Output Feedback Tracking Control for Spacecraft ... - Semantic Scholar
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Output feedback tracking control for spacecraft relative translation subject to input constraints and partial loss of control effectiveness Lin Zhao, Yingmin Jia, Junping Du, Jun Zhang Abstract— In this paper, an adaptive output feedback tracking control scheme is proposed for spacecraft formation flying (SFF) in the presence of external disturbances, uncertain system parameters, input constraints and partial loss of control effectiveness. The proposed controller incorporates a pseudo-velocity filter to account for the unmeasured relative velocity, and the neural network (NN) technique is implemented to approximate the desired nonlinear function and bounded external disturbances. In order to guarantee that the output of the NN used in the controller is bounded by the corresponding bound of the approximated nonlinear function, a switch function is employed to generate a switching between the adaptive NN control and the robust controller. Moreover, a fault tolerant part is included in the controller to compensate the partial loss of actuator effectiveness fault. It is shown that the derived controller not only guarantees the tracking error in the closed-loop system to be uniformly ultimately bounded (UUB) but also ensures the control input can rigorously enforce actuator magnitude constraints. Simulation results are provided to demonstrate the effectiveness of the proposed method.
I. I NTRODUCTION Spacecraft formation flying (SFF) techniques have many advantages in dealing with some space and Earth science missions, such as being more economical, flexible and reliable, etc. In the early days, results about the spacecraft formation control problem were presented based on the linear 3 degree-of-freedom (DOF) translational motion model [1], [2]. Later, the 3 DOF nonlinear model, which was derived for arbitrary orbital eccentricity and with added terms for orbital perturbations, was built for the problem of SFF [5]–[10]. For example, a nonlinear adaptive tracking control scheme was developed in [6], which relied on the restrictive assumptions of a circular orbit, and the availability of exact measurements of all relative velocity and position in translational motion. In practice, precise measurements of velocity are not always satisfied because of either cost limitations or implementation constraints [15]. Therefore, it is highly desirable to design a partial-state feedback controller that does not depend on the measurements of velocity. An adaptive output feedback This work was supported by the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201) and the NSFC (61134005, 61221061, 61327807). Lin Zhao and Yingmin Jia are with the Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]; [email protected]). Junping Du is with the Beijing Key Laboratory of Intelligent Telecommunications Software and Multimedia, School of Computer Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]) Jun Zhang is with the School of Electronic and Information Engineering, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]).
978-1-4799-3271-9/$31.00 ©2014 AACC
controller was proved in [7]. A more general result about the adaptive output feedback control for spacecraft formation control was given in [9]. It should be noted that all the results mentioned above did not consider the problem of input constraints, which cannot be ignored in practice for the existence of limitation on actuators [11]. Another important problem encountered in practice for spacecraft formation control system design is that of control failures, which can cause control system performance deterioration and lead to instability and even catastrophic accidents. Hence, it is necessary to maintain high reliability for spacecraft formation control system design against possible control faults [12]. Fault tolerant control (FTC) is an area of research that emerges to increase availability by specifically designing control algorithms capable of maintaining stability and performance despite the occurrence of faults, and it has received considerable attention from the control research community and aeronautical engineering [13]–[15]. Motivated by the above discussions, in this paper, we investigate the 3 DOF spacecraft formation control design by developing a partial-state feedback control scheme in the presence of external disturbances, uncertain system parameters, input constraints, and partial loss of actuator effectiveness fault. The main contributions of this paper are stated as follows: 1) A FTC scheme is proposed to compensate for the partial loss of actuator effectiveness fault without any fault detection and isolation mechanism. Moreover, the proposed output feedback control approach can guarantee the closedloop tracking system is uniformly ultimately bounded (UUB) stable, when the external disturbances and uncertain system parameters are bounded. 2) The adaptive neural network (NN) approximation technique is used to approximate the desired nonlinear function and bounded external disturbances. A switch function is employed to generate a switching between the adaptive NN control and the robust controller to guarantee that the output of the NN used in the controller is bounded. Moreover, the proposed control approach can explicitly account for input constraints, even in the case of actuator faults. Thus, the objective of efficient, low-cost, and reliable spacecraft formation control design can be met. The rest of this paper is organized as follows: The mathematical model of 3 DOF SFF and the control problem formulation are summarized in Section II. The control design is presented in Section III. Simulation results and conclusion are given in Sections IV and V, respectively. Throughout this paper, Rn and Rn×m denote the n-
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can be regarded as a time-varying potential force, n(rl , rf ) is the nonlinear term defined as n(rl , rf ) = rl − r12 rf3 l . The composite perturbation force Fd mf µ 0 0 m is given by Fd = fdf − mfl fdl , where fdl (fdf ) denotes the perturbation term of leader(follower) spacecraft due to external effects. The rate of the true anomaly ofqthe leader 2 µ is the spacecraft is v˙ = nl (1+el 2cos3 v) , where nl = a3
e iz
eh
er Leader
rl
p iy
ix
rf Follower
(1−el ) 2
Fig. 1: Reference coordinate frames [3].
dimensional and the n × m-dimensional Euclidean spaces, respectively; In denote the n × n identity matrices; X < 0 represents that X is the symmetric negative-definite matrix; the superscripts T and −1 stand for matrix transposition and matrix inverse; tr{·} refers to the trace, | · | refers to the absolute value, and k · k refers to the Euclidean vector norm or the induced matrix 2-norm; sgn(·), tanh(·) and cosh(·) are the sign, standard hyperbolic tangent and cosine functions, respectively. II. M ATHEMATICAL MODEL OF SFF
AND PROBLEM
where αv˙ , βv˙ , βv¨ are known positive constants. With the diversification and complication of spacecraft task and structure, faults of actuators would occurred frequently. Consider the actuators partial loss of effectiveness and represent the fault in product factor, system (1) can be described as
