Active Optics of Large Segmented Mirrors: Dynamics and Control R. Bastaits, G. Rodrigues, B. Mokrani, A. Preumont Active Structures Laboratory, Universit´e Libre de Bruxelles CP. 165-42, 50 Av. F.D. Roosevelt, B-1050 Brussels, Belgium e-mail:
[email protected] February 18, 2009
Abstract This paper examines the active optics of future ELT segmented mirrors from the point of view of dynamics simulation and control. A reduced order structural model, fully representative of the quasi-static behavior and the low frequency dynamics, is obtained by CraigBampton reduction. Assuming an array of edge-sensors and an array of normal sensors, two control options are discussed: (1) A dual loop controller combining a SVD controller of the Jacobian of the edge sensor, and a Zernike modes controller based on the normal sensors, controlling the low order aberrations which are unobservable from the edge sensors. (2) A SVD controller of the extended Jacobian combining the edge sensors and the normal sensors. Both of these controllers are based on the quasi-static behavior of the mirror. The control-structure interaction is addressed through a frequency domain robustness test for MIMO systems; this test separates the dynamic response (residual) from the quasi-static response (primary) of the mirror. This allows to clearly identify the flexible modes which are critical for stability and to study the influence of structural changes in the supporting structure (stiffness and damping). This discussion is intended to be generic rather than targeted to a specific telescope.
1
Introduction
The objectives of this study is threefold: (1) develop a representative numerical model of the segmented mirror and its supporting truss to evaluate the control-structure interaction and its impact on the optical performances (this model should have a minimum complexity to allow extensive parametric studies), (2) develop a control strategy based on two sensor arrays, the edge sensors and the normal to the segments, and (3) examine the interaction between the controller and the structural dynamics and evaluate the requirements in terms of frequency and damping for the supporting truss in order to guarantee the performance, control bandwidth, and an appropriate degree of robustness for the control system. Modelling large complex active structures such as telescopes poses several challenges: (1) Finite element techniques classically used in structural modelling tend to use a very large number of degrees of freedom which reflect the complex geometry of the structural components. (2) Prior to coupling the structural model with the control, the number of 1
degrees of freedom must be drastically reduced, and the reduced control model must still carry the main features of the system, and account for the static as well as dynamic behavior of the system. (3) The control model must give access to all actuator inputs, sensor outputs, and optical performance metrics.
6 edge sensors
3 position actuators (2 stages) Figure 1: Primary mirror (M1) of the future E-ELT telescope; it consists of 984 segments, each of them equipped with 6 edge sensors and 3 two-stage position actuators. The segments are connected to the actuators by a whiffle tree. Figure 1 shows the primary mirror (M1) of E-ELT [1]; it consists of 984 aspherical segments; every segment is equipped with 3 two-stage actuators, controlling the piston and the two tilts, and 6 edge sensors measuring the position of the segment with respect to its six neighbors. The motion of the position actuators is transmitted to the segment through a whiffle tree. Overall, there are 2952 position actuators a and 5604 edge sensors y1 . The system can be represented schematically as in Fig.2. In this figure ka represents the stiffness of the whiffle tree and the unconstrained displacement is given by a = Fa /ka . The position actuators rest on a supporting truss carrying the whole mirror. The disturbances d applied to the system come from thermal gradients, changing gravity vector with the elevation of the telescope, and wind. Figure 3, describes the temporal and spatial frequency distribution of the various layers of the control system involved in the wavefront correction of a large telescope [2]; the spatial frequency is expressed in terms of Zernike modes. In general, adaptive optics operates on a smaller deformable mirror and the amplitudes are small, typically a few microns. Our discussion is focused on M1; the amplitudes to be corrected by the active optics are typically several hundred microns [3]. Currently, the control strategy envisaged by ESO for co-phasing the segments [4] assumes that the supporting truss is rigid and that the natural frequency of the whiffle tree (connecting the actuators to the segments) is well above the bandwidth of the control system (which is realistic). In this case, the behavior of the segmented mirror is assumed quasi-static and the relationship between the actuator displacements a and the edge sensors output y1 is simply y1 = Je a 2
(1)
n Atmospheric Normal to segment
turbulence
+
Normal to wavefront (Shack-Hartmann)
y2 y1
Edge Sensors
d (gravity, wind)
m Segment
a ka ca
Position Actuators
Fa
H(s)
Supporting truss f i
Figure 2: Active optics control flow for large segmented mirrors. The axial d.o.f. at both ends of the actuators are kept in the reduction process.
