Feedback Controller Design for the Basic Science Mode of the LISA Pathfinder Mission

I: L::>I: V I Cl"\.

Copyright © IFAC Automatic Control in Aerospace, Saint-Petersburg, Russia, 2004

IFAC PUBLICATIONS www.elsevier.comllocatefifac

FEEDBACK CONTROLLER DESIGN FOR THE BASIC SCIENCE MODE OF THE LISA PATHFINDER MISSION Peter Gath l , Waiter Fichter l , Tobias Ziegler l , Alexander Schleiche~, Daniele Bortoluzzi3

2

lEADS Astrium GmbH. Friedrichshafen. Germany ZARM. Center for Applied Space Technology and Microgravity. Bremen. Germany 3 University ofTrento. Trento. Italy

Abstract: The paper presents the feedback controller design for the basic science mode of the USA Pathfinder Mission. During the scientific measurements, two test masses on board of the spacecraft are put into a pure gravitational free fall around the sensitive axis with an accuracy of about 3xlO- 14 rn!s2/vHz in the measurement bandwidth between 1 mHz and 30 mHz. This paper describes the controller design, including the electrostatic suspension control loops, the drag-free control loops, and the spacecraft attitude control loops. The original MIMO problem is reduced to a set of SISO designs by applying an input decoupling that only relies on well known geometric properties as well as on the mass properties of the spacecraft and the test masses. Each resulting SISO design is then performed using H... techniques and the overall performance is finally shown in a fully coupled linear simulation, both in frequency and time domain. Copyright © 2004IFAC Keywords: Control analysis, control design, control system, satellite control applications.

1. INTRODUCTION

More recent LISA specific documentation can be found in (Maghami, 2003).

1.1 Background 1.2 Principle. USA Pathfinder is an ESA technology mission (see Vitale et al., 2002) that aims at the verification of key-technologies that are required for the joint ESAlNASA mission USA, a space based gravitational wave detector. The top level requirement that shall be verified in the framework of the USA Pathfinder mission is that two test masses on board of the spacecraft can be put into a pure gravitational free fall with an accuracy of about 3xlO- 14 mls2/VHz in the measurement bandwidth between I mHz and 30 mHz. One of the keytechnologies for this demonstration is a high performance control system. Currently, Phase A studies are completed and the initial and full implementation phase starts at the beginning of 2004.

The core of the LISA Pathfinder satellite is the Lisa Technology Package (LTP). It contains the two (partially) free flying test masses, each surrounded by a sensor cage. The cage is rigidly attached to the spacecraft. In order to demonstrate the top science requirement, the relative motion of the test masses with respect to the spacecraft has to be measured and controlled with extremely high precision. The position and attitude of each of the two test masses relative to the spacecraft are measured electrostatically, as well as by using laser interferometry along the most sensitive axis. This is schematically depicted in Figure 1. In Figure 2, the coordinate system defmitions are shown . . Capacb.oe and

In this paper, the control system design of the basic science mode of USA Pathfinder is described. This is the main scientific operational mode in which the verification of the top-level science requirement is performed.

Optical

Mau:uretner':t 1

Early publications of drag-free control system designs are given e.g. in (Lange, 1964). A rather general application is given in (Pedreiro, 2003).

1Fig. I . Spacecraft with test masses.

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Internal Disturbances: The major contributor to the internal disturbance is suspension noise on the test masses themselves . These are specified in detail in (Vitale, 2002). The effect of both disturbances is taken into account for the design and analysis.

3. DESIGN MODEL AND CONTROL STRUCTURE

3.1 Control Tasks

8

Fig. 2. Coordinate system definition. There are three separate control tasks to be fulfilled: 2. REQUIREMENTS AND CONSTRAINTS

•

2.1 Drag-Free and Attitude Control Requirements In total, there are six test mass coordinates to be controlled by drag-free controller feedback. The coordinates to be controlled are distributed between the two test masses and are depicted in Table 1. In addition to that, there are requirements for the spacecraft's inertial attitude and attitude rate. The inertial attitude shaH not exceed 5 degrees from Sun pointing. The rate requirements are listed in Table 2. Table 1 Drag-Free requirements Axis

Requirement (1 mHz:S f:S 30 mHz) 5x1O'9 mI"Hz 66x 10,9 mI"Hz 50x 10,9 mI"Hz 243x 10,9 radl"Hz 47x 10,9 mI"Hz 69x 10'9 mI"Hz

Table 2 Spacecraft inertial rate requirements Axis COe,Slc ~,SlC

COH SIC

Requirement 7xlO,5 radls 1.3 x 10,5 radls 1.3xlO'5 radls

•

•

First, the six drag-free coordinates have to be controlled within the specifications given above. To this end, the spacecraft is controlled to follow the test masses (along the drag-free coordinates). Second, the remaining six coordinates of the test masses (of the total twelve relative coordinates between the two test masses and the spacecraft) have to be suspension controlled. I.e., the test masses follow the spacecraft, which is accomplished by the suspension control loops (electrostatic actuation system). Third, the inertial attitude of the spacecraft has to be controlled.

