Abstract

In this work the Preisach classical and nonlinear models are used to model the hysteretic response of a piezoceramic deformable mirror for use in adaptive optics. Experimental results show that both models predict the mirror behavior to within 5% root-mean-squared (rms) error . An inversion algorithm of the Preisach classical model for linearization of the mirror response was implemented and tested in an open-loop adaptive optics system using a Shack-Hartmann (SH) sensor. Measured errors were reduced from 20% rms to around 3%.

© 2005 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]

App. Opt.

Osamu Ikeda and Takuso Sato, �??Comparison of deformability between multilayered deformable mirrors with a monomorph or a bimorph actuator,�?? App. Opt. 25, 4591�??4597 (1986).
[CrossRef]

IEEE Transactions on Signal Processing

François Chapeau-Blondeau and Abdelilah Monir, �??Numerical evaluation of the Lambert W function and application to generation of generalised gaussian noise with exponent 1/2,�?? IEEE Transactions on Signal Processing 50, 2160�??2165 (2002).
[CrossRef]

IEEE/ASME Transactions on Mechanotronics

Samir Mittal and Chia-Hsiang Menq, �??Hysteresis compensation in electromagnetic actuators through Preisach model inversion,�?? IEEE/ASME Transactions on Mechanotronics 5, 394�??409 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mechanical Systems and Signal Processing

H. Hu and Ben Mrad, �??A discrete-time compensation algorithm for hysteresis in piezoceramic actuators,�?? Mechanical Systems and Signal Processing 18, 169�??185 (2003).
[CrossRef]

Opt. Eng.

Alexis V. Kudryashov, �??Semipassive bimorph flexible mirrors for atmospheric adaptive optics applications,�?? Opt. Eng. 35, 3064�??3073 (1996).
[CrossRef]

Oxford series in optical and imaging sci

J.W. Hardy, �??Adaptive optics for astronomical telescopes,�?? Oxford series in optical and imaging sciences, (Oxford University Press, New York, 1998).

Precision Eng.

Ping Ge and Musa Jouaneh, �??Modeling hysteresis in piezoceramic actuators,�?? Precision Eng. 17, 211�??221 (1995).
[CrossRef]

Ping Ge and Musa Jouaneh, �??Generalised preisach model for hysteresis nonlinearity of piezoceramic actuators,�?? Precision Eng. 20, 99�??111 (1997).
[CrossRef]

Series in Electromagnetism

Isaak MayerGoyz, �??Mathematical models of hysteresis and their applications,�?? Series in Electromagnetism. Elsevier, Oxford, United Kingdom, 2003.

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Figures (8)

Fig. 1.
Fig. 1.

(a) Elemental hysteresis operator γ ^ αβ and (b) anatomy of a hysteresis loop. Notice that the single parameter usually referred to as hysteresis, is the ratio of the maximum possible output difference for any input (Δa) divided by the output range (Δb).

Fig. 2.
Fig. 2.

Geometrical interpretation of α - β plane (shown on the right), where each point corresponds to a γ ^ operator. The evolution of the input shown on the left, maps onto the triangle in the α - β plane through the dashed and dotted lines. The dashed lines correspond to the maxima and the dotted lines to the minima of the reduced memory sequence. Each pair maximum-minimum defines a corner on the α - β plane.

Fig. 3.
Fig. 3.

(a) Schematic of the single-actuator PZT DM studied using Preisach hysteresis models, and (b) x-displacement hysteresis curves recorded by a SH sensor in response to a driving saw-tooth voltage. The coordinates on top of each plot indicate the lenslet coordinate in the SH array.

Fig. 4.
Fig. 4.

Measured and predicted output of the piezoceramic mirror driven with a sinusoidal sequence. The top graph plots the output measured by the SH and the bottom graph shows the prediction errors. Note the different scale in the plots.

Fig. 5.
Fig. 5.

As in Fig. 4. In this case, the input voltage sequence is a series of uniformly distributed random numbers.

Fig. 6.
Fig. 6.

Example of the creep observed in the DM when driven with an alternating step function.

Fig. 7.
Fig. 7.

Open-loop output of the piezoceramic mirror driven using the Preisach classical hysteresis compensation algorithm and no compensation for a sinusoidal sequence. The top graph plots the output measured by the SH. The middle graph shows the error in the measured output. The central graph plots the difference between the desired behavior and the measured one for both models. The bottom graphs, show the input-output diagrams for each model and the respective correlation coefficients.

Fig. 8.
Fig. 8.

As in figure 7. In this case, the input signal sequence is a signal of uniformly distributed random numbers.

Equations (7)

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f ( t ) = ∫∫ α β μ α β γ ̂ αβ u ( t ) dαdβ ,
f ( t ) = f min + k = 1 n ( t ) [ F M k m k 1 F M k m k ]
F α β = f α f α , β
F α β = ∫∫ T ( αβ ) μ α β dαdβ ,
P α β u = f α , u f α , β , u
P α β u = ∫∫ R ( α β ) μ α β u dαdβ ,
f ( t ) = f u ( t ) + k = 1 n ( t ) [ P ( M k + 1 , m k , u ( t ) ) P ( M k , m k , u ( t ) ) ] .

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