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Newton and Secant Methods for Iterative Remnant Control of Preisach Hysteresis Operators

Jurrien Keulen, Bayu Jayawardhana

TL;DR

This work analyzes the remnant behavior of Preisach hysteresis operators and leverages the remnant curve, under a positive weight assumption, to design Newton- and Secant-based iterative remnant-control strategies. By proving monotonicity and differentiability of the remnant curve, the authors derive update laws that achieve faster convergence to a desired remnant value $y_d$ than previous constant-gain methods. Numerical experiments demonstrate faster convergence for the Secant method and validate the theoretical results under a controlled Preisach setup. The approach offers energy-efficient remnant control in memory-enabled actuators by minimizing input after attaining the target remnant. Practical impact lies in improved transient performance for high-density hysteretic actuators in mechatronics and optical-mechatronic applications.

Abstract

We study the properties of remnant function, which is a function of output remnant versus amplitude of the input signal, of Preisach hysteresis operators. The remnant behavior (or the leftover memory when the input reaches zero) enables an energy-optimal application of piezoactuator systems where the applied electrical field can be removed when the desired strain/displacement has been attained. We show that when the underlying weight of Preisach operators is positive, the resulting remnant curve is monotonically increasing and accordingly a Newton and secant update laws for the iterative remnant control are proposed that allows faster convergence to the desired remnant value than the existing iterative remnant control algorithm in literature as validated by numerical simulation.

Newton and Secant Methods for Iterative Remnant Control of Preisach Hysteresis Operators

TL;DR

This work analyzes the remnant behavior of Preisach hysteresis operators and leverages the remnant curve, under a positive weight assumption, to design Newton- and Secant-based iterative remnant-control strategies. By proving monotonicity and differentiability of the remnant curve, the authors derive update laws that achieve faster convergence to a desired remnant value than previous constant-gain methods. Numerical experiments demonstrate faster convergence for the Secant method and validate the theoretical results under a controlled Preisach setup. The approach offers energy-efficient remnant control in memory-enabled actuators by minimizing input after attaining the target remnant. Practical impact lies in improved transient performance for high-density hysteretic actuators in mechatronics and optical-mechatronic applications.

Abstract

We study the properties of remnant function, which is a function of output remnant versus amplitude of the input signal, of Preisach hysteresis operators. The remnant behavior (or the leftover memory when the input reaches zero) enables an energy-optimal application of piezoactuator systems where the applied electrical field can be removed when the desired strain/displacement has been attained. We show that when the underlying weight of Preisach operators is positive, the resulting remnant curve is monotonically increasing and accordingly a Newton and secant update laws for the iterative remnant control are proposed that allows faster convergence to the desired remnant value than the existing iterative remnant control algorithm in literature as validated by numerical simulation.
Paper Structure (8 sections, 6 theorems, 24 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preisach hysteresis operators and remnant control problem
  3. Preisach hysteresis operator
  4. Remnant control problem for Preisach operators
  5. Remnant curve and its properties
  6. Newton & Secant Methods
  7. Numerical simulation
  8. Conclusions

Key Result

Proposition III.1

Consider a Preisach operator $\mathcal{P}$ as in eq:Preisach with a given weight function $w$ and initial interface $L_0$. For any initial interface $L_0$, the corresponding remnant curve $\rho_{L}$ is invariant to the particular form of $v(A,t)$ satisfying R1 -- R4.

Figures (2)

  • Figure 1: Preisach domain with a particular staircase interface $L_0$, consisting of the sequence $\{\alpha_{1,k},\ \alpha_{2,k},\ \alpha_{3,k},\ \alpha_{4,k},\ \alpha_{5,k}\}$ and $\{\beta_{1,k}$, $\beta_{2,k}$, $\beta_{3,k}$, $\beta_{4,k}$, $\beta_{5,k}\}$, after application of input signal with $A_k=\alpha_{2,k+1}$ some horizontal and vertical lines of $L_k$ are wiped out, resulting in the new interface $L_{k+1}$ consisting of the sequence $\{\alpha_{1,k+1}, \ \alpha_{2,k+1},\ \alpha_{3,k+1},\ \alpha_{4,k+1},\ \alpha_{5,k+1}\}$ and $\{\beta_{1,k+1},\ \beta_{2,k+1},\ \beta_{3,k+1},\ \beta_{4,k+1}\}$.
  • Figure 2: Numerical simulation results, where the proposed secant-based iterative remnant control method is compared with the iterative remnant control method of vasquez2020recursive with different values for $\lambda$.

Theorems & Definitions (6)

  • Proposition III.1
  • Proposition III.2
  • Proposition III.3
  • Proposition III.4
  • Proposition III.5
  • Theorem IV.1