Nonlinear Optimal Control Techniques for Vibration Attenuation Using Magnetostrictive Actuators

Abstract

This article addresses the development of a nonlinear control design for attenuating structural vibrations using magnetostrictive transducers operating in nonlinear and highly hysteretic operating regimes. We consider as a prototype a thin plate subjected to exogenous pressure waves and controlled via Terfenol-D transducers at the plate edges; however, the methodology is sufficiently general to encompass a wide range of structures and magnetic transducer designs. Hysteresis inherent to the transducer materials is quantified using a homogenized energy framework and the resulting nonlinear constitutive relations are used to construct a PDE representation and corresponding finite dimensional model of the structural system. We employ optimal control theory to construct nonlinear open loop control inputs which accommodate the hysteresis inherent to the transducers but are not robust with regard to unmodeled dynamics or disturbances. Robustness is incorporated by employing perturbation techniques to provide linear feedback laws acting on measured disturbances. As illustrated via numerical examples, the resulting hybrid control design provides excellent control authority and robustness for transducers operating in hysteretic and nonlinear regimes.

Open loop nonlinear optimal tracking control of a magnetostrictive terfenol-D actuator

“…The form of the state space model given by (12) can incorporate finite element nodal solutions; however, in the present case a lumped parameter model is used which is sufficient for the operating regime considered. For more general operating regimes that may include spatial variations in displacement along the rod, finite element formulations should be used, see [4,10].…”

“…The nonlinear optimal control design is based on a previous numerical analysis for vibration control and tracking control [3,4,15]. The nonlinear control design is briefly summarized and experimental results are given.…”

“…For more general reference displacements and disturbance loads that excite higher order harmonics, a finite element model may be necessary. Finite element models can be directly incorporated into the model and control design as discussed in [3,4,10].…”

“…The experimental analysis presented here employs a nonlinear control design previously analyzed numerically for controlling a magnetostrictive transducer operating in current control mode [3,4]. In Section 2, the experimental set-up is described.…”

Nonlinear optimal tracking control of a piezoelectric nanopositioning stage

“…Development of the nonlinear optimal tracking control design follows a previous approach focused on negligible relaxation hysteresis of magnetostrictive actuators for vibration attenuation of beam and plate structures and tracking control of rod structures [10,11,15]. We summarize here key equations associated with optimal tracking control and its application in compensating ferroelectric nonlinearities, hysteresis and relaxation behavior.…”