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, macmardg@cds.caltech.edu.

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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