Controllers as fixed points of set-valued operators

Considers the problem of constructing a "controller" for a hybrid system which will solve the viability problem that all points of plant trajectories stay inside a given "viability set". Here, a "controller" is a network of three successive devices, a digital to analog converter, a digital program (a computer together with its control software), and an analog to digital converter. The authors model a controller as an input-output automaton, which they call a "control automaton". The authors give a necessary and sufficient condition that must be satisfied in order that a finite state control automaton solves a viability problem. The authors' results apply to plants modelled by vector differential equations with control and disturbance parameters, or to plants modelled by differential inclusions with a control parameter. The authors represent imprecise sensing of plant state. The main restrictions on the range of applicability of the authors' results are that the set of admissible control laws is finite, and that they can neglect delays when control laws are reset. The latter restriction seems to be inessential.