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The 17th JAXA Workshop on Astrodynamics and Flight Mechanics
Target
Quasar
Observation
Observation
Calibration
Calibration
Attitude
Control
Observation Sequence
1cycle
time
< 3 deg
< 3 deg
Figure 1. Overview of VLBI switching
observation Figure 2. Overview of VSOP (VLBI
Space Observatory Program)
I. Introduction
A. Background of Research
Aradio telescope satellite gASTRO-Gh is expected to be launched in 2012 for VSOP (VLBI Space Observatory
Program)-2.1 VSOP-2 is a space VLBI mission, which expands VLBI observation to space. Since
the resolution in VLBI observation is proportional to baseline (the length between antennas), cooperation
among ground radio telescopes and ASTRO-Gfs antenna can attain such a high resolution that only ground
radio telescopes never attain. Overviews of VLBI switching observation and VSOP are shown in Figs. 1 and
2.
Since ground radio telescopes require calibration of phase fluctuation by air in VSOP-2 mission, all
antenna is required to observe target celestial object and quasar for calibration alternately in one minute
cycle. ASTRO-G realize this fast switching observation by attitude maneuvering. Therefore, ASTRO-G have
to maneuver in order that the loaded antenna may orient two celestial objects alternately. The switching
observation requires ASTRO-Gfs attitude control system to maneuver 3 deg and to converge attitude angle
error within }0.002 deg in 15 sec.
In addition, the antenna loaded into ASTRO-G should be treated as flexible structure. The 8 m diameter
antenna is too huge to identify its modal parameter exactly. Because of this reason, robust control theory
is desirable to feedback control system. Therefore, ASTRO-Gfs control system is required to have both
robustness to modeling error and fast transient response performance.
B. Purpose of This Research
ASTRO-Gfs attitude control system is now researched in cooperation with a satellite manufacturing company.
Current control system strategy is to design feedforward reference that excites no flexible modes, called Nil-
Mode-Exciting (NME) profiler.2 Because NME profiler is designed by using sinc function which doesnft have
frequency component over its cutoff frequency, feedforward control with NME profiler is considered to work
out.
Authors, however, have anxiety of feedforward main control designing. For example, ASTRO-G will load
CMGs as actuator, which rarely used in past satellites. It is anxious if CMGs excite some flexible modes, and
if VLBI observation cannot be done because of this excited vibration. Therefore, whether feedback control
is applicable or not is examined in this paper.
II. Modeling of Satellite Attitude System
A. Equation of Motion
Motion equation of satellite attitude is represented as simultaneous equation consists of rigid body motion
equation and flexible part motion equation.
I NƒÆ + QT NƒÅ = T (1)
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The 17th JAXA Workshop on Astrodynamics and Flight Mechanics
ƒÅN+ 2ƒÄƒÐƒÅ? + ƒÐ2ƒÅ + QƒÆN = 0 (2)
where ƒÆ, ƒÅ, T are attitude angle, flexible variable, and torque respectively. I, ƒÄ, ƒÐ, Q are moment of inertia,
damping matrix, stiffness matrix, and interference matrix respectively. Controlled variable and observable
variable is attitude angle ƒÆ, and control input is torque T. After coordinate transformation for decoupling
flexible modes, transfer function can be written as eq.(3)
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Magnitude (dB)
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??????
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Phase (deg)
Frequency (rad/sec)
Figure 3. Bode diagram of x-axis
P(s) =
ƒ¦(s)
T(s)
=
1
Is2 +
ƒ°N
i=1
?2i
s2 + 2ƒÄiƒÖis + ƒÖ2
i
(3)
Plant can be written as three transfer functions: x-axis,
y-axis, and z-axis transfer functions. Although actual system
has three-axis interference, ASTRO-G has the features listed
below.
1. Since z-axis (= antenna axis) rotation doesnft effect the VLBI
observation, z-axis rotation can be ignored
2. The interference between x-axis and y-axis is negligible
Because of these features, SISO controller was made in this
paper. Figure 3 shows bode diagram of x-axis plant.
