thanks to voice-coil actuators. This DM type, called Adaptive. Secondary Mirror (ASM), is intended for the Large Binocular. Telescope (LBT), located in Mt.
Automatic tuning of the internal position control of an Adaptive Secondary Mirror D. Mari, G. Agapito, S. Baldi, G. Battistelli, E. Mosca, A. Riccardi Abstract— One of the key component of an Adaptive Optics system is the deformable mirror. This mirror can correct the atmospheric turbulence effects by changing its shape. In the last years Adaptive Secondary Mirrors (ASM) have been developed and the Large Binocular Telescope (LBT) will be soon equipped with two ASMs. Each LBT ASM unit has 672 voice-coil force actuators to change the shape of the mirror shell. Because the actuators apply force, an internal position control is required. The LBT ASM internal control uses a local feedback of position and velocity, then the control law for each actuator has two characteristic parameters which define the closed-loop shell dynamics. In this paper, an analysis of the dynamical behavior of the ASM under the proposed control law is provided. Then an algorithm is suggested that, based on a simplified model of the shell dynamics, tunes the controller parameters by means of an automatic procedure, as a replacement for a manual procedure based on the operator skill.
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Below, the DMs generally used in adaptive optics are listed. •
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I. INTRODUCTION Adaptive Optics (AO) is a technology for the realtime correction of rapidly changing optical distortion. It is used in astronomical telescopes[1] and laser communication systems[2] to remove the effects of light wave front distortions induced by atmospheric turbulence (seeing) and in retinal imaging system to reduce optical aberrations[3]. Adaptive optics works by measuring the distortions in a wavefront light beam, with a Wave Front Sensor (WFS), and compensating for them, with a Deformable Mirror (DM). Moreover, a typical AO system (Fig.1) comprises an AO system controller in order to work in closed loop. The AO controller determines the reference signals which define the desired shape for the DM, thanks to the information obtained by the WFS and for each measurement update. The DM is the active part of the AO system, it compensates with its shape (correction phase) for the wavefront deformations (turbulent phase) induced by atmospheric turbulence.
Fig. 1.
AO control loop.
The WFSs generally used in adaptive optics are the following.
Shack-Hartmann WFS: It measures the wavefront slope (first derivative) on an aperture array. It is the most common WFS[4], [5]. Pyramid WFS: It measures, like the Shack-Hartmann WFS, the wavefront slope on an aperture array, but moreover it can modulate its sensitivity[4], [6]. Curvature WFS: It measures the wavefront second derivative (the curvature) and it is generally used with a bimorph DM[4].
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Piezoelectric DMs with discrete actuators: They are continuous faceplate mirrors deformed by an array of piezoelectric actuators. There are the most common DMs[4], [7]. Bimorph DMs: They are mirrors made from two thin layers of piezoelectric material bonded together that can bend. They are generally used in square or hexagonal arrays[4]. Deformable membrane mirrors: They are membrane mirrors deformed by electrostatic electrode actuators positioned under the membrane[8], [4]. Liquid crystal DMs: They make wavefront correction by fixing the actual path length and tuning the refractive index thanks to liquid crystals[4]. Ferrofluid DMs: They are mirrors made of a magnetic liquid whose surface is actuated by an array of current carrying coils[9]. Tilt Mirrors: They are plane mirrors build to compensate beam tilt, or, in astronomical telescopes, a tracking error of the telescope. They have generally actuators with large stroke[4].
The latter ones do not deform its shape, hence in astronomical telescopes equipped with AO systems tilt mirrors and deformable mirrors are always assembled in series because the DMs cannot compensate for beam tilt, or their tilt compensation is too small for the AO requirement[4]. In Fig.2, a scheme of an AO system for astronomical telescopes is shown. The reference source is a so called Guide Star (GS), which can be a natural bright star, or an artificial star[4]. The light beam from the GS goes through the atmospheric turbulence, which distorts the light wavefront. The distorted light wavefront reaches the telescope, which brings it to the actual AO system. The corrected light beam reaches the Scientific Instrument thanks to a Beam Splitter, which split the beam on the basis of the wave-length. This article will deal with a new type of DM for Astronomical Ground Based Telescope, a voice-coil DM. Osservatorio
The final goal of this paper is to suggest an automatic tuning approach for the controller parameters, in order to remove the reliance on the operator, and to allow the calibration of future 8000-10000 actuators ASMs. The rest of the paper is organized as follows. The structure of the ASM of LBT is briefly explained in Section II, where the advances with respect to other DMs are pointed out. In Sections III and IV, a theoretical analysis of the control law is carried on based on a distributed and, respectively, lumped parameter model. The control scheme acting on each single actuator is described more in details in Section V. Then, Section VI presents an experimental analysis of the deformable mirror dynamics that it is the starting point of the automatic tuning procedure proposed in Section VII. Finally, experimental results on a prototype of ASM are presented in Section VIII showing the effectiveness of the proposed procedure. II. A DAPTIVE S ECONDARY M IRRORS
Exapod Interface flange and structural support Fixed exapod Cooled electronics boxes Astatic levers Cold plate Reference plate Thin shell
Fig. 2.
Scheme of an AO system for astronomical telescopes.
Astrofisico di Arcetri (Firenze) is collaborating with A.D.S. International s.r.l., Microgate s.r.l., Aerospatial Engeneering Department of Politecnico di Milano, and Steward Observatory - University of Arizona to the production of a DM, that unlike other ones, is able to tilt and deform its surface thanks to voice-coil actuators. This DM type, called Adaptive Secondary Mirror (ASM), is intended for the Large Binocular Telescope (LBT), located in Mt. Graham (Arizona, USA). LBT ASM has a 911mm diameter and a 1.6mm thick elliptical surface, and it is deformed by 672 force actuators [10]. Each actuator is driven by a dedicate control law which consists of both a decentralized control loop and a centralized feedforward action. Such a control scheme has been designed on the basis of the dynamical characteristics of the mirror and it is already placed on the ASM. The control law depends on a certain number of design parameters that have to be tuned in order to achieve the desired control specifications. At the present time, the parameters calibration procedure is executed by a human operator. Hence, the dynamical performance of the ASM is sensitive to the operator skill.
Fig. 3. Left: adaptive secondary mirror (ASM) for LBT. Right: exploded view of the system.
