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Robust reliable control for autonomous spacecraft rendezvous with limited-thrust
Aerospace Science and Technology 24 (2013) 161–168
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Robust reliable control for autonomous spacecraft rendezvous with limited-thrust ✩ Xuebo Yang ∗ , Huijun Gao Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150080, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 April 2009 Received in revised form 8 June 2011 Accepted 3 November 2011 Available online 22 November 2011 Keywords: Limited-thrust Reliable control Spacecraft rendezvous
This paper investigates the problem of robust reliable control for the spacecraft rendezvous with limitedthrust. Based on the Clohessy–Wiltshire (C–W) equations and by considering the uncertainties and the possible failures, the dynamic model for spacecraft rendezvous is proposed, and the orbital transfer control problem is transformed into a stabilization problem. Then, by a Lyapunov approach, the existence conditions for admissible controllers are formulated in the form of linear matrix inequalities (LMIs), and the controller design is cast into a convex feasibility problem subject to LMI constraints. With the obtained controllers, the rendezvous can be accomplished with the limited-thrust in spite of the possible thruster failures. The effectiveness of the proposed approach is illustrated by simulation examples. © 2011 Elsevier Masson SAS. All rights reserved.
1. Introduction As is well known, autonomous rendezvous is a crucial phase for many important astronautic missions such as intercepting, repairing, saving, docking, large-scale structure assembling and satellite networking. During the last few decades, the problem of autonomous rendezvous has attracted considerable attention and many results have been reported. For example, the optimal impulsive control method for spacecraft rendezvous is studied in [13, 16,21]; adaptive control theory is applied to the rendezvous and docking problem in [20]; an annealing algorithm method for rendezvous orbital control is proposed in [15]; a new rendezvous guidance method based on sliding-mode control theory can be found in [5]; and in [22], the problem of rendezvous is cast into a stabilization problem analyzed by Lyapunov theory. Although there have been many results in this field, the rendezvous orbital control problem has not been fully investigated and still remains challenging. During the rendezvous, many uncertain factors, such as the inaccuracies of the aero-parameters, the errors of the equipment and the variation of the mass, degrade the safety and the precision of the rendezvous. In particular, due to the errors of the detection equipment and the external perturbations, it is hard to determine ✩ This work was partially supported by the 973 Project (2009CB320600), the National Natural Science Foundation of China (60825303, 60834003, 90916005), the Key Laboratory Open Foundation of HIT (HIT.KLOF.2009099), the Foundation for the Author of National Excellent Doctoral Dissertation of China (2007E4) and the Key Laboratory of Integrated Automation for the Process Industry (Northeastern University). Corresponding author. E-mail addresses: [email protected] (X. Yang), [email protected] (H. Gao).
*
1270-9638/$ – see front matter doi:10.1016/j.ast.2011.11.003
© 2011
Elsevier Masson SAS. All rights reserved.
the accurate angular velocity of the target, which is a very important parameter for the calculation of the control input force. Therefore, guaranteeing robustness for the uncertainties is a challenge in the study of the rendezvous orbital control problem. In recent years, some results have been reported to deal with the uncertainties, see, for instance [9,7,18,19,25]. Nevertheless, for the spacecraft rendezvous problem, the uncertainties are always studied separately from other requirements, and it is necessary to take the design requirements into consideration simultaneously. On the other hand, the autonomic systems are always vulnerable to various failures in practical applications. Due to the complexity of spacecraft rendezvous process, it is hard to completely avoid the failures existing in the thrusters, which have much to do with the safety and accuracy of the rendezvous. Hence, in addition to considering the uncertainties mentioned above, reliability against the possible thruster failures is also a major challenge in the autonomous rendezvous problem. In the last decades, reliable control has attracted many researchers and a number of results have been reported. For example, [23,24] present the reliable controller design methods for linear systems, such that the systems can be stabilized and the performances are ensured in spite of some admissible control component outages; the controller design problem for network with random packet losses or missing measurements is studied in [27–30]; in [32], a pre-compensator is utilized to design a reliable state feedback controller for the system with actuator redundancies. In most of the studies, it is often assumed that the signals of the sensors or the outputs of the actuators become zero when failures occur. This modeling method can simplify the controller synthesis. However, it is significant to adopt a more general model to describe the failures with scaling factors with upper and lower bounds. It is worth noticing that such a
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model is seldom utilized in the studies of reliable controller design problem for spacecraft rendezvous. Besides the uncertainties and the possible failures, it is also necessary to take the constraints of limited-thrust into consideration in the study of spacecraft rendezvous problem. In most of the practical aerospace missions, there are many hard constraints for the weight of the equipment and the quantity of the fuel. Therefore, orbital transfer thrusters of the spacecraft have limited power and limited thrust. In recent years, the problem of orbital transfer with limited-thrust has been studied by many researchers. For instance, a control-theoretic framework for low-thrust orbital transfers using orbital elements is derived in [11]; the problem of aero-assisted or gravity-assisted orbital transfers with limitedthrust is studied in [1,4]; evolutionary programming and nonlinear collocation method have been investigated for the problem of orbital transfer with low-thrust [3,12]. It should be noticed that most of the studies in this field focus on the control methods for sole spacecraft, and these methods are not suitable for spacecraft rendezvous because of the relative motion between the chaser and the target during the rendezvous. Therefore, the rendezvous orbital controller design with limited-thrust is still an important problem to be solved. Motivated by the above discussions, in this paper we study the reliable controller design problem for spacecraft rendezvous with parameter uncertainties and limited-thrust. Based on the Clohessy–Wiltshire (C–W) equations, the uncertain rendezvous models with possible thruster failures are established. The autonomous rendezvous problem is transformed into a stabilization problem for the relative motion system. Then, the reliable state feedback controller design method is developed by a Lyapunov approach. The existence conditions for the admissible reliable controllers, with which the uncertain relative motion system can be stabilized with the limited control input in spite of the failures, are formulated in the form of linear matrix inequalities (LMIs), and the controller design problem is cast into a convex feasibility problem subject to LMI constraints. If the feasibility problem is solvable, the desired controller can be constructed. An illustrative example is provided to show the effectiveness and advantage of the proposed control design method. The rest of this paper is organized as follows. In Section 2, the dynamic model of spacecraft rendezvous is established, and the robust reliable controller design problem is formulated. Section 3 presents controller design method. Then, an example is given to illustrate the applicability of the proposed approach in Section 4. Finally, Section 5 draws the conclusion. Notations: The notation used throughout the paper is fairly standard. The superscript “T ” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space and Rn×m denotes the set of all n × m real matrices; · refers to either the Euclidean vector norm or the induced matrix 2-norm. For a real symmetric matrix W , the notation W > 0 ( W < 0) is used to denote its positive- (negative-)definiteness. diag{. . .} stands for a block-diagonal matrix. For any matrix S, sym{ S } means S + S T . In symmetric block matrices or complex matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry. I and 0 denote the identity matrix and zero matrix with compatible dimensions, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. Problem formulation In this section, based on the C–W equations, the relative motion model is established by considering the norm-bounded parameter uncertainties and the possible thruster failures. Then, the reliable
Fig. 1. Spacecraft rendezvous.
