- Email: [email protected]
Predictive control for satellite formation keeping
Journal of Systems Engineering and Electronics Vol. 19, No. 1, 2008, pp.161–166
Predictive control for satellite formation keeping He Donglei & Cao Xibin Research 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, formation keeping control is a key technology because the geometric formation of these satellites must be strictly controlled when they are finishing some specific studies. However, satellites in space often receive disturbances coming from the perturbation sources, such as, the earth’s oblateness perturbation, the atmospheric drag perturbation, the solar pressure perturbation, and so on. It must be noted that the most destabilizing perturbation affecting the formation is the J2 earth oblateness effect, which causes a rotation of the line of apsides or argument of perigee[2] . Therefore, measures must be taken to keep effective formation control, to eliminate this negative influence of satellite formation flying. Predictive control[3] is a newly developed control method, which is capable of tracking the desired response based on minimization of predicted tracking errors. Low-thrust technology[4] is a long-term, continous, impulsive technology, and is of late, being applied to practical flying missions. In this article, based on the description of relative motion of satellites in the Hill equation and the nonlinear equation[5] , a lowthrust formation keeping predictive control method, eliminating disturbances such as J2 perturbation, has
been brought forward to finish the task of formation keeping. Finally, by comparing the simulation results of the proposed controller with those of the traditional proportional-differential one, the effectiveness of the given approach is shown.
2. Relative motion dynamics Consider two satellites in orbit around the spherical earth. Suppose the leader satellite has position r relative to the center of the earth and the follower has position ρ relative to the leader. The unperturbed dynamics of the satellite is given by µ r¨¯ + 3 r¯ = (1) r µ (¯ r + ρ¯) = 0 (2) r¨¯ + ρ¨¯ + |¯ r + ρ¯|3 Where r = r¯ and µ is the earth’s gravitational constant. Taking the difference of these equations yields µ µ (¯ r + ρ¯) − 3 r¯ = 0 (3) ρ¨¯ + |¯ r + ρ¯|3 r 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 to the leader satellite, where X is in the radial direction, Y is in the direction of motion, and Z is normal to the orbital plane. Allowing the relative position vector to be written as ρ¯ = xX + yY + zZ, and assuming the distance
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He Donglei & Cao Xibin
between the satellites is small, and noting the mean motion of the leader satellite n = µ/a3 yields the Hill equations
z¨ + zg(x, y, z, R) = uz + dz
(14)
Where R is the radius of the leader satellite’s circular orbit and 3
g(x, y, z, R) = {[(R + x)2 + (y 2 + z 2 )2 ]/R2 }− 2 (15) Then the real motion dynamics of the formation is described by X=[ x
Fig. 1
Relative motion reference frame 2
x ¨ − 2ny˙ − 3n x = 0 y¨ + 2nx˙ = 0 z¨ + n2 z = 0 Define a state vector as Xd = [ xd
yd
zd
x˙ d
y˙ d
(4) (5) (6) T z˙d ]
(7)
The subscript d signifies the desirable conditions. Then Eqs.(4)-(6) can be written in the following form ⎤ ⎡ 03×3 I3×3 ⎦ Xd X˙ d = ⎣ (8) A B or
X˙ d1 = Xd2
(9)
X˙ d2 = AXd1 + BXd2
(10)
Where Xd1 and Xd2 are desirable relative position vector and desirable relative velocity vector. The matrix A and B are ⎡ ⎤ ⎡ ⎤ 3n2 0 0 0 2n 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A=⎢ 0 0 ⎥ ⎣ 0 ⎦, B = ⎣ −2n 0 0 ⎦ 0 0 −n2 0 0 0 (11)
3. Nonlinear dynamics equation of satellites formation flying In this article, the real relative motion dynamics is described 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)
y
z
x˙ y˙
T z˙ ]
(16)
X˙ 1 = X2
(17)
X˙ 2 = f2 (X) + u + d
(18)
where X1 and X2 are the real relative position state vector and relative velocity state vector, d = [ dx dy dz ]T are the disturbances, including the earth’s oblateness perturbation, the atmospheric drag perturbation, the solar pressure perturbance, and so on. 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 ⊂ U 3 is the control input. Obviously, the real system of satellite formation including Eqs.(17) and (18) satisfy the condition of application of the predictive controller design.
