Robust orbital transfer for low earth orbit spacecraft with small-thrust

Journal of the Franklin Institute 347 (2010) 1863–1887 www.elsevier.com/locate/jfranklin

Robust orbital transfer for low earth orbit spacecraft with small-thrust$ Xuebo Yanga, Huijun Gaoa,, Peng Shib,1 a

Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, China b Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd CF37 1DL, United Kingdom Received 17 July 2010; received in revised form 21 September 2010; accepted 11 October 2010

Abstract This paper studies the problem of robust orbital control for low earth orbit (LEO) spacecraft rendezvous subjects to the parameter uncertainties, the constraints of small-thrust and guaranteed cost during the orbital transfer process. In particular, the rendezvous process is divided into in-plane motion and out-plane motion based on C–W equations, and the relative motion models with parameter uncertainties are established. By considering the property of null controllable with vanishing energy (NCVE), the problem of orbital transfer control with small thrust and bounded control cost is proposed. A new Lyapunov approach is introduced, and the controller design problem is cast into a convex optimization problem subjects to linear matrix inequality (LMI) constraints. With the obtained controller, the orbit transfer process can be accomplished with small thrust and the control cost has an upper bound simultaneously. Different possible initial states of the transfer orbit are also analyzed for the controller design. An illustrative example is provided to show the effectiveness of the proposed control design method, and the different performances caused by different initial states of the transfer orbit are illustrated. & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

$ This work was partially supported by the National Science Foundation for Distinguished Young Scholars of China (60825303), the Doctoral Program of Higher Education of China (20070213084), National Natural Science Foundation of China (90916005), National Natural Science Foundation of China (60804011). Corresponding author. E-mail addresses: [email protected] (X. Yang), [email protected], [email protected] (H. Gao), [email protected] (P. Shi). 1 He is also with the School of Engineering and Science, Victoria University, Melbourne Vic 8001, Australia.

0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2010.10.006

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1. Introduction Autonomous rendezvous of low earth orbit (LEO) spacecraft has been well recognized as an essential problem for many advanced aerospace missions such as inspecting, repairing, saving, large-scale structure assembling and formation flying. During the last few decades, the problem has attracted considerable attentions and many results have been reported, see for example, [4,10,14,15,23,24], and the references therein. However, some correlative issues have not yet been fully investigated because of their complexity, and many existing studies have left considerable room for improvement. Generally, in order to ensure the safety of the spacecraft docking, there should be an orbital transfer process before the docking action, whose goal is to transfer the chaser into a coplanar vicinity orbit of the target. The relative motion during this process should be considered carefully and the orbital accuracy is very important because the chaser is approaching to the target gradually during the orbital transfer and the two spacecraft will move closer after the process. Thus, it is a big challenge to utilize some advanced closed-loop control laws to enhance the orbital control accuracy, which leads to the traditional open-loop control methods are not applicable while they are often utilized during the long-distance navigation process. In the past decades, the relative motion problems for the LEO spacecraft are often studied based on C–W equations, which were derived by Clohessey and Wiltshire in 1960 [3]. Usually, _ ¼ AxðtÞ þ BuðtÞ, the equations are transformed into a general state function in terms of xðtÞ where x(t) is the relative position and velocity states vector and u(t) is the control input vector. This description has been used widely to study the spacecraft rendezvous problems [7,10,14,15,24]. Nevertheless, it should be noticed that the state matrix A and the control input matrix B are difficult to be determined accurately because of many uncertain factors. In particular, the elements of matrix A depend heavily on the angle velocity of the target spacecraft, which is subjected to many inevitable affections such as the continuous external perturbations and the errors of detection. On the other hand, the possible mass variation or the errors of the thrusters may bring the inaccuracy of the control input, which can be regarded as the uncertainty of the input matrix B. These uncertainties may degrade the precision, the stability or even the safety of the rendezvous missions. Recent years, there have been many studies on the problems of uncertain systems, see, for instance [1,2,6,20,21,25,28]. However, the parameter uncertainties have not attracted enough attentions in the existing studies on spacecraft rendezvous control problems. This leads to our desire to take these uncertainties into consideration and find a proper method to deal with them. The energy consumption and the thrust constraint are also important issues for spacecraft rendezvous. Recent years, many studies on the correlative problems have been reported [8,12,13,16,17,19,26]. It is worth noting that the relative motion between two spacecrafts, which can be described by C–W equations, has a property of null controllability with vanishing energy (NCVE) [9,22]. This is a property for linear system that any states of the system can be steered to the origin with an arbitrary small amount of control energy in the L2 sense [18]. Although the L2-norm of control input dose not represent fuel consumption or any physical energy in the definition, it can be seen that the asymptotic convergence of the system states can be completed by very small control input if the terminal time is not restricted. According to this property, it is reasonable to introduce a proper optimization criteria for the control input, and the orbital controller with small thrust can be obtained by solving the optimization problem. The feasibility and the effectiveness of this method are investigated in this paper.

