Optimal and adaptive control of aero-optical wavefronts for adaptive optics

doi:10.1364/JOSAA.29.001625

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Optimal and adaptive control of aero-optical wavefronts for adaptive optics

Jonathan Tesch and Steve Gibson »View Author Affiliations

JOSA A, Vol. 29, Issue 8, pp. 1625-1638 (2012)
http://dx.doi.org/10.1364/JOSAA.29.001625

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Abstract

This paper compares two control methods to predict and correct aero-optical wavefronts derived from recent flight-test data. The first is an optimal linear time-invariant controller constructed from an identified state-space model of the turbulence flow. The second control method is an adaptive controller based on a recursive least-squares lattice filter. The performance of these control schemes versus classical integrator methods is investigated in an adaptive optics experiment that reproduces the aberrations from in-flight measurements of aero-optical turbulence. Experimental results show the improvement in wavefront correction achieved by both prediction methods. Altering the flow characteristics of the disturbance wavefront during the control process illustrates the ability of the adaptive controller to track changes in the aberration statistics.

1. INTRODUCTION

In recent adaptive optics (AO) research, improved wavefront correction has been achieved by adaptive filtering and control [1

B. L. Ellerbroek and T. A. Rhoadarmer, “Real-time adaptive optimization of wavefront reconstruction algorithms for closed-loop adaptive optical systems,” Proc. SPIE 3353, 1174–1185 (1998). [CrossRef]

J. S. Gibson, C. C. Chang, and B. L. Ellerbroek, “Adaptive optics: wave-front correction by use of adaptive filtering and control,” Appl. Opt. 39, 2525–2538 (2000). [CrossRef]

T. A. Rhoadarmer, L. M. Klein, J. S. Gibson, N. Chen, and Y.-T. Liu, “Adaptive control and filtering for closed-loop adaptive-optical wavefront reconstruction,” Proc. SPIE 6306, 63060E (2006). [CrossRef]

Y. T. Liu and J. S. Gibson, “Adaptive control in adaptive optics for directed-energy systems,” Opt. Eng. 46, 046601 (2007). [CrossRef]

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

C. M. S. Corley, M. Nagashima, and B. N. Agrawal, “Beam control and a new laboratory testbed for adaptive optics in a maritime environment,” in Proceedings of 2010 IEEE Aerospace Conference (IEEE, 2010), pp. 1–13.

–7

C. M. S. Corley, “Maritime adaptive optics beam control,” Ph.D. thesis (Naval Post Graduate School, 2010).

] and by optimal linear time-invariant (LTI) filtering and control [8

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, F. Chemla, and D. Rabaud, “Off-axis adaptive optics with optimal control: experimental and numerical validation,” Proc. SPIE 5903, 59030P (2005). [CrossRef]

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464–7476 (2006). [CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven $H_{2}$ optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007). [CrossRef]

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven $H_{2}$ optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008). [CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, and H.-F. Raynaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26, 1307–1325 (2009). [CrossRef]

J. Tesch, J. S. Gibson, S. Gordeyev, and E. Jumper, “Identification, prediction and control of aero optical wavefronts in laser beam propagation,” in 42nd AIAA Plasmadynamics and Lasers Conference (AIAA, 2011), paper 2011-3276.

J. Tesch and J. S. Gibson, “Optimal and adaptive correction of aero-optical wavefronts in an adaptive optics experiment,” Proc. SPIE 8165, 816502 (2011). [CrossRef]

–15

J. Tesch, “High-performance control and prediction for adaptive optics,” Ph.D. thesis (University of California, Los Angeles, 2011).

]. Both methods aim to extend the limited bandwidth of a classical AO control loop by explicitly predicting wavefront aberrations to mitigate loop latency. Without a prediction capability, the only method to improve the bandwidth of wavefront correction is to increase the sample rate of the AO loop.

This paper compares the effectiveness in an AO experiment of two high-performance controllers for reducing phase aberrations resulting from aero-optical turbulence: (1) an optimal (i.e., minimum-variance) LTI controller based on an LTI minimum-variance prediction filter with infinite impulse response and (2) an adaptive controller based on an adaptive prediction filter with finite impulse response (FIR) [5

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

]. The LTI controller has significant advantages in computational efficiency. Experimental results here demonstrate that, for stationary or quasi-stationary turbulence, an optimal LTI controllers can be as effective as a fully adaptive controller. For the experiment described in this paper, aero-optical wavefronts measured in the University of Notre Dame’s Airborne Aero-Optics Laboratory (AAOL) [16

C. Porter, S. Gordeyev, M. Zenk, and E. Jumper, “Flight measurements of aero-optical distortions from a flat-windowed turret on the Airborne Aero-Optics Laboratory (AAOL),” in 42nd AIAA Plasmadynamics and Lasers Conference (AIAA, 2011), paper 2011-3280.

] were mapped to the geometry of University of California, Los Angeles’s AO experiment and added to the laser beam to provide wavefront disturbance.

The LTI prediction filter, which is the main component of the LTI optimal controller, has the form of a Kalman predictor, but it is not obtained by the standard methods of Kalman filter/predictor design. Rather, the prediction filter is identified directly from measured wavefront data sequences by a subspace system identification method, with no required a priori knowledge of the turbulence statistics. Related state-space approaches to modeling, prediction, and control of optical wavefronts have been demonstrated in [10

,11

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven $H_{2}$ optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008). [CrossRef]

,17

R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010). [CrossRef]

A. Beghi, A. Cenedese, and A. Masiero, “A Markov-random-field-based approach to modeling and prediction of atmospheric turbulence,” in 16th Mediterranean Conference on Control and Automation (MCA, 2008), pp. 1735–1740.

A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008). [CrossRef]

–20

A. Beghi, A. Cenedese, and A. Masiero, “Multiscale stochastic approach for phase screens synthesis,” Appl. Opt. 50, 4124–4133 (2011). [CrossRef]

], although several aspects of the controller design and system identification are different here, including the integration of optimal control with a classical AO loop, the frequency-weighted deformable mirror modes used here to reduce the number of control channels, and the lattice-filter-based subspace system identification algorithm. The design of the optimal LTI controller in this paper is particularly related to the approach for optimal LTI control of optical tilt jitter in [21

P. K. Orzechowski, N. Chen, J. S. Gibson, and T. C. Tsao, “Optimal suppression of laser beam jitter by high-order RLS adaptive control,” IEEE Trans. Control Syst. Technol. 16, 255–267 (2008). [CrossRef]

], where both plant and disturbance models for two uncoupled control channels were identified by subspace system identification. Here the plant, disturbance model, and optimal controller are multichannel, although the plant model does not need to be identified.

The adaptive controller is included in this paper for performance comparison with the optimal LTI controller. With stationary wavefront statistics, a perfect plant model, and a sufficiently large order of the adaptive filter at the heart of the adaptive controller, the adaptive and optimal LTI controllers theoretically should perform identically in steady state. However, AO applications frequently involve nonstationary wavefront statistics due to changing flow velocities, turbulence strength, and changing the optical paths. For instance, results in [16

,22

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010). [CrossRef]

S. Gordeyev, J. A. Cress, E. Jumper, and A. B. Cain, “Aero-optical environment around a cylindrical turret with a flat window,” AIAA J. 49, 308–315 (2011). [CrossRef]

–24

M. Weng, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012). [CrossRef]

] reveal a strong relationship between the spatial statistics of aero-optical wavefronts and the turret viewing angle. Thus, a slew maneuver to track a moving target in the AAOL experiment or similar application should produce time-varying wavefront statistics. In such scenarios, a fixed identified LTI prediction filter quickly becomes suboptimal, and the performance of an LTI controller degrades significantly. In contrast, an adaptive controller tracks changes to the wavefront statistics by identifying an optimal prediction filter online using current wavefront data. Even in some applications with stationary turbulence and flow, there may not be an opportunity to collect a sample wavefront sequence and then perform system identification before a high-performance control loop needs to be closed. In such cases, an adaptive controller is useful because it can identify near-optimal gains in real time, although at the expense of greater real-time computational complexity than that of an optimal LTI controller.

