We compare two integral control strategies: a âcentralizedâ one, where each .... rately control piston, tip and tilt (PTT) of each segment (i.e. considering that the ...
1
Integral control from distributed sensing: an Extremely Large Telescope case study Alain Sarlette, Member, IEEE, Christian Bastin, Martin Dimmler, Babak Sedghi, Toomas Erm, Bertrand Bauvir, and Rodolphe Sepulchre, Fellow, IEEE,
Abstract—This paper reports on issues arising when applying distributed control to a fully defined practical system, which is the segmented primary mirror of the European Extremely Large Telescope. The controller must stabilize and maintain 984 segments composing the mirror in a fixed nm-smooth surface. Measurements available for feedback are relative positions of neighboring segments, so the output is spatially localized coupling. We compare two integral control strategies: a “centralized” one, where each segment input is based on measurements taken all over the mirror; and a “distributed” one, where a segment’s input is based on measurements at neighboring segments only. The latter is tuned with LTSI, a spatial frequency domain method especially suited for distributed systems; the use of LTSI explicitly in a MIMO context (linked to the three degrees of freedom of each segment) is a novelty of the paper. We highlight that the localized system nature, due to local relative measurement, causes a fundamental robustness issue for integral, independently of the choice of distributed or centralized controller. We illustrate how integrator “leakage” improves robustness at the cost of performance, and tune the leakage according to our robustness analysis. Simulations on a full realistic model of telescope and disturbances, show that centralized and distributed controllers perform equally well in practice. Index Terms—Distributed system, Centralized control, Distributed control, MIMO systems, Output feedback, Robustness, Frequency domain synthesis, Mirrors, Telescopes, Active arrays
I. I NTRODUCTION
D
ISTRIBUTED SYSTEMS are composed of a large number of subsystems coupled through “local” interactions between “neighboring” subsystems. A centralized control architecture considers the system as a whole and each local control action is determined as a function of measurements or states all over the system. However, this approach can lead to prohibitively or unnecessarily complex controllers. The alternative new paradigm of distributed control ([1], [2], [3], [4], [5], [6], [7], [8]) designs controllers in order to maintain the distributed system architecture in closed-loop: each local input is allowed to depend on neighboring subsystems only. Note the important difference between distributed plant or system, which is given, and distributed controller, which A. Sarlette, C. Bastin and R. Sepulchre were with the Department of Electrical Engineering and Computer Science, University of Li`ege, Institut Montefiore (B28), 4000 Li`ege Sart-Tilman, Belgium. M. Dimmler, T. Erm, B. Sedghi and B. Bauvir are with the European Organization for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Strasse 2, D-85748 Garching bei M¨unchen, Germany. This paper presents research results of the Belgian network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Program, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. The first author was supported as an FNRS fellow (Belgian Fund for Scientific Research).
is an independent design choice. One of the few industrial implementations of distributed control is for paper machines (see [9]). The purpose of the present paper is to report on a comparison of centralized and distributed control for a fully specified distributed system application: the stabilization of the 984 segments composing the primary mirror of the European Extremely Large Telescope (E-ELT). In particular: 1) We highlight that poor rejection of long-range deformations is inherent to the distributed plant architecture; more precisely, it is induced by the measurement which gives local relative positions of the segments; we insist that this limitation is independent of the choice of centralized or distributed controller. 2) We compare a centralized control design using modal decomposition, and a distributed control design using the Linear Time-Invariant and Spatially-Invariant method (LTSI, see [1], [5], [10], [7]) based on spatial Fourier transforms. We simulate both controllers on a realistic model of full telescope and disturbances. This allows us to confidently assess that both controllers perform equally well in practice, despite the many simplifications made for distributed control. The present paper also illustrates • the tuning of “integrator leakage”, see [5], [7], as an essential parameter to optimize the robustness/performance tradeoff with integral control; • explicit consideration of the MIMO character, corresponding to the three degrees of freedom of segment motion, in the LTSI framework. Observations 1 and 2 support two current conjectures in the literature on distributed control. First, the measurement architecture (relative position sensing) determines properties of distributed systems in a crucial way; this has also recently been studied in [11] discussing implications of local coupling on coherence in lattices, [12], and is still under investigation. Second, for (some) distributed systems, a distributed controller will perform as well as a centralized one. In fact, we even observe that a centralized controller is “naturally spatially distributed”: the gain from measurement at sensor j to input at segment i quickly decreases with the distance between j and i. This goes along the lines of [1], [13], which argue that the distributed nature of a plant carries over to its optimal controllers. Several recent publications specifically address the control of segmented telescope mirrors. The authors of [14]
2
use numerical optimization toolboxes to investigate H2 optimal control for particular settings. They avoid the main issue that comes from relative sensing (i.e. outputs being local relative measures of the states) by assuming full state knowledge.While our work was under review, [15] has also considered relative sensing and the related robustness issue, carrying out a numerical µ-analysis. As a complement to these papers, we provide a semi-analytical investigation. We thereby want to give insight into fundamental factors governing limitations, advantages, disadvantages and “optimality” of distributed controllers and systems, that might carry over to other applications. The conclusions of our robustness analysis, given in more detail in [16], agree with the numerical results of [15]. The paper is organized as follows. Section II describes the E-ELT primary mirror system and the main disturbances to be countered with a stabilizing feedback controller. Section III analyzes how a main limitation is induced by the fact that measurements give relative positions of neighboring segments; this characterization of distributed system coupling is one of our main contributions. Sections IV and V respectively describe the centralized and distributed integral controllers. They cover general design, tuning, and some tutorial discussion (illustration of leakage in Section IV, MIMO adaptation of LTSI in Section V). Section VI adds implementation details and presents results of simulations on a realistic E-ELT model; the validation on a realistic model is a main contribution of the present work. Section VII concludes with a broad comparison of centralized vs. distributed control. Notation: R≥0 is the set of nonnegative real numbers. Unless otherwise specified, x(i) denotes component i of vector x and A(i, j) denotes component on row i, column j of matrix A. For temporal frequency domain we use uppercase letters e.g. x(t) becomes X(s). To emphasize spatial frequency domain we add ∗ , e.g. x∗ (t, ξ, λ) or X ∗ (s, ξ, λ). I is the identity matrix; when clear from the context, we sometimes write s instead of s I for s scalar. II. T HE E-ELT PRIMARY MIRROR SYSTEM The observing capabilities of a telescope are essentially determined by the diameter of its primary mirror. With current technology, one-piece mirrors meeting the surface accuracy required for optical observations are limited to ∼ 9 m diameters. Therefore larger mirrors must be built from several segments, whose positions are actively stabilized to achieve a smooth overall mirror surface.
