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Douglas MacMynowski (Caltech)Peter Thompson (Systems Tech.)Mark Sirota (TMT Observatory)
Analysis of TMT Primary Mirror Control-Structure Interaction(SPIE 7017-41)
TMT.SEN.PRE.08.046.REL01 2
Control Problems
TMT.SEN.PRE.08.046.REL01 3
Primary Mirror Control System (M1CS)
Sensors – measure relative displacement between segments,
estimate segment position:
Actuators: approach TBD, but candidates include – “Soft” (e.g. voice-coil), stiffened by servo loop– “Hard” (e.g. screw-type actuator), here I assume open loop
Control– Servo control loop (if soft actuators used),
~20 Hz bandwidth to obtain low frequency desired stiffness (see 7012-58)
– Global control loop: need ~1 Hz bandwidth to reject wind (see 7017-31)
Mirror cell introduces structural coupling that affects both of these
TMT.SEN.PRE.08.046.REL01 4
Outline
Primary Mirror Control System
Control-Structure Interaction
TMT structural dynamics
Actuator servo control loop
Global M1 control loop
TMT.SEN.PRE.08.046.REL01 5
Control-Structure Interaction (CSI)
Design without knowledge of structure – clear potential for problems– Control (both global & servo) can couple into structural modes
Potential for interacting control loops, e.g. each loop stable on its own, but the combination is unstable– Aubrun & Lorell study on ASCIE in early 90’s: destabilization
proportional to number of control loops– Aubrun & Lorell Keck CSI analysis: 0.5 Hz maximum bandwidth
Start with a really simple problem to build intuition (see paper), then apply to TMT M1CS– It is the ratio of total segment mass n × m to mirror cell mass M that
determines stability (not n alone)
TMT.SEN.PRE.08.046.REL01 6
M1 Dynamics
ui
… m
k
m m m
k kk
xi
zi
fi
Mirror cell
492
copi
es!
TMT.SEN.PRE.08.046.REL01 7
Computational Simplification
Use structural modes: good intuition for isolated mirror cell, but actual modes are not orthogonal when evaluated at segment loc’ns.– Use to show that coupling depends on total mass ratio μ=(n×m)/M
Analyze dynamics in Zernike basis– Coupling converges with relatively few basis vectors
More convenient to describe patterns of segment motion rather than individual segment motion
– Dynamics will be coupled; MIMO analysis is required
Z2,0 Z2,+2 Z2,-2
TMT.SEN.PRE.08.046.REL01 8
Telescope Structure
Finite element model of full telescope without segments- Compliance is highest at
low spatial frequency- Assume 0.5% damping
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 (LR)Segment support
TMT.SEN.PRE.08.046.REL01 9
Modeling Approach
Telescope FEM has no segment dynamics (intentionally)
Allows relevant segment dynamics to be added as needed for analysis using separate (detailed) FEM of segment assembly– Segment model reduced to 33 modes for CSI analysis
Telescope model and segment model connected at actuator nodes.– Projected into Zernike basis
segment (w/ rigid body modes)
cell
Connect withstiff spring
TMT.SEN.PRE.08.046.REL01 10
Idealized Actuator Assumptions
Soft (voice-coil) actuator:– Local position (servo) loop between force
and collocated displacement– Offload spring in parallel– Potential to add damping in parallel
Leads to a simple servo control
Hard actuator (e.g. screw + piezo):– Perfect displacement device (output =
command at all frequencies)– Pure stiffness (1e7 N/m) with zero damping
m
k=1.6e5 N/m
m
k=1e7 N/m
d
f, y
TMT.SEN.PRE.08.046.REL01 11
Servo open-loop TFon structure: Zernike basis
Transfer function from voice-coil force to position– In Zernike basis,
mounted on structure– Note maximum structure
resonance included is 36 Hz (static correction retained to 2000 modes, ~70 Hz)
– Note that response on different basis functions is NOT orthogonal; multivariable stabilityrobustness analysis is required! 100 101 102
10-8
10-6
10-4
Frequency (Hz)
Res
pons
e (m
/N)
Z0,0Z1,+1Z1,-1Z2,+2Z2,-2Z2,0Rigid
Coupling totelescopestructure
100
101
102
10-8
10-6
10-4
Frequency (Hz)
Res
pons
e (m
/N)
Rigid
Segmentsupportmodes
Segmentpiston/tip/tilt, onoffload spring
TMT.SEN.PRE.08.046.REL01 12
