An incremental harmonic balance-based approach for harmonic analysis of closed-loop systems with Prandtl-Ishlinskii operator

Analyzing hysteretic systems presents a significant challenge due to the memory effect of hysteresis. In this paper we present an incremental harmonic balance (IHB)-based approach to compute the steady-state response of a closed-loop system with hysteresis under a sinusoidal excitation, where the hysteresis element is modeled by the Prandtl-Ishlinskii (PI) operator. While the describing function method (DFM) can be used to obtain an approximate solution for the closed-loop system based on first-order harmonics, the proposed IHB-based approach iteratively calculates the harmonic components of the hysteretic system up to an arbitrary order. The main challenge is the harmonic calculation of the periodic output of the PI operator for a multi-harmonic input. In order to address this problem, an alternative definition of the play operator is utilized as the hysteron for the PI operator. By using the alternative definition, a set of switching time instants, when the play operator enters or exits the boundary region, are determined by a bisection method. The calculation of the incremental harmonic components is finally reformulated as a linear matrix equality that can be solved efficiently. As an illustration, numerical results for a system involving a proportional-integral feedback controller are presented to demonstrate the advantage of the IHB-based approach over the DFM in approximating the harmonic response of the hysteretic system.