Spatial frequency
where Je is the Jacobian of the edge sensors of the segmented mirror. The pseudo-inverse of
M1 Shape
Adaptive Optics temp. avg
temp. avg
M2 Rigid Body
temp. avg
temp. avg
Main axes 0.01
0.1
1
10
100
Bandwidth [Hz] Figure 3: Temporal and spatial frequency distribution of the various control layers of a large telescope (adapted from [2]). the Jacobian J+ e is best obtained by singular value decomposition (SVD): Je = UΣVT
(2)
where the column of U are the orthonormalized edge sensor modes, the column of V are the 3
orthonormalized actuator modes, and Σ contains the singular values on its diagonal. The control system works according to Fig.4, called SVD controller. d is the disturbance applied to the mirror and the set of filters H(s) provide (hopefully) adequate disturbance rejection and stability margins (we will come back to this). H(s) may be a scalar function if the same loop shaping is applied to all SVD modes; in this case, all the loops have essentially the same control gain. Figure 5 shows a typical distribution of the singular values of Je ,
d
Primary Mirror
a Position Actuators
J = US V
-
Edge Sensors -1
UT
S
H(s)
V
y
T
Figure 4: Block diagram of the co-phasing control system (SVD controller).
10
…
s
0
Coma Trifoil Astigmatism
10
-5
Defocus 10
-15
0
Tilt Piston 25
50
75
Index
Figure 5: Singular values of the Jacobian Je of the edge sensors, ranked by increasing order. Piston, tilt and defocus cannot be controlled from the edge sensors. ranked by increasing order; one sees that the lowest singular values correspond to the lowest optical modes, piston, tilt and defocus, which are unobservable from the edge sensors, and the modes with the lowest singular values next to them are astigmatism, trifoil and coma. This is not very good from an optical control viewpoint and suggests that the full control system should include another set of sensors, y2 , measuring the normal to the segments (Fig.2). Once again, if the supporting truss is rigid, there is a linear relationship between the actuator displacements and normal to the segments y2 = Jn a
(3)
where Jn is the Jacobian of the normal to the segments; it is block-diagonal, because the two components of the normal to a segment depend only on the three displacement actuators 4
under that segment. A practical way to measure the normal to the segments is to use a ShackHartman sensor (one micro lens per segment); however, the Shack-Hartman array measures the normal to the wavefront which, in addition to the normal to the segment, includes the atmospheric turbulence which appears as noise (Fig.2).1
2
Structural dynamics
The dynamics of the mirror consists of global modes involving the supporting truss and the segments, and local modes involving the segments alone. The global modes are critical for the control-structure interaction; the segments are designed in such a way that their local modes have frequencies far above the critical frequency range, but their quasi-static response (to the actuator as well as to gravity and wind disturbances) must be dealt with accurately. In order to handle large optical configurations, it is important to reduce the model as much as possible, without losing the features mentioned before. The starting point is a standard finite element (FE) model ¶µ ¶ µ ¶ ¶ µ µ ¶µ ¨1 K11 K12 x1 f1 x M11 M12 + = (4) ¨2 K21 K22 x2 0 x M21 M22 where the degrees of freedom (d.o.f.) have been partitioned in boundary d.o.f., x1 , including all the loaded ones, especially the axial d.o.f. at both ends of the actuators, at the connection with the truss and the segments (they are highlighted in Fig.2), and the internal d.o.f., x2 which are not loaded and may be eliminated by appropriate reduction.