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Furthermore, in many existing studies, the optimal feedback controllers are often designed by solving algebraic Riccati equation (ARE) based on the linear quadratic regulator (LQR) Rt theory, and the quadratic cost function 0 ½uT ðtÞRuðtÞ þ xT ðtÞQxðtÞ dt is usually introduced to evaluate the performance of the closed-loop systems. However, this method has some disadvantages. In order to solve ARE, we must predetermine the control weight matrices R and Q. But unfortunately, there has not been a maturity method which can be used to determine these matrices, and the assumptions or contrived fixtures of these matrices surely bring much conservatism or even make the problem unsolvable. Thus, another motivation of this paper is to adopt another optimal controller design method which with less conservatism can deal with the autonomous spacecraft rendezvous problem. In our previous study [7], the orbital transfer is considered as a whole process, which actually can be divided into in-plane motion and out-plane motion according to the C–W equations and the analytical solutions. The analysis of the divided motions can make the transfer process clearer, and it is convenient for further study of the transfer orbit. Thus, in this paper, we consider the in-plane motion and out-plane motion separately. And different from the work in [7], where the control input constraint was not considered, the requirements of the guaranteed control cost and the minimal 2-norm of the control input which is introduced based on the property of NCVE are considered. Then, a new orbital control design problem is formulated. By a Lyapunov approach, the controller design problem is cast into a convex optimization problem subject to liner matrix inequality (LMI) constraints. If the optimization problem is solvable, a desired controller can be readily designed, such that the orbit transfer process can be accomplished with small thrust and the control cost meets an upper bound simultaneously. Furthermore, considering the fact of different initial and terminal states of the transfer process cause different performances, some possible patterns of the relative positions of the two spacecrafts are also analyzed in this paper. An illustrative example is provided to show the effectiveness of the proposed control design method, and different performances caused by possible transfer orbits with different initial and terminal relative positions are investigated. The rest of this paper is organized as follows. Section 2 presents the dynamic model of spacecraft rendezvous, and the robust control design problem is formulated. In Section 3, the controller design method is proposed and the different initial states cases are analyzed, respectively. Then, an example is given to illustrate the applicability of the presented 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; J  J2 refers to either the Euclidean vector norm or the induced matrix 2-norm. For a real symmetric matrix W, the notation W 40 ðW o0Þ is used to denote its positive- (negative-) definiteness. diagf. . .g stands for a block-diagonal matrix. 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. Relative motion analysis and problem formulation In this section, the relative motion dynamics is analyzed based on the C–W equations. According to different motion specialities, the whole rendezvous process is divided into in-plane motion and out-plane motion. Then, the relative orbital transfers of in-plane and

