A Robust Modification of a Predictive Adaptive-Optic Control Method for Aero-Optics

Transcription

1 AIAA Aviation June 2016, Washington, D.C. 47th AIAA Plasmadynamics and Lasers Conference AIAA A Robust Modification of a Predictive Adaptive-Optic Control Method for Aero-Optics Robert Burns 1, Eric Jumper 2, Stanislav Gordeyev 3 University of Notre Dame, Notre Dame, IN, 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/m 3 ) 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. 1 Copyright 2016 by William Robert Burns, Eric Jumper, Stanislav Gordeyev. Published by the, Inc., with permission.

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/m 2 ) I 0 = unaberrated beam intensity (W/m 2 ) K GD = Gladstone Dale constant (m 3 /kg) M = Mach number (-) or embedding dimension (-) N d = 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 problem 1. Near-field Oturbulence 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 shown 2 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 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 relationship 3, n( x, y, z) 1 KGD( ) ( x, y, z), where n is the index of refraction, λ is the wavelength, and K GD is a constant. This relationship can be used to compute the exact aberration along a path by first computing the optical path length Figure 1: Optically-aberrating flow over a turret 1. 2

3 z 1 from some point in the aperture, OPL( x, y, t) n( x, y, z, t) dz, and then subtracting the aperture-averaged mean to compute the optical path difference, 0 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 system 1. 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 approximation 4,5,6, shown in Eq OPDrms ( t) It () S() t e. [2] I () t 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, db OPD rms, corrected 10log. [3] 10 OPDrms, uncorrected 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 work 7. 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 Laboratory 8 (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. 3

4 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 expressed 8 in terms of azimuth and elevation as shown in Figure 4, cos cos( )cos( ), and [4] 1 Az El 1 tan( El) tan. [5] sin( Az) OPD It has been demonstrated 8 that aero-optic aberrations can be normalized using the following equation, norm OPDrms, [6] 2 M / D 0 where M is the freestream Mach number, ρ/ρ 0 is the ratio of freestream density to sea-level density, and D is the turret diameter. Figure 5 shows OPD norm 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 Figure 4: Definition of α and β in terms of azimuth and elevation 7 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 cylinder 10 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 jets 11 have been used to reduce the optical aberrations present in the wake of a hemisphere. Later, this group 12 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 al 10 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 detail 13 using Particle Image Velocimetry (PIV) and Schlieren imagery. 4

5 Figure 5: Collapse of OPD rms as a function of viewing angle from AAOL flight tests. 8 A modification of proper orthogonal decomposition 14 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 research 15 to improve flow control using these new methods. One method 16 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 programming 17 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: fd St, [7] V 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 delay 18 in a highspeed adaptive optic system. 5

6 This latency problem has previously been addressed using various prediction methods such as dynamic mode decomposition 19, receding-horizon adaptive control 20, and simple frozen-flow advection prediction 21. We recently developed a method 7 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 opticallyactive 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 work 7. 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 H 1, the lagged mirror sensor by H 2, and the unlagged mirror sensor by H 3. 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 H 3. 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 details 7 ). s H 1 may be decomposed as T t H H e, where ΔT is the cumulative latency in the feedback loop, δt is the 1 1 uncertainty in this latency, and H is the latency-free sensor. Practically, the lagged and unlagged deformable 1 s T mirror sensors may be decomposed as H2 H3e since H 2 is simply a digitally-delayed copy of H 3. An important design goal is to match the latency between the lagged deformable mirror sensor and the wavefront sensor. The latency in H 2 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. 6 Figure 6: Conventional AO loop augmented by wavefront predictor 1.