Fig. 1.
tilt tip
ES measurement PACT locations ES locations
piston
Fig. 2. Segment actuation and sensing system. Position actuators (PACT) operate the segment’s piston, tip and tilt degrees of freedom. Edge sensors (ES) measure local relative displacement of the segments.
operational telescopes (8−10 m diameter). The E-ELT primary mirror (M1) is composed of 984 hexagonal segments of 0.7 m edge length, see Figure 1. Each segment is supported by 3 unidimensional position actuators which move — in first approximation, neglecting small local curvature of M1 — perpendicularly to the mirror surface. This allows to separately control piston, tip and tilt (PTT) of each segment (i.e. considering that the telescope points to the zenith, the vertical position of a segment and its rotation around horizontal axes, see Figure 2). Discontinuities in the mirror’s reflecting surface are accurately monitored by 5604 so-called Edge Sensor pairs, which measure the relative vertical displacement at two positions on adjacent segment edges (see Figure 2). The induced coupling of adjacent segments in the measurement values is a fundamental property giving rise to a distributed system. Explicitly, let •
•
•
A. System description The European Organization for Astronomical Research in the Southern Hemisphere (ESO) is currently planning the construction of the world’s largest telescope, called European Extremely Large Telescope (E-ELT, see [17], [18], [19], [20]. Note that other teams in the world are developing similar ELT projects). Its primary mirror diameter of 42 m will allow it to gather ∼ 25 times more light than the most efficient
The segmented primary mirror of E-ELT is made of 984 segments.
• •
x ∈ R2952 the concatenation of vectors x(i) = (x1 (i), x2 (i), x3 (i)) ∈ R3 , where x1 (i, t), x2 (i, t), x3 (i, t) respectively denote piston, tip and tilt of segment i at time t, for i = 1, 2, ...984 (states); u ˜ ∈ R2952 the similar concatenation of u ˜(i) ∈ R3 with u ˜1 (i), u ˜2 (i), u ˜3 (i) the elongations of the three position actuators of segment i (control inputs); y ∈ R5604 the vector of edge sensor values (control outputs); d ∈ R2952 the segment displacements induced by perturbances, expressed in the same basis as x; n ∈ R5604 the noise on each edge sensor.
Then the static model
x(t) = A u ˜(t) + d(t) y(t) = C x(t) + n(t)
(1)
3
Wind disturbance
M1 dynamics A(s)
d
actuators coupled to backstructure
segments piston, tip, tilt
edge sensors
x
C+∆
Quasi−static disturbances u
digital controller
y
+
n +
Sensing noise
Fig. 3. M1 (E-ELT primary mirror system) with its main disturbances: quasi-static biases and wind (d), edge sensor noise (n), measurement model uncertainty (∆), actuator and backstructure dynamics (A(s)).
These very low temporal frequency but high amplitude (≃ 1 mm) deformations are induced by gravity and thermal effects. 3) Edge sensor noise n is modeled as temporally white, independently distributed on the individual sensors. 4) Measurement model uncertainties represent the (small) difference between assumed measurement model C and the real relation between states and outputs, Creal : Creal = C + ∆ .
captures the essential features required for control design. More dynamical details will be added for practical implementation, see Section VI. Linearity is justified by the very small motions allowed in the system. Matrix A is blockdiagonal, with each invertible 3-by-3 block expressing the coordinate change from actuator lengths to piston, tip and tilt of a segment. For simplicity we define a new control input u = A−1 u ˜. Matrix C expresses the state-to-output map, in which neighboring segments are coupled. It is very sparse, having at most 6 nonzero elements per row. The task of the controller studied in the present work is to maintain the primary mirror close to an imposed shape characterized by x = 0, under the various disturbances. Quantitatively, we must maintain less than 10 nm root-meansquared (RMS) on the wavefront error. The latter measures the difference between the ideal light wave and the one reflected by the mirror, modulo filtering by adaptive optics elsewhere in the telescope. It is expressed by 1 X w1 (˜ x1 (i) − h˜ x1 i)2 + w2 (˜ x2 (i))2 + w3 (˜ x3 (i))2 984 i=1 (2) where w1 , w2 , w3 > 0 are normalizing factors; x ˜ denotes variable x after filtering to accountP for the correcting effect of 1 ˜1 (i). Main difficulties adaptive optics1 ; and h˜ x1 i = 984 i x of the control task come from huge system dimension (2952 inputs, 5604 outputs), extreme accuracy required under various disturbances, and the particular distributed system induced by local coupling in output map C. 984
W =
B. Main disturbances acting on the system Characteristics of expected disturbances govern controller possibilities/limitations and justify the model used for its study and design. Figure 3 summarizes the main disturbances acting on the E-ELT primary mirror system, M1. 1) Wind force on the segments is the strongest varying part of disturbance d to be rejected by feedback. Wind variations have low temporal frequency, concentrated below few tenths of Hz. Wind effect is spatially correlated on different segments, i.e. induced surface deformations are dominated by low spatial frequencies; higher frequency components must also be rejected to reach the required accuracy. 2) Quasi-static disturbances constitute the second main effect in d which necessitates feedback control of M1. 1 More details on the influence of adaptive optics can be found in [16], [20] and related documents.
(3)
This quasi-static systematic error comes from variations in gain and in geometric positions and orientations of edge sensors, due to the residual effect — after calibration — of mounting tolerance as well as secondorder influence of deformations. 5) Actuators and backstructure (the mechanical structure supporting M1) have finite stiffness, whose effect can become significant at the required accuracy. This introduces a coupled dynamical system, denoted by A(s) on Figure 3. Its characterization is difficult, so we treat it as dynamic model uncertainties. The first vibration mode of the backstructure has a temporal frequency of ∼ 3 Hz. The dominance of low temporal frequencies in disturbances to reject, in association with the difficulty to characterize highfrequency plant dynamics A(s), justify the use of static model (1) for controller design. To validate this approximation, we will have to appropriately restrict the controller’s bandwidth. Uncertainties affecting the measurement system — that is noise n and model errors ∆ — are a particular concern because they directly affect the feedback loop. The following section investigates fundamental features of this issue. III. L OCAL RELATIVE MEASUREMENTS ARE A MAIN LIMITING FACTOR
State estimation from local relative measurements has been recognized as a major issue for segmented mirrors in [21]. In particular, [21] highlights the naturally better behavior of local state estimators in presence of noise. The present section further analyzes the fundamental implications of local relative measurements, in the E-ELT primary mirror context and more generally. Thanks to static the approximation, the simple model (1) is diagonalized by a standard singular value decomposition of the measurement matrix: C = R Σ QT , with R ∈ R5604×5604 and Q ∈ R2952×2952 orthonormal, and Σ ∈ R5604×2952 a diagonal matrix where Σ(i, i) =: σi ∈ R≥0 are the singular values of C. In modal basis xM uM
:= :=
QT x QT u
yM nM
:= RT y := RT n ,
(4)
the measurement equation of (1) becomes yM = Σ xM + nM .