The experimental results in this paper illustrate the similarities and differences between the performance of the two controllers for both quasi-stationary wavefront statistics and suddenly changing flow direction. The control and system identification methods are implemented in an AO experiment containing two membrane deformable mirrors and a Shack–Hartmann wavefront sensor. The disturbance wavefront sequence used in the experiment is derived from aberration data obtained during recent flight tests that measured the aero-optical effects on a laser beam propagating between two airplanes. It is noteworthy that the adaptive controller, which is the same as in [5

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

], performs as well in the experiment here with aero-optical wavefronts measured by a Shack–Hartmann sensor as it performed in the experiment in [5

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

] with Kolmogorov wavefronts measured by a self-referencing interferometer.

The LTI optimal prediction filter and the adaptive filter have very different forms, but both are multi-input–multi-output (MIMO) filters that capture the statistics of the aero-optical wavefront sequences. Unlike the LTI prediction filter, the adaptive filter tracks the change in flow direction and rapidly reidentifies near-optimal gains. In principle, an optimal LTI prediction filter could be reidentified when the flow direction changes, but the amount of off-line computation required to reidentify the optimal LTI prediction filter would prevent the process from being performed fast enough for the optimal LTI controller to track a sudden change in wavefront statistics nearly as rapidly as the adaptive controller tracks it.

Section 2 describes the optimal LTI controller and minimum-variance prediction filter along with the identification of the prediction filter from wavefront data. Section 3 describes the adaptive controller. Next, Section 4 discusses the hardware and layout of the AO experiment as well as some important details about the deformable mirrors and wavefront sensor. Section 5 presents a mathematical control model of the experiment and the classical AO control loop. In the experiment, the optimal LTI controller and the adaptive controller (separately) augmented the classical AO loop. Section 6 describes the process of mapping the original aero-optical wavefronts from the AAOL to geometry of the experiment here. Then Section 7 presents the experimental results comparing the performance of the classical AO loop alone, the optimal LTI controller and the adaptive controller.

2. OPTIMAL LTI CONTROL

Because classical AO control loops do not compensate for loop latency, they often are inadequate to reject the broadband atmospheric disturbances encountered by AO systems. When the statistics of the disturbance wavefront are approximately stationary, the bandwidth of the AO control system can be increased by implementing an optimal LTI controller. The primary component of the LTI controller is a minimum-variance prediction filter that predicts future disturbances from past and present wavefront sensor measurements.

Figure 1 shows the structure of the optimal LTI controller and resulting closed-loop system. The transfer function

G (z)

represents the plant, which is an AO system in the applications of interest here. For the discussion in this section,

G (z)

may represent either an open-loop AO system or, as in the experiment described subsequently in this paper, an AO system with a classical AO control loop already closed. The optimal LTI controller consists of the filter

F (z)

and the internal plant model

\hat{G} (z)

. The output disturbance signal

w

, the output error signal

e

, the control command

u

, and the disturbance estimate

\hat{w}

in Fig. 1 are vector signals related by

e = w + G (z) u, \hat{w} = e - \hat{G} (z) u, u = F (z) \hat{w} .

(1)

Fig. 1. Block diagram for optimal LTI control.

It is assumed that a sample sequence

w_{id}

of length

t_{id}

is available for identification of a wavefront disturbance model and design of the filter

F (z)

. This sample sequence is collected with the optimal LTI control loop open, i.e., with

u = 0

. The filter

F (z)

is chosen to minimize the criterion

J = \sum_{t = 1}^{t_{id}} {\hat{e}}^{*} (t) \hat{e} (t),

(2)

where the vector error signal

\hat{e} (t)

\hat{e} = w_{id} + \hat{G} (z) F (z) w_{id} .

(3)

G (z) = \hat{G} (z)

, then

e = \hat{e}

, and if the disturbance has the same statistics as the sample sequence

w_{id}

when the optimal control loop is closed, then the optimal controller minimizes the mean-square value of

e

The filter

F (z)

that minimizes

J

is an LTI filter with infinite impulse response. In general, with common stabilizability and observability conditions on the plant model

\hat{G} (z)

, the optimal control problem of choosing

F (z)

to minimize

J

follows from existing methods of linear-quadratic optimal control. However, the problem is relatively simple here because the linear transfer function

\hat{G} (z)

in this paper and many AO problems can be factored as

\hat{G} (z) = z^{- m} G_{0} (z),

(4)

where

m

is the number of delays and

G_{0} (z)

is a stable transfer function with a stable inverse

G_{0}^{- 1} (z)

. This allows the definition of a filter

\tilde{F} (z)

such that

F (z) = - G_{0}^{- 1} (z) \tilde{F} (z) .

(5)

Substituting Eq. (5) into Eq. (3) yields

\hat{e} = w_{id} - z^{- m} \tilde{F} (z) w_{id},

(6)

so that the optimal control problem of finding

F (z)

to minimize

J

becomes the

m

-step prediction problem with the objective of choosing the prediction filter

\tilde{F} (z)

to minimize the criterion

J

in Eq. (2) for the sample sequence

w_{id}

. The optimal filters

F (z)

and

\tilde{F} (z)

depend only on the statistics of

w_{id}

. In the application in this paper,

m = 1

, but the method here for designing

\tilde{F} (z)

is generalized easily to more delays. In most AO problems,

m

is either 1 or 2.

The filter

\tilde{F} (z)

that minimizes

J

in Eq. (2) for

m = 1

is a one-step Kalman predictor. However, this predictor is not computed by the standard methods for Kalman filter design, which require a priori knowledge of the disturbance statistics, represented by a known or assumed stable filter in state-space form driven by white noise with known or assumed covariance. The prediction filter

\tilde{F} (z)

is computed directly from the data sequence

w_{id}

by the state-space subspace system identification algorithm in [25

N. Y. Chen and J. S. Gibson, “Subspace system identification using a multichannel lattice filter,” in Proceedings of the 2004 American Control Conference (IEEE, 2004), Vol. 1, pp. 855–860.

], which also produces a disturbance model in the state-space form

x (t + 1) = A x (t) + K ε (t), w_{id} (t) = C x (t) + ε (t) .

(7)

The one-step prediction of

w_{id} (t)

{\tilde{w}}_{id} (t) = C x (t),

(8)

which is the estimate of

w_{id} (t)

based on measurements

w_{id} (τ)

τ = t - 1, t - 2, \dots

. Rewriting the second equation in Eq. (7) as

ε (t) = w_{id} (t) - C x (t) = w_{id} (t) - {\tilde{w}}_{id} (t)

(9)

shows that

ε

is the prediction error or innovations sequence.

The system identification procedure produces the matrices

A

K

, and

C

along with the covariance matrix of

ε

. The components of the state vector

x (t)

are internal states in the disturbance model, and they generally have no direct relationship to physical quantities. However, the second-order statistics of the sequence

w_{id}

and an estimate of flow velocity can be computed with the solution to a certain Lyapunov equation and other algebraic equations involving

A

K

C

, and the covariance matrix of

ε

The disturbance model in Eq. (7) can be rearranged by using Eq. (9) to yield

x (t + 1) = [A - K C] x (t) + K w_{id} (t), {\tilde{w}}_{id} (t + 1) = C [A - K C] x (t) + C K w_{id} (t) .

(10)

The system in Eq. (10) is the state-space realization of the filter

\tilde{F} (z)

in Eq. (5) and Fig. 1. This system has the standard form of an LTI state-space model; at sample time

t

, the input is

w_{id} (t)

and the output is

{\tilde{w}}_{id} (t + 1)

, which is the one-step prediction of

w_{id} (t + 1)

When the optimal control loop is closed, the disturbance signal

w

is not available, so the signal

\hat{w}

is used as the input to

\tilde{F} (z)

, as shown in Fig. 1. At sample time

t

, the input is

\hat{w} (t)

and output is

\tilde{\hat{w}} (t + 1)

, the one-step prediction of

\hat{w} (t + 1)

. If

G (z) = \hat{G} (z)

, then

\hat{w} (t) = w (t)

and

\tilde{\hat{w}} (t + 1) = \tilde{w} (t + 1)

, the one-step prediction of

w (t + 1)

In most subspace system identification methods [26

P. Van Overschee and B. De Moor, Subspace Identification for Linear Systems (Kluwer Academic, 1996).

T. Katayama, Subspace Methods for System Identification (Springer-Verlag, 2005).

–28

M. Verhaegen and V. Verdult, Filtering and System Identification, a Least Squares Approach (Cambridge University, 2007).

], the biggest computational step is a

Q R

factorization of a large data matrix often containing many thousands of data points. The unique feature of the subspace algorithm in [25

N. Y. Chen and J. S. Gibson, “Subspace system identification using a multichannel lattice filter,” in Proceedings of the 2004 American Control Conference (IEEE, 2004), Vol. 1, pp. 855–860.

