1
Dynamic analysis of the actively-controlled
segmented mirror of the Thirty Meter
TelescopeDouglas G. MacMartin, Peter M. Thompson, M. Mark Colavita and Mark J. Sirota
Abstract—Current and planned large optical telescopesuse a segmented primary mirror, with the out-of-planedegrees of freedom of each segment actively controlled. Theprimary mirror of the Thirty Meter Telescope (TMT) con-sidered here is composed of 492 segments, with 1476 actua-tors and 2772 sensors. In addition to many more actuatorsand sensors than at existing telescopes, higher bandwidthsare desired to partially compensate for wind-turbulenceloads on the segments. Control-structure-interaction (CSI)limits the achievable bandwidth of the control system.Robustness can be further limited by uncertainty in theinteraction matrix that relates sensor response to segmentmotion. The control system robustness is analyzed here forthe TMT design, but the concepts are applicable to anysegmented-mirror design. The key insight is to analyze thestructural interaction in a Zernike basis; rapid convergencewith additional basis functions is obtained because thedynamic coupling is much stronger at low spatial-frequencythan at high. This analysis approach is both computationalefficient, and provides guidance for structural optimizationto minimize CSI.
Index Terms—Telescopes, Control-structure-interaction
I. INTRODUCTION
Optical telescopes with primary mirror (M1) diameters
larger than about 8.5 m use a segmented primary mirror,
relying on active control of the out-of-plane degrees
of freedom to maintain a smooth optical surface; an
approach pioneered by the Keck telescopes [1], [2].
While the Keck telescopes each have 36 segments, the
design for the Thirty Meter Telescope (Fig. 1 and 2)
has 492 [3], while the 39 m European Extremely Large
Telescope (E-ELT) design has 798 [4].
The primary mirror control system (M1CS) for these
designs builds on the approach used at Keck, with
feedback from edge sensors used to control position
Manuscript submitted to IEEE TCST.D. MacMartin (formerly MacMynowski) is with Control & Dynam-
ical Systems, California Institute of Technology, Pasadena, CA 91125USA, [email protected].
P. Thompson is with Systems Technology Inc., Hawthorne CA.
M. Colavita is with the Jet Propulsion Laboratory, Pasadena CA.M. Sirota is with the TMT Observatory Corporation, Pasadena, CA.
Sensors (12)
Actuators (3)
Fig. 1. Conceptual image of the Thirty Meter Telescope design (left),
and detail of one primary mirror segment (right).
Fig. 2. The 492-segment primary mirror of TMT (left), and segment
actuator and sensor locations (right). Each segment has three positionactuators (‘+’) and two sensors on each inter-segment edge (‘•’) that
measure relative displacement, for a total of 1476 actuators and 2772sensors.
actuators on each segment (see Fig. 2), with an overall
surface precision of order 10 nm rms (though low spatial
frequency motion can be larger). However, for future
telescopes, the problem is more challenging because
of the greater number of segments, sensors and actu-
ators, higher desired control bandwidth, and stringent
performance goals. Aubrun et al. [1], [5] conducted the
dynamic control-structure-interaction (CSI) analysis of
the Keck observatory primary mirror control system,
and furthermore suggested that for a given structure,
the destabilizing effects scale linearly with the number
of control loops [6]; a potential concern given the
large number of segments in planned optical telescopes.
The purpose of this paper is to describe the dynamic
analysis of segmented-mirror control for large arrays of
segments, and for TMT in particular, 25 years after the
corresponding analysis for Keck was published [1].
In addition to the quasi-static gravity and thermal de-
formations controlled at Keck, M1CS at both TMT and
E-ELT will provide some reduction of the response to
unsteady wind turbulence forces on the primary mirror.
The increased bandwidth required to do so also requires
more careful attention to CSI than was required for
Keck. Furthermore, in addition to the “global” feedback
from edge-sensors, TMT will use voice-coil actuators
to control each segment; these are stiffened with a
relatively high-bandwidth servo loop using collocated
encoder feedback within the actuator; CSI must also be
analyzed for these control loops.
Finally, analysis would be incomplete without address-
ing one further complication that results from the large
number of segments. The edge-sensor based feedback
relies on knowledge of the interaction matrix that relates
sensor response to segment motion, in order to estimate
the latter from the former [7]. The condition number
of this matrix increases with the number of segments,
and thus small errors can result in large uncertainty in
the control system gain [8], [9]. Additional analysis is
required to ensure simultaneous stability in the presence
of both this effect and CSI.
Scaling effects for both dynamics and control of large
arrays of segments have been addressed in [10], [11], and
multivariable CSI robustness of the global control loop in
[12], using a more conservative test than the one applied
here (noted later). Progress in CSI analysis for TMT has
been described in a sequence of papers [9], [13]–[17],
and similar analyses for the European ELT in [18]–[21].
The key observation that allows for both rapid analysis
and design intuition is that the segment dynamics can be
analyzed in any basis. For a realistic control bandwidth,
the coupling with the telescope structure is primarily
an issue at low spatial frequencies. As a result, using
a Zernike basis (or something similar) yields rapid
convergence of stability and robustness predictions and
does not require analysis with all 492 segments of the
primary mirror. A higher control bandwidth may require
more basis vectors to predict robustness.
