American Institute of Aeronautics and Astronautics

1

A Robust Modification of a Predictive Adaptive-Optic

Control Method for Aero-Optics

Robert Burns1, Eric Jumper

2, Stanislav Gordeyev

3

University of Notre Dame, Notre Dame, IN, 46556

Abstract

A modification of a previous predictive adaptive-optic controller is presented in this

paper. Conventional adaptive-optic controllers suffer from bandwidth limitations caused by

latency in their control loops. This latency severely limits their capabilities in aero-optic

applications that cannot be overcome with conventional feedback techniques. Our method

uses prior knowledge of flow behavior to predict future behavior, and thus overcome

deadtime. We have modified our previous neural network controller to use a linearized

predictor, which we demonstrate to be more accurate, more robust to noise and flow

disturbances, and less computationally expensive. Our previous neural network method

showed disturbance rejection in the range of 35-55% in simulation over our test conditions

in the most optically-active regions, while the improved method shows disturbance rejection

between 45-75% over the same range. Additionally, we demonstrate that the predictive

control method is stable, even in the presence of latency uncertainty.

Nomenclature

α = viewing angle (rad)

β = modified elevation angle (rad) or integrator gain (-)

ε = vector of wavefront prediction residuals

λ = laser wavelength (m)

ρ = air density (kg/m3)

Σ = diagonal matrix of singular values

Φ = matrix of POD modes (-)

φ = eigenmode vector (-)

A = linear prediction matrix

Az = azimuth (rad)

a = modal coefficients

C = compensator transfer function (-)

D = aperture diameter (m)

d = single-aperture aero-optic disturbance (m)

E = root-mean-square residual error

El = elevation (rad)

f = disturbance frequency (Hz)

1 Graduate Student, Department of Aerospace and Mechanical Engineering, Hessert Laboratory for Aerospace

Research, Notre Dame, IN 46556, AIAA Student Member.

2 Professor, Department of Aerospace and Mechanical Engineering, Fitzpatrick Hall of Engineering, Notre

Dame, IN 46556, AIAA Fellow.

3 Research Associate Professor, Department of Aerospace and Mechanical Engineering, Hessert Laboratory for

Aerospace Research, Notre Dame, IN 46556, AIAA Associate Fellow.

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47th AIAA Plasmadynamics and Lasers Conference

13-17 June 2016, Washington, D.C.

AIAA 2016-3529

Copyright © 2016 by William Robert Burns, Eric Jumper, Stanislav Gordeyev. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA Aviation

American Institute of Aeronautics and Astronautics

2

f = vector-valued nonlinear prediction function

g = function describing flow evolution

h = observer function of special flow properties

I = on-target beam intensity (W/m2)

I0 = unaberrated beam intensity (W/m2)

KGD = Gladstone Dale constant (m3/kg)

M = Mach number (-) or embedding dimension (-)

Nd = number of timesteps of delay in the feedback loop (-)

n = index of refraction (-)

OPD = Optical Path Difference (m)

OPL = Optical Path Length (m)

S = Strehl ratio (-)

St = Strouhal number (-)

U = coefficient matrix

V = freestream velocity (m/s)

v = wavefront vector (m)

W = wavefront vector (m)

x = vector of coefficients (-)

y = wavefront residual output (m)

z = vector of flow properties

I. Introduction

ne significant challenge to the implementation of airborne laser systems is the aero-optics problem1. Near-field

turbulence over a turret aperture causes density fluctuations that are “imprinted” on the laser wavefronts

through the relationship between index of refraction and density. These effects can greatly reduce a beam

control system’s performance in the far-field, especially when pointing through the wake region of a turret. Turret

configurations are desirable in beam control systems for their large fields of regard; however, this comes at the

expense of significant aero-optic aberrations, especially in the wake region. A schematic of the flow structures that

impact aero-optics is shown2 in Figure 1. Pressure and temperature fluctuations contribute to significant density

fluctuations as the boundary layer thickens downstream. Flow separation over the top of the turret results in large

separation regions that roll up into “horn” and secondary vortices. So-called “necklace” vortices form at the base of

the turrets and can be encountered when pointing the laser near the surface of the aircraft. Shocks can form at the top

of the turret at Mach numbers as low as 0.55. All of these flow features cause density fluctuations that adversely

impact optical performance.

The aero-optics problem in the wake region of a turret

(without compensation) is depicted schematically in

Figure 2. The beam’s unaberrated, planar wavefronts

encounter turbulence near the turret’s aperture. The

turbulence-related density variations are effectively

“imprinted” on the wavefront. As the beam propagates to

the far field, these density fluctuations will tend to scatter

the beam. The strength of these aberrations scales with the

squared Mach number of the emitting aircraft, the

freestream density, and the diameter of the turret.

The index of refraction is related to the density of the

medium through the Gladstone-Dale relationship3,

( , , ) 1 ( ) ( , , )GDn x y z K x y z , where n is the index of

refraction, λ is the wavelength, and KGD is a constant. This

relationship can be used to compute the exact aberration

along a path by first computing the optical path length

O

Figure 1: Optically-aberrating flow over a

turret1.

