Adaptive state observer and PD control for dynamic perturbations in optical systems

H. Gilbergs; H. Fang; K. Frenner; W. Osten

doi:10.1364/OE.23.004002

1. Introduction

High performance optical systems are capable of reproducing diffraction limited images over a large area. In order to maintain this performance, tolerances for deviations from the ideal design of the objective are very tight. On the one hand, the surfaces of the optical components must be machined to match the desired form down to a fraction of the design wavelength [1]. On the other hand, the parts must be assembled such that all the components are located on their designated positions.

External influences can alter the positions of the optical elements, introducing errors in the system. Most of this influences will be time dependant, either on a slow scale (drift, thermal expansion) or on a fast scale (structural vibrations). In this paper the focus lies on errors that originate from external vibrations that mechanically couple into the system [2].

Structural vibrations in optical systems have a negative impact on the imaging quality. There are several concepts to mitigate this problem, commonly used in photographic and video cameras. Some aim at using the motion of a single optical element in the system to counter the negative effects of the vibrations [3]. Others are designed to actively or passively move the detector in the image plane to account for the dominant tilt [4] to stabilize the image over multiple exposures in software[5]. In this paper we demonstrate a new method for the detection of the positions of all lenses of an optical system exposed to structural vibrations. This information is used to apply a closed loop control to hold the lenses steady in their designated positions countering the structural vibrations. The identification process is based on measurements of the wavefront error present in the system. In this paper only the x-tilt of the wavefront is used as a data source for the reconstruction, as a decentration of an optical element causes a shift of the image as the leading effect [6].

2. System description

The dynamic perturbances of the lens positions are modelled as damped harmonic oscillators. A damped harmonic oscillator is described by the equation of motion

\ddot{α} + δ \dot{α} + ω^{2} α = 0

where α is the amplitude as a function of time, δ is the damping coefficient and ω is the oscillator frequency.

With the initial conditions of the angle α₀ and a phase shift ϕ at a starting time t = 0 the motion of the lens is described by

α = α_{0} e^{- δ t} \sin (ω t + ϕ) .

An optical system consists of n optical elements, each oscillating in x-direction according to Eq. (2). The observable is the total tilt error in the wavefront, which is induced by the motion of the lenses. The tilt error does not change the shape of the wavefront, but changes its propagation direction. Figure 1 shows the influence of tilt on a plane wave as well as its effect on the Shack-Hartmann wavefront sensor [7].

Fig. 1 (a) A lenslet array is illuminated by a plane wave and generates spot images on the CCD detector. (b) A tilt in the incident wave leads to an evenly distributed shift in the spot positions. The spots positions of a) are marked with red dots.

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The contributions of the individual lenses to the wavefront x-tilt are linearly dependent on the angle and in the first order additive for small decentrations. The total x-tilt as a function of time is represented by

T = \sum_{i = 1}^{n} S_{i} α_{i} = \sum_{i = 1}^{n} S_{i} α_{0, i} e^{- δ_{i} t} \sin (ω_{i} t + ϕ_{i})

where S_i are sensitivities of the x-tilt to the amplitudes α_i. The frequencies ω_i and the damping coefficients δ_i are considered to be known a-priori from simulation or independent measurements.

As the individual oscillations all add up to a single observable, there is no easy way of discriminating how much each lens contributes to the total. The corresponding forward problem of calculating the tilt as a function of the three angles is trivial, but to solve the inverse problem of identifying the angles additional assumptions based on a-priori knowledge have to be introduced.

The measurement data is a time series of wavefront tilt values. The first assumption that is made, is that there are three lenses contributing to the total. Additionally we consider the eigenfrequencies and damping coefficients as known. Their values have been determined by independent measurements. This leaves the problem of reconstructing three amplitudes and three phase shifts of the oscillations from the measurement data.

For the analysis of the time series data, a state observer, which calculates a new reconstruction estimate for each new timestep, is well suited.

3. Adaptive state observer

In this section a state observer with time varying input and output matrices is introduced and its states as well as transition and output matrices are defined. The method is derived from the Kalman filter [8], with the key difference that the process noise is neglected. The general linear, discrete-time state space representation describes the system using a state x_k at timestep k, a state transition to the next timestep and an output to calculate observable properties. The transition from k to k + 1 is governed by

x_{k + 1} = A_{k} x_{k} + B_{k} u_{k},

where A_k is the system matrix that describes the transition without external control and B_k is the input matrix that governs the effect of an control input u_k. The observable output y_k can be extracted from the state using

y_{k} = C_{k} x_{k},

where C_k is the output matrix.

