(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

AOO-37154 AIAA-2000-4456AIAA Guidance, Navigation, andControl Conference and Exhibit14-17 August 2000 Denver, CO

Fixed-Structure Robust CMCMomentum Manager Design

for the International Space StationFen Wu*

Dept. of Mechanical and Aerospace EngineeringNorth Carolina State University

Raleigh, NC 27695In this paper, we study the problem of robust CMG momentum manager design

for the International Space Station (ISS). The CMG momentum manager is the primarycontrol law that provides combined momentum management and attitude control ca-pability. However, the robustness property of this controller have not been addressedadequately in design stage other than post-design verification analysis. Within constraintof the existing controller structure, the newly developed CMG momentum manager de-sign is demonstrated to robustly stabilize the space station against moment of inertialand aerodynamic uncertainties without much loss of performance.

Introduction

CMG momentum manager is the primary non-propulsive controller for International Space Sta-

tion (ISS) flight 5A and subsequent configurations.The purpose of momentum manager is to provide mi-crogravity environment for space station operations.The driving requirements on the CMG attitude con-trol system design include pointing accuracy, operat-ing within the momentum capacity of the CMG system(currently the momentum budget for four CMG sys-tem is 14,400 ft-lb-sec), and accommodating movingpayloads, articulating solar arrays, mass property andaerodynamic uncertainties as well as environmentaldisturbances.

Control of a spacecraft using momentum storage de-vices, like CMGs, requires that the average externaltorque on the spacecraft be zero when viewed froman inertial frame. If the net external torque is non-zero, then the momentum of CMGs would continuallyincrease until they are saturated. By excreting an ex-ternal torques, the CMGs for attitude control wouldunload accumulated angular momentum to preventsaturation. The external torques can be provided bymagnetic torquers, thrusters, gravity gradient torques,and aerodynamic torques. The momentum managerwould utilize the gravity gradient and aerodynamictorques generated from spacecraft maneuver and elim-inate thruster firings.

The CMG momentum manager design problem hasattracted many researchers' attention. The design ob-jective here is to establish a proper tradeoff betweenstation pointing accuracy and CMG momentum usage,

'Assistant Professor, Campus Box 7910. E-mail:fwu@eos.ncsu.edu, Phone: (919) 515-5268, Fax: (919) 515-7968

Copyright © 2000 by Fen Wu. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.

while satisfying the specific mission requirements. Theintegrated momentum management and attitude con-trol was found achievable through continuous controlapproach. In this category, early CMG control designrelied on the decoupled spacecraft dynamics betweenpitch and roll/yaw channels.5'16'17 The simplified dy-namic behavior allowed application of various controltechniques, such as, classical PID, Linear QuadraticRegulator (LQR) and 'Hoo control, to achieve de-sired control performance. It also revealed funda-mental design limitations associated with right-halfplane zeros in the spacecraft dynamics. On the otherhand, the possibility to have a single LQR controllerwas explored in8 by taking advantage of coupling be-tween axes. By incorporating cyclic-disturbance re-jection filter8'17 or including such frequency compo-nents into weighting functions,1'5 the integrated at-titude/momentum controller from LQR and H<x, ap-proaches could suppress the effects of sinusoidal exter-nal torques on the CMG system. Hence, the CMGmomentum manager designs have utilized known dis-turbance frequency information and accommodatedthat into the controller design.

A major concern about CMG momentum manageris its robustness property in the presence of uncer-tainties. In,1 the uncertainties in plant moment ofinertia was considered explicitly using /f synthesis tech-nique.10 However, the resulting controller had unnec-essarily high order states due to dynamic scaling andincreased the complexity in control implementation.The robust design was also sought in the framework ofmixed %2/%oo-4 But only nominal performance wasconsidered there.

The condition at which the sum of all inertial ex-ternal torques has an average value of zero (over anorbit) is called a Torque Equilibrium Attitude (TEA).

1 OF 7

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2000-4456

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

Momentum management without using thrusters re- with rigid/flex body uncertainties and external distur-quires: bances

1. the space station flies at a TEA,

2. that reasonable attitude motions about TEA becapable of producing sufficient external torquesboth in and normal to the orbit plane.

