An Equivalence Result between Linear Hybrid Automata and Piecewise Affine Systems

In this paper we examine a relationship existing among linear hybrid automata (LHA) and piecewise affine (PWA) systems. While a LHA is an autonomous non-deterministic model, a PWA system is a deterministic model with inputs. By extending continuous-time PWA models to include the dynamics of discrete states and resets, we show in a constructive way that a LHA can be equivalently represented as a PWA system, where equivalent means that the two systems generate the same trajectories. The key idea is to model the uncertainty associated with LHA transitions as an additional vector of input disturbances in the corresponding PWA model. By linking the LHA modelling framework (popular in computer science) with the PWA modelling framework (popular in systems science), our equivalence result allows one to expand the use of several existing control theoretical tools (for stability analysis, optimal control, etc.) developed for PWA models to a much wider class of hybrid systems.