Clearly, the three control tasks cannot be accomplished separately, but there are couplings between the axes. Therefore, in input decoupling is performed to reduce the original MIMO design task to simple SI SO designs. Such an approach also yields an extended insight into the system and physical design constraints can easily be identified.

3.2 Design Model The dynamics of the inertial attitude and the dragfree and attitude coordinates can be described as

(I)

2.2 Disturbances External Disturbances, Solar Pressure: Solar pressure is the major external disturbance force . The Sun is assumed to be incident normal to the solar array of the spacecraft. The resulting solar pressure acts along the negative spacecraft z-axis at an assumed level of 10 j.lN. In addition to this constant part of the solar pressure, a random variation has to be accounted for that is due to sub-Hz solar constant fluctuations. The level of this variation is in the order of a fraction of a percent of the constant pressure level. Thruster Noise : For the micro-propulsion thrust noise PSD, a constant magnitude of 0.1x 10-6 N/"Hz is assumed for frequencies over 30 mHz. At I mHz, 1Ox I 0-6 N/"Hz are assumed.

with the following definitions:

(2)

and S selects six drag-free controlled axes out of the twelve test-mass degrees of freedom. is the vector of dragq DF -- {xI y I zI (1I y 2 Z)T 2 free coordinates that consists of the linear displacements of test-mass I, the relative rotation angle of test-mass 1 w.r.t. the spacecraft, and the linear displacements of test-mass 2 in the y and z

axis. I and fare (3x I) external torques and forces, i.e. forces and torques to be generated by the micropropulsion system and disturbances. Accordingly, the (6xl) input vector (eTI I11 f22 f)T contains 2) the controls and disturbances generated by the electrostatic suspension system. Msc and MTM are mass matrices of the spacecraft and the test-masses, respectively .

relationship S+T= I must hold, the specifications for Sand T must satisfy the requirement Sspec +Tspec> I. disturbance

(.:).[G\G(~' G~l(::) (~)}G.- " r.;'

rOI and r02 are skew-symmetric cross product matrices built from the reference positions of the test-masses w.r.t. the spacecraft's center of gravity.

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3.3 Control Strocture

With a decoupling scheme, the design plant for dragfree and attitude control can be re-arranged and described in the frequency domain as follows

Fig. 4. Standard SISO control loop.

(3)

4.1 Drag-Free Controller Design

with Gsc = diag{l/s2} and G TM = diag{1I(s2 + roi 2)} . The ro? are parasitic stiffnesses caused mainly by self-gravity gradient and electrostatic suspension.

A standard control loop is illustrated in Figure 4. Assuming that output variations caused by both, the input disturbance z and the measurement noise '11, must be smaller than the corresponding drag-free requirement on the system output yp' requirements for S and T can be derived as

The control structure for the drag-free and attitude coordinates is given by

(u:u) [(B •• (SBD/f')' diag(K." ..l 0

diag(oKd/.J](q'PD{)

=

Z

N

S <-.!!!L and T <~ req-WG roq- W

(4)

z

Note that no suspension control is applied in axes with drag-free control requirements. The superscript "I" denotes a pseudoinverse operation. The "antidiagonal" structure of the controller, together with the low closed loop bandwidth requirement of the attitude control loop results in extremely simple design plants. In fact, 1Is2 SISO plants can be used for both, drag-free and attitude controller. Suspension control applies only to those axes without drag-free requirements.

(5)

~

where Z=WGS and N=WT z ~

(6)

are the transfer functions from z to YP and from '11 to yp, respectively. Iz and 1" are white noise sources with a power of one. Wz and W" are shaping filters for the input disturbances and for the readout noise, respectively. G is the design plant which is represented by a simple double-integrator 1/s2 for the drag-free controller design. This is justified since the parasitic stiffness Olp2 are much lower than the dragfree closed-loop bandwidths.

A rigorous derivation and justification of this approach can be found in (Fichter, 2004).

4. CLOSED LOOP REQUIREMENTS, CONTROLLER DESIGN, AND RESULTS

In order to achieve controllers with sufficient margins and robustness, an additional requirement is introduced such that S and T must be smaller than +3 dB at all frequencies (not only within the measurement bandwidth). If necessary, additional requirements can be included.

