Automatica 35 (1999) 1201}1214

Fully magnetic attitude control for spacecraft subjectto gravity gradient1

Rafa" WisH niewski, Mogens Blanke*Aalborg University, Department of Control Engineering, Fredrik Bajers Vej 7, DK-9220 Aalborg ", Denmark

Received 17 September 1996; revised 26 May 1998; received in "nal form 25 December 1998

Interaction between control-current in coils and the geomagnetic xeld can be the means to stabilizea satellite about three axes. Nonlinear time-varying analysis shows how this new result is obtained.

Abstract

This paper presents stability and control analysis of a satellite on a near polar orbit subject to the gravity torque. The satellite isactuated by a set of mutually perpendicular coils. The concept is that interaction between the Earth's magnetic "eld and a magnetic"eld generated by the coils results in a mechanical torque used for attitude corrections. Magnetic actuation due to its simplicity andpower e$ciency is attractive on small, inexpensive satellites. This principle is however inherently nonlinear, and di$cult to use sincethe control torque can only be generated perpendicular to the geomagnetic "eld vector. This paper shows that three-axis control canbe achieved with magnetorquers as sole actuators in a low Earth near polar orbit. It considers the problem from a time-varying,nonlinear system point of view and suggests controllers for three-axis stabilization. A stability analysis is presented, and detailedsimulation results show convincing performance over the entire envelope of operation of the Danish "rsted satellite. ( 1999 ElsevierScience Ltd. All rights reserved.

Keywords: Attitude control; Satellite control; Time-varying systems; Periodic motion; Lyapunov stability; Quaternion feedback

1. Introduction

Many valuable control methods have been developedover the past years since the "rst satellite was launched in1957. Generally speaking all those techniques may beclassi"ed as active or passive. Active control means thatan external source of energy is applied to drive anactuator. Typical actuators are: thrusters, reaction ormomentum wheels and electromagnetic coils.

Magnetic torquing is attractive for small, cheap satel-lites in low Earth orbits. Magnetic control systems arerelatively lightweight, require low power and are inex-pensive. These were the main reasons to suggest thisactuation principle for the Danish "rsted satellite in an

*****

* Corresponding author. Tel.: #45 96 35 87 02; fax: #45 9815 17 39; e-mail: raf@control.auc.uk.

1 This paper was not presented at any IFAC meeting. This paperwas recommended for publication in revised form by Associate EditorJ. Z. Sasiadek under the direction of Editor K. Furuta.

early phase when a spin-stabilized mission, i.e. two-axiscontrol was foreseen. Later rede"nition of the scienti"cobjectives required an alternation of the control require-ments to three-axis stabilization. The challenge was thatthree-axis control was not possible with an actuationprinciple that leaves the system controllable in only twodegrees of freedom because the control torque can onlybe generated perpendicular to the local magnetic "eld ofthe Earth.

There is extensive literature covering satellite attitudecontrol analysis and design. Most of the algorithms pre-sented assume application of reaction wheels and/orthrusters. A geometric control approach to a satelliteactuated by a set of two thruster jets was investigated byByrnes and Isidori (1991). It was concluded that a rigidspace-craft with actuators producing torque in only twodirections independently cannot be locally asymp-totically stabilized using continuously di!erentiablestate feedback. A general framework for the analysis of the attitude tracking control problem usingLyapunov theory was presented by Ting-Yung Wen and

0005-1098/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reservedPII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 2 1 - 7

Krentz-Delgado (1991) and salient features of propor-tional-derivative controllers for attitude control wereestablished. The approach by Ting-Yung Wen andKrentz-Delgado (1991) was extended by Egeland andGodhavn (1994), where an adaptive control scheme forsatellite attitude control was derived using passivitytheory.

The magnetic attitude control has the signi"cant lim-itation that the control torque is always perpendicular tothe local geomagnetic "eld vector, therefore, the mag-netic torquing was mainly used for momentum desatura-tion. The number of internationally published papers onattitude control with use of magnetorquers is still rathersmall. A con"guration with two magnetic coils and a re-action wheel was investigated by Cavallo et al. (1993).The proposed attitude control law was designed based onthe sliding mode control. The problem of sole three-axismagnetic control was addressed by Musser and Ward(1989), where local stabilization of the satellite wasachieved via implementation of an in"nite-time-horizonlinear qadratic regulator. An energy optimal solutionwith use of Riccati periodic equation was presented byWisniewski (1995b). Another linear approach was givenby Martel et al. (1988), where the linearized time-varyingsatellite motion model was approximated by a lineartime-invariant counterpart.

Three-axis stabilization with use of magnetic torquingof a satellite without appendages was treated by Wis-niewski (1994), where a sliding mode control law wasprovided and was shown to stabilize a tumbling satellite.Last but not least, a magnetic attitude control wastreated by Steyn (1994). The author proposed a rule-based fuzzy controller and compared it with an adaptiveLQR control. The references listed above did not con-sider global three-axis stability of a magnetic actuatedsatellite, which is covered in this paper.

Complete comprehension of the nature of the satellitecontrol problem requires a new approach merging thecontrol theory with physics of the rigid body motion.A linear analysis by Hughes (1986) showed that motionof a tri-inertial satellite in the gravitational "eld has fourstable equilibria, all having the axis of minimum mo-ments of inertia in the direction of the local vertical andmaximum along orbit normal. In this paper theLyapunov stability theory is employed to obtain globalresults. The Lyapunov function based on mechanicalenergy is formulated such that its minima coincide withthe four equilibria described by Hughes (1986). One ofthese is chosen to be the reference for the control system.To obtain overall stability in all mission phases, severalcontrollers are needed. A velocity controller is proposed"rst, that contributes to dissipation of both kineticand potential energy. This controller is used for recoveryfrom tumbling motion. Second, this controller isexpanded to include attitude information for thelocal use in a normal science observation. Finally, an

extension is made to obtain global stability by removingundesired equilibria including one with the boom upsidedown. This controller is used for boom upside downrecovery.