4. Formation keeping predictive controller Formation keeping of formation flying satellites comes under the action of control input, where the real states in Eq.(16) will track the desired states in Eq.(7). The tracking error is defined as follows T e = eT = Y − Yd = X − Xd (21) eT 1 2 e1 and e2 are relative position and relative velocity tracking error. Consider a performance index that penalizes the tracking error at the next instant and current control expenditure J(u) =
1 [X(t + h) − Xd (t + h)]T Q× 2
Predictive control for satellite formation keeping 1 [X(t + h) − Xd (t + h)] + uT Ru (22) 2 Where Q, R are positive matrices and h is a small constant. By expanding the Tayor series of X(t + h) the following is obtained X(t + h) = X(t) + v[X(t), h] + Λ(h)W X(t)u (23) Minimization of J with respect to u by setting ∂J(u)/∂u = 0 yields an optimal predictive controller
163 outer space flying missions. Therefore, in this article, low-thrust technology is used to control the satellite formation. Compared to the magnitude of the thrust action of the traditional thrust engine, the low-thrust engine can supply a relatively smaller thrust with smaller magnitude. Assuming the magnitude of the lowthrust is umax , modify Eq.(25) to Eq.(27) as follows u = sat(ui , umax )
u = −{[Λ(h)W (X)]T Q[Λ(h)W (X)] + R}−1 × {[Λ(h)W (X)]T Q[e(t) + ν(X, h) − d(t, h)]}
(24)
The meaning of matrices v[X(t), h], Λ(h),W (X) and d(t, h) are explained in detail in Ref. [3]. Let ri , i = 1, · · · , 6, be the lowest order of derivative of Xi in which any component of u first appears at X(t). Denoting ri = 2 for i = 1, 2, 3. Let ⎛ ⎞ Q1 0 ⎠ Q=⎝ 0 h2 Q 2 Where Q1 and Q2 are both positive definite matrices of 3 × 3.Then it follows that, F11
∂f1 ∂X2 ∂f1 ∂X2 = = = 0, F12 = = =I ∂X1 ∂X1 ∂X2 ∂X2
and a low-thrust formation keeping predictive controller can be obtained 1 h2 T [F11 f1 + (F B ) Q u = −P 12 2 1 e1 + he˙ 1 + 2 2h 2 1 F12 f2 − X˙ d2 (t)] + B2T Q2 {e2 + h[f2 − Xd2 (t)]} = h Q1 Q1 Q1 + Q2 [f2 − X˙ d2 (t)] + 2 e1 + e˙ 1 + −P 4 2h 2h Q2 e2 h (25) where −1 1 Q1 + Q2 + h−4 R (26) P = 4 Although the traditional thrust of the engine can give rise to thrust by use of chemical fuel, the control precision of this method is not high and the duration of the engine is limited, because it is useless when the fuel resource on the satellite is exhausted. In recent years, low-thrust technology has received great attention because it can control the orbit muchly, continuously, and precisely. And of late it is being applied to
(27)
where
⎧ ⎨ u , |u | u i i max , sat(ui , umax ) = ⎩ umax , |ui | > umax .
(28)
5. Stability analysis Considering Eqs.(9),(10) and (17),(18), the error relative position vector and error relative velocity vector equations are e˙ 1 = e2 (29) e˙ 2 = f2 (X) − (AXd1 + BXd2 ) + u
(30)
applying Eq.(25) to Eq.(30), it follows that e˙ 2 = −
1 1 P Q1 e1 − P (Q1 /2 + Q2 )e2 2h2 h
(31)
To study the stability of Eqs.(29) and (30), consider the following Lyapunov function candidate V =
1 T 1 e P Q1 e1 + eT e2 4h2 1 2 2
(32)
Take Eq.(31) into account, then the differential of V is 1 V˙ = 2 eT P Q1 e˙ 1 + eT 2 e˙ 2 = 2h 1 (33) 1 T − e2 P (Q1 /2 + Q2 )e2 0 h ˙ V = 0 if and only if e2 =0 . From Eq.(31) and taking P, Q > 0 into consideration , it follows that e1 =0. T Therefore, e = [ eT eT 1 2 ] = 0 is globally, asymptotically 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)
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He Donglei & Cao Xibin
e˙ 2 = f2 (X) + ∆f2 (X) − (AXd1 + BXd2 ) + u
(34)
(35)
where a is a positive constant. For e2 , the controller is stated to be robust in the presence of the modeling uncertainty (35) if the tracking error can be made to 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 (Q1 + 2Q2 ) 1 e˙ 2 = −P e2 + ∆f2 − 2 P Q1 e1 (36) 2h 2h As ∆f2 (X) and e1 are all bounded for all X ⊂ R6 , the quality inside the brace is bounded. In fact, for any given ε > 0, one can always find a certain value 1 η1 (ε) > 0, making ∆f2 − 2 P Q1 e1 < η1 (ε). 2h It is known that (Q1 + 2Q2 ) e2 (37) e˙ 2 = −P 2h is a asymptotically stable system. Then based on Malkin’s theorem it can be proved that there exists a 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 ||(P Q1 )−1 ||
Orbital elements of the leader and follower satellite
Assuming for all X ⊂ R6 ||∆f2 (X)|| < a
Table 1
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 gravitational constant is 3.98 × 105 km3 /S2 . The initial desired relative position is [400,600,692.8] m, the relative velocity is [0.3,-0.8,0.519 6] m/s. The actual relative position 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 matrix. The measurement precision of distance and that of velocity is 5 cm and 2 mm/s. Under the action of the proposed controller, the geometry of the formation in space is shown in Fig. 2.