control problem under study in this paper is formulated based on the model. 2.1. Coordinate of the spacecraft rendezvous A basic model for the study of relative motion is given by C–W equations, derived by Clohessy and Wiltshire in 1960 [2]. The model based on C–W equations has been widely used to study the relative motion between two neighboring spacecraft when the target orbit is approximately circular and the distance between them is much smaller than the orbit radius [10,13,15–17,21,31]. The spacecraft rendezvous system is illustrated in Fig. 1. We assume that the two spacecraft (Target and Chaser) are adjacent, and the orbital coordinate frame in our study is a right-handed Cartesian coordinate, with origin attached to the target spacecraft center of mass, x-axis along the vector from Earth’s center to the target’s center of mass, y-axis along the target orbital circumference in the direction of target velocity, and z-axis completing the right-handed frame. According to the C–W equations, the relative dynamic motion between the chaser and the target can be depicted as:
⎧ 1 ⎪ ⎪ x¨ − 2n y˙ − 3n2 x = u x , ⎪ ⎪ m ⎪ ⎨ 1 y¨ + 2nx˙ = u y , ⎪ m ⎪ ⎪ ⎪ ⎪ ⎩ z¨ + n2 z = 1 u z ,
(1)
m
where x, y, z are the components of the relative position, n is the constant angular velocity of the target spacecraft moving around the earth, m is the mass of the chase spacecraft, u i (i = x, y , z) is the ith component of the specific control force acting on the chaser. 2.2. Thruster failure model with uncertainty According to (1), by assuming that the initial orbits of the chaser and the target are coplanar (x = 0, y = 0, z = 0), the state vector can be defined as q(t ) = [x, y , x˙ , y˙ ] T . Then, the whole rendezvous process can be described by the transformation of state vector q(t ) from nonzero initial state q(0) to the terminal state q(tm ) = 0, where tm is the rendezvous time. Furthermore, by defining control input vector u f (t ) = [u x , u y ] T and output vector f(t ) = [x, y ] T , we have
q˙ (t ) = ( A + A )q(t ) + Bu f (t ), f(t ) = C q(t ),
(2)
X. Yang, H. Gao / Aerospace Science and Technology 24 (2013) 161–168
where
⎡
0
0
1 0 0 0 −2n
⎢ 0 0 A=⎢ ⎣ 3n2 0 ⎡
0 1 0
⎤T
⎤
0 1 ⎥ ⎥, 2n ⎦ 0
⎡
0 1 ⎢0 ⎢ B= m ⎣1 0
(2) the needed thrust along each axis below the required bounds, that is u i (tk ) u i ,max (i = x, y ).
⎤
0 0⎥ ⎥, 0⎦ 1
3. Controller design In this section, the robust reliable state-feedback controller design problem will be investigated. The following lemmas will be used in later development. Their proofs and applications can be found in [6,8,14,26].
⎢0 1⎥ ⎥ C =⎢ ⎣0 0⎦ . 0
0
The matrix A in (2) are introduced to illustrate the uncertainties which are norm-bounded as
A = D F (t ) E ,
R y = [0, 1] T [0, 1]. Then, the thrust input constraint can be written as
u x (t ) = R x u f (t ) < u x,max ; 2 y-axis: u y (t ) = R y u f (t )2 < u y ,max .
q˙ (t ) = Aq(t ), f(t ) = C q(t ),
R Φ S + S T Φ T R T ε R V R T + ε −1 S T V S . Lemma 3. Let M and N be real matrices of appropriate dimensions, for any scalar ε > 0,
NMT 0
0 MNT
εN N T 0
0 ε −1 M M T
.
(4) M a0 = diag{ma0x , ma0 y },
(5)
where K a is the actuator failure-tolerant feedback controller which needs to be determined; the failure matrix M a = diag{max , may }, where 0 mali mai (t ) maui < ∞ (i = x, y), mali and maui are known real constraints; mai (t ) (i = x, y) represent the possible failures of the actuator along x- and y-axis of the chaser. Thus, mali = maui = 0 (which means mai (t ) = 0) represents the complete actuator invalidation. Reversely, mali = maui = 1 (that is mai (t ) = 1) represents the case of no actuator failure. Moreover, 0 < mali < maui and mai (t ) = 1 mean that there exists a partial failure in the corresponding thruster. According to (2) and (5), the uncertain closed-loop system with state-feedback controller can be written as
Lemma 2. For a time-varying diagonal matrix Φ(t ) = diag{σ1 (t ), σ2 (t ), . . . , σ p (t )} and two matrices R and S with appropriate dimensions, if |Φ(t )| V , where V is a known diagonal matrix, then for any scalar ε > 0,
For the actuator failure case, we introduce the following matrices which will be used in our later development.
By considering thruster failures, we introduce a state-feedback controller in the form of
u f (t ) = M a K a q(t ),
E Σ F + F T Σ T E T ε −1 E E T + ε F T F .
R x = [1, 0] T [1, 0],
x-axis:
Lemma 1. Let E, F and Σ be matrices of appropriate dimensions with Σ 1. Then, for any scalar ε > 0,
(3)
where D and E are the constant matrices with proper dimension, F (t ) is an unknown real-time varying matrix with Lebesgue measurable elements bounded by F T (t ) F (t ) I . Due to the limited thrust constraint, the control thrust along the x- and y-axis should both have given upper bound, which are defined as u x,max and u y ,max . In order to deal with each condition along every axis, we introduce the following two matrices R x and R y to divide the vector u f (t ) into u x (t ) and u y (t ) for the corresponding axis.
163
(6)
where A = A + A + B K , and K = M a K a . 2.3. Problem of robust reliable controller design According to the analysis above, the problem to be studied in this paper can be formulated as: The relative motion between two spacecraft is described by the uncertain model (2). For the thruster failure case in (5), determine the state-feedback gain matrices K a such that (1) the closed-loop system (2) is asymptotically stable, which means that the rendezvous of the two spacecraft can be accomplished in spite of the model uncertainty and the possible failure cases;
L a = diag{lax , lay }, J a = diag{ jax , jay }, where ma0i = (mali + maui )/2, lai = [mai (t ) − ma0i ]/ma0i and jai = (maui − mali )/(maui + mali ) with i = x, y. Then, we have M a = M a0 ( I + L a ) and L aT L a J aT J a I . Then, according to (6) and the definitions above, the closedloop system with possible thruster failure can be described as
q˙ (t ) = A + A + B M a0 ( I + L a ) K a q(t ).
(7)
As we discussed before, the spacecraft rendezvous process can be regarded as an asymptotic stabilization process. Then, we first study the stability of the system (7) next. Consider the Lyapunov function V (q) = q T P q, where P is a positive symmetric matrix. The derivation of V (q) can be obtained as
V˙ (q) = q T sym P A + D F (t ) E + B M a0 ( I + L a ) K a
q.
As is well known, the asymptotic stability can be ensured by V˙ (q) < 0. Thus, the asymptotic stability can be ensured by the following inequality
sym P A + D F (t ) E + B M a0 ( I + L a ) K a
< 0.
