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Adaptive optics wavefront controllers are tractably synthesized using loopshaping and internal model control principles. Modal responses at low and high temporal frequencies can thus be specified distinctly using pairs of open-loop reconstructors. Theoretical examples and parametric simulations illustrate the stability, robustness, and flexibility of the method, which incorporates several existing techniques.
©2006 Optical Society of America
1. Introduction
Adaptive optics (AO) controllers that use zonal wavefront control with integral linear feedback are known to be susceptible to instability in unobservable modes such as piston, checkerboard, and even tilt (e.g., when tracking and higher-order correction loops use separate beacons). In multi-conjugate AO (MCAO) systems, many more such modes abound [1, 2]. Mere regularization of reconstructors away from unobservable modes does not preclude redundant integral control states and actuators from drifting in practice. In control-theoretic terms, such a control system lacks detectability [3] (i.e., open-loop stability of unobservable states) and thus violates a basic assumption of modern state-space control design methods.
A traditional solution to this problem applies a forgetting factor to make the integrator “leaky.” By the internal model principle, this is equivalent to assuming a finite aberration correlation time. Detectability is thus achieved, but the resulting lag compensator exhibits only finite loop gain at DC and thus less compensation of static aberrations.
Alternatively, detectability can be achieved through suppression of degenerate modes by simply structured inner feedback loops that retain both pure integral action and open-loop reconstructor design. For example, the Altair AO controller [4] explicitly feeds back unobservable modes. The “absolute form” of the Palomar minimum-variance compensator [5] also suppresses unobservable modes by feeding back actuator commands. There is also the “pseudo-open-loop control” (POLC) [6] technique, whose closed-loop stability and robustness have been demonstrated using both theory [7] and simulation [6, 8].
These inner-loop methods are similar to the earlier Blankinship reconstructor [9], in which a “modal feedback” gain matrix provides not only stable integral control but also a separation property with which the controller mimics the open-loop reconstructor’s modal response at all temporal frequencies [9]. This kind of reconstructor thus smoothes dynamic and static estimates equally and is robust to dynamic and static measurement errors equally. POLC, which resembles modal feedback with generalized filter dynamics [7], possesses the same separation property. Such behavior is not always optimal, though; indeed, one often needs to reconstruct static aberrations more aggressively than dynamic ones [10].
Here, expanding on an earlier brief paper [11], we develop a framework for linear wavefront controller design using more general internal model control (IMC) [12], by which feedback loops are shaped using internal models of actual and desired dynamics. Modal feedback, POLC, and unobservable-mode feedback are thus extended by a loopshaping technique in which a pair of open-loop reconstructors can be used to specify distinct modal behaviors at high and low temporal frequencies. Not only modal bandwidths (e.g., optimized with respect to noise) [13] but also steady-state modal responses are prescribed. This added flexibility is obtained with little of the computational burden and design complexity that full-scale optimal control techniques demand. Closed-loop stability and robustness properties are discussed, and theoretical examples and parametric simulations provide illustration.
2. Theoretical analysis and synthesis
We assume discrete-time signals represented by z-transform vectors and linear time-invariant systems represented by rational z-transform transfer function matrices [14]; analogous results exist for continuous-time signals and systems. Vectors and matrices are implicitly dimensioned; in MCAO, they are also block-structured. Signals such as measurements (e.g., from wavefront sensors), control actions (e.g., actuator commands), measurement error (e.g., noise), disturbances (e.g., aberrations), and residual (wavefront) errors are respectively denoted by y(z), u(z), v(z), w(z), and e(z). Input-output behaviors of actuators and sensors are respectively denoted by G(z) and H(z); the plant H(z)G(z) (i.e., the end-to-end process to be controlled) is assumed to be open-loop stable and have at least one frame delay. The control and inner-loop feedback filters are respectively denoted by C(z) and M(z). Reconstructor estimation and fitting matrices, including any slaving, are respectively denoted by E and F. The identity matrix is denoted by I. The operator diag() returns either a diagonal matrix with given entries or the diagonal entries of a given matrix.
The diagrams in Fig. 1 depict the feedback control system considered here, whose canonical open-loop transfer function relationships
are revealed by inspection. Given the effective compensator transfer function
as well as the closed-loop reconstructor transfer function R(z) = F Q(z)E, derivation of the canonical closed-loop transfer function relationships
is straightforward. The poles [3,14] of the closed-loop system, which for stability must all lie within the open unit disk, are the roots of the rational characteristic equation
Consider an example involving leaky integral control with gain k ∈ (0,1] and unit-delay plant dynamics with unobservable modes. Specifically, C(z) = (1-z -1)-1 kI, M(z) = z -1 mI, H(z)G(z) = z -1 HG, and HG is rank-deficient. In this case, the closed-loop poles are the eigenvalues of (1 - km)I - kEHGF; without integrator leak (m = 0) some poles equal one and are unstable, but with integrator leak (0 < m << 1) these are less than one and thus stabilized.
