American Institute of Aeronautics and Astronautics

1

Robust Tracking Control of Attitude Satellite with Using NewEKF for Large Rotational Maneuvers

Mehrdad JafarbolandMalak-Ashtar University of Technology, Shahin Shahr, Iran 83145-115

Nasser SadatiSharif University of Technology, Tehran, Iran, 11365-8639

Hamidreza MomeniTarbiat Modares University, Tehran, Iran, 14155-4838

ABSTRACTControl of a class of uncertain nonlinear systems

which estimates unavailable state variables isconsidered. A new approach for robust tracking controlproblem of satellite for large rotational maneuvers ispresented in this paper. The features of this approachinclude –a strong algorithm to estimate attitude, basedon discrete extended kalman filter combined with acontinuous extended kalman filter and attitudenonlinear model –a robust controller based on sliding-mode with perturbation estimation.

The estimation result of interval kalman filtering isa sequence of interval estimates that encompasses allpossible optimal attitude estimates, which the intervalsystem may generate. The controller combined with anobserver will be stable by using sequence of intervalestimates in sliding condition of controller. At the end,the simulation results are showing the ability of thismethod in spite of different uncertainty (Up to %50).

INTRODUCTIONToday's world for a main part of its services and

technology is indebted to commercial,communications, agrology, military satellites. In orderto present these services, satellites must retain theirattitude, and in spite of external disturbances and theirparametric uncertainties must have a minuteorientation. In some cases such as special mission thatsatellite need to have large rotation maneuvers. Robustcontrollers should be designed in order to control theuncertain nonlinear multi-input multi-output system.1The accuracy of this controller is depended on theobserver’s accuracy. Therefore, other problem isdesigning an observer to estimate the attitude withsufficient accuracy from observed data, which iscontaminated with noise.

Military satellite may need to be able to maintainorbit and attitude knowledge in the face of jamming,destruction of orbit determination infrastructure such asthe global positioning system (GPS), or both. Civiliansatellite might not want to depend on GPS due topolitical reasons.2

In reference 3, the attitude is estimated using onlyan onboard magnetometer and sun sensor, in reference4, a star Image is added. In reference 5, an optimalbatch estimation and smoother, based on the minimummodel error approach, is developed. In reference 6, a

computational efficient, sequential method is presentedfor attitude estimation using gyro and vectormeasurements. This method is based on minimum-parameter method.

The linear quadratic-Gaussian (LQG) method waspresented for linear models, but John Doyle shows thatLQG solutions, unlike LQ solutions, provide no globalsystem independent guaranteed robustness properties.7The control approach, uses loop shaping techniquesbased on the LQG and the loop transfer recovery(LQG/LTR) methodology, is one suggestion for thisproblem.8,9

A new algorithm based on sliding-mode control (SMC)with adaptive boundary layer (ABL) combined byextended kalman Filter (EKF) observer which usesobserved data of 3-axis magnetometer and one sunsensor. The result of approach has an attitudeestimation accuracy of 0.2 degree and an attitudetracking accuracy of a bit more than 0.2 degree.

BACKGROUNDIn order to determine the attitude of the satellite

body, two frames are defined as follows:i) Orbit frame (a-frame): The frame which is

superposed on the orbit in satellite position and has axisa1 from center of earth towards the satellite masscenter, axis a2 at a tangent with orbit towards motiondirection and axis a3 which is perpendicular to thesatellite orbit plan.10,11

ii) Body frame (b-frame): The frame which issuperposed on the principal axis of the satellite bodywhere b1, b2, and b3 are unit vectors in Figure (1) and iindicate the orbit inclination.SATELLITE ORIENTATIONSatellite is turning on circular orbit by angular velocityΩ . The angular velocity of satellite’s body related toinertial frame in body frame is

T]321[ ωωωω = where in desired conditions

021 == ωω , Ω=3ω and 0321 === ψψψ .Figure 1 shows the angle of satellite attitude; ψ1, ψ2,and ψ3, which define the yaw, roll and pitch angles,respectively. The sequence of rotation is in a way thatfirst ψ3 rotates around 3a , and then ψ2 around new 2band ultimately ψ1 around new 1b axis.