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out-plane are illuminated and their norm-bounded uncertain models which will be studied in this paper are established by considering the parameter uncertainties. Finally, the performance indices which will be considered in the next sections are proposed. 2.1. Description of relative motion Firstly, we review the C–W equations and propose the mathematical description of the relative motion for two vicinal spacecrafts. Assume that the two spacecrafts are target and chaser, respectively, and the orbital coordinate frame is a right-handed Cartesian coordinate. As shown in Fig. 1, the origin attaches to the mass center of the target, x-axis is along the vector from earth center to the origin, y-axis is along the target orbit circumference, and z-axis completes the right-handed frame. Define R0 is the radius of the target circular orbit, n is the angle velocity of the target which is equal with ðme =R30 Þ1=2 , where me is the gravitational parameter of the earth. Thus, according to Newton’s motion theory, the relative dynamic model can be described by C–W’s equations as x€ ¼ 2ny_ þ n2 ðR0 þ xÞ y€ ¼ 2nx_ þ n2 y z€ ¼ 

me ðR0 þ xÞ ½ðR0 þ xÞ2 þ y2 þ z2 3=2 me y 2

½ðR0 þ xÞ þ y2 þ z2 3=2

me z 2

½ðR0 þ xÞ þ y2 þ z2 3=2

þ ax ;

þ ay ;

þ az ;

ð1Þ

ð2Þ ð3Þ

where x, y and z are the components of the relative position in corresponding axes, ai (i=x, y, z) is the ith component of the control input acceleration. The linearized equations around the null solution are given by 8 2 € _ x ¼ ax ; y3n > < x2n €y þ 2nx_ ¼ ay ; ð4Þ > : z€ þ n2 z ¼ az : It can be seen that the first two equations in Eq. (4) describe the in-plane motion and the third equation describes the out-plane motion, respectively, and the two motions are independent. Next, these two motions will be analyzed separately.

Fig. 1. Reference coordinate system for rendezvous.

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2.2. In-plane motion and out-plane motion 2.2.1. In-plane motion _ _ T and control input vector By defining the state vector xðtÞ ¼ ½xðtÞ; yðtÞ; xðtÞ; yðtÞ T u(t)=[ax(t), ay(t)] , we have _ ¼ A1 xðtÞ þ B1 uðtÞ; xðtÞ

ð5Þ

where 2

0

6 0 6 A1 ¼ 6 2 4 3n 0

2

3

0

1

0

0 0

0 0

1 7 7 7; 2n 5

0

2n

0

60 6 B1 ¼ 6 41

0

0

0

3

07 7 7: 05 1

To study the free motion of the chaser without control input thrust, we deduce the transition matrix F1 of system (5) as 3 2 1 2 sinnt ð1cosntÞ 7 0 6 43cosnt n n 7 6 7 6 6 6ðsinntntÞ 1 2 ðcosnt1Þ 4 sinnt3t 7 A1 t F1 ¼ e ¼ 6 7; n n 7 6 6 3nsinnt 0 cosnt 2sinnt 7 5 4 6nðcosnt1Þ 0 2sinnt 4cosnt3 which yields the solutions of the free motion   2 2 1 xðtÞ ¼ 4x0 þ y_ 0  3x0 þ y_ 0 cosnt þ x_ 0 sinnt; n n n

ð6Þ

  2 2 4 yðtÞ ¼ y0  x_ 0 þ x_ 0 cosnt þ 6x0 þ y_ 0 sinntð6nx0 þ 3y_ 0 Þt: n n n

ð7Þ

If y_ 0 ¼ 2nx0 , Eqs. (6) and (7) can be transformed to two periodic varying equations as 1 xðtÞ ¼ x0 cosnt þ x_ 0 sinnt; n

ð8Þ

2 2 yðtÞ ¼ y0  x_ 0 þ x_ 0 cosnt2x0 sinnt: n n

ð9Þ

By defining 2 d ¼ y0  x_ 0 ; n

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi x_ 0 ; ar ¼ x20 þ n

Eqs. (8) and (9)can be rewritten as xðtÞ ¼ ar sinðnt þ aÞ; yðtÞ ¼ d þ 2ar cosðnt þ aÞ:

cosa ¼

x_ 0 ; na

sina ¼

x0 ; ar

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It follows that x2 ðtÞ ½yðtÞd2 þ ¼ 1: a2r ð2ar Þ2