In Fig.3, an ASM for LBT is depicted. The general layout of each ASM (Fig.4) consists of a ∼2mm thick deformable mirror (shell) with ∼1m diameter, a thick plate that provides the reference surface (reference plate), a third plate that supports and cools the actuators (cold plate), the actuators and the capacitive sensors [10]. The shell has a ∼5cm diameter central hole; here a mechanical support is placed, it gives a constraint which blocks the motions of the shell on its plane and is weak in the orthogonal direction. The actuators are voice-coil force actuators and they are distributed in 14 concentric rings (Fig.5). They keep the shell in magnetic levitation and so, thanks to the weak constraint given by the support, the shell can do rigid deformations, which are known as piston, tip and tilt. For each actuator, there is a colocated capacitive sensor which measures the local distance between the shell and the reference plate. In ASMs, unlike the other DMs type, the shell dynamics is controlled by an internal position control loop (Fig.7) which, working at 72 KHz, allows to achieve the right shape at least
in the AO system sampling time (1 ms). Such a internal loop is needed in order to control the force actuators. Otherwise, classical position (i.e. piezoelectric) actuators are controlled by the AO loop itself, like in [7]. In LBT, the ASM internal control is implemented by the following components. • 672 voice-coil force actuators. • 672 capacitive position sensors, each one co-located with one actuator. • 168 Digital Signal Processors (DSPs), each one connected to four Analogical to Digital Converters (ADCs) and four Digital to Analogical Converters (DACs) which work at 72 KHz. So each DSP acquires the measurements from four capacitive sensors and controls the colocated actuators. Moreover, each DSP board receives all reference signals from the AO controller at each measurement update.
Fig. 6.
ASMs placement in LBT: each one serves three focal station.
Fig. 7.
Fig. 4.
Details of ASM for LBT.
Fig. 5. Front view of reference plate in LBT. The 672 actuators are distributed in 14 concentric rings: the armatures of capacitive sensor colocated with the actuators are visible.
The ASM internal position control uses a set of DSPs, working in parallel, to set up a decentralized control strategy in order to be scalable to the future generation of telescopes (30-40m diameter Extremely Large Telescope), which will need 8000-10000 actuators and are currently under design phase [11], [12], [13]. In these telescopes, at the moment, the centralized approach for all actuators is not sustainable to give the necessary computational power and transmitting
ASM internal loop in AO control loop.
bandwidth. In fact, in both approaches the transmitting bandwidth is proportional to actuator number, instead the computational power is proportional to actuator number in decentralized case and to square of the actuator number in centralized case [14]. ASMs have some great advantages respect to the other DMs, introduced in the previous section. • Substituting the existing secondary mirror, ASMs allow a drastic reduction in the number of optical surfaces along the optical train and then a compact system with high transmission and low emissivity [15]. • Voice-coil force actuators have a large stroke (∼100µ m)1 [10]. • They have no contact actuators (less problems in case of actuator break/fault) [16]. • They have a high actuator density that allow the system to operate also with short light wavelength for 8-m class telescope [17]. • They can be used as ground level conjugated mirror in Multi-Conjugated Adaptive Optics (MCAO) and in Ground Layer Adaptive Optics (GLAO) [18], [19], [20]. • They serve all the focal stations of the telescope (Fig.6). Different techniques to control large DM are currently under study in other groups [7], [21], [22]. Before concluding this section, it is pointed out that for the experimental developments a prototype with 45 actuators, 1 The most common actuators, piezoelectric ones, have currently a stroke of ∼10µ m.
called P45, has been available. Such prototype reproduces the three more internal rings of ASM for LBT. In Fig. 8 the geometric layout of the actuators of P45 is showed.
P(k, s) =
19
21
100
16
22 17
9
26
29
50
11
12
18
10
14
15
1
2
3
25
0 33 −50
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−50
36
31 41
0
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42 7
−100
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45
39
6
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27
30 13
43 50
100
Fig. 8. Geometric layout of the actuators of P45: the control points are counted from 1 to 45.
III. D ESCRIPTION AND THEORETICAL ANALYSIS OF THE CONTROL LAW: DISTRIBUTED PARAMETER MODEL In this section, a preliminary analysis of the dynamical behavior of the used control law is carried on by resorting to a simplified distributed parameter model of the shell (thin mirror). To this end, the following simplifying assumptions are made [23], [24]: A1 A2
the shell is assumed to be flat and infinitely extended; the damping is assumed to be homogeneous.
Under such assumptions, the dynamical behavior of the shell can be approximately described by the general dynamical equation of a thin plate
ρb
1 Y (k, s) . = U(k, s) ρ bs2 + γ s + B|k|4
(2)
28
20
8
where | · | denotes the Euclidean norm, Y (k, s) = L {F {y(r,t)}} and U(r,t) = L {F {u(r,t)}}.2 Then, the transfer function from the applied pressure to the shell displacement takes the form
∂ 2 y(r,t) ∂ y(r,t) +γ + B∆2y(r,t) = u(r,t) , ∂ t2 ∂t
(1)
where r ∈ R2 is the position vector, t ∈ R+ denotes time, y(r,t) is the local shell displacement (i.e., the distance between the shell and the reference plate at location r), u(r,t) represents the local applied pressure, ∆ is the LaplaEb3 is the plate cian operator, γ is the damping, B = 12(1− σ 2) flexural rigidity, and E, σ and ρ are respectively the Young’s modulus, Poisson ratio, density of the material used for the shell and b its thickness. Let now s ∈ C be the Laplace variable and k ∈ R2 the spatial frequency. Then, by applying the Laplace transform L {·} with respect to time t and the Fourier transform F {·} with respect to position r, equation (1) can be rewritten in the equivalent form
ρ bs2Y (k, s) + γ sY (k, s) + B|k|4Y (k, s) = U(k, s)
Recall now that each spatial frequency k represents one point in the modal space, corresponding to a deformation mode of the shell of the type e j k·r = cos (k · r) + j sin (k · r) where j is the imaginary unit and · denotes the Euclidean product. Hence, once the spatial frequency k is fixed, the transfer function P(k, s) describes the dynamical response of the shell in one point of the modal space and, therefore, can be regarded as a modal transfer function. Further, it can be seen from (2) that each modal dynamics can be characterized by its stiffness B|k|4 , damping γ , and mass ρ b. In particular, for this simplified model, the modal stiffness increases with the spatial frequency of the mode, whereas the modal mass and damping remain constant. For the reader’s convenience, the Bode diagrams of the modal transfer functions for different values of the spatial frequency k are depicted in Fig. 9. Remark 1: It is worth pointing out that assumptions A1 and A2 are needed here so as to make the dynamical analysis of the shell more streamlined and provide a preliminary description of the basic features of the proposed control law. Of course, since the real mirror is not infinitely extended, a more accurate analysis should take into account also the boundary conditions which play an important role on defining the plate dynamics. Further, in the real system, as will be discussed in Section VI, the damping in the modal transfer functions (2) should not be constant for all the modes but rather mode-dependent. In the following sections, such assumptions will be relaxed and edge boundary effects will be taken into account. In order to track a reference signal y◦ (r,t), a two-degreesof-freedom dynamical controller is used to generate the control law applied to the shell. More specifically, the control action can be decomposed into two terms u(r,t) = uFB (r,t) + uFF (r,t) where uFB (r,t) and uFF (r,t) are the feedback and, respectively, feedforward actions. Such actions are obtained as uFB (r,t) = FF
u (r,t) =
−k p y(r,t) − kd y(r,t) ˙ , ◦ 2 k p + B∆ (y (r,t) ∗ h(t))
(3) (4)
where k p is the proportional gain, kd is the derivative gain, h(t) is the impulse response of some low-pass filter and ∗ stands for time convolution. In the Laplace-Fourier transformed domain, the control law takes the form U(k, s) = −C(s)Y (k, s) + F(k, s)Y ◦ (k, s) 2 In the following, we shall always use the corresponding capital letter to denote the Laplace-Fourier transform of a lower-case letter function.