For a constant orbital altitude, sun inclination and so-lar activity, the atmospheric disturbance torques, aero-dynamic torques on the spacecraft, and the torquescaused by articulating solar arrays are mainly sinu-soidal functions at orbital and twice orbital frequency.Due sinusoidal nature of all external torques the ac-tual TEA is the combination of a constant attituderelative to its reference frame with small harmonic mo-tion. Note that the TEA is not unique, in general aninfinite set of TEAs can be found that satisfy the re-quirement of zero net inertial external torques.

The current design of CMG momentum managerfrom Boeing PG-1 is based on LQR techniques.8 Dif-ferent from the robust design approaches describedabove, it addressed the robustness problem of theCMG momentum manager from post-design analysisperspective using various system analysis tools. Unfor-tunately, this would extend the controllers design cycleand significantly increase the workload to verify con-troller performance. Moreover, there is a design con-straint from the current flight software (FSW). Thatis, the CMG controller was specified in the FSW usingfixed-order, state-space control architecture. There-fore, any modification on control systems would onlybe meaningful within this design boundary.

This paper will mainly focus on the developmentof robust CMG momentum manager design frame-work for the ISS. Specifically, the rigid-body massuncertainty and aerodynamic uncertainty will be in-cluded explicitly into the design process. The lin-ear quadratic performance index will be specified in"worst-case" scenario, which covers all possible vari-ations from nominal plant description. Therefore theresulting controller will have desired robustness prop-erties. Using a full-block 5 procedure, the implicitcontrol synthesis condition based on the uncertain sys-tem data can be converted to a convex optimizationproblem with uncertainty set explicitly specified. Thisconvex condition is in the form of Linear Matrix In-equality (LMI), which can be solved efficiently usinginterior-point algorithms.3'7 It is believed that robustCMG controller design strategy would significantly re-duce or eliminate most of the post-design analysiswork, and accelerate the control design cycle for theISS program.

Robust LQR Design MethodologyA standard uncertain linear system formulation

will be used to model the space station dynamics

'*(*)'q(t)z(t)

'ACo

B0

AnDw

B2~Dm

12.

P(t)w(t).«(*).

(1)

(2)p(t) = A(t) q(t)

where x,x € R", q,p € R>, u e R™", w € R™"-and z e R™* . All the state-space matrices are of ap-propriate dimensions. A is a structured time-varyingparametric uncertainty obeys the following block di-agonal structure

A =

\6i\<l,rt , As+i , , As+/}

So the uncertainty set A defines all possible continuouscurves ($(•) : [0, oo) — > A.

Assuming the system is well-posed, i.e., / - A ADOis nonsingular for all admissible A. Then the actualstate-space matrices of the uncertain system will de-pend on the parametric uncertainty in linear fractionalrepresentation

)2

U(A) Bi(A)Co (A) AH (A)Ci(A) 2?n(A)

+~B0-DQO

Dw

(I-t

Ba(A) "A)2(A)X»i2(A)_

='A BI 1Co A)i I

_Ci An L

^oo)"^ [C0 Doi A>2]

It is assumed that £>n(A) = 0 to make the problemwell defined.

We would like to design a state-feedback controlleru = Fx such that the closed-loop system is robustlystable for all A € A, and the closed-loop performanceindex

J = max / z zdt

= max

.

/

oo(xTC

_dt

is minimized. Note that the performance criterion Jis denned as the worst-case performance with respectto possible parametric uncertainty.

The uncertain system is quadratically stable if thereexists a symmetric matrix X = XT > 0 such that

<0 (3)

This sufficient condition is formulated in terms ofuncertain system matrices, which consists of infinitenumber of constraints and is hard to verify. As shownin,9'13 the full-block 5-procedure would effectivelytranslate the stability and performance tests for uncer-tain systems into their equivalent formulations using a

2 OF 7

AMERICAN INSTITUTE OP AERONAUTICS AND ASTRONAUTICS PAPER 2000-4456

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

full-block multipliers. For this purpose, we introducethe full-block multiplier as

p := {P £ S2n'**n>> : P = PT,

[AT 7] P [A] > 0, VA e A j

Moreover, The dual set of P can be easily derived as

P =

[I -A] P [_* T] < 0, VA 6

/ 0A B00 /

CQ DQQ

1' 0 X

X 00

0P

'I 0 "A B00 J

Co DQQ

Using a full-block scaling matrix P € P, the orig-inal stability condition (3) can be converted into itsequivalent form

<0

The above matrix inequality is specified using nominalsystem and the interconnection data. It is in the formof LMI and can be solved by LMI optimization toolsefficiently. Although only stability problem presentedhere, the full-block 5-procedure procedure can be ap-plied to solve robust LQR control synthesis problemfor CMG momentum manager design, as shown in thefollowing theorem.