] is the use of a multichannel least-squares lattice filter to perform the

Q R

factorization. This permits real-time updating in the

Q R

factorization as data are sampled and significant parallel processing. However, also as in most state-space subspace system identification methods, there remain substantial computational steps after the

Q R

factorization. Hence, the system identification algorithm in [25

N. Y. Chen and J. S. Gibson, “Subspace system identification using a multichannel lattice filter,” in Proceedings of the 2004 American Control Conference (IEEE, 2004), Vol. 1, pp. 855–860.

] can be implemented in a quasi-real-time fashion, but it cannot be made completely real time. While relatively fast among state-space system identification algorithms, the algorithm in [25

N. Y. Chen and J. S. Gibson, “Subspace system identification using a multichannel lattice filter,” in Proceedings of the 2004 American Control Conference (IEEE, 2004), Vol. 1, pp. 855–860.

] is more computationally intensive than the fully adaptive lattice filter that is the main component of the adaptive controller used in this paper.

3. ADAPTIVE CONTROL

The structure of the adaptive controller is shown in Fig. 2. The implementation of the adaptive controller is essentially the same here as in [5

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

]. As in Section 2, the AO problem is formulated as a disturbance rejection problem. The signals, transfer functions, and relationships in Eq. (1) are the same for adaptive control except that the filter

F (z)

, which is the primary component in the adaptive controller, is a recursive least-squares (RLS) FIR lattice filter from [29

S. B. Jiang and J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995). [CrossRef]

]. With stationary wavefront statistics, a perfect plant model (i.e.,

\hat{G} = G

), and a sufficiently large order of the adaptive filter, the adaptive and optimal LTI controllers theoretically should perform identically in the steady state.

Fig. 2. Block diagram for adaptive control.

The adaptive controller contains two copies of the filter

F

. The copy of

F

in the middle of Fig. 2 takes the input

\hat{w}

and generates the adaptive control command

u

. This filter uses the gains determined by the fully adaptive copy of

F

in the bottom part of Fig. 2, which takes the input signal

\hat{G} \hat{w}

and, as indicated by the diagonal arrow in Fig. 2, updates the filter gains to minimize the mean-square value of the tuning signal

{\hat{e}}_{a} = \hat{w} + F (z) \hat{G} (z) \hat{w} .

(11)

In single-input–single-output adaptive control problems, a similar choice of the tuning signal for the adaptive filter often is referred to as the “filtered-x” method [30

B. Widrow and E. Walsh, Adaptive Inverse Control (Prentice Hall, 1996).

]. Although the filters

F

and

\hat{G}

here are MIMO, the linear transfer function

\hat{G} (z)

in this paper, as in many AO problems, is a diagonal transfer function with each diagonal element equal to the same scalar transfer function, so that

F

and

\hat{G}

commute. Hence, if

\hat{G} = G

, then

\hat{w} = w

and

{\hat{e}}_{a} = e

Stability robustness for this class of adaptive controllers hinges on

{‖ G (z) - \hat{G} (z) ‖}_{1}

{‖ G (z) - \hat{G} (z) ‖}_{\infty}

{‖ F (z) ‖}_{1}

, and

{‖ F (z) ‖}_{\infty}

, as discussed in [31

N. O. Arancibia, S. Perez Gibson, and T.-C. Tsao, “Frequency-weighted minimum-variance adaptive control of laser beam jitter,” IEEE/ASME Trans. Mechatron. 14, 337–348 (2009). [CrossRef]

]. For example, a sufficient condition for closed-loop bounded-input–bounded-output stability is

{‖ G (z) - \hat{G} (z) ‖}_{1} \cdot {‖ F (z) ‖}_{1} < 1

. The complexity of AO systems, with numerous hardware components and large numbers of control channels, makes it impossible to obtain useful estimates of any norm of

G (z) - \hat{G} (z)

F (z)

, but the adaptive controller here has exhibited no tendency to go unstable in the experimental AO applications in this paper and [5

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

] or in the high-fidelity AO simulations in [4

Y. T. Liu and J. S. Gibson, “Adaptive control in adaptive optics for directed-energy systems,” Opt. Eng. 46, 046601 (2007). [CrossRef]

The common transversal form of an FIR filter of order

N

F (z) = \sum_{n = 0}^{N} F_{n} z^{- n},

(12)

where the

F_{n}

are matrix gains. In many AO applications, the control channels are uncoupled before the adaptive control loop is closed, as discussed in Subsection 5.B. This implies that the problem of updating the gains in

F

reduces to a set of independent RLS problems for identifying the rows of

F_{n}

. In principle, the standard FIR realization in Eq. (12) could be used in the adaptive controller along with any multichannel RLS algorithm for updating the filter gains, including the classical RLS algorithm [32

A. H. Sayed, Fundamentals of Adaptive Filtering (Wiley, 2003).

]. However, the more complex lattice filter realization and associated RLS updating algorithm used here possesses superior numerical stability and efficiency due to orthogonalization of the data channels in [29

S. B. Jiang and J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995). [CrossRef]

]. This is particularly important in AO applications due to the large number of inputs and outputs. A variable-order version of the adaptive controller, with two control channels but much higher orders, has been used for control of laser beam jitter [21

,31

N. O. Arancibia, S. Perez Gibson, and T.-C. Tsao, “Frequency-weighted minimum-variance adaptive control of laser beam jitter,” IEEE/ASME Trans. Mechatron. 14, 337–348 (2009). [CrossRef]

]. The lattice filter used for the experiments reported in this paper had fixed order

N = 3

(i.e., four taps), 15 channels, and an RLS forgetting factor of 0.99999.

4. AO EXPERIMENT

A. Experimental Layout

Figure 3 depicts the optical layout of the experiment, shown in the photograph in Fig. 4. The primary components are the laser source, two membrane deformable mirrors, a Shack–Hartmann wavefront sensor, and target camera.

Fig. 3. Optical layout. The primary components are the laser source, two membrane deformable mirrors (DM1 and DM2), a Shack–Hartmann wavefront sensor (SHWFS), and target camera.

Fig. 4. Photograph of the experiment.

The illumination source in the experiment is a 0.8 mW, continuous-wave, helium–neon laser with a wavelength of

λ = 634 nm

. After exiting a spatial filter, the beam is expanded to approximately 20 mm in diameter and directed toward DM1, a membrane deformable mirror (DM) with 31 actuators, which alters the optical path difference of the beam. A set of relay lenses image the beam onto DM2, a second membrane DM with 61 actuators. The beam is then split, with one branch directed to a Shack–Hartmann wavefront sensor and the other focused onto a target camera representing a far-field intensity pattern.

The mirror DM1 is the control actuator in the AO system. The mirror DM2 is the disturbance actuator that puts the wavefront aberrations onto the beam after the original aero-optical data are mapped to the geometry of DM2 as discussed in Section 6.

A PC processes the wavefront sensor measurements and runs all control loops to drive DM1. The experiment runs at a sample rate of approximately 40 Hz due to limitations of the hardware used; however, the spatial and temporal statistics of the open-loop and closed-loop wavefront sequences are determined by the 16 KHz frame rate at which the aero-optical wavefronts were collected. Hence, the performance of the control loops, the open-loop, and the closed-loop wavefront errors and the target camera images should be the same as if the experiment were running at the 16 KHz frame rate.

B. DM Control Commands

Both DMs, manufactured by Active Optical Systems, consist of a membrane mirror covering electrostatic pads arranged in a hexagonal grid with a maximum throw of approximately 10 μm. DM1 is identical to the DM used in [5

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

] for adaptive control in AO. For each DM actuator, a command voltage

v

generates an electric field that produces a DM displacement (i.e., phase shift)

d

at that actuator. The displacement and command are related by

d = a v^{2},

(13)

where

a

is a constant related to the actuator’s proximity to the clamped mirror boundary. The values of the coefficients

a

for the various actuators are not needed for the work described in this paper because they are subsumed in the poke matrices discussed in Subsection 5.A.

