有限维线性系统的质量度量：一个框架理论的观点
Measure of quality of finite-dimensional linear systems: A frame-theoretic view
AI总结：
本文从框架理论的角度研究线性控制系统的质量，证明所有LTI系统自然生成状态空间上的框架，并展示三个标准质量度量在框架紧时达到最优值，进而提出基于框架紧度量的连续时间LTI系统质量度量方法。
摘要：
控制系统的质量度量是经典可控性二元概念的定量扩展。本文从框架理论的角度研究线性控制系统的质量。我们证明所有LTI系统自然生成其状态空间上的框架，并且三个标准质量度量，涉及迹、最小特征值和可控性格拉姆矩阵的行列式，在生成的框架为紧时达到最优值。受此启发，并考虑到框架理论信号处理的一些最新发展，我们提出了一种基于由系统生成的框架紧度量的连续时间LTI系统质量度量方法，并讨论了该框架理论质量度量的一些性质。
英文摘要：
A measure of quality of a control system is a quantitative extension of the classical binary notion of controllability. In this article we study the quality of linear control systems from a frame-theoretic perspective. We demonstrate that all LTI systems naturally generate a frame on their state space, and that three standard measures of quality involving the trace, minimum eigenvalue, and the determinant of the controllability Gramian achieve their optimum values when this generated frame is tight. Motivated by this, and in view of some recent developments in frame-theoretic signal processing, we propose a natural measure of quality for continuous time LTI systems based on a measure of tightness of the frame generated by it and then discuss some properties of this frame-theoretic measure of quality.
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