Nonlinear state-space identification of nonlinear systems exhibiting hysteresis dynamics

Hysteresis is a phenomenology commonly encountered in very diverse engineering and science disciplines. The defining property of a hysteretic system is the persistence of an input-output loop as the input frequency approaches zero. Hysteretic systems are inherently nonlinear, as the quasi-static existence of a loop requires an input-output phase shift different from 0 and 180 degrees, which are the only two options offered by linear theory. The identification of hysteresis is challenging, primarily because it is a dynamic kind of nonlinearity governed by internal, and so unmeasurable, variables. The present contribution aims at demonstrating the capability of nonlinear state-space models to accurately identify hysteretic behaviour. Nonlinear terms in the state-space model are constructed as a multivariate polynomial in the states, and their parameters are estimated by minimising a maximum likelihood cost function. Technical issues like the selection of the model order and the polynomial degree are discussed, and model validation is achieved in both broadband and sine conditions. The study is carried out numerically by exploiting synthetic data generated via the Bouc-Wen equations. The Bouc-Wen theoretical model consists in a first-order differential equation encoding the memory associated to hysteresis. This equation is coupled herein to a second-order differential equation representing a mechanical system with a single resonance. The coupled set of equations leads to a nonlinear dynamic system exhibiting rate-dependent hysteresis and hardening of its resonance frequency.