The electronic driver for the DM provides eight-bit resolution for the DM commands, which take integer values between 0 and 255. For the control loops, the command voltages are parameterized as

v = round (\sqrt{c + v_{b}^{2}}), v_{b} = 180 ≃ \sqrt{255^{2} / 2},

(14)

where

c

is a control command and

v_{b}

is an applied bias voltage. The bias

v_{b}

, which is the same for all actuators, produces a concave shape on the DM that can be corrected by adjusting subsequent imaging optics. Each command

c

takes values in the interval

- v_{b}^{2} \leq c \leq v_{b}^{2}

. If the quantization error due to the rounding is neglected, then Eqs. (13) and (14) yield

d = a (c + v_{b}^{2}) .

(15)

Henceforth,

c_{1}

and

c_{2}

will denote, respectively, the vectors containing the 31 actuator commands for DM1 and the 61 actuator commands for DM2.

C. Wavefront Sensor Measurements

The Shack–Hartmann wavefront sensor, also manufactured by Active Optical Systems, consists of a CMOS imaging array overlaid by a regular grid of lenslets. The image plane is subdivided into a

12 \times 12

grid of subapertures. Each subaperture contains the focal point of a single lenslet in an

11 \times 11

grid of pixels. The measurement of the average wavefront slope over each lenslet is determined by a standard intensity centroid calculation for the subaperture image. If

(Δ ξ, Δ η)

is the location of the intensity centroid within a given subaperture, relative to a reference location corresponding to a planar wavefront, then the average spatial derivatives of the wavefront over that subaperture are given by

\begin{array}{cc} s_{ξ} = \frac{2 ν π}{λ f} Δ ξ, & s_{η} = \frac{2 ν π}{λ f} Δ η \end{array},

(16)

where

s_{ξ}

and

s_{η}

are the local wavefront slopes in the

ξ

and

η

directions, respectively,

ν = 6.7 μm

is the pixel pitch, and

f = 6.7 mm

is the lenslet focal length. The reference centroid locations were generated by replacing DM1 and DM2 by flat mirrors and illuminating the sensor with a collimated beam possessing a planar wavefront. The wavefront sensor output is a slope vector

y

of dimension 288, containing

x

and

y

slopes for each of the 144 subapertures. This slope vector is used directly for control as discussed in Section 5. For performance analysis, a

12 \times 12

wavefront image

ϕ

with zero mean was generated by a zonal least-squares reconstructor matrix

R

, which was computed as discussed in [33

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]

5. AO CONTROL PROBLEM

A. Control Model of the AO Hardware

The block diagram in Fig. 5 shows the signals and control loops for the AO problem. The

z^{- 1}

block indicates that the AO system has a one-sample loop delay, which is the overall latency due to DM response and wavefront sensor measurement readout.

Fig. 5. Block diagram of the AO experiment.

The physical wavefront error that reaches the wavefront sensor is

φ = φ_{b} + φ_{2} - φ_{1},

(17)

where

φ_{b}

is a static bias due to DM bias and optical misalignment,

φ_{2}

is the wavefront disturbance added to the beam by the disturbance mirror DM2, and

φ_{1}

is the phase correction provided by the control mirror DM1. For control design, the portion of the block diagram between control and disturbance commands

c_{1}

and

c_{2}

and the measured slope vector

y

is modeled by

y = y_{b} + Γ_{2} c_{2} - z^{- 1} Γ_{1} c_{1},

(18)

where

y_{b}

is the contribution to the slope vector from the wavefront bias

φ_{b}

, the vectors

c_{1}

and

c_{2}

are the command vectors to DM1 and DM2, respectively, and

Γ_{1}

and

Γ_{2}

are the corresponding poke matrices mapping actuator command vectors to slope vectors. The poke matrix

Γ_{2}

is not used in either designing or implementing any of the controllers. Although significant nonlinearities exist in the experiment, the system is tuned to remain largely in a linear regime, and the model given by Eq. (18) is usually sufficient for tuning the AO control system [4

Y. T. Liu and J. S. Gibson, “Adaptive control in adaptive optics for directed-energy systems,” Opt. Eng. 46, 046601 (2007). [CrossRef]

The poke matrix

Γ_{1}

for the 31-actuator mirror DM1 was identified as follows. First, with both

c_{1}

and

c_{2}

zero, the bias

y_{b}

was measured. Then a zero-mean, uniformly distributed random sequence of 400 command vectors

c_{1}

was applied to DM1, with

c_{2} = 0

. This produced a sequence of slope vectors

y

, and

Γ_{1}

was determined by a least-squares fit to data based on Eq. (18). Similarly, the poke matrix

Γ_{2}

was determined by a least-squares fit to data obtained by driving the 61-actuator mirror DM2 with a sequence of 1000 random commands

c_{2}

with

c_{1} = 0

B. Modal Control Channels

The frequency-weighted DM modes shown in Fig. 6 serve as control channels in the all of the control loops. These modes are computed for the particular geometry and actuator influence functions of DM1. The frequency-weight modes have two important properties: they are orthogonal with respect to the actuator influence functions of DM1, and their spatial frequency content increases with mode number. Since DM1 has 31 actuators, there are 31 modes. The modes are constructed by solving a certain eigenvalue problem involving the actuator influence functions of DM1. Details about this construction are given in [4

Y. T. Liu and J. S. Gibson, “Adaptive control in adaptive optics for directed-energy systems,” Opt. Eng. 46, 046601 (2007). [CrossRef]

S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]

Fig. 6. Images of the frequency-weighted DM modes for DM1.

In the control system, the modes used by the control loops are stored as the columns of the matrix

V

, which maps modal commands into DM1 actuator commands. The modal reconstructor matrix

E

, which maps the slope vector to modal coordinates, is the pseudo inverse of the modal poke matrix

Γ_{1} V

; i.e.,

E = {(Γ_{1} V)}^{+} .

(19)

The vector signal

e = E y

(20)

represents the projection of the wavefront error onto the DM modes used in the control loops.

It follows from Eq. (19) that

E Γ_{1} V = I .

(21)

Because of Eq. (21), the modal control channels are uncoupled when only the classical AO loop is closed. Hence, when either the optimal LTI control loop or the adaptive control loop is closed, each element of the control command vector

u

affects only the corresponding modal channel. However, the optimal LTI controller and the adaptive controller couple the channels by using sensor data from all channels used for control to determine each modal control command in

u

In the experiment, the DM actuators often saturate when all of the modal control channels are used. This is partly due to long-term drift in the DM characteristics, resulting from the sensitivity of membrane DMs to environmental effects such as temperature [34

J. D. Mansell and B. Henderson, “Temporal and spatial characterization of polymer membrane deformable mirrors,” Proc. SPIE 7466, 74660D (2009). [CrossRef]

]. Saturation occurs primarily when the higher-order DM modes were included in the control loops. As shown by results in Section 7, the dominant wavefront disturbance lies in modes 1 through 15. Hence, for the experiments reported in this paper, only the first 15 DM modes were used in the control loops. Other experiments showed no performance improvements with more controlled modes.

C. Classical AO Control Loop and Associated Transfer Functions

The classical AO control loop shown in Fig. 5 consists of the least-squares reconstructor matrix

E

, the modal matrix

V

, and the low-pass digital filter

C (z) = β \frac{z}{z - α},

(22)

with gain

β

and positive real pole

α

. If

α = 1

C (z)

is an integrator; if

α < 1

C (z)

is a low-pass filter, sometimes called a leaky integrator. In the AO system,

C (z)

is a diagonal MIMO transfer function with each diagonal element equal to the scalar transfer function in Eq. (22).

The controller

U (z)

in Fig. 5, which augments the classical AO loop, is either the optimal LTI controller inside the large dashed box in Fig. 1 or the adaptive controller inside the dashed box in Fig. 2. The performance of the adaptive controller approximates that of the optimal LTI controller in the steady state.

As in Sections 2 and 3, the true transfer function from the control command

u

to the output error

e

with only the classical AO loop closed is denoted by

G (z)

. The plant model

\hat{G} (z)

used by the optimal LTI controller and the adaptive controller is

\hat{G} (z) = \frac{- β}{z + β - α},

(23)

which is derived from Fig. 5 without the

U (z)

block and with

C (z)

given by Eq. (22). The plant model

G (z)

should be interpreted as a diagonal transfer function with each diagonal term given by Eq. (23) because the signals

u

and

e

both are in DM modal coordinates and Eq. (21) implies that the modal control channels are uncoupled with only the classical AO loop closed. If there is no actuator saturation or quantization and the identified poke matrix

Γ_{1}

accurately represents the mapping from DM1 commands to the wavefront sensor measurements, then

G (z) = \hat{G} (z) .

