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

Matthew Abate, Corbin Klett, Samuel Coogan, Eric Feron

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

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, point-wise-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 point-wise-in-time behavior for LTI systems with reduced conservatism, 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 or additional algebraic developments. Numerous examples are provided to illustrate the findings of this work.
Original languageEnglish (US)
Title of host publication2021 American Control Conference (ACC)
PublisherIEEE
Pages3550-3555
Number of pages6
ISBN (Print)9781665441971
DOIs
StatePublished - May 25 2021
Externally publishedYes

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