reference signal y◦ (r,t) and shell displacement y(r,t): 0
10
W (k, s)
−2
10
−4
10
=
−6
Amplitude [m/N]
10
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10
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−2
10
0
2
0
2
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10 Frequency [Hz]
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0
Phase [deg]
−45
−90
−135
−180
=
−2
10
10
10 Frequency [Hz]
4
10
6
10
Fig. 9. Bode plots of the open-loop modal transfer functions for the case of ASM for LBT: b = 2 × 10−3 m, l = 0.03m, γ = 3.82 × 104 Ns/m. The material is Zerodur: ρ = 2530Kg/m, E = 91 × 109 N/m2 , σ = 0.24 [25].
with C(s)
=
F(k, s)
=
k p + kd s ,
k p + B |k|4 H(s) . 1
(1 + s/ωH )2
The main features of the proposed control law can be summarized as follows. i) The feedforward action is designed so as to achieve offset-free tracking of constant (in time) reference signals in that all the modal transfer functions (6) have unity gain at zero frequency, i.e., W (k, 0) = 1, ∀k. ii) The proportional gain k p modifies the modal stiffness whereas the derivative gain kd acts on the modal damping. In particular, for this simplified model, it turns out that applying the same feedback control law in each point r of the shell corresponds to applying the same feedback control law to each modal transfer function, i.e., in each point k of the modal space. Then, when both k p and kd are strictly positive, all the modal transfer functions are stable. iii) The derivative action is applied only to y(r,t) (and not to y◦ (r,t) − y(r,t)) in order to avoid very large values of the control action due to rapid variations of the reference signal. iv) The low-pass filter H(s) in the feedforward action serves the purpose of smoothing the time evolution of the reference signal, which typically is piece-wise constant in time and so is characterized by step-like transitions. v) The position gain k p , the velocity gain kd , and the cutoff frequency ωH are design parameters that can be tuned so as to shape the frequency responses of the modal transfer functions W (k, s) and achieve the desired control objectives. As to the last point, consider a reference signal consisting of a linear combination of N deformation modes e j ki ·r , i = 1, . . . , N, i.e., ! y◦ (r,t) =
N
∑ ai e j ki ·r
y◦ (t) .
i=1
A possible choice for the transfer function H(s) is a secondorder overdamped filter H(s) =
P(k, s) Y (k, s) = F(k, s) ◦ Y (k, s) 1 + P(k, s)C(s) B |k|4 + k p H(s) . (6) ρ bs2 + (γ + kd )s + B |k|4 + k p
(5)
with cut-off frequency ωH . It can be seen that the feedback action, generated by a proportional-derivative (PD) controller, is decentralized in that the value of uFB (r,t) at the generic position r depends only on the time-evolution of y(r,t) at r (coherently, the feedback controller transfer function C(s) does not depend on the spatial frequency k). On the other hand, the feedforward action is centralized due to the presence of the biharmonic operator ∆2 . It is an easy matter to see that the proposed control law yields the following closed-loop transfer function between
Then, in each point r of the shell, the closed-loop zonal transfer function (in the Laplace domain) from y◦ (r,t) to y(r,t) can be obtained as a linear combination of the modal transfer functions W (ki , s), i = 1, . . . , N, in that N L {y(r,t)} = ∑ αiW (ki , s) L {y◦ (r,t)} i=1
with
αi = ai e
j ki ·r
N
/
∑ ai e
i=1
j ki ·r
!
(7)
.
For the sake of clarity, it is pointed out that the ratio in (7) actually corresponds to a fictitious transfer function (that clearly depends on the particular reference signal due to the interactions) that however it is useful to describes the dynamical behavior at point r for a certain reference signal.
IV. D ESCRIPTION
AND THEORETICAL ANALYSIS OF THE CONTROL LAW: LUMPED PARAMETER MODEL
Besides assumptions A1 and A2, the theoretical analysis carried on in the previous section relied on the hypothesis that both sensors and actuators are continuously distributed on the shell (so that the feedback and feedforward actions (3)-(4) can be applied). However, in the ASM, there is only a finite number of co-located sensor/actuator pairs that are arranged on a grid of N points with circular geometry (e.g., see Fig. 8). Further, each actuation area can be considered negligible with respect to the area pertaining each control input. In view of these considerations, it seems reasonable to consider, instead of the distributed parameter model (1), a
0
10
−2
Amplitude
10
−4
10
−6
10
−8
10
0
10
1
10
2
3
2
3
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4
10
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0 −45 −90 −135 Phase [deg]
The fact that such a transfer function can be written as a linear combination of the modal ones implies that one can focus on the latter for controller design purposes. Thus, by choosing the design parameters so that all the closed-loop modal frequency responses W (k, jω ), not only satisfy the desired control objectives, but also have (approximately) the same bandwidth ωB and the same shape for ω < ωB , one can ensure that the closed-loop zonal transfer functions (7) provide a satisfactory dynamical response for each r. In practice, by choosing k p and kd so that all the loop gains P(k, jω )C( jω ) have a unity gain crossing frequency greater than the desired bandwidth ωB and an adequate phase margin, then one can ensure that all the frequency responses P(k, jω )C( jω )/ [1 + P(k, jω )C( jω )] are almost flat for ω < ωB . In this connection, since the open-loop transfer function for k = 0, representing the plate rigid mode, is the one characterized by both the smallest phase and the smallest bandwidth (see Fig. 9), it can be seen that fulfilling the (bandwidth and phase margin) specifics for such a rigid mode automatically ensure that such specifics are satisfied also by all the remaining modal transfer functions (at least for ω < ωB ). Further, by choosing the cut-off frequency ωH of the low-pass filter H(s) slightly greater than the desired bandwidth ωB , one can ensure also that all the closed-loop modal frequency responses decrease with a certain minimal slope for ω > ωB . For the reader’s convenience, an example of the achievable closed-loop modal frequency responses is provided in Fig. 10. Remark 2: Notice that, in principles, offset-free tracking of constant reference signals could be achieved by a purely decentralized controller provided that an integral action be included in the feedback loop. However, this solution has not been pursued here in order to avoid as much as possible the overshoot in the modal step responses. In fact, the presence of an integrator in the controller would imply that the loop gain P(0, s)C(s), associated with the rigid mode k = 0, has a double pole in 0 and, as well known, an open-loop double integrator always leads to a (possibly large) overshoot in the closed-loop step response (see [26, Chap. 8]). Thus, as discussed above, static errors are corrected with a centralized feedforward action. It will be shown in Section V that such a choice is well-suited to being implemented in the real controller architecture.
