Adaptive Time-Varying Sliding Mode Control for ... - Semantic Scholar
52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy
Adaptive Time-Varying Sliding Mode Control for Autonomous Spacecraft Rendezvous Lin Zhao, Yingmin Jia and Fumitoshi Matsuno Abstract— In this paper, two types optimal adaptive timevarying sliding mode control (ATVSMC) for a class of spacecraft rendezvous systems are proposed. The sliding surfaces pass through the initial point of system at the beginning and then move towards the origin of the error state space. By designing the ATVSMC law, the reaching phase is eliminated, and the system’s robustness to unknown bounded parameter uncertainties and external disturbances is guaranteed from the beginning. The existence of global siding mode is proved by Lyapunov method. Furthermore, the parameters of sliding surfaces are obtained by minimizing the integral of the infinite norm of relative position tracking error. An example is included to illustrate the obtained results.
I. INTRODUCTION Many astronautic missions, such as space transporting, docking and satellite networking, rely heavily on successful rendezvous. During the last few decades, the autonomous spacecraft rendezvous control problem has attracted considerable attention in the literature, see [1]-[10] and references therein. When the target orbit is approximately circular and the distance between them is much smaller than the orbit radius, the C-W equations were used to describe the linear relative motion between two neighboring spacecraft [1]. Based on CW equations, many design methods for rendezvous control have been developed, such as the optimal impulsive rendezvous approaches were proposed in [2] and [3]; the new switch control laws for active collision avoidance manoeuvre were given in [4]. All of the above control methods are open-loop control, which means that the control thrust is previously designed. However, the open-loop control system can be easily effected by disturbances or other uncertain factors, it is necessary to design the closed-loop control feedback controller for spacecraft rendezvous [7]-[9]. For instance, a multi-objective robust H∞ control for spacecraft rendezvous was investigated in [7]; a non-fragile robust H∞ control algorithm for uncertain spacecraft rendezvous system with pole and input constraints was studied in [8]; a mixed H2 /H∞ reliable control scheme for trajectory-tracking of rendezvous was proposed in [9]. In [7]-[9], they all assume that the parameter uncertainties and external disturbances This work was supported by the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201) and the NSFC (61134005, 60921001, 61327807). L. Zhao and Y. Jia are with the Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA), Beijing 100191, China [email protected];[email protected] F. Matsuno is with the Department of Mechanical Engineering and Science, Kyoto University, Kyoto 606-8501, Japan
[email protected]
978-1-4673-5716-6/13/$31.00 ©2013 IEEE
are bounded and their bounds are available to design the control law. However, sometimes bounds on the uncertainties or disturbances may not be easily obtained because of the complexity of the structure of uncertainties in astronautic missions. Sliding mode control (SMC), as an effective robust control strategy, has been successfully applied to a wide variety of complex systems [10]-[14]. Recently, SMC has been used in spacecraft formation flying in the presence of unknown but bounded disturbances [15]-[16]. The main advantage of this technique is the system dynamics’s insensitive to the parameter uncertainties and disturbances satisfied the matching conditions when the system state reaches the sliding surface. However, robustness is guaranteed only after the system state reaches the sliding surface. To solve this problem, the time-varying sliding mode control was proposed in [17][19]. They eliminate the reaching phase, and guarantee the robustness all the time. The time-varying sliding mode control has been effectively allied in rigid spacecraft attitude tracking control [20]-[21], however, for the time-varying sliding mode control for the rendezvous problem of two neighboring spacecraft in the presence of unknown bounded parameter uncertainties and external disturbances, to the best of our knowledge, there are no such results in the existing literature, which motivates our present study. In this paper, we introduce two types adaptive timevarying sliding mode control (ATVSMC) design methods for the rendezvous problem of two neighboring spacecraft in the presence of input constraint, unknown bounded parameter uncertainties and external disturbances. The sliding surfaces pass the initial values of system at the initial moment. Then they move towards the origin of the error state space. By mean of ATVSMC law, the reaching phase is eliminated, and the robustness against lumped uncertainties is guaranteed from the initial moment. Lyapunov method is adopted to prove the existence of global siding mode. The parameters of the sliding surfaces are obtained by minimizing the integral of the infinite norm of relative position tracking error which is taken as the performance criterion. An illustrative example is provided to show the effectiveness of the proposed methods. II. P ROBLEM FORMULATION The spacecraft rendezvous system is illustrated in Fig.1. We assume that the target and chaser spacecrafts are adjacent, and that the orbital coordinate frame is a right-handed Cartesian coordinate with origin attached to the target spacecraft center of mass, where the x-axis is along the vector from
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Chaser Spacecraft Orbit of Target Spacecraft
m0 ∆A2 q2 (t)+F (t)I3×3 u(t)+(I3×3 +F (t)I3×3 )w(t), then, (3) can be further rewritten as q˙1 (t) =q2 (t) (4) 1 q˙2 (t) =A1 q1 (t) + A2 q2 (t) + (u(t) + φ(t)) m0
z
y
x Target Spacecraft
where φ(t) is the lumped uncertainty. Assumption 1: There exist two unknown positive constants c0 and c1 such that the norm of φ(t) satisfies
Earth Center
n
kφ(t)k ≤ c0 + c1 kη(t)k Fig. 1: The spacecraft rendezvous system.
