From discrete specifications to hybrid control

A great challenge for modern systems theory is the design of controllers for continuous systems but with logical specifications. In this paper, we are interested in developing algorithmic methods which given a discrete-time controllable linear system and a discrete specification (in the form of a finite transition system or a temporal logic formula), automatically design controllers resulting in desired, closed-loop behavior. This can be achieved using a natural approach involving three steps. In the first step, given a controllable linear system and discrete specification, we extract a finite transition system model which is equivalent (bisimilar) to the continuous system. The second step solves the controller synthesis problem for finite transition systems using well known and well developed algorithms. The third step, which is the focus of this paper, refines the discrete controller of the finite transition system, to a (necessarily) hybrid controller for the original continuous system. The hybrid controller composed with the continuous plant results in a closed-loop hybrid system that, by construction, satisfies the desired, discrete specification.