AIAA Guidance, Navigation, and Control Conference and Exhibit11-14 August 2003, Austin, Texas

AIAA 2003-5785

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

American Institute of Aeronautics and Astronautics

2

Fig. 1. Satellite attitude angles from a-frame to b-frame

THE SATELLITE MODEL The satellite nonlinear model without noise on acircular orbit in an inverse square gravitational field insatellite body frame is similar to the relationsmentioned in (1).12

Assume the moment-of-inertia matrix is I = diag(I1,I2, I3) in the principal-axis frame and the angularvelocity of the satellite’s orbit is Ω .

)()()()()( tTtuxgxftx d++=& (1a)

))(()( txhtY = (1b)][)( 321321 ωωωψψψ=tx (1c)

=)(xf

Ω−−Ω−−Ω−−

Ω−+−

++

3/)212321)(12(2/)312331)(31(1/)232323)(23(

))2cos())1cos(3)1sin(2(()1sin(3)1cos(2

)2tan())1cos(3)1sin(2(1

IzzIIIzzIIIzzII

ωωωωωω

ψψωψωψωψω

ψψωψωω

(1d)

=

3322

11

/)(/)(/)(

000

)(

ItTItTItTtT

dd

dd

=

−1

0)(

Ixg (1e)

+

−=

)()()()()(

)()()()()(

)()(

zzz

1sin3sin1cos2sin3cos1cos3sin1sin2sin3cos

2cos3cos

321

ψψψψψ

ψψψψψ

ψψ

(1f)

Where dTux ,, and Y are state vectors, inputvector that contains three control moments in thedirection of the body axes, vector of externalperturbation and output vector that must follow thedesired quantity of ],,[ 321 dddd YYYY = , respectively.

dd YYY −=~ is vector of tracking error.

In normal conditions, dY is zero but during themaneuver can has every other desired quantity.MODEL OF SENSOR MEASUREMENTS

The sensors available for the system are a 3- axesmagnetometer and a 1-axes sun sensor. bB is themeasured magnetic field vector (MMFV) in b-framethat is given by

Ttttaabb BRB ][ )(3)(2)(1 ηηη+= (2)

++−=

123112313

23

3 ψψψψψψψψψψ

ψψ

cscssssccs

ccabR

+−+

−

1212313

1232113223

ψψψψψψψψψψψψψψ

ψψψ

cccssscscssscc

scs

where aB is magnetic field vector(MFV)in a-frame,abR is the transformation matrix from a-frame to b-

frame, and the components of whit noise measurementerror, )(3)(2)(1 ,, ttt ηηη , are random, Gaussian, zero-

mean, uncorrelated and discrete-time.13 2mδ is variance

of noise of MMFV. α∆ is the angle measured by sunsensor at the same sample and satisfies

)(432 )cos()cos()cos( tηψψα +=∆ (3)where )(4 tη is noise of sun sensor similar to )(1 tη

with variance 2sδ .

NONLINEAR MODEL OF CONTROLLER ANDOBSERVER FOR TRACKING

The configuration of a reduced order observer andcontroller, as depicted in figure (2), can be consideredas being a cascade connection of (a) an observer whoseoutput is the output estimate Y , and (b) the input-output linearization and sliding-mode controller whichtranslates Y~ into the controller output u .

THE SLIDING-MODE CONTROLLERThe controller is designed base on SMC according toreference 13. Eq. (1) based on input-outputlinearization is rewritten as controllable canonical form(4).