2.1
Model reduction
A Craig-Bampton reduction [5,6] is conducted in two steps: First a Guyan reduction [7] assumes a static relationship between the internal and the boundary d.o.f. µ ¶µ ¶ µ ¶ K11 K12 x1 f1 = (5) K21 K22 x2 0 leading to x2 = −K−1 22 K21 x1 . In a second step, this solution is enriched by a set of fixed boundary modes, solution of ¨ 2 + K22 x2 = 0 M22 x (6) [obtained by setting x1 = 0 in (4)]; they constitute the column of the matrix Ψ2 and are normalized according to ΨT2 M22 Ψ2 = 1. Note that these modes are in a plane orthogonal to the actuators. The number of fixed boundary modes included in the reduction will be discussed shortly. Overall, the coordinate transformation reads µ ¶ µ ¶µ ¶ µ ¶ I 0 x1 x1 x1 = =T (7) x2 α α −K−1 22 K21 Ψ2 where α is the vector of modal amplitude of the fixed boundary modes. Using the transforˆ and the stiffness matrix K ˆ of the reduced system are mation matrix T, the mass matrix M readily obtained according to the classical formulae ˆ = TT MT M
ˆ = TT KT K
(8)
1 The bandwidth of the active optics being much lower than the frequency content of the atmospheric turbulence, n can be significantly attenuated by low-pass filtering.
5
leading to the final equation in µ ¶µ ˆ 11 M ˆ 12 M ˆ 12 M I
reduced coordinates: ¶ µ ¶µ ¶ µ ¶ ˆ 11 0 ¨1 x x1 Sa Fa + d K + = α ¨ α 0 0 Ω2
(9)
ˆ 11 = K11 − K12 K−1 K21 being In this equation, the stiffness matrix is block diagonal, with K 22 2 the Guyan stiffness matrix and Ω being a diagonal matrix with entries equal to the square of the natural frequencies of the fixed boundary modes. It has been assumed that these are not loaded by the external disturbance d (wind, gravity, etc...) nor by the control forces Fa = ka a. Sa is the matrix describing the topology of the actuators. To be complete, a damping matrix should be included; it will be included when we transform into modal coordinates. For a segmented mirror with N segments, the size of x1 is typically ∼ 6N . Because of the way they have been selected, the reduced coordinates x1 describe fully the rigid body motion of the segments and the sensor output can be expressed by y1 = Sy1 x1
(10)
y2 = Sy2 x1
(11)
where the matrices Sy1 and Sy2 describe the topology of the sensor arrays.
2.2
Modal analysis
In order to illustrate our analysis, consider the segmented mirror of Fig.6, consisting of 91 segments supported by a truss. The mass is assumed to be distributed equally between the segments, 50 kg each, and the truss, 4550 kg. The actuator stiffness ka has been selected in order that the first piston mode of the segments is 100 Hz. The truss stiffness has been chosen so that the first global mode is f1 = 20 Hz. The modal damping will be assumed uniformly ξi = 0.01 unless otherwise specified. Figure 7 shows the eigen frequency distribution of the full FE model; the first 20 modes or so are global modes; their mode shapes are a combination of optical aberration modes of low order. Next follow the local modes of the segments (tilt near 75 Hz and piston near 100 Hz). Figure 8 shows the natural frequencies obtained after the Craig-Bampton reduction described above, for various numbers of fixed boundary modes. They are solution of (9) after setting the right hand side to 0. One sees that a small number of fixed boundary modes improves the agreement of the eigenfrequency distribution. The mode shapes are displayed in Fig.9; the frequencies listed at the top of the figure are the natural frequencies of the model; those on the left side are the frequencies of the fixed boundary modes; their relationship with the frequencies of the missing modes in the reduced model is interesting. For a large segmented mirror, even a reduced model is far too complex to be included in full in a control-structure interaction analysis, and only the low frequency modes (which can potentially jeopardize the stability) really matter. The reduced model is thus truncated, but one must make sure that the static behavior is not altered by the truncation.
6
B
z y
x
A
Figure 6: Segmented mirror consisting of 91 segments supported by a truss. The figure shows the support conditions assumed in this study, and the first 3 modes.
