Journal of Systems Engineering and Electronics

Vol. 19, No. 1, 2008, pp.161–166

Predictive control for satellite formation keeping

He Donglei & Cao XibinResearch Center of Satellite Technology, Harbin Institute of Technology, Harbin 150080, P. R. China

(Received October 28, 2006)

Abstract: Based on a Hill equation and a nonlinear equation describing the desired and real dynamics of

relative motion separately, a predictive controller is brought forward, which makes the real state track the desired

ones to keep satellite formation. The stability and robustness of the controller are analyzed. Finally, comparing the

simulation results of the proposed controller with that of the traditional, proportional-differential controller shows

that the former one is capable of keeping the satellite formation more favorably, considering the disturbances such

as the J2 perturbations.

Keywords: predictive control; satellites formation; low-thrust technology; Lyapunov theorem

1. Introduction

For satellite formation flying[1] missions, formationkeeping control is a key technology because the ge-ometric formation of these satellites must be strictlycontrolled when they are finishing some specific stud-ies. However, satellites in space often receive distur-bances coming from the perturbation sources, such as,the earth’s oblateness perturbation, the atmosphericdrag perturbation, the solar pressure perturbation,and so on. It must be noted that the most desta-bilizing perturbation affecting the formation is the J2earth oblateness effect, which causes a rotation of theline of apsides or argument of perigee[2]. Therefore,measures must be taken to keep effective formationcontrol, to eliminate this negative influence of satel-lite formation flying.

Predictive control[3] is a newly developed controlmethod, which is capable of tracking the desired re-sponse based on minimization of predicted trackingerrors. Low-thrust technology[4] is a long-term, con-tinous, impulsive technology, and is of late, being ap-plied to practical flying missions. In this article, basedon the description of relative motion of satellites inthe Hill equation and the nonlinear equation[5], a low-thrust formation keeping predictive control method,eliminating disturbances such as J2 perturbation, has

been brought forward to finish the task of formationkeeping. Finally, by comparing the simulation resultsof the proposed controller with those of the traditionalproportional-differential one, the effectiveness of thegiven approach is shown.

2. Relative motion dynamics

Consider two satellites in orbit around the sphericalearth. Suppose the leader satellite has position r rel-ative to the center of the earth and the follower hasposition ρ relative to the leader. The unperturbeddynamics of the satellite is given by

¨r +µ

r3r = (1)

¨r + ¨ρ +µ

|r + ρ|3 (r + ρ) = 0 (2)

Where r = r and µ is the earth’s gravitational con-stant. Taking the difference of these equations yields

¨ρ +µ

|r + ρ|3 (r + ρ) − µ

r3r = 0 (3)

Assuming the leader satellite is in a circular orbit,then the radius r is a constant. As is shown in Fig.1, consider a moving coordinating system attached tothe leader satellite, where X is in the radial direction,Y is in the direction of motion, and Z is normal tothe orbital plane.

Allowing the relative position vector to be writtenas ρ = xX + yY + zZ, and assuming the distance

162 He Donglei & Cao Xibin

between the satellites is small, and noting the meanmotion of the leader satellite n =

õ/a3 yields the

Hill equations

Fig. 1 Relative motion reference frame

x − 2ny − 3n2x = 0 (4)y + 2nx = 0 (5)z + n2z = 0 (6)

Define a state vector as

Xd = [ xd yd zd xd yd zd ]T (7)

The subscript d signifies the desirable conditions.Then Eqs.(4)-(6) can be written in the following form

Xd =

⎡⎣ 03×3 I3×3

A B

⎤⎦Xd (8)

orXd1 = Xd2 (9)

Xd2 = AXd1 + BXd2 (10)

Where Xd1 and Xd2 are desirable relative position vec-tor and desirable relative velocity vector. The matrixA and B are

A =

⎡⎢⎢⎣

3n2 0 0

0 0 0

0 0 −n2

⎤⎥⎥⎦, B =

⎡⎢⎢⎣

0 2n 0

−2n 0 0

0 0 0

⎤⎥⎥⎦(11)