−180 −225 −270 −315 −360 0 10
1
10
10
10 Frequency [Hz]
4
10
5
10
6
10
Fig. 10. Bode diagrams of the closed-loop modal transfer functions with k p = 1.3 × 109 N/m, kd = 5.7 × 104 Ns/m, ωH /(2π ) = 1.43 KHz.
lumped parameter model wherein the spatial functions y(r,t) and u(r,t) are replaced by vectors y(t) ∈ RN and u(t) ∈ RN whose i-th components yi (t) and ui (t) represent the shell displacement and, respectively, the applied pressure at the ith point of the grid. As consequence, in place of (1), hereafter we shall consider the lumped-parameter system M¨y(t) + D˙y(t) + Ky(t) = u(t)
(8)
where M ∈ RN×N , D ∈ RN×N and K ∈ RN×N are, respectively, the mass, damping and stiffness matrices of the shell. For such a model, the mass matrix M is diagonal since the total shell mass is lumped at the grid points. On the other hand, D and K are full matrices that account for the interactions among the grid points. Notice that such a model naturally overcomes assumptions A1 and A2 since both the shape of the shell and the non-homogeneity of the damping can be taken into account by suitably choosing the matrices D and K. Notice also that system (8) corresponds to a MIMO plant characterized by the open-loop transfer matrix between u(t) and y(t) −1 . P(s) = Ms2 + Ds + K
In the following of this section, it will be shown that, also for the lumped parameter model (8), it is possible to design a two-degrees-of-freedom controller with decentralized feedback that jointly stabilizes the shell dynamics and achieves offset-free tracking. To this end, let the shell be controlled by a two-degrees-of-freedom controller with control law ◦
U(s) = −C(s)Y(s) + F(s)Y (s)
(9)
where Y◦ (s) is the Laplace transform of the reference vector y◦ (t) ∈ RN , C(s) is the transfer matrix of the feedback controller, and F(s) the transfer matrix of the feedforward controller. The feedback action is supposed to be generated by a decentralized PD controller C(s) = K p + Kd s
(10)
where K p and Kd are diagonal matrices whose i-th diagonal elements k p,i and kd,i correspond to the proportional and, respectively, derivative gains applied at the i-th grid point. Notice that, unlike the previous section, here the gains are supposed to be location-dependent in order to account for possible edge boundary effects. As to the feedforward action, it is generated by a centralized controller with transfer matrix F(s) = H(s) (K p + K) where H(s) is the same as in (5). Then, the closed-loop transfer matrix resulting from application of (9) to (8) turns out to be −1 (K + K p) . W(s) = H(s) M s2 + (D + Kd ) s + K + K p (11) Note now that a necessary condition for closed-loop stability is that the matrix K + K p be invertible (otherwise W(s) would have at least one pole in the origin). Thus, whenever the feedback controller C(s) stabilizes the shell, the proposed control law ensures also offset-free tracking of constant references in that W(0) = H(0) (K + K p)−1 (K + K p) = I with I the identity matrix of dimension N × N. As to the possibility of stabilizing the shell via decentralized feedback, it is convenient to consider separately each point of the grid by rewriting (8) as N
N
j=1
j=1
mi y¨i (t) + ∑ di j y˙ j (t) + ∑ ki j y j (t) = ui (t), i = 1, 2, ..., N , (12) where mi is the i-th diagonal element of M, di j the (i, j)-th element of D, and ki j the (i, j)-th element of K. With this respect, (8) can be seen as an interconnected system made up of N mutually coupled subsystems, each one with state yi (t) xi (t) = (13) y˙i (t) and state equation "
N
#
x˙ i (t) = Ai xi (t) + Bi ui (t) + ∑ Ei j x j (t) j=1
(14)
where Ai
=
Ei j
=
h
0 − mkiii
1 − dmiii
k
− mi ij
d
− mi ij
i
,
Bi =
0 1
.
Further, it can be easily seen that the feedback control action associated with the decentralized controller (10) corresponds to the application of a decentralized static state feedback of the type Γi xi (t) , i = 1, . . . , N, ui (t) = −Γ with
Γi =
k p,i
kd,i
(15)
Notice now that the interactions from other subsystems affect each subsystem in a very structured way. In fact, since each interconnection matrix takes the form Bi Ei j , one can see that each interaction Ei j x j (t) affects the i-th subsystem in the same way the local input ui (t) does; this special interconnection structure is known in the literature as matching condition [27]. Then, since all the pairs (Ai , Bi ) are controllable being in controller canonical form, one can invoke well-known results on stabilization of interconnected systems via decentralized static state feedback [28] and conclude that, regardless of the form of the matrices D and K, it is always possible to stabilize system (8) by suitably tuning the feedback gains Γ i (or, equivalently, the diagonal matrices K p and Kd in (10)). V. I MPLEMENTATION OF THE CONTROL LAW As anticipated in Section II, a digital realization of the proposed two-degrees-of-freedom controller is implemented on the real system by means of a set of DSPs. In Fig. 11 a block scheme is provided that describes the operations performed for each actuator-sensor pair of the shell by the corresponding DSP. The resulting discrete-time controller operates with a sampling period Ts of about 1.4 · 10−5 s. Hereafter, with a little abuse of notation, ui (k), yi (k), and y◦i (k) will indicate the values taken on by, respectively, the control input, the measured displacement, and the reference signal at the k-th discrete-time instant kTs . In order to apply the proposed control law, an experimental ˆ of the shell stiffness matrix K has been preestimate K liminary determined via calibration [29], [30]. Then, after receiving the desired reference signal y◦ (k) from the highlevel AO controller, the DSP computes the feedforward term N
ci (k) =
∑ kˆ i j y◦j (k)
j=1
ˆ Afterwards, both where kˆ i j denotes the (i, j)-th element of K. y◦i (k) and ci (k) are filtered by a digital filter H(z), obtained from H(s) in (5) by means of Tustin transformation (here z denotes the forward shift operator). Summing up, the control signal provided to the i-th actuator by the dedicated DSP takes the form ui (k) = k p,i [H(z)y◦i (k) − yi (k)] − kd,i D(z)yi (k) + H(z)ci (k) (16)
y◦ (k)
y◦i (k)
H(z)
ˆi K y◦f ,i (k) +
ci (k) H(z) ei (k)
c f ,i (k) Plant + ui (k)
+
k p,i
−
D/A
i-th Actuator
− i-th command D(z)
ydi (k) kd,i
Shell i-th displacement +
yi (k)
A/D
i-th Sensor − ni (t)
Fig. 11. Control law acting on the i-th actuator-sensor pair placed on the shell of LBT ASM: the control law includes a decentralized PD feedback action and a centralized feedforward action.