250
Piston
Tilt
Eigen frequency [Hz]
200
150
100
50
global modes
0
50
100
150
200
250
Mode index
Figure 7: Eigen frequency distribution of the full FE model.
7
300
80
250
70 60
200
50
Eigen frequency [Hz]
40 30
150
20 10 0
10
20
100
Guyan C-B 1 mode C-B 2 modes C-B 3 modes Full FE model
50
0
0
50
100
150
200
250
300
Mode index
Figure 8: Eigen frequency distribution of reduced models (Guyan and Craig-Bampton) for an increasing number of fixed boundary modes.
19.8 Hz 21.2 Hz 26.5 Hz 26.6 Hz 33.2 Hz 42.5 Hz C-B 3 modes (f3 =55.1Hz) Full FE model
51 Hz
53.4 Hz 57.9 Hz 58.9 Hz 69.6 Hz 10
11
8
9
10
6
7
8
9
5
6
7
1
2
3
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7
8
1
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1
2
3
4
5
1
2
3
4
9
C-B 2 modes (f2 =43 Hz)
C-B 1 mode (f1 =27 Hz) 8
Guyan
Figure 9: Eigen modes of the full model and various reduced models (Craig-Bampton) with an increasing number of fixed boundary modes.
8
2.3
Static response
From Equ.(9), the static response of the system is ˆ −1 [Sa Fa + d] x1 = K 11
(12)
ˆ −1 Sa ka a + Sy K ˆ −1 d ˆ −1 [Sa Fa + d] = Sy K y1 = Sy1 K 1 1 11 11 11
(13)
Comparing this equation with (1), one gets ˆ −1 Sa ka Je = Sy1 K 11
(14)
Using the second output equation, one gets similarly ˆ −1 [Sa Fa + d] = Sy K ˆ −1 Sa ka a + Sy K ˆ −1 d y2 = Sy2 K 2 2 11 11 11
(15)
ˆ −1 Sa ka Jn = Sy2 K 11
(16)
and
2.4
Dynamic response
Assume that the eigenvalue problem has been solved for the reduced system and that the eigen modes have frequencies ωi and have been normalized to a unit modal mass. Let φi be the partition of the eigen modes corresponding to the boundary d.o.f. x1 . Because the control force and the disturbance is only applied to those, the equation governing the dynamic response of mode i is z¨i + 2ξi ωi z˙i + ωi2 zi = φTi Sa Fa + φTi d
(i = 1, . . . , m)
(17)
However, only the m lowest frequency modes, within or close to the bandwidth of the disturbance respond dynamically; the higher ones (i > m) respond in a quasi-static manner and may be regarded as a singular perturbation. The static response of the previous section includes all modes, and if a flexible modes is accounted for dynamically, one must remove its contribution ˆ −1 the contribution Fm of the to the flexibility matrix; this is obtained by subtracting from K 11 m modes which respond dynamically according to (17): Fm =
m X φi φT i
i=1
ωi2
(18)
ˆ −1 − Fm (depending on m) and Thus, the flexibility matrix of the high frequency modes is K 11 the overall dynamic response at the d.o.f. x1 reads ˆ −1 − Fm ][Sa Fa + d] x1 = Φm z + [K 11
(19)
where the components of z are solutions of (17) (dynamic response), Φm = (φ1 , . . . , φm ) is the matrix of mode shapes at the boundary d.o.f., and the second term is the quasi-static response (singular perturbation) of all modes beyond m. From (10), the edge sensor output y1 is ˆ −1 − Fm ]d y1 = Sy1 Φm z + [Je − Sy1 Fm Sa ka ]a + Sy1 [K (20) 11
9
a
GR +
y Flexible modes i = 1,…,m
+
GO
...