)()(),()( tuXbtXftX +=& (4a)

)3,2,1(),(),( == jXhLtXf jrfjj (4b)

)3,2,1,(),()( 1, == − kjXhLLXb j

rfgkjj

j (4c)

];;;;;[

363524331211

ψµψµψµψµψµψµ

&&&

=======X (4d)

where drrt xjDjS )(0~2)( ∫+= λ is time-varying

sliding-surface, ))/(ˆ(1)ˆ( ϕSSatKSPUbU −−−= is

control law, )321( PPPdiagP = is constant matrix

Satellite

ψ2

b1

ψ2

a3

ψ1

ψ3

Earth

a1

b3

b2

ψ3

ψ1

a2

Orbit

American Institute of Aeronautics and Astronautics

3

Figure 2. Block diagram represents the attitude control close-loop system for desired trajectory tracking

with positive component, T]321[ ϕϕϕϕ = is ABLT

vector, ][ 321 SSSS = is sliding vector and

][ 321 KKKK = and ][ 321 KKKK = are defined inEqs. (5) and (7), respectively.

The symbol j indicates jth component of b-frame. kis defined as

)3,2,1(;/0/0

=−=⇒≤

−=⇒≥

jBKjKIFBKjKIF

Minjjjjj

Maxjjjjj

ϕϕ

ϕϕ&&

&&

(5a)

where )3,2,1,( =jiBij are components of 1ˆ−= bbBand under scripts min and max are minimum andmaximum of component, respectively. The ABLTdynamics is

)3,2,1(;)(

)1)((0

=

=−+⇒>

jjdKdMaxjjB

jPdMaxjjBjjjIF

ϕλϕϕ&

&

(5b)

)3,2,1(;)(

)1)(()()(

0

=

=−+

⇒<

jjdKdMaxjjB

jPdMaxjjBjdMaxjjBdMinjjB

j

jIF

ϕλϕ

ϕ

&

&

Indeed, we now need to guarantee that the distanceto the boundary layer always decreases thus, we requiresliding condition:

)3,2,1(;)()/()2/1(

21

2

=−−≤

jSPSdtSd

jjjj

jηϕ&

(6)

by the flowing k1, k2 and k3, the sliding condition issatisfied

MinMaxMax

Max

BUBUBUBFK

11313212

111111

/)ˆˆˆ)1((

+

+−++≥ η (7a)

MinMaxMax

Max

BUBUBUBFK

22323121

222222

/)ˆˆˆ)1((

+

+−++≥ η (7b)

MinMaxMax

Max

BUBUBUBFK

33232131

333333

/)ˆˆˆ)1((

+

+−++≥ η (7c)

THE EFFECTS OF P AND λ For example insidethe boundary layer, S-dynamics is

3ˆ))(13(2ˆ))(12(1))(111(

111

11111 )(

udMinBudMinBudMaxB

fSddMax

KPBS

+

+−

−∆−=++

ϕ&

The respective break frequency is expressed by the

eigenvalue of 1

1111 )(

ϕd

dMaxKPB + , where E is unit

matrix. To attenuate the perturbations in the wide bandthese break frequencies should be designed as high aspossible flowing Elmali and Olgac.14 However, theremust be an upper bound for these break frequencies.This bound is introduced mainly by the capacity ofloop closure speed.

dMaxdMaxd BPBK )(/))(( 1111111 −= λ defines

the desired time-history of ABLT 1φ . Now, since 1λ isthe break-frequency of tow-order low pass filter, itmust be " small " with respect to high-frequencyunmodeled dynamics. If 1λ is increased, attitudeaccuracy will be also increased.12 However, 1λ mustbe chosen by trial and error.THE OBSERVATION STRUCTURE

In this section we develop the attitude estimationalgorithm. The design algorithm consists of replacingthe standard kalman filtering (KF) with new KF. Thisnew KF uses an algorithm based on discrete extendedkalman filter (DEKF) in combination with continuousextended kalman filter (CEKF) and nonlinear attitudemodel. The attitude and the sensors measurementmodels accompanying system’s and observation’snoise, can be given as:

Input noise vector )(tξ Input perturbation Td Sliding-mode Uncertain Nonlinear Output

Controller Plant

)(tη Measurement

Nonlinear Noise vector observer

Sensors

Yd

American Institute of A

)()()())(())(()( txtutxbtxftx ξΓ++=& (8a)

kkkk xhY η+= )( (8b)where, the sampling period be denoted by kk ttT −= +1

for sensors measurements.