(38)
where || • ||means norm value of a matrix or a vector. Therefore, the proposed formation keeping, low-thrust predictive controller (25) can also maintain tracking accuracy of real states (18) for the desired states (7) as long as the unmodeled dynamics satisfy the conditions in Eq.(35), which proves the robustness of the proposed 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 propulsion of 20 mN to make the mathematical simulation. The altitude of the leader satellite is 800 km. Suppose a, e, i, ω, Ω , M are orbital semimajor-axis, eccentricity, inclination, argument of perigee, the right ascension and the mean anomaly, respectively, the parameters of leader satellite and follower are shown in Table 1.
Fig. 2
Controlled formation with J2 perturbation
From Fig. 2, it can be seen that the proposed controller restrain the influence of the J2 perturbation, that is to say, the real relative position and relative velocity track the corresponding desired ones respectively. At the same time, under the consumption that other parameters keep invariable, a traditional proportional-differential controller is used to control the formation, the parameters of proposed controller are
Predictive control for satellite formation keeping 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 controller and the proportional-differential one are shown in Figs. 3,4. The duration of the simulation is one period of the orbit. Comparing the simulation results in Fig. 3 with those in Fig. 4, it can be seen that the system convergence time under the proposed controller is 500 s, the relative position and velocity precision is 1.2 cm and 0.9 mm/s2 , and the amount of fuel consumption is 0.354 m/s. These are all better than the corresponding indexes of the traditional proportional-differential controller whose values are
Fig. 3
165 750 s, 1.6 cm, 1.1 mm/s2 , and 0.736 m/s. That is to say, with respect to the formation keeping problem, the control performance of the proposed predictive controller is better than that of the traditional proportional-differential controller, and it is capable of precisely keeping the formation more favorably under the disturbances, such as, the J2 perturbation.
8. Conclusions In this article, based on the Hill equation and a nonlinear model describing relative motion, a lowthrust, formation keeping predictive controller is brought forward. The stability and robustness of the proposed controller is proven. Comparing the simulation results of the proposed controller with those of the
The control results of the proposed Predictive controller
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He Donglei & Cao Xibin
Fig. 4
The control results of the traditional proportional-differential controller
traditional proportional-differential controller it can be seen that the proposed controller is capable of keeping the satellite formation more favorable, with regard to the disturbances such as J2 perturbation.
References [1] Kapila V, Sparks A G. Spacecraft formation flying: dynamics and control. Journal of Guidance, Controlan Dynamics, 2000 23(3): 561–564. [2] Michael T, Jonathan P H. Advanced guidanced algorithms
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 student of spacecraft design and engineerin, the Harbin Institute of Technology (HIT). His research interest is designing and control of small satellites for mation. E-mail: hedonglei2@ sina.com
for spacecraft formation keeping. Spacesystems Laboratory of MIT. 2002
Cao Xibin was born in 1963.