(Residual response)
T ii
f s + 2xii wii s + wii2 ... 2
ka Sa
Fm x1
+ -
Sy1 Sy2
Fm
y1 y2
Jacobian
a
((
J J= e Jn
y1 y2
(Primary response)
+
+
y
Figure 10: Input-output relationship of the segmented mirror. The nominal plant G0 (s) = J accounts for the quasi-static response (primary response) and the dynamic deviation GR (s) is regarded as an additive uncertainty (residual response). Similarly, from (11), the normal sensor output y2 reads ˆ −1 − Fm ]d y2 = Sy2 Φm z + [Jn − Sy2 Fm Sa ka ]a + Sy2 [K 11
(21)
In these equations, Sy1 Φm and Sy2 Φm are the modal components, for the first m modes, of respectively the edge sensor output and the normal sensor output. Equations (20) and (21) are reduced to (13) and (15) if none of the modes respond dynamically (m = 0). Figure 10 shows a decomposition of the input-output relationship between between the quasi-static contribution (primary response) G0 (s) = J and its dynamic deviation, GR (s); the latter is not taken into account in the controller design and may be regarded as an additive uncertainty. We will use this representation below to examine the robustness of the controller.
3
Control strategy
Given the size and complexity of the system, its variability with temperature and the elevation angle of the telescope, and the large amount of real-time calculations, it seems reasonable to base the control on the quasi-static behavior of the system, and address the dynamic amplification as a perturbation to the nominal system. The baseline co-phasing strategy currently envisaged by ESO is the SVD controller of Fig.4. The edge sensor output y1 is projected into the sensor modes, UT y1 , then scaled according to the inverse of the singular values σi−1 , shaped in the frequency domain according to the diagonal controller H1 (s) (possibly variable with the order of the mode), and then applied to the actuator modes through the matrix V: a1 = VH1 (s)Σ−1 UT y1 (22)
10
This approach reflects the fact that, for each mode, the loop gain must be adjusted according to the singular value. If the same scalar controller is used for all loops, they will have the same gain. This part of the control will include all the modes with significant singular values. However, as mentioned earlier, the lowest singular values of the Jacobian of the edge sensors, Je , correspond to the lowest optical modes (Fig.5), which are the most important from an optical viewpoint. This problem may be solved by using a second set of sensors measuring the normal to the segments, y2 . Two controller structures coupling this information are examined below.
3.1
Dual loop controller
The idea consists of using the SVD controller based on the edge sensors for all the modes of the Jacobian Je with significant singular values, and to add a second loop based on the normal sensors y2 to control a selected set of low order Zernike modes. This is done as follows: Let A = (a1 , . . . , ai , . . .) be the matrix the column of which are the actuator displacements for the Zernike modes of increasing order. They are assumed to be normalized according to AT WA = I
(23)
(the matrix W depends on the geometry of the actuator network). Let N = (n1 , . . . , ni , . . .) be the output of normal sensors for the same low order Zernike modes (with the same normalization). The control input is constructed as follows: the sensor output y2 is first projected in the space of the selected Zernike modes, NT y2 ; next, the proper frequency shaping with the diagonal controller H2 (s) is applied (possibly variable with the order of the mode), and the correction amplitude of the various Zernike modes is then applied to the segment actuators: a2 = AH2 (s)NT y2
(24)
This part of the control includes all the important optical modes of the system. The problem with this dual loop approach is that the two set of modes (singular value decomposition of the Jacobian of the edge sensors on the one hand, and Zernike modes on the other) are not orthogonal and that the control input a1 based on the edge sensors may excite the low order Zernike modes and vice versa. In fact, the control input a1 can be made orthogonal to the Zernike modes by passing it through a spatial filter2 I − AAT W
(25)
a1 = [I − AAT W] VH1 (s)Σ−1 UT y1
(26)
Finally, and the control input is a = a1 +a2 . Generated in this way, the control input of the co-phasing controller will not excite the optical modes controlled by the dual controller. The co-phasing 2 It is easy to see that this filter will remove any component of the control which belongs to the column space of A: (I − AAT W)A = 0
while any vector orthogonal to A will be unaltered, since, if AT WB = 0 (I − AAT W)B = B
11
d Segment Position Actuators
Normal to segments
a
Primary Mirror M1
-
y2
Edge sensors
y1 + T + a1 I-AA W Spatial filter
a2
-1
S UT
H1(s)
V
+
n2 +
Edge sensors controller
A
H2(s)
N
T
Zernike modes controller
Figure 11: Block diagram of the dual loop control strategy for the Active Optics of M1. loop, however, may be excited by control input a2 , although weakly, if only those optical modes corresponding to the lowest singular values in Je are controlled. Figure 11 shows the block diagram of the dual loop control strategy described above. Note that the piston mode of the mirror is not observable, neither by the edge sensors, nor by the normal sensors. In practice, however, the central mirror is removed and its position can be considered as fixed (the edge sensors in the center are attached to the supporting truss).
