1. Introduction

Piezoelectric bimorph mirrors have been used as wavefront correctors since the early days of adaptive optics (AO) for their low-cost, robustness and large stroke [1–4]. However, the performance of AO systems using bimorph mirrors is hindered by the presence of hysteresis in the piezoceramic materials, reducing the closed-loop bandwidth and degrading the open-loop correction.

 

Fig. 1. (a) Elemental hysteresis operator $\widehat{\gamma }$_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).

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Mathematical models of hysteresis are commonly used to describe the response of magnetic [5, 6] and piezoceramic [7–9] materials and devices. However, to our knowledge, only the work by Yang et al. [10] tested the performance of one such model in a curvature bimorph mirror. This work used the Coleman-Hodgdon model, in which the fitting of the experimental hysteresis curve is carried out by using just 8 parameters, thus limiting the quality of the hysteresis calibration. Moreover, the inversion of this model requires a numerical evaluation of the Lambert W function, which is an iterative process that could be computationally demanding [11].

In this work we test the suitability of Preisach classical and nonlinear hysteresis models to describe the behavior of piezoceramic deformable mirrors. Results of the implementation and testing of an inversion algorithm of the Preisach classical model in open-loop are also presented.

2. Mathematical models of hysteresis

In this section we briefly recall the main ideas behind the Preisach classical and nonlinear scalar hysteresis models. An extensive description of these models and their numerical implementation can be found in the book by Mayergoyz [5].

Let us start by assuming we have a causal system h with scalar input u and scalar output f. When the output at a given time T depends not only on the state of input u(T) but also on its previous history, we can say that the system suffers from hysteresis. More succinctly, hysteresis is a rate independent memory effect. By rate independent it we mean that f(T) depends on u(t) for t < T but not on its derivatives.

In order to introduce the mathematical hysteresis models, it is convenient to define an elementary hysteresis operator $\widehat{\gamma }$_αβ, as a discontinuous switch between two values γ _- and γ ₊. The switch from γ _- to γ ₊ is produced when the input takes the value α while increasing and the reverse switch occurs when the input reaches the value β (< α) while decreasing, as depicted in Fig. 1(a).

 

Fig. 2. Geometrical interpretation of α - β plane (shown on the right), where each point corresponds to a $\widehat{\gamma }$ 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.

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2.1. Preisach classical model

In the Preisach classical model the behavior of a system h is described as a weighted integral of elementary hysteresis operators,

and assumes the congruency and wiping-out properties. The congruency property states that all minor loops in the input-output diagram, such as those shown in Fig. 1(b), with the same extreme input values are congruent. The wiping-out property means that only the alternating series of dominant input extremes, referred to as the input reduced memory sequence, is needed to determine the current behavior of the system.

In this description, it is convenient to look at the system in an α - β plane (Fig. 2), where each point on the half-plane α > β corresponds to a $\widehat{\gamma }$_αβ operator. In practice, the input values of a system only take values within a range [u_min , u_max ], and therefore the support of the function μ,(α,β) can be non-null only within a triangular region. The interpretation of the evolution of the system in terms of the $\widehat{\gamma }$_αβ operators can be thought of as follows. Let us assume the system starts in its minimum input value u_min as in Fig. 2, therefore all the elementary hysteresis operators are in the lower γ _--state (indicated in gray). Then, by increasing the input u(t), say to M ₁, all the operators with α < M ₁ (corresponding to the area below α = M ₁ in the α - β plane) will be switched to the γ ₊-state (indicated in green). Then, as the input decreases to say m ₁, all operators with β >m ₁ (corresponding to the region to the right of β =m ₁) are switched to the lower γ _- -state, and so forth for each change of direction of the input. Therefore, to know which operators are in the γ _-- or γ ₊-states, it is only necessary to know the extreme values that form the reduced memory sequence. The reduced memory sequence is given by the corners of the boundary (red) between the γ ₊ (green) and γ _- (gray) states of the operators on the α - β plane.

