Modal control for an adaptive optics system using LQG compensation

Computers Elect. Engng Vol. 18, No. 6, pp. 421-433, 1992

0045-7906/92 $3.00+ 0.00 Pergamon Press Ltd

Printed in Great Britain

MODAL

CONTROL

FOR AN ADAPTIVE OPTICS SYSTEM

USING LQG COMPENSATION MARK A . VON BOKERN I, RANDALL

N . PASCHALL2 and BYRON M . WELSH 2

Iphillips Laboratory PL/LIMI, Kirtland AFB, NM 87117 and 2Air Force Institute of Technology AFIT/ENG, Wright-Patterson AFB, OH 45433, U.S.A.

(Received 14 October 1991; accepted in final revised form 10 January 1992) Abstract--This paper demonstrates the application of modal control for a phase-correcting, groundbased adaptive optical telescope. Specifically, it addresses the design of a Linear Quadratic Gaussian (LQG) controller for generating actuator command voltages to a deformable mirror, given a set of slope measurements from a wave front sensor. A baseline mathematical model is constructed which forms the basis of a computer simulation. The actuator influence functions are not decoupled as is often done in zonal control.

1. I N T R O D U C T I O N It is well known that the atmosphere severely limits the performance of up-looking, ground-based imaging systems. These ground-based imaging systems include, most notably, telescopes used for astronomical observations. Real-time wave-front-reconstruction systems, commonly called adaptive-optics systems, have been shown to improve imaging performance of these systems [1]. Adaptive optical systems compensate for atmospheric turbulence effects by removing the wave front phase perturbations induced on the optical wave as it travels through the atmosphere. Atmospheric compensation is achieved by using wave front sensor measurements to control a wave front compensation device (i.e. a deformable mirror). The performance of these adaptive optical systems has been the subject of much research over the last 15 to 20 years [1-22]. Many of these past performance analyses efforts have concentrated on specific elements of the overall adaptive optical system and have not considered the overall performance of a complete adaptive optical system [2-10,12-14,17-19]. Only recently have analyses of complete adaptive optical systems begun to appear in the literature [15,20-23]. Not explicitly modelled are the dynamic of the tip/tilt mirrors in the te!escope's optical train. This research assumes these mirrors remove 95 %of the mean square tip/tilt phase error. These more recent analyses efforts have included realistic models of the wave front sensor and deformable mirror performance, as well as accurately modelling the performance of the control algorithm connecting the two. The control algorithms studied in these works fall into one of two descriptive categories: least squares and minimum variance. The simpler and more often studied of the two control algorithms is the least squares approach. In this approach the control signals are chosen to minimize the square of the difference between the wave front sensor measurement vector and the same vector that would be obtained from a measurement of the mirror surface. This approach does not take into account the statistical characteristics of the atmosphere or the measurements. These statistical characteristics include the spatial correlation of the atmospheric phase perturbations as well as the temporal correlation properties of the measurements. Since this statistical information is not used, the least squares approach results in suboptimal performance [15]. The second algorithm studied in the literature is commonly described as the minimum variance approach. In contrast to the least squares approach, the minimum variance approach makes use of the information contained in the spatial correlations of the atmospheric phase perturbations as well as the temporal correlations of the measurements. In this approach the control signals are chosen to minimize the mean square difference between the incident wave front and the surface of the deformable mirror. In this paper we study the performance of a complete adaptive optical telescope using a Linear Quadratic Gaussian (LQG) controller. As in the minimum variance approach, LQG control minimizes the mean square difference between the incident wave front and the surface of the mirror. 421

422

MARKA. VONBOKERNet al.

LQG control generates optimal inputs to the deformable mirror given (1) a time series of wave front sensor measurements, (2) the statistical characteristics of those masurements (namely second order time and spatial correlations) and (3) the statistical characteristics of the measurement noise. In contrast to past analyses using the minimum variance approach, LQG control uses the information contained in the temporal correlations of the measurements to improve the control of the deformable mirror. These temporal correlations are especially important for controlling the system during the time intervals between wave front samples. Much of the past work on the control of deformable mirrors have not considered the time varying nature of the atmosphere. As such, these efforts have implicitly assumed that the adaptive optical system is capable of correcting the wave front aberrations at speeds much faster than the characteristic time of the atmosphere. The effect of the finite time delay between wave front sensing and wave front correction has generally not been considered. In Section 2 we discuss the use of Zernike polynomials to describe the atmospheric phase distortions of the wave front traversing the aperture of the optical system. Section 3 contains descriptions of the models used to describe dynamical behavior of the atmosphere, mirror and the wave front sensor. These models are combined in Section 4 to design the L Q G controller. The result of a computer simulation of the overall system behavior are discussed in Section 5, and the design factors that most affect performance are pointed out. 2. Z E R N I K E