The paper is organized as follows. Section 2 describesthe "rsted satellite con"guration, Sections 3 and 4 givesmathematical model used for the attitude control design.Section 5 considers attitude stabilization at large. A con-troller with angular velocity feedback is introduced andproved to be asymptotically stable around four equilib-ria. This control law is shown to be especially e!ective forrecovery of a satellite from tumbling motion with highvalue of the angular velocity. The next step taken is tomodify the rate controller such that the desired referenceattitude is the only stable equilibrium. The rate feedbackis shown to be only locally asymptotically stable aroundthe reference. To achieve global three-axis stabilization,attitude information is shown to be needed as a part ofthe control law. This extension is made in Sections 6and 7. In Section 6 the focus is on local performance ofthe attirude controller for the normal operation. Thesatellite model is approximated by a linear periodicmodel, and the controller is designed employing Floquetstability analysis. A control algorithm for recovery fromthe inverted boom contingency operation is consideredin Section 7. Simulation examples are provided to illus-trate the theoretical "ndings.

The family of controllers presented in this paper withjoint control action for detumbling, nominal operationand a boom upside-down condition provides anattitude control system which can achieve global three-axis attitude stabilization using solely magnetorquersfor actuation. The Danish "rsted satellite will bethe "rst to #y three-axis stabilized using these newresults.

2. ""rsted satellite con5guration

The work has been motivated by the Danish "rstedsatellite mission. The purpose of the mission is to conducta research program in the magnetic "eld of the Earth.The "rsted satellite is a 60 kg auxiliary payload sched-uled to be launched by a MD-Delta II launch vehicle inJanuary 1999 into a 650]850 km orbit with a 963 incli-nation. When ground contact is established in orbit, an8 m long boom is deployed by a ground command. Theboom carries scienti"c instruments that must be dis-placed from the electro-magnetic disturbances present inthe main body of the satellite. The "rsted satellite con"g-uration is depicted in Fig. 1. After boom deployment thenormal operation phase controller is activated. The satel-lite will be three axis stabilized with its boom pointingoutwards from the Earth. Formally, the control objectiveis that a certain coordinate system "xed in the satellitestructure shall coincide with a reference coordinate

1202 R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214

Fig. 1. The "rsted satellite consists of a main body and an 8 m long scienti"c boom.

system "xed in the orbit. The pointing accuracy is re-quired to be within 103 in pitch, roll, and yaw. Thesatellite is expected to be in the vicinity of the referencefor most of the operational time, however, thereare contingency phases due to some malfunctions inthe subsystems causing the instrument boom to pointupside down or the satellite to tumble. Requirementsto the attitude control system include recovery fromsuch contingency by turning the satellite back to thereference.

Stabilization of the "rsted satellite is accomplished byactive use of a set of mutually perpendicular magnetor-quer coils. The coils are mounted in the x, y, and z facetsof the satellite main body. A maximum producible mag-netic moment is 20 A m2. The interaction between ex-ternal magnetic "eld of the Earth and the magnetic "eldgenerated in the coils produces a mechanical torque,which is used to correct the attitude. The maximummechanical torque produced by the coils is approx-imately 0.6]10~3 N m above the equator, and1.2]10~3 N m above the Poles.

3. Notation

The notation used throughout the paper is listed inTable 1. Furthermore, the following terminology isintroduced:

Control CS: right orthogonal coordinate system con-trol spanned on principal axes of the satellite, the y axisalong the axis of maximal moment of inertia, the z axisalong the axis of the minimal, see Fig. 2,

Table 1Notation

#v, 0v, 8v general notation for vectors: v resolved in Control CS,Orbit CS or World CS, respectively

X#8

angular velocity of Control CS w.r.t. World CSX

#0angular velocity of Control CS w.r.t. Orbit CS

X08

angular velocity of Orbit CS w.r.t. World CS

u0

orbital rateI inertia tensor of the satelliteIx, I

y, I

zmoments of inertia about x-, y- and z-principal axes

N#53-

control torqueN

''gravity gradient torque

N$*45

disturbance torques

m magnetic moment generated by a set of the coilsB magnetic "eld of the EarthB3 matrix representation of product B]B33 matrix representation of double cross product

!B](B])#0q quaternion representing rotation of Control CS

w.r.t. Orbit CSq, q

4vector part and scalar part of #

0q

A(#0q) attitude matrix based on #

0q

i0, j

0, k

0unit vectors along x-, y-, z-axis of Orbit CS

dq small perturbation of vector part of attitude quater-nion, #

0q

dX small perturbation of angular velocity #X#0

n#0*-

number of coil windingsi#0*-

current in coilA#0*-

coil areaj) quaternion gain in operational mission phase

algorithm

j3 limit of stability of quaternion gainj1 quaternion gain in boom upside-down algorithmE 3]3 identity matrix

R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214 1203

Fig. 2. Control and orbit coordinate systems.

Orbit CS: orbit (reference) right orthogonal coordi-nate system "xed in orbit, the y axis in the orbital planenormal direction, the z axis along zenith,=orld CS: (inertial) right orthogonal coordinate

system,Pitch, Roll, >aw: the angle describing satellite atti-

tude, pitch referred to the rotation about the y axis of theOrbit CS, roll to the x axis, and yaw to the z axis,

Zenith: a unit vector in the Control CS along the lineconnecting the satellite centre of gravity and the Earthcentre pointing away from the Earth. The vector with theopposite sense is nadir.

4. Modeling

The mathematical model of a satellite is described bywell-known dynamic and the kinematic equations ofmotion, see, e.g. Hughes (1986) or Wertz (1990). Thedynamics relates torques acting on the satellite to thesatellite's angular velocity in the World CS. The kin-ematics provides integration of the angular velocity.

Satellite attitude is parameterized by four componentsof a quaternion describing rotation of the Control CSwith respect to the Orbit CS. The use of quaternions iscommon for spacecraft's description motion, since itgives a singularity-free representation, see, e.g. Wertz(1990).

4.1. Equations of dynamics

The dynamic equation of motion for a satellite in lowEarth orbit is

I #X0#8

(t)"!#X#8

(t)]I #X#8

(t)##N#53-

(t)

##N''

(t)##N$*45

. (1)

There are three torques represented in Eq. (1): the controltorque, the gravity gradient torque and disturbancetorques. The latter includes the aerodynamic drag, anda residual torque due to eccentricity of the orbit (it isassumed that the orbit is circular in the model used forcontrol synthesis).