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from some point in the aperture,1

0( , , ) ( , , , )

z

OPL x y t n x y z t dz , and then subtracting the aperture-averaged mean to

compute the optical path difference,

( , , ) ( , , ) ( , , )OPD x y t OPL x y t OPL x y t [1]

Figure 2: Schematic of the aero-optics problem for a beam control system1.

The RMS optical path difference (OPD) is a convenient metric for estimating the performance of aero-optic

systems since it enables the estimation of the ratio of the Strehl ratio: aberrated on-axis beam intensity to the

diffraction-limited on-axis beam intensity. Strehl ratios can be estimated for each aberration source in a beam

control system: for example, near-field aero-optics, far-field atmospherics, aero-mechanical jitter, optical heating,

etc. These Strehl ratios can be multiplied together to obtain the total loss of intensity at the target. In this paper, we

focus on the aero-optic aberrations only. For instantaneous aero-optic aberrations following a Gaussian spatial

distribution, the Strehl ratio can be estimated through the Marèchal large aperture approximation4,5,6

, shown in Eq. 2.

22 ( )

0

( )( )

( )

rmsOPD tI t

S t eI t

. [2]

It is desirable to maximize the Strehl ratio by mitigating aero-optical aberrations. It is conventional to quantify

reductions in aero-optical aberration using the following definition,

,

10

,

10logrms corrected

rms uncorrected

OPDdB

OPD . [3]

Since laser wavelength is application-dependent, it is useful to consider the impact of wavefront aberration

mitigation in non-dimensional terms by rearranging the equation in terms of a corrected and uncorrected Strehl ratio,

as discussed in our previous work7. A modest -3dB improvement in aperture-averaged disturbance rejection can

improve Strehl ratio from 0.2 to 0.67, for example.

A large body of work has been performed to characterize aero-optical effects on the Airborne Aero-Optics

Laboratory8 (AAOL), which is shown in Figure 3, and its transonic successor AAOL-T

9. AAOL consists of a

pointing and a tracking aircraft. The pointing aircraft emits a diverging 532 nm beam toward the tracking aircraft’s

turret, so that the beam width at the exit aperture is small relative to the flow structures. This means that the optical

aberrations at the exit aperture are primarily tip/tilt and do not significantly impact the wavefronts. Aero-optical

aberrations near the aperture of the tracking turret are imprinted on the laser wavefronts and measured by a Shack-

Hartmann wavefront sensor in the tracking aircraft.

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Figure 3: Picture of the Airborne Aero-Optics Laboratory (AAOL). The pointing aircraft is depicted on

the left and the tracking aircraft on the right.8

Normalized aero-optical aberrations tend to collapse onto a curve that is a function of two quantities: a lookback

angle, α, and a modified elevation angle, β. These two quantities can be expressed8 in terms of azimuth and

elevation as shown in Figure 4,

1cos cos( )cos( )Az El , and [4]

1 tan( )tan

sin( )

El

Az

. [5]

It has been demonstrated8 that aero-optic aberrations can be normalized using the following equation,

2

0/

rmsnorm

OPDOPD

M D , [6]

where M is the freestream Mach number, ρ/ρ0 is the ratio

of freestream density to sea-level density, and D is the

turret diameter. Figure 5 shows OPDnorm as a function of

viewing angle and modified elevation angle from AAOL.

Notably, the optical aberrations are highest in the

separated flow regime, corresponding to viewing angles

of greater than 110 degrees. Laser-based systems are

often tolerant of small aberrations encountered at

forward-looking angles, but not at aft-looking viewing

angles.

Aero-optical aberrations must be mitigated using

some technique for many applications involving either

communications or directed energy. There are two

primary ways of achieving this: flow control and adaptive

optics. Flow control devices are generally very

application-dependent. Active flow control can be used to mitigate aero-optic distortions either by suppressing

aberrations or relocating them away from an optical aperture. Some of these methods are open-loop (operating on

pre-determined, feedback-free control), while others are closed-loop (sensing and reacting to disturbances to

mitigate them).

Active flow control has been employed in transonic flows to manipulate shock formation over a cylinder10

using

pulsed jets in open-loop configurations. This type of manipulation could be used to relocate optically-active flow

features away from an optical aperture.

High-frequency synthetic jets11

have been used to reduce the optical aberrations present in the wake of a

hemisphere. Later, this group12

used a hybrid of passive and active flow control: by using a forward-protruding plate

to decouple the turret wake from the necklace vortices, the effectiveness of the oscillating jets upstream of the

optical aperture to control the turret wake was enhanced. Vukasinovic et al10

have also tested control jets in the

transonic regime to control separation both upstream and downstream of the shock to reduce the sharp velocity and

density gradients present in the shear layer. This type of flow control was studied in more detail13

using Particle

Image Velocimetry (PIV) and Schlieren imagery.

Figure 4: Definition of α and β in terms of

azimuth and elevation7

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Figure 5: Collapse of OPDrms as a function of viewing angle from AAOL flight tests.8

A modification of proper orthogonal decomposition14

has been developed to model closed-loop flow control.