POLYNOMIALS

A modal approach is used to represent the phase distortions of the incident wave front. The modal approach considers the overall aperture phase distortions to be the weighted sum of various modes. The spatial domain of each mode is the entire aperture. The collection of the weights for all modes comprises a statement of the overall distortion. In an adaptive optics control context, the wave front sensor measurements are used to estimate the amount of each mode present in the distorted wave front. In contrast, a zonal approach would treat the phase error as a set of spatial samples or "zones" of phase error, and would not account for actuator coupling in the same way this research does. The deformable mirror actuators are in turn commanded to shape the mirror to produce the appropriate response of each mode. The Zernike functional space is a common means to quantify the wave front phase distortion traversing an aperture [13,24-26]. In this paper, phase distortion is defined as the deviation of the phase from the aperture averaged phase (i.e. piston mode removed). The modal approach used with Zernike basis functions expresses the wave front phase distortion ~ at a location (r, @) and time t within an aperture as the sum: N

qS(r, O, t) = ~ ai(t)Zi(r, 69) ~=0

(1)

where r and O are polar coordinates within the aperture, t is time, i is the mode-ordering number, a~(t) is the ith Zernike coefficient, and Zi(r, 6)) is the ith Zernike function. In general, an arbitrary wave front, requires N to be infinity to obtain an exact representation. Most of the contributions to the phase distortions caused by the atmosphere are accounted for in the lower-order Zernike modes [25]. Due to Nyquist concerns, the finite numbers of wavefront sensors and actuators limit the observability and controllability of higher-order (higher spatial frequency) modes. For the hardware configuration modelled in this paper, N = 14 was chosen as a conservative upper limit on Zernike mode modelled. Table 1 shows the first 14 Zernike functions (less the zeroth mode corresponding to piston) for a circular aperture of radius R. The Zernike functions form an orthogonal basis set which satisfies [25]:

lfo "j'R

~R 2

d@

dr rZi(r, 6))Zj(r, @) = 6 U

(2)

0

where 6~ is the Kronecker delta. A consequence of equation (2) is that the root-mean-square (rms) value of the phase distortion in an aperture is the square root of the sum of the squares of the corresponding Zernike coefficients: 4~ms = x/a 2 + a~ + . - .

+ a~.

(3)

Modal

423

control using LQG compensation

Table 1. Zernike functions

i

Zi(r, O) Table 2. Atmospheric distortion state-space model

i

Fa,,

1

0.07 0.15

r 2 -

r 2

5

cos,2o,

0.085

4

- 0.023

6

~ I 3 ( R ) 3 - 2(R)] sin(~9)

5

0.035

7

N/8[3(R)3 - 2(R)] cos(O)

6

0.036

-

r 3

0.02

9

r 3 xf8(~) cos(30)

8

10

~516(R)4 -- 6(R): + I]

190

0.023 0.02 0.12

II

N/~[4(R)' - 3(R)2 ] cos(20)

II

0.018 12

~014(R)4 - 3(R)2 ] sin,20,

13

r 4 ~10(~) cos(40)

14

12 0.015 13

r 4

~ ( ~ ) sin,40)

14

0.067 0.015

Qa,, 0.1605 0.075 0.405 0.8 0.509 0.124 0.223 0.21 0.241 0.363 0.0694 0.0833 0.0645 0.117

The expansion coefficients ai(t) for an arbitrary phase distortion ~b(r, O, t) can be obtained by [25]:

fdOfdrrW(r,O)¢(r,O,t)Zi(r,O) a~(t) =

(4)

fdOfdrrW(r,O) where W(r, 6)) is the aperture weighting function defined by:

W(r,6)) =

r>R.

(5)

It should be noted that this definition of W(r, 6)) is selected to be consistent with the scaling coefficients in Zi(r. 6)) (see Table 1). The strength of each mode of atmospheric distortion, as well as its time-varying nature, is contained in the Zernike coefficient, a~(t). In addition, the response of the deformable mirror to a set of command voltages can be expressed as a similar time-varying set of Zernike coefficients. This modeling, as well as the modeling of the wave front sensor is discussed in the following section. 3. M O D E L S In order to design a control law for the adaptive optical system, models describing the dynamical behavior of the atmosphere, mirror and wave front sensor are required. Such models represent the time-varying nature of the Zernike coefficients for both the light incident on the deformable mirror and the mirror's counter-distortion. Due to the random nature of atmospheric turbulence and the random noise in the measurement device, their models are stochastic state space models. In a zonal, as opposed to modal, approach the average phase distortion in each zone would need to be modelled as a stochastic process. The LQG approach could be implemented zonally knowing such models.

424

MARKA. VON BOKERNet al.

