CSEM Scientific and Technical Report 2008

European Large Telescope M5 Field Stabilization Unit: Mirror-control
O. Chételat, E. Onillon, Y. Welte
The control design and performance analysis of the M5, one of the mirrors of the "Adaptative European Extremely Large Telescope", is presented
in this paper as a companion to another article of this report entitled "European Large Telescope M5 Field Stabilisation Unit: Opto-mechatronics".
Figure 1 shows the M5 mirror attached to its base by three
actuators and a membrane in the center. The actuators are
made of a piezo stack with its elongation being amplified by a
parallelogram flexure.
Figure 1: M5 mirror and detail of one of the three piezo actuators
The M5s dedicated task is to reject – in a tip-tilt motion – the
disturbances due to the wind load on the telescope structure
and to the atmosphere. Figure 2 shows the power spectrum of
the total disturbance for one axis.
PSD (arcsec 2 /Hz)
Figure 2: Power spectrum of wind and atmospheric disturbance nφ
Figure 3 depicts the dynamic modeling of the M5 in closed
loop. The optics of the telescope sees the two tip-tilt angles
described by the vector φ. The difference between these
angles and the ones of the mirror is due to the disturbance nφ.
0
M5 mirror
10 10
10 0
10 -10
10 -2
10 -20
K E
z=0
1-z -1 s -1
K o
10 -1
M * T
M *
10 0
10 1
frequency (Hz)
actuator
piezo
Figure 3: Dynamic modeling of M5 in closed loop
u
n u
K i
1/R
na s2 a
n x
F
1/Cs ke The mirror angles are linearly calculated from the elongation
of the three actuators (vector x) by a matrix M, because the
angles are assumed small. The mechanical transfer function
G between the piezo force F and the actuator and piezo
elongations x and p (respectively) is obtained by a detailed
FEM modal frequency response analysis as shown in Figure 4.
The electro-mechanical action of the piezo transducers is
modeled taking into account the force constant ke, the piezo
capacitance C and the supply line resistance R (the Laplace
variable is denoted by s). It is assumed that the actuator
elongation x as well as the acceleration a of the piezo
sk e
10 2
G
x
x
p
M
p
10 3
nφ φ
elongation are measured. The optical tip-tilt angles are also
measured with a wave-front sensor modeled as (1−z −1 )/s,
where z = e sT (with T the sample time).
Three control loops are considered. The inner-loop based on
the acceleration a is used to "damp" the system in a robust
way, i.e., based on the passivity (emulation of a viscous
friction thanks to the velocity feedback obtained by integration
of a by Ki and acting on the piezo voltage u). At 1 kHz sample
frequency, the middle loop controls the mechanism in the
angles-piston domain using the controller Ko as well as the
transformation matrices M* (similar to the matrix M, but
including the piston) and its transposed M* T . Finally, the outer
loop is used with the controller KE at a lower sample frequency
(100 Hz) imposed by the technology of the wave-front sensor.
x 2 / F 2 (m/N)
10 -6
10 -8
10 -10
10 0
10 -12
10 1
frequency (Hz)
Figure 4: Frequency response representing one input of the
mechanical transfer function matrix G obtained by finite-element
computation.
The dynamic model of Figure 3 optimizes the controllers of the
three loops and computes the noise in closed loop. Table 1
shows in the first column the maximum noise requirements, in
the second column the residual noise from the rejected wind
and atmospheric disturbances, and in the three last columns,
the noise that would be obtained if 10-bit quantizing for the
voltage u, the acceleration a, and the position x, respectively
were assumed.
Table 1: Computed performances of M5 (all units in milli-arcsec)
Requirements output volt. u acc. a pos. x
dist. nφ 10 bits 10 bits 10 bits
< 70 rms (0…∞Hz) 27.4 1.6 0.001 0.6
< 4 rms (9…∞Hz) 0.67 1.6 0.001 0.6
< 8 max (9…∞Hz) 3 7 0.005 2.4
Table 2 lists the final specifications allowing the M5 to be
within the requirements. These specifications are derived from
the control design for the range, resolution, and analogue
noise of the voltage u, the acceleration a, and the position x.
Table 2: Specifications of electronics derived from control design
Specifications voltage u acceler. a position x
Full range (with margin) 1000 V 2 mm/s 2 0.6 mm
ADC/DAC resolution ≥ 13 bits ≥ 2 bits ≥ 11 bits
Analogue noise (rms) 69 mV 0.24 mm/s 2 88 nm
Thanks to the methodology described in this paper, it has
been possible to specify an electronics that fulfils the
customer requirements.
10 2
91