B. Modeling Error
In general, model identification of flexible structures is getting harder as the eigen frequency of the flexible
modes goes higher, and modeling error becomes more considerable. The flexible appendages loaded into
satellite are a 8 m diameter antenna and a solar paddle. Because of their own weight, these flexible structures
cannot maintain their shapes without any support. Therefore, parameter identification would be done with
partial structures of flexible structures. Then, the characteristics of the whole flexible structures would be
numerically synthesized. After launching, modal parameters would be identified by shaking test on-orbit.
These two model identifications, i.e. on the ground and on-orbit identification, differs from each other. The
difference between ground identification and on-orbit identification in eigen frequency was about 20% at a
maximum.3 Therefore, feedback control system should be robust to modeling error of modal parameters.
III. 2 DOF Control System
A. Robust Feedback Control
y
Wt Ws
u
z2 z1
+
+ P
C
Generalized Plant
w
Figure 4. Block diagram of generalized plant
In this paper, H‡ control was applied as robust feedback control.
5?7 Figure 4 shows generalized plant to synthesis H‡ controller.
A stabilizing controller C could be obtained by designing
singular value of weighting function Wt larger than conceivable
perturbation ƒ¢, and ||GƒÖz||‡ < 1.
1. Guidance for Controller Synthesis
In general, lower flexible modes can be identified almost exactly.
In this paper, the lowest mode was included in nominal
plant, and the other flexible modes were represented as additive
perturbation. This is considered not very challenging controller
synthesis, because Engineering Test Satellite (ETS)-6 demonstrated
H‡ controller synthesis that include the lowest flexible
mode as nominal plant.4
There are two constraints when designing attitude controller
of ASTRO-G. First, calculator performance on board
should be considered because the performance is not very high. Hence, the order of controller is limited.
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The 17th JAXA Workshop on Astrodynamics and Flight Mechanics
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Frequency (rad/sec)
Singular Values (dB)
Wt
Plant
Weighting function Wt which
covers every mode except for 1st Mode
2nd Mode
1st Mode
Figure 5. Singular value of wighting
function for robustness
0 20 40 60 80 100
-1
0
1
2
3
4
time[sec]
Attitude Angle[deg]
Figure 6. Simulation result of feedback
control
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Frequency (rad/sec)
Singular Values (dB)
Controller
Plant
2.61 [rad/sec]
2.14 [rad/sec]
Figure 7. Singular values of H‡ controller
and Plant
Therefore, nominal plant and weighting function should be decided with consideration of their order. Second,
sampling frequency of attitude control calculator should be considered. Under current strategy of attitude
control system, the sampling frequency is considered to be 32 Hz. Hence, there is a region on s-plane the
poles of controller should be.
2. Synthesis of H‡ Controller
H‡ controller was synthesized based on the guidelines mentioned above. Nominal plant includes the lowest
flexible mode, and represented as eq.(4)
Pn(s) =
1
Is2 + ?21
s2 + 2ƒÄ1ƒÖ1s + ƒÖ2
1
(4)
Figure 5 shows weighing function Wt for robustness. Weighting function Wt is designed that singular value
of Wt is to cover over singular value of additive perturbation ƒ¢. Hereby, robust controller which stabilize
if high order modes have modeling error, can be obtained. Weighting function Ws for sensitivity function
is designed as four order low pass filter, in consideration of order of synthesized controller. Taking into
account of constraint about sampling frequency of attitude control calculator, controller is designed with
pole placement constraint by LMI.
3. Simulation Result of Closed-Loop System
Numerical simulation was done with closed-loop system which consists of synthesized H‡ controller and
full-order plant. Full-order plant consists of rigid mode, eight flexible modes of antenna, and four flexible
modes of solar paddle. Step response to the closed-loop was shown in Fig. 6. H‡ controller was discretized
by 32 Hz tustin conversion. A vibration can be seen in Fig. 6 from 5 sec to 20 sec. The frequency of
this vibration corresponds to 1st flexible mode which is included in nominal plant when H‡ controller was
synthesized. Figure 7 shows the frequency characteristics of the controller and plant. The notch effect of the
controller correspond to 1st flexible mode is different in the respect of frequency. This is due to LMI pole
placement constraint in the H‡ control synthesis.