3. I. Wave front phase distortion

The dynamics equation describing the random, time-varying behavior of the Zernike coefficients of the incident wave front due to atmospheric turbulence can be written in the following state space representation [27]: ~a(t) = raXa(/) + Wa(t )

(6)

where Xa(t) is a vector of the first 14 Zernike coefficients describing the wave front phase distortion in the aperture of the optical system, Fa is a 14-by-14 dynamics matrix, and w~(t) is a 14-by-1 vector of zero mean, white Gaussian noise processes. The covariance matrix of the noise Wa(t ) is given by: g{wa(t)w~(t')} = aa6(t - t') (7) where 8{,} is the expectation operator, Qa is the noise strength matrix and 6(t) is the dirac delta. The elements of Fa and Q~ are related to the temporal auto- and cross-correlation properties of the Zernike coefficients. Elements of Fa and Qa were obtained by analyzing the results obtained by integrating the auto-correlation equation found in [26]. The resulting F~ and Q, matrices are diagonal with elements as given in Table 2. The covariances Qa,~ and Q~2 have been reduced by 95% to represent the situation where tilt mirrors are used for performing gross tilt adjustment in the system. 3.2. Mirror counter-distortion

The deformable mirror is assumed to have M regularly-spaced, identical actuators. Each actuator response is assumed linear over the control voltage input range. Linear superposition of responses from different actuators is also assumed. Based on data provided by the Air Force Weapons Laboratory [28] for a typical deformable mirror (Itek 129 actuator monolithic mirror) the dyanamics of each actuator is modeled as a first-order lag, with a time constant of zm. The steady-state phase influence of each actuator is determined in terms of Zernike coefficients using equation (4). The result is a deterministic mirror model of the form: Xm(t)=~Fmxm(t) + BmU(t)

(8)

where x m(t) is a vector of 14 Zernike coefficients describing the phase counter-distortion produced by the mirror, Fm is a 14-by-14 dynamics matrix, Bm is a 14-by-M input distribution matrix, and u(t) is a M-by-1 vector of control voltages. Bm is determined by the projection of each actuator's influence function onto each Zernike mode. This produces a multiple input/multiple output coupled design model as opposed to a single input/single output decoupled model normally used in the design of optical control systems. As a result, modal control is possible instead of zonal control. Fm is a diagonal matrix with entries 1/z where z is the time constant associated with the actuators. Combining the state-space models for the atmosphere and the mirror results in an augmented truth model whose state vector is x ( t ) = [XaX(t) XVm(t)]T. 3.3. Measurement device

The measurement device is a Hartmann-type slope sensor with K spatial samples. Each of the sensor's K subapertures measures the phase slope of a spatial sample of the reflected wave front in the x and y direction, resulting in a total of 2K measurements. The light reflected from the deformable mirror and subsequently measured by the sensor consists of the sum of both the atmospheric distortion and the mirror counter-distortion. Each slope measurement is assumed to be corrupted by additive, independent zero-mean Gaussian white noise based on actual data collected. A measurement equation can be constructed:

z(ti) = n'[xa(t,) + Xm(ti) ] + V(t,)

(9)

where z(t;) is a 2K-by-1 vector of slope measurements, H' is a 2K-by-14 measurement matrix, and v(t~) is a 2K-by-1 noise vector. The/jth element of the H' matrix relates the slopes measured by

Modal control using LQG compensation

425

the ith wave front sensor element to the j t h Zernike mode and is determined by integrating the weighted partial derivatives of the Zernike functions over the area of each subaperture:

H'(i,j) = f A W~(x,y) OZj(x, -~y y) dx dy

(10)

i

where i, j are the row and column of H', A~ is the area of the ith subaperture Wi(x, y) is the phase slope weighting function of Peterson and Cho [23], and Zj(x,y) is the j t h Zernike function, expressed in rectangular coordinates. Equation (10) is only valid for those sensors used for sensing the y directed slope. The rows of H ' corresponding to the x directed slope sensors are calculated by replacing a/dy with d/dx in equation (10). The final matrix is premultiplied by a scale factor such that the units of z(ti) are wavelengths. The slope measurement noise v(ti) is modeled as zero-mean white Gaussian noise sequence, uncorrelated between subapertures and independent of the Zernike coefficients. The variance is:

~{v(ti)vX(ti+j)} = Rf(j)

(11)

where R is a diagonal matrix containing elements proportional to the noise power of each wave front sensor measurement. 4. L Q G C O N T R O L Given a set of measurements z(ti), a model of the stochastic behavior of the system, and performance objectives, the goal of LQG control is to generate a set of optimal control signals u*(ti) that will minimize the variance of the difference between the actual state of the dynamical system and the estimated state. The measurements along with the model of the stochastic behavior are used by a Kalman filter to estimate the state of a dynamic system, ~(ti). With this estimate the control system seeks to minimize the rms phase distortion reflected from the deformable mirror. This is accomplished by generating a control gain matrix, G*, which determines the optimal control u*(ti) [29]: u*(t,) = -- G * ~(t~).