Several additional aspects to the segmented-mirror
control problem are worth noting. For the desired closed-
loop bandwidths, the computational burden of the real-
Segment
(492)
Telescope
structure
Segment
motionA
Edge
sensor
Wind
forces
Kact
Actuator
encoder
Actuator
force
A#
Kglobal
Position
command
Fig. 3. Block diagram showing control loops, both “local” actuator
servo loops (Kact) and “global” edge-sensor based feedback (Kglobal,
A#); the input and output of both Kact and Kglobal have dimension
1476. The dynamics of the segments and control loops will be coupledto the telescope structure (coupling points marked by solid circles) in
a different basis as described in Sec. III and IV.
time controller is not an issue; if it were, then approaches
developed for adaptive optics can easily be extended
to this problem, e.g. [22], [23]. The analysis herein
focuses only on the out-of-plane degrees of freedom
of each segment; in-plane motion does couple with the
out-of-plane control [24], but the effects are essentially
quasi-static and can be separately analyzed. Sensor noise
propagation can also be separately understood [7], [25],
although this may also limit the desired bandwidth of
poorly observed modes.
The next section introduces the control problem in
more detail, followed by analysis in Sec. III of a simpli-
fied problem that contains the most important features
of the full problem. The insights obtained are then used
in Sec. IV to compute CSI robustness for TMT. Finally,
Sec. V introduces interaction-matrix uncertainty and the
analysis required to prove simultaneous stability to this
and CSI.
II. CONTROL PROBLEM
A block diagram for the control problem is shown
in Fig. 3. Each segment of the mirror is controlled by
three position actuators (see Fig. 2), leading to a total of
1476 actuators for TMT. Several different actuator tech-
nologies have been considered, and voice-coils selected
based in part on low transmission of higher-frequency
vibrations to the mirror surface. Stiffness is obtained
using feedback from a local encoder with a bandwidth
of 8–10 Hz; each actuator uses the same controller. The
interaction of these 1476 control loops with the structural
dynamics is the most challenging CSI concern for TMT.
For an individual segment mounted on a rigid base
(rather than on the telescope structure), the uncontrolled
segment behaves roughly as a mass (mirror segment)
on a spring (actuator open-loop spring stiffness), with
2
100
101
102
10−8
10−7
10−6
10−5
10−4
Ma
gn
itu
de
(m
/N)
100
101
102
−200
−150
−100
−50
0
Frequency (Hz)
Ph
ase
(d
eg
)
Fig. 4. Open-loop actuator frequency response (force to collocatedencoder position) for a segment mounted on a rigid base, with (dashed)
and without passive damping. The high frequency resonance resultsfrom internal dynamics within the segment assembly. The largest
compliance that determines the lower resonant frequency comes froman offload spring within the voice-coil actuator. The piston response(three actuators on a segment driven together) is shown; the tip and
tilt responses are similar.
a resonance near 8 Hz (the segment piston and tip/tilt
resonances are not quite at the same frequency), with
the frequency response shown in Fig. 4. The addition of
eddy-current based passive damping within the actuator
makes control design much more straightforward, as will
be seen when the dynamics of the telescope structure are
accounted for. The segment first resonance damping ratio
in Fig. 4 is ζ = 0.75.
With the local actuator control loops closed, they
behave as position actuators for the global control loop.
The global control uses feedback from edge-sensors
between neighbouring segments to maintain the optical
continuity of the mirror, with a bandwidth of order 1 Hz.
Differential capacitive sensors [26] measure the relative
edge height discontinuity, similar to the approach used at
Keck; with two segments per edge there are 2772 sensors
for TMT.
The relationship between the segment motion at the
actuator locations, x, and the edge-sensor response y,
can be expressed through geometry [7] as
y = Ax + η (1)
with sensor noise η. The global control loop involves
first estimating x from y using the pseudo-inverse of
A, and then computing control commands. At Keck the
control is calculated for each actuator (“zonal control”);
for future telescopes, control will be calculated in a
modal basis such as that obtained from the singular value
decomposition of A (e.g. [12], [14]).
(a)
(b)
… …
m
k
pi
qi
Fig. 5. Schematic (a) of n identical oscillators coupled through asupporting structure, with disturbance forces fi and control inputs ui ;this simplified system captures important features of the full telescope
problem. With simplifying assumptions, a change of basis leads to ndecoupled systems of the form (b), where M and Ki are associatedwith the support structure.
Any sensor set that measures relative segment motion
results in the global rigid-body motion of the full mirror
(piston, tip and tilt) being unobservable (A is rank
deficient). The edge sensors at TMT are also sensitive
to the dihedral angle θ between segments (rotation about
the shared edge): the sensor output is a linear combina-
tion of the relative height between segments, and Leffθ,
where the effective moment arm Leff has units of length.
Without dihedral sensitivity, “focus-mode” would also
be unobservable; this pattern corresponds to a uniform
dihedral change for all segments, resulting in a change in
the overall M1 radius of curvature. With practical values
of Leff and with many segments, focus-mode estimation
in particular is sensitive to uncertainty in the matrix
A, and thus control of this mode in particular will be
constrained to a lower bandwidth.