3.2
Extended Jacobian SVD controller
An alternative way to solve the problem of the poorly observable modes associated with the Jacobian Je of the edge sensor consists of building an extended Jacobian coupling the two sets of sensors: µ ¶ µ ¶ y1 Je y= = a = Ja (27) y2 Jn The singular values of the Jacobian of the edge sensor Je and the extended Jacobian J are compared in Fig.12. One sees that, except for the global piston mode which is not observable (see above), the singular values of the extended Jacobian are quite well conditioned and ready for the implementation of Fig.4. After conducting a SVD of the extended Jacobian J = UΣVT , the controller reads a = VH(s)Σ−1 UT y = K(s)y
(28)
with as many modes as necessary. This control strategy is followed in the sequel of this paper.
4
Control-structure interaction
One can see from Fig.4 that if the mirror would respond in a quasi-static manner, the controller transfer matrix would essentially invert that of the mirror. However, because 12
Trifoil, Defocus Trifoil, Coma 10
0
Astigmatism Tilt Coma 10
10
Trifoil
-2
Astigmatism
-15
Defocus Piston
edge sensor
Tilt Piston 0
edge + normal
25
50
250
275
Figure 12: Comparison of the singular values of the Jacobian Je of the edge sensors with the extended Jacobian J. Residual
(a)
(b)
G = KGO
GR -
-1
L = GO GR
+
+ Primary
+ +
GO
-
I+L
G
K Control
Figure 13: Block diagram of the control system (a) Mirror represented by its primary and residual dynamics. (b) Classical representation with multiplicative uncertainty. the response of the mirror includes a dynamic contribution at the frequency of the lowest structural modes and above, the system behaves according to Fig.10 and the robustness with respect to control-structure interaction must be examined with care [8,9]. The structure of the control system is that of Fig.13(a), where the primary response G0 (s) corresponds to the quasi-static response described earlier and the residual response GR (s) is the deviation resulting from the dynamic amplification of the flexible modes; K(s) is the controller. This structure can be changed into the classical form of multiplicative uncertainty of Fig.13(b) with G(s) = K(s)G0 (s) and L = G−1 0 GR . For the structure of Fig.13(b), the general robustness theory of MIMO systems [10,11,12] shows that a sufficient condition for stability is that σ ¯ [L(jω)] < σ[I + G−1 (jω)] 13
ω>0
(29)
(¯ σ and σ stand respectively for the maximum and the minimum singular value), which is transformed here into −1 σ ¯ [G−1 0 GR (jω)] < σ[I + (KG0 ) (jω)]
ω>0
(30)
This test is quite meaningful and will be illustrated below. The left hand side is independent of the controller; it starts from 0 at low frequency where the residual dynamics is negligible and increases gradually when the frequency approaches the flexible modes of the mirror structure, which are not included in the nominal model G0 ; the amplitude is maximum at the resonance frequencies where it is only limited by the structural damping. The right hand side starts from unity at low frequency where | KG0 |À 1 (KG0 controls the performance of the control system) and grows larger than 1 outside the bandwidth of the control system where the system rolls off (| KG0 |¿ 1). The robustness test looks like Fig.14; the critical point A corresponds to the closest distance between these curves. The vertical distance between A and the upper curve should not be smaller than the requested gain margin GM . When the natural frequency of the structure changes from f1 to f1∗ , point A moves horizontally according to the ratio f1∗ /f1 (increasing the frequency will move A to the right). Similarly, changing the damping ratio from ξ1 to ξ1∗ will change the amplitude according to ξ1 /ξ1∗ (increasing the damping will decrease the amplitude of A).