(8)
κ1 > 0 and κ2 > 0, we have sym P A + D F (t ) E + B M a0 ( I + L a ) K a = Υ + sym D F (t ) E + sym B M a0 ( I + La ) K a By Lemma 1 and Lemma 2, for
< Υ + κ1 P D D T P + κ1−1 E T E + κ2 P B F a B T P T + κ2−1 K aT M a0 F a M a0 K a ,
where Υ = sym{ P A + P B M a0 K a }. Then, (9) is satisfied if
(9)
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Υ + κ1 P D D T P + κ1−1 E T E + κ2 P B F a B T P −1
+ κ2
K aT
T M a0 F a M a0 K a
< 0.
(10)
By Schur complement, the inequality in (10) equals to
T ] [ E T K aT M a0 < 0, diag{−κ1 I , −κ2 F a−1 }
Π ∗
(11)
where Π = Υ + κ1 P D D T P + κ2 P B F a B T P . Obviously, the inequality in (11) cannot be dealt with directly. Thus, we make some transformations for (11) next. Define X = P −1 , Y = M a0 K a X . Pre- and post-multiplying (11) by diag{ X , I , I }, the inequality can be transformed as
[X ET Y T ] < 0, diag{−κ1 I , −κ2 F a−1 }
Ψ ∗
(12)
where Ψ = sym{ A X + B Y } + κ1 D D T + κ2 B F a B T . Next, we consider the thrust limitation during the rendezvous process. The thrust constraint described in (4) can be written by
u i (t ) = R i u f (t ) < u i ,max , 2
i = x, y .
(13)
Then, we have [u i (t )]2 = [ R i u f (t )] T [ R i u f (t )] < u 2i ,max , which equals to
q T (t ) K aT M aT R iT R i M a K a q(t ) < u 2i ,max .
(14)
According to the analysis of the asymptotic stabilization, it has been known that V˙ (q) < 0 can be ensured by the condition in (). Thus, we have V (q) < V (0), where V (0) is the value of the energy function for initial relative state between two spacecraft. It is reasonable to introduce a positive scalar ρ satisfying
V (0) = q T (0) P q(0) < ρ .
√
√
ρ Y T R iT + ρ Y T LaT R iT −μi I
(17)
< 0.
(18)
By Lemma 3, for two scalars ε1 > 0 and ε2 > 0, (18) can be ensured by 0 − X + (ε1 + ε2 )ρ I < 0. −1 −1 T T T T T ∗ −μi I + ε1 R i Y Y R i + ε2 R i La Y Y La R i
(19) By Lemma 2, the inequality condition in (19) can be further transformed as
Γ1 + Γ2 + diag Ξ1 , Ξ2 , −ε1 I , −ε2 I , −κ3 F a−1 < 0,
(20)
where
Γ1 = [ 0 I 0 0 0 ]T R i Y [ 0 0 I 0 0 ], Γ2 = [ 0 0 0 I 0 ]T Y T [ 0 0 0 0 I ], Ξ1 = − X + (ε1 + ε2 )ρ I ,
⎡
Ξ2 = −μi I + κ3 R i F a R i .
It can be seen that the inequalities in (12), (15) and (20) are linear matrix inequalities if the scalars ε1 and ε2 are given. Thus,
mali = γ ,
0 0 2 0
⎤
0 0 0 0⎥ ⎥, 0 4⎦ 4 0
⎡
1 0 ⎢ 0 1 E =⎢ ⎣ 0 0.25 −2.25 0
0 0 1 0
⎤
where α is a scalar which denotes the degree of the model uncertainty. For the failure level, we introduce another scalar γ which satisfying
where μi = According to Schur complement and the definition of M a , (17) can be transformed as
−X ∗
In this section, we provide an example to illustrate the usefulness and advantages of the failure-tolerant controller design methods proposed in the above section. Here, we consider a target spacecraft which is moving in a geosynchronous circular orbit with radius r = 42 241 km and its angular velocity is n = 7.2722 × 10−5 rad/s. Assume the mass of the chaser spacecraft m = 200 kg. In the coordinate based on target frame, assume the initial state q(0) = [−1000, −2000, 10, 10] T . According to the derivation of the C–W equations, assume that the model uncertainty of the rendezvous process is given by the following matrices:
(16)
u 2i ,max .
4. Illustrative example
(15)
Then, for (14), (15) and (16), we can see that the thrust constraint can be ensured by (15) and 1 T μ− K a M aT R iT R i M a K a < ρ −1 P , i
Theorem 4. Consider the rendezvous problem between two adjacent spacecraft, whose relative motion can be described as the uncertain model in (2). The possible thruster failure is described as (5) and the maximum allowed thrust along three axes are given by u i ,max (i = x, y ). If there exist proper scalars κ1 > 0, κ2 > 0 and κ3 > 0, a proper matrix Y and a positive symmetric matrix X satisfying the inequalities (12), (15) and (20) simultaneously, then a state-feedback control law in the form of (5) exists, such that the rendezvous can be accomplished in spite of the model uncertainty and the thruster failure, and the needed thrust along each axis below the given upper bounds. The desired control gain is given −1 Y X −1 . by K a = M a0
0 ⎢0 D =α×⎢ ⎣0 2
Thus, we have V (q) < ρ which can be written by
q T (t ) P q(t ) < ρ .
we can solve them by standard software tools and the desired controller can be calculated based on the feasible solution of them. Summarizing above analysis, we give the following theorem as our result of this paper.
0 0⎥ ⎥, 0⎦ 1
maui = 1 + γ .
Then, the failure level can be adjust by changing γ when calculating the controller. In order to simulate the failure case in practice, we assume that the failures occur periodically, and the percentage scalar φ of the signal loss is defined as
φ=
υ, 0%,
ν T < t ν T + ft , ν T + f t < t (ν + 1) T ,
ν = 0, 1, 2, 3 , . . . ,
(21)
where T is a given period, f t is a fraction of T (0 f t < T ), υ is the percentage of the signal loss when failures exist. Then, by (21), the failures condition can be described as: during the front f t seconds in every period T , the failures exist and the percentage of the signal loss is υ ; during the rest of the period, there are no fault and no signal loss occur. We can see that larger f t means longer effective time of the failures and higher υ means more signal loss. Next, to calculate the controller, we assume the uncertainty degree α = 0.0001, the thrust upper bound u i ,max = 50 N and γ = 0.4, which means that mali = 0.4 and maui = 1.6, i = x, y. By solving the LMIs feasibility problem in Theorem 4, we obtain the following associated matrices (for brevity, only the matrices necessary for the construction of the admissible controller are listed):
⎡
⎤
2.7594 −0.3810 −0.0074 −0.0032 ⎢ − 0.3810 7.3128 0.0066 −0.0096 ⎥ ⎥, X = 103 × ⎢ ⎣ −0.0074 0.0066 0.0001 −0.0000 ⎦ −0.0032 −0.0096 −0.0000 0.0001
X. Yang, H. Gao / Aerospace Science and Technology 24 (2013) 161–168
165
Fig. 2. Relative position along x- and y-axis. Fig. 4. Transfer orbit of the chaser spacecraft during the rendezvous process.
Fig. 3. Thrust along x- and y-axis.
Y=
Fig. 5. Relative position along x-axis for different failure cases.