(5)
The columns of R and Q are respectively called edge sensor modes and deformation modes. By convention, the modes are sorted in order of decreasing singular values σi . The latter characterize the respective observability of the associated
4
i
singular values σ of C
3.34
0.334
0.0017 1
1000 2000 2693 2948 deformation mode index i
Fig. 4. Singular values σi of measurement matrix C, covering a wide range in logarithmic scale. The deformation modes associated to the last 260 values are called “poorly observable”. The last 4 modes are not observable at all, i.e. σi = 0 for i = 2949 to 2952 (beyond plot limits).
deformation mode. Our measurement architecture leads to a broad range of mode observabilities, depicted on Figure 4. We call poorly observable modes the last 260 deformation modes, whose singular values are less than 0.1 σmax . The last 4 modes are not observable at all with relative measurements: σi = 0 for i = 2949, ..., 2952. These unobservable deformations consist of full mirror translations and rotations, and mirror “defocus”. The present control loop necessarily leaves them completely “floating”, but they will be addressed by other telescope subsystems based on wavefront sensing (see e.g. [20] and related). A. A fundamental cause for performance limitation As shown in [16], there is a strong correlation between mode indices i, ranked by observability, and spatial frequency: poorly observable modes correspond to low spatial frequency (long-range) deformations. In fact, we identify the measurement of local relative positions as the fundamental cause of this spatial frequency-dependent sensitivity. Explicitly, edge sensor measurements can be interpreted as a spatial derivative: for a pure piston deformation for instance, y(i) = x1 (i + 1) − 1 x1 (i) is the discretization of y(r) = ∂x ∂r for a continuum of infinitesimal surface elements and sensors. Then a deformation like a cos(ξr) yields a measurement a ξ sin(ξr), proportional to ξ the spatial frequency of the deformation. This observation and its interpretation in terms of spatial derivatives can be generalized to many distributed systems, see [12]. We emphasize that it is a fundamental plant property, that cannot be compensated by control design (whatever centralized or distributed). The broad dispersion of singular values (i.e. observabilities) has consequences in terms of state estimation and robustness, as examined in the following. The particular distributed sensing architecture thus determines spatial frequency limitations of the control problem. B. Low signal-to-noise ratios and local vs. global state estimation Because R is orthonormal, noise nM in modal basis still has identically independently distributed statistics. Signal-tonoise ratio from state xM (i) to output yM (i) is thus low for
small σi , that is for deformation modes with a large index i. This inherently limits the performance of the control loop on these modes. Since objective (2) explicitly concerns states x, as opposed to outputs y, a controller’s performance will ultimately depend on the way it extracts state information from the outputs. A standard “centralized estimation” x ˆc of states x from outputs y uses the pseudo-inverse of C. In modal basis, this conveniently writes x ˆcM = Σ+ yM where Σ+ ∈ R2952×5604 is + diagonal, with Σ (i, i) = σi−1 for σi > 0 and Σ+ (i, i) = 0 for σi = 0. Assuming n = 0, this yields x ˆcM = xM except that c x ˆM (i) = 0 for i = 2948 to 2952. For n 6= 0, noise σi−1 nM (i) is added in x ˆcM (i), so centralized estimation strongly amplifies noise on the poorly observable modes. This strong noise on mainly low spatial frequency modes basically reflects that with a localized relative measurement scheme, long-range deformations must be estimated by adding and subtracting a large number of sensor values. The pseudo-inverse of C does not retain the sparsity of C; therefore computing x ˆc indeed requires “centralized” knowledge of all sensor values. In contrast, a distributed controller will require a “distributed estimation” x ˆd of states, based on local outputs. The motion of a generic hexagonal segment influences 12 edge sensors, see Figure 2. A standard distributed estimation is obtained when each segment i produces estimate x ˆd (i) of its state x(i) by assuming neighboring segments j to have x(j) = 0, ∀j 6= i. This requires pseudo-inverting, for each segment, the 12 × 3 local matrix linking the 12 sensors at its edges (see Figure 2) to its own three degrees of freedom only. An adapted procedure is used for segments on the boundary of M1, due to lack of sensors on boundary edges. The local matrix inversion restricts by construction the number of sensor values combined to estimate a deformation. Accordingly, explicit computations confirm that distributed estimation indeed avoids strong noise amplification. The price to pay for this reduced noise amplification is the introduction of a “systematic error” which mainly consists in underestimating the amplitude of poorly observable modes. In other words, distributed estimation naturally filters deformations, such that noise amplification is avoided but poorly observable modes are multiplied by gain (much) lower than 1. This point will be further detailed in the spatial frequency context in Section V. More alternatives for state estimation are proposed in [21]. In light of our interpretation, they all correspond to further filtering alternatives in the trade-off between noise amplification and systematic errors for the estimation of x on the basis of local relative measurements. C. A fundamental robustness issue A key consequence of the poor observability of certain modes for control design is that pure integral control is not robust with respect to (very) small measurement model uncertainty ∆. Consider integral controller in modal basis d dt uM
= −K yM ,
2952×5604
(6)
with some gain matrix K ∈ R . Discarding noise and disturbances but including model uncertainties, (6) and
5
(1) yield the first-order closed-loop MIMO system2 d dt xM
+
= −K Σ (I + Σ ∆M ) xM
(7)
where ∆M := RT ∆ Q. Stability is ensured by imposing negative closed-loop gain. Controller tuning, assuming ∆ = 0, therefore assigns positive real parts to eigenvalues of K Σ. The feedback system is then stable if and only if the eigenvalues of I + Σ+ ∆M have positive real part. For the E-ELT, relative model uncertainty is expected to be less than 1%, i.e. |∆(i, j)| < 0.01 |C(i, j)| ∀ i, j. Because C is ill-conditioned, this low level of uncertainty is still sufficient to destabilize the system if very small Σ+ (i, i) amplify moderate ∆M (i, i). If −∆M (i, i) > 0 dominates the remaining coefficients of ∆M , then I + Σ+ ∆M possesses an eigenvalue ∆M (i, i) α≈1+ . (8) σi This suggests that worst-case uncertainties are those maximizing −∆M (i, i) for a poorly observable mode i. Maximizing for last observable deformation mode −∆M (2948, 2948) with independently chosen 1% uncertainties on all elements of C, yields an actual eigenvalue α = −8.7 clearly indicating instability. According to (8), the closed-loop system remains stable 1 % accuracy; only if C coefficients are known within ∼ 9.7 exact computations confirm this very restrictive value. It is important to note that poor robustness, like noise amplification, is independent of centralized/distributed controller choice through K. It is inherently implied by the local relative measurement architecture. Robustness to ∆ is further investigated in [16]; we observe that (a) variations only in edge sensor or actuator gains is not critical (eigenvalues change by the same relative error as the components of C) and (b) critical situations as above very rarely appear in Monte-Carlo simulations. The same conclusions are reached in [15], where numerical µ-analysis of a smaller system with the same properties is performed. IV. C ENTRALIZED ( MODAL ) CONTROLLER This section builds a centralized controller along standard lines of modal decomposition; the approach is similar to [22] for segmented mirrors. As a tutorial contribution, we illustrate the tuning of leakage to get robustness, inspired from [5]. We apply a controller in the decoupled basis of deformation modes, writing for each deformation mode i UM (s, i) = G(s, i) YM (s, i) ,
i = 1, 2, ...2952
(9)
where UM (s, i) and YM (s, i) are modal inputs and outputs in temporal frequency domain and G(s, i) is the controller transfer function for mode i. (Components of YM for i > 2952 cannot result from physical segment motions; they are called “unphysical modes” and used only for sensor failure detection, see [16].) The main element to reject the expected lowfrequency disturbances is an integral controller, G(s, i) = −kI (i) with kI (i) ≥ 0. This yields a closed-loop system s 2 We here neglect the four unobservable modes, which are not controlled, to simplify notations: e.g. we drop the part Σ+ (i, i) Σ(i, i) = 0 for unobservable modes.