This method attempts to model both the controlled and natural states of the flow, as well as the transition between

these states. The research group demonstrated their temporal POD (TPOD) approach using dielectric discharge

barrier (DBD) plasma actuators to control the first TPOD mode of flow over a circular cylinder. In practice it would

be necessary to control more than the first mode, but nonetheless there are several ideas from this work that could be

incorporated into a model for the physical behavior of various turbulent flows.

An explosion of research in the field of machine learning has led to an effort in fluids research15

to improve

flow control using these new methods. One method16

uses optical sensors for feedback, slotted jets for control

actuators, and a genetic algorithm to implement a control law to delay flow separation. Genetic algorithms are used

in control systems to search for control actions that minimize a cost function for a given set of observations. The

results of their experiment were an 80% reduction in the recirculation zone behind a backward-facing step. Genetic

algorithms are often referred to as model-free control. A similar approach based on genetic programming17

was used

to control a mixing layer flow and compared to open-loop forcing. In addition to achieving better performance than

the open-loop forcing, it is also adaptable and automatically tunable.

Another means of aero-optic mitigation is to use adaptive optics. These systems imprint the conjugate of the

currently-present aero-optic aberration onto a deformable mirror, and then send the pre-corrected laser beam out

from the turret. When the corrected beam then encounters the aero-optic aberration, it emerges from the flow as

nearly planar. These techniques are currently employed on ground-based telescopes to correct for atmospheric

optical effects. However, these aberrations occur on the order of 1-10 Hz and thus the computational and system

requirements are not extremely demanding for currently-available technology. However, sub-sonic aero-optical

phenomena consisting of turbulent boundary layers, shear layers, attached vortices, etc. occur on the order of

kilohertz on full-scale turrets at realistic Mach numbers. In particular, the dominant frequency content of separated

flow on a turret in the wake region typically occurs near a Strouhal ratio of about 1:

fDSt

V , [7]

where f is the frequency of the disturbance in hertz, D is the turret diameter, and V is the freestream velocity. For the

Airborne Aero-Optics Laboratory, the disturbance frequencies observed in the wake region were concentrated near 1

kHz. Thus, a real-time adaptive optics system appropriate for AAOL must be able to perform corrections at a

minimum of 10kHz, assuming 10 discrete corrections per disturbance cycle. This is a demanding computational

requirement given that wavefronts must be represented by matrices that are at least 10x10, and sometimes as large as

30x30. A significant obstacle is cumulative latency in the control system. Time spent on digital calculations,

physical sensor response, analog-to-digital conversion of sensor signals, electronic amplifiers, and

electromechanical response of deformable mirrors will all contribute to a significant degree of delay18

in a high-

speed adaptive optic system.

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This latency problem has previously been

addressed using various prediction methods such as

dynamic mode decomposition19

, receding-horizon

adaptive control20

, and simple “frozen-flow” advection

prediction21

. We recently developed a method7 for

overcoming this latency problem using a predictive

neural network controller in conjunction with a

conventional feedback loop as shown in Figure 6. The

neural network controller uses prior knowledge of

flow behavior to predict future behavior over a short

temporal horizon equal to the known delay in the

feedback loop. We simulated an adaptive-optic control

loop operating at 25kHz with varying amounts of

latency using aero-optic wavefront disturbance data

from the AAOL flight tests. In the most optically-

active fully separated flow regime, disturbance

rejection between 35% and 55% was achieved in our

simulations. As discussed previously, even 35%

reduction in RMS wavefront aberration can have a

large effect on a real system’s performance. The

presumption for this method was that the nonlinear

character of the flow would require a nonlinear

predictive element such as a neural network.

In this work, we develop a modification for the neural network method that uses a linearized prediction model.

We demonstrate that not only is this an acceptable assumption over the amounts of latency considered, but it is

actually scalable to a larger number of predictable modes and leads to better prediction accuracy. Additionally, we

investigate its favorable robustness to perturbed flow conditions and stability. For further details on dimensionality

reduction using POD and the impact of latency on controller bandwidth, refer to our previous work7.

II. Predictive Control System Architecture

A block diagram of the predictive controller is shown in Figure 7. The controller is denoted by C, the mirror

plant model by G, the aero-optic disturbance by d, the residual wavefront error y, the prediction model P, the lagged

wavefront sensor H1, the lagged mirror sensor by H2, and the unlagged mirror sensor by H3.

For consistency and to avoid excessive tedium, we assume that all of the latency in the control system occurs in

wavefront sensing, reconstruction, and control law computation and is accounted for in the model H3. The wavefront

predictor itself is sensitive only to cumulative latency and its predictions are independent of the individual sources of

latencies. Lagged and unlagged mirror sensors are required to reconstruct an estimate of the total wavefront

disturbance from the sensed residual wavefront and the deformable mirror position.

For the purposes of our analysis, we will assume that the conventional controller is a simple integrator and the

deformable mirror has unity transfer function. In the absence of latency this is a trivial control problem, but the

bandwidth limitations imposed by pure latency are severe enough to investigate in isolation of mirror mechanics,

amplifier electrical dynamics, etc. (see our previous paper for more details7).

