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.
On Path-Complete Lyapunov Functions: Geometry and Comparison / Philippe, Matthew; Athanasopoulos, Nikolaos; Angeli, David; Jungers, Raphael M.. - In: IEEE TRANSACTIONS ON AUTOMATIC CONTROL. - ISSN 0018-9286. - STAMPA. - (2018), pp. 1-1. [10.1109/TAC.2018.2863380]
On Path-Complete Lyapunov Functions: Geometry and Comparison
Angeli, David
;
2018
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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