3. Nonlinear dynamics equation of sate-

llites formation flying

In this article, the real relative motion dynamics isdescribed by the following nonlinear model

x − 2y + (R + x)[g(x, y, z, R) − 1] = ux + dx (12)

y + 2x + y[g(x, y, z, R)− 1] = uy + dy (13)

z + zg(x, y, z, R) = uz + dz (14)

Where R is the radius of the leader satellite’s circularorbit and

g(x, y, z, R) = [(R + x)2 + (y2 + z2)2]/R2− 32 (15)

Then the real motion dynamics of the formation isdescribed by

X = [ x y z x y z ]T (16)

X1 = X2 (17)

X2 = f2(X) + u + d (18)

where X1 and X2 are the real relative positionstate vector and relative velocity state vector, d =[ dx dy dz ]T are the disturbances, including theearth’s oblateness perturbation, the atmospheric dragperturbation, the solar pressure perturbance, and soon.

Let the state vectors be the outputs of the system.For the desired and the real systems they are

Yd = Xd (19)

Y = X (20)

X ⊂ R6 is the state vector and u ⊂ U3 is the controlinput. Obviously, the real system of satellite forma-tion including Eqs.(17) and (18) satisfy the conditionof application of the predictive controller design.

4. Formation keeping predictivecontroller

Formation keeping of formation flying satellites comesunder the action of control input, where the real statesin Eq.(16) will track the desired states in Eq.(7). Thetracking error is defined as follows

e =[

eT1 eT

2

]T

= Y − Yd = X − Xd (21)

e1 and e2 are relative position and relative velocitytracking error.

Consider a performance index that penalizes thetracking error at the next instant and current controlexpenditure

J(u) =12[X(t + h) − Xd(t + h)]TQ×

Predictive control for satellite formation keeping 163

[X(t + h) − Xd(t + h)] +12uTRu (22)

Where Q, R are positive matrices and h is a smallconstant. By expanding the Tayor series of X(t + h)the following is obtained

X(t + h) = X(t) + v[X(t), h] + Λ(h)WX(t)u (23)

Minimization of J with respect to u by setting∂J(u)/∂u = 0 yields an optimal predictive controller

u = −[Λ(h)W (X)]TQ[Λ(h)W (X)] + R−1×[Λ(h)W (X)]TQ[e(t) + ν(X, h) − d(t, h)] (24)

The meaning of matrices v[X(t), h], Λ(h),W (X) andd(t, h) are explained in detail in Ref. [3].

Let ri, i = 1, · · · , 6, be the lowest order of derivativeof Xi in which any component of u first appears atX(t). Denoting ri = 2 for i = 1, 2, 3. Let

Q =

⎛⎝ Q1 0

0 h2Q2

⎞⎠

Where Q1 and Q2 are both positive definite matricesof 3 × 3.Then it follows that,

F11 =∂f1

∂X1=

∂X2

∂X1= 0, F12 =

∂f1

∂X2=

∂X2

∂X2= I

and a low-thrust formation keeping predictive con-troller can be obtained

u = −P

(1

2h2(F12B2)TQ1

e1 + he1 +

h2

2[F11f1+

F12f2 − Xd2(t)]

+1h

BT2 Q2e2 + h[f2 − Xd2(t)]

)=

−P

( [Q1

4+ Q2

][f2 − Xd2(t)] +

Q1

2h2e1 +

Q1

2he1+

Q2

he2

)

(25)where

P =(

14Q1 + Q2 + h−4R

)−1

(26)

Although the traditional thrust of the engine cangive rise to thrust by use of chemical fuel, the controlprecision of this method is not high and the durationof the engine is limited, because it is useless when thefuel resource on the satellite is exhausted. In recentyears, low-thrust technology has received great atten-tion because it can control the orbit muchly, continu-ously, and precisely. And of late it is being applied to

outer space flying missions. Therefore, in this article,low-thrust technology is used to control the satelliteformation.