He is a profes-
[3] Lu Ping. Nonlinear predictive controllers for continous sys-
sor of Research Center of Satellite Technology, HIT.
tems. Journal of Guidance, Control and Dynamics, 1994,
His research interest is genernal desingning and simulation technology of small satellite. E-mail: xb-
17(3): 553–560. [4] David E G.Analysis of low thrust orbit transfers using the
[email protected]
Predictive control for satellite formation keeping He Donglei & Cao Xibin Research 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, formation keeping control is a key technology because the geometric formation of these satellites must be strictly controlled when they are finishing some specific studies. However, satellites in space often receive disturbances coming from the perturbation sources, such as, the earth’s oblateness perturbation, the atmospheric drag perturbation, the solar pressure perturbation, and so on. It must be noted that the most destabilizing perturbation affecting the formation is the J2 earth oblateness effect, which causes a rotation of the line of apsides or argument of perigee[2] . Therefore, measures must be taken to keep effective formation control, to eliminate this negative influence of satellite formation flying. Predictive control[3] is a newly developed control method, which is capable of tracking the desired response based on minimization of predicted tracking errors. Low-thrust technology[4] is a long-term, continous, impulsive technology, and is of late, being applied to practical flying missions. In this article, based on the description of relative motion of satellites in the Hill equation and the nonlinear equation[5] , a lowthrust formation keeping predictive control method, eliminating disturbances such as J2 perturbation, has
been brought forward to finish the task of formation keeping. Finally, by comparing the simulation results of the proposed controller with those of the traditional proportional-differential one, the effectiveness of the given approach is shown.
2. Relative motion dynamics Consider two satellites in orbit around the spherical earth. Suppose the leader satellite has position r relative to the center of the earth and the follower has position ρ relative to the leader. The unperturbed dynamics of the satellite is given by µ r¨¯ + 3 r¯ = (1) r µ (¯ r + ρ¯) = 0 (2) r¨¯ + ρ¨¯ + |¯ r + ρ¯|3 Where r = r¯ and µ is the earth’s gravitational constant. Taking the difference of these equations yields µ µ (¯ r + ρ¯) − 3 r¯ = 0 (3) ρ¨¯ + |¯ r + ρ¯|3 r 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 to the leader satellite, where X is in the radial direction, Y is in the direction of motion, and Z is normal to the orbital plane. Allowing the relative position vector to be written as ρ¯ = xX + yY + zZ, and assuming the distance
162
He Donglei & Cao Xibin
between the satellites is small, and noting the mean motion of the leader satellite n = µ/a3 yields the Hill equations
z¨ + zg(x, y, z, R) = uz + dz
(14)
Where R is the radius of the leader satellite’s circular orbit and 3
g(x, y, z, R) = {[(R + x)2 + (y 2 + z 2 )2 ]/R2 }− 2 (15) Then the real motion dynamics of the formation is described by X=[ x
Fig. 1
Relative motion reference frame 2
x ¨ − 2ny˙ − 3n x = 0 y¨ + 2nx˙ = 0 z¨ + n2 z = 0 Define a state vector as Xd = [ xd
yd
zd
x˙ d
y˙ d
(4) (5) (6) T z˙d ]
(7)
The subscript d signifies the desirable conditions. Then Eqs.(4)-(6) can be written in the following form ⎤ ⎡ 03×3 I3×3 ⎦ Xd X˙ d = ⎣ (8) A B or
X˙ d1 = Xd2
(9)
X˙ d2 = AXd1 + BXd2
(10)
Where Xd1 and Xd2 are desirable relative position vector and desirable relative velocity vector. The matrix A and B are ⎡ ⎤ ⎡ ⎤ 3n2 0 0 0 2n 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A=⎢ 0 0 ⎥ ⎣ 0 ⎦, B = ⎣ −2n 0 0 ⎦ 0 0 −n2 0 0 0 (11)
3. Nonlinear dynamics equation of satellites formation flying In this article, the real relative motion dynamics is described 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)
y
z
x˙ y˙
T z˙ ]
(16)
X˙ 1 = X2
(17)
X˙ 2 = f2 (X) + u + d
(18)
where X1 and X2 are the real relative position state vector and relative velocity state vector, d = [ dx dy dz ]T are the disturbances, including the earth’s oblateness perturbation, the atmospheric drag perturbation, the solar pressure perturbance, and so on. 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 ⊂ U 3 is the control input. Obviously, the real system of satellite formation including Eqs.(17) and (18) satisfy the condition of application of the predictive controller design.