III. PRELIMINARY ANALYSIS
Before considering the control system dynamics with
the full telescope structural model, it is useful to first
use some simplifying approximations to explore some
general characteristics of the problem. The schematic in
Fig 5(a) illustrates important features of the dynamics:
there are many identical subsystems (mirror segments)
coupled to each other through the telescope structure.
The key observation that simplifies analysis is that a
3
diagonal system of identical subsystems remains diag-
onal under any change of basis. Thus if the dynamics of
an individual segment are written as g(s), then for any
unitary matrix φ:
φT G(s)φ = G(s) where G(s) =
g(s) 0 · · ·
0. . .
... g(s)
where we assume that the dynamics of each segment are
identical (this is a very good approximation).
As an example, consider the case where the dynamics
of the support structure can be described solely by the
n displacements zi at the segment locations (so that
it has exactly n degrees of freedom and n structural
modes), and has uniform mass distribution. While this
is not a realistic assumption for design, it is sufficient
to illustrate some key scaling laws. For this case, the
modes of the structure evaluated at the segment mounting
locations provide an orthogonal basis for transforming
the segment dynamics. The transformation results in
n decoupled systems that each describe the coupling
between one structural mode and the corresponding
pattern of segment motion. That is, in this case, there
exists a basis that simultaneously diagonalizes both the
supporting structure and the segment dynamics.
We start by ignoring damping for simplicity, although
it will of course be critical to the control design problem,
and we represent each segment by a single degree of
freedom rather than three. Define x, z ∈ Rn as the
vectors of segment and structure displacement, and u,
f ∈ Rn the control inputs and disturbance forces. The
dynamics of the ith segment are described by
mxi + kxi = fi + ui + kzi (2)
The coupling structure dynamics are described by
Mz + Kz = −u + k(x − z) (3)
where K is the stiffness matrix, and the mass matrix
M = (M/n)In×n because of the assumed uniform mass
distribution, with M the total support structure mass, nthe number of segments, and In×n the identity matrix.
For any orthonormal basis φ ∈ Rn×n, with p = φT x,
q = φT z, f = φT f and u = φT u, then
mpi + kpi = fi + ui + kqi (4)
Furthermore, if φ are the modes shapes of the support
structure, so that φ diagonalizes K, then φTi Kφi = Ki
and
Mqi + Kiqi + nkqi = −nui + nkpi (5)
That is, the dynamics decouple into n independent
coupled-oscillator systems, as shown in Fig. 5(b).
Define ω =√
k/m as the oscillator natural frequency
if mounted on a rigid support, and the mass and fre-
quency ratios
µ =nm
Mand Ω =
(Ki/M)1/2
ω(6)
Then for each basis function i (dropping the subscript
for clarity) we have:
[
pq
]
+ ω2
[
1 −1−µ µ + Ω2
][
pq
]
=1
m
[
10
]
f +1
m
[
1−µ
]
u (7)
Scaling frequency by ω, the transfer function from a
displacement input (u/k) to output p is:
s2 + Ω2
s4 + (1 + Ω2 + µ) s2 + Ω2(8)
and the two modes are at frequencies
1 + µ + Ω2±
p
µ2 + 2µ + 2µΩ2 + 1 − 2 Ω2 + Ω4
2
!1/2
(9)
corresponding to in-phase and out-of-phase oscillation
between the structural mode and the corresponding pat-
tern of oscillator motion. If the support structure stiffness
is small compared to the oscillator stiffness (Ki nk),
then to first order the lower resonant frequency (normal-
ized by ω) isΩ
√
(1 + µ)(10)
which is just mass-loading of the telescope structure
resonance. For small mass ratio µ (support structure
massive compared to the total mass of the oscillators),
then the systems decouple. With damping b added in
parallel with the actuator, as in the TMT actuator design,
then the zeros of the transfer function are unaffected
(these correspond to zero motion across the actuator). An
approximate formula for the damping of the two modes
can be derived by neglecting the shift in the imaginary
part of the eigenvalues relative to their undamped values:
2ζ 'b
2
1±1 − µ − Ω
2
p
µ2 + 2µ + 2µΩ2 + 1 − 2 Ω2 + Ω4
!
(11)
For small µ, the mode involving mostly segment mo-
tion is significantly damped, while the mode involving
primarily mirror cell motion is only slightly damped.
Fig. 6 compares the frequency response from eq. (8)
with the frequency response for Zernike focus for TMT,
4
100
101
102
10−5
Ma
gn
itu
de
(m
/N)
Frequency (Hz)
100
101
102
−200
−150
−100
−50
0
Ph
ase
(d
eg
)
Frequency (Hz)
Fig. 6. Actuator open-loop frequency response for TMT focusmode (without added passive damping, solid), compared with theapproximate response from eq. (8) (dashed); the amplitude of the latter
is scaled to match the static gain.
using the models described in the next section. Actuator
damping is not included for ease of comparing the
resonances. There is a single resonance of the telescope
structure (shown in Fig. 9) that predominantly projects
onto Zernike focus. The mass and stiffness values Ω =0.81 and µ = 0.26 provide a good fit to the behavior for
the projection onto this Zernike.