GM
10
A
-1
s [ I + (KG0) ] Structural uncertainty
1 -1
s [ G0 GR ]
0.1
0.01
0.1
1 [Hz]
f1
10
100
Figure 14: Robustness test (30). σ[I + (KG0 )−1 ] refers to the nominal system used in the controller design. σ ¯ [G−1 0 GR ] is an upper bound to the relative magnitude of the residual dynamics. The critical point A corresponds to the closest distance between these curves. The vertical distance between A and the upper curve should exceed the requested gain margin.
5
Loop shaping of the SVD controller
The SVD controller is based on the extended Jacobian, with the same compensator applied to all loops, H(s) = Ih(s) in (28). Because of the SVD controller structure, KG0 = Ih(s)
(31)
and all the singular values of KG0 are identical, σ[KG0 ] = | h(jω) | 14
(32)
Similarly, the sensitivity matrix has also identical singular values, σ[(I + KG0 )−1 ] =
1 | 1 + h(jω) |
(33)
and the controller design can be done as a SISO controller according to classical techniques [13] or frequency shaping using the Bode integrals [14]. Figure 15 shows the Bode plots and the Nichols chart of the controller used in this study; the compensator h(s) consists of an integrator, a lag filter, followed by a lead and a second order Butterworth filter. The crossover is fc = 0.25 Hz and the attenuation at the earth rotation frequency is 125 dB. The robustness margins are clearly visible on the Nichols chart [the exclusion zone around the critical point (−1800 , 0 dB) corresponds to (P M = ±450 , GM = ±10 dB)]. [dB]
f 100
[dB]
125dB
-100
10
45° 0 dB
GM = 25dB
0 f c = 0.25Hz
Earth rotation (1.16 10 -5 Hz) -5
10
-4
10
-3
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-2
[Hz]
10 dB
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-1
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[°] -100 -150
[°]
PM = 70°
-200 -250 -180° 10
-5
10
-4
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-3
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-2
[Hz]
10
-1
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Figure 15: Compensator h(s) common to all loops of the SVD controller. Right: Nichols chart [the exclusion zone around the critical point (−1800 , 0 dB) corresponds to (P M = ±450 , GM = ±10 dB)]. Left: Bode plots showing the gain of 125 dB at the earth rotation frequency and a cut off frequency of fc = 0.25 Hz.
6
Discussion
To illustrate the foregoing discussion, consider again the segmented mirror of Fig.6. Although this example does not correspond to a particular existing telescope, it is representative of the current generation of large telescopes (Keck, VLT); the parameters have been chosen so that f1 = 20 Hz and ξi = 0.01 uniformly. The SVD controller is that just discussed. Figure 16(a) shows the robustness test (30); with a ratio f1 /fc ' 80 the system exhibits a substantial gain margin of 45 dB with respect to control structure interaction. In order to examine more critical configurations, more representative of ELTs, the supporting truss is gradually softened by reducing the Young’s modulus of the bars forming the truss. Figure
15
60
GM=45dB
[dB]
40
A
20 0
f1 = 20 Hz
-20
60
[dB]
40
Limit of stability
A
20 0
f1 = 2.2 Hz
-20
10 -1
10 0
10 1
10 2
[Hz]
Figure 16: Robustness test (30) (a) Stiff supporting truss (f1 = 20 Hz); the gain margin is GM =45 dB. (b) Soft supporting truss at the stability limit. 16(b) shows the stability limit when A touches the upper curve; f1 = 2.2 Hz in this case3 . Figure 17 shows the evolution of the gain margin with the frequency ratio f1 /fc for various values of the damping ratio ξ. According to this plot, if the first structural modes of the supporting structure has a damping ratio of 1%, a gain margin GM =10 requires a frequency separation f1 /fc significantly larger than one decade.