4. Formation keeping predictive controller Formation keeping of formation flying satellites comes under the action of control input, where the real states in Eq.(16) will track the desired states in Eq.(7). The tracking error is defined as follows T e = eT = Y − Yd = X − Xd (21) eT 1 2 e1 and e2 are relative position and relative velocity tracking error. Consider a performance index that penalizes the tracking error at the next instant and current control expenditure J(u) =
1 [X(t + h) − Xd (t + h)]T Q× 2
Predictive control for satellite formation keeping 1 [X(t + h) − Xd (t + h)] + uT Ru (22) 2 Where Q, R are positive matrices and h is a small constant. By expanding the Tayor series of X(t + h) the following is obtained X(t + h) = X(t) + v[X(t), h] + Λ(h)W X(t)u (23) Minimization of J with respect to u by setting ∂J(u)/∂u = 0 yields an optimal predictive controller
163 outer space flying missions. Therefore, in this article, low-thrust technology is used to control the satellite formation. Compared to the magnitude of the thrust action of the traditional thrust engine, the low-thrust engine can supply a relatively smaller thrust with smaller magnitude. Assuming the magnitude of the lowthrust is umax , modify Eq.(25) to Eq.(27) as follows u = sat(ui , umax )
u = −{[Λ(h)W (X)]T Q[Λ(h)W (X)] + R}−1 × {[Λ(h)W (X)]T Q[e(t) + ν(X, h) − d(t, h)]}
(24)
The meaning of matrices v[X(t), h], Λ(h),W (X) and d(t, h) are explained in detail in Ref. [3]. Let ri , i = 1, · · · , 6, be the lowest order of derivative of Xi in which any component of u first appears at X(t). Denoting ri = 2 for i = 1, 2, 3. Let ⎛ ⎞ Q1 0 ⎠ Q=⎝ 0 h2 Q 2 Where Q1 and Q2 are both positive definite matrices of 3 × 3.Then it follows that, F11
∂f1 ∂X2 ∂f1 ∂X2 = = = 0, F12 = = =I ∂X1 ∂X1 ∂X2 ∂X2
and a low-thrust formation keeping predictive controller can be obtained 1 h2 T [F11 f1 + (F B ) Q u = −P 12 2 1 e1 + he˙ 1 + 2 2h 2 1 F12 f2 − X˙ d2 (t)] + B2T Q2 {e2 + h[f2 − Xd2 (t)]} = h Q1 Q1 Q1 + Q2 [f2 − X˙ d2 (t)] + 2 e1 + e˙ 1 + −P 4 2h 2h Q2 e2 h (25) where −1 1 Q1 + Q2 + h−4 R (26) P = 4 Although the traditional thrust of the engine can give rise to thrust by use of chemical fuel, the control precision of this method is not high and the duration of the engine is limited, because it is useless when the fuel resource on the satellite is exhausted. In recent years, low-thrust technology has received great attention because it can control the orbit muchly, continuously, and precisely. And of late it is being applied to
(27)
where
⎧ ⎨ u , |u | u i i max , sat(ui , umax ) = ⎩ umax , |ui | > umax .
(28)
5. Stability analysis Considering Eqs.(9),(10) and (17),(18), the error relative position vector and error relative velocity vector equations are e˙ 1 = e2 (29) e˙ 2 = f2 (X) − (AXd1 + BXd2 ) + u
(30)
applying Eq.(25) to Eq.(30), it follows that e˙ 2 = −
1 1 P Q1 e1 − P (Q1 /2 + Q2 )e2 2h2 h
(31)
To study the stability of Eqs.(29) and (30), consider the following Lyapunov function candidate V =
1 T 1 e P Q1 e1 + eT e2 4h2 1 2 2
(32)
Take Eq.(31) into account, then the differential of V is 1 V˙ = 2 eT P Q1 e˙ 1 + eT 2 e˙ 2 = 2h 1 (33) 1 T − e2 P (Q1 /2 + Q2 )e2 0 h ˙ V = 0 if and only if e2 =0 . From Eq.(31) and taking P, Q > 0 into consideration , it follows that e1 =0. T Therefore, e = [ eT eT 1 2 ] = 0 is globally, asymptotically 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)
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He Donglei & Cao Xibin
e˙ 2 = f2 (X) + ∆f2 (X) − (AXd1 + BXd2 ) + u
(34)
(35)
where a is a positive constant. For e2 , the controller is stated to be robust in the presence of the modeling uncertainty (35) if the tracking error can be made to 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 (Q1 + 2Q2 ) 1 e˙ 2 = −P e2 + ∆f2 − 2 P Q1 e1 (36) 2h 2h As ∆f2 (X) and e1 are all bounded for all X ⊂ R6 , the quality inside the brace is bounded. In fact, for any given ε > 0, one can always find a certain value 1 η1 (ε) > 0, making ∆f2 − 2 P Q1 e1 < η1 (ε). 2h It is known that (Q1 + 2Q2 ) e2 (37) e˙ 2 = −P 2h is a asymptotically stable system. Then based on Malkin’s theorem it can be proved that there exists a 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 ||(P Q1 )−1 ||
Orbital elements of the leader and follower satellite
Assuming for all X ⊂ R6 ||∆f2 (X)|| < a
Table 1
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 gravitational constant is 3.98 × 105 km3 /S2 . The initial desired relative position is [400,600,692.8] m, the relative velocity is [0.3,-0.8,0.519 6] m/s. The actual relative position 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 matrix. The measurement precision of distance and that of velocity is 5 cm and 2 mm/s. Under the action of the proposed controller, the geometry of the formation in space is shown in Fig. 2.