FORMULATION
A. Cartesian coordinate frames To form the basis of the relative position model, we use the standard definition of the Earth-Centered Inertial (ECI) frame Fi , with z-axis towards celestial north. We also employ a standard LVLH-definition of the leader orbit reference frame Fl , with unit vectors defined as er = rrll , eθ = eh × er and eh = hh , where h = rl × r˙l is the angular momentum vector of the orbit, and h = khk. In addition, we define a follower orbit reference frame Ff with origin specified by the relative orbit position vector p = rf − rl = xer + yeθ + zeh and with unit vectors aligned with the unit vectors in Fl at all times, as shown in Fig. 1. B. Relative translational motion The relative position dynamics can be written as [10] ¨ + C(v) ˙ p˙ + N(p, rl , rf , v, ˙ v¨, ul ) = uf + Fd mf p
l
mean motion of the leader, al is the semimajor axis of the leader orbit, and el is the orbit eccentricity. Differentiation v˙ results in the rate of change of the true anomaly as −2n2l el (1+el cos v)3 sin v v¨ = . (1−e2l )3 Assumption 1: The true anomaly rate v˙ and true anomaly rate of change v¨ are bounded by constants, i.e. for all t ≥ t0 ≥ 0 αv˙ ≤ v˙ ≤ βv˙ , |¨ v | ≤ βv¨ (3)
¨ + C(v) ˙ p˙ + N(p, rl , rf , v, ˙ v¨, ul ) = E(t)uf + Fd mf p
(4)
where E(t) = diag{τ1 (t), τ2 (t), τ3 (t)} with e0 ≤ τi (t) ≤ 1(i = 1, 2, 3) being the actuator health indicator for the ith actuator. The case 0 < τi (t) < 1 represent the ith actuator partially loses its actuating power but can work while τi (t) = 1 denote actuator can work normal. C. Control problem statements Let pd ∈ R3 denotes the desired relative position, and the relative position error is described as e = p − pd
(5)
then the relative position error equation can be given as ˙ e + N∗ (e, rl , rf , v, ˙ v¨, ul ) = E(t)uf + Fd (6) mf ¨e + C(v)˙
(1)
where ml (mf ) denotes the leader(follower) spacecraft mass, rl (rf ) denotes the distance of the leader(follower) spacecraft to the center of the earth, ul (uf ) denotes the leader(follower) spacecraft actuator force, and v is the true anomaly of the leader spacecraft. mf N(p, rl , rf , v, ˙ v¨, ul ) = D(v, ˙ v¨, rf )p + n(rl , rf ) + ul (2) ml 0 −v˙ 0 C(v) ˙ = 2mf v˙ 0 0 is a skew-symmetric Coriolis0 0 0 µ − v˙ 2 −¨ v 0 rf3 µ − v˙ 2 0 v¨ like matrix, D(v, ˙ v¨, rf ) = mf rf3 µ 0 0 r3
where N∗ (e, rl , rf , v, ˙ v¨, ul ) is defined as µ − v˙ 2 −¨ v rf3 µ − v˙ 2 0 v¨ N∗ (e, rl , rf , v, ˙ v¨, ul ) = mf e+ rf3 µ 0 0 rf3 µ 2 rl 1 − v˙ −¨ v − r2 rf3 rf3 l µ + mf − v˙ 2 0 v¨ mf µ pd + 0 rf3 µ 0 0 0 rf3 f ¨d + m u . To facilitate the controller design, C(v) ˙ p˙ d + mf p l ml the following assumptions are also made. Assumption 2: The desired trajectory pd and its secondorder derivatives are known and bounded. Assumption 3: All three components of the control force uf are constrained by a bounded value, and expressed by
f
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|uf i (t)| ≤ umax , ∀t > 0, i = 1, 2, 3
(7)
Assumption 4: The external disturbance Fd is bounded such that kFd k ≤ F¯d (8) where F¯d is a known positive constant. The control objectives of this paper can be stated as: for the spacecraft formation tracking error system (6), determine a control law uf , which is a function of about q and qd only, such that the tracking errors e and e˙ are UUB, and the input constraints in (7) are satisfied. III. M AIN R ESULTS A. Velocity filter design To achieve the control objective without the relative veloc˙ a pseudo-velocity filter [16], [17] is introduced, which ity p, consists of the following two dynamic equations g˙ = −(K + L)g + (K2 + KL − K)e
(9)
q = −Ke + g
(10)
and 3
where g ∈ R is a pseudo-velocity tracking error signal, q ∈ R3 is the filtered velocity error signal. K = diag{k1 , k2 , k3 } and L = diag{l1 , l2 , l3 } are control gain matrices. Note that li and ki (i = 1, 2, 3) are selected according to the demands of Theorem 1. Define an auxiliary error variable r as r = e˙ + e + q
(11)
In view of (9)-(11), it follows that q˙ = −Lq − Kr
(12)
Take the time derivation of (11) and combine with (12) yields r˙ = ¨e + r − e − (L + I3 )q − Kr
(13)
Multiplying both side of (13) by mf yields mf r˙ = −Cr − mf (K − I3 )r + Fd + E(t)uf + Φ
(14)
where Φ = −N∗ + Ce − mf e + Cq − mf (L + I3 )q. Remark 1: If we design a control law uf such that e, q, r are UUB, then e˙ is also UUB from (11). B. Controller design It is clear that if p and p˙ converge to the desired trajectories, then e = 0, e˙ = 0, and the uncertain function Φ will converge to the desired nonlinear function ∗
Φd = −N (0, rl , rf , v, ˙ v¨, ul ) rl where N∗ (0, rl , rf , v, ˙ v¨, ul ) = mf µ mf
µ kpd +rl k3
− v˙ 2
−¨ v µ kpd +rl k3
kpd +rl k3
−
0 0
1 rl2
pd
directly used in controller design. Due to the advantages of neural networks (NNs) for approximating unknown system dynamics and their powerful representation capabilities for nonlinear functions [18], [19], this work employs the singlelayer NN approximation technique to learn Φd and Fd . ¯ = Φd + Fd , using the approximation property Denote Φ ¯ can be approximated by of NN, the unknown function Φ ¯ = WT Θ(p , p˙ , p Φ d d ¨d )
where W ∈ Rn×3 is the weight matrix of the output layer, ¨d ) ∈ Rn is the NN basis function with n and Θ(pd , p˙ d , p ¯ ∗ be the denoting the number of hidden-layer neurons. Let Φ optimal function approximation using an ideal NN, then we have ¯ =Φ ¯ ∗ + ² = (W∗ )T Θ + ² (17) Φ where W∗ ∈ Rn×3 is the optimal approximation weight and ² ∈ R3 is the approximation error. For any given constant ²¯ > 0, an NN with the ideal weight matrix can always be found such that k²k ≤ ²¯ holds. Further, because the optimal ¯ is difficult weight needed for the best approximation of Φ to determine, its estimate function is defined as follows ˆ¯ = W ˆ TΘ Φ
(18)
ˆ is the estimate of W∗ . where W Property 1: The optimal function approximation is bou¯ ∗ , where W ¯ ∗ is a positive nded so that tr{(W∗ )T (W∗ )} ≤ W constant. Since the masses of the leader and follower spacecrafts, the desired signals, and the external disturbances are bounded, ¯ is bounded such that the nonlinear function Φ ¯ Assumption 5: kΦk ≤ φ, where φ is a positive constant. e0 umax , where κ > 1 is a Assumption 6: φ < (1−e 0 )κ+e0 constant. Based upon the above analysis, denote the following controller and adaptive law as uf = uf a + uf b
(19)
ˆ¯ − (1 − m(t))φsgn(r), u = uf a = K tanh(λq) − m(t)Φ fb (1−e0 )κ − e0 kuf a ksgn(r), Z
t
˙ T (θ))dθ (ΘeT (θ) + ΘqT (θ) − Θe Z t ˆ W(θ)dθ (20) +m(t)γΘeT − m(t)%γ
ˆ = m(t)γ W
(15)
0
+
− v˙ 2 0 v¨ µ 0 0 kpd +rl k3 mf ¨d + ml ul . In the practical SFF, the nonlinear +C(v) ˙ p˙ d +mf p function Φd is always uncertain because of the system parameters in v˙ and v¨, meanwhile, the external disturbance Fd are always uncertain and unknown, thus it can not be
(16)
0
where λ, γ, % are positive constants, tanh(λq) = [tanh(λq1 ), tanh(λq2 ), tanh(λq3 )]T and T sgn(r) = [sgn(r1 ), sgn(r2 ), sgn(r 3 )] . The switching func 0, if kW ˆ T Θk > φ , which tion m(t) is given by m(t) = 1, if kW ˆ T Θk ≤ φ generates a switching between the adaptive NN control term and the robust control term φsgn(r).
