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Research Papers

# A Polynomial-Based Linear Mapping Strategy for Feedforward Compensation of Hysteresis in Piezoelectric Actuators

[+] Author and Article Information
Saeid Bashash

Smart Structures and Nanoelectromechanical Systems Laboratory, Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921sbashas@clemson.edu

Smart Structures and Nanoelectromechanical Systems Laboratory, Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921jalili@clemson.edu

PZT is a short abbreviation for lead zirconate titanate $(PbZrTiO3–PbTiO3)$. PZT ceramics are the most widely used materials in a variety of today’s piezoelectric-based positioning applications because of their excellent properties (20).

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Corresponding author.

J. Dyn. Sys., Meas., Control 130(3), 031008 (Apr 25, 2008) (10 pages) doi:10.1115/1.2907372 History: Received May 17, 2006; Revised January 23, 2008; Published April 25, 2008

## Abstract

A set of memory-based properties is employed in this paper for modeling multiple-path hysteresis response of piezoelectric actuators. These properties, namely, targeting turning points, curve alignment, and wiping-out effect, are applied in a linear mapping strategy to develop a mathematical framework for modeling the hysteresis phenomenon. More specifically, the locations of turning points are detected and recorded for the prediction of future hysteresis trajectory. An internal trajectory is assumed to follow a multiple-segmented path via a continuous connection of several curves passing through every two consequent turning points. These curves adopt their shapes via a linear mapping strategy from the reference hysteresis curves with polynomial configurations. Experimental implementation of the proposed method demonstrates slight improvement over the widely used Prandtl–Ishlinskii hysteresis operator. However, to maintain the level of precision during the operation, a sufficient number of memory units must be included to record the turning points. Otherwise, in the event of memory saturation, two memory-allocation modes, namely, “open” and “closed” strategies, can be implemented. It is shown that the closed memory-allocation strategy demonstrates better performance by keeping the most important target points. The proposed modeling framework is adopted in an inverse model-based control scheme for feedforward compensation of hysteresis nonlinearity. The controller is experimentally implemented on a three-dimensional nanopositioning stage for surface topography tracking, a problem typically encountered in scanning probe microscopy applications.

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## Figures

Figure 1

Figure 2

Targeting turning point and curve-alignment property

Figure 3

Demonstration of wiping-out property

Figure 4

Linear mapping of the hysteresis curve between two initial and target points

Figure 5

A typical multisegment hysteresis path

Figure 6

Figure assisting the proof of the curve-alignment satisfaction

Figure 7

3D PZT-driven nanopositioning stage comprising of the Physik Instrumente P-753.11c Z-stage on top of a PI P-733.2CL 2D XY-stage

Figure 8

Experimental verification of the proposed hysteresis model: (a) input profile, (b) experimental nanopositioning stage response (solid line) and model response (dashed line), (c) nanopositioning stage hysteresis response, and (d) model hysteresis response

Figure 9

Prandtl–Ishlinskii (PI) and linear model comparisons: (a) PI model response, (b) linear model response, (c) PI model error comparison, and (d) linear model error comparison

Figure 10

Memory-allocation strategy: (a) recording a turning point and (b) wiping-out effect

Figure 11

Saturated memory function: (a) open and (b) closed memory-allocation strategies

Figure 12

Open memory-allocation performance: (a) input profile, (b) hysteresis with full memory units; model responses with (c) one memory unit and (d) two memory units; hysteresis responses with (e) one memory unit and (f) two memory units

Figure 13

Closed memory-allocation performance; model responses with full and (a) one memory unit and (b) two memory units; hysteresis responses with (c) one memory unit and (d) two memory units

Figure 14

Inverse model-based feedforward control strategy

Figure 15

Inverse hysteresis response (corresponding to hysteresis response of Fig. 8)

Figure 16

Surface topography tracking with Physik Instrumente 753.11c Z nanopositioning stage mounted atop a P-733.2CL XY-stage: (a) X-axis tracking, (b) Y-axis tracking, (c) Z-axis tracking, and (d) XYZ surface topography tracking

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