A representative root locus for these values of Ωand µ is sketched in Fig. 7, using a PID controller.
Control design is straightforward for the uncoupled
system, however this controller destabilizes the coupled
structural mode when the segment is mounted on the
flexible telescope structure. The extent of destabilization
depends on the frequency separation of the pole and zero,
which again depends on the mass ratio (Eq. 10). Adding
passive damping to the actuator damps both modes and
increases the maximum stable gain of simple controllers,
but the gain will always be limited by the destabilizing
interaction with the coupled structural dynamics.
The case in Fig. 7 corresponds to Ω = 0.82, for
a structural resonance relatively close to the segment
resonance. Fig. 8 illustrates the behaviour for higher
frequency structural modes (using Ω = 1.6). With no
damping, the root locus topology is similar to before,
although now it is the lower frequency pole that involves
more segment motion, and thus the order of the pole
and zero introduced by the coupling to the structure is
flipped relative to before. With passive damping added,
both modes now have more damping, following from
Eq. (11) and the higher value of Ω. The added damping
and the shift in pole-zero order lead to resonances above
the segment support resonance being less of a robustness
−80 −60 −40 −20 0−100
−80
−60
−40
−20
0
20
40
60
80
100
Real part of pole
Imagin
ary
part
of pole
−2 −1 0 125
30
35
−80 −60 −40 −20 0−100
−80
−60
−40
−20
0
20
40
60
80
100
Real part of pole
Imagin
ary
part
of pole
−2 −1 0 125
30
35
Fig. 7. Root locus for actuator servo loop, using mass and stiffness
from TMT focus mode, and a PID controller (which yields the dampedzeros). Without any passive damping added, the closed-loop systemwith these parameters would be unstable (see inset). The addition
of passive damping in parallel with the actuator makes the controlproblem easier (bottom panel).
challenge than lower frequency resonances.
The main observations from this simple analysis are as
follows. First, that much can be gained by analysis in an
appropriate basis set (as opposed to considering individ-
ual segment motion). Second, recall that the analysis in
[6] suggested that destabilization due to CSI was approx-
imately linear in the number of control loops. While this
is true for a given structure, it is the mass ratio (nm)/Mthat is the relevant parameter. Increasing the number of
segments while keeping the areal density constant does
not affect stability. Third, the lowest frequency support
structure resonances will decrease in frequency relative
to their uncoupled values by an amount that again
depends on the mass ratio, leading to a pole-zero pair that
5
−80 −60 −40 −20 0−100
−80
−60
−40
−20
0
20
40
60
80
100
Real part of pole
Ima
gin
ary
pa
rt o
f p
ole
Fig. 8. Root locus as in Fig. 7 but with structure stiffness increased bya factor of four (corresponding to a structural mode at higher frequencythan the segment resonance). The case with no damping is shown
in black and is qualitatively similar to before. However, with passivedamping (red), then there is now more damping on both modes.
is a challenge for robust control design. The addition of
passive damping simplifies the control problem. Finally,
higher frequency structural resonances are both better
damped by added actuator passive damping, and the
order of the zero and pole are flipped in frequency, and
thus these present less of a challenge for CSI.
IV. CSI ANALYSIS FOR TMT
A. Structural models
We will rely on the previous analysis to provide guid-
ance in understanding the characteristics of the actual
telescope system. We first briefly introduce the structural
models we use, describe the shift to a different basis for
control, and then analyze CSI for both the actuator servo
loops and the global control loop.
The full CSI analysis for TMT relies on the finite-
element model (FEM) of the telescope structure. For
ease of model reduction while retaining both accuracy
and flexibility in modeling the segment dynamics, the
segments are not included in the telescope FEM. A
modal model is obtained from the FEM; 5000 modes
(up to nearly 100 Hz) are extracted, although only a few
dozen low frequency modes matter for CSI. Typically
500 modes (up to 30 Hz) are retained, with the static
correction included for truncated modes; convergence
with the number of modes retained has been verified.
Because the segment model is replicated up to 492
times, a simple lumped-mass model is fit to the detailed
FEM of an individual segment before coupling with
the main telescope model. This approach ensures that
the desired segment dynamics are retained regardless
of any model reduction performed on the main tele-
scope structural model, and allows flexibility in choosing
what segment dynamics to include – only the dynamics
associated with retained basis vectors are needed, as
described below. Model validation has been conducted
by constructing two fully independent models, one in-
terconnecting the component models in state-space, and
the other in the frequency domain; both yield identical
results for CSI predictions.
The structural damping is assumed to be 0.5% (e.g.
Keck damping is in this range [27]). From Fig. 7 this is
a critical assumption, since it determines the damping of
the zeros, which are unaffected by any actuator passive
damping.