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Consider the following Lyapunov function
Substituting the controller (19) into (26) yields V˙ ≤ s1 eT K(r − e − q) + s2 qT (−Lq − Kr) h ˜ T Θ + ²) +rT K tanh(λq) + Φ − Φd + m(t)(W i −mf (K − I3 )r + uf b − (I3 − E(t))uf
1 1 1 1 ˜ T W} ˜ V = s1 eT Ke + s2 qT q + mf rT r + tr{W 2 2 2 2γ p T p ln(cosh(λq1 )) pln(cosh(λq1 )) p (21) + pln(cosh(λq2 )) Λ−1 pln(cosh(λq2 )) ln(cosh(λq6 )) ln(cosh(λq3 )) ˜ = W∗ − W, ˆ Λ = λI3 and s1 > 0, s2 > 0 are where W constants given in Theorem 1. For ∀Vmax > 0, we construct ˜ the set ΩV as ΩV = {(e, q, r, W)|V ≤ Vmax }. Further, based on Assumption 2, we define the following compact set Ωd = ¨Td ]T k ≤ Vd }, where Vd is a positive ¨d )|k[pTd , p˙ Td , p {(pd , p˙ d , p constant. Since the sets ΩV and Ωd are compact, respectively, then the variable Φ − Φd satisfies kΦ − Φd k ≤ σ on the compact set ΩV × Ωd , where σ is a positive constant. Then one has the following stability and performance results. Theorem 1: Consider the spacecraft formation tracking error system described by (14), and suppose Assumptions 1-6 are satisfied. The filter is defined by (9)-(10), and the controller is provided by (19) with the adaptive law (20), if the initial conditions satisfy 1 1 1 T s1 e (0)Ke(0) + s2 qT (0)q(0) + mf rT (0)r(0) 2 2 2 1 ˜ T (0)W(0)} ˜ + tr{W ≤ Vmax (22) 2γ
1 ˜ T W} ˆ˙ + q˙ T tanh(λq) − tr{W γ ¯ − (1 − m(t))φkrk +(1 − m(t))krkkΦk ≤ s1 eT K(r − e − q) + s2 qT (−Lq − Kr) −qT L tanh(λq) + m(t)rT ² + rT (Φ − Φd ) ˜ T W} ˆ −mf rT (K − I3 )r + %m(t)tr{W +rT E(t)uf b − rT (I3 − E(t))uf a
where −rT Cr = 0 and rT sgn(r) ≥ krk are applied. Note ˜ T W} ˆ = −m(t)tr{W ˜ T W} ˜ + m(t)tr{W ˜ T W∗ } that m(t)tr{W T m(t) ∗ T ∗ 1 ¯ ∗ ˜ ˜ ≤ − m(t) 2 tr{W W} + 2 tr{(W ) W } ≤ 2 W . In particT ular, using the Young’s inequality, we have e Kr ≤ 4ζ11 eT Ke + ζ1 rT Kr, −eT Kq ≤ 4ζ12 eT Ke + ζ2 qT Kq, −qT Kr ≤ 1 1 T T T 2 2 T 4ζ3 q Kq + ζ3 r Kr, m(t)r ² ≤ 4ζ4 krk + ζ4 k²k , r (Φ − 1 2 2 Φd ) ≤ 4ζ5 krk + ζ5 kΦ − Φd k , where ζi (i = 1, 2, · · · , 5) are constants satisfied (24). Thus, if K and L are chosen s2 satisfied (23), we have s2 L − s1 ζ2 K − 4ζ K ≥ 0, mf (K − 3 1 1 I3 )− 4ζ4 I3 − 4ζ5 I3 −s2 ζ3 K−s1 ζ1 K > 0, then we can obtain V˙ ≤ −λmin (λ−1 L) tanhT (λq) tanh(λq) − rT [mf (K − I3 ) 1 1 − I3 − I3 − s2 ζ3 K − s1 ζ1 K]r − s1 (1 4ζ4 4ζ5 1 1 T s2 − )e Ke − qT (s2 L − s1 ζ2 K − K)q − 4ζ1 4ζ2 4ζ3 3 3 X X |ri |(1 − e0 )kuf a k |ri |(1 − e0 )κkuf a k + −
and the control gain matrices K and L are chosen as ki >
mf +
1 4ζ4
+
1 4ζ5
mf − s2 ζ3 − s1 ζ1
, li ≥
s1 ζ2 ki +
s2 4ζ3 ki
s2
, i = 1, 2, 3
(23) where si (i = 1, 2), ζi (i = 1, 2, · · · , 5) are constants satisfied 1 1 si > 0, ζi > 0, 1 − − >0 4ζ 4ζ 1 2 mf − s2 ζ3 − s1 ζ1 > 0 mf + 1 + 1 e0 umax 4ζ4 4ζ5 < −φ mf − s2 ζ3 − s1 ζ1 (1 − e0 )κ + e0
i=1
(24)
+
V˙ = s1 eT K(r − e − q) + s2 qT (−Lq − Kr) +rT [−Cr − mf (K − I6×6 )r + fdf + uf − (I3 1 ˜ T W} ˆ˙ + q˙ T tanh(λq)(26) −E(t))uf + Φ] − tr{W γ
3 X
1 ¯∗ |ri |(1 − e0 )kuf a k + ζ4 ²¯2 + ζ5 σ 2 + %W 2 i=1
1 ¯∗ ≤ −$kzk2 + ζ4 ²¯2 + ζ5 σ 2 + %W 2
(25)
Take the derivative of V , and from (11), (12) and (14), we have
i=1
1 ¯∗ +ζ4 ²¯2 + ζ5 σ 2 + %W 2 ≤ −λmin (L) tanhT (λq) tanh(λq) − rT [mf (K − I3 ) 1 1 I3 − I3 − s2 ζ3 K − s1 ζ1 K]r − s1 (1 − 4ζ4 4ζ5 3 X 1 1 − |ri |(1 − e0 )κkuf a k − )λmin (K)eT e − 4ζ1 4ζ2 i=1
then the uniform ultimate boundedness of tracking errors e and e˙ can be guaranteed. Proof: Based on (20), we have ˆ˙ = m(t)(γΘrT − %γ W) ˆ W
(27)
(28)
T T where tanh n (λq) tanh(λq) ≤ λq tanh(λq) is applied, 1 1 $ = min s1 (1 − 4ζ1 − 4ζ2 )λmin (K), λmin (λ−1 L), o λmin (mf (K − I3 ) − 4ζ14 I3 − 4ζ15 I3 − s2 ζ3 K − s1 ζ1 K) , z = q ¯∗ ζ4 ²¯2 +ζ5 σ 2 + 21 %W T T T T , it is [e , r , tanh (λq)] . Denote Υ = $ seen from (28) that V˙ < 0 when e, r, q are outside of the set
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S = {(e, r, q)|kek ≤ Υ, krk ≤ Υ, k tanh(λq)k ≤ Υ} (29)
which is a small set containing the origin (e, r, q) = 0. Hence, the tracking error e, auxiliary error variable r, filtered output q are UUB as limt→∞ kek ∈ S, limt→∞ krk ∈ S, limt→∞ kqk ∈ S, which implies that larger ki will yield better tracking control performance. Then, based on Remark 1, the tracking error e˙ is also UUB. This completes the proof. Remark 2: To guarantee the input constraints in (7) is satisfied, the gain ki (i = 1, 2, 3) should be selected such that (1 − e0 )κ (ki + φ) ≤ umax (30) ki + φ + e0 which means ki should be chosen as mf +
1 4ζ4
+
1 4ζ5
mf − s2 ζ3 − s1 ζ1
< ki ≤
e0 umax −φ (1 − e0 )κ + e0
˙ q(k∆T ) = −K
(32)
Accordingly, it is obtained from (12) that r = −K−1 Lq − ˙ Hence, r can be numerically derived during the impleK−1 q. mentation of the controller (19), because q˙ is obtained with (32). It can be summarized that uf a and uf b are independent on velocity measurements, although r is involved. Remark 4: In general, the NN has poor approximation capability to the unknown function in the system dynamics at its learning phase [19], and hence, the output of NN may be very large and may exceed the bound of the approximated nonlinear function. In order to make the output of the NN bounded by φ, a switch function m(t) is used to generate a switching between the adaptive NN control and the robust control φ. Next, we consider two special cases under Theorem 1: Case 1: e0 is unknown. In this case, we can only use the controller uf 1 to guarantee the uniform ultimate boundedness of tracking errors e and e˙ . If the inequality −rT (1 − E(t))uf a ≤ 4ζ16 rT r + ζ6 (1 − e0 )2 kuf a k2 ≤ 4ζ16 rT r + ζ6 (1 − e0 )2 (kKk + φ)2 is applied in the proofing of Theorem 1, where ζ6 is a positive constant, we can choose the following control gains to guarantee the uniform ultimate boundedness of tracking errors e and e˙ , and the input constraints in (7), where mf +
1
+
1
+
1
mf +
4ζ4 4ζ5 4ζ6 , li are chosen as in umax − φ ≥ ki > mf −s 2 ζ3 −s1 ζ1 (23), si , ζi are constants satisfied (24). Case 2: The actuators are fault-free, i.e., E(t) = I3 .