(12)

4.1. Quadratic cost The linear quadratic Gaussian (LQG) controller attempts to minimize the expected value of a quadratic cost associated with states x(t~) and control inputs u(t~) [29]:

J = ~{i~o½[XT(ti)Xx(ti) + uX(t~)Uu(t~)]} .

(13)

The cost weighting matrices X and U are chosen to optimize overall system performance. x(ti) is the augmented state vector given by: x=

[Xa1 Xm

.

(14)

For this problem, we desire to minimize the rms phase distortion in the wave reflected from the deformable mirror. The reflected wave contains atmospheric distortion and mirror counterdistortion. Note that the mean square residual phase distortion of the wave reflected from the deformable mirror will be, using equation (3): (])ms= ~{[Xa .Jr-Xm]T[Xa..[-Xm]}"

(15)

Thus, an appropriate quadratic cost term could be assigned as: [xa + Xm]T[C][Xa + Xm]

(16)

with C being a diagonal matrix of identical elements. We arbitrarily desired the maximum value of any Zernike coefficient of the corrected wave front to be 1/20 of a wavelength; we therefore assigned the diagonal elements of C to be square of the reciprocal of this value [29], or 400. With

426

MARK A. VON BOKERN et al.

the augmented state vector x defined in equation (14) it is easily shown that the corresponding X matrix for equation (13) is the double-banded diagonal matrix:

Since it is desired to keep the actuators within the linear range of operation, a quadratic cost is also assigned to the control voltages. If the maximum allowed magnitude of the control signals is A; the diagonal elements of the U matrix are 1/A 2. 4.2. K a l m a n filter

State estimates from measurements are accomplished by a Kalman filter. With the system "state" defined by equation (14), we have the problem of estimating the atmospheric and mirror Zernike coefficients, aa(ti) and Xm(ti). Since this controller will be implemented on a digital computer, a discrete-time version of equation (6) can be expressed as: Xa(ti+ 1) = tPa(ti+ l -- t i ) X a ( t i ) + Wad(ti)

(18)

where t~ a is the state transition matrix, exp{(Fa)(ti+~- ti)}, and Wad(t~) is a noise sequence of variance Qad [30]:

Qad ij = e{wad(ti)W

ad(tj)}

=

-- z ) Q ~ ~ ( t i + , i

A discrete-time version of equation (8) can also be expressed as [30]: Xm(t,+ 1) = q~n,(t~+, -- t,)Xm(ti) + BmdU(t,)

(20)

where Bmd is the discrete-time input distribution matrix defined by [30]: li + I

Bmd =

~t

~m(ti+ J -- z)Bm dz.

(21)

i

With the discrete-time versions of the atmospheric and mirror dynamical behavior defined, the estimation of the system state can be accomplished. The Kalman filter accepts as inputs the phase slope measurements z(t~) and generates an estimate of the atmospheric distortion states, ~,(t3. It accomplishes this by using an internal model of atmospheric distortion dynamics to propagate this state estimate and its covariance, P,, and then updating its estimate by appropriate processing of measurements. The filter covariance is: Pa~g{[Xa - ~a] [Xa -- ia]X} •

(22)

The Kalman filter actually estimates only the atmospheric distortion; the deterministicallymodeled mirror counter-distortion needs no such sophistication. The "estimate" of the mirror counter-distortion, ~m in equation (27), is obtained independent of the Kalman filter from: Xm (t ?t ) = (I)m (ti --

ti-,

)Xrn(t + , ) + BadU(t~+, ).

(23)

Kalman filter operation can be divided into two sequential processes involving the propagation and measurement update of the filter state estimate and covariance [30]. The governing equations are [29]: ~a ( t / - ) = CrPa(t i - - t,+, )~a(t/+_ ,)

(24)

P a ( t ? ) = dPa(t i -- ti+ , )Pa(t +_ 1)¢Ta (t i -- t,+ , ) + Qad

(25)

Ka (ti) = P~ (t i+ )H'TR --1

(26)

f q ( t + ) = fc~(t; ) + K~(t~) { z , - [ H ' f q ( t [ - ) + H'~m(t. )]}

(27)

Pa(t~+ ) = [ea~(tF) + H'VR-~H'] -'.

(28)

Modal control using LQG compensation

427

The values at time t7 represent the propagated value just prior to a measurement update. Values at t~+ represent the updated values just after a measurement has been processed. 4.3. Controller gain

To calculate the steady-state controller gain matrix is G* in equation (12), the augmented model is used:

]~xa(t)]=[ Fa L~m(t)..