the earth center to the target center of mass, the y-axis is along the target orbit circumference and the z-axis completes the right-handed frame. The relative dynamic model can be described by the following C-W equations 1 (ux + wx ) x ¨ − 2ny˙ − 3n2 x = m0 − mt ˙ 1 (uy + wy ) y¨ + 2nx˙ = (1) m − mt ˙ 0 1 z¨ + n2 z = (uz + wz ) m0 − mt ˙ where x, y and z are the components of the relative position, n is the angular velocity of the target moving around the earth, m0 is the initial mass of the chaser, m ˙ = const is the mass-flow-rate of propellant of the chaser’s thrusters, ui (i = x, y, z) is the ith component of the control input force and wi (i = x, y, z) is the ith component of the external disturbance. 1 about the point The Taylor series expansion of 1 − θt t = 0 is given as follows
where η(t) = [q1T (t), q2T (t)]T and k·k refers to the Euclidean vector norm or the induced matrix 2-norm. Remark 1: In [7]-[9], the mass-flow-rate of propellant of the chaser’s thrusters is not considered and the parameter uncertainties are assumed to have known bounds. In this paper, more general uncertain model for rendezvous system is considered. Taking into consideration of the capacity of the power of actuator, the actual control input force is subject to the boundedness constraint as specified below: ku(t)k ≤ U
where A1 = n20 A¯1 , A2 = 2n0 A¯2 , ∆A1 = n20 (2δ + 3 0 0 δ 2 )A¯1 , ∆A2 = −2δn0 A¯2 , A¯1 = 0 0 0 , A¯2 = 0 0 −1 0 1 0 −1 0 0 , B = 1 I3×3 , ∆B = F (t) I3×3 , n = m0 m0 0 0 0 n0 (1 + δ), n0 is the theoretical angle velocity, and δ denotes the magnitude of uncertainty. Denote φ(t) = m0 ∆A1 q1 (t)+
(6)
where U denotes the maximum input force. Given the desired states (q1d (t), q2d (t)), where q˙1d (t) = q2d (t), q1d (t) = [xd , yd , zd ]T , then the error-state vector of the system (4) is defined as e(t) = e1 (t) = q1 (t) − q1d (t), e(t) ˙ = e2 (t) = q2 (t) − q2d (t) (7) and e(0) ˙ is assumed to satisfy e(0) ˙ = e2 (0) = 0. Lemma 1: 2 1 t2 (1 + )[1 − exp(−at)] + 2 ab ab b 1 t −2(1 + ) + 1 > 0 ab b 1 t 1 g(t) = − exp(−at) + + 1 + >0 ab b ab
f (t) =
1 (θ)n+1 = 1+θt+θ2 t2 +· · ·+θn tn + tn+1 (2) 1 − θt (1 − θξ)n+2 m ˙ where θ = m , 0 < ξ < t. Let q1 (t) = q(t) = 0 T [x, y, z] , q2 (t) = q(t), ˙ w(t) = [wx , wy , wz ]T , u(t) = T [ux , uy , uz ] and F (t) = θt + θ2 t2 + · · · + θn tn + θ n+1 n+1 . Then, (1) can be rewritten as (1−θξ)n+2 t q˙1 (t) =q2 (t) q˙2 (t) =(A1 + ∆A1 )q1 (t) + (A2 + ∆A2 )q2 (t) (3) + (B + ∆B)(u(t) + w(t))
(5)
(8) (9)
for t ∈ [0, b], where a > 0 and b > 0 are constants. Proof: Lemma 1 can be easily proved by analyzing the monotonicity of f (t), g(t), and thus is omitted for brevity. The objective of this paper can be given as follows: For the system (7) and Assumption 1, find the control law to drive the states (q1 (t), q2 (t)) from the initial states (q1 (0), q2 (0)) to the desired states (q1d (t), q2d (t)) , while the constraint ku(t)k ≤ U is met. III. M AIN R ESULTS The design of the ATVSMC mainly has three steps. First, choose a time-varying sliding surface with a certain scheme, and then derive the ATVSMC law by Lyapunov method. At last, design the parameters of sliding surface such that the performance criterion is minimized.
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A. Constant-acceleration sliding mode (CASM) Let us consider the sliding surface s(t) = 0, where s(t) ∈ R3 is defined as ( At2 + Bt + C, t ≤ T s(t) = e2 (t) + ke1 (t) + (10) 0, t > T where k > 0 is the slope of the predetermined sliding surface and T > 0 is the switch time, and both are design parameters. Let A ∈ R3 , B ∈ R3 and C ∈ R3 be constant vectors such that 1) The sliding surface passes the initial value of system at the initial moment, that is, at t = 0, e2 (0) + ke1 (0) + C = 0. 2) the right-hand side of (10) and its derivative are both continuous at t = T , that is, ( AT 2 + BT + C = 0 (11) 2AT + B = 0 1 (0) Based on these assumptions, we have A = − e2 (0)+ke = T2 ke(0) e2 (0)+ke1 (0) ke(0) − T2 , B = 2 = 2 , C = −e (0) − 2 T T ke1 (0) = −ke(0). Without the requirement of the information of upper bound of the lumped uncertainty φ(t), we introduce the following adaptive sliding mode control law ( − ke2 (t) − 2At − B, t ≤ T u(t) = m0 u0 (t)+m0 (12) − ke2 (t), t > T
where u0 (t) = q˙2d (t) − A1 q1 (t) − A2 q2 (t) − Ks(t) s(t) − [¯ c0 (t, η(t)) + c¯1 (t, η(t))kη(t)k] ks(t)k , c¯0 (t, η(t)) and c¯1 (t, η(t)) are the adaptive parameters about c0 and c1 , K > 0 is a little constant. Now, consider the following adaptive laws for the upper bound of the norm kφ(t)k such that ( c¯˙0 (t, η(t)) = l0 ks(t)k (13) c¯˙1 (t, η(t)) = l1 ks(t)kkη(t)k where l0 and l1 are adaptive gains with positive constants. Theorem 1: Given system (4) and (7), if Assumption 1 is satisfied, the sliding mode exist in a given initial state, and the system has to stay on the sliding surface all the time by employing the control law (12) with the adaptive laws (13). Proof: Consider the following Lyapunov function 1 1 1 V = sT s + l0−1 c˜20 + l1−1 c˜21 (14) 2 2 2 where c˜0 (t, η(t)) = c¯0 (t, η(t)) − c0 and c˜1 (t, η(t)) = c¯1 (t, η(t)) − c1 are parameter adaptive errors. Differentiating V with respect to time yields T s [e˙ 2 (t) + ke2 (t) + 2At + B], t ≤ T V˙ = l0−1 c˜0 c˜˙0 + l1−1 c˜1 c˜˙1 + T s [e˙ 2 (t) + ke2 (t)], t > T (15) Substituting (5), (12)-(l3) into the above equation gives V˙ = −KsT s − ksk[¯ c0 + c¯1 kηk] + sT φ(t) +l0−1 c˜0 c˜˙0 + l1−1 c˜1 c˜˙1 ≤ −KsT s + c˜0 (l0−1 c˜˙0 − ksk) + c˜1 (l1−1 c˜˙1 − kskkηk) = −KsT s < 0 (16)
