Newton and Bouligand derivatives of the scalar play and stop operator

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable.

Optimal control of reaction-diffusion systems with hysteresis

This paper is concerned with the optimal control of hysteresis-reaction-diffusion systems. We study a control problem with two sorts of controls, namely distributed control functions, or controls which act on a part of the boundary of the domain. The state equation is given by a reaction-diffusion system with the additional challenge that the reaction term includes a scalar stop operator. We choose a variational inequality to represent the hysteresis. In this paper, we prove first order necessary optimality conditions. In particular, under certain regularity assumptions, we derive results about the continuity properties of the adjoint system. For the case of distributed controls, we improve the optimality conditions and show uniqueness of the adjoint variables. We employ the optimality system to prove higher regularity of the optimal solutions of our problem. The specific feature of rate-independent hysteresis in the state equation leads to difficulties concerning the analysis of the solution operator. Non-locality in time of the Hadamard derivative of the control-to-state operator complicates the derivation of an adjoint system. This work is motivated by its academic challenge, as well as by its possible potential for applications such as in economic modeling.

A Generalized Inertial-Dependent Prandtl-Ishlinskii Model for Wide-Band Frequency Piezoelectric Actuator

Smart materials such as piezoceramics used in industrial applications, are subjected to nonlinear phenomenon of hysteresis which degrades the tracking performance of the actuator in operation. Depending on the operating frequency and input magnitude applied to the piezoelectric actuators (PA), the symmetric and asymmetric hysteresis loops can be observed. A generalized inertial-dependent Prandtl-Ishlinskii (PI) model is proposed in term of the stop operator to compensate the effects of nonlinearity in asymmetric hysteresis loops for PA. A nonlinear envelope function is assigned for the threshold of the stop operator to minimize the error between the model response and the measured response obtained through the experiments at frequency range from 200 Hz to 500 Hz.

Modeling Hysteresis with Inertial-Dependent Prandtl-Ishlinskii Model in Wide-Band Frequency-Operated Piezoelectric Actuator

One of the major problems occurring in many technical applications is the presence of the hysteretic behavior in sensors and actuators, which causes a nonlinear relationship between input and output variables in such devices. Since the nonlinear phenomenon of hysteresis degrades the performance of the piezoelectric materials and piezoelectric drive mechanisms, for example, in positioning control framework, it has to be characterized in order to mitigate the effect of the nonlinearity in the devices. This paper is aimed to characterize and model the hysteresis in typical piezoelectric actuators under load-free and preloaded circumstances incorporating the inertial effect of the system. For this purpose, the piezoelectric actuator is modeled as a mass-spring-damper system, which is expressed in terms of a stop operator as one of the essential yet efficient hysteresis operators in the Prandtl-Ishlinskii (PI) model. The reason of utilizing the stop operator in this study is for the sake of control purposes, as the stop operator plays as the inverse of the play operator in the PI model and can be used in a feed-forward controller scheme to suppress the effect of hysteresis in general control framework. The results reveal that this model exhibits better correspondence to the measurement output compared to that of the classical PI model.