H1 may be decomposed as 1 1

s T tH H e

, where ΔT is the cumulative latency in the feedback loop, δt is the

uncertainty in this latency, and 1H is the latency-free sensor. Practically, the lagged and unlagged deformable

mirror sensors may be decomposed as 2 3

s TH H e since H2 is simply a digitally-delayed copy of H3. An

important design goal is to match the latency between the lagged deformable mirror sensor and the wavefront

sensor. The latency in H2 is considered to be a tunable parameter matched to the nominal value of the latency in the

wavefront sensor, while the latency in the wavefront sensor, ΔT, is measured a priori with some uncertainty, δt. Due

to simple measurement uncertainty as well as digital jitter, this uncertainty will always be non-zero; however, the δt

= 0 case is considered first.

Figure 6: Conventional AO loop augmented

by wavefront predictor1.

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Figure 7: Predictive controller block diagram.

Consider the single-input-single-output (SISO) version of this system. This system gives the following general

transfer function from the output to the reference signal,

1 2 31

C s G sys

r C s G s P s H s C s G s P s H s C s G s H s

. [8]

The rationale for matching the delays between H2 and H1 is now clear: in the absence of uncertainty, the

complex dynamics of the predictor vanish and the closed loop transfer function becomes simply,

31

C s G sys

r C s G s H s

, [9]

which can typically be stabilized by a P-I controller in an adaptive-optics control loop. Given the simplicity of our

idealized component models, the stability of this loop is trivial to assess in the absence of latency uncertainty.

Now, consider the multidimensional case. Define the transfer function matrix T(s),

1 1 1

1 2

2 2 2

1 2

1 2

N

N

N N N

N

y y ys s s

r r r

y y ys s s

r r rs

y y ys s s

r r r

T

, [10]

which in the multidimensional case can be solved for as follows if the latency uncertainty is zero,

1

3s s s s s s

T C G I C G H . [11]

As in the single-dimensional case, this system can easily be controlled (with varying performance) with

classical methods. Recall that adaptive optics system components are in general open-loop stable.

Similarly, the disturbance rejection function,

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1 1 1

1 2

2 2 2

1 2

1 2

N

N

N N N

N

y y ys s s

d d d

y y ys s s

d d ds

y y ys s s

d d d

S, [12]

becomes (in the absence of delay uncertainty),

1

2 3 3s s s s s s s s s s s

S I C G P H C G H I C G H . [13]

Hence, the prediction filter must be open-loop stable to ensure closed-loop stability. The interpretation for this

necessity is essentially that the predictor acts as a feed-forward element.

We will now detail the algorithm of the predictor itself, and then re-assess the stability problem using this

framework.

III. Prediction Method and Estimation

We will first briefly summarize the key assumptions from our nonlinear neural network predictor7 and then

discuss the necessary modifications. We assume that we can approximate the evolution of an aero-optical wavefront

vk by some discrete function f that depends on prior wavefronts vk-1…vk-M+1, with some amount of truncation error

denoted by εk+1.

1 1 1 1, ,...,k k k k M k v f v v v ε [14]

A wavefront predictor should approximate f such that the mean-square of the residual error is minimized,

2

1

L

kkE

ε ,

where L is the total number of wavefront observations in a dataset. This is has the structure of a non-linear

autoregressive (NAR) problem.

For practical applications, it is desirable to reduce the order of the prediction model. Practical applications of

adaptive optics in aero-optics will have wavefront measurements on the order of 15x15 up to 30x30 subapertures,

ranging from ~200 subapertures to ~700 subapertures for a circular beam inscribed in a rectangular sensor array.

Direct wavefront prediction is computationally-intensive and prone to measurement noise. A simple linear

prediction matrix of the form vk+1=Avk alone would contain in the worst case between 105 and 10

6 elements, which

is impractical for real-time applications. This problem becomes substantially worse if more than a single prior

wavefront is needed for a prediction; which, in the general case is true due to the nonlinear nature of the flow.

A number of methods can be used for dimensionality reduction. A natural choice of basis functions for many

optical applications are the Zernike modes, since they describe physically meaningful optical aberrations such as

focus/defocus, coma, astigmatism, etc. and are orthogonal on the unit disk. However, these modes are physically

unrelated to the physics of the flow itself. A better modal decomposition for aero-optics is the Proper Orthogonal

Decomposition22

, which can be computed via a snapshot method that produces the most rapidly-converging set of

orthogonal modes possible.

These modes, also known as Karhunen-Loève (K-L) or Principle Component Analysis (PCA) modes, can be

calculated for multidimensional discrete-time data using the Singular Value Decomposition (SVD) or by solving the

eigenvalue problem on the data matrix’s autocorrelation matrix. In this case, let V be a matrix of measurement

snapshots of wavefronts organized by column vectors of samples ordered by increasing time as shown,

T

1 2 NV v v v . [15]

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The data matrix, V, is decomposed using SVD,

HV UΣΦ . [16]