Theorem 1 Given the uncertain system (l)-(2) andperformance index J, if one of the following equivalentconditions are satisfied

1. there exist X, Z > 0, such that for all A € A,

T0 XX 0

0

0

-/ 00 0

/ 0

0 I_ (Ci(A) + £>12(A)F) Du(A) _

<0

-X

<0

Tr[Z] < 7

(4)

(5)

(6)

2. there exist Y,Z > 0, and PI , A G P satisfying,

or ali

***

X

***

X

[r[Z]

Ae A,T ' 0 Y

Y 0 ° °0 PI 00 0 -7 _

/ 0-BO -DOO > o

0 /

(7)T " -Y 0 0 "

0 P2 0.0 0 Z _I 'f _L n j?\T t r< i n j?\T ~— (OQ T Uozr ) — (Gi + Ui2-T )

00 ~^*^10 *s. r\I 0 >0

0 I(8)

<7 (9)

then the closed-loop system rendered by the state-feedback controller is robustly stable and has its per-formance J < 7 for all A e A.

Theorem 1 demonstrated that the equivalence be-tween conditions 1 and 2 through a full-block multi-plier P. It is easy to see that both conditions areindeed LMI constraints. However, the synthesis con-dition 1 is given in terms of uncertain system matriceswhich usually involves infinite number of constraintsand is hard to verify. Alternative condition 2 is prefer-able form for control design purpose because of itsseparation between nominal system and parametricuncertainty constraint.

The multiplier set P £ P (or P 6 P) itself presentssome interesting questions in terms of computationalcomplexity. First, P is clearly a convex set. But itis generally semi-infinite dimensional in the sense thatevery possible elements in the parameter set A needsto be checked. Seemingly computational demanding,however, the multiplier set can be simplified consider-ably by imposing additional assumptions on the set A.Given the block diagonal assumption on the parame-ter A, it is possible to confine the search of multipliersto a smaller set defined as

•pel _

Obviously, Pcl C P. In addition, it can easily beshown9' 13 that the multiplier set Pcl is fully character-ized by finitely many (2s) LMIs specified at the verticesof A, i.e., P € Pcl if and only if for any A« 6 {A, A}

Q < 0 and [A^ /] P > 0

3 OF 7

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2000-4456

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

The restriction of search of multipliers to Pcl generallywould cause conservatism in the corresponding analy-sis condition.

The scaling set P can be further refined by exploringthe block-diagonal structure of the parametric uncer-tainty. Instead of restricting Q negative definite, onemay confine the diagonal elements of Q associated withthe uncertainty block as negative definite, that is

Qi<0, i = ! , • • • , « + /

Then it becomes a less conservative full-block scalingset "Pc2 with its elements as

Qi^*

•k

*

*

^

QsST

s

R

Consequently, the constraints over the element of theset Pc2 is specified over 2s corners of the parametricuncertainty set. This again reduces to finite numberof LMI constraints. Clearly, we have Pcl C Pc2 C Pheld.

Note that the number of LMI constraints specifiedby the sets Pcl , Pc2 increase very rapidly as the num-ber of uncertainty blocks increases. So they shouldonly be used with small number of parametric uncer-tainties.

Linearized Space Station ModelThe spacecraft which rotates with angular rate u>o

in circular orbit is expected to maintain local ver-tical, local horizontal (LVLH) orientation during itsnormal operation. When its TEA deviate from LVLHby small angles, the spacecraft dynamics can be lin-earized around the TEA as

1 (Ae6ndn + Tc") (10)

Note that all variables are expressed in LVLH frame(notation "n"). where 0n represents the small anglesfrom body to LVLH orientation, and its derivative iswn. Jb is the moment of inertia matrix expressed by

</22 < 2 3 in body frame. Tn is the control

torque provided by CMGs, while attitude invariantgravity gradient, gyroscopic and aerodynamic torquesis lumped into Tn. Moreover,

3Jl2

-Jl3

-^T-tiJ ——

-4Jl3

3(J33 - Jll) -3/23

/23 (Jn — Jy.0 2J23 — (J22 — Jll ~ Jss)