B. Zernike basis
The structural modes of the telescope do not give an
orthonormal basis for describing segment dynamics (that
approximation might be reasonable if the mirror cell
supporting the segments was the only flexible component
of the telescope). However, it is still useful to project
the dynamics onto a different basis. Instead of modes,
we choose a Zernike basis (the natural basis on a
circle; polynomials of degree p in radius, and sines
or cosines azimuthally), which we modify slightly to
orthonormalize at the 492 segment locations to give a
unitary transformation. If we included 492 basis vectors,
there would be no computational savings relative to the
original untransformed system. However, the stability
characteristics can be accurately predicted with relatively
few basis vectors because the coupling is dominated by
the most compliant and hence lowest frequency modes
of the supporting structure. These are also the lowest
wavenumber modes, and thus predominantly project onto
the lowest order Zernike basis vectors. Fig. 9 shows the
mode shape for a representative low-frequency (9.3 Hz)
structural mode. Although this particular mode is not
exactly Zernike-focus, the mode is extremely well cap-
tured by its projection onto the lowest 15 Zernike basis
vectors (up to radial degree 4). For high wavenumber
motion that involves significant relative motion between
neighbouring segments, the support structure is relatively
stiff (see Fig. 10).
Note that, as in Fig. 9, any structural mode will project
onto multiple basis vectors, and conversely, any basis
vector will include dynamics associated with multiple
modes, and thus multivariable analysis is still required.
Although we do not rely on this, for TMT the Zernike-
basis nearly diagonalizes the structural dynamics, and
6
1 2 3 4 5 6 7 8 9 10−0.5
0
0.5
1
Zernike radial degree
Pro
jectio
n a
mp
litu
de
1 2 3 4 5 6 7 8 9 10−0.2
0
0.2
0.4
0.6
0.8
Zernike radial degree
Pro
jectio
n a
mp
litu
de
Fig. 9. Mode shape, evaluated on the primary mirror, of two
representative structural modes of the telescope, and their projectiononto a Zernike basis (rms of each component normalized by the overallrms across M1). The first is the predominant mode associated with
Zernike focus (e.g. in Fig. 6); over 93% of the rms displacement iscaptured by the projection onto Zernike-focus, and over 99% of therms captured by the projection onto basis vectors of radial degree 4
and lower. The second illustrates that not all modes project entirelyonto a single Zernike, nonetheless over 90% of the rms is associatedwith either astigmatism or coma, and again, 99% of the rms is captured
by the projection onto basis vectors of radial degree 4 and lower.
indeed SISO analysis for each Zernike is a good predic-
tor of the multivariable analysis. It is not immediately
obvious why this should be true. However, the mirror
cell that supports the segments is a truss that can be
reasonably approximated at low spatial frequencies as a
uniform circular plate, with corresponding flexible mode
shapes similar to Zernike basis functions.
C. Actuator servo loop
The transfer function between voice-coil force and
the nearly-collocated encoder position for a segment on
a rigid base was shown in Fig. 4, with and without
additional passive damping. The damping results in a
significantly easier problem for control, as suggested
by the root locus for the simplified system in Fig. 7;
any structural mode that has non-zero motion across
the actuator will be at least slightly damped (and those
modes that do not, do not matter). For a single segment
mounted on the telescope structure, the transfer function
is similar to the rigid-base case, and indeed it might not
be obvious that there is any potential stability problem.
However, the coupling is clear when the dynamics are
transformed into a Zernike basis, as shown in Fig. 11.
The multivariable robustness metric used here is to
require the maximum singular value of the sensitivity
1 2 3 4 5 6 7 8 9 1010
−8
10−7
10−6
Com
plia
nce
(m
/N)
Zernike basis function radial degree
Structure
Segment support
Fig. 10. Static compliance of telescope structure on Zernike basis.
The horizontal line illustrates the segment static compliance for com-parison; at low spatial frequencies the structure is soft compared to thesegments, at high the reverse is true and the coupling is small.
100
101
102
10−8
10−7
10−6
10−5
Frequency (Hz)
Ma
gn
itu
de
100
101
102
−200
−150
−100
−50
0
Frequency (Hz)
Ph
ase
(d
eg
)
Fig. 11. Actuator frequency response on telescope structure, Zernikebasis, including the first 21 basis elements (up to radial degree 5).This includes focus-mode, for which the frequency response without
actuator damping was shown in Fig. 6. The solid black line correspondsto a segment mounted on a rigid base (the damped case from Fig. 4).
to be less than two; this is a reasonable margin in the
absence of a specific understanding of the structure and
magnitude of the uncertainty (e.g. gain margin of two).
Note that [12] considers the dynamics to be an uncertain
perturbation on the static response, and uses the dynamic
model to estimate the size of the uncertainty bound,
while here we include the dynamics as part of the best
estimate of the plant, and require robustness to additional
uncertainty on the model. Either approach is reasonable
for the global control loop (considered in [12]) where the
7
bandwidth is much lower than the structural resonances,
but the approach of [12] is too conservative to allow any
control design for the higher bandwidth servo loop [20].
Because the encoder is nearly collocated with the
actuator, the transfer function will be phase-bounded
regardless of the structural coupling. Thus, rather than
relying solely on the model-predicted sensitivity, we
rely on collocation and phase stability between 15 and
300 Hz, and a high gain margin above 300 Hz where
collocation may not hold. The control design used here
is a simple PID with high-frequency roll-off, tuned so
that the desired robustness margin is satisfied; it is not
the details of gain choices that is important, but rather
the lessons learned.