Using magnetorquer coils for control, a torque is gen-erated by an interaction of the geomagnetic "eld with themagnetorquer current i

#0*-(t) giving rise to a magnetic

moment m(t)

m(t)"n#0*-

i#0*-

(t)A#0*-

. (2)

With the coils placed perpendicular to the x, y and z-axesof the Control CS, the control torque acting on thesatellite is

#N#53-

(t)"#m(t)]#B(t). (3)

The magnetic moment given in the Control CS, #m, isconsidered as the control signal throughout the paper.The torque originating from the gravity gradient (Wertz,1990) is given by

#N''

(t)"3u20(#k

0]I #k

0). (4)

Disturbance torques are considered fairly small com-pared with the gravity gradient torque, and can be disre-garded in the later analysis. This is a consequence ofaltitude of orbit chosen (Wisniewski, 1995a) and holds formost low earth polar orbit satellite missions.

4.2. Equation of kinematics

The attitude of the satellite is de"ned as an orient-ation of the Control CS relative to the Orbit CS. Theattitude quaternion #

0q consists of four parameters

[q1

q2

q3

q4]T. Therefore, it does not provide a minimal

attitude representation, which is locally 3 for SO(3), buto!ers global representation of kinematics without singu-larities. The "rst three components [q

1q2

q3]T form the

vector part of the quaternion and q4

is regarded as thescalar part. The construction of the unit quaternionarises from the Euler theorem claiming that the rotationof coordinate systems can be uniquely described by a unitvector e"[e

1e2

e3]T giving an axis of rotation as well

as its sense, and an angle of rotation /. The four para-meters of the quaternion #

0q are then

[q1

q2

q3

q4]T,

Ce1sin

/

2e2sin

/

2e3sin

/

2cos

/

2 DT. (5)

The kinematics is, when parameterized by the attitudequaternion

q5 "12#X

#0q4#1

2#X

#0]q,

qR4"!1

2#X

#0) q. (6)

1204 R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214

Fig. 3. Mechanical torque generated by magnetorquers is alwaysperpendicular to the geomagnetic "eld vector. This implies thatyaw is not controllable over poles, and roll is not controllable overequator.

The satellite's angular velocity relative to the World CSand the angular velocity with respect to the Orbit CS isgiven by the relation

#X#0"#X

#8!u

0#j0. (7)

Eqs. (1), (3), (4), (6) and (7) constitute the entire model ofthe satellite, where the magnetic moment #m(t) is thecontrol variable.

4.3. Control limitations

The satellite actuated by a set of magnetorquershas a serious limitation. The mechanical torque,produced by the interaction of the geomagnetic "eldand the magnetic "eld generated by the magnetorquers,is always perpendicular to the geomagnetic "eld vector.Thus, the direction parallel to the local geomagnetic"eld vector is not controllable. However, the geomag-netic "eld changes its orientation in the Orbit CSwhen the satellite moves in orbit. This implies that ina linear time-invariant setting yaw is not controllableover the poles but only a quarter of an orbit later overthe equator, reaches a region with controllability. Thisis illustrated in Fig. 3. This problem has so far pre-vented three-axis control with only magnetorqueractuation. In the following, the control problem isapproached as the nonlinear and time-varying problem.This will be shown to be a fruitful advance. The "rststep is to look at stabilization of angular velocity. Thesecond to consider local stabilization of attitude and thelast to attempt an extension to obtain global attitudestability.

5. Rate control

A tumbling satellite is one which rotates with relativelyhigh value of angular velocity. In a satellite mission, thisis a condition that can be experienced in various phasesincluding contingencies where attitude sensors may beunavailable. It is vital to be able to stabilize such tumbl-ing motion towards zero angular velocity. This requiresdissipation of energy.

The aim of this section is to provide an algorithm forenergy dissipation of the tumbling satellite. To achievethis, a rate detumbling controller is proposed whichincorporates information about the angular velocityand time propagation of the geomagnetic "eld. Whengenerating control currents, it is desirable that the pro-duced magnetic moment is perpendicular to the localgeomagnetic "eld vector, because this component is theonly one that gives a nonzero control torque.

A candidate for generation of the magnetic moment isan angular velocity feedback

#m(t)"(H #X#0

(t) )]#B(t), (8)

where H is a positive-de"nite constant matrix. There aretwo main reasons to suggest this feedback:

(i) It contributes to dissipation of kinetic energy of thesatellite.

(ii) It provides four stable equilibria depicted in Fig. 4.The equilibria are such that the z axis of the ControlCS (the axis of the minimal moment of inertia) pointsin the direction of the z axis of the Orbit CS (zenith),and the unit vector of the y axis of the Control CS(the axis of the largest moment of inertia) is parallel tothe y axis of the Orbit CS.

Global stability of the control law (8) can be expressed inthe following theorem.

Theorem 1. Consider the control law

#m(t)"(H #X#0

(t) )]#B(t),

then the satellite, Eqs. (1)}(7), in a nonequatorial orbit has4 asymptotically stable equilibria

M(#X#0

, #k0, #j

0) : (0,$0k

0,$0j

0)N. (9)

Magnetic torquing following Eq. (8) obviously introduc-es time dependency in the equations of the satellitemotion. This time variation is periodic by nature, whicharises from two superimposed periodic #uctuations of thegeomagnetic "eld. One is due to revolution of the satelliteabout the Earth with the period corresponding to theorbital period, the second due to rotation of the Earth.The orientation of the orbit is "xed in an inertial coordi-nate system. Thus the rotation of the Earth is visible as#uctuation of the geomagnetic "eld vector's y componentwith frequency 1/24 h. The Earth's magnetic "eld seenfrom a satellite in a near polar orbit is illustrated in

R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214 1205

Fig. 4. Four locally stable equilibria of the angular velocity feedback(8): M (#X

#0, #k

0, #j

0) : (0, 0k

0, 0j

0), (0, 0k

0, !0j

0), (0, !0k

0, 0j

0), (0, !0k

0,

!0j0)N.

Fig. 5. The geomagnetic "eld vector seen from the Orbit CS calculatedby the 10th order spherical harmonic model (24 h period in January1999).

Fig. 5. The periodic character of the geomagnetic "eld isused in the proof of Theorem 1.

Proof. Consider the following scalar function expressingthe total energy of the satellite. The total energy is thesum of kinetic energy of rotary motion, potential energygenerated by the gravity gradient and the energy origin-

ating from the revolution of the satellite around theEarth (Hughes, 1986),

E505"E

,*/#E

''#E

':30.