-2J23 0 2J12(</22 — Jll — Js3) — 2Ji2 0

where Agaero denotes the matrix of derivative with re-spect to the aerodynamic torque. The rate of attitudeerror is simply

6n = ujn (11)

Finally, the dynamics of CMC momentum hc is de-scribed by

Wo

hn — -O™ hn — Tnri — " ' J (12)

and fi™n = skew {0, — WQ, 0}.To count for the uncertainties in the space station

inertial tensor and aerodynamics, it is assumed thatJ6 = Jb + Aj, Ae = Ae + &Al) + &Ae<iero , and A^ =Aw + &Aa , then we have perturbed equation (10) as

1 {AB6n + + T? + T?

As shown in,15 the above uncertain equation can bere-written in standard LFT form

n _ i jb\-lf A on• n _ / Tw \"

,n i n-m , rr\n~ ~

3 i

(13)

lij = Cgti

Pij = "ijQij

3 i

(14)(15)(16)

where for 1 < i < 3, 1 < j <

BPij = [o3x3 o3x3 (Jbrl

rb\-l 9Jb

- Jb\-l

p? = Bpll,Cq7 = I3.Therefore, the complete linearized spacecraft equa-

tions of motion (including uncertainty) derived from(13)-(16), (11) and (12) is given by

' h™ 'en

d>n

. i .

" -n?n o o0 0 I0 (J6)"1^ (J6)~MW

C9

X

' hnrlcen

un

p+

00

(J6)-1

D9

Tn +

Bp

Dg\-I '0

(J-O)-iDg

p= Aq

3x3

4 OF 7

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2000-4456

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

with the structured uncertainty A in the set

A = {diag{<$n,--- ,<533,A7} :|%| < 1, 1 < i < 3,1 < j < i, ||A7|| < 1}

The feedback controller has fixed structure chosenas

BcDt

(17)

where Ac matrix reflects dynamics of disturbance re-jection filters at zero, one and twice orbit frequency.Bc dictates the blending effect of attitude and momen-tum emphasis designs. In particular,

'03X30000

" AohAnhAl2hA2ihA-llh

00

-Wo/300

AoeAngAl20A%10

•^•22S

0W01

000

0"0000

0000

B,=

The controller takes the difference between statesh,9,u> and corresponding commands as input, andgenerates CMC torque commands as its outputs.The filter states /o,/i,/2 in the controller formulahas their derivatives defined by (3 x 3) submatri-ces Aoh,Aoe,Anh,Ai2h,Aie,A2h- fo provides threeLVLH frame filters, which are used to achieve zero atti-tude error or zero momentum at zero frequency in anyaxis, fi and /2 provides three undamped second or-der filters at orbital and twice orbital frequencies, andperforms disturbance rejection at specified frequencies.

By incorporating cyclic rejection filters, the finalplant model used for control design of the Space Sta-tion attitude and momentum management controlleris a 24-state uncertain linear system, which containsCMC momentum, attitude, angular rate, and integra-tors, filters at orbit rate and twice orbit rate. Thecontrol design problem boils down to find appropriatematrices Cc and Dc using Theorem 1.

Design ExampleTo demonstrate the proposed robust LQR design

method, we study the robust CMC controller designfor the assembly stage c29-5a (assembly sequence 7).The multi-body dynamics for this configuration consistof 5 rigid body components, which are core body, FGBand SM PV arrays, and two PV solar arrays. DuringLVLH TEA operation, the core body is trying to holdits TEA as close as possible, and the two solar arraystrack the sun radiation. The aerodynamic model isthe combination of the specular and diffuse flat plate

effects. The specular force considers the pure normalforce on the plate resulting from molecules bouncingoff, while the diffuse force describes drag effect alongvelocity caused by molecules stick. The percent ofaero coefficient represents the percentage of diffuse vs.specular forces.

The uncertain parameters considered are due torigid-body effect, which includes the moment of iner-tia, the atmospheric density and the percentage diffuseaero coefficient. Assuming percentage variations fordiagonal and off-diagonal elements of the inertia ma-trix leads to 6 uncertain parameters. The atmosphericdensity varies from its maximum to minimum basedon altitude, solar and geomagnetic activity. Thus,the aerodynamic uncertainty is crudely modeled asan (3 x 3) full block uncertainty. For mathemati-cal convenience, all uncertain elements will be treatedas time-varying parametric uncertainty to fit into thecontrol design paradigm.