With any particular choice of controller, nominal sta-
bility could be established by taking eigenvalues of the
full system with all segments, but this is computationally
intensive and does not provide useful design guidance.
Sedghi et al. [20] instead use characteristic transfer
functions or CTFs [28] to prove stability: taking the
eigenvalues qi(jω) of the transfer function matrix at each
frequency, then the multivariable system is stable if the
closed-loop system is stable for each qi. However, rather
than computing eigenvalues of the full 3nseg × 3nseg
system as in [20], in Fig. 12 we show that these CTFs
converge rapidly if the system is first transformed into a
Zernike basis. This amounts to a two-step procedure for
proving stability: retaining relatively few Zernike basis
elements results in a system with many fewer inputs and
outputs; a second frequency-dependent diagonalizing
transformation is then used to evaluate nominal stability
for this smaller subset, since the Zernike-transformed
system is still not diagonal. The effect of the neglected
higher-order Zernike basis elements on the first few
CTFs is small (i.e., diagonal dominance is satisfied),
and it is these first few CTFs that matter most for
stability. Starting with a Zernike transformation to isolate
the structural dynamics that couple most strongly with
the segment control system thus results in a substantial
computational savings that is essential during design.
The most important result obtained from transforming
to the Zernike basis is shown in Fig. 13. If the servo
loops are closed on a segment by segment basis, taking a
subset of segments distributed uniformly over the mirror,
then the peak sensitivity increases nearly linearly with
the number of loops closed, as suggested by [6], and
control of all 492 segments needs to be simulated in
order to accurately predict the peak sensitivity. However,
this simply reflects a gradual increase in the projection of
the control loops onto the low-spatial-frequency modes
that dominate the structural coupling. Using a Zernike
−220 −200 −180 −160 −140 −120 −100 −8010
−1
100
Phase (deg)
Magnitude
Fig. 12. Nichols plot for characteristic transfer functions (CTFs)of servo loop, illustrating convergence of stability and robustnesscalculations with Zernike basis. Blue lines show the Nichols plots of
the CTF for the full 1476×1476 system, while magenta lines showthe CTF Nichols plots calculated only for the first 6 Zernike basiselements (radial degree p ≤ 2); these are similar for the least-stable
elements of the full CTF. The Nichols plot corresponding to a singlesegment mounted on a rigid base is also shown for comparison (black,thick line). The red oval indicates a peak sensitivity of two.
basis, results converge almost immediately, since the
worst-case structural modes project almost entirely onto
low-order Zernike basis functions (mostly radial degree
one, and some onto radial degree two), and there is only
a small increase in the peak sensitivity with further basis
functions added.
The multivariable peak sensitivity is shown in Fig. 14
where only Zernike basis vectors up to a given radial
degree p are included. The peak sensitivity is remarkably
well predicted by SISO analysis with each Zernike
separately, shown in Fig. 15. While the system is not
sufficiently diagonally dominant to directly infer stability
without relying on the CTFs shown in Fig. 12, it is
nonetheless useful to consider SISO analysis of each
Zernike, as the correspondance between each peak in
the sensitivity and a particular Zernike can be used as
a guide to optimizing the telescope structural dynamic
characteristics.
If the control bandwidth is increased to 20 Hz (requir-
ing an increase in the frequency to which collocation
is satisfied), then the convergence behavior in Fig. 13
remains. The structural modes that result in the peak of
the sensitivity are still the lower spatial frequency and
thus also lower temporal frequency modes, which project
primarily onto the lowest Zernike modes. Not only are
higher frequency modes stiffer, and hence couple less
with the control, the pole-zero ordering is flipped as
8
0 100 200 300 400 5001
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Number of segments
||S
||∞
0 10 201.6
1.8
2
Fig. 13. The Zernike basis (red squares, and inset) is much moreefficient for predicting the maximum over frequency of the largest sin-
gular value of the sensitivity, ‖S‖∞. Results converge with relativelyfew basis vectors, while simply increasing the number of segmentsconsidered in the analysis increases the maximum singular value almost
linearly (blue circles).
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Frequency (Hz)
σm
ax(S
)
P
P
P T
TFF
p=0
p=0,1
p=0,1,2
p≤ 7
Rigid base
Fig. 14. Maximum singular value as a function of frequency for servoloop, for increasing number of basis vectors added by Zernike radial
degree p; the legend shows the maximum radial degree included, andthe dominant peaks are labeled with “P” if the peak is due to modesthat predominantly project onto piston, “T” if predominantly tip/tilt
modes, or “F” if predominantly focus and astigmatism.
seen in Fig. 8 and 11, and the damping of these modes
is higher, leading to the smoother sensitivity at high
frequencies in Fig. 14.
The low-order basis approach motivated by the sim-
plified analysis in Sec. III is thus useful both for intuition
about which structural modes matter, and for fast design
iterations enabled by the rapid convergence with the
−250 −200 −150 −100 −50 0 5010
−2
10−1
100
101
Phase (deg)
Magnitude
Fig. 15. SISO Nichols plot for servo loop when mounted on a rigidbase (black, thick line) and for each of the first 21 Zernike basis vectors
(up to radial degree 5) plotted separately. The peak SISO sensitivityis 1.85, only slightly lower than the peak multivariable sensitivity inFig. 14. The red oval indicates a peak sensitivity of two.
number of bases included.
