Bounding the State Covariance Matrix for Switched Linear Systems with Noise

Klett, Corbin; Abate, Matthew; Yoon, Yongeun; Coogan, Samuel; Féron, Éric

doi:10.23919/acc45564.2020.9147787

Cited by 3 publications

(1 citation statement)

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“…the Chapman-Kolmogorov equations [15,Chapter 16]. See also [16] for the discrete-time parallel to (10).…”

Section: Preliminariesmentioning

confidence: 99%

Performance Analysis and Non-Quadratic Lyapunov Functions for Linear Time-Varying Systems

Abate¹,

Klett²,

Coogan³

et al. 2020

Preprint

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Performance analysis for linear time-invariant (LTI) systems has been closely tied to quadratic Lyapunov functions ever since it was shown that LTI system stability is equivalent to the existence of such a Lyapunov function. Some metrics for LTI systems, however, have resisted treatment via means of quadratic Lyapunov functions. Among these, pointwise-in-time metrics, such as peak norms, are not captured accurately using these techniques, and this shortcoming has prevented the development of tools to analyze system behavior by means other than e.g. time-domain simulations. This work demonstrates how the more general class of homogeneous polynomial Lyapunov functions can be used to approximate pointwise-in-time behavior for LTI systems with greater accuracy, and we extend this to the case of linear time-varying (LTV) systems as well. Our findings rely on the recent observation that the search for homogeneous polynomial Lyapunov functions for LTV systems can be recast as a search for quadratic Lyapunov functions for a related hierarchy of time-varying Lyapunov differential equations; thus, performance guarantees for LTV systems are attainable without heavy computation. Numerous examples are provided to demonstrate the findings of this work.

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“…the Chapman-Kolmogorov equations [15,Chapter 16]. See also [16] for the discrete-time parallel to (10).…”

Section: Preliminariesmentioning

confidence: 99%