0 Xa(t ) 0 Fm][Xm(t)]q-[Bm]U(t)q-[Wa(ot) ].

(29)

Using the discrete-time version of this augmented model: x(t,+ 1) = ~(t,+l - ti)x(ti) + Bau(t,) + wa(t,)

(30)

the steady-state controller gain matrix G* is calculated from [29]:

G~* = [U + BJ/~B~]

'BJKoe(t,+l -- t,)

(31)

where the controller Riccati matrix Kc is propagated via the backward Riccati difference equation [29] which, at steady state, is: Kc = X + ~v(ti+ i - ti)KJP(t~+, - t~)

-~T(t~+, -- t,)KcB~[U + B J K c B a ] - ' × 8JK~(t,+,

- t,).

(32)

Substituting this K~ matrix into equation (31) yields the steady-state controller gain G*. 5. P E R F O R M A N C E We simulated the performance of the L Q G contolled system using the multimode simulation for optimal filter evaluation (MSOFE) software [31]. This software implements a truth model of the dynamics and measurement models to generate simulated time histories of measurement samples. The true states of the system, Xa and Xm, are simultaneously available. A Kalman filter, also implemented in the simulation software, processes the simulated measurements and produces state estimates i , and ~m" For this simulation, the mirror states were carried in the Kalman filter for convenience. The L Q G controller in the simulation reads in the steady-state regulator gain G* from an external file and multiplies it by the state estimate vector to yield the simulated mirror control voltages. Steady-state Kalman filter gains are not used in the simulation. The Kalman filter gains are recomputed at each measurement update. Outputs from the simulation include plots of true states and corresponding filter estimates, plots of rms phase distortion of the incident and reflected wave (i.e. corrected image), and plots of the minimum and maximum mirror control voltages. Several parameters associated with the optical system under study and the atmosphere are required to perform the simulation. The simulations are carried out assuming that the optical system is imaging a stationary point source. The wave length of the source is 514 nm. The telescope entrance aperture has a radius of 0.75 m. The deformable mirror is a 97 actuator (i.e. M = 97) monolithic mirror with Gaussian like influence functions. Eleven actuators span the diameter of the entrance aperture, corresponding to a spacing of 6.8 cm. The time constant rm of each actuator response is 0.45 ms and the maximum drive signal allowed for each actuator A is 10. The wave front sensor consists of 69 subapertures (i.e. K = 69) arranged to fill the entrance aperture. Nine sensor subapertures span the diameter of the entrance aperture corresponding to a subaperture spacing of 8.33 cm. The sampling rate of the wave front sensor is 143 Hz, corresponding to a sampling period of 7 ms. The fluctuations of the index of refraction are described by Kolmogorov statistics and the vertical profile of the structure constant of the refractive index fluctuations Cn is described by the Hufnagel-Valley 21 model [32]. The wind profile is described by the Bufton model [32]. Using these atmospheric parameters the Fried coherence cell size r0 is 5.0 cm [32]. For this value of r0 we have an aperture diameter of 15r0, an actuator spacing of 1.36r0 and a wave front sensor subaperture spacing of 1.66ro.

428

MARK A. VON BOKERN et al.

It should also be pointed out that the model for this paper assumes that tip/tilt mirrors ahead of the deformable mirror remove 95% of the mean square distortion due to Zernike modes 1 and 2 (tilts). Additionally we assume the integrated flux incident on the wave front sensor is 1000 photon counts per subaperture (low noise condition). Using the flux level and the Hartmann wave front sensor noise model given in Ref. [21] the elements of R are calculated to be 0.01755 square wavelengths [21]. Other noise levels are examined in Ref. [27]. To evaluate the performance of the Kalman filter, Fig. 1 shows a time-history of a sample realization of the y-tilt Zernike coefficient of the incident light x,~ as well as the filter estimate, ~,,. Filter estimation error is defined as the true state minus the filter estimate of the state. The filter covariance P~ is the filter's indication of uncertainty in its estimates, as in equation (22). The square root of the (1,1) element of the P~ matrix is what the filter believes to be the 1-o value of its error in estimating x,,. Although Fig. 1 shows that the filter appears to be tracking the true state value, it does not indicate how the actual filter error compares with the filter's estimate of the 1-o values. Figure 2 shows a time history of the filter error for state 1, as well as these filter-computed 1-o values. Looking at the single realization of filter error of Fig. 2 and assuming the error is an ergodic random process, it appears that + P x / / ~ , 1) is a reasonalbe 1-0 value for the error, and that the error is zero mean. In order to obtain the "true" error mean and variance, a Monte Carlo analysis of ten simulation runs was accomplished. The equations used to process the results of these ten runs are: 1