like (6) with K diagonal, prone to instability with small model uncertainties as explained in Section III-C. Following [5], robustness is recovered by introducing integrator leakage: 1 such that simple integrator 1s of mode i is replaced by s+b(i) G(s, i) =
−kI (i) s + b(i)
(10)
with b(i), kI (i) ≥ 0. The closed-loop equation then becomes d dt xM
= −(B + K Σ (I + Σ+ ∆M )) xM
instead of (7), with diagonal B : B(i, i) = b(i) the “leakage”. Now building a bad model uncertainty ∆iM with expected unstable eigenvector close to mode i, as described in Section III-C, the worst closed-loop eigenvalue is approximated by αi′ ≃ −(b(i) + kI (i)σi (1 +
∆iM (i,i) )) σi
(11) ∆iM (i,i)
so b(i) guards against potential sign change in (1 + σi ). Leakage is a fundamental tuning parameter because it increases robustness at the cost of static performance. Indeed, (1),(9) without noise yields disturbance-to-state sensi1 M (s,i) tivity transfer function L(s, i) := X DM (s,i) = 1−G(s,i) σi = s+b(i) s+b(i)+kI (i)σi .
b(i) Then L(0, i) = b(i)+k equals zero only I (i)σi if leakage b(i) = 0. Moreover, sensitivity L(0, i) approaches 1 as σi goes to 0, for large i (see also Figure 5). Thus on poorly observable modes, we get very poor static disturbance rejection as a tradeoff for robustness. Controller tuning shall minimize leakage subject to a robustness constraint.
A. Centralized controller tuning The following criteria govern controller tuning. • Unobservable modes cannot be controlled with the present subsystem: kI (i) = 0 for i = 2949 to 2952. • Based on centralized state estimator x ˆcM = Σ+ yM , closed-loop gain of mode i is kI (i) σi . But limiting noise amplification requires limited kI (i) (see discussion in Section III-B). We therefore impose the same kI (i) to all modes for which σi < σmax 10 , that is the poorly observable modes i =2693 to 2948. This unavoidably implies decreasing closed-loop gain for poorly observable modes (σi ≪ 1 for i large). Modes i < 2693 can all be given the same closed-loop gain kI (i) σi . • The controller must avoid exciting actuator-backstructure dynamics, see Section II-B. A good rule of thumb to avoid exciting undesirable dynamics at x Hz is to take x . However, a control closed-loop bandwidth below 10 despite a lowest backstructure vibration frequency at 3 Hz, it is not necessary to restrict control bandwidth to 0.3 Hz for all deformation modes i. Indeed, there is a natural correlation such that low temporal frequency vibration modes also have low spatial frequency and are thus closer to the least observable deformation modes. In fact, vibration modes that are significantly coupled to deformation modes i = 1 to 2692 have frequencies above ∼ 30 Hz; thus we can allow 3 Hz maximal control bandwidth on those modes, and restrict it to 0.3 Hz only for modes i > 2693.
6
j
1
a:
b: (0,1)
Magnitude(L(0,i))
0.8
(1,0)
(-1,1)
γ
i
δ
(0,0) (1,-1)
(-1,0)
0.5
(0,-1)
δ
β γ
Fig. 6. a: Numbering of segments in a 2D hexagonal array. b: symmetric indices used in control parameters
0 2800
β α
2846
2896 2927 2948 deformation mode index i
Fig. 5. DC gain of the disturbance-to-state transfer function, |L(0, i)|, resulting from integrator leakage in the centralized controller.