This leads to

E505"1

2#XT

#0I #X

#0#3

2u2

0(#kT

0I #k

0!I

z)

#12u2

0(I

:!#jT

0I # j

0). (10)

The satellite's maximal moment of inertia is Iy, and I

zis the minimal. This implies that the function E

505is

positive de"nite. The potential energy due to the gravitygradient is zero if the satellite boom is pointing towardszenith or nadir, whereas the energy due to motion of thesatellite in orbit is zero if the unit vectors j

0and j

#are

aligned.The time derivative of E

505will now be shown to be

negative semide"nite. With

EQ505"#XT

#0I #X0

#0#3u2

0#kT

0I #k0

0#u2

0#jT0I #j0

0(11)

Eqs. (1)} (7) are substituted into Eq. (11) yielding

EQ505"#XT

#0(!#X

#8]I #X

#8#3u2

0#kT

#]I #k

0##N

#53-)

!u0#XT

#0I(#j

0]#X

#0)#3u2

0#kT

0I(#k

0]#X

#0)

!u20#jT0I(#j

0]#X

#0). (12)

Recognize that from Eq. (7) the following equality holds:

#XT#0

(#X#8

]I #X#8

)"u0#XT

#0(#j

0]I #X

#0)

#u20#XT

#0(#j

0]I#j

0). (13)

Hence Eq. (12) is reduced to the simple expression

EQ505"#XT

#0#N

#53-. (14)

If the proposed control law (8) is applied then Eq. (14)becomes

EQ505"!(#B]#X

#0)T(#B]H #X

#0), (15)

or

EQ505"!#XT

#0B3 TB3 H #X

#0. (16)

Here B3 is a skew symmetric matrix representing the crossproduct operator: #B]. The matrix B3 TB3 is positivesemide"nite and H is positive de"nite. The derivative ofthe total energy is thus negative semide"nite.

The Krasovskii}LaSalle lemma (see Lemma A.1 fordetails) is applicable in this proof since both the systemand the control law (8) are periodic. The set S in Lemma 5

S"M#X#0

, #0q : &t50 such that EQ

505"0N

contains elements #X#0

, #0q such that either of the vectors

#X#0

or H #X#0

is parallel to #B (#B]#X#0"0 or

#B]H #X#0"0)

S"M#X#0

, #0q : &t50 #B parallel to #B

parallel to H #X#0

N. (17)

1206 R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214

It will be shown that the only invariant set contained inthe set S is the trivial trajectory X

#0(t),0, which corres-

ponds to one of the equilibria (9).Assume that the vector #B (t) was parallel either to

#X#0

(t) or to H #X#0

(t) for each t't0

then from Eqs. (3)and (8) the control torque is equal to zero.

The angular velocity, #X#0

, is given by the dynamicequations (1), (3) and (7) for #N

#53-"0. The geomagnetic

"eld determines the vector 0B, the time propagation ofwhich in the Orbit CS was depicted in Fig. 5. It is obviousthat the vector #X

#0(or H #X

#0) can only be parallel to 0B

instantaneously.Notice that this is true for the geomagnetic "eld ob-

served in all orbits with nonzero inclination. The onlyexception is the case of an equatorial orbit, where thegeomagnetic "eld vector is always parallel to the orbitnormal. Now, the angular velocity #X

#0can always be

parallel to the geomagnetic "eld since the satellitemotion, such that the angular velocity #X

#0and the

principal axis j#

(coinciding with the maximummoment of inertia) are both perpendicular to the orbitalplane, corresponds to a stable equilibrium (Hughes,1986).

For all other orbits the largest invariant set containedin S is the trivial trajectory #X

#0,0. The angular velocity

is thus zero for all t50 and the trajectory of the torque-free motion coincides with a stable equilibrium. There arefour stable equilibria for the torque free motion (Hughes,1986). M (#X

#0, #k

0, #j

0): (0,$0k

0,$0j

0)N. Now, with the

use of control law (8) these stable equilibria beome alsoasymptotically stable. K

Remark 2. The control law (8) can be used for three-axismagnetic stabilization of the satellite in a neighbourhoodof one of the 4 equilibria stated in Theorem 1, thus alsoin the neighbourhood of the reference M(#X

#0, #k

0, #j

0):

(0, 0k0, 0j

0)N, if I

y'I

x'I

z.

Remark 3. If the velocity controller (8) is applied fora limited time interval during an orbit, then the solutiontrajectory of the satellite motion still converges to one ofthe four equilibria in Theorem 1, since the total energyfrom Eq. (14) is constant if the control torque is zero, andis dissipated when the controller is active.

This control strategy is very useful if the attitude con-trol can only take place via telecommand from a groundstation, and time of radio contact is limited. The control-ler is also bene"cial for a satellite with an attitude deter-mination algorithm based on aa sun sensor, since theattitude information may not be available when the satel-lite is in eclipse. In both cases controller (8) is activatedwhen the feedback signals are available and switched o!otherwise. The control law is still stable.

It was proved that the satellite with the control law (8)is asymptotically stable around four equilibria (9). This

Fig. 6. Simulation using the angular velocity controller, #k0;

character-izes convergence of #k

0towards 0k

0, whereas #j

0:characterizes converges

of #j#to 0j

0. The satellite trajectory converges towards the equilibrium

M(#X#0

, #k0, #j

0) : (0, !0k

0, !0j

0)N. This is seen from #k

0;and #j

0:converging to !1.

simple control law was found well applicable for energydissipation of a tumbling satellite. Simulation results ofthe detumbling phase controller based on Eq. (8) arepresented in the next subsection.

5.1. Simulation results for detumbling controller

The performance of the detumbling phase controlleris illustrated in Fig. 6. The initial angular velocity is#X

#0"[0.003 0.005 !0.003]T rad/s. Pitch, roll and

yaw are 403, !403 and 803, respectively. The velocitygain applied in this simulation study isH"diag([0.3 1.4 0.8]T)]108A ms/T. The satellite tra-jectory is seen to converge to the equilibriumM(#X

#0, #k

0, #j

0) : (0,!0k

0,!0j

0)N, with the boom upside-

down. Other initial conditions could lead to a boom-upequilibrium.