where D(z) is a discrete approximation of the derivative action. In particular, D(z) is obtained by applying the Tustin transformation to a continuous time filtered derivative action of the type s D(s) = , (17) (1 + s/ωD )2 where the cutoff frequency ωD /(2π ) is set equal to 15 KHz. Remark 3: It must be pointed out that the theoretical analysis carried on in the previous sections only takes into account the shell dynamics by, implicitly, considering negligible the dynamics of both the sensor and the actuator. As a matter of fact, the sensor and actuator transfer functions can be well-approximated by low-pass filters with cutoff frequency 27 KHz and 100 KHz, respectively. Thus, since the desired closed-loop bandwidth is approximately 1 KHz, one can argue that such transfer functions do not substantially modify the closed-loop dynamics of the shell (for example, referring to the analysis of Section III, the phase margins of the loop gains remain strictly positive and closed-loop stability is preserved). The same considerations hold also for the effect of replacing the ideal derivative action with its filtered approximation (17). It can be seen that each DSP applies a local PD control law that, however, exploits also the global information about ˆ the static behavior of the shell provided by the matrix K. Further, the proposed ASM internal control for the LBT has a fixed structure wherein some design parameters (the cutoff frequency ωH , the proportional and derivative gains k p,i and kd,i , i = 1, . . . , N) can be tuned in order to meet specific requirements on the step response of the shell. Typical specifications are settling time of about 1 ms and overshoot less than 10% (by settling time we mean the amount of time necessary for the tracking error to remain within a ±10% interval). On the basis of the theoretical developments of section III, a first attempt consisted in applying the same proportional and derivative gains for all the actuator/sensor pair. However, it was found experimentally on P45 that, even if stability
of the shell can be ensured, it is not possible to achieve the desired dynamical behavior (in terms of step response specifications). This state of affairs can be understood by recalling that, as already pointed out in Remark 1, the simplifying assumptions A1 and A2 do not hold for the real deformable mirror. For example, the local damping is not the same for each actuator because the shell is finitely extended. In fact, as we said previously, between the shell and the reference plate there is trapped a thin (∼ 60µ m) film of air producing viscous damping when the shell is moved away from or towards the reference plate and so it is reasonable to assume that this damping is much smaller at the edges than in the corner. Further, due to the boundary conditions, also the structural damping of the shell is not the same over the whole area.3 Hence, it has been made the conjecture that the damping varies along the radius of the shell. Experimentally it has been checked that the last conjecture is correct, because the step response specifications can be met by setting different proportional and derivative gains on the three rings of P45. VI. DYNAMICAL
ANALYSIS OF THE SHELL
In this section, a dynamical analysis of the shell behavior is presented that represents the starting point of the automatic tuning approach proposed in the next section. A. Static decoupling of the shell dynamics To this end, it is convenient to consider a singular value ˆ decomposition (SVD) of the estimated stiffness matrix K: ˆ = ΛΞ ϒT , K
(18)
where Ξ is a diagonal matrix and Λ , ϒ are orthogonal matrices, i.e., Λ Λ T = I and ϒ ϒ T = I. All the three matrices are ˆ The real-valued, square, and with the same dimensions of K. columns λ i , i = 1, . . . , N of Λ form an orthogonal basis of the control action space, whereas the columns υ i , i = 1, . . . , N 3 Notice, however, that the structural damping is believed to be negligible with respect to the viscous one.
of ϒ form an orthogonal basis of the shell displacement space. In particular, each column of ϒ represents a static deformation mode of the shell. Clearly, since the matrix ϒ is square, the number of excitable static modes equals the number of grid points or, equivalently, of actuators. By exploiting the SVD (18), it is possible to rewrite the shell dynamics as Ms y¨ s (t) + Ds y˙ s (t) + Ks ys (t) = us (t) ,
(19)
combination of all the mirror modes. In fact, the closed-loop modal transfer matrix between ys ◦ (t) = ϒ T y◦ (t) and us (t) takes the form −1 T Λ ϒ. Qs (s) = Ps −1 (s) Ms s2 + Ds s + Ks + k p Λ T ϒ Thus, unless conditions a)-c) are satisfied, Qs (s) is a full matrix and, consequently, one has ! N
U(s)
where
=
∑ Qsiℓ (s)λ i
R(s)
i=1
ys (t) = ϒ T y(t) , us (t) = Λ T u(t) Ms = Λ T M ϒ , Ds = Λ T D ϒ , Ks = Λ T K ϒ . Here, ys (t) and us (t) are the vectors of the modal coefficients of the shell displacement and, respectively, control action. Then, the transfer matrix of the shell in the space of the static modes turns out to be −1 = ϒT P(s) Λ . Ps (s) = Ms s2 + Ds s + Ks
A few comments on the modal dynamical equation (19) are in order. Supposing that a good calibration has been ˆ ≈ K, by construction the matrix Ks performed so that K is approximately diagonal and equal to Ξ . Thus, by means of the change of variables arising from the SVD (18), it is possible to statically decouple the shell behavior. Further, whenever the following additional conditions are satisfied a) K is symmetric; b) M = mI for some scalar m; c) D = α M + β K for some scalars α and β (Rayleigh hypothesis); it is immediate to see that the very same change of variables also dynamically decouples the shell behavior in that the matrices Ms and Ds (and consequently the modal transfer matrix Ps (s)) turn out to be diagonal. Notice that in this case an analysis similar to the one of Section III could be carried on, with the diagonal elements of Ps (s) being equivalent to the open-loop modal transfer functions (2). If, instead, at least one of conditions a)-c) is not fulfilled, the SVD (18) does not provide a clean decoupling of the shell dynamics. In fact, even applying a control input u(t) = λ ℓ u(t) that excites only the ℓ-th static mode, the shell displacement y(t) will in general be a combination of all the static modes, in that ! N
Y(s) =
∑ Piℓs (s)υ i
U(s) .
i=1
where Piℓs (s) is the (i, ℓ)-th element of Ps (s). We name this phenomenon as cross-talk. It is worth pointing out that, when a feedback control law is applied, the effect of modal cross-talk becomes even more evident. In order to see this, consider a simple proportional control law of the type u(t) = k p [y◦ (t) − y(t)] and let the reference signal be y◦ (t) = υ ℓ y◦ (t). Then, even if the reference signal only excites the ℓ-th static mode, due to modal cross-talk the actual control action turns out to be a
N
Y(s)
=
N
∑∑
Pjis (s)Qsiℓ (s)υ j
j=1 i=1
!