The diagram in Fig. 2 depicts an internal model control system for which M(z) = J(z) - E H 0(z)G 0(z)F. The prediction filter H 0(z)G 0(z) models the plant, and the loopshaping filter J(z) is a design variable. The closed-loop reconstructor and characteristic equation become
where Δ(z) = H(z)G(z) - H 0(z)G 0(z) represents model uncertainties such as misregistration and unmodeled phase lag. By the small gain principle of robust control [3], robustness to model uncertainty can be pursued by judiciously tuning E, F, and/or C(z) in order to avoid exciting the modes and dynamics of Δ(z).
The loopshaping filter J(z) is assumed to be stable, square, and minimum-phase (i.e., full-rank wherever |z| = 1), so that its modes are visible to the controller whether or not they are observable by the plant. One can specify values R 1 = limz→∞ R(z) and R 0 = limz→1 R(z) that the closed-loop reconstructor ought to satisfy via the z-transform initial value and final value theorems [14]. The matrix R 1 governs the one-step behavior at temporal frequencies well above the bandwidth, R 0 governs the steady-state behavior at temporal frequencies well below the bandwidth, and R(z) should stably interpolate between the two. Open-loop (e.g., minimum-variance) reconstruction techniques [15,16] can be used to generate R 1 and R 0. We illustrate this design method using the following simple examples that involve pure integral control with gain k ∈ (0,1] and no model error: C(z) = (1 - z -1)-1 kI and H 0(z)G 0(z) = H(z)G(z).
Consider the special case of Blankinship modal feedback [9] or POLC [5], in which J(z) = z -1 I. (If F and E comprise a pseudo-inverse reconstructor, then this simply feeds back unobservable modes [4].) In this case, the separability of the modal and temporal-frequency responses of the closed-loop reconstructor
is clear. Also, the closed-loop poles all lie at z = 1 - k and are thus stable for all k ∈ (0,1].
Consider also the general case in which J(z) = z -1 J, where one-step and steady-state estimation matrices are related by E 1 = JE 0 and the fitting matrix is F. In this case,
has one-step response R 1 = FE 1 and steady-state response R 0 = FE 0, and the closed-loop poles are the eigenvalues of I - kJ. In particular, suppose J = A 0 for symmetric positive-definite matrices A 0 and A 1; if A 1 ≥ A 0, then the eigenvalues of I - kJ lie in the real interval [1-k,1), hence the closed-loop system is stable for all k ∈ (0,1]. These sufficient conditions are satisfied by any pair of minimum-variance estimators E 0 = B and E 1 = B that differ only in that the one-step estimator E 1 is regularized more than the steady-state estimator E 0 is.
3. Theoretical illustration and comparisons
The following contrived example illustratively compares the closed-loop behavior of leaky integration, POLC or Blankinship IMC, and loopshaped IMC schemes. For ease of presentation, we assume continuous-time dynamics without transport delays. Let G(s) = G = I 4×4 and H(s) = H = diag(1,0.1,0.01,0) define a 4-DOF plant having strong, moderate, weak, and null (i.e., unobservable) modes. Pseudoinverse, mildly regularized, and strongly regularized open-loop estimators are respectively denoted by E -1 = diag(1,10,100,0), E 0 = diag(1,10,10,0), and E 1 = diag(1,3,10,0). These estimators reflect design goals by which E 1 (which attenuates the controller gain for both medium and weak modes) estimates “fast” turbulent aberrations with robustness to noise, while E 0 (which attenuates the controller gain for only weak modes) estimates “slow” static aberrations with robustness to misregistration and other static modeling errors. Several stabilizing controller designs are considered:
- leaky integration, pseudoinverse estimator E = E -1, and leak factor m = 0.002
- loopshaped IMC, estimator E = E 1, and loop shaper J = diag(1,0.3,1,1)
- leaky integration, mildly regularized estimator E = E 0, and leak factor m = 0.002
- POLC or Blankinship IMC, mildly regularized estimator E = E 0, and loop shaper J = I 4×4
- leaky integration, strongly regularized estimator E = E 1, and leak factor m = 0.002
- POLC or Blankinship IMC, mildly regularized estimator E = E 1, and loop shaper J = I 4×4
Fig. 3. Frequency responses of noise rejection by pseudoinverse leaky-integral (upper left), regularized loopshaped-IMC (upper right), mildly and strongly regularized leaky-integral (middle and lower left), and mildly and strongly regularized POLC/Blankinship-IMC (middle and lower right) controllers for various modes.