DISCRETE EXTENDED KALMAN FILTER The discretenonlinear form of process in the state-space is

txxtxbbtxfxxf

xubxfx

kkk

kkk

kkkkkkkk

∆Γ=Γ∆=′∆+=′

Γ+′′+′=+

)()(;)(;)()(

)()(1 ξ

(9)

Taylor approximation of )( kkxf ′ at kx and that of

)( kk xh at 1ˆ −kkx ; that is,

)]ˆ([

);ˆ()()(

kk

kk

kkkkkkk

xx

fA

xxAxfxf

∂

′∂=

−+′≅′

(10a)

)]ˆ([

);ˆ()ˆ()(

1|

1|1|

−

−−

∂

∂=

−+≅

kkk

kk

kkkkkkkkk

xxh

C

xxCxhxh (10b)

Using the following assumes and (10), (9) gives

1|1|

||

ˆ)ˆ(ˆ)ˆ(

−− +−=

′+−′=

kkkkkkkk

kkkkkkkkk

xCxhYWubxAxfu

(11a)

kkkk

kkkkkkxCW

uxAxη

ξ+=

Γ++=+1 (11b)

where Ak and Ck are matrices and k

ξ and k

η are

system and observation, respectively.The statistics of the noise vectors are- 0 ==

kkEE ηξ

- Gaussian white noise such that kkQVar =)( ξ )

and kkRVar =)(η are positive definite matrices and

0)( =Tjk

E ηη for all k and l .14

The sequential algorithm of DEKF can be describedas follows

))ˆ(ˆˆ

ˆˆ

)(

)(

1|(1||

11|111|

11111,111,

1,,

11,1,

−−

−−−−−

−−−−−−−−

−

−−−

−+=

+=

ΓΓ+=

−=

+=

kkkkkkkkk

kkkkkk

Tkkk

Tkkkkkk

kkkkkk

kTkkkk

Tkkkk

xhYGxx

uxAx

QAPAP

PCGIP

RCPCCPG

where G and P are kalman gain and covariancmatrices, respectively. 1ˆ −kkx is optimal "predication

vector at time tk based on all data at times t0, t1, …, tk-

and kkx is "estimation" vector at time tk based on dat

time t0, t1, …, tk-1 .CONTINUOUS EXTENDED KALMAN FILTER The timevarying continuous nonlinear system is considered. ThP, Q matrices and the variations of noises of continuousystem (for small T) are

tRRtQQ

tRE

tQE

kk

Tt

Tt

∆=∆=

−=

−=

/;

)()(

)()(

)()(

)()(

τδηη

τδξξ

τ

τ

(13)

by replacing(13) in (12a) and neglectingTkkkk CPC 1, − , (14) will be obtained

tGGRCPG k ∆== − ;1 (14)by using (12c) to kkP ,1+ and replacing kkP from

(12b), using )( tAIAk ∆+= and neglecting higher-order terms in t∆ , (15a) will be obtained by replacing(12d) in (12e), (15b) can be obtained .15

Ttttttt

Ttt

Tttttt

QPCRCP

APPAP

)()()()()(1)()()(

)()()()()(

Γ′Γ′+

−+=−

& (15a)

))ˆ(()ˆ(ˆ )()()()()()()( ttttttt xgYGubxfx −++=& (15b)SUGGESTED OBSERVER The attitude angles will beestimate in discontinuous-time form because sensormeasurements are discrete (the sample period is T).But, the controller needs continuous-time attitudeestimation. This is the typical problem encounteredwhen processing noise measurement data, off-line.

- The algorithms of extended kalman filterThree types of observers are presented in the

flowing:A-approach) Using only DEKF: The sampling

period T divided to N steps. The observation vector isnot accessible, for N-1 steps of this N steps. In order tosolve this problem, pseudo-measurement is determinedbased on data estimation of previous steps. Results ofthis N-1 steps are smooth, because stochastic system ischanged to deterministic system.16

B-approach) Using CEKF: Because (15a) and(15b) equations calculated to discontinuously from incomputer, the sampling T is divided to N steps andpseudo-measurement is used again.17

C-approach) This paper's suggestion: The observershould be designed base on combining flowing items,according to figure (4).