6. Local attitude stability

The rate control achieved stabilization and energydissipation as desired, but reaching a desired attitudewith boom-up could not be guaranteed. In the opera-tional mission phase of the "rsted satellite the boomshould be upright with the boom tip above the horizon.The attitude controller designed for this mission phasemust thus be stable towards the reference

M(#X#0

, #k0, #j

0) : (0, 0k

0, 0j

0)N. (18)

R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214 1207

This will be shown to be achievable by adding perturba-tions of the attitude to the control law (8). This is intuit-ively feasible since the maximum of the potential energyE':30

in Eq. (10) is three times smaller than the maxi-mum potential energy due to the gravity gradient.This implies that, a shift from the equilibriaM(#X

#0, #k

0, #i

0) : (0,$0k

0,!0j

0)N to (0,$0k

0, 0j

0) re-

quires less energy than the move from (0,!0k0,$0j

0) to

(0, 0k0,$0j

0).

A proposed control law for the three-axis attitudestabilization is

#m(t)"H #X#0

(t)]#B(t)!Kq(t)]#B(t), (19)

where attitude information is added to the rate controllaw. Here H and K are positive-de"nite matrices. Theproperties of control law (19) will be analyzed usinglinear control theory.

In Eq. (19) a small perturbation of the vector partof the attitude quaternion is added to the controller inEq. (8). For small K the satellite is stable in the neigh-bourhood of the reference M(#X

#0, #k

0, #j

0) : (0, 0k

0, 0j

0)N,

since the di!erential equations describing the motion ofthe satellite are well posed. At this point of the analysisthe gain H is assumed to be constant. The next step of thedesign is to compute the gain K for a given H such thatthe system is still locally stable and the domain of localstability can be extended.

The system is "rst linearized. The satellite motion isconsidered in a neighbourhood of the following refer-ence: the angular velocity of the satellite rotation w.r.t.the Orbit CS is zero (#X

#0"0), and the attitude is

such that the Control CS coincides with the Orbit(#0q"[0 0 0 1]T).The linearization of the angular velocity is additive.

From Eq. (7) the angular velocity #X#8

is

#X#8"A(#

0q )[0 u

00]T#dX, (20)

where dX is a small perturbation.Linearization of the attitude quaternion is multiplica-

tive. Two successive rotations are needed. The "rst is thetransormation from the Orbit CS to a reference coordi-nate system, the second from the reference coordinatesystem to the Control CS. The reference coordinate sys-tem is the Orbit CS in the paper, thus the rotation fromthe Orbit CS to the reference coordinate system is triv-ially given by the identity operation.

#0q"d3 q ) [0 0 0 1]T"d3 q, (21)

where d3 q is a small perturbation of the attitude quater-nion, #

0q, and && ) '' is an operation of quaternion multipli-

cation, for details see Wertz (1990). Accordingto the physical interpretation of a quaternion given inEq. (5)

d3 q"Ce1sin

d/

2e2sin

d/

2e3sin

d/

2cos

d/

2 DT, (22)

and for a small angle /

d3 q+[dq1

dq2

dq3

1]T,Cdq

1 D. (23)

The linearized equations of motion (1)}(7) are then (formore details see Wisniewski, 1996)

d

dt CdXdq D"AC

dXdq D#B33 (t) (H dX#K dq), (24)

where

A"

0 0 u0px

!6u20px

0 0

0 0 0 0 6u20py

0

u0pz

0 0 0 0 012

0 0 0 0 !u0

0 12

0 0 0

0 0 12

u0

0 0

,

px"

Iy!I

zIx

, py"

Iz!I

xIy

, pz"

Ix!I

yIz

,

B33 "I~1

!0B2y!0B2

z0B

x0B

y0B

x0B

z

0Bx0B

y0B

x0B

z!0B2

x!0B2

z0B2

y0B

z0B

y0B

z!0B2

x!0B2

y

0

.

As discussed in Section 5, variation of the local geo-magnetic "eld observed in the Orbit CS re#ects both theorbital motion and the rotation of the Earth. However,for the near polar orbits the main contribution to thegeomagnetic "eld propagation is caused by the orbitalmotion as illustrated in Fig. 5. In order to simplify theanalysis, an approximation of the geomagnetic "eld iscalculated with a period equivalent to one orbit. This isdone by averaging the geomagnetic "eld. The in#uence ofthe di!erence between the real geomagnetic "eld and thisapproximation is evaluated in a subsequent simulationstudy.

In the sequel local stability of the satellite is analysedusing Floquet theory (Mohler, 1991). Consider the fol-lowing form of Eq. (24)

dx

dt"A< (j, t)x, (25)

where

A< (K, t)"A#B (t)[H K],

H is considered to be a constant matrix, whereas K isa matrix of parameters. Furthermore,

A3 (K, t)"A< (K, t#¹ ), for ¹"2n/u0.

According to Floquet theory system (25) is asymp-totically stable if all characteristic multipliers, i.e. theeigenvalues of the monodromy matrix W

AK (t0

, K) for a cer-tain value of K lie in the open unit circle.

1208 R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214

Fig. 7. The locus for the characteristic multiplier forj3[0, 7]105] Am2/T, h"1]108 A ms/T using Eq. (19) as thecontrol law.

Now, such K are found for which the linearized satel-lite system (24), with a certain "xed value of the positivede"nite matrix H, is stable. This can be done by plottinga locus for the characteristic multipliers as a function ofK. An example of a scalar case H"hE and K"jE isshown in Fig. 7. The value of the velocity gain, h, waschosen equal 1]108 Ams/T. Then the locus of the char-acteristic multiplier for j3[0, 7]105] A m2/T was plot-ted. The gain j3 "5]105 Am2/T is the limit of stability.Hence, a certain j)(j3 shall be chosen for which thesatellite motion is guaranteed to be asymptotically stable.For j"0 controller (19) is equivalent to the velocitycontroller, which is stable.

6.1. Simulation results for normal operationphase controller

A Monte Carlo simulation was made to investigatestability towards the reference (18). The initial state israndom with the nominal condition as the mean value.The envelope for the Monte Carlo simulation includesvalues of the attitude above the horizon, i.e. #k

0z'0 and

X#0"0. The chosen attitude controller has velocity

gain, H"1]108EAms/T, and the quaternion gain isK"j) E"3]105E A m2/T.

An example of the simulation test for the satellite ona circular orbit is depicted in Fig. 8. The simulation startsat the equilibrium with the yaw axis turned 1803 awayfrom the desired direction, M (#X

#0, #k

0, #j

0) : (0, 0k

0,

!0j0)N. The trajectory is within the margin of 103 from

the reference already after one orbit. The e!ect of theattitude term in the controller is seen to make the equilib-rium M (0, 0k

0, !0j

0)N unstable, thus extending the region

of convergence to the initial values of the attitude for

Fig. 8. The satellite trajectory converges from the equilibriumM(#X

#0, #k

0, #j

0) : (0, 0k

0)N to the reference.