10

~E(t) ~ - - ~ ek(t)

(33)

10k=l

1

l0

02E(t) ~ 1---0-~__ 1 k~, [e~(/) -- #2e(/)]

(34)

where: ek(t)= Xk(t)- 2k(t) (wavelengths); k = sample realization number; /~e(t)= mean of random process E(t) (wavelengths); and o ~ ( t ) = variance of random process E(t) (wavelengths2). Figure 3 shows a plot of the mean and standard deviation of the filter error for state 1, as calculated from the ten sample realizations using equations (33) and (34). Visual inspection of this plot reveals the error process is approximately zero mean, with standard deviations approximately equal to the Px//~, 1) values from Fig. 2. Plots similar to those of Figures 1-3, can be found in Ref. [27] for the remaining 13 Zernike modes. For all modes, the square root of the filter variance was approximately the one-sigma value for the filter error. 1.5 I

T Filter

I

--System 1.0

0.5

o

-0.5

-1.0 -1.5

1

-0.2

0

1

1

[

~

005

,

L.I

~

0.10

i

i

i

i

0.15

i

i

i

~

020

i

i

i

q

,

0.25 Time (sec)

Fig. 1

i

J

i

0.30

i

0 35

,

,

0.40

0.45

0.50

Modal control using LQG compensation 0.10

i

429

i

Error ---_+VF-~, I)

f;?

,

J

,,

,

-0.05

i

I

- 0 . 1 0

i

0

T

I

I

I

005

0.10

I

035

I

i

I

i

~

0.20

i

i,

p

I

0.30

0.25

I

0.35

I

q

0.40

i

l

~

0.45

~

~

i

0.50

Time (sec)

Fig. 2

The performance of the L Q G controller can be expressed in terms of residual rms phase distortion of the corrected wave front as compared to the rms phase distortion of the incident wave front. Again, incident here means after the tip/tilt mirrors have removed most of the gross tilt. Recall from equation (29) that states 1-14 of the overall system model are the Zernike coefficients for atmosphere-induced phase distortion, and states 15-28 are Zernike coefficients for the mirror-induced counter-distortion. Since reflection from the deformable mirror is modeled as addition of atmospheric and mirror phase distortions, the Zernike coefficients of the reflected (i.e. corrected) wave front are: a(t) = xa(t) + Xm(t)

(35)

= x~_,4(t) + x,5_28(t)

(36)

= [III]x(t).

(37)

It has been shown in equation (3) that the rms phase distortion is the square root of the sum of the squares of the Zernike coefficients [27]. The rms phase error of the incident wave front is given by ~brms(t) = . / ~

(xi)2.

(38)

~ / / = 1

The residual rms phase of the corrected wave front is given by ~b~ms(t) =

X/'~ (x~ +

x,+ ,,)2.

(39)

i

A time history of a sample realization of the rms phase distortion is shown in Fig. 4 for both the incident and correlated wave front. The plot clearly shows the reflected light has less rms phase error than the incident light. Moreover, this single realization shows the rms phase error of the corrected wave front tends to stay in the vicinity of about 0.1 wavelengths, whereas the rms phase error of the incident wave front varies between about 0.1 wavelength and 0.5 wavelength. Calculating similar results for each of the ten Monte Carlo realizations and generating the usual statistics results in values plotted in Fig. 5. The upper three lines of the graph show the mean and the [mean + l-a] values on the rms phase distortion of the incident wave front. The apparent transient in these three lines during the initial 0.05 s of the simulation was caused by unrealistic initial conditions in the truth model (all states zero). It would be a more realistic simulation if the initial true states were random. The lower three lines of the graph represent the mean and the [mean + l-a] values on the rms phase distortion of the corrected wave front. One feature of note

430

MARK A. VON BOKERN et al.

r ~/~E(t)

0.10 I

0.05

] ----~E(t)

r ±~Elt)

,

i;:, ': i

0

-0.05

-0.10 0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Time (sec)

Fig. 3

is that the 1-a values on the corrected wave front are tighter than on the incident wave front. This seems to suggest the quality of the corrected wave front is somewhat constant, despite wide variation in the amount of atmospheric distortion. Another feature is the sawtooth appearance of the lower set of plots. The period of the sawtooth is the Hartmann sampling period (7 ms). This suggests the correction is most effective just after a measurement, and degrades as the atmosphere changes between measurements. One may wonder if even better performance is possible. In other words, one can ask, what is limiting the performance shown in Fig. 5? Besides the obvious limitations imposed by the temporal sampling rate of the wave front sensor and the spatial sampling of the wave front sensor and deformable mirror, at least three answers are possible: (1) saturation of mirror actuators; (2) improperly chosen weighting matrices in the LQ regulator; or (3) the filter's estimation errors. The first possible reason, actuator saturation, is immediately eliminated based on Fig. 6. This plot shows the absolute envelope for the actuator control voltages for the ensemble of ten Monte Carlo realizations. Given that saturation occurs when the magnitude of the command voltage exceeds 10 V, and the maximum actual excursion was only about ± 1 V, saturation did not occur. In fact, one could say the mirror had quite a bit of dynamic range left. The second reason,

06

Incident - - - Corrected

.