-For one step of N steps (with sensorsmeasurement): Using DEKF.

-For N-1 steps of N steps (without sensorsmeasurement): The covariance will be calculated from(15a) according to one part of robust CEKF and statevector from (14a) according to attitude nonlinearmodel.

(12a

(12b

(12c

(12d

(12e

)

)

)

)

)

eronautics and Astronautics

4

e"

1,a

-es

-The disadvantage of A-approachSince system model is nonlinear, in order to use

DEKF, Taylor approximation should be used. Also, itis clear that the only time-consuming operation in theDEKF process is the real-time computation of thekalman gain matrices Gk form Eq. (12a). Whether itwould involve matrix inversions directly or sequentialalgorithms are used, therefore essentially, N will besmall and error of Taylor approximation increases,

American Institute of Aeronautics and Astronautics

5

Figure 3. The suggested algorithm for EKF (observer)

subsequently.

-The disadvantage of B-approachThis approach does not need matrix inversion, and

N can be selected greater than A-approach. But thecovariance matrix calculation has greater error,because of neglected higher-order terms in t∆ .-The advantages of C-approach

At each step of kalman filtering there are, twostages, we will derive a recursive formula that givesthe "predication" 1ˆ −kkx from the estimate

111 ˆˆ −−− = kkk xx firstly and then the "estimate"

kkk xx ˆˆ = from the predication 1ˆ −kkx .In the N-1 step of kalman filtering, that

observation vector in not accessible:-For predication stage, using eq. (8a), attitude

predication accuracy will be increased.-For estimation stage, using eq. (15a), the

covariance matrix calculation will be fast, and N canbe selected greater.Because in only one step of kalman filtering anobservation vector exists, in order to have moreaccuracy DEKF in used. SIMULATION RESULTS OF OBSERVER Simulation isdone for explained satellite in reference 14. There aretwo sensors available for attitude estimation: a 3-axismagnetometer, a 1-axis sun sensor. Let the sensormeasurement noises be a sequence of zero-mean

Gaussian white noise such that ][103 8 Tm −×=δ and.][deg005.0=sδ , respectively.

The moment of inertia matrix is changed to2]18317553.70[)( NmsIdiag = for inherent

stability (without controller). The true and estimatedof attitude angles are shown in fig. (5)-(8). When theinitial error is damped, the maximum estimation errorwill be less than 0.2 degree, (the estimation error willbe less than 0.005 degree in the without maneuvercase).

The estimation accuracy is reported least than 1degree in references 16 and 17 therefore theestimation accuracy is improved by a factor of ten.

Figure 5. The real value of the attitude angles

Integra- tor

Nonlinear dynamic model

Stateestimat- ion

Kalman Gain

Update

Covariance Update

Integra- tor

Covariance continuous

model

From Sensor Discrete Continuous

)(ˆ tx& kt

kt

t

t

)(tu

)(tu)(ˆ tP

)(ˆ tx

kkx |ˆ

kkx |ˆ

1|ˆ −kkx

1|ˆ −kkx

1|ˆ −kkx

1|ˆ

−kkP

1|ˆ

−kkP

kkP |ˆ kkP |

ˆ

)(ˆ tP&

kY

kG

0 2000 4000 6000 8000 10000 12000-15

-10

-5

0

5

10

15

time [ sec. ]

Yaw

, Rol

l and

Pitc

h

yaw Roll Pitch

American Institute of Aeronautics and Astronautics

6

Figure 6. Yaw angle real and estimation values

Figure 7. Yaw angle estimation error

Figure 8. Pitch angle estimation error

THE ROBUST NONLINEAR CONTROLLER ANDOBSERVER FOR TRACKING PROBLEM

If uncertain parameters and uncertain variables ofsystems are estimated desirably, and the slidingconditions are satisfied, then the sufficient conditionfor Lyapunov stability will exit. But, only uncertainparameter considered in standard sliding-mode and the