Thus, the sliding surface (10) exists, and the system is insensitive to the lumped uncertainty for any t ∈ [0, +∞). Remark 2: Because of the initial value of the system belongs to the sliding surface, and Theorem 1 guarantee that the system has to stay on the sliding surface all the time, so the system is insensitive to the lumped uncertainty φ(t) for any t ∈ [0, +∞) and the robustness is guaranteed. Theorem 2: Given system (4) and (7), if the control law (12) with the adaptive laws (13) are given, the system is globally asymptotically stable. Proof: Based on Theorem 1, the sliding surface (10) exists for any t ∈ [0, +∞), then the following relationship ( − At2 − Bt − C, t ≤ T e2 (t) + ke1 (t) = (17) 0, t > T always holds. Therefore, solve (17), we can obtain C B 2A A − 2 + 3 ] exp(−kt) − t2 k k k k 2A − kB kB − 2A − Ck 2 + t+ k2 k3
e(t) =[e(0) +
(18)
for t ∈ [0, T ], and e(t) = e(T ) exp[−k(t − T )]
(19)
for t ∈ (T, +∞). Substitute t = T into (18), we can obtain e(T ) and obviously e(T ) is bounded definitely, so ∀ e(0), limt→+∞ e(t) = 0, that is, the overall system is globally asymptotically stable. Denote ˜ = max{m0 [kq˙2d (t) − A1 q1 (t) − A2 q2 (t)k + Kks(t)k U + c¯0 + c¯1 kηk]}, t ≥ 0 Before giving the optimal design of parameters k and T , the following assumption should be satisfied for the constraint U , that is ˜. Assumption 2: U > U Then, substituting (13) into (17)-(18), we have 2 1 t2 (1 + )[1 − exp(−kt)] + 2 kT kT T 1 t − 2(1 + ) + 1} kT T
e(t) =e(0){
(20)
for t ∈ [0, T ], and 2 exp[−k(t − T )] k2 T 2 2 1 − 2 (k + ) exp(−kt)} k T T
e(t) =e(0){
(21)
for t ∈ (T, +∞). Therefore e¨(t) =
2e(0) 1 { [1 − exp(−kt)] − k exp(−kt)} T T
(22)
for t ∈ [0, T ], and
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e¨(t) =
2e(0) [exp(kT ) − kT − 1] exp(−kt) T
(23)
for t ∈ (T, +∞). It follows from Assumption 2 that there ˜ is a constant Umax = Um−0U such that the constraint in (6) is satisfied if the following condition holds Umax k¨ e(t)k∞ ≤ √ 3
(24)
Without loss of generality, we assume that ke(0)k∞ = |ex (0)| = |x(0) − xd (0)|, where | · | refers to the absolute value and k · k∞ refers to the infinite vector norm or the row matrix norm. Thus based on Lemma 1, we have 1 2 (1 + kT )[1 − exp(−kt)] + ke(t)k∞ = |ex (0)|{ kT 1 −2(1 + kT ) Tt + 1}
t2 T2
(25)
for t ∈ [0, T ], and 2 exp[−k(t − T )] k2 T 2 2 1 − 2 (k + ) exp(−kt)} k T T for t ∈ (T, +∞). Meanwhile,
(26)
2 1 =|ex (0)|| { [1 − exp(−kt)] − k exp(−kt)}| T T (27)
for t ∈ [0, T ], and 2 k¨ e(t)k∞ =|ex (0)|| [exp(kT ) − kT − 1]| exp(−kt) (28) T for t ∈ (T, +∞). Note that the monotonicity of (27) and (28), so from (24) the following constraints should be satisfied 2k Umax k¨ e(0)k∞ =|ex (0)|| | ≤ √ T 3 2 1 (29) k¨ e(T )k∞ =|ex (0)|| { [1 − exp(−kT )] T T Umax − k exp(−kT )}| ≤ √ 3 Now, we consider the criterion Z +∞ J= ke(t)k∞ dt (30) 0
Theorem 3: When s √ s 6 3|e (0)| 3U x √ max T = T¯ = , k = k¯ = Umax 2 3|ex (0)|
(31)
the criterion in (30) is minimized and the constraints of (29) can be guaranteed. Proof: From (25), (26) and (30), we have Z +∞ Z +∞ sgn(ex (t))ex (t)dt (32) ke(t)k∞ dt = J= 0
T 1 T 1 + ) = |ex (0)|( + ) (34) 3 k 3 k Therefore, we consider the constraints in √(29) and choose x (0)| max T k = 2√U3|e yields J = |ex (0)|( T3 + 2 U3|e ), which max T x (0)| ¯ Then, substituting k¯ has its minimum value for T¯ and k. and T¯ into (29) yields J = ex (0)sgn(ex (0))(
Umax 1 (35) k¨ e(T = T¯)k∞ = √ Umax [1 − 4 exp(−3)] < √ 3 3 3 Thus, k¯ and T¯ yield the minimum value of J under the constraints of (29). B. Constant-velocity sliding mode (CVSM)
ke(t)k∞ =|ex (0)|{
k¨ e(t)k∞
Then, from (20) and (21), we have
0
Lemma 1 implies that ex (t) expressed in (20) and (21) dose not change its sign, that is sgn(ex (t)) = sgn(ex (0)), thus Z T J =sgn(ex (0)) ex (t)dt 0 (33) Z +∞ + sgn(ex (0)) ex (t)dt
as
The Constant-velocity sliding surface function is defined ( At + B, t ≤ T s(t) = e2 (t) + ke1 (t) + (36) 0, t > T
where A ∈ R3 and B ∈ R3 are constant vectors such that 1) The sliding surface passes the initial value of system at the initial moment, that is, at t = 0, e2 (0) + ke1 (0) + B = 0 2) the right-hand side of (36) and its derivative are both continuous at t = T , that is, AT + B = 0. Based on these assumptions, we have A = ke1T(0) = ke(0) T , B = −ke1 (0) = −ke(0). Next, we also give the following adaptive sliding mode control law ( − ke2 (t) − A, t ≤ T u(t) = m0 u0 (t) + m0 (37) − ke2 (t), t > T where u0 (t) is defined in (12). Theorem 4: Given system (4) and (7), if Assumption 1 is satisfied, the sliding mode exist in a given initial state, and the system has to stay on the sliding surface all the time by employing the control law (38) with the adaptive laws (13). Proof: The proof of Theorem 4 is similar to that of Theorem 1 and thus is omitted for brevity. Theorem 5: Given system (4) and (7), if the control law (37) with the adaptive laws (13) is given, the system is globally asymptotically stable. Proof: Based on Theorem 4, the sliding surface (36) exists for any t ∈ [0, +∞), therefore, we have A A B A B − 2 ] exp(−kt) − t − + 2 k k k k k for t ∈ [0, T ], and e(t) = [e(0) +
e(t) = e(T ) exp[−k(t − T )]
(38)
(39)
for t ∈ (T, +∞). Similar with the proof in Theorem 1, we can obtain ∀ e(0), limt→+∞ e(t) = 0. Now, substituting (37) into (38)-(39), we have 1 1 1 exp(−kt) + t + 1 + ] kT T kT
(40)
1 exp[−k(t − T )] − exp(−kt)} kT
(41)
e(t) = e(0)[− for t ∈ [0, T ], and
T
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e(t) = e(0){
for t ∈ (T, +∞). Therefore ke(0) e¨(t) = − exp(−kt) T for t ∈ [0, T ], and
TABLE I: The parameter values for two types control strategies
(42)
Sliding mode types
k
T
l0
l1
K
CASM
0.1
30 √ 10 3
0.01
0.009
0.00001
0.01
0.008
0.00001
ke(0) [exp(kT ) − 1] exp(−kt) (43) T for t ∈ (T, +∞). Then, the constraint in (6) is satisfied if the conditions (24) holds. We also assume that ke(0)k∞ = |ex (0)| = |x(0) − xd (0)|, thus based on Lemma 1, we have e¨(t) =
800
Relative positions / m
600
500
400
300
200
100
0
0
(45)
10
20
30
40 Time / s
50
60
70
80
Fig. 2: CASM: state response signals.