.

.

.

.

.

.

.

.

.

.

04

~112

0.2

0

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improperly chosen weighting matrices X and U, was eliminated by a tuning experiment. The nonzero elements of the X matrix of the cost equation were increased by 50%. This penalized the distortion of the corrected wave front more heavily. The simulation was re-run using the new steady-state controller gain, G*, and performance did not improve. The third reason, filter estimation error, was analyzed in the following manner. The filter's function is essentially to estimate the Zernike coefficients of the incident, uncorrected wave front. Assume the regulator/ mirror combination can perfectly implement the filter's estimate (without a sign change). The rms phase error due solely to the filter's estimation error is the lower bound on rms phase error attainable and is calculated using:

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similarity o f this g r a p h a n d the lower three plots o f Fig. 5 s t r o n g l y suggest t h a t filter e s t i m a t i o n e r r o r is the p e r f o r m a n c e - l i m i t i n g factor. 6. C O N C L U S I O N S This p a p e r presents the design o f an L Q G c o n t r o l l e r for a n a d a p t i v e optics systems. A t r u t h m o d e l is d e v e l o p e d b a s e d on theoretical d e v e l o p e m e n t s a n d a c t u a l test data. T h e resulting state-space m o d e l has as its states the 14 Z e r n i k e coefficients a s s o c i a t e d with the a t m o s p h e r i c t u r b u l e n c e a n d the 14 Z e r n i k e coefficients a s s o c i a t e d with the d e f o r m a b l e m i r r o r . T h e reflected i m a g e can be described by c o m b i n i n g the Z e r n i k e coefficients. A s a result o f the a p p r o a c h t a k e n in this research, m o d a l c o n t r o l is a c c o m p l i s h e d as o p p o s e d to z o n a l c o n t r o l since the c o u p l i n g o f a c t u a t o r influence functions is t a k e n into a c c o u n t in d e v e l o p i n g the design models. T h e results o f c o v a r i a n c e a n d M o n t e C a r l o analysis d e m o n s t r a t e s the limits o n p e r f o r m a n c e i m p o s e d by the K a l m a n filter. T h e c o m p u t e r s i m u l a t i o n s p r e d i c t that the c o n t r o l l e r p r e s e n t e d in this p a p e r can reduce p h a s e d i s t o r t i o n to an rms value o f a p p r o x i m a t e l y 0.1 wavelengths. F o r o p e r a t i o n in a variety o f noise c o n d i t i o n s , one c o u l d possibly use a b a n k o f filters with each filter t u n e d to a desired noise level. H o w e v e r , to i m p l e m e n t such an a p p r o a c h lower o r d e r elemental filters m a y be required. F i n a l l y , to h a n d l e time delays in the m e a s u r e m e n t s , one c o u l d use a p r e d i c t o r to o v e r c o m e the d e l a y impact. REFERENCES 1. J. H. Hardy, Active optics: a new technology for the control of light. Proc. IEEE 66, 651-697 (1978). 2. D. L. Fried, Least-square fitting a wave front distortion estimate to an array of phase-difference measurements. J. opt. Soc. Am. 67, 370-375 (1977). 3. R. H. Hudgin, Wave front reconstruction for compensated imaging. J. opt. Soc. Am. 67, 375-377 (1977). 4. R. H. Hudgin, Optimal wave front estimation. J. opt. Soc. Am. 67, 378-382 (1977). 5, R. H. Hudgin, Wave front compensation error due to finite corrector-element size. 3. opt. Soc. Am. 67, 393-395 (1977). 6. J. Y. Wang, Optical resolution through a turbulent medium with adaptive phase compensations. J. opt. Soc. Am. 67, 383-390 (1977). 7. J. Y. Wang and J. K. Markey, Modal compensation of atmospheric turbulence phase distortion. J. opt. Soc. Am. 68, 78-87 (1978). 8. J. Y. Wang, Effect of finite bandwidth on far-field performance of modal wavefront-compensative systems. J. opt. Soc. Am. 69, 819-828 (1979). 9. R. J. Noll, Phase estimates from slope-type wave front sensors, J. opt. Soc. Am. 68, 139-140 (1978). 10. D. P. Greenwood, Mutual coherence function of a wave front corrected by zonal adaptive optics. J. opt. Soc. Am. 69, 549-554 (1979). 11. D. P. Greenwood, Bandwidth specification for adaptive optics systems. J. opt. Soc. Am. 67, 390-393 (1977). 12. J. Herrmann, Least-squares wave front errors of minimum norm. J. opt. Soc. Am. 70, 28-35 (1980). 13. R. Cubalchini, Modal wave-front estimation from phase derivative measurements. J. opt. Soc. Am. 69, 972-977 (1979). 14. W. H. Southwell, Wave-front estimation from wave-front slope measurements. J. opt. Soc. Am. 70, 998-1006 (1980).