1
+
1
4ζ5 4 , li are chosen as where umax − φ ≥ ki > mf −s4ζ 2 ζ3 −s1 ζ1 in (23), and si , ζi are constants satisfied (24), then the uniform ultimate boundedness of tracking errors e and e˙ can be guaranteed, and the input constraints in (7) can be satisfied. Remark 5: The controller (19) is independent on the precise knowledge of the true anomaly rate v, ˙ the true anomaly rate of change v¨, the spacecrafts mass ml , mf , and the external disturbance Fd . Therefore, from the standpoint of uncertainties and external disturbances rejection, the derived controller has great stability robustness.
(31)
combined with (23). Remark 3: Note that the term r in (19) is not physically measurable since it contains e˙ , however, the controller is implemented with digital computer in practical aerospace engineering. The value of q˙ in the time of (k + 1)∆T can be approximately estimated by using the one-step pre˙ vious information from attitude sensors as q(k∆T ) = q((k+1)∆T )−q(k∆T ) , where ∆T is the control update period. ∆T By using (10), we can calculate the value of g˙ by e((k + 1)∆T ) − e(k∆T ) ∆T g((k + 1)∆T ) − g(k∆T ) + ∆T
In this case, we can also only use the controller uf a to guarantee the uniform ultimate boundedness of tracking errors e and e˙ . If we choose the following control gains,
IV. S IMULATION RESULTS In this section, simulation results for a leader-follower SFF are presented to illustrate the performance of the presented design method. The two spacecraft SFF problem having the following parameters [9]: Me = 5.974 × 1024 kg, G = 6.673 × 10−11 m3 /kg · s2 , ml = 100 kg, mf = 100 kg, el = 0.6, al = 1.6103 × 107 m, ul = 0 N, v(0) = 0 rad, p(0) = ˙ [20, 10, −20]T m, p(0) = [0, 0, 0]T m/s. The desired relative 4π trajectory is selected as pd = [−10 cos( 3π Tl t), 10 sin( Tl t), 5π 5 cos( Tl t)]T m, where Tl is the orbital period of the leader spacecraft. Orbital perturbation force due to J2 disturbance is included in the simulations. Suppose that the maxmum input control force umax is 40N, then the design parameters si (i = 1, 2) and ζi (i = 1, 2, · · · , 5) are chosen as s1 = 0.03, s2 = 0.6, ζ1 = 0.5, ζ2 = 0.6, ζ3 = 10, ζ4 = 1, ζ5 = 1 and the main control parameters of uf 1 for simulation are selected as K = diag{12, 12, 12}, L = diag{2, 2, 2}, λ = 1, φ = 3, γ = 0.5, % = 0.1. In order to implement the adaptive neural network based controller (19), the radial basis ¯ function of the neural network is applied to approximate Φ, and the vectors Θ ∈ Rn used in the simulation are chosen as 2 −
kx−ci k 2
σ i , i = 1, 2, · · · , n, Gaussian-type functions Θi (x) = e ¨Td ]T , σi ∈ R, and ci denote the vector where x = [pTd , p˙ Td , p with the same dimension as x. Moreover, n = 10 and σi = 15 are selected in the simulation, and the elements of ci are chosen between -1 and 1 randomly. The chattering effect that may be induced by the use of the discontinuity function sgn(·) in (19). In order to reduce the chattering, the function sgn(·) is replaced by a continuous saturation function sat( ε· ) [15], where ε = 0.01 is chosen in this simulation. We first consider the scenario of actuators fault-free, in which a relative ideal situation is simulated. Applying the controller uf a with the adaptive update law (20), we illustrate the simulation results as Fig. 2, from which it can be concluded that the objective of 3 DOF spacecraft relative position tracking is achieved in the presence of uncertain parameters and external disturbances. Indeed, due ¯ can to employment of the neural network, the effect of Φ be accommodated as soon as possible, and then fast tracking manoeuvring can be achieved. Moreover, under this control law, the relative position tracking error and velocity error
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the desired nonlinear function and bounded external disturbances. In order to guarantee that the output of the NN is bounded, a switch function is employed. The solution also applies a fault tolerant control part to compensate for the partial loss of actuator effectiveness fault. It is proven that the tracking error is UUB, and the control input can rigorously enforce actuator magnitude constraints.
50 m
p1 p2
0
p3 −50
0
50
100
150
200 250 300 1. Relative position error
350
400
450
500
5 m/s
dp1/dt dp2/dt
0
dp3/dt −5
0
50
100
150
200 250 300 2. Relative velocity error
350
400
450
500
20 N
uf1 uf2
0
R EFERENCES
uf3 −20
0
50
100
150
200
250 300 3. Control force
350
400
450
500
Fig. 2: Actuators fault-free.
50 m
p1 p2
0
p3 −50
0
50
100
150
200 250 300 1. Relative position error
350
400
450
500
5 m/s
dp1/dt dp2/dt
0
dp3/dt −5
0
50
100
150
200 250 300 2. Relative velocity error
350
400
450
500
N
40 20
uf1
0
uf2 uf3
−20 −40
0
50
100
150
200
250 300 3. Control force
350
400
450
500
Fig. 3: Actuators faults.
signals converge to zero in about 350 s as shown in Fig. 2-1-Fig. 2-2. It is interesting to note that the control force of each axis in this case is less than the required maximum bound, as shown in Fig. 2-3. Next, we consider the following actuators partial loss of control effectiveness in the SFF: 1) the actuator uf 1 decreases 30% of its normal value after 15 s; 2) the actuator uf 2 loses its power of 40% in 25 s; and 3) the actuator uf 3 undergoes 40% loss of effectiveness in 30 s. Then, apply the controller uf with the adaptive update law (20), we illustrate the simulation results as Fig. 3, where the main control parameters of uf b for simulation are selected as: e0 = 0.6, κ = 1.1. The time responses of the relative position tracking error and velocity error are presented in Fig. 3-1Fig. 3-2, respectively. The control force is shown in Fig. 3-3. As expected, we clearly see that the proposed fault-tolerant control scheme managed to compensate for the partial loss of effectiveness fault without velocity measurement. As shown in Fig. 3-3, the control force of each axis is still within its maximum allowable limit even in the presence of external disturbances, uncertain system parameters and actuator faults. V. C ONCLUSION We have presented a solution to the problem of output feedback control to relative translation tracking in a leaderfollower spacecraft formation, when there are uncertain system parameters, external disturbances, input constraints and partial loss of actuator effectiveness. The solution incorporates a pseudo-velocity filter to account for the unmeasured relative velocity, and uses the NN technique to approximate
[1] G. W. Hill, Researches in the lunar theory, American Journal of Mathematics, 1(1), 1878, 5–26. [2] W. H. Clohessy, and R. S. Wiltshire, Terminal guidance system for satellite rendezvous, Journal of Aerospace Sciences, 27(9), 1960, 653– 658. [3] H. Schaub, and J. L. Junkins, Analytical Mechanics of Space Systems, AIAA Education Series, American Institute of Aeronautics and Astronautics, Reston, VA, 2003. [4] R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education Series, Reston, VA: American Institute of Aeronautics and Astronautics, 1999. [5] V. Kapila, A. G. Sparks, J. M. Buffington, and Q. Yan, Spacecraft formation flying: dynamics and control, Jouranl of Guidance, Control, and Dynamics, 23(3), 2000, 561–564. [6] M. De. Queiroz, V. Kapila, and Q. Yan, Adaptive nonlinear control of multiple spacecraft formation flying, Journal of Guidance, Control and Dynamics, 23(3), 2000, 385–390. [7] H. Wong, V. Kapila, and A. G. Sparks, Adaptive output feedback tracking control of spacecraft formation, International Journal of Robust and Nonlinear Control, 12(2-3), 2002, 117–139. [8] H. Liu, and J. Li, Terminal sliding mode control for spacecraft formation flying, IEEE Transactions on Aerospace and Electronic Systems, 45(3), 2009, 835–846. [9] R. Kristiansen, A. Loria, A. Chaillet, and P. J. Nicklasson, Adaptive output feedback control of spacecraft relative translation, in Proceedings of the 45th IEEE Conference on Decision and Control, 13-15 December 2006, San Diego, CA, USA, pp. 6010-6015. [10] R. Kristiansen, and P. J. Nicklasson, Spacecraft formation flying: a review and new results on state feedback control, Acta Astronautica, 65(11-12), 2009, 1537–1552. [11] H. J. Gao, X. B. Yang, and P. Shi, Multi-objective robust H∞ control of spacecraft rendezvous, IEEE Transactions on Control Systems Technology, 17(4), 2009, 794–802. [12] Z. F. Gao, B. Jiang, R. Y. Qi, and Y. F. Xu, Robust reliable control for a near space vehicle with parametric uncertainties and actuator faults, International Journal of Systems Science, 42(12), 2011, 2113–2124. [13] J. H. Jin, S. H. Ko, and C. K. Ryoo, Fault tolerant control for satellites with four reaction wheels, Control Engineering Practice, 16(10), 2008, 1250–1258. [14] W. C. Cai, X. H. Liao and Y. D. Song, Indirect robust adaptive faulttolerant control for attitude tracking of spacecraft, Journal of Guidance Control and Dynamics, 31(5), 2008, 1456–1463. [15] B. Xiao, Q. L. Hu, and P. Shi, Attitude stabilization of spacecrafts under actuator saturation and partial loss of control effectiveness, IEEE Transactions on Control Systems Technology, 21(6), 2013, 2251–2263. [16] T. Burg, D. Dawson, J. Hu, and M. De-Queiroz, An adaptive partial state feedback controller for RLED robot manipulators, IEEE Transactions on Automatic Control, 41(7), 1996, 1024–1030. [17] S. Purwar, I. N. Kar, and A. N. Jha, Adaptive output feedback tracking control of robot manipulators using position measurements only, Expert Systems with Applications, 34(4), 2008, 2789–2798. [18] B. Xiao, Q. L. Hu, and G. F. Ma, Adaptive quaternion-based output feedback control for flexible spacecraft attitude tracking with input constraints, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 225(2), 2011, 226–240. [19] A. M. Zou, K. D. Kumar, Z. G. Hou, and X. Liu, Finite-time attitude tracking control for spacecraft using terminal sliding mode and chebyshev neural network, IEEE Transactions on System, Mam, and Cybernetics-Part B: Cybernetics, 41(4), 2011, 950–963.