Given the trivial fitter F = I 4×4 along with integral control with arbitrary gain ω, the Laplace-transform transfer function of the effective compensator Q(s) can be expressed as
for the leaky integration schemes and
for the IMC schemes. Temporal frequency responses for rejection of measurement error (e.g., noise) and wavefront disturbances are then R(j2πf) = Q(j2πf)E and S(j2πf) = I - R(j2πf)H, respectively. These are plotted vs. the bandwidth-normalized frequency 2πf/ω in Fig. 3 and Fig. 4 (respectively). Note that the null mode’s reconstruction gain is zero in all cases.
The graphs in Fig. 3 and Fig. 4 illustrate how each control loop is shaped in pursuit of the stated design goals. For example, the nominal controller with pseudoinverse estimation and leaky integration behaves as expected; the non-degenerate modes have equal bandwidths, the disturbance gains level off at low frequencies, and the noise gains are uniformly proportional to their estimation gains. The regularized controllers depart from this response variously.
Fig. 4. Frequency responses of disturbance rejection by pseudoinverse leaky-integral (upper left), regularized loopshaped-IMC (upper right), mildly and strongly regularized leaky-integral (middle and lower left), and mildly and strongly regularized POLC/Blankinship-IMC (middle and lower right) controllers for various modes.
With regularized leaky integration, the bandwidths/poles scale down, the noise gains scale down at high frequency, and the low-frequency disturbance gains scale up, all according to the regularization of E. However, at low frequencies the reconstructor response necessarily resembles that of the pseudoinverse estimator. If E = E 1, this scheme performs as desired at high frequencies but is too aggressive at low frequencies. If E = E 0, this scheme is too aggressive at both high and low frequencies.
With regularized POLC/Blankinship IMC, the bandwidths/poles are unchanged and the noise gains necessarily scale down uniformly vs. frequency (per the separation property). However, the low-frequency disturbance rejection is necessarily degraded by the attenuated reconstructor response. If E = E 1, this scheme performs as desired at high frequencies but is too conservative at low frequencies. If E = E 0, this scheme performs as desired at low frequencies but is too aggressive at high frequencies.
With regularized loopshaped IMC, the distinct design goals expressed by E 0 and E 1 are both successfully met. For example, the controller responds less to the medium mode at high frequencies yet corrects it well at low frequencies, while the response to the weak mode is attenuated at all frequencies. The bandwidths/poles assume whatever values are needed to be consistent with the desired gains. Thus, only loopshaped IMC is flexible enough to implement the desired responses at both low and high temporal frequencies.
Fig. 5. Frequency responses of numerical error rejection by pseudoinverse leaky-integral (upper left), regularized loopshaped-IMC (upper right), mildly and strongly regularized leaky-integral (middle and lower left), and mildly and strongly regularized POLC/Blankinship-IMC (middle and lower right) controllers for various modes.
Post-estimator numerical errors such as finite-precision roundoff can cause unobservable modes to drift in traditional AO control loops. Temporal frequency responses for rejection of such errors are given by Q(j2πf), are plotted vs. the normalized frequency 2πf/ω in Fig. 5, and variously illustrate the suppression of unobservable modes. For the leaky integration schemes, the effective rejection bandwidth is relatively low (mω = 0.002ω), and the closed-loop error gain at steady state is relatively large (1/m = 500). For the POLC/Blankinship IMC and loopshaped IMC schemes, on the other hand, the rejection bandwidth is higher (J 44 ω = ω), and the closed-loop error gain at steady state is lower (1/J 44 = 1).
4. Simulation exercise
The following scalable (though by no means universal) Monte Carlo simulation exercise demonstrates a 2-DOF technique for tuning loopshaped IMC wavefront controllers. Consider a basic AO system with aperture diameter D = 8 m, a 216-subaperture Hartmann wavefront sensor (WFS), a 257-actuator continuous deformable mirror (DM), and Fried geometry with subaperture & actuator spacing d = 0.5 m. The WFS ideally measures analytical G-tilt, and the DM has spline-based influence functions.
Wavefront controllers for this system are structured as follows. The control law uses unit-gain discrete-time integral compensation, whose nominal bandwidth is 1/2π times the frame rate. The actuator model G(z) = z -1 G includes the trivial double-pass matrix G = 2I 257×257, and the fitting matrix F = ½I 257×257 is equally trivial. The sensor model H(z) consists of the 432×257 interaction matrix H. Desired behaviors at high and low temporal frequencies are represented by the one-step and steady-state open-loop estimation matrices
(respectively) using the Wiener form for minimum-variance reconstructors [15,16], where the weighting matrix W and regularization matrix D are positive semi-definite. The IMC estimator and loop shaper are thus E = E 1 and J(z) = z -1 J (respectively), where
The regularization style (as represented by D) that is chosen here smoothes the wavefront estimation response by penalizing various discretized spatial second derivatives as well as unobservable modes. The positive scalar tuning parameters q 1 and q 0 quantify the amount of regularization in each estimator; indeed, a weighted pseudoinverse reconstructor is approximated if q 1 and q 0 are very small. Also, D is conveniently normalized according to
so that q 1 > 1 and q 0 > 1 can be called “large” while q 1 < 1 and q 0 < 1 can be called “small.” As shown previously, closed-loop stability is ensured if q 1 ≥ q 0.