1000
ux
0
(46)
−1000 −2000
Input forces / N
k¨ e(t)k∞
k = |ex (0)| exp(−kt) T
x y z
700
1 1 1 exp(−kt) + t + 1 + ] (44) ke(t)k∞ = |ex (0)|[− kT T kT for t ∈ [0, T ], and k ke(t)k∞ = |ex (0)| [exp(kT ) − 1] exp(−kt) T for t ∈ (T, +∞). Meanwhile,
√ 3 15
CVSM
for t ∈ [0, T ], and k k¨ e(t)k∞ = |ex (0)| [exp(kT ) − 1] exp(−kt) (47) T for t ∈ (T, +∞). Noticing that the monotonicity of (46) and (47), we obtain the following constraints k Umax e(0)k∞ = |ex (0)|| | ≤ √ k¨ T 3 (48) k Umax k¨ e(T )k∞ = |ex (0)|| [1 − exp(−kT )] ≤ √ T 3 Now, we consider the criterion in (30), the following Theorem 6 are used to decide the optimal parameters k and T . Theorem 6: When s √ 2 3|ex (0)| ¯ T = T = Umax s (49) 2Umax ¯ k = k = √ 3|ex (0)| the criterion in (30) is minimized and the constraints of (48) can be guaranteed. Proof: The proof of Theorem 6 is similar to that of Theorem 3 and thus is omitted for brevity. IV. S IMULATION RESULTS In this section, we give an example to illustrate the effectiveness of the above control law design methods. Here, we consider a pair of adjacent spacecraft, and make the following assumptions. The mass of the chaser is 300 kg, and the target is moving along a geosynchronous orbit of radius 42241 km with an orbital period of 24 h. Thus, the angle velocity n = 7.2722×10−5 rads/s. The mass-flow-rate of propellant of the chasers thrusters £is 30 g/s. Assume that ¤T . the initial relative position q1 (0) = 800 600 500 Furthermore, for simplicity, we assume that the initial state £ ¤T . Assume that the input control force q2 (0) = 0 0 0
0
10
20
30
40
50
60
70
80
1000
u
y
0 −1000 −2000
0
10
20
30
40
50
60
70
80
500
u
z
0 −500 −1000
0
10
20
30
40 Time / s
50
60
70
80
Fig. 3: CASM: control input signals.
constraint is U = 4000N . The desired trajectory is q1d (t) = q2d (t) = 0, thus we have e2 (0) = 0. With the assumptions, √ we have Umax = 163 3 . In simulation, we choose δ = 0.1 and the following external disturbance signal: ( 10 sin 0.2t, 0 < t < 30s w(t) = 0, otherwise To prevent the control input signals from chattering, we s(t) s(t) replace ks(t)k with ks(t)k+0.1 . Next, we simulate the dynamic behavior of system (7) controlled according to two types ATVSMC laws presented in this paper. The optimal values of k, T , the adaptive gains l0 , l1 and the gain K are given in Table 1. The state response trajectories of the closed-loop system under two types ATVSMC are depicted in Fig. 2 and Fig. 5, respectively. It can be seen from From Fig. 2 and Fig. 5 that the closed-loop systems are both globally asymptotically stable subject to parameter uncertainties and external disturbances from the initial moment. The control input forces in three axes are depicted in Fig. 3 and Fig. 6, respectively. It can be seen that the largest input force of three axes is bellow the maximum allowed force 4000N . The adaptive parameters trajectories are depicted in Fig. 4 and Fig. 7, and they are all bounded. The simulation results verify that the desired requirements of the closed-loop system have been all achieved, and we can further see that the state convergence rates under CVSM is faster than the state convergence rates under CASM.
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obtained by minimizing the integral of the infinite norm of position tracking error. An illustrative example has shown the effectiveness of the proposed methods.
−4
8
x 10
6
Adaptive parameters
4 2
c
0
0
0
10
20
30
40
50
60
70
R EFERENCES
80
0.1 0.08 0.06 0.04 0.02
c
1
0
0
10
20
30
40
50
60
70
80
Time / s
Fig. 4: CASM: adaptive parameters.
800
x y z
700
Relative positions / m
600
500
400
300
200
100
0
0
10
20
30
40
50
60
70
80
Time / s
Fig. 5: CVSM: state response signals.
2000
u
x
0
Input forces / N
−2000
0
10
20
30
40
50
60
70
80
2000
u
y
0
−2000
0
10
20
30
40
50
60
70
80
1000
u
z
0
−1000
0
10
20
30
40
50
60
70
80
Time / s
Fig. 6: CVSM: control input signals.
−4
6
x 10
Adaptive parameters
4
2
c
0
0
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
80
0.08 0.06 0.04 0.02 0
c1 80
Time / s
Fig. 7: CVSM: adaptive parameters.
V. C ONCLUSIONS This paper proposes two types optimal ATVSMC for autonomous spacecraft rendezvous systems subject to input constraint. All two sliding surfaces pass the initial values of system at the initial moment, and then move towards the origin of the error state space. By designing the ATVSMC law, the reaching phase is eliminated, and the robustness against unknown bounded lumped uncertainty is guaranteed from the initial moment. Two parameters of the sliding surfaces are
[1] W. H. Clohessy, R. S. Wiltshire, Terminal guidance system for satellite rendezvous, Journal of Aerospace Sciences, vol. 27, no. 9, pp. 653658, 1960. [2] D. J. Jezewski, J. D. Donaldson, An analytical approach to optimal rendezvous using Clohessy-Wiltshire equations, Journal of the Astronautical Sciences, vol. 27, no. 3, pp. 293-310, 1979. [3] H. Y. Li, Y. Z. Luo, J. Zhang, G. J. Tang, Optimal multi-objective linearized impulsive rendezvous under uncertainty, Acta Astronautica, vol. 66, no. 3-4, pp. 439-445, 2010. [4] Y. Q. Qi, Y. M. Jia, Active collision avoidance manoeuvre design based on equidistant interpolation, Acta Astronautica, vol. 69, no. 11-12, pp. 987-996, 2011. [5] P. Singla, K. Subbarao, J. L. Junkins, Adaptive output feedback control for spacecraft rendezvous and docking under measurement uncertainty, Journal of Guidance, Control and Dynamics, vol. 29, no. 4, pp. 892902, 2006. [6] S. Lu, S. J. Xu, Adaptive control for autonomous rendezvous of spacecraft on elliptical orbit, Acta Mechanica Sinica, vol. 25, no. 4, pp. 539-545, 2009. [7] H. J. Gao, X. B. Yang, P. Shi, Multi-objective robust H∞ control of spacecraft rendezvous, IEEE Transactions on Control Systems Technology, vol. 17, no. 4, pp. 794-802, 2009. [8] X. Y. Gao, K. L. Teo, G. R. Duan, Non-fragile robust H∞ control for uncertain spacecraft rendezvous system with pole and input constraints, International Journal of Control, vol. 85, no. 7, pp. 933-941, 2012. [9] L. C. Man, X. Y. Meng, Z. Z. Liu, L. F. Du, Multi-objective and reliable control for trajectory-tracking of rendezvous via parameterdependent Lyapunov functions, Acta Astronautica, vol. 81, no. 1, pp. 122-136, 2012. [10] J. U. Park, K. H. Choi, S. Lee, Orbital rendezvous using two-step slidhg mode control, Aerospace Science and Technology, vol. 3, no. 4, pp. 563-569, 1999. [11] B. Ebrahimi, M. Bahrami, J. Roshanian, Optimal sliding-mode guidance with terminal velocity constraint for fixed-interval propulsive maneuvers, Acta Astronautica, vol. 60, no. 10, pp. 556-562, 2006. [12] V. Utkin, Sliding Modes in Control Optimization, Berlin: SpringerVerlag, 1992. [13] D. J. Yoo, M. J. Chung, A variable structure control with simple adaptation laws for upper bounds on the norm of the uncertainties, IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 860–865, 1992. [14] N. Y. Gang, W. C. Ho. Daniel, J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica, vol. 41, no. 5, pp. 873-880, 2003. [15] L. Hui, J. F. Li. Terminal sliding mode control for spacecraft formation flying, IEEE Transactions on Aerospace and Electronic Systems, vol. 45, no. 3, pp. 835-846, 2009. [16] J. H. Bae, Y. D. Kim. Adaptive