Tracking error analysis for singularly perturbed systems preceded by piecewise linear hysteresis

Abstract

Abstract-In this paper we introduce a new method for analyzing the closed-loop control system that involves a singularly perturbed plant preceded by hysteresis nonlinearity with piecewise linear characteristics, an example of which is a piezoactuated nanopositioners. Different methods are compared to quantify the tracking error and examine how the change in slopes from one segment to another interacts with the controller parameters and hence affects the tracking error. These methods all involve the combination of inverse hysteresis compensation and feedback control, but the points of insertion for the hysteresis inverse vary. A proportional-integral feedback controller is used throughout this comparison. The presented analysis is important because it provides an explicit expression for the tracking error, where the feedback controller parameters can be adjusted for the desired performance. Simulation and experimental results are presented for tracking control of a piezo-actuated nanopositioner, where the hysteresis is modeled with a Prandtl-Ishlinskii (PI) operator. The analysis carried out in this paper is applicable to other operators, such as the modified PI-operator and the Krasnoselskii-Porkovskii (KP) operator among others, since they all demonstrate piecewise linear hysteresis characteristics.

Tracking Error Analysis for Feedback Systems With Hysteresis Inversion and Fast Linear Dynamics1

“…The analysis in this work is a continuation of our work presented in Ref. [28]. The previous work [28], was limited to the analysis of the driving terms of the tracking error when the hysteresis is on a particular linear segment.…”

“…[28]. The previous work [28], was limited to the analysis of the driving terms of the tracking error when the hysteresis is on a particular linear segment. As such, it did not provide insight into the cumulative behavior when the hysteresis traverses different linear segments, as well as the frequency-scaling behavior that is presented in the current work.…”

Closed-loop analysis for systems with fast linear dynamics preceded by hysteresis

“…In [11], we assume that the plant has an infinite bandwidth and is represented by a DC gain, which corresponds to the case ε = 0. In that case, the tracking error e is fully determined by the slow modelẋ.…”

“…Furthermore, we will be able to study how the tracking error scales with the reference frequency, a question that is of practical interest but remains largely open. The analysis in this work is a continuation of our work presented in [11]. In [11], we used a piecewise linear operator to model the hysteresis and performed preliminary analysis of three different control methods which differ in the point of insertion for the inverse operator.…”

“…The analysis in this work is a continuation of our work presented in [11]. In [11], we used a piecewise linear operator to model the hysteresis and performed preliminary analysis of three different control methods which differ in the point of insertion for the inverse operator. The previous work was limited to the analysis of the driving terms of the tracking error when the hysteresis is on a particular linear segment.…”