B. Feedforward System Designing
In this section, feedforward control system was designed. To design feedforward control, reference called
NME (Nil-Mode-Exciting) profiler was used. Figure 8 shows block diagram of 2 DOF control system. In
this paper, reference is given as torque dimension.
Nominal plant in Fig.8 was studied after simulation. For a time, nominal plant was considered to be
rigid mode and 1st flexible mode, and represented as eq.(5)
Pn =
1
Is2 + ?21
s2 + 2ƒÄ1ƒÖ1s + ƒÖ2
1
(5)
1. NME Profiler
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The 17th JAXA Workshop on Astrodynamics and Flight Mechanics
+
+
+ -
P
Pn C
R
Attitude Error
Attitude Angle
FF torque y(t)
FB reference
Torque
e(t)
r(t)
u(t)
Figure 8. Block diagram of 2 degree of freedom
control, with reference(R) and feedback
controller(C)
The reference R of 2 DOF control was designed by NME profiler.
Subsequent paragraphs show concept and design procedure
of NME profiler.
Concept of NME Profiler The concept of NME profiler
is to excite no flexible modes. NME profiler is based on sinc
function represented as eq.(6)
x(t) =
sin(ƒÖst)
ƒÖst
(6)
Sinc function doesnft have frequency component higher than
cut off frequency ƒÖs rad/sec. Hence, being designed ƒÖs lower
than the eigen frequency of the lowest flexible modes, the sinc
function excites no flexible modes.
Designing Procedure of NME profiler The designing procedure of NME profiler is listed as blow.
1. Adding two sinc functions: sinc function for acceleration and deceleration
2. Multiplying the result of (1) and a hamming window in time domain
Figure 9 shows sinc functions for acceleration and deceleration, and Fig.10 shows hamming window. Hamming
window was multiplied because sinc function have infinity time section. In the cause of windowing,
NME profiler have some frequency component over ƒÖs, the cut off frequency of sinc function. However, effect
of windowing is negligible, i.e. frequency component of NME profiler over cut off frequency can be ignored.
Thus, NME profiler can be mentioned as reference which excites no flexible modes.
C. Numerical Simulation
This section shows the results of numerical simulations of designed 2 DOF control system consists of H‡
feedback controller and NME profiler. Block diagram of 2DOF system is shown in Fig.8. Feedback controller
C was H‡ controller designed in section 2, nominal plant in 2 DOF system was rigid mode 1
Is2 , and reference
R was NME profiler. Controlled plant includes rigid mode and twelve flexible modes.
First, simulation with nominal plant was done. Figure 11 shows the result of attitude angle. As illustrated
in Fig. 11, attitude angle converge within 3 } 0.005 deg about 16 or 17 sec. This result almost satisfy the
mission demand.
Next, simulation with perturbed plant was done. In this simulation, frequency of all flexible modes are
designed to be (1) nominal values, (2) 0.8 times of nominal values, (3) 0.9 times of nominal values, and (4)
1.2 times of nominal values. Figure 12 shows the results of simulation. As illustrated in Fig. 12, the worst
case was when frequency error was 20% lower than nominal values. In this worst case, however, the control
system has stability and convergence attitude angle about 20 sec.
Finally, feedforward controller was studied with simulation. In this simulation, how many flexible modes
nominal plant in feedforward controller 1
Pn
should include, was studied. Simulation cases are that (1) Pn was
0 5 10 15 20 25 30
-1
-0.5
0
0.5
1
time [sec]
Torque [a.u.]
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Figure 9. NME profiler designing procedure 1:
sinc functions for acceleration and deceleration
0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
time [sec]
Torque [a.u.]
