Inverse feedforward controller for complex hysteretic nonlinearities in smart-material systems

Undesired complex hysteretic nonlinearities are present to a varying degree in virtually all smart-material-based sensors and actuators provided they are driven with sufficiently high amplitudes. In motion and active vibration control applications, for example, these nonlinearities can excite unwanted dynamics, which leads in the best case to reduced system performance and in the worst case to unstable system operation. This necessitates the development of purely phenomenological models that characterize these nonlinearities in a way that is sufficiently accurate, amenable to a compensator design for actuator linearization, and efficient enough for use in real-time applications. To fulfil these demanding requirements, this article describes a new compensator design method for invertible complex hysteretic nonlinearities that is based on the so-called Prandtl-Ishlinskii hysteresis operator. The parameter identification of this model can be formulated as a quadratic optimization problem, which produces the best L22-norm approximation for the measured output-input data of the real hysteretic nonlinearity. Special linear inequality constraints for the parameters guarantee the unique solvability of the identification problem and the invertability of the identified model. This leads to a robustness of the identification procedure against unknown measurement errors, unknown model errors, and unknown model orders. The corresponding compensator can be directly calculated and thus efficiently implemented from the model by analytical transformation laws. Finally, the compensator design method is used to generate an inverse feedforward controller for the linearization of a magnetostrictive actuator. In comparison to the conventionally controlled magnetostrictive actuator, the nonlinearity error of the inverse controlled magnetostrictive actuator is lowered from about 30% to about 3%.