−0.0001 0.0000 −0.0365 −0.0016 . −0.0000 −0.0000 −0.0016 −0.0355
Therefore, the state-feedback gain matrix is calculated as −1 K a = M a0 × Y × X −1
−0.0077 0.0016 −2.8162 −0.3950 . = −0.0036 −0.0023 −0.4218 −1.8901
With the control law u(t ) = K a q(t ), the chaser spacecraft starts rendezvous process from the initial relative state. First, we consider the case without thruster failure. The relative position along x- and y-axis is shown in Fig. 2. The needed thrust along x- and y-axis during the rendezvous process is shown in Fig. 3. From Fig. 2, we can see that the relative position between two spacecraft converges to zero eventually, which means that the rendezvous is accomplished. From Fig. 3, the maximum needed thrust along x- and y-axis are 38.0908 N and 18.6066 N respectively, which are both less than the thrust upper bound 50 N. The trans-
fer orbit of the chaser spacecraft during the rendezvous process is shown in Fig. 4. Next, we consider the failure cases during the rendezvous process. Assume T = 100 s and consider the following two cases:
Case 1:
υ = 20%,
f t = 50 s;
Case 2:
υ = 70%,
f t = 90 s.
To compare the relative position and needed thrust under these two failure cases, we give the following four figures. Fig. 5 and Fig. 6 show the relative position along x- and y-axis, and Fig. 7 and Fig. 8 give the needed thrust along the two axes respectively. Fig. 9 shows the transfer orbits of different cases. We can see that the rendezvous can also be accomplished even there exists serious thruster failure. Of course, the transfer orbit of the chaser spacecraft under serious failure case would be prolonged, and longer rendezvous time is needed. The more consumption of time and orbit range can be acceptable if there is no strict requirements of them.
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X. Yang, H. Gao / Aerospace Science and Technology 24 (2013) 161–168
Fig. 6. Relative position along y-axis for different failure cases. Fig. 8. Thrust along y-axis for different failure cases.
Fig. 7. Thrust along x-axis for different failure cases. Fig. 9. Transfer orbit of the chaser spacecraft for different failure cases.
In addition, we must note that the feasibility of the LMIs in Theorem 4 depends on the degree of the failure and the upper bound of the thrust, that is small γ or u i ,max might cause infeasibility of the LMIs and thus the proper controller cannot be obtained by the proposed approach in this paper. Table 1 and Table 2 give the minimum allowed u i ,max (min u i ,max ) for different given γ and minimum allowed γ (min γ ) for different given u i ,max when solving the feasibility of the LMIs in Theorem 4. From Table 1, we can see that the thrust upper bound can be reduced if the failure degree becomes smaller. However, it can be seen that the minimum thrust upper bound u i ,max is reduced more and more slowly with increasing γ , and it is near to 20 N when γ > 0.9. Thus, in order to ensure that the proper controller can be obtained by solving the LMIs in Theorem 4, the given thrust upper bound should be greater than 20 N, which means that the proposed controller design method requires the minimum thrust of the chaser spacecraft should be near to 20 N. On the other hand, it can be seen from Table 2 that, if the thrusters have higher thrust upper bound, the desired controller can be obtained under
smaller γ . However, it is hard to reduce γ to less than 0.2. That is because the minimum allowed γ is still more than 0.2 even if u i ,max = 200 N, and it cannot be reduced continuously when thrust upper bound keep increasing. Based on Table 1 and Table 2 and according to practical requirements of the rendezvous mission, proper parameters can be chosen to make sure the proper controller can be obtained by solving the LMIs in Theorem 4. With the designed controller, the needed thrust during the practical rendezvous process is smaller than the given upper bound. Next, we assume the given thrust upper bound is 50 N and test the practical needed thrust during the rendezvous process. According to Table 2, we can see the minimum allowed γ is 0.3416 for u i ,max = 50. That is why we choose γ = 0.4 when doing our first simulation. Then, Table 3 lists the maximum needed thrust along x- and y-axis (max T x and max T y ) under γ from 0.4 to 0.9. Next, we change the thrust upper bound. Assume u i ,max = 30 N. According to Table 2, the minimum allowed γ for solving the
X. Yang, H. Gao / Aerospace Science and Technology 24 (2013) 161–168
Table 1 Minimum allowed u i ,max for different given
167
γ when solving the LMIs in Theorem 4.
γ
0.3
0.4
0.5
0.6
0.7
0.8
0.9
min u i ,max
58.0253 N
48.9435 N
28.4890 N
26.9026 N
22.9586 N
21.4412 N
20.0432 N
Table 2 Minimum allowed
γ for different given u i ,max when solving the LMIs in Theorem 4.
u i ,max
20 N
30 N
40 N
50 N
60 N
70 N
80 N
100 N
200 N
min γ
0.9012
0.4799
0.3954
0.3416
0.2781
0.2405
0.2274
0.2004
0.2003
Table 3 Maximum needed thrust along x- and y-axis under different
γ for u i ,max = 50 N.
γ
0.4
0.5
0.6
0.7
0.8
0.9
max T x max T y
20.9398 N 10.6623 N
18.4426 N 9.6391 N
16.5031 N 8.7209 N
15.7301 N 8.3772 N
13.4184 N 7.1330 N
12.3779 N 6.6789 N
Table 4 Maximum needed thrust along x- and y-axis under different
γ for u i ,max = 30 N.
γ
0.5
0.6
0.7
0.8
0.9
max T x max T y
14.9805 N 7.3489 N
13.4970 N 6.9908 N
12.4946 N 6.6465 N
11.4194 N 6.1485 N
11.2666 N 5.9401 N
LMIs in Theorem 4 is 0.4799. Table 4 shows the maximum needed thrust along x- and y-axis under different γ from 0.5 to 0.9. We can see that the practical needed thrust during the rendezvous process is also reduced if the given upper bound is reduced when calculating the controller. However, from Table 3 and Table 4, we can also found that the practical thrust is much smaller than the given upper bound. For instance, the practical needed thrust along x-axis under γ = 0.5 and u i ,max = 50 N is 18.4426 N, which is even smaller than half of the upper bound. And the similar phenomenon also exists in the case of u i ,max = 30 N. This is because the proposed approach has some conservatism during dealing with the matrix inequalities. Thus, reliable controller design method with less conservatism deserve further study in the future. 5. Conclusions This paper has presented a robust reliable control design method for the autonomous spacecraft rendezvous with parameter uncertainties and limited-thrust. Based on the relative motion model established by C–W equations, and by considering the possible thruster failure of the chaser spacecraft, the problem of rendezvous has been transformed into a stabilization problem of the closed-loop system. Then, by using Lyapunov method, the controller design problems have been further transformed into the convex optimization problems with linear matrix inequality constraints. The designed controllers can stabilize the uncertain systems by limited input in spite of the possible failures. An illustrative example has shown the effectiveness of the proposed controller design method. References [1] H. Baumann, Thrust limited coplanar aeroassisted orbital transfer, Journal of Guidance Control and Dynamics 24 (4) (2001) 732–738. [2] W.H. Clohessy, R.S. Wiltshire, Terminal guidance system for satellite rendezvous, Journal of Aerospace Science 27 (9) (1960) 653–658. [3] V. Coverstone-Carrol, Near-optimal low-thrust trajectories via micro-genetic algorithms, Journal of Guidance, Control, and Dynamics 20 (1) (1997) 196–198. [4] T.J. Debban, T.T. McConaghy, J.M. Longuski, Design and optimization of lowthrust gravity-assist trajectories to selected planets, in: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 5–8 August 2002, Monterey, California, 2002.
