Approximate bisimulations for nonlinear dynamical systems

Antoine Girard (1, 2) , George J. Pappas (1)

1 Department of Electrical and Systems Engineering
2 LMC - IMAG - Laboratoire de Modélisation et Calcul

The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunov-like inequalities, which for deterministic systems can be computed using recent sum-of-squares techniques. Our framework is illustrated on a safety verification example.

Domaines: Optimisation et contrôle [math.OC]

Antoine Girard, George J. Pappas. Approximate bisimulations for nonlinear dynamical systems. Conference on Decision and Control and European Control Conference, Dec 2005, Seville, Spain. pp.684-689, ⟨10.1109/CDC.2005.1582235⟩. ⟨hal-00171559⟩