Breaking Dense Structures: Proving Stability of Densely Structured Hybrid Systems

Abstraction and refinement is widely used in software development. Such techniques are valuable since they allow to handle even more complex systems. One key point is the ability to decompose a large system into subsystems, analyze those subsystems and deduce properties of the larger system. As cyber-physical systems tend to become more and more complex, such techniques become more appealing. In 2009, Oehlerking and Theel presented a (de-)composition technique for hybrid systems. This technique is graph-based and constructs a Lyapunov function for hybrid systems having a complex discrete state space. The technique consists of (1) decomposing the underlying graph of the hybrid system into subgraphs, (2) computing multiple local Lyapunov functions for the subgraphs, and finally (3) composing the local Lyapunov functions into a piecewise Lyapunov function. A Lyapunov function can serve multiple purposes, e.g., it certifies stability or termination of a system or allows to construct invariant sets, which in turn may be used to certify safety and security. In this paper, we propose an improvement to the decomposing technique, which relaxes the graph structure before applying the decomposition technique. Our relaxation significantly reduces the connectivity of the graph by exploiting super-dense switching. The relaxation makes the decomposition technique more efficient on one hand and on the other allows to decompose a wider range of graph structures.Comment: In Proceedings ESSS 2015, arXiv:1506.0325

Pre-orders for Reasoning about Stability

Pre-orders between processes, like simulation, have played a central role in the verification and analysis of discrete-state systems. Logical characterization of such pre-orders have allowed one to verify the correctness of a system by analyzing an abstraction of the system. In this paper, we investigate whether this approach can be feasibly applied to reason about stability properties of a system. Stability is an important property of systems that have a continuous component in their state space; it stipulates that when a system is started somewhere close to its ideal starting state, its behavior is close to its ideal, desired behavior. In [6], it was shown that stability with respect to equilibrium states is not preserved by bisimulation and hence additional continuity constraints were imposed on the bisimulation relation to ensure preservation of Lyapunov stability. We first show that stability of trajectories is not invariant even under the notion of bisimulation with continuity conditions introduced in [6]. We then present the notion of uniformly continuous simulations — namely, simulation with some additional uniform continuity conditions on the relation—that can be used to reason about stability of trajectories. Finally, we show that uniformly continuous simulations are widely prevalent, by recasting many classical results on proving stability of dynamical and hybrid systems as establishing the existence of a simple, obviously stable system that simulates the desired system through uniformly continuous simulations