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Contents lists available at SciVerse ScienceDirect
Aerospace Science and Technology www.elsevier.com/locate/aescte
Robust reliable control for autonomous spacecraft rendezvous with limited-thrust ✩ Xuebo Yang ∗ , Huijun Gao Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150080, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 April 2009 Received in revised form 8 June 2011 Accepted 3 November 2011 Available online 22 November 2011 Keywords: Limited-thrust Reliable control Spacecraft rendezvous
This paper investigates the problem of robust reliable control for the spacecraft rendezvous with limitedthrust. Based on the Clohessy–Wiltshire (C–W) equations and by considering the uncertainties and the possible failures, the dynamic model for spacecraft rendezvous is proposed, and the orbital transfer control problem is transformed into a stabilization problem. Then, by a Lyapunov approach, the existence conditions for admissible controllers are formulated in the form of linear matrix inequalities (LMIs), and the controller design is cast into a convex feasibility problem subject to LMI constraints. With the obtained controllers, the rendezvous can be accomplished with the limited-thrust in spite of the possible thruster failures. The effectiveness of the proposed approach is illustrated by simulation examples. © 2011 Elsevier Masson SAS. All rights reserved.
1. Introduction As is well known, autonomous rendezvous is a crucial phase for many important astronautic missions such as intercepting, repairing, saving, docking, large-scale structure assembling and satellite networking. During the last few decades, the problem of autonomous rendezvous has attracted considerable attention and many results have been reported. For example, the optimal impulsive control method for spacecraft rendezvous is studied in [13, 16,21]; adaptive control theory is applied to the rendezvous and docking problem in [20]; an annealing algorithm method for rendezvous orbital control is proposed in [15]; a new rendezvous guidance method based on sliding-mode control theory can be found in [5]; and in [22], the problem of rendezvous is cast into a stabilization problem analyzed by Lyapunov theory. Although there have been many results in this field, the rendezvous orbital control problem has not been fully investigated and still remains challenging. During the rendezvous, many uncertain factors, such as the inaccuracies of the aero-parameters, the errors of the equipment and the variation of the mass, degrade the safety and the precision of the rendezvous. In particular, due to the errors of the detection equipment and the external perturbations, it is hard to determine ✩ This work was partially supported by the 973 Project (2009CB320600), the National Natural Science Foundation of China (60825303, 60834003, 90916005), the Key Laboratory Open Foundation of HIT (HIT.KLOF.2009099), the Foundation for the Author of National Excellent Doctoral Dissertation of China (2007E4) and the Key Laboratory of Integrated Automation for the Process Industry (Northeastern University). Corresponding author. E-mail addresses: [email protected] (X. Yang), [email protected] (H. Gao).
*
1270-9638/$ – see front matter doi:10.1016/j.ast.2011.11.003
© 2011
Elsevier Masson SAS. All rights reserved.
the accurate angular velocity of the target, which is a very important parameter for the calculation of the control input force. Therefore, guaranteeing robustness for the uncertainties is a challenge in the study of the rendezvous orbital control problem. In recent years, some results have been reported to deal with the uncertainties, see, for instance [9,7,18,19,25]. Nevertheless, for the spacecraft rendezvous problem, the uncertainties are always studied separately from other requirements, and it is necessary to take the design requirements into consideration simultaneously. On the other hand, the autonomic systems are always vulnerable to various failures in practical applications. Due to the complexity of spacecraft rendezvous process, it is hard to completely avoid the failures existing in the thrusters, which have much to do with the safety and accuracy of the rendezvous. Hence, in addition to considering the uncertainties mentioned above, reliability against the possible thruster failures is also a major challenge in the autonomous rendezvous problem. In the last decades, reliable control has attracted many researchers and a number of results have been reported. For example, [23,24] present the reliable controller design methods for linear systems, such that the systems can be stabilized and the performances are ensured in spite of some admissible control component outages; the controller design problem for network with random packet losses or missing measurements is studied in [27–30]; in [32], a pre-compensator is utilized to design a reliable state feedback controller for the system with actuator redundancies. In most of the studies, it is often assumed that the signals of the sensors or the outputs of the actuators become zero when failures occur. This modeling method can simplify the controller synthesis. However, it is significant to adopt a more general model to describe the failures with scaling factors with upper and lower bounds. It is worth noticing that such a
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model is seldom utilized in the studies of reliable controller design problem for spacecraft rendezvous. Besides the uncertainties and the possible failures, it is also necessary to take the constraints of limited-thrust into consideration in the study of spacecraft rendezvous problem. In most of the practical aerospace missions, there are many hard constraints for the weight of the equipment and the quantity of the fuel. Therefore, orbital transfer thrusters of the spacecraft have limited power and limited thrust. In recent years, the problem of orbital transfer with limited-thrust has been studied by many researchers. For instance, a control-theoretic framework for low-thrust orbital transfers using orbital elements is derived in [11]; the problem of aero-assisted or gravity-assisted orbital transfers with limitedthrust is studied in [1,4]; evolutionary programming and nonlinear collocation method have been investigated for the problem of orbital transfer with low-thrust [3,12]. It should be noticed that most of the studies in this field focus on the control methods for sole spacecraft, and these methods are not suitable for spacecraft rendezvous because of the relative motion between the chaser and the target during the rendezvous. Therefore, the rendezvous orbital controller design with limited-thrust is still an important problem to be solved. Motivated by the above discussions, in this paper we study the reliable controller design problem for spacecraft rendezvous with parameter uncertainties and limited-thrust. Based on the Clohessy–Wiltshire (C–W) equations, the uncertain rendezvous models with possible thruster failures are established. The autonomous rendezvous problem is transformed into a stabilization problem for the relative motion system. Then, the reliable state feedback controller design method is developed by a Lyapunov approach. The existence conditions for the admissible reliable controllers, with which the uncertain relative motion system can be stabilized with the limited control input in spite of the failures, are formulated in the form of linear matrix inequalities (LMIs), and the controller design problem is cast into a convex feasibility problem subject to LMI constraints. If the feasibility problem is solvable, the desired controller can be constructed. An illustrative example is provided to show the effectiveness and advantage of the proposed control design method. The rest of this paper is organized as follows. In Section 2, the dynamic model of spacecraft rendezvous is established, and the robust reliable controller design problem is formulated. Section 3 presents controller design method. Then, an example is given to illustrate the applicability of the proposed approach in Section 4. Finally, Section 5 draws the conclusion. Notations: The notation used throughout the paper is fairly standard. The superscript “T ” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space and Rn×m denotes the set of all n × m real matrices; · refers to either the Euclidean vector norm or the induced matrix 2-norm. For a real symmetric matrix W , the notation W > 0 ( W < 0) is used to denote its positive- (negative-)definiteness. diag{. . .} stands for a block-diagonal matrix. For any matrix S, sym{ S } means S + S T . In symmetric block matrices or complex matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry. I and 0 denote the identity matrix and zero matrix with compatible dimensions, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. Problem formulation In this section, based on the C–W equations, the relative motion model is established by considering the norm-bounded parameter uncertainties and the possible thruster failures. Then, the reliable
Fig. 1. Spacecraft rendezvous.