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Output feedback tracking control for spacecraft relative translation subject to input constraints and partial loss of control effectiveness Lin Zhao, Yingmin Jia, Junping Du, Jun Zhang Abstract— In this paper, an adaptive output feedback tracking control scheme is proposed for spacecraft formation flying (SFF) in the presence of external disturbances, uncertain system parameters, input constraints and partial loss of control effectiveness. The proposed controller incorporates a pseudo-velocity filter to account for the unmeasured relative velocity, and the neural network (NN) technique is implemented to approximate the desired nonlinear function and bounded external disturbances. In order to guarantee that the output of the NN used in the controller is bounded by the corresponding bound of the approximated nonlinear function, a switch function is employed to generate a switching between the adaptive NN control and the robust controller. Moreover, a fault tolerant part is included in the controller to compensate the partial loss of actuator effectiveness fault. It is shown that the derived controller not only guarantees the tracking error in the closed-loop system to be uniformly ultimately bounded (UUB) but also ensures the control input can rigorously enforce actuator magnitude constraints. Simulation results are provided to demonstrate the effectiveness of the proposed method.
I. I NTRODUCTION Spacecraft formation flying (SFF) techniques have many advantages in dealing with some space and Earth science missions, such as being more economical, flexible and reliable, etc. In the early days, results about the spacecraft formation control problem were presented based on the linear 3 degree-of-freedom (DOF) translational motion model [1], [2]. Later, the 3 DOF nonlinear model, which was derived for arbitrary orbital eccentricity and with added terms for orbital perturbations, was built for the problem of SFF [5]–[10]. For example, a nonlinear adaptive tracking control scheme was developed in [6], which relied on the restrictive assumptions of a circular orbit, and the availability of exact measurements of all relative velocity and position in translational motion. In practice, precise measurements of velocity are not always satisfied because of either cost limitations or implementation constraints [15]. Therefore, it is highly desirable to design a partial-state feedback controller that does not depend on the measurements of velocity. An adaptive output feedback This work was supported by the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201) and the NSFC (61134005, 61221061, 61327807). Lin Zhao and Yingmin Jia are with the Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]; [email protected]). Junping Du is with the Beijing Key Laboratory of Intelligent Telecommunications Software and Multimedia, School of Computer Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]) Jun Zhang is with the School of Electronic and Information Engineering, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected]).
978-1-4799-3271-9/$31.00 ©2014 AACC
controller was proved in [7]. A more general result about the adaptive output feedback control for spacecraft formation control was given in [9]. It should be noted that all the results mentioned above did not consider the problem of input constraints, which cannot be ignored in practice for the existence of limitation on actuators [11]. Another important problem encountered in practice for spacecraft formation control system design is that of control failures, which can cause control system performance deterioration and lead to instability and even catastrophic accidents. Hence, it is necessary to maintain high reliability for spacecraft formation control system design against possible control faults [12]. Fault tolerant control (FTC) is an area of research that emerges to increase availability by specifically designing control algorithms capable of maintaining stability and performance despite the occurrence of faults, and it has received considerable attention from the control research community and aeronautical engineering [13]–[15]. Motivated by the above discussions, in this paper, we investigate the 3 DOF spacecraft formation control design by developing a partial-state feedback control scheme in the presence of external disturbances, uncertain system parameters, input constraints, and partial loss of actuator effectiveness fault. The main contributions of this paper are stated as follows: 1) A FTC scheme is proposed to compensate for the partial loss of actuator effectiveness fault without any fault detection and isolation mechanism. Moreover, the proposed output feedback control approach can guarantee the closedloop tracking system is uniformly ultimately bounded (UUB) stable, when the external disturbances and uncertain system parameters are bounded. 2) The adaptive neural network (NN) approximation technique is used to approximate the desired nonlinear function and bounded external disturbances. A switch function is employed to generate a switching between the adaptive NN control and the robust controller to guarantee that the output of the NN used in the controller is bounded. Moreover, the proposed control approach can explicitly account for input constraints, even in the case of actuator faults. Thus, the objective of efficient, low-cost, and reliable spacecraft formation control design can be met. The rest of this paper is organized as follows: The mathematical model of 3 DOF SFF and the control problem formulation are summarized in Section II. The control design is presented in Section III. Simulation results and conclusion are given in Sections IV and V, respectively. Throughout this paper, Rn and Rn×m denote the n-
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can be regarded as a time-varying potential force, n(rl , rf ) is the nonlinear term defined as n(rl , rf ) = rl − r12 rf3 l . The composite perturbation force Fd mf µ 0 0 m is given by Fd = fdf − mfl fdl , where fdl (fdf ) denotes the perturbation term of leader(follower) spacecraft due to external effects. The rate of the true anomaly ofqthe leader 2 µ is the spacecraft is v˙ = nl (1+el 2cos3 v) , where nl = a3
e iz
eh
er Leader
rl
p iy
ix
rf Follower
(1−el ) 2
Fig. 1: Reference coordinate frames [3].