ð10Þ

Obviously, the trajectory of the chaser’s in-plane free motion can be regarded as an ellipse pffiffiffi with center (0, d) and eccentricity e ¼ ð 3=2Þ. If d=0, the chaser is moving along the elliptical orbit of Chaser 1, whose origin coincides with the position of the target as shown in Fig. 2. If da0 and with the same radius, the chaser’s in-plane free motion orbit is the elliptical orbit of Chaser 2 as shown in Fig. 2. 2.2.2. Out-plane motion For the out-plane motion, define the state vector xðtÞ ¼ ½zðtÞ; z_ ðtÞT and control input vector u(t)=az(t). Then, we have _ ¼ A2 xðtÞ þ B2 uðtÞ; xðtÞ

ð11Þ

where 

0 A2 ¼ n2

 1 ; 0

  0 : B2 ¼ 1

The transition matrix F2 is given by   cosnt sinnt A2 t F2 ¼ e ¼ ; sinnt cosnt which yields the solutions of the free motion 1 zðtÞ ¼ z0 cosnt þ z_ 0 sinnt: n By defining sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 z_ 0 2 ; b ¼ z0 þ n

cosb ¼

ð12Þ

z0 ; b

sinb ¼

_z 0 ; bn

Fig. 2. The free in-plane motion of the chaser.

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the motion function (12) can be written as zðtÞ ¼ bcosðnt þ bÞ:

ð13Þ

This means that the relative motion in z-axis is a fluctuation around the origin. 2.3. Relative orbital transfer 2.3.1. In-plane motion For the in-plane motion control of the spacecraft rendezvous, the chaser should not move to the target point directly due to the security requirement. Instead, there should be a relative orbit transfer process before the final docking action. The initial orbit and the terminal orbit of this process can be shown in Fig. 3. The goal of this paper is to design a proper controller for the chaser, such that the chaser can be asymptotically transferred into the terminal orbit which is close to the target. To solve this problem conveniently, we introduce two vectors xi ðtÞ ¼ ½xi ðtÞ; yi ðtÞ; x_ i ðtÞ; y_ i ðtÞ and xt ðtÞ ¼ ½xt ðtÞ; yt ðtÞ; ðx_ t ðtÞ; y_ t ðtÞ to represent the chaser’s states in the initial orbit and the terminal orbit, respectively. Then, we have x_ i ðtÞ ¼ A1 xi ðtÞ þ B1 uðtÞ;

ð14Þ

x_ t ðtÞ ¼ A1 xt ðtÞ:

ð15Þ

Define the state error vector xe ðtÞ ¼ xi ðtÞxt ðtÞ, and its state equation can be obtained as x_ e ðtÞ ¼ A1 xe ðtÞ þ B1 uðtÞ:

ð16Þ

Obviously, if we find a proper control law u(t) which can steers system (16) stable, then the state error vector xe(t) will asymptotically converge to zero, which means that the chaser can be transferred into the terminal orbit. Furthermore, by considering the parameter uncertainties exist in the matrices A1 and B1, we introduce the uncertain matrices DA1 and DB1 , and the equation of the error state can be described as x_ e ðtÞ ¼ ðA1 þ DA1 Þxe ðtÞ þ ðB1 þ DB1 ÞuðtÞ:

Fig. 3. Initial orbit and terminal orbit of the in-plane orbit transfer.