A state observer uses a stream of measurement data to estimate the internal state of a system, which, combined with a model, describes the system. Judging from the model Eq. (3), using the state

x = (\begin{matrix} α_{0, 1} \\ ⋮ \\ α_{0, n} \\ ϕ_{1} \\ ⋮ \\ ϕ_{n} \end{matrix})

seems straightforward, but leads to a nonlinear system output, as the ϕ_i are within the sinus functions. A better solution is to decompose Eq. (3) using the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), which leads to the x-tilt

T (t) = \sum_{i = 1}^{n} S_{i} α_{0, i} e^{- δ_{i} t} [\sin (ω_{i} t) \cos (ϕ_{i}) + \cos (ω_{i} t) \sin (ϕ_{i})]

T (t) = \sum_{i = 1}^{n} S_{i} α_{0, i} \cos (ϕ_{i}) e^{- δ_{i} t} \sin (ω_{i} t) + \sum_{i = 1}^{n} S_{i} α_{0, i} \sin (ϕ_{i}) e^{- δ_{i} t} \cos (ω_{i} t)

= \sum_{i = 1}^{n} (c_{i} e^{- δ_{i} t} \sin (ω_{i} t) + s_{i} e^{- δ_{i} t} \cos (ω_{i} t))

with the new state variables c_i = S_iα₀_,i cosϕ_i and s_i = S_iα₀_,i sinϕ_i.

With the state x and the time dependent output matrix C(t) defined as

x = (\begin{matrix} c_{1} \\ ⋮ \\ c_{n} \\ s_{1} \\ ⋮ \\ s_{n} \end{matrix}) C (t) = {(\begin{matrix} e^{- δ_{1} t} \sin (ω_{1} t) \\ ⋮ \\ e^{- δ_{n} t} \sin (ω_{n} t) \\ e^{- δ_{1} t} \cos (ω_{1} t) \\ ⋮ \\ e^{- δ_{n} t} \cos (ω_{n} t) \end{matrix})}^{'},

the model Eq. (9) can be rewritten as T(t) = C(t)·x. As the initial conditions are time invariant, the system matrix A = I is the identity matrix.

The first step of the adaptive state observer is the projection, where the state estimate ${\hat{x}}_{k - 1}$ and its estimated covariance matrix ${\hat{P}}_{k - 1}$ at the timestep k − 1 are projected to intermediate estimates of the state ${\hat{x}}_{k | k - 1}$ and the covarianve matrix ${\hat{P}}_{k | k - 1}$ .

{\hat{x}}_{k | k - 1} = A {\hat{x}}_{k - 1} + Bu = {\hat{x}}_{k - 1} + Bu

{\hat{P}}_{k | k - 1} = A {\hat{P}}_{k - 1} A^{'} = {\hat{P}}_{k - 1},

where B is the control input matrix which maps the control input u to a change in the observable. For the verification of the state observer without closed loop control the input u is initially set to zero. A closer look at B and u will follow at the introduction of PD control in Section 4.2

As the system matrix A is the identity matrix, the intermediate estimation for the next timestep without control input is, that the state stays constant.

The projected state estimate can be refined using the information that the newly arriving measurement y_k carries. The updated estimate will have better accuracy and robustness to external influences not described by the model. Therefore a correction step

{\hat{x}}_{k} = {\hat{x}}_{k | k - 1} + {\hat{K}}_{k} {\tilde{y}}_{k}

{\hat{P}}_{k} = {\hat{P}}_{k | k - 1} - {\hat{K}}_{k} S_{k} {\hat{K}}^{'}_{k}

based on measurement data y_k follows the projection, with the observer gain

{\hat{K}}_{k}

, measurement residual ỹ_k and residual covariance S_k at a given measurement noise R_k defined as

{\tilde{y}}_{k} = y_{k} - C_{k} {\hat{x}}_{k | k - 1}

S_{k} = C_{k} P_{k | k - 1} {C^{'}}_{k} + R_{k}

{\hat{K}}_{k} = P_{k | k - 1} C^{'}_{k} S_{k}^{- 1} .

4. Experimental verification

4.1. Oscillating lenses

The first experiment demonstrates the tracking of independently oscillating lenses using the adaptive state observer.

The experimental setup is a simplified optical system consisting of three lenses (n = 3), each mounted on a pendulum to allow for a circular motion in the x-y-plane (Fig. 2). Configurable weights at the bottom of each pendulum can be used to generate different vibration frequencies. In order to reduce friction, the swings are mounted using ball bearings.

