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.
Philippe, M., Athanasopoulos, N., Angeli, D., & Jungers, R. M. (2018). On Path-Complete Lyapunov Functions: Geometry and Comparison. IEEE Transactions on Automatic Control. https://doi.org/10.1109/TAC.2018.2863380