(24)

When only the classical AO loop closed and Eq. (24) holds, the transfer function from the effect of the disturbance

φ_{2}

on an individual controlled DM mode to the wavefront error for the same mode is the sensitivity transfer function, or error-rejection function,

S (z) = \frac{z - α}{z + β - α} .

(25)

For the experiments reported in Section 7, the pole in

C (z)

was chosen as

α = 0.95

to prevent integrator windup and saturation of the actuator commands to DM1. The gain in

C (z)

was

β = 0.3

. Choosing

β

involves a trade-off between maximizing disturbance rejection bandwidth and amplifying high-frequency noise. Figure 7 shows that gains between

β = 0.3

and

β = 0.5

approximately maximize the disturbance rejection bandwidth without significantly amplifying high-frequency noise.

Fig. 7. Bode plot of the sensitivity transfer function

S (z)

for

α = 0.95

and various controller gains

β

. For the experiments in this paper,

β = 0.3

was used.

Figure 7 predicts and additional experimental results not presented in this paper have shown that the performance of the classical AO loop alone depends significantly on the values of

α

and

β

. However, if

G (z) = \hat{G} (z)

, it follows from Eqs. (5) and (6) that, for any

0 < α < 1

and

β > 0

, the optimal LTI controller in this paper will yield the minimum wavefront error variance achievable by any combination of control loops. Experimental results for several choices of

α

and

β

have shown that, when the DM1 commands do not saturate and closed-loop stability is maintained, the steady-state performance of the optimal LTI and that of the adaptive controller indeed are largely independent of

α

and

β

The disturbance signal

w

in Figs. 1 and 2 represents the projection onto the controlled DM modes of the residual wavefront error with only the classical AO loop closed (i.e., with neither the optimal LTI controller nor the adaptive controller engaged). The optimal LTI controller and the adaptive controller attempt to correct the portion of the wavefront disturbance represented by

w

. If

G (z) = \hat{G} (z)

, then

w = S (z) E (y_{2} + y_{b}),

(26)

where

y_{2}

and

y_{b}

are the contributions to the open-loop slop vector from the disturbance and bias wavefronts

φ_{2}

and

φ_{b}

, respectively.

6. GENERATING WAVEFRONT DISTURBANCES FOR THE AO EXPERIMENT

A. AAOL Aero-Optical Data

The disturbance wavefronts used in the experiment came from Notre Dame’s AAOL [16

]. The original aero-optical wavefronts were produced by turbulence over a flat-windowed turret [16

,22

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010). [CrossRef]

S. Gordeyev, J. A. Cress, E. Jumper, and A. B. Cain, “Aero-optical environment around a cylindrical turret with a flat window,” AIAA J. 49, 308–315 (2011). [CrossRef]

–24

M. Weng, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012). [CrossRef]

] during a flight test in which a continuous-wave laser was transmitted between two planes flying in a constant formation at an altitude of 4570 m. The planes were separated by approximately 50 m to ensure aero-optical turbulence was the primary source of wavefront aberrations. Other significant parameters for the flight test are give in Table 1.

Table 1. Experimental Details for the Notre Dame AAOL Data Set

Turret azimuthal angle	119°
Turret elevation angle	57°
Freestream mach	0.36
Altitude	4570 m
Target distance	50 m
Aperture size	10.1 cm
Sampling rate	16 kHz

While the predominant source of wavefront aberrations in the AAOL data is the turbulent boundary layer around the turret, some small electronic noise likely is present in the data. A detailed analysis of the statistics of the AAOL data can be found in [16

]. Since the control loops in this paper attempt to minimize the total phase error that would be encountered in similar airborne environments, no attempts were made to distinguish between the sources of the measured turbulence.

For the AO experiment in this paper, a sequence of 8000 disturbance command vectors

c_{2}

for the 61-actuator mirror DM2 was generated by mapping a sequence of 8000 aero-optical wavefronts from AAOL to the geometry of the experiment here, as described below. This was done with the goal of replicating the statistics of the aero-optical wavefront aberrations as closely as possible. The temporal statistics dictated by the 16 KHz frame rate of the original AAOL data were maintained in the final wavefront disturbance sequence applied in the experiment here, even though the hardware limited the frame rate for the experiment to 40 Hz.

B. Least-Squares Estimation of Missing Wavefront Data

Each aero-optical wavefront image from AAOL contains 591 pixels in a roughly circular array with a 78 pixel hole in the center due to an obscuration produced by the secondary mirror in the AAOL optics. Since the AO experiment in this paper has no obscuration, the hole in the original data was filled by using a multichannel spatial–temporal prediction filter to preserve the statistics and flow of the original data. This approach appears reasonable in view of the characteristic flow of the wavefront data.

The missing pixel values were estimated as linear combinations of past and future pixel values from outside the hole. The coefficients in this prediction filter are determined by a least-squares fit to data for pixels outside the hole. The top half of the hole consisting of obscured pixels was filled first, and then the bottom half was filled. Figure 8 shows the geometry for the procedure to fill the top half of the hole. The semicircular region denoted by

S

in Fig. 8 contains the top half of the obscured pixels. The vector

s (t)

contains the estimated values of the pixels in

S

at time

t

, and

r (t)

contains the pixels in the regions

R_{1}

and

R_{2}

. Also shown in Fig. 8 is the mean flow velocity

\hat{v}

, which was estimated by an image correlation method. The direction of

\hat{v}

suggests that the pixel values in

S

should be correlated primarily with past pixel values in

R_{2}

and future pixel values in

R_{1}

. Hence the prediction model

s (t) = a_{0} r (0) + \sum_{k = 1}^{p} a_{k} r (t - k) + \sum_{ℓ = 1}^{q} b_{ℓ} r (t + ℓ),

(27)

where

a_{k}

and

b_{ℓ}

are matrix coefficients and

p

and

q

are integers, was used to supply the missing pixel values in

S

Fig. 8. Layout of the AAOL wavefront data, with regions used to fill the central, circular obscuration.

\hat{v}

is the approximate flow velocity estimated using an image correlation analysis and is equal to

- 0.96 pixels

per time step in the horizontal direction and 0.46 pixels per time step in the vertical direction.

Since there were no data available from

S

for use in determining

a_{k}

and

b_{ℓ}

, the coefficients

a_{k}

and

b_{ℓ}

were chosen to minimize the 2-norm of the error sequence

ϵ

in the prediction model

\tilde{s} (t) = a_{0} \tilde{r} (0) + \sum_{k = 1}^{p} a_{k} \tilde{r} (t - k) + \sum_{ℓ = 1}^{q} b_{ℓ} \tilde{r} (t + ℓ) + ϵ (t),

(28)

where

\tilde{r}

and

\tilde{s}

contain values of the pixels in regions

{\tilde{R}}_{1}

{\tilde{R}}_{2}

, and

\tilde{S}

, which, as shown in Fig. 8, have the same shapes and sizes as

R_{1}

R_{2}

, and

S

, respectively. Although the spatial and temporal statistics of the data varied somewhat over the aperture, it was assumed that the statistics in

R_{1}

R_{2}

, and

S

are sufficiently similar to statistics in

{\tilde{R}}_{1}

{\tilde{R}}_{2}

, and

\tilde{S}

to allow identification of

a_{k}

and

b_{ℓ}

that produce realistic missing data for the obscured pixels. It was found that values of

p

and

q

greater than 5 produced no further reduction in the fit-to-data error

ϵ

, so

p = q = 5

was used.

The bottom portion of the obscuration was filled by the same approach, with one modification to prevent a discontinuity along the line between the two halves of the hole. For filling the bottom half of the hole, the diagram in Fig. 8 was reflected about the horizontal diameter. Then the resulting regions

R_{1}

and

R_{2}

were expanded to include the pixels in the bottom row of the top half of the hole so that the values determined for these pixels when the top half of the hole was filled were used along with the other pixel values in the reflected

R_{1}

and

R_{2}

to estimate values for the pixels in the bottom half of the hole. The reflected regions

{\tilde{R}}_{1}

and

{\tilde{R}}_{2}

were expanded similarly for determining coefficient matrices

a_{k}

and

b_{ℓ}

by solving the least-squares problem for Eq. (28), this time for the bottom half of the hole. Since the wavefront statistics vary over the aperture, the optimal values of

a_{k}

and

b_{ℓ}

for the bottom half of the hole are different from those for the top half (even for the elements of

a_{k}

and

b_{ℓ}

from the two halves that correspond to each other through reflection about the horizontal axis).

