Design of linear feedback controllers for dynamic systems with hysteresis

This paper proposes an approach to deal with the control of a class of dynamical systems affected by hysteresis, which is particularly common in applications of smart materials to motion control. The controlled plant is assumed to be a combination of a linear system with a hysteretic operator that can appear either in series or in a feedback path with respect to the linear component, while the controller is defined as a linear combination of the tracking error, its integral and derivatives. This paper mainly focuses on tracking behavior with constant references, and formulates the output regulation as a problem of stability of a polytopic linear differential inclusion, which does not require the identification of an accurate (direct or inverse) model of the hysteresis. The resulting conditions allow the user to seek for controller parameters that guarantee the achievement of a predefined control goal by solving a linear matrix inequality problem. Beside validation through numerical simulation, the method is successfully applied to control a challenging and innovative system, which uses two bars of magnetic shape memory alloy as the active elements of a multistable precise positioning device.