R(s)
where Qsiℓ is the (i, ℓ)-element of Qs (s). B. Experimental results In order to analyze the dynamical behavior of the shell and in particular the effects of the modal cross-talk, a batch of experiments has been performed on the P45 prototype. To this end, a proportional controller with a small positionindependent gain k p has been used for all the actuators (it has been observed experimentally that such a controller ensure stability and guarantees, for each mode, a high Signal to Noise Ratio (SNR)). In each experiment, a single static mode, say ℓ, has been excited by applying the modal reference signal (20) y◦ (k) = υ ℓ w(k) where w(k) is a discrete-time signal with a constant Power Spectral Density (PSD) in a frequency range including the working bandwidth. Then, the data set of the ℓ-th modal experiment is generated as {uℓ (k), yℓ (k), us ℓ (k), ys ℓ (k), k = 0, 1, . . . , T } where T is the simulation time. This makes it possible to compute an empirical transfer function estimate (EFTE) of the ℓ-th modal transfer function s (s) as the ratio of the discrete Fourier transform (DFT) Pℓℓ of the ℓ-th component of ys ℓ (k) to the DFT of the ℓ-th component of us ℓ (k). Such a transfer function approximately describes the dynamics associated to the ℓ-th shell static mode. Further, it is also possible to obtain, for each actuator i, an EFTE of the zonal transfer function when the ℓ-th static mode is excited by computing the ratio of the DFT of the i-th component of yℓ (k) to the DFT of the i-th component of uℓ (k). Such a transfer function approximately describes the dynamics associated to the actuator/mode pair (i, ℓ) (i.e., the dynamics associated to i-th actuator when the ℓ-th shell static mode is excited). In Fig. 12, a comparison of the dynamics of one static mode and of the corresponding actuator/mode pairs is provided (for the sake of clarity, only one actuator per ring is considered). A first, important observation is that all the transfer functions have the same dominant second order dynamics, thus indicating that: the SVD (18) decouple the
shell dynamics, at least to a first approximation; and, coherently with the theoretical analysis of Section III, the zonal dynamics of the ring follows the modal one. However, it can also be seen that, due to the fact that conditions a)-c) are not satisfied by the shell and modal cross-talk occurs, the considered transfer functions show different dynamical features (especially at high frequencies). This was predictable since, as already pointed out, the local mass and damping are different over the shell area due to the presence of the edges. −4
10
satisfying the overshoot and settling time requirements is applied. First, in the identification step, an estimate of the plant is determined. Then, an Internal Model Control (IMC) design is performed in order to determine the controller gains so that the closed-loop transfer function be close to the reference one. Since the plant characteristics can differ greatly from the estimated model, particularly during the initial learning stage, an iterative strategy, the so-called windsurfer approach, is adopted. The learning process progressively increases the cut-off frequency of the reference model, so that the plant characteristics can be learned more accurately over an increasing bandwidth and with less risk of implementing a destabilizing controller.
−5
10
Amplitude [m/N]
A. Complexity reduction procedure −6
10
−7
10
external ring dynamics central ring dynamics internal ring dynamics static mode
−8
10
2
3
10
10 Frequency [Hz]
0
−50
Phase [deg]
−100
−150
−200
−250
−300
external ring dynamics central ring dynamics internal ring dynamics static mode
2
3
10
10 Frequency [Hz]
Fig. 12. P45.
Experimental Bode plots concerning to a single static mode of
VII. I NTERNAL POSITION
By means of the theoretical analysis of Section III, it was shown that the fulfillment of the specifications for the mirror most critical mode, characterized by the smallest phase and the smallest bandwidth, ensures that they are also fulfilled for all the other modal transfer functions. This observation suggests that, instead of considering all the mirror static modes, one can restrict the attention only to the most critical ones. Further, since for a given mode the zonal dynamics generally varies radially due to cross-talk (see Section VI-B), it seems reasonable to consider one transfer function for each radial region by looking for the most critical actuator/mode pair for each radial region. This can be easily done when a data set generates as in Section VI-B is available. However, some precautions have to be taken. In fact, the shell geometry and the actuators arrangement are such that for each control point there are modes whose SNR is too low, because those modes have a node near such points. Thus, not all the modes excite with a sufficiently high SNR each control point of the shell. In order to take into account this effect, the significant modes are selected by applying the test
CONTROL TUNING PROCEDURE
In this section, an automatic approach for designing the control parameters of a generic ASM is presented. Since the experimental results show that the greatest dynamical differences are among the center and the edges of the mirror and the requirements can be met by varying the gains radially, the procedure focus on radial dynamics of the shell. To this end, it is convenient to cast all rings in a certain number of radial regions (for example, for P45 one can consider three regions corresponding to the three rings). At first, the procedure reduces the complexity of control design by selecting one actuator/mode pair for each radial region. These pairs, corresponding to SISO plants, represent the most critical cases in terms of stability. Next, for each selected SISO plant, an identification for the control procedure based on a reference closed-loop model
|ϒiℓ | ≥ η pvℓ ,
(21)
where η ∈ [0, 1] is some suitable threshold and pv is the mode peak-to-valley: pvℓ = max υiℓ − min υiℓ . i=1,...,N
i=1,...,N
(22)
where υiℓ is the i-th element of υ ℓ . Next, the Nyquist criterion is applied on the ETFEs of significant modes to find the most stability critical mode for each control point. In particular, for each point of the grid, the most critical mode is determined as the one whose corresponding zonal transfer function when such a mode is excited (determined as described in Section VI-B) has the minimum gain margin. Finally, for each radial region, a SISO plant is selected corresponding to the most critical actuator/mode pair. B. Controller parameters tuning algorithm Let us denote by P0 (z) the open-loop transfer function of a critical actuator/mode pair (determined as suggested in the previous section). Referring to Fig. 11, the goal is to
find two parameters, Γ = (k p , kd ) such that the closed-loop system satisfies the step response specifications. The latter ones can be expressed in terms of a reference model Mr (z) representing the desired closed-loop dynamical behavior. Then, in line of principles, the controller parameters can be determined via a IMC design technique [31] based on the solution of the optimization problem Γopt = argmin kJ(P0, Γ)k2 , Γ
(23)
where k · k2 denotes the H2 norm of a transfer function and J(P0 , Γ) = W0 (z) − Mr (z)
(24)
represents the discrepancy between the reference model Mr and the closed-loop system (k p + g0 )P0 (z) = (k p + g0)P0 (z)S0 (z) , 1 + (k p + kd D(z))P0 (z) (25) where S0 (z) is the sensitivity transfer function and g0 = 1/P0(1) is a gain used to ensure W0 (1) = 1 (corresponding to the term introduced by the centralized feedforward action). The reference model is chosen as a discretization of the second order overdamped filter 1 , (26) Mr (s) = (1 + s/ωM )2 W0 (z) =
which satisfies the step-response specifications for suitable values of the cut-off frequency ωM . Unfortunately, the minimization in (23) cannot be directly implemented since the open-loop transfer function P0 (z) is unknown. Further, at the moment, a method yielding a solution to the joint problem of identification and control design is not available in the literature. Nevertheless, it is possible to separately solve a closed-loop identification problem, given the controller, and a control design problem, given the plant model. Thus, iterative identification for control procedures have been proposed wherein stages of identification and stages of control design follow one after the other until convergence to the optimal controller has been achieved [31]. b In fact, denoting by P(z) a nominal model of the SISO plant P0 (z), application of triangular inequality yields b Γ)k2 + kJ(P0, Γ) − J(P, b Γ)k2 . kJ(P0 , Γ)k2 ≤ kJ(P,
(27)
Then, instead of directly minimizing kJ(P0 , Γ)k2 , a suboptimal controller can be obtained by minimizing separately the two performance functions in (27). In fact, supposing that from a previous step of the iterative procedure a stabilizing controller Γ− be available (albeit not satisfying the specifications), a novel nominal model Pb+ (z) and a novel controller parameters Γ+ can be obtained according to Pb+
Γ+
=
argmin kJ(P0 , Γ− ) − J(P, Γ−)k2 ,
(28)
=
argmin kJ(Pb+ , Γ)k2 .
