Fast inverse compensation of Preisach-type hysteresis operators using field-programmable gate arrays

Preisach-type operators model hysteresis via weighted superposition of a large (and even infinite) number of basic hysteretic elements (called hysterons), and they have proven capable of capturing various complicated hysteretic behaviors. While inverse compensation is an effective approach to control of hysteretic systems, inversion of Preisach-type operators is a bottleneck in demanding, high-speed applications due to the high computational cost. In this paper a novel and general framework is proposed for fast inversion of a wide class of Preisach-type operators, by exploiting the massive parallelism offered by field-programmable gate arrays (FPGAs) to process the inherently parallel hysteresis operators. The theory, algorithm, and implementation of the inversion are presented. The inversion output is computed iteratively with guaranteed convergence (up to machine precision) provided the hysteresis operator is piecewise monotone and Lipschitz continuous. For an operator consisting of m hysterons, the proposed approach shows a computational complexity of O(log m), in contrast to O(m) for methods using general DSPs. The effectiveness of the fast inversion approach is demonstrated by implementation results on open-loop tracking of kHz reference signals, based on inversion of a Krasnosel´skii-Pokrovskii operator.