The design objective is to minimize the performanceindex with respect to parameter variations. Morespecifically, the control law is required to stabilize thespacecraft against unstable gravity gradient torques,to maintain attitude errors and keep CMC momentumaccumulation within acceptable bounds, and rejectstrong periodic disturbances caused by the rotationof solar panels and variation in atmospheric densityassociated with diurnal bulge. It is also required thatthe system is robust against uncertainties in the sta-tion's momentum of inertia and aerodynamics. Notethat the flexible modes do not cause significant ro-bustness problems because of the separation of closed-loop bandwidth from the lowest flexible mode in thefrequency domain. The time-domain performance re-quirements include:

• maintain attitude ±15° for roll/yaw, +15° to-20° for pitch wrt LVLH,

• maintain attitude stability of 3.5° per axis perorbit,

• maintain angular rate wrt LVLH less than0.02°/sec.

As the first step of design, CMG controller was syn-thesized using PG-1 design tool based on the nominalplant model. An identity matrix scaled by O.OSwo wasadded to the open-loop A matrix as suggested in.1The addition of the scaled identity matrix pushes theuncontrollable marginal stable pole to the stable re-gion, and forces all the closed-loop poles in the lefthalf plane of -0.05w0 line for faster closed-loop sys-tem response. The nonlinear time-domain controlledperformance was then simulated using Station/OrbiterMultibody Berthing Analysis Tool (SOMEAT). SOM-BAT is a high-fidelity rigid/flexible multi-body dy-namics simulation tool which was developed for NASAspace station program. It also incorporates several

5 OF 7

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2000-4456

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time(sec)

(a) attitude in Y-P-R sequence: roll (solid), pitch (dashdot), yaw (dot)

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

spacecraft control subsystems, such as, PG-1 fight con-trol software, Russian motion control systems (MCS)and rotary joint control algorithms for accurate simu-lation use. Fig. 1 presents the nominal performance ofthis CMG momentum manager under nominal aerody-namic condition. The simulation lasts about two orbitcycles and all the performance requirements (attitudeerror, angular rate and momentum usage) are satis-fied. This will be used as the benchmark to evaluaterobust CMG momentum manager performance.

For robust CMG controller design, we consider 10%and 5% parameter variation bounds on diagonal andoff-diagonal elements of moment of inertia matrix, andatomspheric density change from maximal to minimal.The multiplier was chosen as block diagonal scalingmatrix to simplify the robust LQR control synthesisconditions. As mentioned before, the full block scalingmatrix requires the number of LMI constraints increas-ing exponentially as the number of uncertainty blocks.This would lead to 215 LMIs on full scaling multi-plier for the problem at hand, and can not be handlednumerically by available LMI optimization tools. How-ever, these constraints can be eliminated completelythrough block diagonal scaling matrix. The drawbackis that the resulting controller could be too conserva-tive to achieve adequate performance. To demonstraterobust performance of synthesized CMG momentummanager, we perturb the nominal mass properties (di-agonal elements of moment of inertia matrix) by 6%,7% and 8% respectively, and simulate its robust per-formance under maximal atomspheric density. Thesimulation results are shown in Fig. 2. Adequatetime-domain performance in terms of maximal atti-tude deviation, angular rate, and momentum peak areobserved compared with Fig. 1.

(b) angular rate

Concluding RemarksRobust LQR synthesis framework was proposed and

applied to the robust CMG momentum manager de-sign problem for a space station fight configuration.Within the constraint of current fight control struc-ture, The robust design approach addresses the defi-ciency of nominal CMG controllers from control syn-thesis point of view and provides the possibility toreduce or eliminate post-design analysis work. Theproposed framework is also useful to evaluate funda-mental tradeoff between achievable performance andcontroller robustness.

Due to slow moving nature of the spacecraft op-erations, the performance of designed robust CMGcontrollers could sacrifice from arbitrarily fast param-eter variation assumption. The continued researchwould try to to evaluate the controller performancewith respect to uncertain parameter variation rates,and developed robust T~L^ performance synthesis pro-

-1000

-2000

-3000

(b) CMG momentum

9000 10000

(b) CMG gimbal angles: inner (solid), outer (dash dot)

Fig. 1 Nominal performance of CMG momentummanager based on conventional LQR technique.