ð17Þ

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The uncertain matrices DA1 and DB1 are assumed to be the following form: ½DA1 DB1  ¼ D1 F1 ðtÞ½E1 G1 ; where D1, E1 and G1 are known constant matrices with proper dimensions, and F1(t) is an unknown real-time varying matrix with the elements bounded by F1T ðtÞF1 ðtÞrI. With the state-feedback control law uðtÞ ¼ K1 xe ðtÞ;

ð18Þ

the closed-loop system can be described as x_ e ðtÞ ¼ ½ðA1 þ DA1 Þ þ ðB1 þ DB1 ÞK1 Þxe ðtÞ;

ð19Þ

where K1 is the gain matrix which needs to be determined. Thus, we can see that the proper controller K1, which makes the close-loop system (19) asymptotically stable, is the desired controller which can accomplish the orbital transfer in spite of the parameter uncertainties. 2.3.2. Out-plane motion For the out-plane motion control of the spacecraft rendezvous, the goal is to transfer the chaser into the target orbital plane, which can be regarded as the asymptotic stabilization problem of system (11). By considering the parameter uncertainties, the uncertain system can be described as xðtÞ ¼ ðA2 þ DA2 ÞxðtÞ þ ðB2 þ DB2 ÞuðtÞ:

ð20Þ

Similarly, the matrices DA2 and DB2 are assumed to have the following description: ½DA2 DB2  ¼ D2 F2 ðtÞ½E2 G2 ; where D2, E2 and G2 are known constant matrices with proper dimensions, and F2(t) is an unknown real-time varying matrix with the elements bounded by F2T ðtÞF2 ðtÞrI. The purpose of the out-plane motion control is to find a proper state-feedback control law uðtÞ ¼ K2 xðtÞ; such that the state of the closed-loop system _ ¼ ½ðA2 þ DA2 Þ þ ðB2 þ DB2 ÞK2 ÞxðtÞ xðtÞ

ð21Þ

asymptotically converge to zero. Remark 1. In this paper, we only consider the state-feedback control problem, and it is assumed that the real-time state signals can be transmitted accurately. It is worth mentioning that the output-feedback control problem is more important in practice. For spacecraft rendezvous, the output-feedback control problem with possible missing measurements deserves to be further studied in the future work. 2.4. Energy performance and control cost In many studies, two kinds of constraints are often considered for the orbital controller design. The first one is the energy constraint with the requirement of minimal propellant, and the other one is the time constraint with the requirement of minimal duration of the maneuver. In this paper, we consider the problem of spacecraft rendezvous with small thrust, and there is not any strict time constraints. Minimization of the propellant and reduction of the energy cost is practical and necessary because of the small capacity of propellant storage.

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The total energy of the control input signal can be considered in the L2-norm sense, which can be described as Z 1 1=2 2 JuðtÞJL2 ¼ juðtÞj dt : 0

Based on this definition, the work in [18] once proposed a property associated with the energy performance for linear systems, which is called as null controllability with vanishing energy (NCVE). For the convenience of further discussion, the definition of this property is listed next, and the followed lemma shows the necessary and sufficient condition of this property. The detailed description and the proof can also be found in [22]. _ ¼ AxðtÞ þ BuðtÞ is said to be null controllability with Definition 1. The system xðtÞ vanishing energy (NCVE), if for any x0 2 H, there exist a sequence of controls un and of times Tn, such that xx0 ;un ðTn Þ ¼ 0 for any n 2 N, and Z Tn jun ðsÞj2 ds ¼ 0: lim n-1

0

_ ¼ AxðtÞ þ BuðtÞ is NCVE if and only if it is controllable and Lemma 1. The system xðtÞ Re lr0 for any l 2 sðAÞ, where sðAÞ is the set of all eigenvalues of A. From the definition of NCVE, we note that the L2-norm of the control input does not represent the fuel consumption or energy expense in the physical sense. However, we can see that the states of the system which has this property can be steered to the origin with small control input. This is a very significant property for spacecraft rendezvous engineering. According to Lemma 1, it can be seen that systems (5) and (11) have the property of NCVE because they are controllable and sðA1 Þ ¼ ð0; 0; ni; niÞ, sðA2 Þ ¼ ðni; niÞ. Thus, when designing the orbital controllers, the admissible control input thrust can be designed as small as possible to ensure the low propellant performance. For this purpose, the 2-norm of the control input is considered in this paper, which is described as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JuðtÞJ2 ¼ uT ðtÞuðtÞ: ð22Þ Then, we introduce a positive scalar m as the bound of JuðtÞJ2 in terms of JuðtÞJ22 rm:

