Harmonic analysis for hysteresis operators with application to control design for systems with hysteresis

Systems with hysteresis present challenges to control designers due to the difficulty in analyzing the effect of hysteresis on the system's behavior. In this paper we propose a novel approach to the analysis of the harmonic content produced by a hysteresis operator when it is subject to a periodic input. The commonly used Prandtl-Ishlinskii (PI) operator is used as an example. In particular, we present an algorithm for evaluating the Fourier series of the outputs of individual elementary hysteresis units (called hysterons) comprising the operator. Each output is reconstructed as a combination of the input and a series of pulses. The formation of these pulse waves requires computation of time instants when the hysteron behavior switches, and these times are evaluated using knowledge of the input signal and hysteresis parameters. Illustrative examples are presented for a play operator (hysteron for the PI operator), when it is subject to a sinusoidal signal and a sawtooth signal, respectively, and we show that the resulting Fourier coefficients are functions of the signal amplitude and the play radius. We also show that these calculations can be used as a control design tool for systems with hysteresis, for example, by providing guidance on selecting the order of the servocompensator for controlling such systems.