control problem under study in this paper is formulated based on the model. 2.1. Coordinate of the spacecraft rendezvous A basic model for the study of relative motion is given by C–W equations, derived by Clohessy and Wiltshire in 1960 [2]. The model based on C–W equations has been widely used to study the relative motion between two neighboring spacecraft when the target orbit is approximately circular and the distance between them is much smaller than the orbit radius [10,13,15–17,21,31]. The spacecraft rendezvous system is illustrated in Fig. 1. We assume that the two spacecraft (Target and Chaser) are adjacent, and the orbital coordinate frame in our study is a right-handed Cartesian coordinate, with origin attached to the target spacecraft center of mass, x-axis along the vector from Earth’s center to the target’s center of mass, y-axis along the target orbital circumference in the direction of target velocity, and z-axis completing the right-handed frame. According to the C–W equations, the relative dynamic motion between the chaser and the target can be depicted as:
⎧ 1 ⎪ ⎪ x¨ − 2n y˙ − 3n2 x = u x , ⎪ ⎪ m ⎪ ⎨ 1 y¨ + 2nx˙ = u y , ⎪ m ⎪ ⎪ ⎪ ⎪ ⎩ z¨ + n2 z = 1 u z ,
(1)
m
where x, y, z are the components of the relative position, n is the constant angular velocity of the target spacecraft moving around the earth, m is the mass of the chase spacecraft, u i (i = x, y , z) is the ith component of the specific control force acting on the chaser. 2.2. Thruster failure model with uncertainty According to (1), by assuming that the initial orbits of the chaser and the target are coplanar (x = 0, y = 0, z = 0), the state vector can be defined as q(t ) = [x, y , x˙ , y˙ ] T . Then, the whole rendezvous process can be described by the transformation of state vector q(t ) from nonzero initial state q(0) to the terminal state q(tm ) = 0, where tm is the rendezvous time. Furthermore, by defining control input vector u f (t ) = [u x , u y ] T and output vector f(t ) = [x, y ] T , we have
q˙ (t ) = ( A + A )q(t ) + Bu f (t ), f(t ) = C q(t ),
(2)
X. Yang, H. Gao / Aerospace Science and Technology 24 (2013) 161–168
where
⎡
0
0
1 0 0 0 −2n
⎢ 0 0 A=⎢ ⎣ 3n2 0 ⎡
0 1 0
⎤T
⎤
0 1 ⎥ ⎥, 2n ⎦ 0
⎡
0 1 ⎢0 ⎢ B= m ⎣1 0
(2) the needed thrust along each axis below the required bounds, that is u i (tk ) u i ,max (i = x, y ).
⎤
0 0⎥ ⎥, 0⎦ 1
3. Controller design In this section, the robust reliable state-feedback controller design problem will be investigated. The following lemmas will be used in later development. Their proofs and applications can be found in [6,8,14,26].
⎢0 1⎥ ⎥ C =⎢ ⎣0 0⎦ . 0
0
The matrix A in (2) are introduced to illustrate the uncertainties which are norm-bounded as
A = D F (t ) E ,
R y = [0, 1] T [0, 1]. Then, the thrust input constraint can be written as
u x (t ) = R x u f (t ) < u x,max ; 2 y-axis: u y (t ) = R y u f (t )2 < u y ,max .
q˙ (t ) = Aq(t ), f(t ) = C q(t ),
R Φ S + S T Φ T R T ε R V R T + ε −1 S T V S . Lemma 3. Let M and N be real matrices of appropriate dimensions, for any scalar ε > 0,
NMT 0
0 MNT
εN N T 0
0 ε −1 M M T
.
(4) M a0 = diag{ma0x , ma0 y },
(5)
where K a is the actuator failure-tolerant feedback controller which needs to be determined; the failure matrix M a = diag{max , may }, where 0 mali mai (t ) maui < ∞ (i = x, y), mali and maui are known real constraints; mai (t ) (i = x, y) represent the possible failures of the actuator along x- and y-axis of the chaser. Thus, mali = maui = 0 (which means mai (t ) = 0) represents the complete actuator invalidation. Reversely, mali = maui = 1 (that is mai (t ) = 1) represents the case of no actuator failure. Moreover, 0 < mali < maui and mai (t ) = 1 mean that there exists a partial failure in the corresponding thruster. According to (2) and (5), the uncertain closed-loop system with state-feedback controller can be written as
Lemma 2. For a time-varying diagonal matrix Φ(t ) = diag{σ1 (t ), σ2 (t ), . . . , σ p (t )} and two matrices R and S with appropriate dimensions, if |Φ(t )| V , where V is a known diagonal matrix, then for any scalar ε > 0,
For the actuator failure case, we introduce the following matrices which will be used in our later development.
By considering thruster failures, we introduce a state-feedback controller in the form of
u f (t ) = M a K a q(t ),
E Σ F + F T Σ T E T ε −1 E E T + ε F T F .
R x = [1, 0] T [1, 0],
x-axis:
Lemma 1. Let E, F and Σ be matrices of appropriate dimensions with Σ 1. Then, for any scalar ε > 0,
(3)
where D and E are the constant matrices with proper dimension, F (t ) is an unknown real-time varying matrix with Lebesgue measurable elements bounded by F T (t ) F (t ) I . Due to the limited thrust constraint, the control thrust along the x- and y-axis should both have given upper bound, which are defined as u x,max and u y ,max . In order to deal with each condition along every axis, we introduce the following two matrices R x and R y to divide the vector u f (t ) into u x (t ) and u y (t ) for the corresponding axis.
163
(6)
where A = A + A + B K , and K = M a K a . 2.3. Problem of robust reliable controller design According to the analysis above, the problem to be studied in this paper can be formulated as: The relative motion between two spacecraft is described by the uncertain model (2). For the thruster failure case in (5), determine the state-feedback gain matrices K a such that (1) the closed-loop system (2) is asymptotically stable, which means that the rendezvous of the two spacecraft can be accomplished in spite of the model uncertainty and the possible failure cases;
L a = diag{lax , lay }, J a = diag{ jax , jay }, where ma0i = (mali + maui )/2, lai = [mai (t ) − ma0i ]/ma0i and jai = (maui − mali )/(maui + mali ) with i = x, y. Then, we have M a = M a0 ( I + L a ) and L aT L a J aT J a I . Then, according to (6) and the definitions above, the closedloop system with possible thruster failure can be described as
q˙ (t ) = A + A + B M a0 ( I + L a ) K a q(t ).
(7)
As we discussed before, the spacecraft rendezvous process can be regarded as an asymptotic stabilization process. Then, we first study the stability of the system (7) next. Consider the Lyapunov function V (q) = q T P q, where P is a positive symmetric matrix. The derivation of V (q) can be obtained as
V˙ (q) = q T sym P A + D F (t ) E + B M a0 ( I + L a ) K a
q.
As is well known, the asymptotic stability can be ensured by V˙ (q) < 0. Thus, the asymptotic stability can be ensured by the following inequality
sym P A + D F (t ) E + B M a0 ( I + L a ) K a
< 0.