ð23Þ

The scalar m could be a predetermined small scalar or a criterion for optimization problem. On the other hand, the control cost is another important index in our consideration. Generally, by defining the control weight matrices R and Q with proper dimensions, the quadratic control cost function can be introduced as Z t J¼ ½uT ðtÞRuðtÞ þ xT ðtÞQxðtÞ dt; ð24Þ 0

which has been usually regarded as an optimization index to evaluate the control performance of the input consumption and the state variation in certain mathematical sense for the closed-loop systems. Instead of the optimization of J, we consider its upper bound which is usually called as guaranteed cost, and the purpose of this paper is to make

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the orbital transfer control cost J meets an upper bound J , which can also be used to evaluate the performances of the closed-loop systems with the designed controllers. 3. Relative motion control In this section, we investigate the robust state-feedback controller design problem for the orbital transfer. Based on Lyapunov stable theory, the controller design requirements such as input constraint and the guaranteed cost are formulated as some LMI conditions, and the controller design problem is cast into a convex optimization problem subject to the LMI constraints. Due to the fact that the controller design and the closed-loop system performance also depend on the initial state of the system, we discuss the different possible initial cases of the transfer orbit in this section. 3.1. Controller design Due to the similar patterns of the state functions of in-plane motion and out-plane motion, we only study the in-plane motion controller design problem here, and the obtained results can also be adopted to deal with the case of out-plane motion. Firstly, we recall the following results which will be used in our later development, and their proofs and the applications can be found in [5,11,27]. Lemma 2. Given matrices F ¼ FT , M and N of appropriate dimensions, F þ MSN þ N T ST M T o0; for all S satisfying S T SrI, if and only if there exists a scalar e40 such that F þ eMM T þ e1 N T No0: Lemma 3. Let M and N be real matrices of appropriate dimensions, for any scalar e40, " # " # 0 NM T 0 eNN T r : 0 e1 MM T MN T 0

Next, we first consider the nominal system which has no parameter uncertainty and study the existence conditions of the proper controller for it. Then, based on the results, the system with uncertainties will be taken into consideration and the results will be extended. Finally, the proper controller design method will be obtained. For the nominal system (16), the existence of a guaranteed cost controller can be easily judged by the following proposition. Proposition 1. Consider system (16) and the controller (18). If there exists a positive symmetric matrix P satisfying Q þ K1T RK 1 þ PðA1 þ B1 K1 Þ þ ðA1 þ B1 K1 ÞT Pr0;

ð25Þ

then the closed-loop system is asymptotic stable and the performance index (24) has an upper bound.

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Proof. For system (16) and the controller (18), the closed-loop system can be described as x_ e ðtÞ ¼ ðA1 þ B1 K1 ÞxðtÞ:

ð26Þ

Define the Lyapunov function V ðxe Þ ¼ xTe ðtÞPxe ðtÞ;

ð27Þ

where P is a positive symmetry matrix. Then, we obtain V_ ðxe Þ ¼ xTe ðtÞ½PðA1 þ B1 K1 Þ þ ðA1 þ B1 K1 ÞT Pxe ðtÞ: According to Eq. (25) and considering Q40 and R40, we can see that V_ ðxÞrxTe ðtÞðQ þ K1T RK 1 Þxe ðtÞo0;

ð28Þ

which means that the closed-loop system (26) is asymptotic stable. Furthermore, the upper bound of the performance index (24) can be obtained by the integral of Eq. (28) as follows: Z 1 Z 1 J¼ ½xTe ðtÞQxe ðtÞ þ uT ðtÞRuðtÞ dt ¼ ½xTe ðtÞðQ þ K1T RK 1 Þxe ðtÞ dt 0 0 Z 1 Z 1 T T r fxe ðtÞ½PðA1 þ B1 K1 Þ þ ðA1 þ B1 K1 Þ Pxe ðtÞg dt ¼ V_ ðxe Þ dtrxe0 Pxe0 : 0

0

ð29Þ The proof is completed.

