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