The problem of calculating the output f(t) of the system for a given reduced memory sequence and input u(t), becomes that of calculating the integral defined in equation 1, over the area below the boundary in the α - β plane. This integral can be taken to a format more convenient for numerical implementation,

where f_min is the output of the system when the input takes its minimum value u_min , n(t) is the number of maxima in the reduced memory sequence at the time t, M_k and m_k are the maxima and minima that form the reduced memory sequence and F is the first-order reversal function. Which is defined

where f_α,β is the system output value after increasing the input from the minimum value up to α and then decreasing it down to the value β, and f_α = f_α,α . It can be shown that this is equivalent to

where the integration is performed over the triangle T(α,β) defined by the points with coordinates (α,β), (β,β) and (α, α). Now, by looking at equations 2 and 3 it can seen that in order to calculate the output of the system at a given time, one only needs to know the values of the system output at the first reversal curves f_α,β . The method suggested in most references [7, 6, 9], and adopted here, consists of measuring the first-order reversal values experimentally at a square grid of points over the α - β plane, and estimating the required value of F by linear interpolation. In principle, there is nothing preventing the use of other interpolation methods for higher accuracy, other than for the increase in number of calculations required.

2.2. Preisach nonlinear model

The fundamental difference between the Preisach nonlinear and classical models is the relaxation of the congruency property assumption. Instead, the more general hypothesis of equal vertical chords is assumed. This hypothesis implies that for all minor loops in the input-output diagram with the same extreme input values, the vertical segments joining the highest and lowest value (vertical chords) have equal lengths for the same input values. This translates into a ”nonlinear” weighting function in formula 1, which also depends on the current input value u, that is, μ = μ(α,β,u(t)). This leads to the replacement of the first-order reversal function F by a second-order reversal function,

where f_α,u is as defined before and f_α,β,u is the system output value after increasing the input from the minimum value up to α, then decreasing it down to the value β and finally increasing it up to u. This definition is equivalent to

where the integration is performed over the rectangle R(α, β) with corners (α, β), (u,β), (α, u) and (u,u). In this model, the system output is calculated as

where $f_{u\left(t\right)}^{\mathit{-}}$ is the output value after the input is taken from its minimum value to its maximum and then to u.

 

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.

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Note that other than the function P, which is measured experimentally over a cubic grid of points in the α - β -u space and interpolated elsewhere, the only information required to calculate the system output is the reduced memory sequence.

3. Experimental Setup

The piezoceramic deformable mirror (DM) used in the following experiments was made in-house using a single layer of lead zirconate titanate (PZT) PIC151, glued to a glass substrate with the dimensions shown in Fig. 3(a). The mirror had a single actuator that was driven with a power supply that can provide up to ±220 V. This voltage range gave a stroke range of 60μm measured at the center of the mirror.

In the experimental setup the mirror under test was illuminated by a collimated monochromatic beam (λ = 632.8 nm) at normal incidence. The mirror response, shown in Fig. figure 3(b) was measured using a SH wavefront sensor. In what follows, the experimental setup will be considered as a system in which the input is the voltage used to drive the mirror and the output is the x-displacement of the lenslet with coordinates (-2,1) on the SH lattice shown in Fig. 3(b).

4. Experimental validation of the models

To test the suitability of both the Preisach classical and nonlinear hysteresis models, we uniformly sampled the functions F and P in their corresponding 2- and 3-dimensional spaces. The number of samples along each dimension N was found by increasing N (starting at N = 5) until no further improvement in performance was obtained. Intermediate values of F and P are linearly interpolated.

Once the hysteresis calibration data was acquired, the mirror was driven using different input sequences, and the resulting SH spot displacements were recorded. Then, the prediction of three different models were compared to these experimental results. The first model assumes the mirror response is hysteresis-free and linear, while the other two models are the Preisach classical and nonlinear. The results for two input sequences, a sinusoid and a sequence of uniformly distributed random values are shown in Figs. 4 and 5. The bottom plots show the difference between each of the models and the recorded data. The rms error between the model prediction and the experimental data is reduced by a factor of 3 – 4. This number depends strongly on the sequence of signals used to drive the mirror, piezoceramic material and mirror geometry. However, we believe the graphs presented here provide a reasonable idea of the improvement that one can expect by using these hysteresis models.