dimensional and the n × m-dimensional Euclidean spaces, respectively; In denote the n × n identity matrices; X < 0 represents that X is the symmetric negative-definite matrix; the superscripts T and −1 stand for matrix transposition and matrix inverse; tr{·} refers to the trace, | · | refers to the absolute value, and k · k refers to the Euclidean vector norm or the induced matrix 2-norm; sgn(·), tanh(·) and cosh(·) are the sign, standard hyperbolic tangent and cosine functions, respectively. II. M ATHEMATICAL MODEL OF SFF
AND PROBLEM
where αv˙ , βv˙ , βv¨ are known positive constants. With the diversification and complication of spacecraft task and structure, faults of actuators would occurred frequently. Consider the actuators partial loss of effectiveness and represent the fault in product factor, system (1) can be described as
FORMULATION
A. Cartesian coordinate frames To form the basis of the relative position model, we use the standard definition of the Earth-Centered Inertial (ECI) frame Fi , with z-axis towards celestial north. We also employ a standard LVLH-definition of the leader orbit reference frame Fl , with unit vectors defined as er = rrll , eθ = eh × er and eh = hh , where h = rl × r˙l is the angular momentum vector of the orbit, and h = khk. In addition, we define a follower orbit reference frame Ff with origin specified by the relative orbit position vector p = rf − rl = xer + yeθ + zeh and with unit vectors aligned with the unit vectors in Fl at all times, as shown in Fig. 1. B. Relative translational motion The relative position dynamics can be written as [10] ¨ + C(v) ˙ p˙ + N(p, rl , rf , v, ˙ v¨, ul ) = uf + Fd mf p
l
mean motion of the leader, al is the semimajor axis of the leader orbit, and el is the orbit eccentricity. Differentiation v˙ results in the rate of change of the true anomaly as −2n2l el (1+el cos v)3 sin v v¨ = . (1−e2l )3 Assumption 1: The true anomaly rate v˙ and true anomaly rate of change v¨ are bounded by constants, i.e. for all t ≥ t0 ≥ 0 αv˙ ≤ v˙ ≤ βv˙ , |¨ v | ≤ βv¨ (3)
¨ + C(v) ˙ p˙ + N(p, rl , rf , v, ˙ v¨, ul ) = E(t)uf + Fd mf p
(4)
where E(t) = diag{τ1 (t), τ2 (t), τ3 (t)} with e0 ≤ τi (t) ≤ 1(i = 1, 2, 3) being the actuator health indicator for the ith actuator. The case 0 < τi (t) < 1 represent the ith actuator partially loses its actuating power but can work while τi (t) = 1 denote actuator can work normal. C. Control problem statements Let pd ∈ R3 denotes the desired relative position, and the relative position error is described as e = p − pd
(5)
then the relative position error equation can be given as ˙ e + N∗ (e, rl , rf , v, ˙ v¨, ul ) = E(t)uf + Fd (6) mf ¨e + C(v)˙
(1)
where ml (mf ) denotes the leader(follower) spacecraft mass, rl (rf ) denotes the distance of the leader(follower) spacecraft to the center of the earth, ul (uf ) denotes the leader(follower) spacecraft actuator force, and v is the true anomaly of the leader spacecraft. mf N(p, rl , rf , v, ˙ v¨, ul ) = D(v, ˙ v¨, rf )p + n(rl , rf ) + ul (2) ml 0 −v˙ 0 C(v) ˙ = 2mf v˙ 0 0 is a skew-symmetric Coriolis0 0 0 µ − v˙ 2 −¨ v 0 rf3 µ − v˙ 2 0 v¨ like matrix, D(v, ˙ v¨, rf ) = mf rf3 µ 0 0 r3
where N∗ (e, rl , rf , v, ˙ v¨, ul ) is defined as µ − v˙ 2 −¨ v rf3 µ − v˙ 2 0 v¨ N∗ (e, rl , rf , v, ˙ v¨, ul ) = mf e+ rf3 µ 0 0 rf3 µ 2 rl 1 − v˙ −¨ v − r2 rf3 rf3 l µ + mf − v˙ 2 0 v¨ mf µ pd + 0 rf3 µ 0 0 0 rf3 f ¨d + m u . To facilitate the controller design, C(v) ˙ p˙ d + mf p l ml the following assumptions are also made. Assumption 2: The desired trajectory pd and its secondorder derivatives are known and bounded. Assumption 3: All three components of the control force uf are constrained by a bounded value, and expressed by
f
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|uf i (t)| ≤ umax , ∀t > 0, i = 1, 2, 3
(7)
Assumption 4: The external disturbance Fd is bounded such that kFd k ≤ F¯d (8) where F¯d is a known positive constant. The control objectives of this paper can be stated as: for the spacecraft formation tracking error system (6), determine a control law uf , which is a function of about q and qd only, such that the tracking errors e and e˙ are UUB, and the input constraints in (7) are satisfied. III. M AIN R ESULTS A. Velocity filter design To achieve the control objective without the relative veloc˙ a pseudo-velocity filter [16], [17] is introduced, which ity p, consists of the following two dynamic equations g˙ = −(K + L)g + (K2 + KL − K)e
(9)
q = −Ke + g
(10)
and 3
where g ∈ R is a pseudo-velocity tracking error signal, q ∈ R3 is the filtered velocity error signal. K = diag{k1 , k2 , k3 } and L = diag{l1 , l2 , l3 } are control gain matrices. Note that li and ki (i = 1, 2, 3) are selected according to the demands of Theorem 1. Define an auxiliary error variable r as r = e˙ + e + q
(11)
In view of (9)-(11), it follows that q˙ = −Lq − Kr
(12)
Take the time derivation of (11) and combine with (12) yields r˙ = ¨e + r − e − (L + I3 )q − Kr
(13)
Multiplying both side of (13) by mf yields mf r˙ = −Cr − mf (K − I3 )r + Fd + E(t)uf + Φ
(14)
where Φ = −N∗ + Ce − mf e + Cq − mf (L + I3 )q. Remark 1: If we design a control law uf such that e, q, r are UUB, then e˙ is also UUB from (11). B. Controller design It is clear that if p and p˙ converge to the desired trajectories, then e = 0, e˙ = 0, and the uncertain function Φ will converge to the desired nonlinear function ∗
Φd = −N (0, rl , rf , v, ˙ v¨, ul ) rl where N∗ (0, rl , rf , v, ˙ v¨, ul ) = mf µ mf
µ kpd +rl k3
− v˙ 2
−¨ v µ kpd +rl k3
kpd +rl k3
−
0 0
1 rl2
pd
directly used in controller design. Due to the advantages of neural networks (NNs) for approximating unknown system dynamics and their powerful representation capabilities for nonlinear functions [18], [19], this work employs the singlelayer NN approximation technique to learn Φd and Fd . ¯ = Φd + Fd , using the approximation property Denote Φ ¯ can be approximated by of NN, the unknown function Φ ¯ = WT Θ(p , p˙ , p Φ d d ¨d )
where W ∈ Rn×3 is the weight matrix of the output layer, ¨d ) ∈ Rn is the NN basis function with n and Θ(pd , p˙ d , p ¯ ∗ be the denoting the number of hidden-layer neurons. Let Φ optimal function approximation using an ideal NN, then we have ¯ =Φ ¯ ∗ + ² = (W∗ )T Θ + ² (17) Φ where W∗ ∈ Rn×3 is the optimal approximation weight and ² ∈ R3 is the approximation error. For any given constant ²¯ > 0, an NN with the ideal weight matrix can always be found such that k²k ≤ ²¯ holds. Further, because the optimal ¯ is difficult weight needed for the best approximation of Φ to determine, its estimate function is defined as follows ˆ¯ = W ˆ TΘ Φ
(18)
ˆ is the estimate of W∗ . where W Property 1: The optimal function approximation is bou¯ ∗ , where W ¯ ∗ is a positive nded so that tr{(W∗ )T (W∗ )} ≤ W constant. Since the masses of the leader and follower spacecrafts, the desired signals, and the external disturbances are bounded, ¯ is bounded such that the nonlinear function Φ ¯ Assumption 5: kΦk ≤ φ, where φ is a positive constant. e0 umax , where κ > 1 is a Assumption 6: φ < (1−e 0 )κ+e0 constant. Based upon the above analysis, denote the following controller and adaptive law as uf = uf a + uf b
(19)
ˆ¯ − (1 − m(t))φsgn(r), u = uf a = K tanh(λq) − m(t)Φ fb (1−e0 )κ − e0 kuf a ksgn(r), Z
t
˙ T (θ))dθ (ΘeT (θ) + ΘqT (θ) − Θe Z t ˆ W(θ)dθ (20) +m(t)γΘeT − m(t)%γ
ˆ = m(t)γ W
(15)
0
+
− v˙ 2 0 v¨ µ 0 0 kpd +rl k3 mf ¨d + ml ul . In the practical SFF, the nonlinear +C(v) ˙ p˙ d +mf p function Φd is always uncertain because of the system parameters in v˙ and v¨, meanwhile, the external disturbance Fd are always uncertain and unknown, thus it can not be
(16)
0
where λ, γ, % are positive constants, tanh(λq) = [tanh(λq1 ), tanh(λq2 ), tanh(λq3 )]T and T sgn(r) = [sgn(r1 ), sgn(r2 ), sgn(r 3 )] . The switching func 0, if kW ˆ T Θk > φ , which tion m(t) is given by m(t) = 1, if kW ˆ T Θk ≤ φ generates a switching between the adaptive NN control term and the robust control term φsgn(r).