Leakage b(i) ∈ R≥0 is minimized under robust stability constraint αi′ < −p∗0 = −0.1 Hz with αi′ given in (11). The resulting values for controller tuning are 14.4 rad/s for i =1 to 2692, kI (i) = 5.69 rad/s for • kI (i) = σi i =2693 to 2948. ∆i (i,i) ∗ • b(i) = max{p0 − kI (i)σi (1 + Mσ ) ; 0} nonzero for i i ≥ 2846, with maximal leakage b(2948) = 0.7 rad/s. The corresponding DC gain of disturbance-to-state transfer function |L(s = 0, i)| is represented on Figure 5. It grows quasi linearly with i > 2846 and reveals very poor rejection of disturbances on high-index deformation modes. To further investigate leakage / bad model uncertainties and approximation (11), we have computed the exact closedloop eigenvalues associated to “bad model uncertainties” ∆iM , built for expected unstable eigenvector close to mode i (see procedure in Section III-C), for i > 2800. This analysis confirms that eigenvalues remain safely below the intended p∗0 = −0.1 Hz with b(i) tuned as above. It should be noted that without leakage, destabilizing positive eigenvalues appear even with ∆iM tailored for 2800 < i < 2846, for which approximation (11) does not predict instability. This indicates that those ∆iM tailored for i < 2846 actually tend to destabilize modes j > i, which are safeguarded by leakage. •
V. D ISTRIBUTED (LTSI) CONTROLLER This section builds a distributed controller using a so-called “LTSI” methodology described in [10], [5], [7]. As a tutorial contribution, we explicitly provide a MIMO adaptation of the LTSI method. For the rest, we closely follow the design method of Gorinevsky et al. A. LTSI controller structure LTSI control (see [1], [5], [7], [10]) is an efficient design method which assumes a Linear Time-invariant and SpatiallyInvariant (LTSI) system and uses a controller with the same LTSI property, allowing controller tuning in spatiotemporal frequency domain. In the present context, this means that (a) we approximate the E-ELT primary mirror as having identical segments (dynamics and coupling) with regular spacing in an infinite or periodic array and (b) we will use the same controller for each segment. The LTSI framework requires a
spatiotemporally factorized system (see [7]), which is automatically satisfied by our static plant. The strongest approximation is of course to consider the system as infinite or periodic. We hope to approach this situation for practical purposes by imposing a distributed controller with very local coupling: we require u(i) to be a function of sensors y(j) close to segment i, so the impulse response from x to u is spatially localized and boundary effects due to finite mirror size only affect a small number of segments. Note that the segmented structure here allows to exactly impose a finite spatial coupling extension, unlike for continuous structures where finite spatial extension is itself an approximation, see e.g. [10], [5], [7]. In the following, for simplicity we directly consider local state estimates x ˆd (see Section III-B) as outputs of the system. d 3 x ˆ (i) ∈ R depends on sensors j at the edges of i only, thus on the states x(k) ∈ R3 of neighboring segments k. We further allow u(i) ∈ R3 to depend on x ˆd (j) for segments j neighbor of i only. This defines a local coupling with radius at most two segments. Explicitly, using the two-dimensional segment numbering system i → (m, n) as depicted on Figure 6a we write U (s, m, n) =
1 1 X X
ˆ d (s, j, l) (12) G(s, m−j, n−l) X
j=−1 l=−1
ˆ d (s, m, n) ∈ C3 are the temwhere U (s, m, n), X poral Fourier transforms of u(t, m, n), x ˆd (t, m, n), and 3×3 G(s, m, n) ∈ C for each m, n ∈ {−1, 0, 1} are transfer ˆ d . Thanks to spatial invariance, functions linking U and X the spatial eigenmodes of the system are Fourier modes. Therefore, applying two-dimensional discrete-to-continuous spatial Fourier transform to (12) yields ˆ d∗ (s, ξ, λ) U ∗ (s, ξ, λ) = G∗ (s, ξ, λ) X
(13)
a set of decoupled 3 × 3 MIMO systems, one for each spatial frequency (ξ, λ) ∈ [−π, π] × [−π, π] and temporal frequency s ∈ C. For illustration purposes, Figure 7 depicts spatial Fourier modes of extreme frequencies. Given the simple model (1) with u = B −1 u ˜, closed-loop behavior is then governed by (13) and the relation between actual state x and its local estimation x ˆd . This relation is also spatially invariant (up to boundaries) and thus writes ˆ d∗ (s, ξ, λ) = H ∗ (ξ, λ) X ∗ (s, ξ, λ) X ∗
3×3
(14)
with H (ξ, λ) ∈ C . The latter can be viewed as the static plant transfer function (in absence of noise and disturbances). Figure 8 graphically represents the components of H ∗ (ξ, λ).
7
Here both b∗ and k ∗ are matrices ∈ C3×3 (actually R3×3 from symmetry, see further) whose elements depend on spatial frequencies ξ, λ. Matrix division must be understood such that ˆ d∗ . The dependence (13),(15) means s U ∗ = −b∗ U ∗ − k ∗ X ∗ ∗ in ξ, λ of b and k is constrained by the restriction to couple segment (m, n) with its 6 neighboring segments only. By symmetry, we can further impose identical coupling for each pair of opposite segments w.r.t. (m, n), see Figure 6b. With these assumptions, we have
ES measure
ES measure
(a) Piston Fourier mode at zero spatial frequency
ES measure
(b) Piston Fourier mode at π spatial frequency
ES measure
(c) Tip/tilt Fourier mode at zero spatial frequency
(d) Tip/tilt Fourier mode at π spatial frequency
k ∗ (ξ, λ) = kα + kβ cos(ξ) + kγ cos(λ) + kδ cos(ξ − λ) (16)
Fig. 7. Side view of spatial Fourier modes. At low spatial frequency, tip/tilt modes yield large edge sensor (ES) signals but piston modes are poorly observable. At high spatial frequency, piston modes are well observable while tip/tilt modes are only observed sideways, through ES not represented here.
b∗ (ξ, λ) = bα + bβ cos(ξ) + bγ cos(λ) + bδ cos(ξ − λ)
where ki , bi , i ∈ {α, β, γ, δ}, are 3 × 3 matrices of control parameters to tune. B. LTSI controller tuning
Fig. 8. Spatial frequency response H ∗ (ξ, λ) from actual state X ∗ (ξ, λ) ∈ ˆ d∗ (ξ, λ) ∈ C3 . See the text and Figure 7 for C3 to its distributed estimate X interpretation of features in these plots.
