Optimal Control of Hysteresis in Smart Actuators: A Viscosity Solutions Approach

Hysteresis in smart materials hinders their wider applicability in actuators. The low dimensional hysteresis models for these materials are hybrid systems with both controlled switching and autonomous switching. In particular, they belong to the class of Duhem hysteresis models and can be formulated as systems with both continuous and switching controls. In this paper, we study the control methodology for smart actuators through the example of controlling a commercially available magnetostrictive actuator. For illustrative purposes, an infinite horizon optimal control problem is considered. We show that the value function satisfies a Hamilton-Jacobi-Bellman equation (HJB) of a hybrid form in the viscosity sense. We further prove uniqueness of the viscosity solution to the (HJB), and provide a numerical scheme to approximate the solution together with a sub-optimal controller synthesis method. Numerical and experimental results based on this approach are presented.