Compared to the magnitude of the thrust action ofthe traditional thrust engine, the low-thrust enginecan supply a relatively smaller thrust with smallermagnitude. Assuming the magnitude of the low-thrust is umax, modify Eq.(25) to Eq.(27) as follows

u = sat(ui, umax) (27)

where

sat(ui, umax) =

⎧⎨⎩

ui, |ui| umax,

umax, |ui| > umax.(28)

5. Stability analysis

Considering Eqs.(9),(10) and (17),(18), the error rel-ative position vector and error relative velocity vectorequations are

e1 = e2 (29)

e2 = f2(X) − (AXd1 + BXd2) + u (30)

applying Eq.(25) to Eq.(30), it follows that

e2 = − 12h2

PQ1e1 − 1h

P (Q1/2 + Q2)e2 (31)

To study the stability of Eqs.(29) and (30), considerthe following Lyapunov function candidate

V =1

4h2eT1 PQ1e1 +

12eT2 e2 (32)

Take Eq.(31) into account, then the differential of V

isV =

12h2

eT1 PQ1e1 + eT

2 e2 =

− 1h

eT2 P (Q1/2 + Q2)e2 0

(33)

V = 0 if and only if e2 =0 . From Eq.(31) and takingP, Q > 0 into consideration , it follows that e1 =0.Therefore, e = [ eT

1 eT2 ]T = 0 is globally, asymptot-

ically stable under the control input action of Eq.(25).

6. Robustness analysis

To analyze the robustness of the proposed controller,first, the authors consider the unmodeled dynamics∆f2(X) in relative velocity error equation Eq.(30)

164 He Donglei & Cao Xibin

e2 = f2(X) + ∆f2(X) − (AXd1 + BXd2) + u (34)

Assuming for all X ⊂ R6

||∆f2(X)|| < a (35)

where a is a positive constant. For e2, the controlleris stated to be robust in the presence of the model-ing uncertainty (35) if the tracking error can be madeto satisfy ||e2|| < ε, with not too much initial error,where ε is a specified small constant.

Supposing R = 0, then the error dynamics (34)becomes

e2 = −P(Q1 + 2Q2)

2he2 +

[∆f2 − 1

2h2PQ1e1

](36)

As ∆f2(X) and e1 are all bounded for all X ⊂ R6,the quality inside the brace is bounded. In fact, forany given ε > 0, one can always find a certain value

η1(ε) > 0, making∣∣∣∣∣∣∣∣[∆f2 − 1

2h2PQ1e1

] ∣∣∣∣∣∣∣∣ < η1(ε).

It is known that

e2 = −P(Q1 + 2Q2)

2he2 (37)

is a asymptotically stable system. Then based onMalkin’s theorem it can be proved that there existsa certain value η2(ε) > 0, such that, for e2(0) < η2(ε),||e2(t)|| < ε and for all t 0.Then from Eqs.(35) and(36), it can be seen that

||e1(t)|| < 2ah2||(PQ1)−1|| (38)

where || • ||means norm value of a matrix or a vector.Therefore, the proposed formation keeping, low-thrustpredictive controller (25) can also maintain trackingaccuracy of real states (18) for the desired states (7)as long as the unmodeled dynamics satisfy the condi-tions in Eq.(35), which proves the robustness of theproposed method.

7. Mathematical simulations

Consider a circular satellite formation flying model,with a radius of 1000 m, under the J2 perturbation,and an assumed maximum value of low-thrust propul-sion of 20 mN to make the mathematical simulation.The altitude of the leader satellite is 800 km. Supposea, e, i, ω,Ω , M are orbital semimajor-axis, eccentricity,inclination, argument of perigee, the right ascensionand the mean anomaly, respectively, the parametersof leader satellite and follower are shown in Table 1.