Nonzero off-diagonal plots reflect that the systematic error introduced by local estimation couples piston, tip and tilt motions, e.g. an actual pure piston motion would be estimated as composed of nonzero piston, tip and tilt. The diagonal plots further illustrate the fundamental state estimation problem (see Section III and also [21]): local state estimation avoids noise amplification, but in turn it systematically underestimates the amplitude of poorly observable modes. Diagonal components of |H ∗ | are therefore close to zero on low spatial frequency piston deformations (top left on Fig. 7) and significantly decreased on high frequency tip/tilt deformations (bottom right on Fig. 7). Note that the definition of tip and tilt coordinates at 90◦ angles, while the hexagonal mirror lattice is invariant under 60◦ rotations, leads to asymmetric behavior of tip and tilt. Like for the centralized controller, we choose as temporal structure an integrator with leakage G∗ (s, ξ, λ) =
−k ∗ (ξ, λ) . s I + b∗ (ξ, λ)
(15)
To reduce design effort, matrices ki , bi , i ∈ {α, β, γ, δ}, are chosen diagonal. This amounts to controlling the MIMO segments with three SISO controllers, a recurrent practice. It leaves 4×3×2 = 24 scalar parameters to tune. SISO controller structure is justified a posteriori by the acceptable performance obtained. Loopshaping based on frequency-domain criteria is performed in two steps. First, the MIMO coupling in the local ˆ d∗ is neglected: stability, estimation system X ∗ → Y ∗ → X robustness and performance constraints are formulated as if the off-diagonal elements of the transfer function on Figure 8 were zero. This approximation allows to apply the efficient SISO tuning methodology of [5], using a large but linear optimization setting. In a second step, we add a stability criterion for the exact MIMO system. This introduces nonlinear constraints at all spatial frequencies in the optimization problem. We use as final parameter values in our controller, the local solution of this nonlinear optimization problem computed by starting from the solution of the first step. 1) First step: using SISO criteria: We apply the SISO tuning method of [5] to piston, tip and tilt separately. Uncertainty ∆ on measurement model C, see ∗ (3), is translated in spatial frequency domain by Creal (ξ, λ) = C ∗ (ξ, λ) + ∆∗ (ξ, λ). For the plant transfer function as defined ∗ in (14), this implies Hreal (ξ, λ) = H ∗ (ξ, λ) + ε∗ (ξ, λ) and defining khk = maxξ, λ (|H ∗ |) and δ = maxi, j ( ∆(i,j) C(i,j) ), we have |ε∗ (ξ, λ)| ≤ δkhk. Then the denominator of all closedloop transfer functions is s + b∗ + (H ∗ + ε∗ ) k ∗ . Like in Section IV, the tradeoff is to get maximal stabilization performance under appropriate robustness. Following [5], we set leakage minimization as an optimization objective Z Z min b∗ (ξ, λ) dξ dλ. (17) ∗ b (ξ,λ)
under several constraints. A stabilization gain constraint — minimal bandwidth hereunder — is explicitly requested only in part of the spatial frequency domain, where observability (more generally that would be system gain) is large enough; we call this the “efficient control domain”. The following constraints, adapted from [5], must be examined at each spatial
8
frequency. In practice the spatial frequency domain is meshed to obtain a finite linear program. ∗ ∗ • Nominal closed-loop stability: H k ≥ 0 ∀ξ, λ. ∗ ∗ ∗ • Robust closed-loop stability: |b + (H ± δkhk) k | ≥ ∗ p0 ∀ξ, λ where, as for tuning the centralized controller, −p∗0 < 0 gives the worst closed-loop eigenvalue. • Minimal temporal bandwidth for ξ, λ in efficient control domain: |b∗ + (H ∗ ± δkhk) k ∗ | ≥ p∗1 where p∗1 is the minimal temporal frequency ω where transfer function ∗ U∗ √1 | U ∗ (0)|. |D ∗ (jω)| = 2 D • Maximal temporal bandwidth to avoid actuator-backstructure dynamics: |b∗ + (H ∗ ± δkhk) k ∗ | ≤ p∗2 ∀ξ, λ. ∗ • Positive leakage: b ≥ 0 ∀ξ, λ. Our parameter choice is p∗0 = 0.1 Hz (as for centralized control), p∗1 = 1 Hz (chosen by iterative simulation for acceptable performance), p∗2 = 3 Hz (argument on backstructure vibrations similar to Section IV-A, but the more stringent 0.3 Hz on low spatial frequency deformations is automatically induced by their local underestimation), δ = 2.5% (to include 1% uncertainty in measurement model, plus the error made by approximating the mirror as infinite), efficient control domain delimited by H ∗ (ξ, λ) ≥ 31 maxξ,λ |H ∗ |. The whole optimization is solved for piston, tip and tilt independently, in a few seconds. 2) Second step: introducing MIMO criterion: We repeat the optimization procedure of the preceding step but adding a stability constraint for the exact nominal MIMO system (also see [23] for details on a simplified model). MIMO stability requires all roots of det(I + (sI + b∗ )−1 k ∗ H ∗ )
(18)
to have negative real part (see e.g. [24]). Formally, (18) leads to (s+z1∗ )(s+z2∗ )(s+z3∗ ) where zi∗ are third order polynomials in the elements of k ∗ and b∗ . Therefore the MIMO stability condition that we impose ℜe(zi∗ (ξ, λ)) ≥ p∗0 ,
i = 1, 2, 3 , ∀ξ, λ
(19)
leads to a set of nonlinear constraints. The resulting nonlinear optimization task is solved locally, starting from the solution of the SISO step. With our particular choice of parameters, it turns out that solution values for k ∗ and b∗ barely change during this second step; somewhat fortuitously in fact, the solution obtained in the first SISO step already satisfies (19). If we increase the value of parameter p∗0 , the second optimization step significantly increases leakage w.r.t. the solution of the first step. Properties of the resulting controller — still theoretical, assuming an infinite mirror — are illustrated on Figure 9. Tip and tilt Fourier modes are sufficiently observable at all frequencies to allow zero leakage, leading to zero disturbance∗ to-state sensitivity transfer function X D ∗ (not represented). On the piston component in contrast, leakage is active at low ∗ spatial frequencies and induces large X D ∗ , see Figure 9a (to compare with the analogous Figure 5 of the centralized controller). Temporal bandwidth is well between 1 Hz and 3 Hz for spatial frequencies in the efficient control domain. The latter
a:
b:
c:
d:
Fig. 9. Properties of the distributed controller resulting from automatic tuning, in LTSI approximation. See the text for interpretations of the features.
most prominently excludes low spatial frequency on piston, also some high spatial frequency on tip/tilt, see Figures 9b, 9c. A slight bandwidth dip for low spatial frequency tip (also tilt, not represented), see Figure 9c, probably reflects coupling with piston motion. Figure 9d shows the maximum real part of closed-loop poles of the MIMO system, as a function of spatial frequency; for completeness, implementation details in the controller (see Section VI-A) have been included. The figure confirms that stability criterion (19) is well satisfied. Figures 9a and 9d together illustrate that low spatial frequencies constitute the critical zone where performance and robustness criteria collide.