(8)
κ1 > 0 and κ2 > 0, we have sym P A + D F (t ) E + B M a0 ( I + L a ) K a = Υ + sym D F (t ) E + sym B M a0 ( I + La ) K a By Lemma 1 and Lemma 2, for
< Υ + κ1 P D D T P + κ1−1 E T E + κ2 P B F a B T P T + κ2−1 K aT M a0 F a M a0 K a ,
where Υ = sym{ P A + P B M a0 K a }. Then, (9) is satisfied if
(9)
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Υ + κ1 P D D T P + κ1−1 E T E + κ2 P B F a B T P −1
+ κ2
K aT
T M a0 F a M a0 K a
< 0.
(10)
By Schur complement, the inequality in (10) equals to
T ] [ E T K aT M a0 < 0, diag{−κ1 I , −κ2 F a−1 }
Π ∗
(11)
where Π = Υ + κ1 P D D T P + κ2 P B F a B T P . Obviously, the inequality in (11) cannot be dealt with directly. Thus, we make some transformations for (11) next. Define X = P −1 , Y = M a0 K a X . Pre- and post-multiplying (11) by diag{ X , I , I }, the inequality can be transformed as
[X ET Y T ] < 0, diag{−κ1 I , −κ2 F a−1 }
Ψ ∗
(12)
where Ψ = sym{ A X + B Y } + κ1 D D T + κ2 B F a B T . Next, we consider the thrust limitation during the rendezvous process. The thrust constraint described in (4) can be written by
u i (t ) = R i u f (t ) < u i ,max , 2
i = x, y .
(13)
Then, we have [u i (t )]2 = [ R i u f (t )] T [ R i u f (t )] < u 2i ,max , which equals to
q T (t ) K aT M aT R iT R i M a K a q(t ) < u 2i ,max .
(14)
According to the analysis of the asymptotic stabilization, it has been known that V˙ (q) < 0 can be ensured by the condition in (). Thus, we have V (q) < V (0), where V (0) is the value of the energy function for initial relative state between two spacecraft. It is reasonable to introduce a positive scalar ρ satisfying
V (0) = q T (0) P q(0) < ρ .
√
√
ρ Y T R iT + ρ Y T LaT R iT −μi I
(17)
< 0.
(18)
By Lemma 3, for two scalars ε1 > 0 and ε2 > 0, (18) can be ensured by 0 − X + (ε1 + ε2 )ρ I < 0. −1 −1 T T T T T ∗ −μi I + ε1 R i Y Y R i + ε2 R i La Y Y La R i
(19) By Lemma 2, the inequality condition in (19) can be further transformed as
Γ1 + Γ2 + diag Ξ1 , Ξ2 , −ε1 I , −ε2 I , −κ3 F a−1 < 0,
(20)
where
Γ1 = [ 0 I 0 0 0 ]T R i Y [ 0 0 I 0 0 ], Γ2 = [ 0 0 0 I 0 ]T Y T [ 0 0 0 0 I ], Ξ1 = − X + (ε1 + ε2 )ρ I ,
⎡
Ξ2 = −μi I + κ3 R i F a R i .
It can be seen that the inequalities in (12), (15) and (20) are linear matrix inequalities if the scalars ε1 and ε2 are given. Thus,
mali = γ ,
0 0 2 0
⎤
0 0 0 0⎥ ⎥, 0 4⎦ 4 0
⎡
1 0 ⎢ 0 1 E =⎢ ⎣ 0 0.25 −2.25 0
0 0 1 0
⎤
where α is a scalar which denotes the degree of the model uncertainty. For the failure level, we introduce another scalar γ which satisfying
where μi = According to Schur complement and the definition of M a , (17) can be transformed as
−X ∗
In this section, we provide an example to illustrate the usefulness and advantages of the failure-tolerant controller design methods proposed in the above section. Here, we consider a target spacecraft which is moving in a geosynchronous circular orbit with radius r = 42 241 km and its angular velocity is n = 7.2722 × 10−5 rad/s. Assume the mass of the chaser spacecraft m = 200 kg. In the coordinate based on target frame, assume the initial state q(0) = [−1000, −2000, 10, 10] T . According to the derivation of the C–W equations, assume that the model uncertainty of the rendezvous process is given by the following matrices:
(16)
u 2i ,max .
4. Illustrative example
(15)
Then, for (14), (15) and (16), we can see that the thrust constraint can be ensured by (15) and 1 T μ− K a M aT R iT R i M a K a < ρ −1 P , i
Theorem 4. Consider the rendezvous problem between two adjacent spacecraft, whose relative motion can be described as the uncertain model in (2). The possible thruster failure is described as (5) and the maximum allowed thrust along three axes are given by u i ,max (i = x, y ). If there exist proper scalars κ1 > 0, κ2 > 0 and κ3 > 0, a proper matrix Y and a positive symmetric matrix X satisfying the inequalities (12), (15) and (20) simultaneously, then a state-feedback control law in the form of (5) exists, such that the rendezvous can be accomplished in spite of the model uncertainty and the thruster failure, and the needed thrust along each axis below the given upper bounds. The desired control gain is given −1 Y X −1 . by K a = M a0
0 ⎢0 D =α×⎢ ⎣0 2
Thus, we have V (q) < ρ which can be written by
q T (t ) P q(t ) < ρ .
we can solve them by standard software tools and the desired controller can be calculated based on the feasible solution of them. Summarizing above analysis, we give the following theorem as our result of this paper.
0 0⎥ ⎥, 0⎦ 1
maui = 1 + γ .
Then, the failure level can be adjust by changing γ when calculating the controller. In order to simulate the failure case in practice, we assume that the failures occur periodically, and the percentage scalar φ of the signal loss is defined as
φ=
υ, 0%,
ν T < t ν T + ft , ν T + f t < t (ν + 1) T ,
ν = 0, 1, 2, 3 , . . . ,
(21)
where T is a given period, f t is a fraction of T (0 f t < T ), υ is the percentage of the signal loss when failures exist. Then, by (21), the failures condition can be described as: during the front f t seconds in every period T , the failures exist and the percentage of the signal loss is υ ; during the rest of the period, there are no fault and no signal loss occur. We can see that larger f t means longer effective time of the failures and higher υ means more signal loss. Next, to calculate the controller, we assume the uncertainty degree α = 0.0001, the thrust upper bound u i ,max = 50 N and γ = 0.4, which means that mali = 0.4 and maui = 1.6, i = x, y. By solving the LMIs feasibility problem in Theorem 4, we obtain the following associated matrices (for brevity, only the matrices necessary for the construction of the admissible controller are listed):
⎡
⎤
2.7594 −0.3810 −0.0074 −0.0032 ⎢ − 0.3810 7.3128 0.0066 −0.0096 ⎥ ⎥, X = 103 × ⎢ ⎣ −0.0074 0.0066 0.0001 −0.0000 ⎦ −0.0032 −0.0096 −0.0000 0.0001
X. Yang, H. Gao / Aerospace Science and Technology 24 (2013) 161–168
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Fig. 2. Relative position along x- and y-axis. Fig. 4. Transfer orbit of the chaser spacecraft during the rendezvous process.
Fig. 3. Thrust along x- and y-axis.
Y=
Fig. 5. Relative position along x-axis for different failure cases.