7 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, C sgs y s. [8] r 1 C s G s P s H s C s G s P s H s C s G s H s The rationale for matching the delays between H 2 and H 1 is now clear: in the absence of uncertainty, the complex dynamics of the predictor vanish and the closed loop transfer function becomes simply, C sg s y s, [9] r 1 C s G s H s 3 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. T s Now, consider the multidimensional case. Define the transfer function matrix T(s), y1 y1 y1 r1 r2 rn y2 y2 y2 r1 r2 r N yn yn yn r1 r2 rn s s s s s s s s s, [10] which in the multidimensional case can be solved for as follows if the latency uncertainty is zero, s s s s s s 1 T C G I C G H 3. [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, 7

8 S s y y y d1 d2 d y y y d1 d2 d y y y d1 d2 d s s s s s s s s s N N N becomes (in the absence of delay uncertainty), N N N, [12] S s I C s G s P s H s C s G s H s I C s G s H s. [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 predictor 7 and then discuss the necessary modifications. We assume that we can approximate the evolution of an aero-optical wavefront v k by some discrete function f that depends on prior wavefronts v k-1 v k-m+1, with some amount of truncation error denoted by ε k+1. k1 k k1 km 1 k1 v f v, v,..., v ε [14] A wavefront predictor should approximate f such that the mean-square of the residual error is minimized, L E ε, k1 k 2 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 v k+1 =Av k alone would contain in the worst case between 10 5 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 Decomposition 22, 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 N V v v v. [15] 8 1

9 The data matrix, V, is decomposed using SVD, H V UΣΦ. [16] The spatial modes are extracted from the columns of Φ, 1 2 R Φ φ φ φ, [17] where R rank( 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, N v x ( k) Φ, [19] k n n n1 where Φ n are the time-invariant POD modes, x n (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 dimension 23. In this case, the NAR problem becomes a function of the modal coefficients, as shown, N k1 1 k k1 km 1 k1 v Φ g x, x,..., x ε, [20] N LxN Φ R are the truncated set of POD modes and : where 1 g R Thus, it is necessary to use some method to estimate the function g. NxM N R is the nonlinear prediction function. 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, xk,,..., x x g x x x A ε xkm1 k 1 k 1 k k 1 k M 1 k, [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, x1 x2 xlm 2 3 LM1 M 1 M 2... L x x x Y x x x ε AX ε A ε xm xm 1 xl 1, [22] 9

10 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, arg min A LM k1 x k 1 xk A x km1 The pseudoinverse method to solve this problem is,. [23] 1 1 H H H Y AX AUΣV A Y UΣV YU Σ V, [24] or equivalently using ordinary least squares, 1 A YX XX H H. [25] This method is known as Vector Auto-Regression 24 (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, xk 1 xk A1 A2 AM xk xk xk 1 I 0 0 xk 1 A aug x x 0 I 0 x k M 2 k M 1 km 1 a stability analysis for this autonomous process simply requires that, sup zc : zi A 0 1. [27] aug VAR estimators will always meet this condition for natural stationary processes, such as mean-removed, tip/tiltremoved 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 t 0. 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 Niculescu 25. The key results of their analysis will be applied here. First, make the simplifying assumptions that s s T s t s s s e e H s I. In simple terms, the assumptions are that the G I and H1 H2 H 1, where 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, T 1 s C s s s s e e I C C P [28] st s t 1 [26] 10

11 The integrator compensator will have the form, reduces to, C s I, where β is the feedback gain. The transfer function s 1 s 1 s T s t T s I P s e e 1. [29] Hence, the characteristic equation that must be studied for stability is, s st s t det I1 P se e 1 0. [30] Now, choose a proper matrix fractional description 1 P s N s D s such that N P (s) and D P (s) are coprime polynomial matrices and D P (s) is Hurwitz (allowing roots on the imaginary axis). This fractional matrix description will exist if the system is both controllable and observable 26. The stability of this characteristic equation can be treated using the theory of delay differential equations 25. If the above characteristic equation is written in the following form, st s t s s Pse e det D / N 1 0, [31] P then its stability can be analyzed by studying the neutral equation 25, 1 x( t) N 1 0D0 x t T t N0D0 x t T 0, [32] where N 0 and D 0 are the column-reduced coefficient matrices of N P s and D s P s /, respectively. The details of this calculation are not important in this case for reasons that will be shortly addressed, but refer to Michiels and Niculescu 25 for additional information. This neutral equation is strongly stable if 25, 1 i1 1 i i sup r N D e N D e : 0, 2, i 1, 2 1. r The stability condition with respect to infinitesimal delay uncertainty may be equivalently stated as, 1 1 ND 0 0. [33] 2 However, because s s s P N D 1 P P / is a strictly proper transfer function, the spectral radius above is 0 (as discussed in Michiels and Niculescu 25 ) 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, x A x B u k 1 P k P k yk CPxk DPu. [34] k P 11