Fig. 2 (a) The setup consists of three lenses, each mounted on a physical pendulum. A solid state laser (λ = 532nm) is focused on a pinhole to act as a point light source. A Shack-Hartmann sensor is used for the detection. For reference measurements each lens pendulum is equipped with an IMU (not depicted). (b) The pendulums hold the lenses at a distance L = 160mm from the rotation axis.

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The wavefront is measured by a Shack-Hartmann sensor [7], a solid state laser (λ = 532nm) focused on a pinhole (d = 10μm) acts as a point light illumination. For the state observer only the dominant component of the wavefront, the x-tilt, is used, all other measured wavefront Zernike terms [9] are omitted.

Due to the high measurement frequency the integration time for each wavefront is 5ms. At this detection rate a wavefront sensor repeatability [10] of 0.03λ has been determined.

As a reference to evaluate the accuracy of the state observer, the motion of the lenses is monitored with inertial measurement units. They are also used to determine the eigenfrequencies and the damping coefficients of the pendulums, which are assumed to be known for the state observer model.

The adaptive state observer relies on a linear relationship between the angle of the lens pendulum and its contribution to the wavefront tilt (see Eq. (3)). As the simulation data in Fig. 3(a) shows, this assumption holds well for angles in the range of −3° to 3°. The gradients of the lines are the sensitivities S_i in the Eqs. (3)–(8).

Fig. 3 (a) Simulation of the wavefront x-tilt in the aperture of the wavefront sensor as a function of the angular displacement of each lens. For small angles a linear dependence can be assumed. (b) Measured angular data from the gyroscopes (green dots) and a damped harmonic oscillator fit (black line). The data is reproduced accurately, with a slight degradation at lower amplitudes, where holding friction becomes noticeable.

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The reference curve used to fit the damped harmonic oscillator model is measured using an inertial measurement unit consisting of a gyroscope sensor with a high angular resolution and an accelerometer [11]. From this measurement data the frequency ω and the damping coefficient δ of the oscillation are determined. Those two parameters are used as a-priori data for the observer. As Fig. 3(b) shows, the measured angles fit the damped harmonic oscillator model reasonably well.

The reconstruction has been conducted on measurement data of the x-tilt recorded over 7s while all lenses were in motion. After two seconds the observer estimate reproduces the reference measurement of the IMUs (Fig. 4). The slight deviations from the control measurement are due to the sensor integration time. The wavefront measurement takes 5ms during which the lenses are still moving, creating a motion blur in the sensor raw data.

Fig. 4 Results of the time varying state observer applied to experimental data. The reference measurement from the IMUs (small markers) is reproduced with slight deviations due to motion blur in the wavefront measurement.

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The next step is to use this observer output to control the oscillations.

4.2. Quasi-statical oscillations and PD control

The second experiment shows the possibility of closed loop control based on the state estimate of the adaptive state observer. For a high repeatability of the oscillations with identical initial conditions and the possibility to include control, a second experimental setup, where the lens decentrations can be controlled using a linear actuation system, is used.

The optical system is a telecentric design with a 4x reduction. It consists of 5 lenses, three of which can be decentrated individually in x direction, whereas the remaining two move as a group on a single stage (n = 4). Figure 5 shows a sketch of the optical system. The trajectories of the optical elements are calculated at discrete time instants (dt = 0.02s) using Eq. (2). To estimate the positions the adaptive state observer is applied.

Fig. 5 Optical setup used for the PD control experiment. The object plane (left side) is projected to the image plane with a 4x reduction. The three single lenses as well as the lens group in the center can be controlled using a linear actuation system. For the detection of the wavefront tilt a Shack-Hartmann sensor located after the image plane is used (not depicted). The distance from the object plane to the image is 857.7mm

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Based on this estimate a closed loop control is applied to counter the oscillations of the optical elements directly. A proportional-derivative controller (PD controller) [12] is a widely used method to control various processes. It aims at minimizing the difference from an estimated process variable, here the estimated decentration ${\tilde{α}}_{i}$ , from its setpoint $({\tilde{α}}_{i} = 0)$ by adjusting a control variable, here the real decentration α_i.

To control all n optical elements of the system, n separate PD controllers have to be used. The PD controllers use the estimated amplitudes from the adaptive state observer to generate the control output

u_{i, k} = K_{p} α_{i, k} + K_{d} \frac{α_{i, k} - α_{i, k - 1}}{Δ t}

which is fed back into the motion control and the observer. The PD gains have been hand tuned to K_p = 0.02, and K_d = 0.007.

