Viewing hybrid systems as products of control systems and automata

The purpose of this note is to show how hybrid systems may be modeled as products of nonlinear control systems and finite state automata. By a hybrid system, we mean a network of consisting of continuous, nonlinear control system connected to a discrete finite state automaton. Our point of view is that the automaton switches between the control systems, and that this switching is a function of the discrete input symbols or letters that it receives. There are several ways in which this may be modeled. The approach we take is a simple one: we show how a nonlinear control system may be viewed as a pair consisting of a bialgebra of operators coding the dynamics, and an algebra of observations coding the state space. We also show that a finite automaton has a similar representation. A hybrid system is then modeled by taking suitable products of the bialgebras coding the dynamics and the observation algebras coding the state spaces. An important advantage of viewing hybrid systems in an operator representation is that it is easy to specify algebraically how the various components of the hybrid system are “glued” together. One can