A game theoretic approach to controller design for hybrid systems

TLDR

Hybrid systems integrate discrete event dynamics that capture linguistic and qualitative information with nonlinear continuous dynamics that model physical processes such as aircraft responses to aileron and throttle, and use input variables to represent both continuous and discrete control and disturbance parameters. The paper proposes a controller design method that ensures safety specifications are met in hybrid systems. By translating safety specifications into constraints on reachable state sets, the authors employ optimal control and game theory to derive Hamilton‑Jacobi equations whose solutions define reachable set boundaries, from which feedback control laws for continuous and discrete variables are computed to keep the system within a safe subset, and they discuss computational aspects of solving these equations. The method is validated on hybrid automata examples including aircraft conflict resolution, autopilot flight mode switching, and vehicle collision avoidance.

Abstract

We present a method to design controllers for safety specifications in hybrid systems. The hybrid system combines discrete event dynamics with nonlinear continuous dynamics: the discrete event dynamics model linguistic and qualitative information and naturally accommodate mode switching logic, and the continuous dynamics model the physical processes themselves, such as the continuous response of an aircraft to the forces of aileron and throttle. Input variables model both continuous and discrete control and disturbance parameters. We translate safety specifications into restrictions on the system's reachable sets of states. Then, using analysis based on optimal control and game theory for automata and continuous dynamical systems, we derive Hamilton-Jacobi equations whose solutions describe the boundaries of reachable 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, we demonstrate out techniques on examples of hybrid automata modeling aircraft conflict resolution, autopilot flight mode switching, and vehicle collision avoidance.