D. Global loop
In the design of Keck, it was the dynamic stability
analysis of the global (edge-sensor feedback) control
loop that was required [1], [5], and integral control was
assumed. Including additional roll-off above the control
bandwidth greatly reduces the CSI; here we use
C(s) =ki
s
1
(1 + s/α)2with α ' 4ki (12)
If the interaction-matrix (A in Eq. (1)) is known per-
fectly, then it can be inverted, giving a perfect estimation
of segment motion, other than the unobservable piston,
tip, and tilt of the overall primary mirror. With this
assumption, the multivariable peak sensitivity for the
global loop is plotted in Fig. 16, using a Zernike basis
for the control, with bandwidths indicated in the caption.
(The sensitivity is evaluated at the output; the input
sensitivity is indistinguishable.) The peak sensitivity
results from the phase lag introduced both by the servo
loop command response and by the roll-off in Eq. (12).
The “ripples” near 2 Hz result from choosing different
bandwidths for different radial degrees.
From Fig. 14 for the servo loop, all of the signif-
icant structural modes that cause coupling are above
5 Hz, where the global control loop has small gain, and
thus there is little interaction with these modes for the
bandwidths considered here. However, robustness cannot
9
0 5 10 150
0.5
1
1.5
2
Frequency (Hz)
σm
ax(S
), σ
ma
x(L
)
5 6 7 80.2
0.3
0.4
0.5
0.6
Fig. 16. Global CSI maximum sensitivity (blue) and principal loopgain (red). Interaction-matrix uncertainty results in an uncertain gain,indicated here with shaded bands; the inset shows the worst-case
principal loop gain after accounting for A-matrix uncertainty. The caseplotted here corresponds to 1.5 Hz bandwidth on radial degrees 4 andhigher, 1.25 Hz on radial degree 3, and 0.75 Hz on radial degree 2,
with the shaded band corresponding to an uncertain gain factor of 1.2to account for A matrix uncertainty (see Fig. 19(a)).
(a)
(b)
! " " #$ % & '
( ) * + , - . + , / . )
0 1 2 3 4 5 6 7 8 9 9 :
; <= > ? @A B > ? @C D
E F G H I J K H I L K F
Fig. 17. Interaction matrix uncertainty: The uncertainty in sensor gainS = I+δs and actuator gain X = I+δx are explicitly separated fromA. The plant and control dynamics are G(s) and K(s) respectively,
with G(0) = I . The unitary matrix Ψ transforms into- and out-of amodal basis with diagonal estimator gain matrix Γ; required stabilitymargins can be reduced by considering the norm of ΨT ΓΨB0SA.
yet be concluded without analysis of interaction matrix
uncertainty. Very small errors in A can result in large
errors in its inverse, and the resulting gain uncertainty
needs to be accounted for in evaluating robustness.
V. INTERACTION MATRIX UNCERTAINTY
Robustness of the global control loop is complicated
by uncertainty in the interaction matrix A in Eq. (1).
The condition number of A scales with the number of
segments, and thus robustness to small errors has the
potential to be a larger challenge for large segmented-
mirror telescopes such as TMT than at Keck. The
Fig. 18. Example of the effect of A-matrix uncertainty, for a 0.1%uncorrelated random uncertainty in every sensor gain. The pattern onthe left results in the estimated pattern on the right; roughly the correct
pattern plus a comparable amplitude of focus-mode.
condition number (and thus quantitative results in this
section) also depend on the sensor sensitivity to dihedral
angle changes; for TMT, Leff = 0.052 m. Uncertainty or
variation in sensor gain we explicitly separate out of Awith a diagonal gain matrix S = I + δS as shown in
Fig. 17, for reasons that will be clear. Define B0 as the
pseudo-inverse of the nominal matrix A0. The product
Q = B0SA is ideally the identity (except for projecting
out global piston/tip/tilt), but will differ for A 6= A0 or
S 6= I.
There are several sources of uncertainty. Uncertain
actuator gain has no significant effect on robustness.
However, uncertain sensor gain can have a significant
effect, if the uncertainty is uncorrelated between sensors,
even for gain errors of order 0.1%. With TMT sensors,
the ratio of dihedral and height sensitivity Leff also varies
with changes in the gap between segments [26], but the
sensors also measure gap, and hence this effect is easily
corrected. Finally, sensor installation tolerances affect
every non-zero element of A independently.
If Q has an eigenvalue less than zero, then regardless
of how small the control bandwidth, the closed-loop
system will be unstable. Uncertainty in sensor gain alone
can never lead to this type of instability (barring a sign
error); if A = A0 then Q will be positive-semi-definite
if S is. Similarly, Leff variations cannot cause this type
of instability as the dihedral and height sensitivity affect
different singular vectors of A, as shown in [8]. This
type of instability can occur for errors that independently
affect every non-zero element of A [8]. This is in
principle possible for sensor installation errors, as noted
above, although we have not observed this at realistic
tolerances [9]; it is also possible if A is measured rather
than calculated.