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Consider the following Lyapunov function
Substituting the controller (19) into (26) yields V˙ ≤ s1 eT K(r − e − q) + s2 qT (−Lq − Kr) h ˜ T Θ + ²) +rT K tanh(λq) + Φ − Φd + m(t)(W i −mf (K − I3 )r + uf b − (I3 − E(t))uf
1 1 1 1 ˜ T W} ˜ V = s1 eT Ke + s2 qT q + mf rT r + tr{W 2 2 2 2γ p T p ln(cosh(λq1 )) pln(cosh(λq1 )) p (21) + pln(cosh(λq2 )) Λ−1 pln(cosh(λq2 )) ln(cosh(λq6 )) ln(cosh(λq3 )) ˜ = W∗ − W, ˆ Λ = λI3 and s1 > 0, s2 > 0 are where W constants given in Theorem 1. For ∀Vmax > 0, we construct ˜ the set ΩV as ΩV = {(e, q, r, W)|V ≤ Vmax }. Further, based on Assumption 2, we define the following compact set Ωd = ¨Td ]T k ≤ Vd }, where Vd is a positive ¨d )|k[pTd , p˙ Td , p {(pd , p˙ d , p constant. Since the sets ΩV and Ωd are compact, respectively, then the variable Φ − Φd satisfies kΦ − Φd k ≤ σ on the compact set ΩV × Ωd , where σ is a positive constant. Then one has the following stability and performance results. Theorem 1: Consider the spacecraft formation tracking error system described by (14), and suppose Assumptions 1-6 are satisfied. The filter is defined by (9)-(10), and the controller is provided by (19) with the adaptive law (20), if the initial conditions satisfy 1 1 1 T s1 e (0)Ke(0) + s2 qT (0)q(0) + mf rT (0)r(0) 2 2 2 1 ˜ T (0)W(0)} ˜ + tr{W ≤ Vmax (22) 2γ
1 ˜ T W} ˆ˙ + q˙ T tanh(λq) − tr{W γ ¯ − (1 − m(t))φkrk +(1 − m(t))krkkΦk ≤ s1 eT K(r − e − q) + s2 qT (−Lq − Kr) −qT L tanh(λq) + m(t)rT ² + rT (Φ − Φd ) ˜ T W} ˆ −mf rT (K − I3 )r + %m(t)tr{W +rT E(t)uf b − rT (I3 − E(t))uf a
where −rT Cr = 0 and rT sgn(r) ≥ krk are applied. Note ˜ T W} ˆ = −m(t)tr{W ˜ T W} ˜ + m(t)tr{W ˜ T W∗ } that m(t)tr{W T m(t) ∗ T ∗ 1 ¯ ∗ ˜ ˜ ≤ − m(t) 2 tr{W W} + 2 tr{(W ) W } ≤ 2 W . In particT ular, using the Young’s inequality, we have e Kr ≤ 4ζ11 eT Ke + ζ1 rT Kr, −eT Kq ≤ 4ζ12 eT Ke + ζ2 qT Kq, −qT Kr ≤ 1 1 T T T 2 2 T 4ζ3 q Kq + ζ3 r Kr, m(t)r ² ≤ 4ζ4 krk + ζ4 k²k , r (Φ − 1 2 2 Φd ) ≤ 4ζ5 krk + ζ5 kΦ − Φd k , where ζi (i = 1, 2, · · · , 5) are constants satisfied (24). Thus, if K and L are chosen s2 satisfied (23), we have s2 L − s1 ζ2 K − 4ζ K ≥ 0, mf (K − 3 1 1 I3 )− 4ζ4 I3 − 4ζ5 I3 −s2 ζ3 K−s1 ζ1 K > 0, then we can obtain V˙ ≤ −λmin (λ−1 L) tanhT (λq) tanh(λq) − rT [mf (K − I3 ) 1 1 − I3 − I3 − s2 ζ3 K − s1 ζ1 K]r − s1 (1 4ζ4 4ζ5 1 1 T s2 − )e Ke − qT (s2 L − s1 ζ2 K − K)q − 4ζ1 4ζ2 4ζ3 3 3 X X |ri |(1 − e0 )kuf a k |ri |(1 − e0 )κkuf a k + −
and the control gain matrices K and L are chosen as ki >
mf +
1 4ζ4
+
1 4ζ5
mf − s2 ζ3 − s1 ζ1
, li ≥
s1 ζ2 ki +
s2 4ζ3 ki
s2
, i = 1, 2, 3
(23) where si (i = 1, 2), ζi (i = 1, 2, · · · , 5) are constants satisfied 1 1 si > 0, ζi > 0, 1 − − >0 4ζ 4ζ 1 2 mf − s2 ζ3 − s1 ζ1 > 0 mf + 1 + 1 e0 umax 4ζ4 4ζ5 < −φ mf − s2 ζ3 − s1 ζ1 (1 − e0 )κ + e0
i=1
(24)
+
V˙ = s1 eT K(r − e − q) + s2 qT (−Lq − Kr) +rT [−Cr − mf (K − I6×6 )r + fdf + uf − (I3 1 ˜ T W} ˆ˙ + q˙ T tanh(λq)(26) −E(t))uf + Φ] − tr{W γ
3 X
1 ¯∗ |ri |(1 − e0 )kuf a k + ζ4 ²¯2 + ζ5 σ 2 + %W 2 i=1
1 ¯∗ ≤ −$kzk2 + ζ4 ²¯2 + ζ5 σ 2 + %W 2
(25)
Take the derivative of V , and from (11), (12) and (14), we have
i=1
1 ¯∗ +ζ4 ²¯2 + ζ5 σ 2 + %W 2 ≤ −λmin (L) tanhT (λq) tanh(λq) − rT [mf (K − I3 ) 1 1 I3 − I3 − s2 ζ3 K − s1 ζ1 K]r − s1 (1 − 4ζ4 4ζ5 3 X 1 1 − |ri |(1 − e0 )κkuf a k − )λmin (K)eT e − 4ζ1 4ζ2 i=1
then the uniform ultimate boundedness of tracking errors e and e˙ can be guaranteed. Proof: Based on (20), we have ˆ˙ = m(t)(γΘrT − %γ W) ˆ W
(27)
(28)
T T where tanh n (λq) tanh(λq) ≤ λq tanh(λq) is applied, 1 1 $ = min s1 (1 − 4ζ1 − 4ζ2 )λmin (K), λmin (λ−1 L), o λmin (mf (K − I3 ) − 4ζ14 I3 − 4ζ15 I3 − s2 ζ3 K − s1 ζ1 K) , z = q ¯∗ ζ4 ²¯2 +ζ5 σ 2 + 21 %W T T T T , it is [e , r , tanh (λq)] . Denote Υ = $ seen from (28) that V˙ < 0 when e, r, q are outside of the set
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S = {(e, r, q)|kek ≤ Υ, krk ≤ Υ, k tanh(λq)k ≤ Υ} (29)
which is a small set containing the origin (e, r, q) = 0. Hence, the tracking error e, auxiliary error variable r, filtered output q are UUB as limt→∞ kek ∈ S, limt→∞ krk ∈ S, limt→∞ kqk ∈ S, which implies that larger ki will yield better tracking control performance. Then, based on Remark 1, the tracking error e˙ is also UUB. This completes the proof. Remark 2: To guarantee the input constraints in (7) is satisfied, the gain ki (i = 1, 2, 3) should be selected such that (1 − e0 )κ (ki + φ) ≤ umax (30) ki + φ + e0 which means ki should be chosen as mf +
1 4ζ4
+
1 4ζ5
mf − s2 ζ3 − s1 ζ1
< ki ≤
e0 umax −φ (1 − e0 )κ + e0
˙ q(k∆T ) = −K
(32)
Accordingly, it is obtained from (12) that r = −K−1 Lq − ˙ Hence, r can be numerically derived during the impleK−1 q. mentation of the controller (19), because q˙ is obtained with (32). It can be summarized that uf a and uf b are independent on velocity measurements, although r is involved. Remark 4: In general, the NN has poor approximation capability to the unknown function in the system dynamics at its learning phase [19], and hence, the output of NN may be very large and may exceed the bound of the approximated nonlinear function. In order to make the output of the NN bounded by φ, a switch function m(t) is used to generate a switching between the adaptive NN control and the robust control φ. Next, we consider two special cases under Theorem 1: Case 1: e0 is unknown. In this case, we can only use the controller uf 1 to guarantee the uniform ultimate boundedness of tracking errors e and e˙ . If the inequality −rT (1 − E(t))uf a ≤ 4ζ16 rT r + ζ6 (1 − e0 )2 kuf a k2 ≤ 4ζ16 rT r + ζ6 (1 − e0 )2 (kKk + φ)2 is applied in the proofing of Theorem 1, where ζ6 is a positive constant, we can choose the following control gains to guarantee the uniform ultimate boundedness of tracking errors e and e˙ , and the input constraints in (7), where mf +
1
+
1
+
1
mf +
4ζ4 4ζ5 4ζ6 , li are chosen as in umax − φ ≥ ki > mf −s 2 ζ3 −s1 ζ1 (23), si , ζi are constants satisfied (24). Case 2: The actuators are fault-free, i.e., E(t) = I3 .