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15. E. P. Walner, Optimal wave front correction using slope measurements. J. opt. Soc. Am. 73, 1771-1776 (1983). 16. G. A. Tyler, Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain. J. opt. Soc. Am. 1, 251-262 (1984). 17. R. C. Smithson et al., Quantitative simulation of image correction for astronomy with a segmented actice mirror. Appl. Opt. 27, 1615-1620 (1988). 18. R. C. Smithson and M. L. Peri, Partial correction of astronomical images with active mirrors. J. opt. Soc. Am. 6, 92-97 (1989). 19. J. P. Gaffard and C. Boyer, Adaptive optics for optimization of image resolution. Appl. Opt. 26, 3772-3777 (1987). 20. J. D. Downie and J. W. Goodman, Optimal wavefront control for adaptive segmented mirrors. Appl. Opt. 28, 5326-5332 (1989). 21. B. M. Welsh and C. S. Gardner, Performance analysis of adaptive optics systems using slope sensors. J. opt. Soc. Am. 6, 1913-1923 (1989). 22. B. M. W. C. S. Gardner and L. A. Thompson, Design and performance analysis of adaptive optical telescopes using laser guide stars. Proc. IEEE 11, 1721-1743 (1990). 23. D. P. Petersen and K. H. Cho, Sampling and reconstruction of a turbulence-distorted wave front. J. opt. Soc. Am. 3, 818-825 (1986). 24. J. Herrmann, Cross coupling and aliasing in modal wave-front estimation. J. opt. Soc. Am. 71, 989-992 (1981). 25. R. J. Noll, Zernike polynomials and atmospheric turbulence. J. opt. Soc. Am. 66, 207-211 (1976). 26. C. B. Hogge and R. R. Butts, Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence. IEEE Trans. Antennas Propagation AP-24, 144-154 (1976). 27. M. A. Von Bokern, Design of a linear quadratic Gaussian control law for an adaptive optics system. Master's thesis, Air Force Institute of Technology (AU), Wright-Patterson AFB OH (1990). 28. A. Lusk and D. Russell, Characterization system, unpublished notes, Air Force Weapons Laboratory, Kirtland AFB NM. 29. P. S. Maybeck, Stochastic Models, Estimation, and Control, Vol. 3. Academic Press, New York (1982). 30. P. S. Maybeck, Stochastic Models, Estimation, and Control. Academic Press, New York (1979). 31. S. H. Musick and N. A. Carlson, User's manual for a multimode simulation for optimal filter evaluation (msofe). Contract F3361586C1047, Integrity System lncorported, Winchester, Mass. (1990). 32. R. R. Parenti, Recent advances in adaptive optics methods and technology. SPIE Vol. lO00---Laser Wavefront Control, pp. 101-109 (1988). AUTHORS' BIOGRAPHIES

Mark A. Von Bokera--Mark Von Bokern received his BSEE degree from Ohio University in 1986. After a three-year tour as an aircraft radar analyst at the Air Force's Foreign Technology Division in Dayton, Ohio, he attended the Air Force Institute of Technology (AFIT) and obtained his MSEE in 1990. His thesis research investigated the application of modern control theory to adaptive optics telescopes. His current assignment is at the Phillips Laboratory, Kirtland AFB, New Mexico, where he is involved with low-light imaging of exo-atmospheric objects.

Randall N. PaschalI--Capt Randall N. Paschall, Ph.D. has been an Assistant Professor of Electrical Engineering at the Air Force Institute of Technology at Wright-Patterson AFB, Ohio since 1988. He has taught courses in fire control, flight control, integrated navigation systems, linear systems theory, digital control and optimal control. He is the director of AFIT's Guidance, Navigation and Flight Control Lab. His research interest include optimal integration of GPS and INS, realtime simulation of nonlinear flight dynamics and optimal control for adaptive optic applications. Prior to AFIT, Capt Paschall was a guidance and control engineer for the Low Level Laser Guided Bomb program at Eglin AFB, Florida. Dr Paschall is a member of Tau Beta Pi, Etta Kappa Nu, the Institute of Navigation and is past president of the Dayton IEEE Controls Systems Society.