VI. I MPLEMENTATION AND SIMULATION A. Addressing high-frequency dynamics The controller designed so far assumes a perfectly static model (1) for the primary mirror system. Section II-B mentions high-frequency dynamics A(s) due to finite stiffness of backstructure and actuators. Control bandwidth limitation, enforced in sections IV and V, avoids exciting backstructure vibration. Actuators have 5.7 · 107 N/m stiffness with notably weak modal damping of 0.75%. This leads to second order dynamics with resonance peaks at high temporal frequencies — 76 Hz for piston-to-piston motion and 58 Hz for tip-tip and tilt-tilt — but of very large magnitude ∼ 40 dB. Rejecting these peaks requires to further adapt our controller for practical implementation. Solutions to damp resonance peaks in the closed loop are either a notch filter or, as proposed in [5], steeper highfrequency control roll-off. We choose the second solution because it requires less accurate knowledge of resonance frequency. Our integral controller (with leakage) is therefore
9
Sensitivity
1 0.5 0 −1 10
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Frequency (Hz): 3 Magnitude (abs): 0.707
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−20
TABLE I S IMULATION RESULTS : WAVEFRONT ERROR IN NM RMS
Complementary sensitivity
1 0.5 0
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Fig. 10. Temporal characteristics of the centralized controller on one mode i < 2693, illustrating robustness and the 3 Hz closed-loop bandwidth. Wind disturbances are attenuated by more than 3 dB for up to 1.9 Hz and are the only signal amplified by actuator resonance.
augmented by a double pole at3 20 Hz, i.e. Centralized control:
G(s, i)
Distributed control:
G∗ (s, ξ, λ)
−kI (i) 1 · s + b(i) ( ps + 1)2 −k ∗ 1 = · s ∗ s + b ( p + 1)2
=
with p = 2π 20 rad/s. Typical temporal properties of the resulting controller are shown on Figure 10, for a wellobserved mode in centralized control (thus in particular, without leakage). Sensitivity functions and Nyquist plot describe a robust system, with gain margin 23.6 dB and phase margin 77◦ . Complementary sensitivity confirms 3 Hz bandwidth. Disturbance(e.g. wind)-to-state sensitivity, that is L(s, i) of Section IV, shows more than 3 dB attenuation for up to 1.9 Hz. As actuator resonance is avoided in the control loop, direct wind disturbance is the only signal amplified by resonance, but such high-frequency disturbances are not expected. B. Simulation results ESO has built a comprehensive finite element model of the whole E-ELT system, including all mechanical structure and connections, actuating elements, any slight differences in individual mirror segment properties (curvature, support,...), possible telescope orientations,... . From this a realistic model for primary mirror (M1) behavior has been extracted. It contains 20000 states, which cover vibration modes from 3 Hz to 130 Hz. The model is subjected to wind (the dominant disturbance) deduced from a realistic experimental profile, and √ includes sensor noise of 1 nm/ Hz. The controllers, built as described in Sections IV and V with the adaptation of Section VI-A, are discretized at 200 Hz. Sensor and actuator quantization is investigated in [16] to give appropriate requirements, but simulations in the present paper do not include their effect. 3 This value is chosen well above the closed-loop bandwidth — to be compatible with the static system assumption used so far — and low enough to sufficiently lower the actuator resonance peak.
Open-loop
Centralized
Distributed
790 NA
785 2.84
790 2.78
Section III-C presents model uncertainties ∆ which destabilize the pure integral controller. Including these particular ∆ in simulations with controller leakage set to zero, we indeed observe unstable behavior, both for centralized and distributed controllers. This illustrates the robustness issue and importance of leakage in practice. Figure 11 and Table I compare open loop, centralized control and distributed control. The main evaluation criterion is to minimize W given by (2), that is wavefront error after adaptive optics correction, with operational requirement W < 10 nm root-mean-square (RMS). To further illustrate controller properties, we also present the wavefront error without adaptive optics correction. The main conclusion is that both controllers achieve very similar performance, finally pushing W down to ∼ 2.8 nm. On long-range deformation modes (a main part of the disturbance), as expected control is very weak but adaptive optics perform efficient filtering. Complementarily, on short-range deformations adaptive optics are inefficient but our controller makes its major contribution. Note that the level of wavefront error observed at all spatial frequencies in open loop would make adaptive optics unusable in practice, hence the symbol NA in Table I. VII. F INAL DISCUSSION This paper compares two strategies for the control of the huge MIMO system consisting of the segmented primary mirror of the E-ELT. The first controller is centralized and based on diagonalization of the system into spatial eigenmodes. The second controller is distributed, using only neighboring measurements to determine local control actions; it is tuned in the spatial frequency domain. We make two main observations, supported by theoretical developments and by simulations on a very realistic system model. 1) Very small uncertainties on the system can lead to instability under pure integral control, independently of the centralized or distributed nature of the controller. The fundamental reason for poor robustness is the distributed plant itself, in particular measurements which only provide relative heights of neighboring segments. This makes integrator leakage an essential tuning parameter to recover robustness at the cost of stabilization accuracy. 2) The distributed controller, designed with sometimes crude approximations, finally performs as well on the realistic system as the centralized controller. The larger freedom offered by the centralized controller — including information transmission over large distances — and the more accurate model used for its design do not allow to (significantly) improve system behavior. The first observation has recently attracted attention in the research community, see e.g. [12], [11] and related publica-
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Fig. 11. Comprehensive simulation of M1 assuming wind and edge sensor noise: in open loop, with centralized control, and with distributed control. The plots show the temporal frequency distribution of wavefront error on (a) deformation mode 1000 – a typical short-range deformation, well-observed mode – and (b) deformation mode 2900 – a typical long-range deformation, poorly-observed mode. Centralized and distributed controllers achieve similar performance. On short-range deformations, the controllers significantly push down the error (at low temporal frequency). On long-range deformations, adaptive optics (AO) further in the telescope filter the error down to acceptable level. Open-loop with AO is just given for reference: in practice the AO system cannot work with a disturbance like the one given by the open-loop simulation.
0
controller gains are naturally “distributed”: the gain from measurement at sensor j to input at segment i quickly decreases with the distance between j and i, see Figure 12. It is encouraging for theoretical investigations that approximations of invariance on an infinite domain seem to be valid for a system with less than 40 segments diameter and a hole in the middle (see Figure 1); of course this is linked to very local coupling, over at most two segment radius.
gain from yj to ui
10
−2
10
−4
10
−6
10
−8
10
0
distance from j to i [meters] 10 20 30 40
Fig. 12. Gain attributed, with centralized controller, to edge sensor j in the control input of segment i. Segment number i is fixed to a random typical value. The gain of all edge sensors, j = 1...5604, is depicted as a function of their distance to segment i. The gain quickly decreases with increasing distance.