Fig. 9. Performance of the constant gain controller for the "rstedsatellite on the elliptic orbit in#uenced by the aerodynamic drag. Theinitial conditions are 403 pitch, !403 roll and 803 yaw. The resultantattitude is within 83.

which the boom is above the horizon and the initialsatellite angular relative to the Orbit CS is zero.

The time constants of the closed-loop system are onthe order of one orbit, therefore if the satellite is subjectto cyclic disturbances with fundamental frequency corre-sponding to the orbital period, this disturbance will onlybe marginally damped. Aerodynamic drag in an ellipticorbit gives exactly this type of disturbances. The simula-tion results for the "rsted satellite on impact ofthe aerodynamic drag and the torque due to eccentricityof the orbit is illustrated in Fig. 9. The attitudeerror is within 83, which is within the required limitof $103.

R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214 1209

The linear attitude controller performs satisfactory forall initial value of attitude #k

0;'0 (boom is upright).

A global stabilizing attitude controller will be investi-gated in the next section.

7. Contingency operation controller for inverted boom

A condition with the boom upside-down, i.e. pointingtowards the centre of the Earth, may likely occur due toconsiderable in#uence of the aerodynamic torque. Suchundesired attitude must be corrected by the attitudecontrol system. Hence, the attitude control problem is toturn the satellite from the equilibria M(0, !0k

0, $3j

0)N

to the upright attitude, #k0;'0. This issue will be ad-

dressed in this section and a control law will be suggestedto cope with the boom-down condition.

The following control law is considered

#m(t)"H #X#0

(t)]#B(t)#g(t)q (t)]#B(t), (26)

where H is a positive constant matrix. g(t) is a piecewisecontinuous positive scalar function satisfying

g(t)"const'0, t3(k¹s, (k#1)¹

s), k"1, 2,2,

g (k¹s)'g ( (k#1)¹

s)'0, (27)

where ¹sis a positive constant. It is noted that control

law (26) has a time-varying attitude gain g(t) comparedwith control law (19).

Before the features of control law (26) are discussed,the following theorem is presented. Its proof is incorpor-ated in Appendix B.

Theorem 4. Consider control law (26) then the satellite,given by Eqs. (1)} (7), has 4 asymptotically stable localequilibria

M(#X#0

, #k0, #j

0) : (0,$0k

0,$0j

0)N.

The stability analysis shows that the equilibrium(0, 0k

0, 0i

0) is locally stable if g (t)"g8 (g8 , where g8 is the

limit of stability for control law (19). On the other hand, ifg(t)"g6 is large such that the quaternion feedback is themost signi"cant component on the r.h.s. of Eq. (1)

I #X0#0+g8 (q]#B)]#B (28)

then the vectors I #X#0

and (q]#B)]#B become parallel,and

I #XT#0

( (q(t)"#B)]#B)'0. (29)

It was assumed in Eq. (29) that the vectors #B and q arenot parallel. This is viable, since the controller can beactivated when the most favourable conditions in orbitalmotion for the boom upside-down algorithm occur, as

Fig. 10. Boom upside-down algorithm is activated in the regions ofNorth or South Poles.

illustrated in Fig. 10. It follows from Eq. (29) that

#XT#0

((q]#B)]#B)'0, (30)

since I is positive de"nite. If Eq. (29) is always satis"ed,then the satellite is asymptotically stable about the refer-ence M(0, 0k

0, 0j

0)N. To prove this statement, the following

Lyapunov candidate function is proposed:

E505"1

2#X

#0I #X

#0#3

2u2

0(#kT

0I #k

0!I

;)

#12u2

0(I

:!#jT

0I #j

0)#g (q2

1#q2

2#q2

3

#(1!q4)2 ), (31)

where g is a positive constant. The Lyapunov candidatefunction (31) resembles the total energy in Eq. (10) (Proofof Theorem 1). An extra attitude quaternion term hasbeen added, however.

The attitude quaternion satis"es the constraint equa-tion q2

1#2#q2

4"1, thus

E505"1

2#XT

#0I #X

#0#3

2u2

0(#kT

0I #k

0!I

;)

#12u2

0(I

:!#jT

0I #j

0)#2g (1!q

4). (32)

The time derivative of the Lyapunov candidate functionis

EQ505"#XT

#0#N

#53-#g#XT

#0q. (33)

In Eq. (33) the following was used:

d

dt2g (1!q

4)"g#X

#0q. (34)

1210 R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214

Finally, EQ505

is

EQ505"!H #XT

#0B3 TB3 #X

#0!g8 #XT

#0B3 TB3 q#g #XT

#0q, (35)

which is negative de"nite for su$ciently large g8 . Noticethat Eq. (29) is satis"ed only until the angular velocityterm in Eq. (1) becomes dominant. The objective of thisinvestigation, however, is not to derive a complete glo-bally stabilizing controller in one step, but rather toget a control law providing the necessary accelerationto turn the satellite from the upside down to uprightattitude.

It was assumed that the vectors #B and q should not beparallel. This implies that the boom upside-down algo-rithm must be triggered in the zones near the North orSouth Poles. As mentioned in Section 4.2, the vector partof the attitude quaternion, q, determines the axis of rota-tion from the Orbit CS to the Control CS. If the boom isupside-down, then q is perpendicular to the z-axis of theOrbit CS (the zenith). The zenith, though, is parallel tothe geomagnetic "eld vector over the poles, and #B andq are perpendicular. The boom upside-down situation isillustrated in Fig. 10.

From the analysis carried out so far it follows that forg(t)"g6 large enough the satellite trajectory is turnedfrom the boom upside-down towards the boom uprightattitude. For practical implementation, g6 shall be chosenlarger than

max(EN''

E )

min(E0BE2)+9]105 Nm/T

such that the control torque is larger than the gravitygradient. Recapitulating, if g(t)"g) the system is asymp-totically stable for all values of attitude such that theboom tip is above the horizon. If g (t)"g6 the satelliteboom axis is turned from upside-down to upright. Fur-thermore, if g (t) satis"es Eq. (27) then the satellite islocally asymptotically stable towards the four equilibriaM(#X

#0, #k

0, #j

0) : (0, $0k

0, $0j

0)N. The following algo-

rithm is now straightforward.If the boom tip is below the horizon, #k

0;40, imple-

ment control law (26) with g (t)"g6 . The satellite reachesa boom upright attitude, #k

0;'0, and g (t) shall gradually

decrease from g6 to g( . Nevertheless, due to possibly largeangular velocity involved the satellite may again arrive ata boom upside-down attitude. However, if #k

0;40 the

magnetic moment, #m, is set to 0, until the boom isupright once again.