Figure 9 compares the power spectral density of the time series produced by the procedure here for a pixel in the middle of region

S

with the corresponding power spectral density for a pixel in the middle of each of the regions

R_{1}

{\tilde{R}}_{1}

{\tilde{R}}_{2}

, and

\tilde{S}

in Fig. 9. The pixel in

S

shows overall bandwidth and shapes in all frequency ranges similar to those for the other pixels. Also, in an animation showing the sequence of wavefronts with the hole filled in, the flow inside the hole appears the same as the flow outside the hole, and there is no obvious boundary around the hole or between the top and bottom halves of the hole. Another noteworthy observation from Fig. 9 is that the temporal statistics for the original data points in regions

R_{1}

{\tilde{R}}_{1}

{\tilde{R}}_{2}

, and

\tilde{S}

in Fig. 8 show significant differences even though the overall shapes of the power spectra are similar. These differences are due to the fact that the boundary-layer turbulence is not homogeneous over the AAOL aperture.

Fig. 9. Power spectral density for various pixels in the AAOL wavefront data compared to the same pixel in the interpolated region

S

C. Constructing Disturbance DM Commands from the Wavefront Data

After the hole in the aero-optical wavefronts was filled, the resulting wavefronts on the original

27 \times 27

grid were resized by the MATLAB imresize function to obtain wavefront images on the

12 \times 12

grid associated with the wavefront sensor in the experiment. Then, for each resulting

12 \times 12

wavefront image

ϕ

, a command vector

c_{2}

for the disturbance mirror DM2 was determined to produce the best approximation to

ϕ

in a least-squares sense.

In this least-squares problem, the matrix

R

is the reconstructor referred to in Subsection 4.C mapping wavefront sensor vectors to wavefront images on the

12 \times 12

sensor grid, and

Γ_{2}

(determined as described in Section 5.A) is the poke matrix mapping the DM2 command vector

c_{2}

to the corresponding wavefront sensor slope vector. For each wavefront image

ϕ

(vectorized),

c_{2}

was chosen to minimize the regularized least-squares criterion

J_{c} = {(γ ϕ - R Γ_{2} c_{2})}^{T} (γ ϕ - R Γ_{2} c_{2}) + c_{2}^{T} Π c_{2},

(29)

where

γ

is a scale factor converting the dimensionless wavefront data into appropriate units. The positive-definite regularization matrix

Π

is diagonal with relatively larger diagonal elements corresponding to actuators likely to saturate. Actuators near the clamped DM boundary have smaller actuator influence functions, and these actuators are prone to saturation if the disturbance commands

c_{2}

are computed without regularization.

7. EXPERIMENTAL RESULTS

This section presents results from four experiments to compare the performance of the optimal LTI control loop, the adaptive control loop, and the classical AO loop alone. During each of these experiments, DM2 added the same disturbance sequence to the laser beam. The four experiments were the following.

• Experiment 1: Open-Loop. The control command $c_{1}$ to DM1 was zero, but DM2 added the disturbance wavefronts to the laser beam (i.e., $c_{2} \neq 0$ ).
• Experiment 2: Classical AO Only. The classical AO loop alone was closed (i.e., $u = 0$ ).
• Experiment 3: Optimal LTI Control. Both the classical and optimal LTI control loops were closed. The optimal control loop was closed for $t > 100$ time steps to allow the classical AO loop to reach steady-state performance.
• Experiment 4: Adaptive Control. Both the classical and adaptive loops were closed. The adaptive controller began a period of 100 learning steps at $t = 100$ , and the adaptive control loop closed at $t = 200$ time steps.

During each of these experiments, DM2 added an 8000-frame disturbance sequence to the laser beam. The first 4000 frames of this sequence were the last 4000 frames from the 8000-frame disturbance sequence described in Section 6. The second half of the disturbance sequence for the four experiments was obtained by rotating the same 4000 frames (i.e., the last 4000 frames in Section 6) by 90°. Thus, the difference between the first and second half of the disturbance sequence for experiments 1–4 was the 90° rotation. This rather extreme, sudden change in the wavefront disturbance was introduced to illustrate the main difference between the capability of the optimal LTI controller and that of the adaptive controller.

Prior to experiments 1–4, the classical AO loop alone was closed, and DM2 was driven with the first 4000 frames of the disturbance commands described in Section 6. The resulting sequence of residual wavefronts was used as the sequence

w_{id}

in Section 2 to identify the prediction filter

\tilde{F} (z)

in the optimal LTI controller. The two halves of the disturbance sequence from Section 6 were used as described here because, in applications,

\tilde{F} (z)

must be identified before the optimal LTI controller is applied, and the sequence

w_{id}

should be statistically similar to the disturbance encountered by the optimal LTI controller.

Figures 10 and 11 compare the effectiveness of the control loops for reducing the wavefront error measured by the wavefront sensor and increasing the intensity measured by the target camera. For these results, the wavefront images were reconstructed on the

12 \times 12

wavefront sensor grid. Figure 10(a) shows the root mean square (RMS) value of the wavefront errors computed over the

12 \times 12

grid and a 200-frame moving window. Figure 10(b) shows the peak intensity of the average target camera image over the 200-frame moving window. Figure 11 shows average target camera images for each half of each of the four experiments. Tables 2 and 3 show performance measures computed for each half of each experiment.

Fig. 10. (a) RMS value of the wavefront error computed over space and time for a 200-frame moving window; (b) maximum intensity in the average target camera image for the 200-frame moving window.

Fig. 11. Mean target camera images for (a) time steps 300 to 4000 and (b) time steps 4000 to 8000. Target camera images are

60 \times 60 pixels

, with pixel

pitch size = 6.7 μ m

Table 2. RMS Values of the Wavefront Error in Experiments 1–4 Computed over Space and Time Intervals

T_{1} = [300 ∶ 4000]

and

T_{2} = [4000 ∶ 8000]

	RMS (μm)
	$T_{1}$	$T_{2}$
Open loop	0.56	0.58
Classical AO	0.41	0.42
Optimal LTI	0.27	0.36
Adaptive	0.26	0.28

Table 3. Target Camera Performance Measures for Experiments 1–4, Computed over Time Intervals

T_{1} = [300 ∶ 4000]

and

T_{2} = [4000 ∶ 8000]

	$I_{\max}$		$I_{avg}$		$σ_{ξ}^{2}$		$σ_{η}^{2}$
	$T_{1}$	$T_{2}$	$T_{1}$	$T_{2}$	$T_{1}$	$T_{2}$	$T_{1}$	$T_{2}$
Open loop	23.5	28.7	85.8	94.9	34.8	13.4	22.1	29.7
Classical AO	32.6	36.9	101.5	109.1	21.7	7.8	12.3	16.3
Optimal LTI	51.3	38.7	128.1	111.3	7.2	8.4	6.1	9.9
Adaptive	49.2	49.3	121.2	122.4	7.6	6.1	5.8	6.1

I_{\max} = maximum intensity

for each average image in Fig. 11;

I_{\max} = average value

of peak intensities over

T_{1}

and

T_{2}

;

σ_{ξ}^{2}

and

σ_{η}^{2} = variances

of positions of peak intensities. Intensity units are pixel values, and position variances are numbers of pixels squared.

Figure 10(a) shows that, during the first 4000 time steps, the optimal LTI controller and the adaptive controller both reduce the wavefront error significantly as compared to the classical AO loop, and Figs. 10(b) and 11 show the corresponding increased target camera intensities. As expected, the performance of the optimal LTI controller and the adaptive controller are very similar in the first half of the experiments, after about the first 500 time steps, during which the adaptive filter is identifying near-optimal gains. Also as expected, Fig. 10 shows that the performance of the optimal LTI controller deteriorates quickly when the flow direction changes after time step 4000, whereas the adaptive filter identifies new gains so that the adaptive controller continues to reduce wavefront error and increase target camera intensity significantly.