)3,2,1( =jf j is estimated as )3,2,1(ˆ =jf j . Theestimation error on jf is assumed to be bounded by

some known function )ˆ( jjjj FffF ≤− . But the

variables of system are uncertain, and this effect is notconsidered. A new approach based on sliding-modeand interval mathematics for robust controller ispresented in this paper. In each step of EKF, based oninterval mathematics, an upper boundary and a lowerboundary are estimated, and then uncertain parametersand uncertain variables are used together indetermination )3,2,1( =jF j . Therefore, if

)3,2,1( =jk j is determined according to (7), thenthe sufficient condition for Lyapunov stability ofcontroller will exit.-Robust kalman filtering

An uncertain matrix can be decomposed intocertain matrix Ak and uncertain matrix kA∆ .

Suppose that the uncertain parameters are onlyknown to be bounded. Then we can write

],[ kkkkkkIk AAAAAAA ∆+∆−=∆+= (16)

where tA∆ is constant bound for uncertainty. A

closed and bounded subset ],[ xx is referred to as an

interval.15 Let ],[ xx , ],[ xx , and ],[ xx be intervals.The basic arithmetic operations of intervals aredefined

];[],[],[ 21212211 xxxxxxxx ±±=± (17)

==⇒

=

,,,min,,,max

],[],[.],[

21212121

21212121

2211

xxxxxxxxyxxxxxxxxy

yyxxxx

∉=∉⇒

=

−

−

−

undefinedisxxthenxxoifxxxxthenxxoif

xxxxxxxx

1

1

122112211

],[],[]/1,/1[],[],[

],[.],[],[/],[

SIMULATION RESULTS OF ROBUST NONLINEARCONTROLLER AND OBSERVER TO TRACKING

PROBLEMS.In this section the satellite in reference 13 is used, forsimulation. Modeling uncertainties appears in theform of )3,2,1( =∆+ jII jj (fig. (9)) and externalperturbation has been considered as eq. (18), anddesired trajectories are shown in figure (10) where themaximum maneuver angle is 18 degrees.

Figure 9 The uncertainty with 10% variation

0 2000 4000 6000 8000 10000 12000-10

-5

0

5

10

15

time [ sec. ]

Yaw

And

Yaw

Est

imat

ion

Yaw EstimatedYaw

Yaw

Yaw Estimated

0 2000 4000 6000 8000 10000 12000-0.06

-0.04

-0.02

0

0.02

0.04

0.06

time [ sec. ]

Yaw

Est

imat

ion

Erro

r

0 2000 4000 6000 8000 10000 12000-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

time [ sec. ]

Pitc

h E

stim

atio

n E

rror

American Institute of Aeronautics and Astronautics

7

Figure 10. Desired trajectory of attitude angles.

Figure 11. Tracking error of attitude angles.

Figure 12. Yaw angles estimation error

).()4/(sin

)cos()sin(

41 mNt

tt

edT

+Ω−ΩΩ−

−=π

(18)

Figure 13. Roll angles estimation error.

Figure 14. Pitch angles estimation error.

Figure 15. S-dynamic and ABL for yaw axis.

In this section, sliding-mode controller andobserver based on our approach is used according tofig. (2). The simulation results containing: trackingerror of attitude angles )(

dψψ − , estimation error of

attitude angles )(m

ψψ − , 1S , 1ϕ and three input

torque of three axes )(u are shown in fig. (11)-(19).