−0.0001 0.0000 −0.0365 −0.0016 . −0.0000 −0.0000 −0.0016 −0.0355
Therefore, the state-feedback gain matrix is calculated as −1 K a = M a0 × Y × X −1
−0.0077 0.0016 −2.8162 −0.3950 . = −0.0036 −0.0023 −0.4218 −1.8901
With the control law u(t ) = K a q(t ), the chaser spacecraft starts rendezvous process from the initial relative state. First, we consider the case without thruster failure. The relative position along x- and y-axis is shown in Fig. 2. The needed thrust along x- and y-axis during the rendezvous process is shown in Fig. 3. From Fig. 2, we can see that the relative position between two spacecraft converges to zero eventually, which means that the rendezvous is accomplished. From Fig. 3, the maximum needed thrust along x- and y-axis are 38.0908 N and 18.6066 N respectively, which are both less than the thrust upper bound 50 N. The trans-
fer orbit of the chaser spacecraft during the rendezvous process is shown in Fig. 4. Next, we consider the failure cases during the rendezvous process. Assume T = 100 s and consider the following two cases:
Case 1:
υ = 20%,
f t = 50 s;
Case 2:
υ = 70%,
f t = 90 s.
To compare the relative position and needed thrust under these two failure cases, we give the following four figures. Fig. 5 and Fig. 6 show the relative position along x- and y-axis, and Fig. 7 and Fig. 8 give the needed thrust along the two axes respectively. Fig. 9 shows the transfer orbits of different cases. We can see that the rendezvous can also be accomplished even there exists serious thruster failure. Of course, the transfer orbit of the chaser spacecraft under serious failure case would be prolonged, and longer rendezvous time is needed. The more consumption of time and orbit range can be acceptable if there is no strict requirements of them.
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Fig. 6. Relative position along y-axis for different failure cases. Fig. 8. Thrust along y-axis for different failure cases.
Fig. 7. Thrust along x-axis for different failure cases. Fig. 9. Transfer orbit of the chaser spacecraft for different failure cases.
In addition, we must note that the feasibility of the LMIs in Theorem 4 depends on the degree of the failure and the upper bound of the thrust, that is small γ or u i ,max might cause infeasibility of the LMIs and thus the proper controller cannot be obtained by the proposed approach in this paper. Table 1 and Table 2 give the minimum allowed u i ,max (min u i ,max ) for different given γ and minimum allowed γ (min γ ) for different given u i ,max when solving the feasibility of the LMIs in Theorem 4. From Table 1, we can see that the thrust upper bound can be reduced if the failure degree becomes smaller. However, it can be seen that the minimum thrust upper bound u i ,max is reduced more and more slowly with increasing γ , and it is near to 20 N when γ > 0.9. Thus, in order to ensure that the proper controller can be obtained by solving the LMIs in Theorem 4, the given thrust upper bound should be greater than 20 N, which means that the proposed controller design method requires the minimum thrust of the chaser spacecraft should be near to 20 N. On the other hand, it can be seen from Table 2 that, if the thrusters have higher thrust upper bound, the desired controller can be obtained under
smaller γ . However, it is hard to reduce γ to less than 0.2. That is because the minimum allowed γ is still more than 0.2 even if u i ,max = 200 N, and it cannot be reduced continuously when thrust upper bound keep increasing. Based on Table 1 and Table 2 and according to practical requirements of the rendezvous mission, proper parameters can be chosen to make sure the proper controller can be obtained by solving the LMIs in Theorem 4. With the designed controller, the needed thrust during the practical rendezvous process is smaller than the given upper bound. Next, we assume the given thrust upper bound is 50 N and test the practical needed thrust during the rendezvous process. According to Table 2, we can see the minimum allowed γ is 0.3416 for u i ,max = 50. That is why we choose γ = 0.4 when doing our first simulation. Then, Table 3 lists the maximum needed thrust along x- and y-axis (max T x and max T y ) under γ from 0.4 to 0.9. Next, we change the thrust upper bound. Assume u i ,max = 30 N. According to Table 2, the minimum allowed γ for solving the
X. Yang, H. Gao / Aerospace Science and Technology 24 (2013) 161–168
Table 1 Minimum allowed u i ,max for different given
167
γ when solving the LMIs in Theorem 4.
γ
0.3
0.4
0.5
0.6
0.7
0.8
0.9
min u i ,max
58.0253 N
48.9435 N
28.4890 N
26.9026 N
22.9586 N
21.4412 N
20.0432 N
Table 2 Minimum allowed
γ for different given u i ,max when solving the LMIs in Theorem 4.
u i ,max
20 N
30 N
40 N
50 N
60 N
70 N
80 N
100 N
200 N
min γ
0.9012
0.4799
0.3954
0.3416
0.2781
0.2405
0.2274
0.2004
0.2003
Table 3 Maximum needed thrust along x- and y-axis under different
γ for u i ,max = 50 N.
γ
0.4
0.5
0.6
0.7
0.8
0.9
max T x max T y
20.9398 N 10.6623 N
18.4426 N 9.6391 N
16.5031 N 8.7209 N
15.7301 N 8.3772 N
13.4184 N 7.1330 N
12.3779 N 6.6789 N
Table 4 Maximum needed thrust along x- and y-axis under different
γ for u i ,max = 30 N.
γ
0.5
0.6
0.7
0.8
0.9
max T x max T y
14.9805 N 7.3489 N
13.4970 N 6.9908 N
12.4946 N 6.6465 N
11.4194 N 6.1485 N
11.2666 N 5.9401 N
LMIs in Theorem 4 is 0.4799. Table 4 shows the maximum needed thrust along x- and y-axis under different γ from 0.5 to 0.9. We can see that the practical needed thrust during the rendezvous process is also reduced if the given upper bound is reduced when calculating the controller. However, from Table 3 and Table 4, we can also found that the practical thrust is much smaller than the given upper bound. For instance, the practical needed thrust along x-axis under γ = 0.5 and u i ,max = 50 N is 18.4426 N, which is even smaller than half of the upper bound. And the similar phenomenon also exists in the case of u i ,max = 30 N. This is because the proposed approach has some conservatism during dealing with the matrix inequalities. Thus, reliable controller design method with less conservatism deserve further study in the future. 5. Conclusions This paper has presented a robust reliable control design method for the autonomous spacecraft rendezvous with parameter uncertainties and limited-thrust. Based on the relative motion model established by C–W equations, and by considering the possible thruster failure of the chaser spacecraft, the problem of rendezvous has been transformed into a stabilization problem of the closed-loop system. Then, by using Lyapunov method, the controller design problems have been further transformed into the convex optimization problems with linear matrix inequality constraints. The designed controllers can stabilize the uncertain systems by limited input in spite of the possible failures. An illustrative example has shown the effectiveness of the proposed controller design method. References [1] H. Baumann, Thrust limited coplanar aeroassisted orbital transfer, Journal of Guidance Control and Dynamics 24 (4) (2001) 732–738. [2] W.H. Clohessy, R.S. Wiltshire, Terminal guidance system for satellite rendezvous, Journal of Aerospace Science 27 (9) (1960) 653–658. [3] V. Coverstone-Carrol, Near-optimal low-thrust trajectories via micro-genetic algorithms, Journal of Guidance, Control, and Dynamics 20 (1) (1997) 196–198. [4] T.J. Debban, T.T. McConaghy, J.M. Longuski, Design and optimization of lowthrust gravity-assist trajectories to selected planets, in: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 5–8 August 2002, Monterey, California, 2002.
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