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Research Papers

Robust Performance Enhancement Using Disturbance Observers for Hysteresis Compensation Based on Generalized Prandtl–Ishlinskii Model

[+] Author and Article Information
Ahmed H. El-Shaer

Hitachi Global Storage Technologies (HGST),
San Jose, CA 95119
e-mail: aheshaer@gmail.com

Mohammad Al Janaideh

Department of Mechatronics Engineering,
The University of Jordan,
Amman, 11942 Jordan
e-mail: aljanaideh@gmail.com

Pavel Krejčí

Mathematical Institute,
Academy of Sciences of the Czech Republic,
Źitná 25,
CZ-11567 Praha 1, Czech Republic
e-mail: krejci@math.cas.cz

Masayoshi Tomizuka

Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720e-mail: tomizuka@me.berkeley.edu

The command fmincon in Matlab is used to compute α in Eq. (A3).

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement and Control. Manuscript received December 20, 2011; final manuscript received December 17, 2012; published online May 29, 2013. Assoc. Editor: Sergey Nersesov.

J. Dyn. Sys., Meas., Control 135(5), 051008 (May 29, 2013) (13 pages) Paper No: DS-11-1399; doi: 10.1115/1.4023762 History: Received December 20, 2011; Revised December 17, 2012

This paper presents an approach employing disturbance observers to enhance the performance of inverse-based hysteresis compensation based on the generalized Prandtl–Ishlinskii model in feedback control reference-tracking applications. It is first shown that the error resulting from inexact hysteresis compensation is an L-bounded signal. Hence, a disturbance observer (DOB) is designed to cancel its effect and improve the closed loop robust tracking performance in the presence of plant dynamics uncertainty. The design of the DOB makes use of an equivalent internal model-based estimation of exogenous disturbances, where the internal model dynamics is designed to have at least an eigenvalue at the origin. The synthesis is then formulated as an H weighted-sensitivity optimization for static output feedback (SOF) gain of a Luenberger observer. A linearization heuristic is then implemented to solve the bilinear-matrix-inequality (BMI) constrained semidefinite program (SDP) for a (sub)optimal static gain. Simulation results indicate that tracking performance is indeed improved using the combined inversion-based compensation and the DOB.

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Figures

Grahic Jump Location
Fig. 1

Inversion-based hysteresis compensation scheme without DOB

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Fig. 2

Inversion-based hysteresis compensation with DOB-based feedback controller (dashed-dot)

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Fig. 3

DOB-based closed-loop system

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Fig. 4

Weighted plant Pw(s)

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Fig. 5

Bode plots of 1/Wp(s) and 1-Q(s) (left), closed-loop sensitivity S(s) (right)

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Fig. 6

Sample realization of the external disturbance signal

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Fig. 7

The input-output relationship between the reference r(t) = 18 sin (2πt) + 28 sin (6πt) and the output of: (a) the Prandtl-Ishlinskii model G[r](t), and (b) the inverse Prandtl–Ishlinskii model G-1[r](t)

Grahic Jump Location
Fig. 8

Example 1: (a) and (b) show the input-output relationship between the reference input r(t) = 18 sin (2πt) + 28 sin (6πt) and the output y(t) of the closed-loop system with (a) the exact inverse Prandtl–Ishlinskii model G∧-1[v] = G-1[v], and without (b) the inverse Prandtl–Ishlinskii model G∧-1[v] = v. (c) and (d) show tracking error e(t) of the closed-loop system for (a) and (b), respectively.

Grahic Jump Location
Fig. 9

Example 2: (a) and (b) show the input-output relationship between the reference input r(t) = 18 sin (2πt) + 28 sin (6πt) and the output y(t) of the closed-loop system with (a) G∧-1 = 0.9G-1, and (b) G∧-1 = 0.5G-1. (c) and (d) show the tracking error e(t) of the closed-loop system with (c) G∧-1 = 0.9G-1, and (d) G∧-1 = 0.5G-1. (e) and (f) show the tracking error e(t) considering G∧-1 = 0.7G-1 with (e) the DOB, and without (f) the DOB.

Grahic Jump Location
Fig. 10

(a) and (b) show the input-output relationship between the reference input r(t)=18 sin (2πt)+28 sin (6πt) and (a) the output of the generalized Prandtl–Ishlinskii model H[r](t), and (b) the output of the inverse generalized Prandtl–Ishlinskii model H-1[r](t). (c) shows the input-output relationship between the reference input r(t) and the output of the inverse compensation H°H∧[r](t), (d) shows the error of the inverse compensation eη(t) = r(t)-H°H∧-1[r](t) with H∧-1 = H-1 (solid line), H∧-1 = 0.9 H-1 (dashed-dotted line), H∧-1 = 0.8 H-1 (dotted line), and H∧-1 = 0.7 H-1 (dashed line).

Grahic Jump Location
Fig. 11

Example 3. (a) and (b) show the input-output relationship between the reference input r(t) = 18 sin (2πt)+28 sin (6πt) and the output of the closed-loop system y(t) with the DOB and with (a) the exact inverse generalized Prandtl–Ishlinskii model H∧-1[v] = H-1[v], and without (b) the inverse generalized Prandtl–Ishlinskii model H∧-1[v] = v. (c) and (d) the tracking error e(t) of the closed-loop system of (a) and (b), respectively. (e) and (f) show the tracking error e(t) of the closed-loop system with the DOB (solid line) and without the DOB (dashed line) with (e) H∧-1 = 0.95 H-1, and with (f) H∧-1 = 0.7 H-1.

Grahic Jump Location
Fig. 12

Example 4. The input-output relationship between the reference input r(t) = 18 sin (2πt)+28 sin (6πt) and the output of the closed-loop system y(t) with (a) H∧-1 = H-1, (b) H∧-1 = 0.9 H-1, (c) H∧-1 = 0.8 H-1, (d) H∧-1 = 0.7 H-1, (e) H∧-1 = 0.4 H-1, and (f) H∧-1 = 0.1 H-1.

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