0 200 400 600 800 1000 1200-5

0

5

10

15

20[deg.]

time [ sec. ]

Des

aire

d Tr

ajec

torie

s [ d

eg. ]

Yaw Roll Pitch

0 200 400 600 800 1000 1200-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

time [ sec. ]

Atti

tude

Est

imat

ion

Erro

r [ d

eg. ]

Y__Axis Z__Axis

X__Axis

0 200 400 600 800 1000 1200-0.2

-0.15

-0.1

-0.05

0

0.05

time [ sec. ]

X__A

xis

Est

imat

ion

Erro

r

0 200 400 600 800 1000 1200-0.2

-0.15

-0.1

-0.05

0

0.05

time [ sec. ]

Y__

Axi

s E

stim

atio

n E

rror

0 200 400 600 800 1000 1200-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

time [ sec. ]

Z__A

xis

Est

imat

ion

Erro

r

0 200 400 600 800 1000 1200-1.5

-1

-0.5

0

0.5

1

1.5

time [ sec. ]

s1 a

nd p

hy1

Dyn

amic

s [ d

eg./s

] S1 +phy1-phy1

American Institute of Aeronautics and Astronautics8

Figure 16. Input torque of Yaw axis (u1)

Figure 18. Fig. (16) for 830[s] to 835[s]according to maximum of u1

Figure 17. Input torque of Roll axis (u2)

Figure 19. Fig. (17) for 810[s] to 815[s]according to maximum of u2

200 400 600 800 1000 1200-0.04

-0.02

0

0.02

0.04

0.06

time [ sec. ]

Torq

ue [

N.m

]u1

0 200 400 600 800 1000 1200-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

time [ sec. ]

Torq

ue [

N.m

]

u2

0

830 831 832 833 834 835-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

time [ sec. ]

Torq

ue [

N.m

]

u1

810 811 812 813 814 815-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time [ sec. ]

Tor

que

[ N.m

]

u2

Table 1 Tracking error, estimation error and u deviations as function of uncertainty and noise variancMaximum of error ( Tracking|Estimation) [degrees] Maximum of u

].[ mNPitchRollYaw

u 2u 1u Tracking|Estima. Tracking|Estima. Tracking|Estima.

NoiseVariance

Present of

uncerta -inty

0. 60.4720.0290.1186| 0.11750.2445| 0.24410.1327 | 0.1324mδ30. 70.4070.0280.1200| 0.11920.2452| 0.24510.1330| 0.1326 mδ300. 203890.0270.1207| 0.11970.2482 | 0.24720.1367| 0.1363 mδ300

%10

0. 70.5760.0370.1518 |0.14970.2549| 0.25490.1465 | 0.1461 mδ30. 50.3480.0340.1528| 0.15080.2564 | 0.25660.1507| 0.1502 mδ300. 80.3830.0410.1520| 0.15020.2514| 0.25150.1472 | 0.1466 mδ300

%20

10.3720.0430.1886| 0.18560.2629| 0.26300.1619| 0.1607 mδ310.4320.0390.1867| 0.18360.2591| 0.26800.1610 | 0.1597 mδ3010.3850.480.1886| 0.18540.2639| 0.26370.1628| 0.1614 mδ300

%30

20.3520.0540.2652| 0.25970.2851| 0.28510.2029| 0.2000 mδ320.3420.0520.2673| 0.26190.2782 | 0.27870.1995 | 0.1970 mδ3020.3440.050.2656| 0.25990.2784 | 0.27910.1977 | 0.1952 mδ300

%50

e

341

52

38

80

84

75

.33

.31

.32

.82

.82

.78

American Institute of Aeronautics and Astronautics

9

Table 1 demonstrates the effect of uncertainty andnoise variance on estimation error and tracking error,for desired trajectories (fig.(10)) and ][101 8 Tm −×=δ .

CONCLUSIONFor uncertain systems, which have controller and

observer, robustness of combination of controller withobservation is the principal problem, becauserobustness of only controller or observer is notsufficient. The filtering result produced by the intervalkalman filtering scheme is a sequence of intervalestimates, I

kx that encompasses all possible optimal

estimates kx of the state vectors kx which theinterval system may generate. The controller incombination with observer is robust by using abovesequence of interval estimation in sliding condition ofsliding-mode control with adaptive boundary layerthickness.

This combination dose not effect on trackingaccuracy and estimation error of combination ofDEKF and CEKF is less than only DEKF.