Servo open-loop TFon structure: Zernike basis
Transfer function from voice-coil force to position– In Zernike basis,
mounted on structure– Note maximum structure
resonance included is 36 Hz (static correction retained to 2000 modes, ~70 Hz)
– Note that response on different basis functions is NOT orthogonal; multivariable stabilityrobustness analysis is required! 100 101 102
10-8
10-6
10-4
Frequency (Hz)
Res
pons
e (m
/N)
Z0,0Z1,+1Z1,-1Z2,+2Z2,-2Z2,0Rigid
Coupling totelescopestructure
Segmentsupportmodes
Segmentpiston/tip/tilt, onoffload spring
TMT.SEN.PRE.08.046.REL01 13
Servo Loop Design
Two strategies:1. No change to plant: difficult to simultaneously navigate CSI with
telescope (most low frequency) and interaction with segment dynamics (high frequency)
2. Add passive damping in parallel to actuator: stabilizes 90 Hz+ segment dynamic modes
– See P. Thompson et al.for details (SPIE 7012-xx)
10-1 100 101 102106
107
108
Frequency (Hz)
Gai
n (N
/m)
TMT.SEN.PRE.08.046.REL01 14
Multivariable (MIMO)Robustness Evaluation
SISO analysis is not sufficient (doesn’t capture coupling)– Generalization is H∞: Want ||S||∞ < 2
(analogous to SISO GM/PM constraints)– Plot the maximum singular value
of multivariable sensitivityat each frequency
Distance tocritical point is|1+L| = |S(jω)|-1min|1+L|>1/2 ⇒GM > 6dB, PM > 30°
TMT.SEN.PRE.08.046.REL01 15
Servo Loop Robustness
Robustness is limited by structural modes that project almost entirely onto Zernike radial degree 2 and 3– Accurately predicted using only 10 (not 492) basis functions
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
Max Zernike radial degree included
Max
Sen
sitiv
ity
||S||∞
10.5 Hz mode
0 10 20 30 40 500
0.5
1
1.5
2
2.5
3
3.5
Frequency (Hz)
σm
ax(S
)
p=0p≤ 1p≤ 2p≤ 3p≤ 4p≤ 5p≤ 6
Current design not sufficiently robust!
TMT.SEN.PRE.08.046.REL01 16
Global Loop
Transfer function from commanded actuator displacement to mirrormotion: Telescope structure modes are visibleGlobal control loop doesn’t depend (much) on hard vs soft actuator
100 101 10210-1
100
101
102
Frequency (Hz)
Res
pons
e (m
/m)
Soft actuatorHard actuator
100 101 102
100
101
Frequency (Hz)
Res
pons
e (m
/m) 10.5
10.313
Z2,+2Z2,-2Z2,0Rigid
Soft actuator
TMT.SEN.PRE.08.046.REL01 17
10-1 100 101 10210-3
10-2
10-1
100
101
Frequency (Hz)
Loop
tran
sfer
fn
Mag. 0.3, Freq. 14.8
Z2,0
Z2,-2Z2,2Rigid Base
Global Control Design
SISO analysis:– With 2-pole roll-off at 7 Hz,
maximum bandwidth ~ 2Hz (6 dB gain margin)
– Even with hard actuator, limited by telescopestructure, not 35 Hzsegment resonance
TMT.SEN.PRE.08.046.REL01 18
0 5 10 150
0.5
1
1.5
2
Frequency (Hz)
σm
ax(S
)
2≤ p≤ 5p=2 onlyRigid base
Global Control LoopMIMO Robustness Evaluation
1.5 Hz control bandwidth is achievable (Keck uses <0.1 Hz)– Higher bandwidth achievable at higher spatial frequency:
>2 Hz for radial degree > 4– Depends on assumed
structural dampingStability boundary predicted using 3 (not 492) basis functionsNo significant difference in global control between hard or soft actuators.
1.5 Hz controlbandwidth: ||S||∞=1.87
TMT.SEN.PRE.08.046.REL01 19
Conclusions
Control-structure interaction limits the achievable bandwidth of primary mirror control system– Zernike-basis analysis simplifies computation
Local control loop (soft actuator):– Challenging design, plausible with passive damping in parallel
Global control loop: – 1 Hz desired control bandwidth appears achievable– Achievable bandwidth is the same for hard or soft actuators– Depends on damping assumptions– Errors in A matrix may also limit bandwidth
TMT.SEN.PRE.08.046.REL01 20
Acknowledgments
The authors gratefully acknowledge the support of the TMT partner institutions. They are the Association of Canadian Universities for Research in Astronomy (ACURA), the California Institute of Technology and the University of California. This work was supported as well by the Gordon and Betty Moore Foundation, the Canada Foundation for Innovation, the Ontario Ministry of Research 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.