The control system’s response to (and inherent trade-offs between) measurement error, static aberrations, and dynamic aberrations can be parametrically characterized using Monte Carlo time-domain AO simulations with 8X-oversampled subapertures. Measurement error is generated by white WFS noise as well as uncorrelated random WFS biases. Static (e.g., beam-train) aberrations are generated by static Kolmogorov turbulence realizations; this choice of turbulence-like spatial statistics is arbitrary but reasonable nevertheless. Dynamic aberrations are generated by a layer of Kolmogorov turbulence that evolves according to Taylor frozen flow. The system pupil, DM, WFS, and aberrations are assumed to lie in conjugate planes so that neither anisoplanatism nor scintillation needs to be considered.
Table 1 shows the dependence on q 0 of the residual wavefront error due to static aberration with r 0 = 1 m at wavelength λ = 0.5 μm. Not surprisingly, increasing the steady-state regularization strength q 0 smoothes the response and increases the error. This fitting error scales with respect to d/r 0 in the usual sense [17].
Table 2 shows the dependence on q 0 of the residual wavefront error due to 1.0-μrad (1-σ) uncorrelated random WFS biases. Not surprisingly, increasing the steady-state regularization q 0 smoothes the response and decreases the error. The trend is also qualitatively indicative of the controller’s first-order sensitivity to static model uncertainties. This bias-induced error scales with respect to the 1-σ standard deviation in the usual sense [17].
Table 3 shows the dependence on (q 1,q 0) of the residual wavefront error due to dynamic turbulence with r 0 = 1 m at wavelength λ = 0.5 μm, given the sampling period T = 0.02 sec and wind speed v = 25 m/sec. Not surprisingly, increasing the one-step regularization strength q 1 and/or the steady-state regularization strength q 0 smoothes the response and generally increases the error. These servo-lag and fitting error components should scale with respect to the wind speed, loop gain, and frame rate as well as d/r 0 in the usual sense [17].
Table 4 shows the dependence on (q 1,q 0) of the wavefront error due to 1.0-μrad (1-σ) WFS noise. Not surprisingly, increasing the one-step regularization strength q 1 smoothes the response and decreases the error; moreover, increasing the steady-state regularization strength q 0 also generally decreases the error, albeit more weakly. This noise-induced error scales with respect to the 1-σ standard deviation in the usual sense [17].
Table 1. Monte Carlo 1-σ RMS wavefront error (μm) due to static aberrations vs. q 1 and q 0
Table 2. Monte Carlo 1-σ RMS wavefront error (μm) due to random WFS biases vs. q 1 and q 0
Table 3. Monte Carlo 1-σ RMS wavefront error (μm) due to dynamic aberrations vs. q 1 and q 0
Table 4. Monte Carlo 1-σ RMS wavefront error (μm) due to WFS noise vs. q 1 and q 0
Although this particular exercise is only one of many possible instances, it shows that designing with two open-loop estimators clearly affords more flexibility than with only one. The one-step and steady-state estimators can be jointly tuned to systematically balance static-fitting, dynamic-fitting, servo-lag, bias-induced, and noise-induced errors without resorting to formally optimal (e.g., Kalman filter-based) control synthesis. Practical implementation of a comparable formally optimal controller would anyway require up to twice as many states (for both static and dynamic aberrations) and similar a posteriori tuning.
In practice, model uncertainties such as misregistration, calibration errors, and cone nonlinearities must also be considered. The particular approach depends on the system architecture and open-loop reconstructor parameterization, but generally speaking these uncertainties each define components of the aforementioned uncertainty operator Δ(z), whose dominant modes should be reflected in penalties such as the matrix D. Optimization of parameters such as q 0 and q 1 (e.g., adjusting the steady-state responses of regularized modes using q 0 and adjusting their relative bandwidths [13,15,16] using q 1) can be guided by further Monte Carlo simulation as well as robust control metrics such as H 2 norms, H ∞ norms, and structured singular values [3].
5. Summary
Wavefront controllers have been tractably synthesized using loopshaping and internal model control principles. Modal responses at low and high temporal frequencies can thus be specified distinctly using pairs of open-loop reconstructors. Several existing techniques, such as modal feedback and pseudo-open-loop control, are incorporated by this method. Designing for stability and robustness has been discussed, and the available flexibility has been illustrated using theoretical examples and parametric AO simulations.
Acknowledgments
The author wishes to thank the anonymous reviewers for their corrections and suggestions. This work was supported by Lockheed Martin independent research and development funds.
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