Table 1 Orbital elements of the leader

and follower satellite

Orbital elements Leader satellite Follower satelite

a/km 7 178.137 7 178.212 4

e 0 0.000 061 116 4

i/o 98.597 5 98.601 425 9

ω/o 0 4.221 3

Ω/o 0 359.994 4

M/o 0 355.779 5

Other parameters are shown as follows: J2=1 082.6×10−6, Re = 6 378.137 km and the earth gravi-tational constant is 3.98 × 105 km3/S2. The initial de-sired relative position is [400,600,692.8] m, the relativevelocity is [0.3,-0.8,0.519 6] m/s. The actual relativeposition and relative velocity is [400.3,600.7,691.55]m and [0.306,-0.792,0.510 6] m/s . Let h = 100,Q1 = Q2 = 10I, R = 100I, and I is the identity ma-trix. The measurement precision of distance and thatof velocity is 5 cm and 2 mm/s. Under the action ofthe proposed controller, the geometry of the formationin space is shown in Fig. 2.

Fig. 2 Controlled formation with J2 perturbation

From Fig. 2, it can be seen that the proposed con-troller restrain the influence of the J2 perturbation,that is to say, the real relative position and relativevelocity track the corresponding desired ones respec-tively.

At the same time, under the consumption thatother parameters keep invariable, a traditionalproportional-differential controller is used to controlthe formation, the parameters of proposed controllerare

Predictive control for satellite formation keeping 165

KP = diag(0.000 8, 0.000 8, 0.000 8)

KD = diag(0.06, 0.06, 0.06)

Then the simulation results of the proposed controllerand the proportional-differential one are shown inFigs. 3,4. The duration of the simulation is one periodof the orbit.

Comparing the simulation results in Fig. 3 withthose in Fig. 4, it can be seen that the systemconvergence time under the proposed controller is500 s, the relative position and velocity precisionis 1.2 cm and 0.9 mm/s2, and the amount of fuelconsumption is 0.354 m/s. These are all betterthan the corresponding indexes of the traditionalproportional-differential controller whose values are

Fig. 3 The control results of the proposed Predictive controller

750 s, 1.6 cm, 1.1 mm/s2, and 0.736 m/s. That isto say, with respect to the formation keeping prob-lem, the control performance of the proposed predic-tive controller is better than that of the traditionalproportional-differential controller, and it is capableof precisely keeping the formation more favorably un-der the disturbances, such as, the J2 perturbation.

8. Conclusions

In this article, based on the Hill equation and anonlinear model describing relative motion, a low-thrust, formation keeping predictive controller isbrought forward. The stability and robustness of theproposed controller is proven. Comparing the simula-tion results of the proposed controller with those of the

166 He Donglei & Cao Xibin

Fig. 4 The control results of the traditional proportional-differential controller

traditional proportional-differential controller it canbe seen that the proposed controller is capable of keep-ing the satellite formation more favorable, with regardto the disturbances such as J2 perturbation.

References

[1] Kapila V, Sparks A G. Spacecraft formation flying: dy-

namics and control. Journal of Guidance, Controlan Dy-

namics, 2000 23(3): 561–564.

[2] Michael T, Jonathan P H. Advanced guidanced algorithms

for spacecraft formation keeping. Spacesystems Laboratory

of MIT. 2002

[3] Lu Ping. Nonlinear predictive controllers for continous sys-

tems. Journal of Guidance, Control and Dynamics, 1994,

17(3): 553–560.

[4] David E G.Analysis of low thrust orbit transfers using the

lagrange planetry equations. IEPC-97–160, Electric Rocket

Propulsion Society, 1997

[5] Yeh H H,Andrew S. Nonlinear tracking control of satellite

formations. Journal of Guidance, Control and Dynamics,

2002,25(2): 376–386.

He Donglei was born in 1979. He is a graduate stu-dent of spacecraft design and engineerin, the HarbinInstitute of Technology (HIT). His research interestis designing and control of small satellites for mation.E-mail: hedonglei2@ sina.com

Cao Xibin was born in 1963. He is a profes-sor of Research Center of Satellite Technology, HIT.His research interest is genernal desingning and sim-ulation technology of small satellite. E-mail: xb-cao@hit.edu.cn