Adaptive Estimation of Play Radii for a Prandtl–Ishlinskii Hysteresis Operator

The Prandtl–Ishlinskii (PI) operator is a mathematical model for hysteresis, and it is comprised of weighted superposition of multiple play (backlash) operators. The PI operator has been widely used in modeling and compensation of hysteresis in smart material actuators, robots, and mechanical systems. The behavior of a PI operator is determined by both the weights and the radii of individual play operators. However, existing modeling work has mostly been focused on the identification of the weight parameters by assuming the play radii to be known. While the latter approach is convenient due to the linear relationship between the operator output and the weight parameters, it often requires a large number of plays with preassigned radii in order to adequately capture the hysteresis in a given application. In this work, for the first time, we propose an adaptive estimation algorithm to identify the play radii of a PI operator, to enable accurate modeling of hysteresis with a small number of plays and, thus, reduce the complexity in control. The major challenge lies in the nonlinear, complex, time-varying relationship between the PI operator output and the play radii. The proposed algorithm utilizes available measurement and information, including the instantaneous slope of the hysteretic input–output graph, to derive a modified estimation error function that is proportional to the parameter error. With a mild condition on the input, we establish the persistent excitation of the resulting regressor vector and the parameter convergence under a gradient algorithm with parameter projection. Both simulation and experimental results are presented to illustrate the proposed approach. In particular, comparison results based on experimental data from a piezoelectric nanopositioner show that, with the proposed method, the resulting PI operator outperforms identified PI operators of larger numbers of plays with preassigned radius values.