Calculation of the modified control matrix for a selected unimorph
deformable mirror to compensate the piezoelectric hysteresis effect using
the inverse Bouc–Wen model

M. A. Aghababayee; M. Mosayebi; H. Saghafifar

doi:10.1364/AO.448707

Applied Optics
Vol. 61,
Issue 9,
pp. 2293-2305
(2022)
•https://doi.org/10.1364/AO.448707

Calculation of the modified control matrix for a selected unimorph deformable mirror to compensate the piezoelectric hysteresis effect using the inverse Bouc–Wen model

M. A. Aghababayee, M. Mosayebi, and H. Saghafifar

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M. A. Aghababayee, M. Mosayebi, and H. Saghafifar, "Calculation of the modified control matrix for a selected unimorph deformable mirror to compensate the piezoelectric hysteresis effect using the inverse Bouc–Wen model," Appl. Opt. 61, 2293-2305 (2022)

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Abstract

The hysteresis behavior of piezoelectric actuators degrades the positioning accuracy and bandwidth of nano-positioning systems. Therefore, considering the hysteresis of piezoelectric deformable mirrors is completely essential and also improves the modeling accuracy of adaptive optics layouts. Because of the unique adaptability and mathematical flexibility of the Bouc–Wen model it has gained popularity, and as a result, in many scientific applications, it is one of the most conventional models typically employed to describe nonlinear hysteretic systems. Among different deformable mirrors, a unimorph piezoelectric deformable mirror is a suitable choice to be used in adaptive optics systems because of its relative convenience and cost-effective production. This paper proposes a new, to the best of our knowledge, approach to determine the influence function and the voltage control matrix of a specific unimorph mirror by considering a simplified inverse Bouc–Wen hysteresis model as a frequency function. Then the results for two selected standard Zernike modes, defocus and astigmatism-x, have been simulated using Comsol Multiphysics and MATLAB at a range of 5 to 100 Hz. For a more comprehensive comparison, the root-mean-square error and the coefficients of the Zernike terms have been applied as two criteria. According to the simulation results, the hysteresis effect of piezoelectric actuators has been significantly compensated by applying the inverse Bouc–Wen model at different frequencies, especially for higher frequencies. The effectiveness of the inverse Bouc–Wen model to compensate the hysteresis has been observed in astigmatism-x mode slightly more than in the defocus mode.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Frequency (Hz)	$k$	$α$	$β$	$γ$
5	0.2215	−0.0473	0.0847	0.4477
10	0.2209	−0.0598	0.2629	0.6506
20	0.2214	−0.0832	0.7183	1.1444
30	0.2198	−0.1146	1.5761	2.0662
40	0.2181	−0.1453	2.8847	3.4124
50	0.2181	−0.1838	2.0272	2.6029
60	0.2176	−0.1877	2.4755	3.0932
70	0.5153	−0.2492	4.0013	4.6221
80	0.2170	−0.2055	2.9687	3.5753
90	0.2114	−0.2970	5.9981	6.6169
100	0.2166	−0.2293	3.6008	4.2298

Layer	Property (unit)	Value
Piezo disk	Material	PZT-5 H
	Density ( $k g \cdot m^{- 3}$ )	7500
	Thickness (µm)	200
	Diameter (mm)	32
	Voltage ( $V$ )	$- 50 \sim + 100$
	Young’s modulus (GPa)	80.2
	Poisson’s ratio	0.35
	Transverse constant d31 ( ${m V}^{- 1}$ )	$- 265 \times 10^{- 12}$
	Relative permittivity $ε_{11}$	3130
	Relative permittivity $ε_{33}$	3400
Mirror glass	Material	silicone
	Density ( $k g \cdot m^{- 3}$ )	2329
	Thickness (µm)	200
	Diameter (mm)	40
	Young’s modulus (GPa)	170
	Poisson’s ratio	0.28
Flex cable	Material	Peek
	Density ( $k g \cdot m^{- 3}$ )	1320
	Thickness (µm)	200
	Young’s modulus (GPa)	3.6

Defocus Mode Residual
	Without Hysteresis		With Hysteresis
Frequency (Hz)	$E_{r m s}$ (µm)	$e_{r r m s}$ (%)	$E_{r m s}$ (µm)	$e_{r r m s}$ (%)	Relative Error (%)
5	0.371	2.61%	0.209	1.42%	1.21%
10	0.467	3.39%	0.226	1.64%	1.75%
20	0.591	4.58%	0.276	1.88%	2.716%
50	0.692	5.91%	0.325	2.21%	3.71%
100	0.779	7.49%	0.381	2.59%	4.916%

Astigmatism-x Mode Residual
	Without Hysteresis		With Hysteresis
Frequency (Hz)	$E_{r m s}$ (µm)	$e_{r r m s}$ (%)	$E_{r m s}$ (µm)	$e_{r r m s}$ (%)	Relative Error (%)
5	0.224	3.15%	0.105	1.45%	1.7%
10	0.279	4.08%	0.117	1.67%	2.41%
20	0.332	5.24%	0.136	1.99%	3.25%
50	0.381	6.61%	0.157	2.48%	4.13%
100	0.429	8.55%	0.172	2.94%	5.61%

Frequency (Hz)	$k$	$α$	$β$	$γ$
5	0.2215	−0.0473	0.0847	0.4477
10	0.2209	−0.0598	0.2629	0.6506
20	0.2214	−0.0832	0.7183	1.1444
30	0.2198	−0.1146	1.5761	2.0662
40	0.2181	−0.1453	2.8847	3.4124
50	0.2181	−0.1838	2.0272	2.6029
60	0.2176	−0.1877	2.4755	3.0932
70	0.5153	−0.2492	4.0013	4.6221
80	0.2170	−0.2055	2.9687	3.5753
90	0.2114	−0.2970	5.9981	6.6169
100	0.2166	−0.2293	3.6008	4.2298

Abstract

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Equations (17)

Applied Optics