Learning fixed-complexity polyhedral Lyapunov functions from counterexamples
(2022) 2022 IEEE 61st Conference on Decision and Control (CDC) — Location: Cancun, Mexico (6.December.2022)
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- Abstract
- We study the problem of synthesizing polyhedral Lyapunov functions for hybrid linear systems. Such functions are defined as convex piecewise linear functions, with a finite number of pieces. We first prove that deciding whether there exists an m-piece polyhedral Lyapunov function for a given hybrid linear system is NP-hard. We then present a counterexample-guided algorithm for solving this problem. The algorithm alternates between choosing a candidate polyhedral function based on a finite set of counterexamples and verifying whether the candidate satisfies the Lyapunov conditions. If the verification fails, we find a new counterexample that is added to our set. We prove that if the algorithm terminates, it discovers a valid Lyapunov function or concludes that no such Lyapunov function exists. However, our initial algorithm can be nonterminating. We modify our algorithm to provide a terminating version based on the so-called cutting-plane argument from nonsmooth optimization. We demonstrate our algorithm on numerical examples.
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Berger, G., & Sankaranarayana, S. (2022). Learning fixed-complexity polyhedral Lyapunov functions from counterexamples. Published. 2022 IEEE 61st Conference on Decision and Control (CDC), Cancun, Mexico. https://doi.org/10.1109/CDC51059.2022.9992338