The spatial modes are extracted from the columns of Φ,

1 2 RΦ φ φ φ , [17]

where rank( )R V . The temporal coefficients are then calculated from a projection of the spatial modes onto the

original observations. Taking advantage of the fact that the pseudoinverse of an orthonormal matrix is its Hermitian

transpose, this can be written as

H x Φ V Φ V , [18]

where x is a matrix of temporal coefficients. This is an important fact in terms of computational efficiency since it

means that the projection of wavefronts onto POD modes is a simple matrix multiplication. The POD modes are

ranked by the importance of their contribution to the overall energy of the system. Quite often, the POD modes

converge quickly to give a good low-dimensional model. Additionally, in the case of naturally occurring fluid flow,

low order modes typically exhibit smooth behavior. If it is assumed that the POD modes do not change, then each

wavefront can be decomposed as follows,

1

( )N

k n n

n

x k

v Φ , [19]

where nΦ are the time-invariant POD modes, xn(k) are the coefficients at timestep k, and N is the desired truncation

dimension. Additionally, we assume that M snapshots are sufficient to approximate the next wavefront in the

sequence: we refer to M as the embedding dimension23

. In this case, the NAR problem becomes a function of the

modal coefficients, as shown,

1 1 1 1 1, ,...,N

k k k k M k v Φ g x x x ε , [20]

where 1

N LxNΦ R are the truncated set of POD modes and : NxM Ng R R is the nonlinear prediction function.

Thus, it is necessary to use some method to estimate the function g.

We recognize at this point that every continuous, differentiable nonlinear function can be linearized at any point

along a system’s trajectory. For the practical application of latency compensation, we assume that the flow dynamics

are approximately linear over the prediction horizon. This assumption allows us to write g as a matrix function,

1

1 1 1

1

, ,...,

k

k

k k k k M k

k M

x

xx g x x x A ε

x

, [21]

where A is referred to as the prediction matrix. The prediction matrix can be calculated by setting up the

overconstrained linear problem using a set of training input and output matrices X and Y,

1 2

2 3 1

1 2

1 1

...

L M

L M

M M L

M M L

x x x

x x xY x x x ε AX ε A ε

x x x

, [22]

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and then solving for the A matrix by calculating the pseudoinverse of the right-hand data matrix or similar

technique. The resulting matrix will be optimal in the least-squares sense for the POD coefficient prediction; that is,

it is desired to solve the convex optimization problem,

1

1

1

arg min

kL M

k

k

k M

A

x

x A

x

. [23]

The pseudoinverse method to solve this problem is,

1

1H H H

Y AX AUΣV A Y UΣV YU Σ V , [24]

or equivalently using ordinary least squares,

1

H H

A YX XX. [25]

This method is known as Vector Auto-Regression24

(VAR), and combined with the POD model reduction we

refer to it as POD-VAR. It is noteworthy that this solution coincides with the maximum likelihood estimate if the

error terms are normally distributed. Additionally, it generates a stable predictor if the modeled process is stationary

with time-invariant covariance. Writing the prediction problem such the prediction matrix is square,

1

1 1

2 1 1

k k k

k k k

aug

k M k M k M

1 2 Mx x xA A A

x x xI 0 0A

x x x0 I 0

[26]

a stability analysis for this autonomous process simply requires that,

sup : 0 1z z augC I A . [27]

VAR estimators will always meet this condition for natural stationary processes, such as mean-removed, tip/tilt-

removed aero-optic disturbances resulting from an unchanging flow condition.

IV. Stability of the Closed-Loop System with Delay Uncertainty

The stability problem of Section 2 becomes more difficult to address if there is a mismatch between the

prediction horizon and the true latency in the system; that is, if 0t . This mismatch will always be present in a

real system – while digital systems can be very good at measuring this sort of latency and matching it closely, the

effect of digital electronics jitter and some small uncertainty cannot be ignored. The remainder of this section will

address the effect of this mismatch.

A general framework for investigating the effect of delay uncertainty is described in Chapter 11 of Michiels and

Niculescu25

. The key results of their analysis will be applied here. First, make the simplifying assumptions that

s G I and 1 2 1s T s ts s s e e H H H , where s H I . In simple terms, the assumptions are that the

deformable mirror responds instantly while the wavefront sensor responds with some pure time delay and some

uncertainty in that delay. The resulting transfer function becomes,

1

1s T s ts s s s s e e

T C I C C P [28]

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The integrator compensator will have the form, ss

C I , where β is the feedback gain. The transfer function

reduces to,

1

1 1s T s tss s e e

T I P . [29]

Hence, the characteristic equation that must be studied for stability is,

det 1 1 0s T s tss e e

I P . [30]

Now, choose a proper matrix fractional description 1

P Ps s s

P N D such that NP(s) and DP(s) are co-

prime polynomial matrices and DP(s) is Hurwitz (allowing roots on the imaginary axis). This fractional matrix

description will exist if the system is both controllable and observable26

. The stability of this characteristic equation

can be treated using the theory of delay differential equations25

. If the above characteristic equation is written in the

following form,

det / 1 0s T s t

P Ps s s e e D N , [31]

then its stability can be analyzed by studying the neutral equation25

,

1 1

0 0 0 0( ) 0t t T t t T x N D x N D x , [32]

where N0 and D0 are the column-reduced coefficient matrices of P sN and /P s s D , respectively. The

details of this calculation are not important in this case for reasons that will be shortly addressed, but refer to

Michiels and Niculescu25

for additional information.

This neutral equation is strongly stable if25

,

1 21 1

0 0 0 0sup : 0,2 , 1,2 1i i

ir e e i

N D N D .