controller design for spacecraft formation flying using sliding mode controller and neural networks, Journal of the Franklin Institute, vol. 349, no. 2, pp. 578-603, 2012. [17] S. B. Choi, D. W. Park, S. Jayasuriya, A time-varying sliding surface for fast and robust tracking control of second-order uncertain systems, Automatica, vol. 30, no. 5, pp. 899-904, 1994. [18] A. Bartoszewicz, Time-varying sliding modes for second-order systems, IEE Proceedings-Control Theory and Applications, vol. 143, no. 5, pp. 455-462, 1996. [19] Y. Q. Jin, X. D. Liu, W. Q, C. Z. Hou, Time-varying sliding mode control for a class of uncertain MIMO nonlinear system subject to control input constraint, Science China F: Information Sciences, vol. 53, no. 1, pp. 89-100, 2010. [20] Y. Q. Jin, X. D. Liu, W. Qiu, C. Z. Hou, Time-varying sliding mode controls in rigid spacecraft attitude tracking, Chinese Journal of Aeronautics, vol. 21, no. 4, pp. 352-360, 2008. [21] B. L. Cong, X. D. Liu, Z. Chen, Exponential time-varying sliding mode control for large angle attitude eigenaxis maneuver of rigid spacecraft, Chinese Journal of Aeronautics, vol. 23, no. 4, pp. 447-453, 2010.
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Adaptive Time-Varying Sliding Mode Control for Autonomous Spacecraft Rendezvous Lin Zhao, Yingmin Jia and Fumitoshi Matsuno Abstract— In this paper, two types optimal adaptive timevarying sliding mode control (ATVSMC) for a class of spacecraft rendezvous systems are proposed. The sliding surfaces pass through the initial point of system at the beginning and then move towards the origin of the error state space. By designing the ATVSMC law, the reaching phase is eliminated, and the system’s robustness to unknown bounded parameter uncertainties and external disturbances is guaranteed from the beginning. The existence of global siding mode is proved by Lyapunov method. Furthermore, the parameters of sliding surfaces are obtained by minimizing the integral of the infinite norm of relative position tracking error. An example is included to illustrate the obtained results.
I. INTRODUCTION Many astronautic missions, such as space transporting, docking and satellite networking, rely heavily on successful rendezvous. During the last few decades, the autonomous spacecraft rendezvous control problem has attracted considerable attention in the literature, see [1]-[10] and references therein. When the target orbit is approximately circular and the distance between them is much smaller than the orbit radius, the C-W equations were used to describe the linear relative motion between two neighboring spacecraft [1]. Based on CW equations, many design methods for rendezvous control have been developed, such as the optimal impulsive rendezvous approaches were proposed in [2] and [3]; the new switch control laws for active collision avoidance manoeuvre were given in [4]. All of the above control methods are open-loop control, which means that the control thrust is previously designed. However, the open-loop control system can be easily effected by disturbances or other uncertain factors, it is necessary to design the closed-loop control feedback controller for spacecraft rendezvous [7]-[9]. For instance, a multi-objective robust H∞ control for spacecraft rendezvous was investigated in [7]; a non-fragile robust H∞ control algorithm for uncertain spacecraft rendezvous system with pole and input constraints was studied in [8]; a mixed H2 /H∞ reliable control scheme for trajectory-tracking of rendezvous was proposed in [9]. In [7]-[9], they all assume that the parameter uncertainties and external disturbances This work was supported by the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201) and the NSFC (61134005, 60921001, 61327807). L. Zhao and Y. Jia are with the Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA), Beijing 100191, China [email protected];[email protected] F. Matsuno is with the Department of Mechanical Engineering and Science, Kyoto University, Kyoto 606-8501, Japan
[email protected]
978-1-4673-5716-6/13/$31.00 ©2013 IEEE
are bounded and their bounds are available to design the control law. However, sometimes bounds on the uncertainties or disturbances may not be easily obtained because of the complexity of the structure of uncertainties in astronautic missions. Sliding mode control (SMC), as an effective robust control strategy, has been successfully applied to a wide variety of complex systems [10]-[14]. Recently, SMC has been used in spacecraft formation flying in the presence of unknown but bounded disturbances [15]-[16]. The main advantage of this technique is the system dynamics’s insensitive to the parameter uncertainties and disturbances satisfied the matching conditions when the system state reaches the sliding surface. However, robustness is guaranteed only after the system state reaches the sliding surface. To solve this problem, the time-varying sliding mode control was proposed in [17][19]. They eliminate the reaching phase, and guarantee the robustness all the time. The time-varying sliding mode control has been effectively allied in rigid spacecraft attitude tracking control [20]-[21], however, for the time-varying sliding mode control for the rendezvous problem of two neighboring spacecraft in the presence of unknown bounded parameter uncertainties and external disturbances, to the best of our knowledge, there are no such results in the existing literature, which motivates our present study. In this paper, we introduce two types adaptive timevarying sliding mode control (ATVSMC) design methods for the rendezvous problem of two neighboring spacecraft in the presence of input constraint, unknown bounded parameter uncertainties and external disturbances. The sliding surfaces pass the initial values of system at the initial moment. Then they move towards the origin of the error state space. By mean of ATVSMC law, the reaching phase is eliminated, and the robustness against lumped uncertainties is guaranteed from the initial moment. Lyapunov method is adopted to prove the existence of global siding mode. The parameters of the sliding surfaces are obtained by minimizing the integral of the infinite norm of relative position tracking error which is taken as the performance criterion. An illustrative example is provided to show the effectiveness of the proposed methods. II. P ROBLEM FORMULATION The spacecraft rendezvous system is illustrated in Fig.1. We assume that the target and chaser spacecrafts are adjacent, and that the orbital coordinate frame is a right-handed Cartesian coordinate with origin attached to the target spacecraft center of mass, where the x-axis is along the vector from
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Chaser Spacecraft Orbit of Target Spacecraft
m0 ∆A2 q2 (t)+F (t)I3×3 u(t)+(I3×3 +F (t)I3×3 )w(t), then, (3) can be further rewritten as q˙1 (t) =q2 (t) (4) 1 q˙2 (t) =A1 q1 (t) + A2 q2 (t) + (u(t) + φ(t)) m0
z
y
x Target Spacecraft
where φ(t) is the lumped uncertainty. Assumption 1: There exist two unknown positive constants c0 and c1 such that the norm of φ(t) satisfies
Earth Center
n
kφ(t)k ≤ c0 + c1 kη(t)k Fig. 1: The spacecraft rendezvous system.
