Control of Smart Actuators: A Viscosity Solutions Approach

Abstract

Hysteresis in smart materials hinders their wider applicability in actuators. In this report we investigate control of smart actuators through the example of controlling a commercially available magnetostrictive actuator. At low frequencies, the magnetostriction can be related to the bulk magnetization through a square law, thus control of the magnetization amounts to control of the magnetostriction. The model we use is the low dimensional Jiles-Atherton model for ferromagnetic hysteresis, which is a hybrid system. For illustrative purpose, we consider an infinite horizon control problem. The approach we take features dynamic programming and Hamilton-Jacobi equations. In particular, we show that the value function of the control problem satisfies a Hamilton-Jacobi-Bellman equation (HJB) of some hybrid form in the viscosity sense. We further prove uniqueness of solutions to the (HJB), and provide a numerical scheme to approximate the solution together with a suboptimal controller synthesis method.