1
+
1
4ζ5 4 , li are chosen as where umax − φ ≥ ki > mf −s4ζ 2 ζ3 −s1 ζ1 in (23), and si , ζi are constants satisfied (24), then the uniform ultimate boundedness of tracking errors e and e˙ can be guaranteed, and the input constraints in (7) can be satisfied. Remark 5: The controller (19) is independent on the precise knowledge of the true anomaly rate v, ˙ the true anomaly rate of change v¨, the spacecrafts mass ml , mf , and the external disturbance Fd . Therefore, from the standpoint of uncertainties and external disturbances rejection, the derived controller has great stability robustness.
(31)
combined with (23). Remark 3: Note that the term r in (19) is not physically measurable since it contains e˙ , however, the controller is implemented with digital computer in practical aerospace engineering. The value of q˙ in the time of (k + 1)∆T can be approximately estimated by using the one-step pre˙ vious information from attitude sensors as q(k∆T ) = q((k+1)∆T )−q(k∆T ) , where ∆T is the control update period. ∆T By using (10), we can calculate the value of g˙ by e((k + 1)∆T ) − e(k∆T ) ∆T g((k + 1)∆T ) − g(k∆T ) + ∆T
In this case, we can also only use the controller uf a to guarantee the uniform ultimate boundedness of tracking errors e and e˙ . If we choose the following control gains,
IV. S IMULATION RESULTS In this section, simulation results for a leader-follower SFF are presented to illustrate the performance of the presented design method. The two spacecraft SFF problem having the following parameters [9]: Me = 5.974 × 1024 kg, G = 6.673 × 10−11 m3 /kg · s2 , ml = 100 kg, mf = 100 kg, el = 0.6, al = 1.6103 × 107 m, ul = 0 N, v(0) = 0 rad, p(0) = ˙ [20, 10, −20]T m, p(0) = [0, 0, 0]T m/s. The desired relative 4π trajectory is selected as pd = [−10 cos( 3π Tl t), 10 sin( Tl t), 5π 5 cos( Tl t)]T m, where Tl is the orbital period of the leader spacecraft. Orbital perturbation force due to J2 disturbance is included in the simulations. Suppose that the maxmum input control force umax is 40N, then the design parameters si (i = 1, 2) and ζi (i = 1, 2, · · · , 5) are chosen as s1 = 0.03, s2 = 0.6, ζ1 = 0.5, ζ2 = 0.6, ζ3 = 10, ζ4 = 1, ζ5 = 1 and the main control parameters of uf 1 for simulation are selected as K = diag{12, 12, 12}, L = diag{2, 2, 2}, λ = 1, φ = 3, γ = 0.5, % = 0.1. In order to implement the adaptive neural network based controller (19), the radial basis ¯ function of the neural network is applied to approximate Φ, and the vectors Θ ∈ Rn used in the simulation are chosen as 2 −
kx−ci k 2
σ i , i = 1, 2, · · · , n, Gaussian-type functions Θi (x) = e ¨Td ]T , σi ∈ R, and ci denote the vector where x = [pTd , p˙ Td , p with the same dimension as x. Moreover, n = 10 and σi = 15 are selected in the simulation, and the elements of ci are chosen between -1 and 1 randomly. The chattering effect that may be induced by the use of the discontinuity function sgn(·) in (19). In order to reduce the chattering, the function sgn(·) is replaced by a continuous saturation function sat( ε· ) [15], where ε = 0.01 is chosen in this simulation. We first consider the scenario of actuators fault-free, in which a relative ideal situation is simulated. Applying the controller uf a with the adaptive update law (20), we illustrate the simulation results as Fig. 2, from which it can be concluded that the objective of 3 DOF spacecraft relative position tracking is achieved in the presence of uncertain parameters and external disturbances. Indeed, due ¯ can to employment of the neural network, the effect of Φ be accommodated as soon as possible, and then fast tracking manoeuvring can be achieved. Moreover, under this control law, the relative position tracking error and velocity error
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the desired nonlinear function and bounded external disturbances. In order to guarantee that the output of the NN is bounded, a switch function is employed. The solution also applies a fault tolerant control part to compensate for the partial loss of actuator effectiveness fault. It is proven that the tracking error is UUB, and the control input can rigorously enforce actuator magnitude constraints.
50 m
p1 p2
0
p3 −50
0
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200 250 300 1. Relative position error
350
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500
5 m/s
dp1/dt dp2/dt
0
dp3/dt −5
0
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20 N
uf1 uf2
0
R EFERENCES
uf3 −20
0
50
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350
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Fig. 2: Actuators fault-free.
50 m
p1 p2
0
p3 −50
0
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200 250 300 1. Relative position error
350
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5 m/s
dp1/dt dp2/dt
0
dp3/dt −5
0
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N
40 20
uf1
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uf2 uf3
−20 −40
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Fig. 3: Actuators faults.
signals converge to zero in about 350 s as shown in Fig. 2-1-Fig. 2-2. It is interesting to note that the control force of each axis in this case is less than the required maximum bound, as shown in Fig. 2-3. Next, we consider the following actuators partial loss of control effectiveness in the SFF: 1) the actuator uf 1 decreases 30% of its normal value after 15 s; 2) the actuator uf 2 loses its power of 40% in 25 s; and 3) the actuator uf 3 undergoes 40% loss of effectiveness in 30 s. Then, apply the controller uf with the adaptive update law (20), we illustrate the simulation results as Fig. 3, where the main control parameters of uf b for simulation are selected as: e0 = 0.6, κ = 1.1. The time responses of the relative position tracking error and velocity error are presented in Fig. 3-1Fig. 3-2, respectively. The control force is shown in Fig. 3-3. As expected, we clearly see that the proposed fault-tolerant control scheme managed to compensate for the partial loss of effectiveness fault without velocity measurement. As shown in Fig. 3-3, the control force of each axis is still within its maximum allowable limit even in the presence of external disturbances, uncertain system parameters and actuator faults. V. C ONCLUSION We have presented a solution to the problem of output feedback control to relative translation tracking in a leaderfollower spacecraft formation, when there are uncertain system parameters, external disturbances, input constraints and partial loss of actuator effectiveness. The solution incorporates a pseudo-velocity filter to account for the unmeasured relative velocity, and uses the NN technique to approximate
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