The stability condition with respect to infinitesimal delay uncertainty may be equivalently stated as,

1

0 0

1

2r

N D . [33]

However, because 1

/P Ps s s

N D is a strictly proper transfer function, the spectral radius above is 0 (as

discussed in Michiels and Niculescu25

) and practical stability is preserved for infinitesimal delay mismatches.

In summary, the delay compensation system outlined in this section is practically stable for infinitesimal delay

mismatch if:

1. P(s) is proper.

2. P(s) is open-loop stable.

3. P(s) is observable and controllable.

These conditions will be validated later in this section.

While Eq. 26 is convenient for practical estimation, it is helpful to put the predictor into state-space form for

application and further analysis. One such state space model is,

1k k k

k k k

P P

P P

x A x B u

y C x D u. [34]

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In this model, the input vector uk is the most recent measurement of POD coefficients, the output vector yk is the

POD coefficient prediction at timestep k, and xk are a set of “stored” POD coefficient vectors that are used in the

prediction. Decompose the prediction matrix, N NMA R as, 1 MA A A where

N N

k

A R . The state

space matrices themselves have dimension ( 1) ( 1)N M N M PA R ,

( 1)N M N PB R , ( 1)N N M PC R , and

N NPD R and are chosen to be,

N N N N N N

N N N N N N

N N N N N N N N

N N N N

N N N N N N N N

P

0 0 0

I 0 0

A 0 I 0 0

0 0

0 0 I 0

, [35]

N N

N N

N N

P

I

0B

0

, [36]

2 3 MPC A A A , [37]

and,

1PD A . [38]

This form of the predictor allows a convenient description of the transfer function matrix in discrete form, P(z),

as follows:

1

( )z z

P P P PP C I A B D . [39]

As discussed previously in this section, this system must be controllable and observable for stability in the

presence of delay uncertainty. Otherwise, internally-cancelling modes will exist in P(z) that appear in both the poles

and zeros, and hence the resulting fractional decomposition P(z) = N(z)D(z)-1

will not be coprime. Controllability is

tested with the following condition26

,

( 1) 1rank[ ] ( 1)N M N M P P P P PA A B A B , [40]

and observability is tested with,

1 1

rank 1

N M

N M

P

P P

P P

C

C A

C A

. [41]

For this system,

( 1)

kN N

k

N N

M k N N

P P

0

A B I

0

for k > 2, [42]

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so controllability is guaranteed analytically through the structure of the system matrices (in particular, 2rank[ ] ( 1)M N M P P P P PA A B A B ). The observability matrix will also be full-rank via a similar

argument if CP is full-rank. Since CP is estimated from a natural process, Cp will in general be full-rank.

Now it is desired to assess the stability of this discrete predictor as it interacts with a real adaptive optics

system. The approach taken here will use conformal mapping to transform discrete space to continuous space in

such a manner that conventional s-plane analysis tools can be applied. A better (but far more complex) approach

would be to use the theory of sampled data systems to accurately model the interaction of the digital predictor with

an analog system. Future research should investigate this.

The exact mapping between the z-plane and the s-plane is Tsz e . The difficulty with this mapping is that

conventional analysis tools cannot be used with non-polynomial expressions. There are several useful

approximations to this mapping that could be chosen to map between the s-plane and the z-plane. One

approximation that is closely related to this is the Padé approximation27

,

2

1 0

2

1 0

1

1

nTs n

n

n

k s k s k se

k s k s k s

, [43]

which is used to approximate time delay in continuous systems. This approximation is derived through a Taylor

series expansion of the exponential.

A similar approximation is the Tustin transformation (or bilinear transformation), defined as28

,

1 / 2

1 / 2

Ts Tsz e

Ts

. [44]

An important characteristic of this mapping is that the stability boundary (the imaginary axis in the s-plane and

the unit circle in the z-plane) is mapped exactly.

Minimum-phase systems in the s-plane are also mapped to minimum phase systems in the z-plane, however the

converse is not guaranteed. In fact, time delay systems are in general non-minimum phase. Minimum phase is not a

requirement for this analysis, but it is useful to understand the limitations of this approximation.

Finally, some frequencies will be significantly distorted as a consequence of this transformation – particularly

those near the Nyquist frequency, 2/T. The dominant disturbance frequencies addressed in this research are between

800 and 2000 Hz, while the control loop samping frequency is 25 kHz. Since the primary disturbances occur at an

order of magnitude lower than the sampling frequency, high-frequency distortion will amplify noise but should not

significantly impact overall predictor response.

Using the bilinear transformation, P(z) can be approximated in continuous space as P(s). Because the stability

boundary is preserved through the transformation and P(z) is stable, P(s) is also stable (but not minimum phase).

Because P(s) is controllable and observable, there exists a proper and stable matrix fractional description P(s) =

NP(s)DP(s)-1

such that NP(s) and DP(s) are co-prime and DP(s) is Hurwitz. Therefore, the stability conditions outlined

in this section are valid for this predictor and the system is stable in both the nominal case and in the presence of

infinitesimal delay uncertainty.

