Digital controller design for Bouc–Wen model with high-order hysteretic nonlinearities through approximated scalar sign function

In the models of piezoelectric actuator (PEA) systems, the hysteresis effects are often described by the concise Bouc–Wen models; however, these models are nonlinear and non-smooth because of the existence of terms which involve the absolute-value function. In particular, a challenging control problem is posed when, due to the properties of certain materials, such absolute-value terms are of high order. This control problem has been rarely studied. This article proposes the use of the approximated scalar sign function (ASSF), which is a numerically stable and differentiable nonlinear function, to represent the hysteresis function, with single-order or high-order absolute-value terms. This innovative step leads to a nonlinear but sufficiently smooth model. Then, a systematic digital design methodology is presented, which involves the following steps: (1) establish an optimal linear model based upon the resulting smooth model, (2) adopt a PI-based analogue controller and (3) apply the prediction-based digital redesign technique for digital implementation. A digital observer is also developed for state reconstruction, and to improve the input disturbance rejection. The positioning control of a PEA system with a high-order hysteretic Bouc–Wen model is implemented to demonstrate the effectiveness of the proposed ASSF based modelling and controller design approaches.