For the actual feedback controller design of the decoupled subsystems, the original drag-free requirements are re-formulated in terms of closed loop frequency response specifications. The use of the a., design technique allows an optimization of the spectral densities. Based on the requirements given for the different drag-free controlled axes (see above), requirements for the sensitivity function S and the complementary sensitivity function T (closed loop) can be derived. These specifications can easily be checked for consistency and physical infeasibilities can be identified. Since the complex

Fig. 5 shows an example for the corresponding Sreq and Treq requirements for the x axis of test-mass 1, including Selrl and Telrl of a controller design which satisfies these requirements. It can also be observed that, even at the critical frequency of I mHz, the requirements are consistent (Tr I while Sreq < 0) and a feasible solution can be found.

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Based on Eq. (8), an additional specification for T can be derived which is valid within the measurement bandwidth, such that (9)

The design plants together with the resulting margins of the suspension controller designs are listed in Table 4.

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Table 4 Stability margins of suspension controllers

Fig. 5. Example for SISO controller specification.

Desi~

PM{d~}

GM {dB} -9.3

43.2


1/( S2 - 8.10-6) 1/(s2-6·10-6)

-5.2

30.0

Xl2

1/( S 2 - 2 .10-6)

5.4

17.8

e2

1/(s2 -8 .10-6)

-3

35.4

Axis

Based on this design approach, controllers for all drag-free axes are designed using an Boo approach. The resulting preliminary designs provide good damping behaviour and sufficient gain and phase margins. The results are listed in Table 3.

111 .2

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Table 3: Stability margins of Drag-Free Controllers Axis Xl Yl Zl

el Y2 Z2

GM (dB) -18 .7 -14.8 -11.9 -13.4 -10.6 -13.8

4.3 Attitude Controller Design

PM (deg) 46.3 48.4 41.2 45.0 41.5 48.6

The design plant of the attitude controllers includes the test-mass dynamics with the corresponding dragfree controllers. Again, a single-axis design can be accomplished on a I1s2 plant due to the decoupling outlined above and since the bandwidth of the attitude controller is far below the bandwidths of the drag-free control loops.

4.2 Suspension Controller Design The bandwidth of the attitude controllers must be reduced as much as possible in order to reduce their effect within the measurement bandwidth. Since any motion of the satellite directly yields a relative motion of the test masses in the opposite direction, the drag-free requirements also apply to the satellite's attitude control within the measurement bandwidth. The resulting controller margins for the design plant configuration are listed in Table 5.

The design approach for the suspension controllers (i .e. for axes that are not drag-free controlled and therefore need to be stabilized with the electrostatic suspensions) is similar to the design of the drag-free controllers. However, the design plant must include the worst-case negative stiffness due to the small bandwidths of the resulting suspension controllers. This makes the Boo design slightly more difficult. The extremely low bandwidth is due to an additional requirement that is introduced on the suspension controller for X2 since any signal generated by this controller directly influences the measurement performance. Therefore, the controller gain is limited according to (Vitale, 2002) such that

Table 5: Stability margins of attitude controllers

(7)

Axis

Gain margin (dB)

8S/c TjS/c

14.4 6.3 4.6

~~c

where 0lp2 is the stiffness. Using the law of cosines and assuming that the phase Cl within the measurement bandwidth is less than 30 deg, this requirement can be transformed such that

margin

Phase {deg) 29.6 25.1 21.9

5. NOMINAL PERFORMANCE RESULTS

5.1 Performance of the Drag-Free Coordinates Using a Transfer-Function Analysis The design of the SI SO controllers is verified with a transfer-function analysis on the fully coupled MIMO system. This linear system includes all crosscoupling and noise effects. The controller design yields drag-free and suspension controllers that satisfy all relevant requirements on the drag-free controlled test mass axes with sufficient margins.

For the other suspension controlled axes, this is not a requirement but serves as a guideline in order to reduce the influence of any suspension controller on the measurement due to cross-coupling effects.

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Table 7 Margins Qn :!us~nsion controlled axes.

The perfonnance margins reported in Table 6 include all cross-coupling effects. The most critical axis is the 9 I axis where the resulting margin is only 20%. The reason for this is the level of the LTP read-out noise (below the closed loop bandwidth) rather than the control design itself. As can be seen in Figs. 6 and 7, this is the major contribution which clearly dominates other noise sources, such as JlN-thruster noise, by an order of magnitude within the measurement bandwidth.

