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首页> 外文期刊>IEEE Transactions on Automatic Control >Simultaneous Triangularization of Switching Linear Systems: Arbitrary Eigenvalue Assignment and Genericity
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Simultaneous Triangularization of Switching Linear Systems: Arbitrary Eigenvalue Assignment and Genericity

机译：切换线性系统的同时三角化：任意特征值分配和通用性

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摘要

A sufficient condition for the stability of arbitrary switching linear systems (SLSs) without control inputs is that the individual subsystems are stable and their evolution matrices are simultaneously triangularizable (ST). This sufficient condition for stability is known to be extremely restrictive and not robust, and therefore of very limited applicability. The situation can be radically different when control inputs are present. Indeed, previous results have established that, depending on the number of states, inputs and subsystems, the existence of feedback matrices for each subsystem so that the corresponding closed-loop matrices are stable and ST can become a generic property, i.e., a property valid for almost every set of system parameters. This note provides novel contributions along two lines. First, we give sufficient conditions for the genericity of the property of existence of feedback matrices so that the subsystem closed-loop matrices are ST (not necessarily stable). Second, we give conditions for the genericity of the property of existence of feedback matrices that, in addition to achieving ST, enable arbitrary eigenvalue selection for each subsystem's closed-loop matrix. The latter conditions are less stringent than existing ones, and the approach employed in their derivation can be interpreted as an extension to SLSs of specific aspects of the notion of eigenvalue controllability for (non-switching) linear systems.
机译：没有控制输入的任意线性切换系统（SLS）稳定性的充分条件是各个子系统稳定，并且其演化矩阵同时可三角化（ST）。已知这种足够的稳定性条件是非常严格的，并且不可靠，因此适用性非常有限。当存在控制输入时，情况可能完全不同。实际上，先前的结果已经确定，根据状态，输入和子系统的数量，每个子系统的反馈矩阵的存在，以便相应的闭环矩阵稳定，并且ST可以成为通用属性，即有效的属性适用于几乎所有系统参数集。本笔记提供了两方面的新颖贡献。首先，我们为反馈矩阵的存在性质的通用性提供了充分的条件，以使子系统的闭环矩阵为ST（不一定稳定）。其次，我们给出了反馈矩阵存在性的一般性的条件，除了实现ST之外，还可以为每个子系统的闭环矩阵选择任意特征值。后一种条件比现有条件没有那么严格，并且在其推导中采用的方法可以解释为对（非切换）线性系统特征值可控性概念的特定方面的SLS的扩展。

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