6 OF 7

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2000-4456

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time(sec)

(a) attitude in Y-P-R sequence: roll (solid), pitch (dashdot), yaw (dot)

(b) angular rate

(b) CMC momentum

0 1000 2000 3000 4000 5000Time<sec)

7000 8000 9000 10000

(b) CMC gimbal angles: inner (solid), outer (dash dot)

Fig. 2 Robust performance using robust CMGmomentum manager.

cedures for slowly varying parametric uncertainties.

Acknowledgement The author would like toacknowledge the partial financial support of thisresearch work from North Carolina Space GrantConsortium (NCSGC).

References1Balas, G.J., Packard, A., and Harduvel, J.T., "Application

of /i-synthesis Techniques to Momentum Management and At-titude Control of the Space Station", in Proc. AIAA Guidance,Nav. Contr. Conf., AIAA paper No. AIAA-91-2662-CP, 1991,pp. 565-575.

2Bernstein, D.S. and Haddad, W.M., "LQG Control withAn T-Loo Performance Bound: A Riccati Equation Approach,"IEEE Trans. Automat. Contr., AC-34:293-305, 1989.

3Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan,V., Linear Matrix Inequalities in Systems and Control Theory,SIAM, Philadelphia, PA, 1994.

4Byun, K.W., Subramaniam, M., Hua, T., and Sunkel, J.,"Robust Attitude Control Design for a Spacecraft with Artic-ulating Appendages," in Proc. AIAA Guidance, Nav. Contr.Conf., 1997.

5Elgersma, M.R., Stein, G., Jackson, M.R., and Yeichner,J., "Robust Controllers for Space Station Momentum Manage-ment," IEEE Contr. Syst. Mag., 14-22, 1992.

6Feron, E., "Analysis of Robust Hi Performance with Mul-tipliers," in Proc. 33rd IEEE Conf. Dec. Contr., 1994, pp. 2015-2020.

7Gahinet, P., Nemirovskii, A., Laub, A., and Chilali, M.,LMI Control Toolbox, Mathworks, Natick, MA, 1995.

8Harduvel, J.T., "Continuous Momentum Management ofEarth-oriented Spacecraft," AIAA J. Guidance, Contr. Dyn.,15(6):1417-1426, 1992.

9Iwasaki, T. and Kara, S., "Well-posedness of Feedback Sys-tems: Insights into Exact Robustness Analysis and ApproximateComputations," IEEE Trans. Automat. Contr., AC-43:619-630, 1998.

10Packard, A. and Doyle, J., "The complex structured singu-lar value," Automatica, 21:71-109, 1993.

11Paganini, F., Sets and Constraints in the Analysis of Un-certain Systems, PhD thesis, California Inst. of Technology,1995.

12Peterson, I.R. and McFarlane, D.C., "Optimal GuaranteedCost Control and Filtering for Uncertain Linear Systems," IEEETrans. Automat. Contr., AC-39:1971-1977, 1994.

13Scherer, C.W., "Robust Mixed Control and LPV Controlwith Full Block Scalings," Recent Advances on LMI Methods inControl (L. El Ghaoui, S. Niculescu ed.), SIAM, 1999.

14Stoorvogel, A., "The Robust % Control Problem: AWorst-Case Design," IEEE Trans. Automat. Contr., AC-38:1358-1370, 1993.

15Tchernychev, A., Robust Analysis Approach for the ISSControl Design, Boeing Memorandum A3-J085-AT-9801314,1998.

16Warren, W., Wie, B., and Geller, D., "Periodic-Disturbance Accommodating Control of the Space Station forAsymptotic Momentum Management," AIAA J. Guidance,Contr. Dyn., 13(6):984-992, 1990.

17Wie, B., Byun, K.W., Warren, W., Geller, D., Long, D.,and Sunkel, J., "A New Momentum Management Controller forthe Space Station," AIAA J. Guidance, Contr. Dyn., 12(5):714-722, 1989.

18Zhou, K., Doyle, J., and Glover, K., Robust and OptimalControl, Prentice-Hall, Upper Saddle River, NJ, 1996.

7 OF 7

AMERICAN INSTITUTE op AERONAUTICS AND ASTRONAUTICS PAPER 2000-4456