To make the observer aware that a control has been applied, the B_k matrix that maps the amplitude changes to x-tilt changes and the corresponding u_k vector in the projection step (Eq. (11)) are defined as

B_{k} = (\begin{matrix} S_{1} \cos (ϕ_{1, k}) \\ ⋮ \\ S_{n} \cos (ϕ_{n, k}) \\ S_{1} \sin (ϕ_{1, k}) \\ ⋮ \\ S_{n} \sin (ϕ_{n, k}) \end{matrix}) u_{k} = (\begin{matrix} u_{1, k} \\ ⋮ \\ u_{n, k} \\ u_{1, k} \\ ⋮ \\ u_{n, k} \end{matrix}),

where the phase shifts of the lens oscillations ϕ_i,k can be directly calculated from

{\hat{x}}_{k}

. Figure 6 shows a schematic of the applied closed loop control.

Fig. 6 Schematic of the full PD control loop. A seperate PD controller is implemented for each of the n optical elements in the system.

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The results of the adaptive state observer with and without PD control are depicted in Fig. 7, with both curves starting at identical initial conditions. As the measurements are quasi-static, the motion blur errors from Fig. 4 have disappeared. The adaptive state observer takes less than a second to lock in on the correct oscillations (Fig. 7(a)). After this time the estimates for $\hat{x}$ begin to stabilize and the PD control is triggered. From here it takes additional 4s to effectively stop the lenses from oscillating (Fig. 7(b)).

Fig. 7 Comparison of the resulting lens oscillations for the same initial conditions with and without PD control. (a) The exact values of the decentrations (small markers) are reproduced accurately after 1s. (b) The controller dampens the vibrations to zero in 5s. The measurements are conducted on a quasi-static setup for high repeatability of the oscillations for identical initial conditions.

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5. Conclusion and outlook

The detection of dynamic perturbations in optical systems from measurements of the x-tilt of the wavefront has been demonstrated. The oscillations of the lenses of a simplified setup are reproduced accurately. Furthermore a closed loop PD control based on the adaptive state observer has been successfully implemented on a quasi-static setup.

As we have shown in simulations in our previous work [13], this method can be expanded to more complex systems with coupled vibrations. To increase the overall accuracy of the observer in the dynamic setup, the measurements can be conducted with a higher sampling rate of a dedicated tip-tilt sensor [14]. Alternatively a high power light source can be used to shorten the integration time. Additionally more Zernike terms can be used for the observer input in more complex optical systems.

Acknowledgments

The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology ( EXC 310/1) at the University of Stuttgart.

References and links

1. K. F. Beckstette, “Ultrapräzise Oberflächenbearbeitung am Beispiel von Lithografieoptiken (Ultraprecise Surface Figuring for Lithography Optics),” Tech. Mess. 69(12), 526 (2002). [CrossRef]

2. J. Holterman, T. J. de Vries, and F. Auer, “Decoupling of collocated actuator-sensor-pairs for active vibration control,” in 21st Benelux Meeting on Systems and Control Book of Abstracts (2002), p. 66.

3. K. Washisu, “Control Apparatus For Image Blur Correction,” US 6757488 B2 (2004).

4. S. Kakiuchi, “Camera Provided With Camera-Shake Compensation Functionality,” US 2006/0008263 A1 (2006).

5. S.-J. Ko, S.-H. Lee, and K.-H. Lee, “Digital image stabilizing algorithms based on bit-plane matching,” IEEE Trans. Consumer Electronics 44(3), 617–622 (1998). [CrossRef]

6. J. H. Burge, “An easy way to relate optical element motion to system pointing stability,” Proc. SPIE 6288, 62880I (2006). [CrossRef]

7. R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61(5), 656 (1971).

8. B. D. Anderson and J. B. Moore, Optimal Filtering (Courier Dover Publications, 2012).

9. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207 (1976). [CrossRef]

10. D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002). [CrossRef]

11. M. M. Morrison, “Inertial Measurement Unit,” US 4711125 (1987).

12. M. Araki, “PID control,” Control Systems, Robotics and Automation 2, 1–23, (2002).

13. H. Gilbergs, N. Wengert, K. Frenner, P. Eberhard, and W. Osten, “Reconstruction of dynamical perturbations in optical systems by opto-mechanical simulation methods,” Proc. SPIE 8326, 83262N (2012). [CrossRef]

14. J. Watson, “Tip-Tilt Correction for Astronomical Telescopes using Adaptive Control,” Wescon - Integrated Circuit Expo (1997), pp. 490–494.

Adaptive state observer and PD control for dynamic perturbations in optical systems

Abstract

1. Introduction

2. System description

3. Adaptive state observer

4. Experimental verification

4.1. Oscillating lenses

4.2. Quasi-statical oscillations and PD control

5. Conclusion and outlook

Acknowledgments

References and links

Cited By

Figures (7)

Equations (19)

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