This procedure uses the fact that the total energy of thesatellite for torque-free motion, #m"0, is preserved asproved in Theorem 1. The above algorithm makes theequilibria M (0, !0k

0, $0j

0)N unstable. The equilibrium

(0, 0k0, !0j

0) is also unstable since g (t) converges to

a constant nonzero value g( . Finally, only one equilibrium(0, 0k

0, 0j

0) remains asymptotically stable, thus it is glo-

bally asymptotically stable.

7.1. Simulation results for contingency operationcontroller

A simulation study has con"rmed our hypothesis thatthe boom upside-down control algorithm providesglobally asymptotically stable satellite motion. Simula-tion results are shown in Figs. 11}13. The initialconditions are such that the satellite has the upside-down attitude corresponding to the equilibriumM(#X

#0, #k0 , #j

0) : (0,!0k

0, 0j

0)N. The velocity gain is

Fig. 11. The velocity gain H is 1]108E Am s/T. The attitude gain g (t)is time varying with an initial value 15 ) 105 Am2/T. It converges tog("3]105 A m2/T.

Fig. 12. #k0;

characterizes convergence of #k0

towards 0k0

(if #k0;(0

satellite is upside-down), whereas #j0:

characterizes convergence of #j0

towards 0j0.

R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214 1211

Fig. 13. The attitude quaternion, #0q converges to [0, , 0, 1]T from an

upside-down attitude.

Fig. 14. The velocity and attitude gains are H"1]108 E Ams/T andg6 "27]105 A m2/T. The attitude controller is activated when #k

0;'0

(if #k,;40 then #m"0).

H"1]108E Am s/T, and the quaternion gain is chosento gradually decrease from g6 "15]105 A m2/T tog("3]105 A m2/T. The controller is seen to have a satis-factory performance. It takes less than half an orbit toturn the satellite up, and it is stabilized to the operationalregion within 6 orbits. This is rather convincing consider-ing the available mechanical torque which is less than1.2]10~3 N m, only three times more than the magni-tude of the maximum gravity gradient torque for thissatellite. A simulation for the same initial conditions but

with an initial attitude gain g("27]105 A m2/T is depic-ted in Fig. 14. The acceleration imposed by the attitudecontroller is high enough to turn the boom upright, butthe controller is not able to decelerate the motion when#k

0;'0, and the satellite turns upside-down again. The

attitude controller is seen to be active only when #k0;'0.

The combined control action is seen to be globallyasymptotically stable.

8. Conclusions

This paper has shown that it is feasible to obtaina three-axis stable platform in low Earth orbits usingmagnetorquers as the only actuators for the attitudecontrol system.

More detailed, this work contributes to the develop-ment of proportional-derivative feedback control basedonly on magnetic torquing for low Earth orbit satellites.A stepwise re"nement from locally to globally stabilizingcontrollers were proposed, and rigorous stability analysiswas carried out. A velocity controller, using a crossproduct between the angular velocity and the localgeomagnetic "eld, was shown to give four stable equilib-ria, one of which was the reference. Methods for pertur-bing the satellite motion from three undesired equilibriawere then investigated. It was shown using resultsfrom the theory of periodic systems that a locally stablesolution could be found. This controller had a propor-tional term with constant gain as part of the nonlinearalgorithm. Full global asymptotic stability was requiredto cope with an extreme condition like a boom upside-down contingency. To handle this, a time-varying gainwas introdued in the proportional term and switchingused to avoid generation of de-stabilizing torques. Thiscontrol algorithm was shown to make the satellite glo-bally asymptotically stable to the reference. Achievingthree-axis stabilization is quite remarkable result fora system that is only controllable in two axes in a lineartime invariant setting.

Simulation results showed the pro"ciency of the newcontrol algorithms for three di!erent mission phases: thedetumbling phase, the normal operation phase, anda contingency operation phase with inverted boom.These controllers are used in the Danish "rsted satellite.This mission is believed to be the "rst to #y three-axisstabilized with magnetorquers as sole actuators for atti-tude control.

Acknowledgements

The support of this work by the Danish "rstedSatellite Project and by the Faculty of Technology andScience at Aalborg University is greatly appreciated.

1212 R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214

Fig. 15. An example of level set ¸v(c). The set M

y(c) consists of two

subsets, whereas the level set, ¸v(c) is the subset of M

v(c), which is

containing the equilibrium.

Appendix A. Periodic extension of Lyapunov stability

The aim of this appendix is to present an importantlemma by Krasovskii}LaSalle lemma (Vidyasagar, 1993).The focus is on the stability of the class of periodicnonlinear systems

x5 "f(t, x (t)) where x3Rn and t3R`

(A.1)

satisfying

∀t50 xf(t#¹, x)"f(t, x).

As an introduction to Krasovskii}LaSalle lemma de"ni-tions of two auxiliary sets is provided. Firstly, the de"ni-tion of a level set is introduced.

To begin with consider a C1 locally positive de"nitefunction v :R

`]RnPR such that v is periodic with the

same period ¹ as the system. Now, a level set ¸v(c) is the

connected subset of a set

Mv(c)"Mx3Rn : &t50 such that v(t, x)4cN (A.2)

containing the equilibrium 0. An example of the level set,¸v(c) is depicted in Fig. 15. The set M

v(c) consists of

two subsets. The level set, ¸v(c) is the subset of M

v(c),

which contains the equilibrium. The second auxiliary setis A

v(c)

Av(c)"Mx3¸

v(c) : ∀t50 v(t, x)4cN. (A.3)

Notice that the di!erence between ¸v(c) (M

v(c)) and A

v(c)

is in the quanti"ers. Now, the stability of a periodicnonlinear system is established in Krasovski}LaSallelemma.

