In this paper, we present a novel abstraction technique and a model-checking algorithm for verifying Lyapunov and asymptotic stability of a class of hybrid systems called piecewise constant derivatives. We propose a new abstract data structure, namely, finite weighted graphs, and a modification of the predicate abstraction based on the faces in the system description. The weights on the edges trace the distance of the executions from the origin, and are computed by using linear programming. Model-checking consists of analyzing the finite weighted graph for the absence of certain kinds of cycles which can be solved by dynamic programming. We show that the abstraction is sound in that a positive result on the analysis of the graph implies that the original system is stable. Finally, we present our experiments with a prototype implementation of the abstraction and verification procedures which demonstrate the feasibility of the approach.

In this paper, we present a hybridization method for stability analysis of switched linear hybrid system (LHS), that constructs a switched system with polyhedral inclusion dynamics (PHS) using a state-space partition that is specific to stability analysis. We use a previous result based on quantitative predicate abstraction to analyse the stability of PHS. We show completeness of the hybridization based verification technique for the class of asymptotically stable linear system and a subclass of switched linear systems whose dynamics are pairwise Lipschitz continuous on the state-space and uniformly converging in time. For this class of systems, we show that by increasing the granularity of the region partition, we eventually reach an abstract switched system with polyhedral inclusion dynamics that is asymptotically stable. On the practical side, we implemented our approach in the tool Averist, and experimentally compared our approach with a state-of-the-art tool for stability analysis of hybrid systems based on Lyapunov functions. Our experimental results illustrate that our method is less prone to numerical errors and scales better than the traditional approaches. In addition, our tool returns a counterexample in the event that it fails to prove stability, providing feedback regarding the potential reason for instability. We also examined heuristics for the choice of state-space partition during refinement.

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