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Research Papers

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

[+] Author and Article Information
Mohamed Edardar

Electrical and Electronic Engineering,
University of Tripoli,
Tripoli, Libya
e-mail: moh_edardar@yahoo.com

Xiaobo Tan

Electrical and Computer Engineering,
Michigan State University,
428 S. Shaw Lane,
East Lansing, MI 48824
e-mail: xbtan@egr.msu.edu

Hassan K. Khalil

Electrical and Computer Engineering,
Michigan State University,
428 S. Shaw Lane,
East Lansing, MI 48824
e-mail: khalil@egr.msu.edu

The work was supported by the National Science Foundation (CMMI 0824830, CMMI 1301243).

For convenience, we will drop the subscript i in the analysis unless necessary and use m and γ to denote the slope and intercept of the line segment under consideration.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 20, 2012; final manuscript received December 27, 2013; published online April 4, 2014. Assoc. Editor: Qingze Zou.

J. Dyn. Sys., Meas., Control 136(4), 041010 (Apr 04, 2014) (12 pages) Paper No: DS-12-1382; doi: 10.1115/1.4026511 History: Received November 20, 2012; Revised December 27, 2013

## Abstract

Analysis of closed-loop systems involving hysteresis is important to both the understanding of these systems and the synthesis of control schemes. However, such analysis is challenging due to the nonsmooth nature of hysteresis nonlinearities. In this paper, singular perturbation techniques are employed to derive an analytical approximation to the tracking error for a system consisting of fast linear dynamics preceded by a piecewise linear hysteresis nonlinearity, which is motivated by applications such as piezo-actuated nanopositioning. The control architecture considered combines hysteresis inversion and proportional-integral feedback, with and without a constant feedforward control. The analysis incorporates the effect of uncertainty in the hysteresis model, and offers insight into how the tracking performance depends on the system parameters and the references, thereby offering guidance in the controller design. Simulation and experimental results on a piezo-actuated nanopositioning system are presented to support the analysis. In particular, the control scheme incorporating the feedforward element consistently outperforms the classical PI controller in tracking a variety of references.

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## Figures

Fig. 1

Illustration of a hysteresis loop with piecewise linear characteristics

Fig. 2

The proposed scheme for singularly perturbed systems preceded by hysteresis with added feedforward branch

Fig. 3

Hysteresis inverse in the feedback path

Fig. 4

Illustration of the time instants when periodic signals cross different linear segments of the hysteresis loops

Fig. 5

Simulations results on tracking a periodic reference composed of two sinusoidal signals of 25 Hz and 50 Hz

Fig. 6

Simulation results on tracking a sawtooth reference of 5 Hz

Fig. 7

Simulation results on tracking a sawtooth reference with and without feedforward compensation

Fig. 8

Simulation results on tracking a sawtooth reference when uncertainty is present/absent in the hysteresis model

Fig. 9

Comparison of simulation and analytical results on the tracking error as the reference frequency is varied

Fig. 10

Comparison of the amplitude of tracking error for different control gains as the reference frequency is varied

Fig. 11

Experimental results on the tracking performance of a 35 Hz sinusoidal reference

Fig. 12

Experimental results on the tracking error for sinusoidal references of different frequencies

Fig. 13

Comparison of simulation, analytical, and experimental results on the tracking error as the reference frequency is varied

Fig. 14

Experimental results on the tracking of a triangular reference

Fig. 15

Experimental results on the tracking of a multisine reference with frequencies of 15 Hz and 30 Hz

Fig. 16

Comparison of tracking errors with and without the feedforward component for (a) a triangular reference and (b) mutisine reference

Fig. 17

Comparison of tracking error with and without feedforward component for a sinusoidal reference

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