Figure 10. NME profiler designing procedure 2:
hamming window
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The 17th JAXA Workshop on Astrodynamics and Flight Mechanics
0 10 20 30 40 50
-1
0
1
2
3
4
q [deg]
0 10 20 30 40 50
2.99
2.995
3
3.005
time [sec]
q [deg]
NME profiler
Attitude Angle
3.01
Convergence
Observation Time
Figure 11. Simulation result: NME
response of 2 DOF system (antenna
gain loss)
0 10 20 30 40 50
-1
0
1
2
3
4
[deg]
0 10 20 30 40 50
2.99
2.995
3
3.005
time [sec]
[deg]
NME profiler
! = !
n
* 1.0
! = !
n
* 0.8
! = !
n
* 0.9
! = !
n
* 1.2
q
q
3.01
Convergence
Observation Time
Figure 12. Simulation result: NME
response of 2 DOF system (antenna
gain loss)
0 10 20 30 40 50
-1
0
1
2
3
4
[deg]
0 10 20 30 40 50
2.99
2.995
3
3.005
[deg]
time [sec]
NME profiler
P
n
= rigid
P
n
= rigid + 1st
P
n
= rigid + 1st + 2nd + 3rd
q
q
3.01
Convergence
Observation Time
Figure 13. Simulation result: NME
response of 2 DOF system (antenna
gain loss)
only rigid mode, (2) Pn was rigid mode and 1st flexible mode, and (3) Pn was rigid mode and 1st, 2nd, and
3rd flexible mode. Fig. 13 shows the simulation results of these cases. Compared to case (1), convergence
time of case (2) is short. However, case (3) had the same convergence time as case (2). Therefore, we
concluded that nominal plant in feedforward controller Pn should be rigid mode and 1st flexible mode.
IV. Conclusion
VSOP-2 project was introduced, and robust and agile control system demand was explained. 2 DOF
control system with H‡ feedback control and NME profiler was designed to fulfill the mission demand. By
numerical simulation, effectiveness of the designed 2 DOF control was shown. Because feedback controller
was designed by H‡ control theory, the closed-loop doesnft lose stability if modeling error of eigen frequency
of flexible modes was about 20%. Because, the designed 2 DOF control system had robustness and fast
transient response, applicability of feedback control was shown.
References
1VSOP-2 Project: http://www.vsop.isas.ac.jp/vsop2e/
2T. Kamiya, K. Maeda, N. Ogura: gPreshaping Profiler for Flexible Spacecraft Rest-to-Rest Maneuvers,h AIAA, Guid.
Navigation, and Control Conf. and Exhibit, Keystone, Colorado, Aug. 21-24, 2006.
3Yamaguchi Y., Kida T., Komatsu K., Sano M., Sekiguchi T., Ishikawa S., Ichikawa S., Yamada K., Chida Y., and Adachi
S.:gETS-VI On-Orbit System Identification Experimentsh, JSME International Journal, Vol. 40, No. 4, 1997.
4Chida Y., Yamaguchi Y., Soga H., Kida T., Yamaguchi I., Sekiguchi T.: gOn-Orbit Attitude Control Experiments for
ETS-VI ? I-PD and Two-Degree-of-Freedom H‡ Control ?h, Proceedings of the 35th Conference on Decision and Control,
Kobe, Japan, 1996.
5Yi L. and Tomizuka M.: gTwo-Degree-of-Freedom Control with Robust Feedback Control for Hard Disk Servo Systemsh,
ISSS/ASME Transactions on Mechatronics, Vol. 4, No. 1, 1999.
6Hirata M., Liu K., Mita T., and Yamaguchi T.: gHead Positioning Control of a Hard Disk Drive Using H‡ Control
Theoryh(in Japanese), SICE, Vol.29, No.1, 1993.
7Chida Y., gAn H‡ Controller Design Method of Integral-Type Servo Systems and Its Application to a Flexible Structure
Systemh, Second IEEE Conference on Control Application, 1993.
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