Achieved 47·10"8 mI"Hz 3.10-7 mI"Hz 3.10-9 mI"Hz

!PI X2

Reguirement 5.10-7 radl"Hz 5.10-7 radl"Hz 6·10-9 mI"Hz 5.10-7 radl"Hz 5.10-7 radl"Hz 5.10- 7 radI"Hz

8.10-7 radl"Hz 48.10-8 mI"Hz 3.10-7 mI"Hz

92 T)2 Cf>2

The perfonnance on the suspension controlled axes is listed in Table 7. It can easily be seen that the design here is much closer to the requirements. For the 92 axis, the original requirement is even violated due to the large suspension noise. Unfortunately, the controller bandwidth cannot be increased on this axis due the strong cross-coupling effects on the measurement axis. However, this axis only contributes to the overall science requirement via cross-coupling effects and therefore, this small violation can be tolerated.

Marg!n 1.06 1.6 2.0 0.625 1.04 1.6

5.2 Absolute Acceleration Performance

Meeting the overall science requirement of the absolute acceleration of one test mass in X direction is the essence of the LISA Pathfinder mission. In Fig. 8 it is shown that this requirement is met. The critical frequency here is the lower end of the measurement bandwidth, i.e. I mHz. The total system noise is a careful balance between all components to meet the ambitious scientific requirements. It can be seen from this figure that the total performance is at present essentially driven by the test mass suspension noise. Nevertheless, the requirement is met even at the lower end of the measurement bandwidth (at 1 mHz) with a margin of about 10 percent. In the above graph this fact is masked by the logarithmic scale.

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Fig. 6. Drag-Free performance on test-mass 1

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Fig. 7. Drag-Free performance on test-mass 2

Xl YI Zl

91 Y2 Z2

2.10-9 mlVHz 30.10-9 mI"Hz 25.10-9 mI"Hz 2.10- 7 radI"Hz 20.10-9 mI"Hz 24.10- 9 mI"Hz

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Margin 2.5 2.3 2.0 1.2 2.3 2.8

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Table 6 Margins on drag-free controlled axes. Reguirement 09 5.10 mI"Hz 69.10-9 mI"Hz 50.10-9 mI"Hz 243.10-9 radl"Hz 47-10-9 mI"Hz 69.10-9 mI"Hz

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Achieved

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Fig. 8. Total acceleration perfonnance.

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Since the derivation of the S and T requirements for the controller design and the transfer function analysis allows a very detailed insight into the physical performance limitations, they can directly be used to evaluate changes in the design and identify areas where improvements (e.g. a noise reduction) is most valuable. ...,"-. ...

Future activities will focus on fine-tuning of the controller designs, non-linear simulations of modetransitions, and more detailed performance and stability analyses.

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REFERENCES

Fig. 10. PSD plots for test-mass 2.

Fichter, W . and P. Gath (2004). DFACS Design Report, LPF-TN-ASG-OOOI, Technical Note, EADS Astrium GmbH. Lange, B. (1964). The Control and Use of Drag-Free Satellites, Ph.D. Thesis, Stanford University, Department of Aeronautical Engineering, Report 194. Lange, B. (1964). The Drag-Free Satellite, A/AA Journal, VoI. 2, No. 9, p . 1950. Maghami, P.G., F.L. Markley, M.B. Houghton, C.J. Dennehy (2003). Design and Analysis of the STI Disturbance Reduction System (DRS) Spacecraft Controller, AAS-03-065, 26 th Annual AAS Guidance and Control Conference, Breckenridge, CO. Pedreiro, N. (2003). Spacecraft Architecture for Disturbance-Free Payload, Journal of Guidance, Control, and Dynamics, VoI. 26, No. 5, pp. 794-804. Vitale, S., et al. (2002). LISA Technology Package Architect, LISA technology package on board SMART-2, ESA contract #15580/0IINLIHB, ReI. 1.3, Unitn-Int 10-2002.

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~.~ I'"~ (kf:

Fig. 11 . PSD plot for total acceleration.

5.3 Verification with Time-Domain Simulation The results of the transfer-function analysis are verified by using a time-domain simulation. This time-domain simulation is based on a Simulink implementation of the linear model including all noise sources, cross-coupling effects and the spacecraft dynamics. In order to generate PSD plots of a reasonable accuracy even at low frequencies, 60 hours of operation are simulated. The PSD plots generated from the time-series are shown in Fig. 9 through 11 and show a very good match to the results obtained with the transferfunction analysis presented above. 6. CONCLUSIONS The control system for LISA Pathfmder is one of the key-technologies for mission success. It consists of drag-free control, test mass suspension control, and attitude control. With the control design approach outlined in this paper all drag-free requirements can be met. In addition, it is shown that also the overall acceleration requirement is met. Since the controller design is reduced to SI SO controller designs, it does not rely on the physical layout of the sensor equipment. All mass and geometric properties are included in the decoupling matrices and therefore, the same approach can be used for different geometries.

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