V. Simulation Results

Our simulation software is written in C++ using LAPACK for linear algebra routines. The wavefront

disturbance data was taken from Notre Dame’s Airborne Aero-Optics Laboratory (AAOL) flight test program. Each

test condition consists of 15,000 total wavefront snapshots. The first 3,000 wavefront snapshots are used to calculate

POD modes and train the coefficient prediction matrix. This value was selected as this is sufficient many snapshots

for well-converged POD modes. Then, the next 12,000 wavefronts are used to evaluate controller performance

independently. Thus, a separate predictive controller is trained for each flow condition.

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An important selection during the training phase of

the controller is to determine a priori the optimal

number of POD modes to include in the controller. The

lowest-frequency (and thus, more predictable) content in

the disturbances tends to occur in the low-order POD

modes. As the prediction window increases, the higher

frequency content becomes harder to predict and thus

diminishing returns are observed. An optimization study

was performed to determine the optimal number of POD

modes to include for varying amounts of latency. The

results are shown in Figure 8. There is not much

practical benefit to including more than about 20 modes

for if larger amounts of latency are anticipated in the

controller. For small amounts of latency, improvements

are observed up to 64 modes. Since there is no

significant performance penalty for using a larger

number of modes, we will select N=16 (to match our

prior work) and N=64. A practical application may

require fewer modes to reduce computational cost.

We may now expand the results of the previous optimization study to further investigate how the controller

behaves. The full aperture-averaged disturbance rejection results for N=16, N=64 and M=4 are shown in Figure 9.

Similar to the results obtained in our previous work7 (labeled as “POD-ANN”), we observe better disturbance

rejection in the highly-separated flow regime around a viewing angle of 120 deg. However, we now observe

disturbance rejection near -6.5 dB for one timestep of latency with a viewing angle beyond 120 deg., whereas our

previous controller achieved roughly -4dB. For a large amount of latency, we observe approximately -3dB of

disturbance rejection while the neural network controller achieved roughly -2dB.

Figure 9: Performance comparison of the linear controller with the prior neural network controller for

several amounts of latency.

Figure 8: Ensembled mean disturbance

rejection of all viewing angles for latencies

between 1 to 5 timesteps (ascending order).

0 10 20 30 40 50 60 70-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

Num Modes (N)

Reje

ction (

dB

)

Ensembled mean rejection for all viewing angles

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That a linear matrix predictor performs better than a nonlinear predictor in this highly-separated flow regime is

consistent with expectations, since a separated shear flow is dominated by linear dynamics. We may also investigate

the structure of the prediction matrices as shown in Figure 10 to gain a better understanding of the dynamic

dependencies between POD modes. Naturally, the strongest dependence of a prediction (timestep k+1) is on the

immediately preceding coefficient (timestep k) as indicated by the diagonals. The next-strongest dependence is on

the diagonals of the previous coefficient (timestep k-1). The reader will observe that this is approximately a discrete

solution of the continuous differential equation, 0ax x . Higher-order derivatives (up to the 3rd

-order) are also

automatically approximated in the matrix as evidenced by the diagonal terms. The influence matrices also indicate

that strong dynamic coupling is present in the fluid flow. This is an important feature of the predictor. Capturing the

coupling between each mode improves prediction quality. This is part of the reason that simple single-input-single-

output conventional PID controllers do not perform well in aero-optic applications. An additional observation is that

the prediction matrix is primarily lower-triangular, meaning that higher-order modes tend to depend on lower-order

modes. A possible modification to the predictive controller would be to force the prediction matrix to be purely

lower-triangular, which would cut the computational cost of the prediction roughly by half.

Figure 10: Influence matrices at various viewing angles.

VI. Robustness and Sensitivity

We may investigate the robustness of the controller by training the predictor on a data set and establishing

baseline performance as per the previous section, and then testing the controller once again using a data set from a

perturbed flight condition.

In order to make this possible, we make the assumption that the flow direction is consistent with respect to the

plane of the optical aperture. We use the spectral four-beam Malley probe technique29

to estimate the mean flow

direction and then rotate the wavefronts with bilinear interpolation such that the flow direction is in a consistent

direction. This step is important because low-order POD modes (i.e., the most energetic and predictable modes) tend

to convey information about the convective nature of coherent structures while higher-order modes contain

information about their evolution. If the flow angle changes substantially, the modes would attempt to predict an

incorrect direction for the convective portions of the disturbances. While the four-beam Malley technique is

computationally expensive, the aperture-plane flow angle could also be estimated geometrically from azimuth and

elevation.

For initial robustness test, we trained the predictive controller on 3,000 wavefront snapshots using flight test

data at α = 120.7 deg, β = 72.4 deg, M = 0.51, and ρ = .78 kg/m3 (14kft altitude). The controller was then evaluated

using the following 12,000 wavefront snapshots to establish baseline performance. We then used exactly the same

controller on a different set of wavefronts from a different flight test. The “robustness” test was performed on a data

set with α = 129.3 deg, β = 65.3 deg, M = 0.50, and ρ = .86 kg/m3 (11.5kft altitude). This is a substantially perturbed

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flow condition from the training condition; however, it is still in the same highly-separated flow regime. The results

of the robustness test are shown in Figure 11 for varying amounts of feedback latency. There is essentially no

difference between the baseline and robust performance for small amounts of latency. For larger amounts of latency,

errors in the training condition build up and disturbance rejection quality drops by up to 7%.