 

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.

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In a noise-free system the nonlinear hysteresis model should be more accurate than the classical model for all input sequences. However, this is not the case from Fig. 4, and may be due to the presence creep in the piezoceramic material. Creep is a slow deformation that follows a change in the input signal driving the mirror and is not accounted for by the hysteresis models. The amplitude of this deformation depends on the amplitude of the input change applied. Figure 6 illustrates the worst case scenario, where the input is changed from its minimum value to its maximum in one step, resulting in a creep of around 5% over a period of 40 seconds. This time-scale makes the creep negligible in a closed-loop configuration.

5. Preisach classical inversion: open-loop test

Having observed that both hysteresis models perform similarly, an inversion algorithm for the simpler classical model was implemented. The inversion algorithm is as described in reference [6] using linear interpolation for the inversion of the F function.

In order to test the inversion algorithm in an open-loop configuration, sequences of desired output values were defined, and then, both the hysteresis-free linear inversion and the Preisach inversion algorithm were used to calculate the sequences of input values with which to drive the mirror and the resulting output sequences were recorded. Finally, the error of each model is calculated as the rms of the difference between the desired and the measured output sequences. The results of two of such processes are shown in Figs. 7 and 8. Both cases show a remarkable improvement with the Preisach model, which can be identified as an error reduction of almost an order of magnitude. The input-output diagrams in these figures, clearly show how the mirror driven with the inversion algorithm can be considered linear to the fourth decimal place of the correlation coefficient.

 

Fig. 5. As in Fig. 4. In this case, the input voltage sequence is a series of uniformly distributed random numbers.

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Fig. 6. Example of the creep observed in the DM when driven with an alternating step function.

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6. Discussion

Preisach models in closed-loop will not be discussed here, because hysteresis is a fundamentally nonlinear process, and therefore, the performance of the system and model are not easily quantifiable in terms of concepts developed for linear systems, such as bandwidth.

Preisach-type algorithms have been previously discarded for use in AO [10], where it was argued that they are computationally demanding, both in terms of memory and calculations. This is not the case, and to see this let us look at an AO system with a SH as a wavefront sensor and a piezoceramic DM as a wavefront corrector. If the system is driven in closed-loop using a simple least-squares control, then the control matrix that needs to be kept in memory has dimensions on the order of N_a N_s , N_a being the number of the DM actuators and N_s the number of SH lenslets (typically 20–100). In comparison, the hysteresis calibration data required for driving a DM using Preisach classical model is of the order of N_a N ², with N being the number samples along one dimension in the α-β plane. As demonstrated here, even a small N (10–15) can produce a significant performance improvement, without increasing memory requirements. In terms of computing power required for the Preisach inversion algorithm, it is comparable to that of other algorithms such as the Coleman-Hodgdon [10]. The number of operations required by Preisach-type models is set by the reduced memory sequence of the signal and the type of interpolation used in the Preisach space. Even though the number of elements of the reduced memory sequence can be infinite, by discretizing the input values, the number of operations required for each inversion would be upper bounded.

 

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.

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7. Conclusions

The Preisach classical and nonlinear models were used to successfully model the hysteretic response of a piezoceramic deformable mirror, with both models performing similarly. It was shown that the response of a piezoceramic DM can be linearized by inverting Preisach classical model. This inversion algorithm reduced the output errors by almost an order of magnitude, to around 3%, which is consistent with results reported for piezoceramic piston actuators [7, 9]. The experimental configuration used in this work is that a of conventional AO system to be used in closed-loop, and there is no particular advantage or technical reason for using a SH to study the mirror hysteresis. We shall then emphasize, that no hardware modifications should be performed to existing AO systems to incorporate hysteresis correction as described in this work. Only a calibration data run and the incorporation of a hysteresis inversion algorithm as part of the control software are required.

 

Fig. 8. As in figure 7. In this case, the input signal sequence is a signal of uniformly distributed random numbers.

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