(29)
P
Γ
With this procedure, a search for the gains (k p , kd ) which minimize kJ(Pb+ , Γ)k2 rather than kJ(P0, Γ)k2 is implemented. So it is necessary that the resulting control law be
robust against the plant perturbations. From the theoretical point of view, robustness of the control law can be analyzed by checking the condition kJ(P0 , Γ+ ) − J(Pb+ , Γ+ )k2 ≪ kJ(Pb+ , Γ+ )k2 .
(30)
However, in practice, this condition cannot be directly checked and, as a consequence, at each iteration the current gains have to be tested directly on the mirror. In the ASM unit there is already a supervisor which observes the shell dynamics and disables the current controller if it is dangerous for the shell (the supervisor is a higher level software algorithm that process the locally collected data: in particular every DSP monitors four actuators and sends the data to the supervisor). In this case the iteration has to be repeated with a more accurate nominal model. On the contrary, the iterations go on and the iterative procedure ends when the settling time specification is obtained. It is a nontrivial problem to solve the identification step (28). This is an identification criterion completely oriented to a particular control performance goal, where the mismatch between plant and nominal model is measured thanks to a control performance function of plant and model. Since the control performance function J(·) regards a feedback connection of the plant with the current controller, the identification criterion has to use data from closed-loop experiments. Referring to (24)-(26), the H2 norm to be minimized in (28) can be expressed in an explicit way as
(k p + g0 )P(z) (k p + g0)P0 (z)
−
=
1 + (k p + kd D(z))P0 (z) 1 + (k p + kd D(z))P(z) 2
1
(P0 (z) − P(z)) = 1 + (k p + kd D(z))P0 (z)
k p + g0
= 1 + (k p + kd D(z))P(z) 2
= S0 (z)(P0 (z) − P(z))(k p + g0)S(z) , (31) 2
which is a robust performance cost measuring the plant− model mismatch for a given controller Γ− = (k− p , kd ). Of course, it makes sense only if the controller stabilizes both P0 (z) and P(z). Therefore, it is possible to use a least-squares prediction error criterion to make identification. In particular, the two-stages method identification is used [32], [33]. 1) Two-stages identification strategy: The two-stages method separates the parameters vector θ , which characb (z) of (25), into two vectors terizes the nominal model W θ = [α β ], which are identified in different steps. (1) Estimate the β -parameters of the sensitivity function Sb+ (z) in the model u(k) = S(z, β )y◦ (k) + G(z)εu (k) ,
(32)
by the Prediction Error Method (PEM) [34], and construct an estimate of the control signal ub+ (k) = Sb+ (z)(k p + g0)y◦ (k) .
(33)
(2) Estimate the α -parameters of the nominal model Pb+ (z) using the model y(k) = P(z, α )b u+ (k) + εy (k) ,
(34)
by the PEM. Notice that the estimate Pb+ (z) minimizes the identification performance function (31) if Sb+ (z) =
1
1 + (k p + kd D(z))Pb+ (z)
.
(35)
The two-stages method is quite robust and rather simple to use. Moreover the complexity, i.e. the order, of S(z) and P(z) can be independently chosen: in particular, the order of S(z) can be higher than the order of P(z), in order to find a pratically unbiased estimate of S0 (z) in the first step [32]. C. Windsurfer approach Γin = (k pin , kdin ), ωHin , ωMin
Modal experiment on (P0 , Γι ) (rf , y, u)-data
Two-stages method identification Pbι +1
ι = ι +1 ωH = (1 + ν )ωH ωM = (1 + ν )ωM
IMC design
Γι +1
Stability test no yes
Robustness test no yes
ts =
2π ωM
< 1ms
no yes
Γopt = Γι +1
Fig. 13.
Diagram block of ASM control design strategy.
The IMC design allows to use the windsurfer approach which suggests to increase the cutoff frequency ωM of the reference model at each iteration [35]. In this way, the shell
dynamics can be led gradually to have the desired settling time, without the risk of damaging the shell with a dangerous transient. In fact, the iterative procedure can begin using a reference model Mr (z) with a small cutoff frequency ωM , which can not achieve the requirements but corresponds to low gains. When the shell reproduces correctly the reference model characteristics around the bandwidth [0, ωM ], the cutoff frequency ωM is slowly increased in the next iteration. With this idea, the gains increase gradually, and so the local overall stiffness and damping too (i.e. those of the closed loop system). In Fig. 13 a block diagram for the ASM control design strategy is presented. There are a lot of observations to make here. • In the stability test the supervisor controls that the shell behavior remains stable while the gains are increased until they reach the values Γι +1 on the respective radial regions. If the shell dynamics becomes dangerous, the supervisor disables the current gains and implements the ones of the previous iteration. In this case, the stability test can be repeated using a smaller ωH ; otherwise other choices are: to identify a more accurate (higher-order) model, to change the identification pre-filter, or to repeat IMC design reducing ωM . • During the robustness test, a set of experiments is made. In each one, the reference signals are steps which command to the shell to reproduce a particular mode with an ∼ 10µ m amplitude. Hence, the supervisor verifies that the shell reaches the desired modal configuration in ts = ω2πM . If the robustness test is not passed satisfactorily, the identification or the IMC design can be repeated. • At each iteration ι , the pre-filter cutoff frequency ωH is increased in order to guarantee the overshoot specification (ν > 0 and small). For this reason, it is necessary to initialize ωHin = (1 + µ )ωMin , with µ ≥ 0 (e.g. µ ≃ 0.4). • Generally speaking, the iterative process typically can continue until the control objective is achieved (desired bandwidth); otherwise the iterative process can prematurely terminate because fundamental performance limitations due to right-half-plane poles and zeros of the plant and/or models or no further improvements in the identified model can be made for a reasonably large set of input-output measurements. Thus, if the windsurfer approach fails, one can conclude that too restrictive control objectives have been imposed. However, even if the iterative process prematurely terminates, a controller is nevertheless obtained possibly characterized by the widest bandwidth (among those compatible with the specifications). Using the same reference model Mr (z) for all radial regions at each iteration, the same local dynamical behavior can be given to the shell. In fact, the gains k p and kd will be diversified on the radial regions in order to make equal the local closed loop stiffness and damping. VIII. E XPERIMENTAL VALIDATION
ON
P45
The identification and control design steps of the proposed procedure have been validated by means of experimental
Ring External
Central
Internal
No. act. per ring 21
15
9
Critical mode 1 2 3 10 11 1 2 3 1
Critical gains [N/µ m] [0.108,0.203] [0.087,0.106] [0.090,0.110] 0.128 0.150 [0.156,0.161] [0.108,0.117] [0.101,0.142] [0.094,0.099]
No. destabilizing act. 5 11 3 1 1 2 7 6 9
TABLE I C RITICAL MODES FOR EACH RING OF P45, WITH η = 0.2.
Ring External Central Internal
na 0 0 0
nb 9 7 7
nc 9 9 7
nd 9 9 7
nf 13 11 11
nk 3 3 3
TABLE III BB MODELS STRUCTURES AT THE SECOND IDENTIFICATION STAGE .