12 In this model, the input vector u k is the most recent measurement of POD coefficients, the output vector y k is the POD coefficient prediction at timestep k, and x k are a set of stored POD coefficient vectors that are used in the N NM prediction. Decompose the prediction matrix, NN A A A where A R. The state space matrices themselves have dimension N N DP R and are chosen to be, I 0 0 A P 0 I 0 0 0NN 0NN 0 0 I 0 NN NN NN NN NN NN NN NN NN NN NN NN NN NN A R as, N ( M 1) N ( M 1) AP R, 1 M N ( M 1) N BP R,, [35] k N N ( M 1) CP R, and B P I 0 0 NN NN NN, [36] 2 3 M C A A A, [37] P and, D A. [38] P 1 This form of the predictor allows a convenient description of the transfer function matrix in discrete form, P(z), as follows: z 1 P() z C I A B D. [39] P P P P 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 condition 26, N( M1) 1 rank[ P P P P P] NM ( 1) A A B A B, [40] and observability is tested with, CP rank CA P P NM 1. [41] NM1 1 CA P P For this system, 0 k AP BP I 0 knn NN ( M k 1) NN for k > 2, [42] 12

13 so controllability is guaranteed analytically through the structure of the system matrices (in particular, M 2 rank[ AP APBP AP B P] NM ( 1) ). The observability matrix will also be full-rank via a similar argument if C P is full-rank. Since C P is estimated from a natural process, C p 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. Ts The exact mapping between the z-plane and the s-plane is z 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 approximation 27, Ts 1 e, [43] 1 2 n k1s k0s kns 2 n k1s k0s kns 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 as 28, Ts 1 Ts / 2 z e. [44] 1 Ts / 2 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) = N P (s)d P (s) -1 such that N P (s) and D P (s) are co-prime and D P (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. 13

14 Rejection (db) 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 Num Modes (N) Figure 8: Ensembled mean disturbance rejection of all viewing angles for latencies between 1 to 5 timesteps (ascending order). 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 work 7 (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. 14

15 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, ax x 0. Higher-order derivatives (up to the 3 rd -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-singleoutput 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 technique 29 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 = 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 = deg, β = 65.3 deg, M = 0.50, and ρ =.86 kg/m3 (11.5kft altitude). This is a substantially perturbed 15

16 Disturbance Rejection (%) 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%. 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Number Timesteps of Feedback Delay Baseline Rejection Robust Rejection 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.) 16

17 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 L yn, rms f L d. [45] n 1 n, rms 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, s s f f. [47] s 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, u f s f perturbed f. [48] baseline The above quantity may be measured directly from experiment and simulation. Each test condition then gives a new set of equations, f f fperturbed fbaseline, [49] which in turn leads to the following system of equations for N test sensitivity tests, fperturbed f baseline f fperturbed f baseline [50] f f Ntest Ntest perturbed f baseline N test The full set of test conditions used for this sensitivity analysis is listed in Table 1. 17

18 Table 1: Sensitivity test conditions (flat window). Condition # α (deg) β (deg) ρ (kg/m3) M (-) 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 f/ β 0.10 (%/deg) 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. References 1 Wang, M., Mani, A., and Gordeyev, S., Physics and Computation of Aero-Optics, Annual Review of Fluid Mechanics, vol. 44, Jan. 2012, pp Gordeyev, S., and Jumper, E., Fluid dynamics and aero-optics of turrets, Progress in Aerospace Sciences, vol. 46, Nov. 2010, pp Gladstone, J. H., and Dale, T. P., Researches on the Refraction, Dispersion, and Sensitiveness of Liquids, Philosophical Transactions of the Royal Society of London, vol. 153, 1863, pp Mahajan, V. N., Strehl ratio for primary aberrations: some analytical results for circular and annular pupils, JOSA, vol. 72, 1982, pp

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