(38)
where || • ||means norm value of a matrix or a vector. Therefore, the proposed formation keeping, low-thrust predictive controller (25) can also maintain tracking accuracy of real states (18) for the desired states (7) as long as the unmodeled dynamics satisfy the conditions in Eq.(35), which proves the robustness of the proposed 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 propulsion of 20 mN to make the mathematical simulation. The altitude of the leader satellite is 800 km. Suppose a, e, i, ω, Ω , M are orbital semimajor-axis, eccentricity, inclination, argument of perigee, the right ascension and the mean anomaly, respectively, the parameters of leader satellite and follower are shown in Table 1.
Fig. 2
Controlled formation with J2 perturbation
From Fig. 2, it can be seen that the proposed controller restrain the influence of the J2 perturbation, that is to say, the real relative position and relative velocity track the corresponding desired ones respectively. At the same time, under the consumption that other parameters keep invariable, a traditional proportional-differential controller is used to control the formation, the parameters of proposed controller are
Predictive control for satellite formation keeping 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 controller and the proportional-differential one are shown in Figs. 3,4. The duration of the simulation is one period of the orbit. Comparing the simulation results in Fig. 3 with those in Fig. 4, it can be seen that the system convergence time under the proposed controller is 500 s, the relative position and velocity precision is 1.2 cm and 0.9 mm/s2 , and the amount of fuel consumption is 0.354 m/s. These are all better than the corresponding indexes of the traditional proportional-differential controller whose values are
Fig. 3
165 750 s, 1.6 cm, 1.1 mm/s2 , and 0.736 m/s. That is to say, with respect to the formation keeping problem, the control performance of the proposed predictive controller is better than that of the traditional proportional-differential controller, and it is capable of precisely keeping the formation more favorably under the disturbances, such as, the J2 perturbation.
8. Conclusions In this article, based on the Hill equation and a nonlinear model describing relative motion, a lowthrust, formation keeping predictive controller is brought forward. The stability and robustness of the proposed controller is proven. Comparing the simulation results of the proposed controller with those of the
The control results of the proposed Predictive controller
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He Donglei & Cao Xibin
Fig. 4
The control results of the traditional proportional-differential controller
traditional proportional-differential controller it can be seen that the proposed controller is capable of keeping the satellite formation more favorable, with regard to the disturbances such as J2 perturbation.
References [1] Kapila V, Sparks A G. Spacecraft formation flying: dynamics and control. Journal of Guidance, Controlan Dynamics, 2000 23(3): 561–564. [2] Michael T, Jonathan P H. Advanced guidanced algorithms
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 student of spacecraft design and engineerin, the Harbin Institute of Technology (HIT). His research interest is designing and control of small satellites for mation. E-mail: hedonglei2@ sina.com
for spacecraft formation keeping. Spacesystems Laboratory of MIT. 2002
Cao Xibin was born in 1963.
He is a profes-
[3] Lu Ping. Nonlinear predictive controllers for continous sys-
sor of Research Center of Satellite Technology, HIT.
tems. Journal of Guidance, Control and Dynamics, 1994,
His research interest is genernal desingning and simulation technology of small satellite. E-mail: xb-
17(3): 553–560. [4] David E G.Analysis of low thrust orbit transfers using the
[email protected]