Decomposition of stability proofs for hybrid systems

The verification of hybrid systems, encompassing both discrete-time and continuoustime behavior, is a problem of rising importance. Hybrid behavior occurs wherever a digital system, operating in discrete time, interacts with a real-world environment, which evolves in continuous time. One desired property of hybrid systems is global asymptotic stability. A globally asymptotically stable system converges toward a pre-defined target state from everywhere in the state space. This property guarantees that the system is robust to temporary external disturbances, correcting their effects on its own accord. Stability proofs are usually conducted with the help of so-called Lyapunov functions, which act as generalized energy functions of the system. A Lyapunov function maps each of the possible system states onto an energy value, such that the energy decreases as the system evolves. The existence of such a function serves as a proof of global asymptotic stability. Furthermore, numerical methods for the computation of such functions exist, allowing for automated stability verification. While these methods work well for smallscale systems, they do, however, not scale up well to systems with large discrete state spaces that appear in many real-world applications. Therefore, it is desirable to conduct these computations in a decompositional manner, solving many small-scale problems instead of one large-scale problem. This thesis bridges this gap by introducing an automatable decomposition methodology that works on hybrid automaton models. A hybrid automaton is viewed as a graph and decomposed into sub-components, for which Lyapunov function computation are conducted individually. The results of these computations are then combined to yield a stability proof for the entire system. These local proofs are not only more lightweight than the larger (and possibly intractable) standard proof, but also allow for the localization of the problem if the computation fails, and the compositional construction of complex stable hybrid automata. The decomposition takes place on two levels: strongly connected components and cycles of the automaton. Our results prove that strongly connected components can be analyzed completely separately. For the cycle-based second level, properties of Lyapunov functions are exploited to combine local proofs into a global proof. Furthermore, the results are extended to the domain of probabilistic hybrid automata, which are a combination of hybrid systems and Markov processes. The decomposition results in this thesis are generalized to this setting, to allow for automatable decompositional stability proofs for probabilistic systems. For both non-probabilistic and probabilistic systems, the decompositional approach is also exploited to yield a set of rules for the structured construction of stable hybrid systems with complex discrete behavior. Furthermore, we derive a general method for the hierarchical component-based design of stabilizing hybrid controllers for a given plant, building on the decomposition results.