Identification and nonlinearity compensation of hysteresis using NARX models

TL;DR

The paper tackles identification and compensation of hysteresis in dynamical systems using NARX models with gray-box constraints that enforce hysteretic behavior. It introduces static and quasi-static analyses to explain loop formation and prescribes parameter constraints to guarantee a continuum of equilibria, enabling hysteresis in identified models. Two compensation strategies are developed: a model-based approach that derives a compensator from a forward model $\mathcal{M}$, and a compensator-identification approach using an inverse model $\breve{\mathcal{M}}$; both are validated numerically and experimentally. Results show substantial hysteresis attenuation and improved tracking, with performance closely tied to the accuracy of the identified models, and highlight a practical framework for applying such compensators online. The work advances gray-box NARX methodologies for hysteresis modeling and provides generalizable guidance for compensator design beyond the specific experiments presented.

Abstract

This paper deals with two problems: the identification and compensation of hysteresis nonlinearity in dynamical systems using nonlinear polynomial autoregressive models with exogenous inputs (NARX). First, based on gray-box identification techniques, some constraints on the structure and parameters of NARX models are proposed to ensure that the identified models display a key-feature of hysteresis. In addition, a more general framework is developed to explain how hysteresis occurs in such models. Second, two strategies to design hysteresis compensators are presented. In one strategy the compensation law is obtained through simple algebraic manipulations performed on the identified models. It has been found that the compensators based on gray-box models outperform the cases with models identified using black-box techniques. In the second strategy, the compensation law is directly identified from the data. Both numerical and experimental results are presented to illustrate the efficiency of the proposed procedures.

Figures (11)

• Figure 1: Schematic representation of hysteresis loop in the $u \times y$ plane. Attracting sets are shown in black continuous lines, whereas the repelling sets are indicated in red dash-dot. The hysteresis loop is indicated by dotted lines.
• Figure 2: Compensator design based on identified NARX models. (a) Model identification, and (b) compensator design based on identified models.
• Figure 3: Signals used to identify system (\ref{['Eq:NumericalResults:System']}). (a) excitation, and (b) simulated output.
• Figure 4: Results of quasi-static analysis for model (\ref{['Eq:NumericalResults:IdentifiedModel:Cms']}) with input $u_k{=}70\sin(2\pi k)\,{\rm V}$. The hysteresis loop indicated with ($\cdots$) is a result of the interaction of (---) attracting ($\tilde{y}_{\rm L}^{\rm a}$, $\tilde{y}_{\rm U}^{\rm a}$) and (-$\,\cdot\,$-) repelling ($\tilde{y}_{\rm L}^{\rm r}$, $\tilde{y}_{\rm U}^{\rm r}$) sets. () indicates the orientation of the hysteresis loop.
• Figure 5: Free-run simulation of model (\ref{['Eq:NumericalResults:IdentifiedModel:Cms']}). This figure is arranged in columns, which have: (a) sinusoidal input of voltage $u_k{=}40\sin(2\pi k)\,{\rm V}$ and in (b) the case where this input becomes constant during a loading ($\bullet$) and unloading ($\blacklozenge$) regime with the final value of $16.8\,{\rm V}$, its temporal responses are shown in (c) and (d) while the hysteresis loops are in (e) and (f), respectively. (---) represents the measured data and (- -) is the estimated output of the model. The full records have $N=50000$ data points.

Theorems & Definitions (5)

• Definition 1
• Lemma 1
• Example 1
• Remark 1
• Example 2