8619087
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发表时间：
2018-12
期刊：
影响因子：
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通讯作者：
Guosong Yang;A. J. Schmidt;D. Liberzon
中科院分区：
文献类型：
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作者：
Guosong Yang;A. J. Schmidt;D. Liberzon
本文引入了切换系统拓扑熵的概念，利用在有限精度内近似所有初始状态所需的最小初始状态数来表示。我们表明，它可以等价地定义使用的最大数量的初始状态可分离的有限精度内，并引入开关相关的量，如每个模式的活动时间，这被证明是有用的，在计算拓扑熵的切换线性系统。对于一般的线性切换系统，我们证明了拓扑熵与初始状态无关，并利用系统矩阵的模和迹的有效时间加权平均值分别建立了上界和下界.对于具有标值状态或同时可对角化矩阵的线性切换系统，利用特征值的有效时间加权平均，导出了拓扑熵的计算公式，并将其推广到同时可三角化矩阵的情形，得到了拓扑熵的上界.在这三种具有特殊矩阵结构的情况下，我们也给出了更一般但更保守的拓扑熵上界。
This paper introduces a notion of topological entropy for switched systems, formulated using the minimal number of initial states needed to approximate all initial states within a finite precision. We show that it can be equivalently defined using the maximal number of initial states separable within a finite precision, and introduce switching-related quantities such as the active time of each mode, which prove to be useful in calculating the topological entropy of switched linear systems. For general switched linear systems, we show that the topological entropy is independent of the set of initial states, and establish upper and lower bounds using the active-time-weighted averages of the norms and traces of system matrices in individual modes, respectively. For switched linear systems with scalar-valued state or simultaneously diagonalizable matrices, we derive formulae for the topological entropy using active-time-weighted averages of eigenvalues, which can be extended to the case with simultaneously triangularizable matrices to obtain an upper bound. In these three cases with special matrix structure, we also provide more general but more conservative upper bounds for the topological entropy.