Table 3 shows that the optimal LTI controller and adaptive controller reduce the variances of the positions of the peak intensities significantly as compared to the classical AO loop. This indicates that, in addition to reducing higher-order wavefront error, the optimal LTI controller and adaptive controller reduce first-order wavefront tilt and jitter of the position of the peak intensity on the target camera image plane. This jitter causes

I_{\max}

, the maximum intensity for the average image, to be less than

I_{\max}

, the average instantaneous peak intensity, in all experiments.

Modal analysis of the wavefront errors yields additional insight into the performance of the control loops. For this analysis, an error sequence

e

in modal coordinates was computed as in Eq. (20) with the modal reconstructor

E

computed as in Eq. (19) but with the modal matrix

V

containing all 31 frequency-weighted DM modes. Thus, the vector sequence

e

has dimension 31, and each component is a modal time series.

Figure 12 shows the RMS values of the individual modal time series for time steps 300–4000 in each of Experiments 1–4. The relatively small amounts of power in modes 16 through 31 suggest that controlling more than the first 15 modes should not improve the performance of the AO system significantly, and this was confirmed by experiments with more modes in the control loops.

Fig. 12. RMS values of all 31 modal time series for time steps 300 through 4000.

Figure 13 compares the power spectral densities (PSDs) of the modal time series for the three modes with the greatest power according to Fig. 12. The PSDs show that the classical AO controller mitigated the wavefront disturbances up to 1000 Hz at most and had negligible effect beyond this bandwidth, as predicted by the sensitivity transfer function in Fig. 7.

Fig. 13. Power spectral densities of select modal time series for (left) time steps 300 through 4000 and (right) 4001 through 8000.

During the initial 4000 time steps, both the LTI and adaptive controllers extended the disturbance rejection bandwidth to nearly 3000 Hz in modes 1 and 2 and more than 2000 Hz in mode 6. During the second 4000 time steps, the LTI controller (which was optimal for the first 4000 steps but not the second 4000 steps) achieved some modest error reduction in modes 1 and 6, but not in mode 2, while the adaptive controller achieved error reduction very similar to that achieved in the first 4000 steps.

The PSDs in Fig. 13 show that the optimal LTI controller and the adaptive controller tend to whiten the residual wavefront errors, a common feature of minimum-variance controllers, whether LTI or adaptive. Sometimes, as in the PSDs for mode 6, low-level disturbance at high frequencies is amplified, though not to a level sufficient to degrade performance. This effect, referred to commonly as the waterbed effect, is common in minimum-variance control and filtering. The narrow spikes at approximately 3300, 6200, and 6600 Hz most likely result from electronic noise in the original flight-test data. These spikes represent a very small part of the disturbance power, most of which lies below 3000 Hz. Therefore, the optimal LTI and adaptive controllers, both of which have the objective of minimizing the variance of the wavefront error, exert only minimal control effort at these frequencies.

8. CONCLUSIONS

In the AO experiment in this paper, the optimal LTI controller based on a wavefront prediction filter determined by subspace system identification was compared to an adaptive controller based on an RLS lattice filter. The optimal LTI controller and the adaptive controller augmented a well-tuned classical AO loop for wavefront correction. These controllers were used in the experiment to predict and correct optical phase aberrations produced by aero-optical turbulence. The optimal controller and the adaptive controller, in separate experiments, augmented the classical AO loop. The prediction capability of the optimal and adaptive controllers resulted in significant performance improvements as measured by reduced residual wavefront error and increased intensity on a target camera.

For the series of experiments reported in this paper, a membrane DM corrected wavefront errors, and a second membrane DM added disturbance wavefronts to the laser beam. These disturbance wavefronts were derived from the original aero-optical flight-test data. Each experiment that generated results for comparing the controllers consisted of two 4000-frame segments. In the second segment of each experiment, the flow direction of the disturbance wavefronts was rotated 90°. The LTI prediction filter for the optimal controller was identified prior to the control experiments from an independent wavefront sequence with statistics similar to the statistics of the disturbance wavefronts used in the control experiments. This sample wavefront sequence used for system identification was measured by the wavefront sensor with only the classical AO loop closed.

Since the sample sequence used for identification of the LTI prediction filter had similar statistics and the same flow direction as those in the first segment of each control experiment, the optimal LTI controller performed well over the first 4000 frames as compared to the classical AO loop alone. Also, during this segment of the experiments, the optimal LTI controller and the adaptive controller produced similar reductions in wavefront error and increases in target camera intensity. The fact that the adaptive controller, which used a four-tap multichannel FIR filter, performed as well as the LTI controller with the optimal multichannel infinite impulse response prediction filter reflects the importance exploiting spatial as well as temporal correlations in wavefront prediction.

During the second 4000-frame segment of the control experiments, the LTI controller (which was optimal only for the wavefront disturbance in the first segment) yielded only marginal improvement over the classical AO loop, whereas the adaptive controller continued to achieve significantly greater performance than the classical AO loop. The LTI prediction filter could be reidentified from a new sample sequence when a change in flow direction or other turbulence characteristics occurs, but the amounts of data and off-line computation required would prevent the LTI prediction filter from tracking changing wavefront statistics as fast as the adaptive controller.

The advantage of the optimal LTI controller as compared to the adaptive controller is the much smaller real-time computational burden of the LTI controller. The results here indicate that, in applications with quasi-stationary turbulence and constant dominant flow velocity and where there is time for the initial identification of the LTI prediction filter, optimal LTI control is an attractive alternative to fully adaptive control for wavefront prediction and correction significantly beyond the bandwidth of classical AO control loops.

ACKNOWLEDGMENTS

This work was supported by the high-energy Laser Joint Technology Office and the U.S. Office of Naval Research under grant N00014 07-1-1063. The authors of this paper are indebted to the authors of [16

,22

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010). [CrossRef]

S. Gordeyev, J. A. Cress, E. Jumper, and A. B. Cain, “Aero-optical environment around a cylindrical turret with a flat window,” AIAA J. 49, 308–315 (2011). [CrossRef]

–24

M. Weng, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012). [CrossRef]

] for the aero-optical data and indispensable assistance with interpreting the data.