the earth center to the target center of mass, the y-axis is along the target orbit circumference and the z-axis completes the right-handed frame. The relative dynamic model can be described by the following C-W equations 1 (ux + wx ) x ¨ − 2ny˙ − 3n2 x = m0 − mt ˙ 1 (uy + wy ) y¨ + 2nx˙ = (1) m − mt ˙ 0 1 z¨ + n2 z = (uz + wz ) m0 − mt ˙ where x, y and z are the components of the relative position, n is the angular velocity of the target moving around the earth, m0 is the initial mass of the chaser, m ˙ = const is the mass-flow-rate of propellant of the chaser’s thrusters, ui (i = x, y, z) is the ith component of the control input force and wi (i = x, y, z) is the ith component of the external disturbance. 1 about the point The Taylor series expansion of 1 − θt t = 0 is given as follows
where η(t) = [q1T (t), q2T (t)]T and k·k refers to the Euclidean vector norm or the induced matrix 2-norm. Remark 1: In [7]-[9], the mass-flow-rate of propellant of the chaser’s thrusters is not considered and the parameter uncertainties are assumed to have known bounds. In this paper, more general uncertain model for rendezvous system is considered. Taking into consideration of the capacity of the power of actuator, the actual control input force is subject to the boundedness constraint as specified below: ku(t)k ≤ U
where A1 = n20 A¯1 , A2 = 2n0 A¯2 , ∆A1 = n20 (2δ + 3 0 0 δ 2 )A¯1 , ∆A2 = −2δn0 A¯2 , A¯1 = 0 0 0 , A¯2 = 0 0 −1 0 1 0 −1 0 0 , B = 1 I3×3 , ∆B = F (t) I3×3 , n = m0 m0 0 0 0 n0 (1 + δ), n0 is the theoretical angle velocity, and δ denotes the magnitude of uncertainty. Denote φ(t) = m0 ∆A1 q1 (t)+
(6)
where U denotes the maximum input force. Given the desired states (q1d (t), q2d (t)), where q˙1d (t) = q2d (t), q1d (t) = [xd , yd , zd ]T , then the error-state vector of the system (4) is defined as e(t) = e1 (t) = q1 (t) − q1d (t), e(t) ˙ = e2 (t) = q2 (t) − q2d (t) (7) and e(0) ˙ is assumed to satisfy e(0) ˙ = e2 (0) = 0. Lemma 1: 2 1 t2 (1 + )[1 − exp(−at)] + 2 ab ab b 1 t −2(1 + ) + 1 > 0 ab b 1 t 1 g(t) = − exp(−at) + + 1 + >0 ab b ab
f (t) =
1 (θ)n+1 = 1+θt+θ2 t2 +· · ·+θn tn + tn+1 (2) 1 − θt (1 − θξ)n+2 m ˙ where θ = m , 0 < ξ < t. Let q1 (t) = q(t) = 0 T [x, y, z] , q2 (t) = q(t), ˙ w(t) = [wx , wy , wz ]T , u(t) = T [ux , uy , uz ] and F (t) = θt + θ2 t2 + · · · + θn tn + θ n+1 n+1 . Then, (1) can be rewritten as (1−θξ)n+2 t q˙1 (t) =q2 (t) q˙2 (t) =(A1 + ∆A1 )q1 (t) + (A2 + ∆A2 )q2 (t) (3) + (B + ∆B)(u(t) + w(t))
(5)
(8) (9)
for t ∈ [0, b], where a > 0 and b > 0 are constants. Proof: Lemma 1 can be easily proved by analyzing the monotonicity of f (t), g(t), and thus is omitted for brevity. The objective of this paper can be given as follows: For the system (7) and Assumption 1, find the control law to drive the states (q1 (t), q2 (t)) from the initial states (q1 (0), q2 (0)) to the desired states (q1d (t), q2d (t)) , while the constraint ku(t)k ≤ U is met. III. M AIN R ESULTS The design of the ATVSMC mainly has three steps. First, choose a time-varying sliding surface with a certain scheme, and then derive the ATVSMC law by Lyapunov method. At last, design the parameters of sliding surface such that the performance criterion is minimized.