Figure 11: Effect of perturbing the flow condition on controller performance for varying amounts of

latency.

We may further investigate what is really happening by comparing the POD modes of each flow condition. The

POD modes are “reordered” such that they are arranged by similarity rather than energetic contribution, as shown in

Figure 12. While some differences are certainly present, most of the modes from the original predictive controller

are still present in the perturbed flow condition. This observation gives insight into why the predictive controller

tends to perform well. Model reduction via POD not only reduces computational cost but also increases robustness

by reducing sensitivity to perturbed flow conditions.

Figure 12: POD mode comparison using mode matching. (Mode 1 comparison at the top-left, Mode 4

comparison at the top-right, Mode 32 comparison at the bottom-right.)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

0 2 4 6

Dis

turb

ance

Re

ject

ion

(%

)

Number Timesteps of Feedback Delay

Baseline Rejection

Robust Rejection

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We now evaluate controller sensitivity further for several combinations of baseline cases and perturbed flow

cases. The method for testing this is to train the predictive controller on a subset of wavefront data in a baseline

condition, evaluate the controller on the subsequent wavefronts from the baseline condition to establish baseline

disturbance rejection, and then evaluate the controller again using the wavefront disturbances from a perturbed flow

condition.

The sensitivity to each parameter can be determined by calculating the change in controller performance as a

function of the change in each parameter. Ideally, the sensitivity to each parameter could be measured by perturbing

one parameter per test case, and calculating the sensitivity directly. Since this data is not available, an indirect

method will be required.

The fully separated flow regime will be examined here since it is of primary interest for real applications. It is

assumed that the sensitivity is constant in this regime; i.e., that performance degradation will follow linearly with

parameter changes. For convenience, define the function f as the mean aperture-averaged RMS disturbance rejection

of the controller over L test wavefronts,

,

,1

1 L

n rms

n rmsn

yf

dL

. [45]

Then, define s as a vector describing the change in viewing angle and modified elevation angle,

s . [46]

The directional derivative of f in the direction of s will define the sensitivity of the controller to a perturbation in

that direction,

f f s

s

s

. [47]

Recognizing that the change in RMS disturbance rejection from the baseline to the perturbed condition is

approximately equal to the LHS times the “length” of the perturbation vector, the following equation can be written,

perturbed baselinef f f u s . [48]

The above quantity may be measured directly from experiment and simulation. Each test condition then gives a

new set of equations,

perturbed baseline

f ff f

, [49]

which in turn leads to the following system of equations for Ntest sensitivity tests,

1 1 1

2 22

test test

test

perturbed baseline

perturbed baseline

N Nperturbed baseline N

f ff

f f

f

f f

. [50]

The full set of test conditions used for this sensitivity analysis is listed in Table 1.

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Table 1: Sensitivity test conditions (flat window).

Condition # α (deg) β (deg) ρ (kg/m3) M (-)

1 126.4

52.0 0.85 0.51

2 129.3 65.3 0.86 0.50

3 135.7 51.1 0.87 0.51

4 141 59.7 0.85 0.51

The resulting estimated sensitivity parameters are shown in Table 2. An important result of this analysis is that

the POD-VAR control algorithm is nearly 4 times more sensitive to changes in modified elevation than to changes

in viewing angle. This is a consequence of the fact that POD modes do not significantly change with α since this

direction is closely aligned with the flow direction. Changes in β can result in encountering different parts of flow

structures such as horn vortices, necklace vortices, secondary vortices, etc.

Table 2: Sensitivity of mean disturbance rejection to perturbations in viewing and modified elevation angles.

∂f/∂α 0.10 (%/deg)

∂f/∂β 0.38 (%/deg)

This has practical importance for system design. If the POD-VAR predictors were to be trained and stored as a

function of α and β rather than dynamically updated in real-time, then it would be necessary to have roughly 4 times

finer resolution in the β lookup than in the α lookup.

On the other hand, if the POD-VAR predictor is updated in real-time in an outer loop, then these values can

establish the “drift” in controller performance as a function of the predictor update loop and turret slew rates.

VII. Conclusions

Latency in adaptive-optic control systems significantly limits controller performance in aero-optic applications

due to the high-frequency nature of disturbances. We have presented a modification to our previous neural network

controller that focuses on mitigating this limitation of adaptive-optic systems using flow prediction. The new linear

POD-VAR controller improves disturbance rejection from 35%-55% in the case of the neural network controller to

45%-75% over the same range of test conditions in simulation. While a nonlinear predictor may in general be more

accurate under ideal conditions, cumulative error in multistep prediction problems tends to build up more rapidly in

a nonlinear predictor while error buildup is not as rapid in the linear model. We have demonstrated good robustness

to perturbed flow conditions for the most optically-active case and discussed some of the physical reasons for this

characteristic. We have also shown that our controller is stable in both the nominal condition and in the presence of

delay uncertainty.

Future research should include realistic mirror and amplifier models, and re-evaluate the performance of the

controller under these conditions. The problem of varying amounts of latencies for each component in the feedback

loop does not affect the performance of the predictor in the nominal condition, but the stability analysis of this paper

should be expanded to address this more realistic case.

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