REFERENCES

1.	B. L. Ellerbroek and T. A. Rhoadarmer, “Real-time adaptive optimization of wavefront reconstruction algorithms for closed-loop adaptive optical systems,” Proc. SPIE 3353, 1174–1185 (1998). [CrossRef]
2.	J. S. Gibson, C. C. Chang, and B. L. Ellerbroek, “Adaptive optics: wave-front correction by use of adaptive filtering and control,” Appl. Opt. 39, 2525–2538 (2000). [CrossRef]
3.	T. A. Rhoadarmer, L. M. Klein, J. S. Gibson, N. Chen, and Y.-T. Liu, “Adaptive control and filtering for closed-loop adaptive-optical wavefront reconstruction,” Proc. SPIE 6306, 63060E (2006). [CrossRef]
4.	Y. T. Liu and J. S. Gibson, “Adaptive control in adaptive optics for directed-energy systems,” Opt. Eng. 46, 046601 (2007). [CrossRef]
5.	S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]
6.	C. M. S. Corley, M. Nagashima, and B. N. Agrawal, “Beam control and a new laboratory testbed for adaptive optics in a maritime environment,” in Proceedings of 2010 IEEE Aerospace Conference (IEEE, 2010), pp. 1–13.
7.	C. M. S. Corley, “Maritime adaptive optics beam control,” Ph.D. thesis (Naval Post Graduate School, 2010).
8.	C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, F. Chemla, and D. Rabaud, “Off-axis adaptive optics with optimal control: experimental and numerical validation,” Proc. SPIE 5903, 59030P (2005). [CrossRef]
9.	C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464–7476 (2006). [CrossRef]
10.	K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven $H_{2}$ optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007). [CrossRef]
11.	K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven $H_{2}$ optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008). [CrossRef]
12.	C. Petit, J.-M. Conan, C. Kulcsár, and H.-F. Raynaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26, 1307–1325 (2009). [CrossRef]
13.	J. Tesch, J. S. Gibson, S. Gordeyev, and E. Jumper, “Identification, prediction and control of aero optical wavefronts in laser beam propagation,” in 42nd AIAA Plasmadynamics and Lasers Conference (AIAA, 2011), paper 2011-3276.
14.	J. Tesch and J. S. Gibson, “Optimal and adaptive correction of aero-optical wavefronts in an adaptive optics experiment,” Proc. SPIE 8165, 816502 (2011). [CrossRef]
15.	J. Tesch, “High-performance control and prediction for adaptive optics,” Ph.D. thesis (University of California, Los Angeles, 2011).
16.	C. Porter, S. Gordeyev, M. Zenk, and E. Jumper, “Flight measurements of aero-optical distortions from a flat-windowed turret on the Airborne Aero-Optics Laboratory (AAOL),” in 42nd AIAA Plasmadynamics and Lasers Conference (AIAA, 2011), paper 2011-3280.
17.	R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010). [CrossRef]
18.	A. Beghi, A. Cenedese, and A. Masiero, “A Markov-random-field-based approach to modeling and prediction of atmospheric turbulence,” in 16th Mediterranean Conference on Control and Automation (MCA, 2008), pp. 1735–1740.
19.	A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008). [CrossRef]
20.	A. Beghi, A. Cenedese, and A. Masiero, “Multiscale stochastic approach for phase screens synthesis,” Appl. Opt. 50, 4124–4133 (2011). [CrossRef]
21.	P. K. Orzechowski, N. Chen, J. S. Gibson, and T. C. Tsao, “Optimal suppression of laser beam jitter by high-order RLS adaptive control,” IEEE Trans. Control Syst. Technol. 16, 255–267 (2008). [CrossRef]
22.	S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010). [CrossRef]
23.	S. Gordeyev, J. A. Cress, E. Jumper, and A. B. Cain, “Aero-optical environment around a cylindrical turret with a flat window,” AIAA J. 49, 308–315 (2011). [CrossRef]
24.	M. Weng, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012). [CrossRef]
25.	N. Y. Chen and J. S. Gibson, “Subspace system identification using a multichannel lattice filter,” in Proceedings of the 2004 American Control Conference (IEEE, 2004), Vol. 1, pp. 855–860.
26.	P. Van Overschee and B. De Moor, Subspace Identification for Linear Systems (Kluwer Academic, 1996).
27.	T. Katayama, Subspace Methods for System Identification (Springer-Verlag, 2005).
28.	M. Verhaegen and V. Verdult, Filtering and System Identification, a Least Squares Approach (Cambridge University, 2007).
29.	S. B. Jiang and J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995). [CrossRef]
30.	B. Widrow and E. Walsh, Adaptive Inverse Control (Prentice Hall, 1996).
31.	N. O. Arancibia, S. Perez Gibson, and T.-C. Tsao, “Frequency-weighted minimum-variance adaptive control of laser beam jitter,” IEEE/ASME Trans. Mechatron. 14, 337–348 (2009). [CrossRef]
32.	A. H. Sayed, Fundamentals of Adaptive Filtering (Wiley, 2003).
33.	W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]
34.	J. D. Mansell and B. Henderson, “Temporal and spatial characterization of polymer membrane deformable mirrors,” Proc. SPIE 7466, 74660D (2009). [CrossRef]

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.7350) Atmospheric and oceanic optics : Wave-front sensing

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: October 31, 2011
Revised Manuscript: May 15, 2012
Manuscript Accepted: June 1, 2012
Published: July 23, 2012

Citation
Jonathan Tesch and Steve Gibson, "Optimal and adaptive control of aero-optical wavefronts for adaptive optics," J. Opt. Soc. Am. A 29, 1625-1638 (2012)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-8-1625

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References

B. L. Ellerbroek and T. A. Rhoadarmer, “Real-time adaptive optimization of wavefront reconstruction algorithms for closed-loop adaptive optical systems,” Proc. SPIE 3353, 1174–1185 (1998). [CrossRef]
J. S. Gibson, C. C. Chang, and B. L. Ellerbroek, “Adaptive optics: wave-front correction by use of adaptive filtering and control,” Appl. Opt. 39, 2525–2538 (2000). [CrossRef]
T. A. Rhoadarmer, L. M. Klein, J. S. Gibson, N. Chen, and Y.-T. Liu, “Adaptive control and filtering for closed-loop adaptive-optical wavefront reconstruction,” Proc. SPIE 6306, 63060E (2006). [CrossRef]
Y. T. Liu and J. S. Gibson, “Adaptive control in adaptive optics for directed-energy systems,” Opt. Eng. 46, 046601 (2007). [CrossRef]
S. Monirabbasi and J. S. Gibson, “Adaptive control in an adaptive optics experiment,” J. Opt. Soc. Am. A 27, A84–A96 (2010). [CrossRef]
C. M. S. Corley, M. Nagashima, and B. N. Agrawal, “Beam control and a new laboratory testbed for adaptive optics in a maritime environment,” in Proceedings of 2010 IEEE Aerospace Conference (IEEE, 2010), pp. 1–13.
C. M. S. Corley, “Maritime adaptive optics beam control,” Ph.D. thesis (Naval Post Graduate School, 2010).
C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, F. Chemla, and D. Rabaud, “Off-axis adaptive optics with optimal control: experimental and numerical validation,” Proc. SPIE 5903, 59030P (2005). [CrossRef]
C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464–7476 (2006). [CrossRef]
K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2 optimal control,” J. Opt. Soc. Am. A 24, 1714–1725 (2007). [CrossRef]
K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2 optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008). [CrossRef]
C. Petit, J.-M. Conan, C. Kulcsár, and H.-F. Raynaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26, 1307–1325 (2009). [CrossRef]
J. Tesch, J. S. Gibson, S. Gordeyev, and E. Jumper, “Identification, prediction and control of aero optical wavefronts in laser beam propagation,” in 42nd AIAA Plasmadynamics and Lasers Conference (AIAA, 2011), paper 2011-3276.
J. Tesch and J. S. Gibson, “Optimal and adaptive correction of aero-optical wavefronts in an adaptive optics experiment,” Proc. SPIE 8165, 816502 (2011). [CrossRef]
J. Tesch, “High-performance control and prediction for adaptive optics,” Ph.D. thesis (University of California, Los Angeles, 2011).
C. Porter, S. Gordeyev, M. Zenk, and E. Jumper, “Flight measurements of aero-optical distortions from a flat-windowed turret on the Airborne Aero-Optics Laboratory (AAOL),” in 42nd AIAA Plasmadynamics and Lasers Conference (AIAA, 2011), paper 2011-3280.
R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010). [CrossRef]
A. Beghi, A. Cenedese, and A. Masiero, “A Markov-random-field-based approach to modeling and prediction of atmospheric turbulence,” in 16th Mediterranean Conference on Control and Automation (MCA, 2008), pp. 1735–1740.
A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008). [CrossRef]
A. Beghi, A. Cenedese, and A. Masiero, “Multiscale stochastic approach for phase screens synthesis,” Appl. Opt. 50, 4124–4133 (2011). [CrossRef]
P. K. Orzechowski, N. Chen, J. S. Gibson, and T. C. Tsao, “Optimal suppression of laser beam jitter by high-order RLS adaptive control,” IEEE Trans. Control Syst. Technol. 16, 255–267 (2008). [CrossRef]
S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010). [CrossRef]
S. Gordeyev, J. A. Cress, E. Jumper, and A. B. Cain, “Aero-optical environment around a cylindrical turret with a flat window,” AIAA J. 49, 308–315 (2011). [CrossRef]
M. Weng, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012). [CrossRef]
N. Y. Chen and J. S. Gibson, “Subspace system identification using a multichannel lattice filter,” in Proceedings of the 2004 American Control Conference (IEEE, 2004), Vol. 1, pp. 855–860.
P. Van Overschee and B. De Moor, Subspace Identification for Linear Systems (Kluwer Academic, 1996).
T. Katayama, Subspace Methods for System Identification (Springer-Verlag, 2005).
M. Verhaegen and V. Verdult, Filtering and System Identification, a Least Squares Approach (Cambridge University, 2007).
S. B. Jiang and J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995). [CrossRef]
B. Widrow and E. Walsh, Adaptive Inverse Control (Prentice Hall, 1996).
N. O. Arancibia, S. Perez Gibson, and T.-C. Tsao, “Frequency-weighted minimum-variance adaptive control of laser beam jitter,” IEEE/ASME Trans. Mechatron. 14, 337–348 (2009). [CrossRef]
A. H. Sayed, Fundamentals of Adaptive Filtering (Wiley, 2003).
W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]
J. D. Mansell and B. Henderson, “Temporal and spatial characterization of polymer membrane deformable mirrors,” Proc. SPIE 7466, 74660D (2009). [CrossRef]

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