Lemma A.1 (Krasovskii}LaSalle). Consider the periodicsystem (A.1). Suppose there exists a C1 function

v :R`

]RnPR such that v is periodic with the same periodas the system, v is locally positive de,nite. Moreover, thereexists an open neighbourhood N of 0 such that

∀t50, ∀x3N vR (t, x)40. (A.4)

Choose a constant c'0 such that the level set ¸v(c) is

bounded and contained in N. Finally let

S"Mx3¸v(c) : &t50 such that vR (t, x)"0N, (A.5)

and let M denote the largest invariant set of the system inhand contained in S. ¹hen

x03A

v(c), t

050Nlim

t?=

d(x (t, t0, x

0), M)"0, (6)

where x (t, t0, x

0) is the solution of Eq. (A.1) which at initial

time t0

passes through the initial point x0. d (y, M) denotes

the distance from the point y to the set M.

Appendix B

Proof of Theorem 4. For simplicity of notation the equa-tion of satellite Eqs. (1)}(7) with controller (8) are repre-sented by

x5 (t)"f (x (t), t), (B.1)

and the equations of satellite motion with controller (26),for constant e (t)"e(k¹ ), are denoted as

x5 (t)"f,(x (t), t). (B.2)

Furthermore, di!erential equation (B.1) for the initialcondition x (t

0)"x

0has the solution x(t, t

0, x

0), and

di!erential equation (B.2) for the initial conditionx(t

0)"x

0has the solution x

,(t, t

0, x

0).

The kinematic and dynamic di!erential equations areLipschitz, and the following is true

if limk?=

fk(x(t), t)"f (x (t), t) then

limk?=

xk(t, t

0, x

0)"x (t, t

0, x

0),

thus

if limt?=

x (t, t0, x

0)"y

fthen

limt?=

xk(t, t

0, x

0)"y

f.

This means, if limt?=

e(t)"0, each trajectory of thesatellite actuated by control law (26) converges to oneof the asymptotically stable equilibria of system (B.1) :M(#X

#0, #k

0, #j

0) : (0, $0k

0, $0j

0)N. K

R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214 1213

References

Ch. I. Byrnes, & Isidori, A. (1991). On the attitude stabilization of rigidspacecraft. Automatica, 27(1), 87}95.

Cavallo, A., De Maria, G., Ferrara, F., & Nistri, P. (1993). A slidingmanifold approach to satellite attitude control. In 12th=orld Con-gress IFAC, Sidney (pp. 177}184).

Egeland, O., & Godhavn, J. M. (1994). Passivity-based adaptive atti-tude control of a rigid spacecraft. IEEE ¹ransactions on AutomaticControl, 39(4), 842}846.

Hughes, P. C. (1986). Spacecraft attitude dynamics. New York: Wiley.Martel, F., Parimal, K. P., & Psiaki, M. (1988). Active magnetic control

system for gravity gradient stabilized spacecraft. In AnnualAIAA/;tah State ;niversity Conference on Small Satellites(pp. 1}10). September 1988.

Mohler, R. R. (1991). Nonlinear systems, Volume Dynamics and Con-trol. Englewood Cli!s, NJ: Prentice-Hall.

Musser K. L., & Ward, L. E. (1989). Autonomous spacecraft attitudecontrol using magnetic torquing only. In Fight Mechanics Estima-tion ¹heory Symposium, NASA.

Steyn, W. H. (1994). Comparison of low-earth orbitting satellite attitudecontrollers submitted to controllability constraints. Journal of Guid-ance, Control, and Dynamics, 17(4), 795}804.

Vidyasagar, M. (1993). Nonlinear systems analysis. Englewood Cli!s,NJ: Prentice-Hall.

Ting-Yung Wen, J., & Kreutz-Delgado, K. (1991). The attitude controlproblem. IEEE ¹ransactions on Automatic Control, 36(10),1732}1743.

Wertz, J. R. (1990). Spacecraft attitude determination and control. Dor-drecht: Kluwer.

Wisniewski, R. (1994). Nonlinear control for satellite detumbling basedon magnetic torquing. In Joint Services Data Exchange for Guid-ance, Navigation, and Control, Arizona (p. 10A).

Wisniewski, R. (1995). In#uence of aerodynamic torque on +rstedsatellite motion. Technical Report TN-251, Aalborg University.

Wisniewski, R. (1995). Three-axis attitude control * linear time-vary-ing approach. In ¹enth IFAC=orkshop on Control Applications ofOptimization, Haifa, Israel, December 1995.

Wisniewski, R. (1996). Satellite attitude control using only electro-magnetic actuation. Ph.D. thesis, Aalborg University, December1996.

Rafael Wisniewski was born in Szczecin,Poland, in 1968. He received his M.Sc.degree in Engineering from Technical Uni-versity of Szczecin, Poland, in 1992 andPh.D. degree in Electrical Engineeringfrom Aalborg University, Denmark, in1997. He stayed with NASA Goddard in1998 as a visiting scholar. He is currentlyan Assistant Professor at the Departmentof Control Engineering, Aalborg Univer-sity. His research interests are in nonlinear

control theory, periodic systems, attitude control and estimation.

Mogens Blanke was born in 1947 inCopenhagen, Denmark. He received theM.Sc. and Ph.D. degrees in automatic con-trol from the Technical University of De-nmark (DTU) in 1974 and 1982, respec-tively. He was a systems analyst at theEuropean Space Agency in the Nether-lands in 1975}1976, assistant and later as-sociate professor control engineering atDTU from 1976 to 1985. He was head ofthe Marine Division of the Automation

Company S+ren T. Lyngs+ A/S from 1985 until 1990. Main activitieswere total ship integrated control and diesel engine control. Since 1990,he has been Professor of Control Engineering at Aalborg University,Denmark, where he started a research group in autonomous systemsand fault-tolerant control, an activity that includes an aerospace sys-tems laboratory.

Mogens Blanke's research interests include identi"cation and controlof continuous, nonlinear systems, estimation and fault detection, anddesign methods for fault tolerant control. Particular application inter-ests are spacecraft attitude control, automatic steering and rudder-rolldamping for ships, and controls for ship propulsion plants.

Mogens Blanke is active within IFAC and started the WorkingGroup on Marine Systems in 1986. He was chairman of the TechnicalCommittee of Marine Systems in 1990}1996 and is currently chairof the Co-ordination Committee for Transportation Systems andVehicles.

1214 R. Wis&niewski, M. Blanke/Automatica 35 (1999) 1201}1214