tions. In particular, [11] also shows that bad long-range behavior in distributed systems is directly related to relative position sensing. An effect of “boundary conditions” reminiscent of PDEs is also present. The issue is still under investigation. For the E-ELT, the whole problem would be circumvented by measuring nm-accurate position w.r.t. a common reference over the whole mirror. A common reference allows to suppress the local coupling in the measurements; in the interpretation of Section III-A, outputs would not be restricted anymore to derivatives of the shape. Unfortunately, this solution seems technologically infeasible. The second observation, made here on a practical, finite, realistic system, supports similar theoretical investigation e.g. in [1], [13] for idealized (infinite) arrays. In fact we even observe, in agreement with [1], [13], that centralized
The similar performance of centralized and distributed controllers in realistic simulations motivates their further comparison. Regarding control design: • Local output-to-state estimation naturally avoids noise amplification. • Distributed control tunes fewer parameters for a comparable performance (it incorporates some “natural tuning”). • Controllers based on structural assumptions (e.g. distributed, LTSI) are often more robust than controllers tailored for the exact state space model (e.g. modal tuning), although this is not observed for ∆ defined by (3) in particular. • The natural formulation of controller criteria in spatiotemporal frequency domain further leads to transparent tuning of LTSI. • Although design with the LTSI method involves approximations, the resulting controller can still be analyzed on the exact full system to guarantee good properties. • Eigenmodes of the centralized controller are closer to optical modes than the Fourier basis used for LTSI design (approximating modes of an infinite mirror). Therefore a centralized primary mirror controller could interact with other telescope subsystems more easily. Regarding general implementation (see [16] for details): • Implementation cost and communication hardware are
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not reduced by interconnecting all segments locally w.r.t. a centralized communication bus. • Controller speed from sensors to actuators can be faster with local interaction than over a centralized bus; for the E-ELT, this issue is not critical. • Distributed systems contain no critical components (e.g. central computer, communication bus) whose failure disables the whole system. Conversely, centralized systems can better exploit redundancy to detect and compensate failures of non-critical components like few edge sensors. As a conclusion, distributed controllers seem to offer a practically appealing alternative for systems like the E-ELT primary mirror, but the way towards their practical implementation will highly benefit from a more comprehensive theory of performance limitations for distributed systems. R EFERENCES [1] B. Bamieh, F. Paganini, and M. A. Dahleh, “Distributed control of spatially invariant systems,” IEEE Transactions on Automatic Control, vol. 47, no. 7, pp. 1091–1107, 2002. [2] G. de Castro and F. Paganini, “Convex synthesis of localized controllers for spatially invariant systems,” Automatica, vol. 38, no. 3, pp. 445–456, 2002. [3] G. E. Stewart, D. M. Gorinevsky, and G. A. Dumont, “Two-dimensional loop shaping,” Automatica, vol. 39, no. 5, pp. 779–792, 2003. [4] R. D’Andrea and G. E. Dullerud, “Distributed control design for spatially interconnected systems,” IEEE Transactions on Automatic Control, vol. 48, no. 9, pp. 1478–1495, 2003. [5] G. Stein and D. Gorinevsky, “Design of surface shape control for large two-dimensional arrays,” IEEE Transactions on Control Systems Technology, vol. 13, no. 3, pp. 422–433, 2005. [6] C. Langbort and R. D’Andrea, “Distributed control of spatially reversible interconnected systems with boundary conditions,” SIAM Journal on Control ans Optimization, vol. 44, no. 1, pp. 1–28, 2005. [7] D. Gorinevsky, S. Boyd, and G. Stein, “Design of low-bandwidth spatially distributed feedback,” IEEE Transactions on Automatic Control, vol. 53, no. 1, pp. 257–272, 2008. [8] A. Sarlette, “Geometry and symmetries in coordination control,” Ph.D. dissertation, University of Liege, 2009, advisor: R. Sepulchre. [9] G. Stewart, D. Gorinevsky, and G. Dumont, “Feedback controller design for a spatially distributed system: the paper machine problem,” IEEE Transactions on Control Systems Technology, vol. 11, no. 5, pp. 612– 628, 2003. [10] D. Gorinevsky, S. Boyd, and G. Stein, “Optimization-based tuning of low-bandwidth control in spatially distributed systems,” Proceedings of The 2003 American Control Conference, vol. 1-6, pp. 2658–2663, 2003. [11] B. Bamieh, M. R. Jovanovi´c, P. Mitra, and S. Patterson, “Coherence in large-scale networks: dimension dependent limitations of local feedback,” IEEE Transactions on Automatic Control, 2009, conditionally accepted. [12] A. Sarlette and R. Sepulchre, “A PDE viewpoint on basic properties of coordination algorithms with symmetries,” Proceedings of the 48th IEEE Conference on Decision and Control, pp. 5139–5144, 2009. [13] N. Motee and A. Jadbabaie, “Optimal control of spatially distributed systems,” IEEE Transactions on Automatic Control, vol. 53, no. 7, pp. 1616–1629, 2008. [14] S.-X. Jiang, P. Voulgaris, L. Holloway, and L. Thompson, “H2 control of large segmented telescopes,” Journal of Vibration and Control, vol. 15, no. 6, pp. 923–949, 2009. [15] D. MacMynowski, “Interaction matrix uncertainty in active (and adaptive) optics,” Applied Optics, vol. 48, no. 11, pp. 2105–2114, 2009. [16] C. Bastin, A. Sarlette, and R. Sepulchre, “E-ELT Programme, M1 control strategies study, final report,” University of Liege, Tech. Rep. E-TREULG-0449-0006 Issue 2.1, 2009. [17] R. Gilmozzi and J. Spyromilio, “The European Extremely Large Telescope (E-ELT),” The Messenger, no. 127, pp. 11–19, March 2007. [18] J. Spyromilio, F. Comern, S. DOdorico, M. Kissler-Patig, and R. Gilmozzi, “Progress on the European Extremely Large Telescope,” The Messenger, vol. 133, pp. 2–8, September 2008. [19] R. Gilmozzi and J. Spyromilio, “The 42m European ELT: Status,” Proc. SPIE, vol. 7012, p. 701201, 2008.
[20] B. Bauvir, H. Bonnet, M. Dimmler, and J. Spyromilio, “E-ELT Programme, Technical Specifications for the study of M1 control strategies,” European Southern Observatory, Tech. Rep. E-SPE-ESO-449-0224 Issue 1.2, 2008. [21] D. MacMynowski, “Hierarchic estimation for control of segmentedmirror telescopes,” Journal of Guidance, Control, and Dynamics, vol. 28, no. 5, pp. 1072–1075, 2005. [22] D. MacMartin and G. Chanan, “Control of the California Extremely Large Telescope primary mirror,” Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 4840, pp. 69–80, 2003. [23] B. Modave, “Distributed control of the M1 mirror of the Extremely Large Telescope,” Diploma Thesis, Universit´e de Li`ege, 2009, (supervisor R. Sepulchre). [24] M. Morari and E. Zafiriou, Robust Process Control, N. J. Englewood Cliffs, Ed. Prentice Hall, 1989.