Ring External Central Internal
h 6 6 8
First stage ωL /2π [Hz] 2000 900 2500
h 8 6 6
Second stage ωL /2π [Hz] 1800 900 1500
TABLE IV Ring
Critical mode
External Central Internal
2 (tip) 3 (tilt) 1 (piston)
Minimum critical gain [N/µ m] 0.087 0.101 0.094
Actuator
CHARACTERISTICS OF THE IDENTIFICATION PRE - FILTERS USED DURING THE VALIDATION ON
31 14 40
TABLE II A CTUATORS ASSOCIATED WITH THE MINIMUM CRITICAL GAINS ON EACH RING OF P45.
data previously collected on P45. More specifically, a set of position error and velocity gains, fulfilling the specifications, was available from a previous manual calibration. Further, an experimental data set consisting of step responses of all control points for each modal experiment using such gains was also available. The validation goal is to verify that the operations of identification and control design, described in the previous section, return gains approximately equal to the manually calibrated ones. A. Complexity reduction First of all, the complexity reduction procedure of Section VI-A has been applied. For P45 it has been found experimentally, with a trial-and-error procedure, that suitable values for the parameters k pin and η are 0.06 N/µ m and, respectively, 0.2. Notice that, for this prototype, the radial regions coincide with the rings. The results after the Nyquist criterion application are summed up in Table I. It can be seen that, for almost all the control points (apart from two), the most critical modes are found among the first three shell modes (piston, tip, and tilt), which are the rigid modes. These results agree with the theoretical analysis of Section III. Table II reports the most critical actuator/mode pairs selected for the identification and control design steps: the actuator numbers refers to Fig. 8. B. Identification and control design After the complexity reduction procedure, the identification and control design steps have been carried out for the three actuator/mode pairs of TABLE II. In the identification step, a Black Box (BB) model has been used both at the first and at the second identification stage for each ring [34]. The complexity of the model has been tuned by resorting to the Akaike’s information criterion. As a result, the BB model used at the first stage is the same for all rings; referring to Chap. 4.2 of [34] it has na = 0, nb = 16,
External ring 0.0174
P45.
Central ring 0.0620
TABLE V P ERCENTAGE DISCREPANCY IN TERMS OF H2 SIDES OF
Manual calibration Validation
External kp 0.200 0.193
ring kd 70 68
Internal ring 0.0310
NORM BETWEEN THE TWO
(35).
Central kp 0.100 0.110
ring kd 35 37
Internal kp 0.100 0.105
ring kd 35 34
TABLE VI C OMPARISON BETWEEN THE MANUALLY CALIBRATED GAINS AND THE ONES OBTAINED VIA THE PROPOSED PROCEDURE .
nc = 9, nd = 9, n f = 16, nk = 1. As to the second stage, the model structures are summed up in Table III. In both stages, the pre-filter for the identification is a discretization of an h order low-pass filter L(s) =
1 , (1 + s/ωL)h
(36)
with cutoff frequency ωL . The pre-filter parameters are given in Table IV. As pointed out at the end of Section VI-B, the accuracy of the two-stages identification technique can be measured by comparing the left-hand and the right-hand side of (35). To this end, their percentage discrepancy in terms of H2 norm is reported in Table V. As to the IMC design, the reference model for each ring has been chosen so as to the reproduce the step response of the corresponding actuator/mode pair of Table II. In Table VI, the already-available gains, obtained by manual calibration, are compared with ones obtained after the identification and control design steps. In view of these results, one can conclude that: • the two-stages identification method allows one to correctly identify the dominant dynamics of each actuator/mode pair; • the IMC design yields gains that are very close to the manually calibrated ones (providing the desired behavior).
IX. CONCLUSIONS ASMs need an internal position control because the actuators used in this technology are force ones (voice-coil actuators). Up to the present, ASM internal position control is calibrated by a human operator whose skill affects the global performance of the mirror. In this paper, in order to eliminate the reliance on operator skills, a procedure has been proposed for the automatic tuning of the controller parameters so as to satisfy the step response specifications. The procedure relies on an iterative strategy, the windsurfer approach, which allows to gradually lead the closed-loop shell dynamics to achieve the desired behaviour, without the risk of damaging the shell with a dangerous transient. The final goal is to find gains such that the closed-loop dynamics meet the requirements and also equalize the local closed-loop stiffness and damping. For a generic ASM, the greatest dynamical differences are among the center and the edges of the shell. For this reason, in order to reduce the problem complexity, all rings have been split into a certain number of radial regions. The automatic tuning procedure finds, for each radial region, the most critical (in terms of stability) actuator/mode pair and applies to this pair an iterative algorithm that alternates identification and control design steps until the specifications are satisfied. Experimental validation on P45 has shown that the identification and control design steps of the automatic procedure are able to find values for the controller parameters that are quite close to the manually calibrated ones (with a difference of 10% for position error gains and 6% for velocity gains). R EFERENCES [1] J. M. Beckers, “Adaptive optics for astronomy - Principles, performance, and applications,” ARA&A, vol. 31, pp. 13–62, 1993. [2] T. Weyrauch and M. A. Vorontsov, “Free-space laser communications with adaptive optics: Atmospheric compensation experiments,” Journal of Optical and Fiber Communications Research, vol. 1, no. 4, pp. 355–379, December 2004. [3] A. Roorda and D. R. Williams, “Retinal imaging using adaptive optics,” in Wavefront Customized Visual Correction: The Quest for Super Vision II, R. r. Krueger, R. A. Applegate, and S. M. MacRae, Eds. SLACK, 2001, ch. 5, pp. 43–51. [4] S. K. Saha, Diffraction-Limited imaging with Large and Moderate Telescopes. World Scientific Publishing Co. Pte. Ltd., 2007. [5] L. Gilles and B. Ellerbroek, “Laser guide star shack-hartman wavefront sensor modeling: matched-filtering wavefront-sensor nonlinearity and impact of sodium layer variability for the thirty meter telescope,” B. L. Ellerbroek and D. B. Calia, Eds., vol. 6272, no. 1. SPIE, 2006, p. 62721A. [Online]. Available: http://link.aip.org/link/?PSI/6272/62721A/1 [6] O. Wulff and D. Looze, “Nonlinear control for pyramid sensors in adaptive optics,” B. L. Ellerbroek and D. B. Calia, Eds., vol. 6272, no. 1. SPIE, 2006, p. 62721S. [Online]. Available: http://link.aip.org/link/?PSI/6272/62721S/1 [7] C. Correia, H.-F. Raynaud, C. Kulcs´ar, and J.-M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A, vol. 27, no. 2, pp. 333–349, 2010. [Online]. Available: http://josaa.osa.org/abstract.cfm?URI=josaa-27-2-333 [8] E. Fernandez and P. Artal, “Membrane deformable mirror for adaptive optics: performance limits in visual optics,” Opt. Express, vol. 11, no. 9, pp. 1056–1069, 2003. [Online]. Available: http://www.opticsexpress.org/abstract.cfm?URI=oe-11-9-1056
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