Of more direct relevance to CSI analysis is that
the maximum singular value σ(Q) can be large even
10
if the eigenvalues are all stable. While this is not a
stability problem in the absence of dynamics, it can lead
to performance issues, and more critically, can couple
with CSI to result in instability. Large singular values
correspond to a large (multi-variable) gain change. To
accomodate this uncertainty naively requires a large gain
margin and a corresponding limit on control bandwidth
to guarantee stability; see Fig. 17(a).
The particular displacement pattern that has the largest
effect depends on the specific errors and is therefore
not predictable; an example is shown in Fig. 18, where
0.1% uncorrelated uncertainty in the sensor gains gives
σ(Q) = 1.32. However, the error is in the least observ-
able, most spatially smooth modes, and focus-mode in
particular. The mechanism by which instability is possi-
ble (though unlikely) is if a focus-mode force command
leads to excitation of a structural resonance that also
includes some of this particular high spatial-frequency
pattern; this in turn would result in a larger erroneous
focus-mode estimate and corresponding control system
correction, and so forth.
To guarantee stability, then rather than constrain the
gain of all patterns of motion by an extra factor of σ(Q),the directionality information can be used, and only the
gain of the lowest spatial frequencies reduced. Define
Q = ΨT ΓΨQ, where Ψ is a unitary transformation into
a Zernike or similar basis and Γ a diagonal matrix to
reduce the estimator gain of low spatial frequencies.
Fig. 19 illustrates the dependence of σ(Q) on focus
and astigmatism gain reductions, for 0.1% and 1%
uncorrelated uncertainty in sensor gains. The highest
singular value is limited first by the focus gain, next
by astigmatism gain, and further reductions below 1.15
or 3.5 in these two cases would require reductions in the
gain of trefoil and coma. Note that these factors are in
addition to any gain reductions on low order modes that
are imposed by the dynamics.
For TMT we expect that 0.1% uncorrelated sensor
gain uncertainty is achievable. From Fig. 19, reducing
focus-mode gain by a third reduces the maximum sin-
gular value to σ(Q) = 1.2. This factor can then be used
as an additional uncertain gain in CSI analysis, as in
Fig. 16. This may still be conservative, but has only a
minor impact on the achievable bandwidth of the global
loop, and hence on the resulting M1CS performance.
VI. CONCLUSIONS
Planned large optical telescopes are enabled by ac-
tive control of the segmented mirror, but the control
bandwidth is limited by control-structure interaction.
Analyzing the dynamics in an appropriate basis results
Focus relative gain
Astig
ma
tism
rela
tive
ga
in
1.1
5
1.2
1.2
5
1.3
0 0.2 0.4 0.6 0.8 10
0.5
1
3.5
4
4.5 5
5.5
Focus relative gain
Astig
ma
tism
rela
tive
ga
in
0 0.2 0.4 0.6 0.8 10
0.5
1
Fig. 19. Maximum singular value of Q = ΨT ΓΨB0SA as a function
of estimator gain reductions (in Γ) on focus and astigmatism, and for0.1% (top) and 1% (bottom) uncorrelated uncertainty in sensor gainS.
in (i) rapid convergence in stability and robustness calcu-
lations with few basis vectors included, reducing com-
putation time and thus time between design iterations,
and (ii) provides intuition regarding important aspects to
the coupling, which can be used for design guidance
for structural optimization. The telescope structure is
only soft at low spatial frequencies, and thus CSI is
only significant for low spatial-frequency patterns of
segment motion. The strength of the coupling depends
on the mass ratio; the total mass of all of the segments
compared with the modal mass of flexible modes. For
TMT, CSI is primarily a concern for the actuator servo
loops, since these operate at a higher bandwidth than the
global edge-sensor feedback that maintains the optical
performance of the primary mirror segment array.
Robustness of the global control loop is also com-
plicated by uncertainty in the interaction matrix that
relates edge-sensors to segment motion. Because of ill-
conditioning (low spatial frequency displacement pat-
terns are less observable), estimation is quite sensitive
to small errors in this matrix. Once again, analysis in an
appropriate basis shows that the gain of the estimator
only needs to be reduced for these poorly observed
low spatial-frequency patterns; this results in only a
small increase in the required stability margins for CSI
analysis.
ACKNOWLEDGMENTS
The TMT Project gratefully acknowledges the support
of the TMT collaborating institutions. They are the Asso-
ciation of Canadian Universities for Research in Astron-
11
omy (ACURA), the California Institute of Technology,
the University of California, the National Astronomical
Observatory of Japan, the National Astronomical Obser-
vatories of China and their consortium partners, and the
Department of Science and Technology of India and their
supported institutes. This work was supported as well
by the Gordon and Betty Moore Foundation, the Canada
Foundation for Innovation, the Ontario Ministry of Re-
search and Innovation, the National Research Council of
Canada, the Natural Sciences and Engineering Research
Council of Canada, the British Columbia Knowledge
Development Fund, the Association of Universities for
Research in Astronomy (AURA) and the U.S. National
Science Foundation.
MMC is employed at the Jet Propulsion Laboratory,
California Institute of Technology, which is operated
under contract for NASA.
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