GM [dB]
80
x .05
60
0
40
2 0.0 1 0.0 05 0.0
20
GM=10dB
0 UNSTABLE -20 10
1
10
2
f /f
1 c
Figure 17: Evolution of the gain margin with the frequency ratio f1 /fc for various values of the damping ratio ξi . 3 Note that the robustness test is only a sufficient condition; its failures implies only that instability is possible.
16
7
Conclusion
The numerical modelling of large segmented mirrors has been addressed. This model has minimum complexity to account for control-structure interaction. Two control strategies have been discussed, both based on a quasi-static behavior of the mirror. The control-structure interaction has been addressed through a frequency domain robustness test for MIMO systems; this test separates the dynamic response (residual) from the quasi-static response (primary) of the mirror. The evolution of the robustness test with the dominant frequency f1 and the damping ratio ξ has been analyzed. The study has been illustrated with an example involving 91 segments, 270 inputs (excluding the central segment) and 654 outputs (edge sensors and normal sensors), and a SVD controller; the procedure is very fast and allows extensive parametric studies. An important conclusion of this study is that, for a structural damping ratio of 1%, the frequency gap between the crossover frequency of the controller, fc , and the critical flexible mode, f1 , must be significantly larger than one decade. This condition may be more and more difficult to fulfill as the size of the telescope grows [15].
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References
[1] Gilmozzi, R., Spyromilio, J., The 42m European ELT: status, In Ground-based and Airborne Telescopes II - SPIE 7012 (2008), Larry M. Stepp, Ed. [2] Angeli, G.Z., Cho, M.K., Whorton, M.S., Active optics and architecture for a giant segmented mirror telescope, in Future Giant Telescopes (Angel, and Gilmozzi, eds.), 2002. Proc. SPIE 4840, Paper No. 4840-22, pages 129-139. [3] European Southern Observatory, The VLT White Book, ESO, 1998. [4] Dimmler, M. E-ELT Programme: M1 Control Strategies, E-TRE-ESO-449-0211, Issue 1, April 2008. [5] Geradin, M., Rixen, D., Mechanical Vibrations, Wiley, 1994. [6] Craig, R.R., Bampton, M.C.C, Coupling of Substructures for Dynamic Analyses, AIAA Journal, Vol.6(7), 1313-1319, 1968. [7] Guyan, R.J., Reduction of Stiffness and Mass Matrices, AIAA Journal, Vol.3(2), p.380, 1965. [8] Aubrun, J.N., Lorell, K.R., Mast, T.S. & Nelson, J.E., Dynamic Analysis of the Actively Controlled Segmented Mirror of the W.M. Keck Ten-Meter Telescope, IEEE Control Systems Magazine, 3-10, December 1987. [9] Aubrun, J.N., Lorell, K.R., Havas & T.W., Henninger, W.C., Performance Analysis of the Segment Alignment Control System for the Ten-Meter Telescope, Automatica, Vol.24, No 4, 437-453, 1988. [10] Doyle, J.C. and Stein, G., Multivariate Feedback Design: Concepts for a Modern/classical Synthesis, IEEE Trans. on Automatic Control, Vol. AC-26, pp.4-17, Feb.1981. [11] Maciejowski, Multivariable Feedback Design, Addison-Wesley, 1989. [12] Kosut, R.L., Salzwedel, H., Emami-Naeini, A., Robust Control of Flexible Spacecraft, AIAA J. of Guidance, Vol.6, No 2, 104-111, March-April 1983. [13] Franklin, G.F., Powell, J.D., Emani-Naemi, A., Feedback Control of Dynamic Systems, Addison-Wesley, 1986. [14] Lurie, B.J. and Enright, P.J., Classical Feedback Control, Marcel Dekker, 2000.
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[15] Preumont, A., Bastaits, R., Rodrigues, G., Active Optics of Large Segmented Mirrors: Scale Effects, Invited Lecture, Int. Conf. on Smart Structures and Materials, SMART’09, Porto, July 2009.
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