Tracking Error Analysis for Singularly Perturbed Systems Preceded by Piecewise Linear Hysteresis Mohamed Edardar, Xiaobo Tan and Hassan K. Khalil Abstract— In this paper we introduce a new method for analyzing the closed-loop control system that involves a singu- larly perturbed plant preceded by hysteresis nonlinearity with piecewise linear characteristics, an example of which is a piezo- actuated 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. I. I NTRODUCTION Piezoelectric actuators are commonly used in nanoposi- tioning applications such as scanning tunneling microscopes and atomic force microscopes. They have large bandwidth and can produce large mechanical forces [1], [2]. How- ever, they exhibit non-desirable behaviors such as hysteresis, creep, and vibrations. Hysteresis nonlinearity heavily affects the positioning precision. Therefore, in the last two decades, research has been widely conducted for both hysteresis modeling and control design. Control methods used for mitigating the hysteresis effect are classified in general under three categories [1], which are feedback control, feedforward compensation, and the integration of both. Figure 1 illustrates the feedforward scheme augmenting feedback control. The feedback control can be as simple as integral control [3] or more sophisticated, such as adaptive control [4]–[7], servo-compensator [8], and sliding mode control (SMC) [9]–[13]. In this paper, we conduct the analysis of the tracking error for several control schemes that combine the hysteresis inversion with feedback control. We consider a plant that consists of linear dynamics preceded by hysteresis nonlinear- ity, as has often been proposed for smart material-actuated The work was supported by the National Science Foundation (CMMI 0824830). M. Edardar, X. Tan, and H. K. Khalil are with the Department of Electrical and Computer Engineering, Michigan State University, 428 S. Shaw Lane, East Lansing, MI 48824. edardarm@msu.edu (M. E.), xbtan@egr.msu.edu (X. T.), khalil@egr.msu.edu (H. K.) Fig. 1. A general control scheme for a nanopositioner that combines feedforward compensation with feedback controller. systems [1], [7], [14]–[16]. In the first scheme (Scheme 1), the hysteresis inversion is placed inside the feedback loop, following the feedback controller and preceding the hysteresis nonlinearity. In the second scheme (Scheme 2), the hysteresis inversion is placed in the feedforward path. Both Scheme 1 and Scheme 2 have been used extensively in the literature with demonstrated success in experiments [17]–[20] ; however, rigorous analysis on these schemes has been limited. We also suggest a third scheme (Scheme 3), where hysteresis inversion is placed in both the feedforward path and the feedback path. Motivated by the properties of piezo-actuated nanoposi- tioning systems, we assume that the linear dynamics of the plant are stable and have large bandwidth. This assumption allows us to use singular perturbation analysis to obtain low- frequency approximation of the solution in which the linear dynamics are neglected. For higher frequency references we can use singular perturbations to improve the accuracy of the approximation. In addition, we assume that the hysteresis nonlinearity has piecewise linear characteristics; in other words, all hys- teresis loops (major loops or minor loops) consist of linear segments, where each segment s i has a slope m i and an intercept γ i with the output axis. See Fig. 2 for illustration. This assumption is used in [21] where an adaptive inverse control scheme is presented to identify the hysteresis slopes and intercepts in order to compute the hysteresis inverse. The assumption also holds true for a wide class of hysteresis op- erators, such as the Prandtl-Ishlinskii (PI) operator [19], the modified PI operator [22], and the Krasnoselskii-Porkovskii (KP) operator among others [14]. In our analysis, we further assume that, when the hysteresis nonlinearity is known, its inverse can be constructed exactly. This assumption holds true for many operators, examples of which include the PI operator [19] and modified PI operator [22]. In our analysis, we consider both the case where the 51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA U.S. Government work not protected by U.S. copyright 3139 978-1-4673-2066-5/12/$31.00 ©2012 IEEE