A game theoretic approach to controller design for hybrid systems

We present a method to design controllers for safety specifications in hybr id systems. The hybrid system combines discrete event dynamics with nonline ar continuous dynamics: the discrete event dynamics model linguistic and qu alitative information and naturally accommodate mode switching logic, and t he continuous dynamics model the physical processes themselves, such as the continous response of an aircraft to the forces of aileron and throttle. I nput variables model both continuous and discrete control and disturbance p arameters. We translate safety specifications into restrictions on the syst em's reachable sets of states. Then, using analysis based on optimal contro l and game theory for automata and continuous dynamical systems, we derive Hamilton-Jacobi equations whose solutions describe the boundaries of reacha ble sets. These equations are the heart of our general controller synthesis technique for hybrid systems, in which we calculate feedback control laws for the continuous and discrete variables, which guarantee that the hybrid system remains in the "safe subset" of the reachable set. We discuss issues related to computing solutions to Hamilton-Jacobi equations. Throughout, w e demonstrate our techniques on examples of hybrid automata modeling aircra ft conflict resolution, autopilot flight mode switching, and vehicle collis ion avoidance.