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A. Constant-acceleration sliding mode (CASM) Let us consider the sliding surface s(t) = 0, where s(t) ∈ R3 is defined as ( At2 + Bt + C, t ≤ T s(t) = e2 (t) + ke1 (t) + (10) 0, t > T where k > 0 is the slope of the predetermined sliding surface and T > 0 is the switch time, and both are design parameters. Let A ∈ R3 , B ∈ R3 and C ∈ R3 be constant vectors such that 1) The sliding surface passes the initial value of system at the initial moment, that is, at t = 0, e2 (0) + ke1 (0) + C = 0. 2) the right-hand side of (10) and its derivative are both continuous at t = T , that is, ( AT 2 + BT + C = 0 (11) 2AT + B = 0 1 (0) Based on these assumptions, we have A = − e2 (0)+ke = T2 ke(0) e2 (0)+ke1 (0) ke(0) − T2 , B = 2 = 2 , C = −e (0) − 2 T T ke1 (0) = −ke(0). Without the requirement of the information of upper bound of the lumped uncertainty φ(t), we introduce the following adaptive sliding mode control law ( − ke2 (t) − 2At − B, t ≤ T u(t) = m0 u0 (t)+m0 (12) − ke2 (t), t > T
where u0 (t) = q˙2d (t) − A1 q1 (t) − A2 q2 (t) − Ks(t) s(t) − [¯ c0 (t, η(t)) + c¯1 (t, η(t))kη(t)k] ks(t)k , c¯0 (t, η(t)) and c¯1 (t, η(t)) are the adaptive parameters about c0 and c1 , K > 0 is a little constant. Now, consider the following adaptive laws for the upper bound of the norm kφ(t)k such that ( c¯˙0 (t, η(t)) = l0 ks(t)k (13) c¯˙1 (t, η(t)) = l1 ks(t)kkη(t)k where l0 and l1 are adaptive gains with positive constants. Theorem 1: Given system (4) and (7), if Assumption 1 is satisfied, the sliding mode exist in a given initial state, and the system has to stay on the sliding surface all the time by employing the control law (12) with the adaptive laws (13). Proof: Consider the following Lyapunov function 1 1 1 V = sT s + l0−1 c˜20 + l1−1 c˜21 (14) 2 2 2 where c˜0 (t, η(t)) = c¯0 (t, η(t)) − c0 and c˜1 (t, η(t)) = c¯1 (t, η(t)) − c1 are parameter adaptive errors. Differentiating V with respect to time yields T s [e˙ 2 (t) + ke2 (t) + 2At + B], t ≤ T V˙ = l0−1 c˜0 c˜˙0 + l1−1 c˜1 c˜˙1 + T s [e˙ 2 (t) + ke2 (t)], t > T (15) Substituting (5), (12)-(l3) into the above equation gives V˙ = −KsT s − ksk[¯ c0 + c¯1 kηk] + sT φ(t) +l0−1 c˜0 c˜˙0 + l1−1 c˜1 c˜˙1 ≤ −KsT s + c˜0 (l0−1 c˜˙0 − ksk) + c˜1 (l1−1 c˜˙1 − kskkηk) = −KsT s < 0 (16)
Thus, the sliding surface (10) exists, and the system is insensitive to the lumped uncertainty for any t ∈ [0, +∞). Remark 2: Because of the initial value of the system belongs to the sliding surface, and Theorem 1 guarantee that the system has to stay on the sliding surface all the time, so the system is insensitive to the lumped uncertainty φ(t) for any t ∈ [0, +∞) and the robustness is guaranteed. Theorem 2: Given system (4) and (7), if the control law (12) with the adaptive laws (13) are given, the system is globally asymptotically stable. Proof: Based on Theorem 1, the sliding surface (10) exists for any t ∈ [0, +∞), then the following relationship ( − At2 − Bt − C, t ≤ T e2 (t) + ke1 (t) = (17) 0, t > T always holds. Therefore, solve (17), we can obtain C B 2A A − 2 + 3 ] exp(−kt) − t2 k k k k 2A − kB kB − 2A − Ck 2 + t+ k2 k3
e(t) =[e(0) +
(18)
for t ∈ [0, T ], and e(t) = e(T ) exp[−k(t − T )]
(19)
for t ∈ (T, +∞). Substitute t = T into (18), we can obtain e(T ) and obviously e(T ) is bounded definitely, so ∀ e(0), limt→+∞ e(t) = 0, that is, the overall system is globally asymptotically stable. Denote ˜ = max{m0 [kq˙2d (t) − A1 q1 (t) − A2 q2 (t)k + Kks(t)k U + c¯0 + c¯1 kηk]}, t ≥ 0 Before giving the optimal design of parameters k and T , the following assumption should be satisfied for the constraint U , that is ˜. Assumption 2: U > U Then, substituting (13) into (17)-(18), we have 2 1 t2 (1 + )[1 − exp(−kt)] + 2 kT kT T 1 t − 2(1 + ) + 1} kT T
e(t) =e(0){
(20)
for t ∈ [0, T ], and 2 exp[−k(t − T )] k2 T 2 2 1 − 2 (k + ) exp(−kt)} k T T
e(t) =e(0){
(21)
for t ∈ (T, +∞). Therefore e¨(t) =
2e(0) 1 { [1 − exp(−kt)] − k exp(−kt)} T T
(22)
for t ∈ [0, T ], and
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e¨(t) =
2e(0) [exp(kT ) − kT − 1] exp(−kt) T
(23)
for t ∈ (T, +∞). It follows from Assumption 2 that there ˜ is a constant Umax = Um−0U such that the constraint in (6) is satisfied if the following condition holds Umax k¨ e(t)k∞ ≤ √ 3
(24)
Without loss of generality, we assume that ke(0)k∞ = |ex (0)| = |x(0) − xd (0)|, where | · | refers to the absolute value and k · k∞ refers to the infinite vector norm or the row matrix norm. Thus based on Lemma 1, we have 1 2 (1 + kT )[1 − exp(−kt)] + ke(t)k∞ = |ex (0)|{ kT 1 −2(1 + kT ) Tt + 1}
t2 T2
(25)
for t ∈ [0, T ], and 2 exp[−k(t − T )] k2 T 2 2 1 − 2 (k + ) exp(−kt)} k T T for t ∈ (T, +∞). Meanwhile,
(26)
2 1 =|ex (0)|| { [1 − exp(−kt)] − k exp(−kt)}| T T (27)
for t ∈ [0, T ], and 2 k¨ e(t)k∞ =|ex (0)|| [exp(kT ) − kT − 1]| exp(−kt) (28) T for t ∈ (T, +∞). Note that the monotonicity of (27) and (28), so from (24) the following constraints should be satisfied 2k Umax k¨ e(0)k∞ =|ex (0)|| | ≤ √ T 3 2 1 (29) k¨ e(T )k∞ =|ex (0)|| { [1 − exp(−kT )] T T Umax − k exp(−kT )}| ≤ √ 3 Now, we consider the criterion Z +∞ J= ke(t)k∞ dt (30) 0
Theorem 3: When s √ s 6 3|e (0)| 3U x √ max T = T¯ = , k = k¯ = Umax 2 3|ex (0)|
(31)
the criterion in (30) is minimized and the constraints of (29) can be guaranteed. Proof: From (25), (26) and (30), we have Z +∞ Z +∞ sgn(ex (t))ex (t)dt (32) ke(t)k∞ dt = J= 0
T 1 T 1 + ) = |ex (0)|( + ) (34) 3 k 3 k Therefore, we consider the constraints in √(29) and choose x (0)| max T k = 2√U3|e yields J = |ex (0)|( T3 + 2 U3|e ), which max T x (0)| ¯ Then, substituting k¯ has its minimum value for T¯ and k. and T¯ into (29) yields J = ex (0)sgn(ex (0))(
Umax 1 (35) k¨ e(T = T¯)k∞ = √ Umax [1 − 4 exp(−3)] < √ 3 3 3 Thus, k¯ and T¯ yield the minimum value of J under the constraints of (29). B. Constant-velocity sliding mode (CVSM)
ke(t)k∞ =|ex (0)|{
k¨ e(t)k∞
Then, from (20) and (21), we have
0
Lemma 1 implies that ex (t) expressed in (20) and (21) dose not change its sign, that is sgn(ex (t)) = sgn(ex (0)), thus Z T J =sgn(ex (0)) ex (t)dt 0 (33) Z +∞ + sgn(ex (0)) ex (t)dt
as
The Constant-velocity sliding surface function is defined ( At + B, t ≤ T s(t) = e2 (t) + ke1 (t) + (36) 0, t > T
