On Path-Complete Lyapunov Functions: Geometry and Comparison

Philippe, Matthew;Athanasopoulos, Nikolaos;David, Angeli;Jungers, Raphaël
(2018) IEEE Transactions on Automatic Control — p. 1 (2018)

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Authors
  • Philippe, MatthewUCLouvain
    Author
  • Athanasopoulos, NikolaosUCLouvain
    Author
  • David, Angeli
    Author
  • Jungers, RaphaëlUCLouvain
    Author
Abstract
We study optimization-based criteria for the stability of switching systems, known as Path-Complete Lyapunov Functions, and ask the question “can we decide algorithmically when a criterion is less conservative than another'”. Our contribution is twofold. First, we show that a Path-Complete Lyapunov Function, which is a multiple Lyapunov function by nature, can always be expressed as a common Lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it. Geometrically, our results provide for each Path-Complete criterion an implied invariant set. Second, we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given Path-Complete Lyapunov functions
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Philippe, M., Athanasopoulos, N., David, A., & Jungers, R. (2018). On Path-Complete Lyapunov Functions: Geometry and Comparison. IEEE Transactions on Automatic Control, 1. https://doi.org/10.1109/tac.2018.2863380 (Original work published 2018)