Adaptive Estimation of Threshold Parameters for a Prandtl-Ishlinskii Hysteresis Operator

The Prandtl-Ishlinskii (PI) operator has been used widely in the modeling and inverse compensation of hysteresis nonlinearity in actuators made of smart materials, such as piezoelectric and magnetostrictive materials. A PI operator consists of weighted superposition of play operators, each of which is characterized by a threshold (also known as radius) parameter that determines the width of the corresponding hysteresis loop. While much work has been reported in identifying the weight parameters for the play operators, the threshold parameters have typically been assigned a priori in an arbitrary fashion. In this paper, for the first time, an adaptive algorithm is proposed for estimating online the unknown thresholds of a PI operator. The key challenge is that the output of the PI operator depends on the play thresholds in a complex, nonlinear, and time-varying manner. To address this challenge, the proposed algorithm utilizes the instantaneous slope of the input-output graph of the PI operator to infer the operating regime of each play, based on which a modified estimation error function is derived that is proportional to the error of threshold parameters. It is further shown, under a mild condition on the input, the regressor vector is persistently exciting and a gradient algorithm (with parameter projection) results in parameter convergence. The approach is illustrated in detail with a two-play PI operator, along with the results for the general case of n-play operators. Simulation results are presented to demonstrate the effectiveness of the proposed approach.