where A ∈ R3 and B ∈ R3 are constant vectors such that 1) The sliding surface passes the initial value of system at the initial moment, that is, at t = 0, e2 (0) + ke1 (0) + B = 0 2) the right-hand side of (36) and its derivative are both continuous at t = T , that is, AT + B = 0. Based on these assumptions, we have A = ke1T(0) = ke(0) T , B = −ke1 (0) = −ke(0). Next, we also give the following adaptive sliding mode control law ( − ke2 (t) − A, t ≤ T u(t) = m0 u0 (t) + m0 (37) − ke2 (t), t > T where u0 (t) is defined in (12). Theorem 4: Given system (4) and (7), if Assumption 1 is satisfied, the sliding mode exist in a given initial state, and the system has to stay on the sliding surface all the time by employing the control law (38) with the adaptive laws (13). Proof: The proof of Theorem 4 is similar to that of Theorem 1 and thus is omitted for brevity. Theorem 5: Given system (4) and (7), if the control law (37) with the adaptive laws (13) is given, the system is globally asymptotically stable. Proof: Based on Theorem 4, the sliding surface (36) exists for any t ∈ [0, +∞), therefore, we have A A B A B − 2 ] exp(−kt) − t − + 2 k k k k k for t ∈ [0, T ], and e(t) = [e(0) +
e(t) = e(T ) exp[−k(t − T )]
(38)
(39)
for t ∈ (T, +∞). Similar with the proof in Theorem 1, we can obtain ∀ e(0), limt→+∞ e(t) = 0. Now, substituting (37) into (38)-(39), we have 1 1 1 exp(−kt) + t + 1 + ] kT T kT
(40)
1 exp[−k(t − T )] − exp(−kt)} kT
(41)
e(t) = e(0)[− for t ∈ [0, T ], and
T
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e(t) = e(0){
for t ∈ (T, +∞). Therefore ke(0) e¨(t) = − exp(−kt) T for t ∈ [0, T ], and
TABLE I: The parameter values for two types control strategies
(42)
Sliding mode types
k
T
l0
l1
K
CASM
0.1
30 √ 10 3
0.01
0.009
0.00001
0.01
0.008
0.00001
ke(0) [exp(kT ) − 1] exp(−kt) (43) T for t ∈ (T, +∞). Then, the constraint in (6) is satisfied if the conditions (24) holds. We also assume that ke(0)k∞ = |ex (0)| = |x(0) − xd (0)|, thus based on Lemma 1, we have e¨(t) =
800
Relative positions / m
600
500
400
300
200
100
0
0
(45)
10
20
30
40 Time / s
50
60
70
80
Fig. 2: CASM: state response signals.
1000
ux
0
(46)
−1000 −2000
Input forces / N
k¨ e(t)k∞
k = |ex (0)| exp(−kt) T
x y z
700
1 1 1 exp(−kt) + t + 1 + ] (44) ke(t)k∞ = |ex (0)|[− kT T kT for t ∈ [0, T ], and k ke(t)k∞ = |ex (0)| [exp(kT ) − 1] exp(−kt) T for t ∈ (T, +∞). Meanwhile,
√ 3 15
CVSM
for t ∈ [0, T ], and k k¨ e(t)k∞ = |ex (0)| [exp(kT ) − 1] exp(−kt) (47) T for t ∈ (T, +∞). Noticing that the monotonicity of (46) and (47), we obtain the following constraints k Umax e(0)k∞ = |ex (0)|| | ≤ √ k¨ T 3 (48) k Umax k¨ e(T )k∞ = |ex (0)|| [1 − exp(−kT )] ≤ √ T 3 Now, we consider the criterion in (30), the following Theorem 6 are used to decide the optimal parameters k and T . Theorem 6: When s √ 2 3|ex (0)| ¯ T = T = Umax s (49) 2Umax ¯ k = k = √ 3|ex (0)| the criterion in (30) is minimized and the constraints of (48) can be guaranteed. Proof: The proof of Theorem 6 is similar to that of Theorem 3 and thus is omitted for brevity. IV. S IMULATION RESULTS In this section, we give an example to illustrate the effectiveness of the above control law design methods. Here, we consider a pair of adjacent spacecraft, and make the following assumptions. The mass of the chaser is 300 kg, and the target is moving along a geosynchronous orbit of radius 42241 km with an orbital period of 24 h. Thus, the angle velocity n = 7.2722×10−5 rads/s. The mass-flow-rate of propellant of the chasers thrusters £is 30 g/s. Assume that ¤T . the initial relative position q1 (0) = 800 600 500 Furthermore, for simplicity, we assume that the initial state £ ¤T . Assume that the input control force q2 (0) = 0 0 0
0
10
20
30
40
50
60
70
80
1000
u
y
0 −1000 −2000
0
10
20
30
40
50
60
70
80
500
u
z
0 −500 −1000
0
10
20
30
40 Time / s
50
60
70
80
Fig. 3: CASM: control input signals.
constraint is U = 4000N . The desired trajectory is q1d (t) = q2d (t) = 0, thus we have e2 (0) = 0. With the assumptions, √ we have Umax = 163 3 . In simulation, we choose δ = 0.1 and the following external disturbance signal: ( 10 sin 0.2t, 0 < t < 30s w(t) = 0, otherwise To prevent the control input signals from chattering, we s(t) s(t) replace ks(t)k with ks(t)k+0.1 . Next, we simulate the dynamic behavior of system (7) controlled according to two types ATVSMC laws presented in this paper. The optimal values of k, T , the adaptive gains l0 , l1 and the gain K are given in Table 1. The state response trajectories of the closed-loop system under two types ATVSMC are depicted in Fig. 2 and Fig. 5, respectively. It can be seen from From Fig. 2 and Fig. 5 that the closed-loop systems are both globally asymptotically stable subject to parameter uncertainties and external disturbances from the initial moment. The control input forces in three axes are depicted in Fig. 3 and Fig. 6, respectively. It can be seen that the largest input force of three axes is bellow the maximum allowed force 4000N . The adaptive parameters trajectories are depicted in Fig. 4 and Fig. 7, and they are all bounded. The simulation results verify that the desired requirements of the closed-loop system have been all achieved, and we can further see that the state convergence rates under CVSM is faster than the state convergence rates under CASM.
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obtained by minimizing the integral of the infinite norm of position tracking error. An illustrative example has shown the effectiveness of the proposed methods.
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Fig. 4: CASM: adaptive parameters.
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Fig. 5: CVSM: state response signals.
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Fig. 6: CVSM: control input signals.
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Fig. 7: CVSM: adaptive parameters.
V. C ONCLUSIONS This paper proposes two types optimal ATVSMC for autonomous spacecraft rendezvous systems subject to input constraint. All two sliding surfaces pass the initial values of system at the initial moment, and then move towards the origin of the error state space. By designing the ATVSMC law, the reaching phase is eliminated, and the robustness against unknown bounded lumped uncertainty is guaranteed from the initial moment. Two parameters of the sliding surfaces are
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