REFERENCES1. Sedlak, J. E., “Improved Spacecraft Attitude FilterUsing a Sequentially Correlated Magnetometer NoiseModel”, Aviation Systems Conference 16th DASC.AIAA/IEEE, PP (8.4-9)- (8.4-16), 1997.2. Crassidis, J. L., Lightsey, E. G. and Markley, F. L.,“Efficient and Optimal Attitude Determination UsingRecursive Global Position System Signal Operations”,Journal of Guidance, Control, and Dynamics, vol. 22,No. 2, pp. 193- 201, March- April 1999.3. Psiaki, M. L., “Autonomous Low-Earth-OrbitDetermination from Magnetometer and Sun SensorData”, Journal of Guidance, Control, and Dynamics,vol. 22, No. 2, pp. 296-304, March- April 1999.4. Wiśniewski, R., “Linear Time-Varying Approach toSatellite Attitude Control Using Only ElectromagneticActuation”, Journal of Guidance, Control, andDynamics, Vol. 23, No. 4, PP 640-647, 2000.5. Crassidis, J. L., and Markley, F. L., “MinimumModel Error Approach for Attitude Estimation”,Journal of Guidance, Control, and Dynamics, vol. 20,No. 6, pp. 1241-1247, November-December 1997.6. Oshman, Y., and Markley, F. L., “Minimal-Parameter Attitude Matrix Estimation from VectorObservations”, Journal of Guidance, Control, andDynamics, vol. 21, No. 4, pp. 595-602, July- August1998.7. John C. Doyle, “ Guaranteed Margins for LQGRegulators”, IEEE Transaction on Automatic Control,Vol. AC-23, No. 4, August 1978.8. Lahdhiri T., Alouani A.T., “LQG/LTR SpacecraftAttitude Control “, IEEE Proceeding of the 32nd

conference on precision and control, San AntonioTexas December 1993.

9. Stein G., Athans M., ” The LQG/LTR procedure formultivariable Feedback Control Design” , IEEETransaction on Automatic Control, Vol. AC-32, No. 2,February 1987.10. Jafarboland M., Sadati N., Momeni, H. R. andghodjebaklou H. “Controlling the Attitude of LinearTime-Varying Model LEO Satellite Using OnlyElectromagnetic Actuation, “ IEEE AerospaceConference Big Sky, Montana, March 9-16, 2002.11. Jafarboland, M., Momeni, H. R., Sadati, N. andGhodjeh baclu, H., “Combining Permanent Magnetand Electromagnet in Momentum Removal Methodfor Earth-Pointing Satellite,” AIAA Guidance,Navigation, and control conference and Exhibit,Montreal, Canada, (A01-37170), pp. 1-9, 2001.12. Yuri B. Shtessel, ”Nonlinear Output Tracking ViaNonlinear Dynamic Sliding Manifolds,” IEEEIntimation Symposium on Intelligent Control, 16-18August 1994, Columbus, Ohio, USA, pp. 297-302.13. Julie, K., and Itzhack, Y., "Evaluation of Attitudeand Orbit Estimation Using Actual Earth MagneticField Data, " Journal of Guidance, Control, AndDynamics, Vol. 24, No. 3, May – June 2001, pp. 619-623.14. Elmali H., and Olgac, N., ”Satellite attitudeControl Via Sliding Mode With Perturbationestimation,” IEE Proceedings-Control TheoryApplication, vol. 143, No. 3 May 1997, pp. 276- 282.15. C. K. Chui, G. Chen, Kalman Filtering WithReal_Time Applications, Springer, Germany, 1999.16. Grewal, M. S. and Shira, M., "Application ofKalman Filtering to Gyro-less Attitude Determinationand control System for Environmental Satellite, “IEEE proceeding of the 34th Conference on Decision& control, New Orleans, LA- December 1995.17. Markle, F. l. and Berman, N., "DeterministicEKF–Like Estimator for spacecraft AttitudeEstimation, " IEEE Proceeding of the Attitude ControlConference, Maryland, June 1994.