&

Besides the guaranteed cost requirement, the thrust constraint (23) is another important issue which is not considered in Proposition 1. Thus, in order to ensure the existence of the proper controller, we should further consider the control input constraint for the closedloop system (26). For Eqs. (22) and (23), we have uT ðtÞuðtÞrm, which can be written as 1 T x ðtÞK1T K1 xe ðtÞr1: m e

ð30Þ

Considering the Lyapunov function (27), we introduce another parameter d, which constrains the cost bound (29) in terms of xe0 Pxe0 rd:

ð31Þ

For V_ ðxe Þo0, which has been ensured by (25), we have xTe ðtÞPxe ðtÞrxTe0 Pxe0 rd, and thus ð1=dÞxTe ðtÞPxe ðtÞr1. Therefore, we can see that the condition (30) can be ensured by 1 T 1 x ðtÞK1T K1 xe ðtÞr xTe ðtÞPxe ðtÞ: m e d

ð32Þ

Obviously, Eq. (32) is equal to 1 T 1 K1 K1  Pr0: m d

ð33Þ

Thus, the input constraint JuðtÞJ22 rm can be guaranteed if the inequations (31) and (33) are satisfied simultaneously. Accordingly, we can see that if there are feasible solutions of the matrix inequations (25), (31) and (33), the proper controller for the nominal system (16) is existent, and the closed-loop system (26) with the controller is asymptotically stable with

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constrained control input and its control cost has an upper bound during the control duration. However, it should be noticed that, the matrix inequality conditions (25), (31) and (33) cannot be used to design the desired orbital transfer controller directly. This is because that the parameter uncertainties are not considered and the inequalities are not LMIs due to the multiplication of some covariates. Therefore, our next objective is to further consider the uncertain system (17), and convert the conditions to LMIs which can be readily solved by standard numerical software. According to Proposition 1, the stability and the guaranteed cost requirement of the closed-loop system (19) can be ensured by PA1 þ PB1 K1 þ AT1 P þ K1T BT1 P þ Q þ K1T RK 1 þ PD1 F1 ðtÞðE1 þ G1 K1 Þ þ ðE1 þ G1 K1 ÞT F1T ðtÞDT1 Pr0:

ð34Þ

By Lemma 2, the inequality (34) holds if and only if PA1 þ PB1 K1 þ AT1 P þ K1T BT1 P þ Q þ K1T RK 1 þ ePDDT P þ e1 ðE1 þ E2 KÞT ðE1 þ E2 KÞr0:

ð35Þ It should be noticed that, for traditional optimal controller design method based on solving the ARE, the weight matrices R and Q in cost index (24) must be determined before the calculations, and this may increase the conservatism of the results or even make the problem unsolvable. In this paper, instead of making assumptions or contrived fixing of R and Q, we regard them as two matrix variables, whose values can be calculated with the ~ ¼ Q1 . By Schur gain matrices of the proper controller. Next, we define R~ ¼ R1 and Q complement, Eq. (35) can be transformed into 2 3 P E1T þ K T G1T K1T I 6 7 eI 0 0 7 6 6 7r0; ð36Þ 6 ~  R 0 7 4 5 ~    Q where P ¼ PA1 þ PB1 K1 þ AT1 P þ K1T BT1 P þ ePD1 DT1 P: Define X=P1 and Y=K1X. Then, by pre- and post-multiplying (36) by diagfX T ; I; I; Ig and its transpose, respectively, we have 3 2 X O XE T1 þ Y T G1T Y T 7 6 6 eI 0 0 7 7r0; 6 ð37Þ 6  R~ 0 7 5 4 ~    Q where O ¼ X T AT1 þ A1 X þ B1 Y þ Y T BT1 þ eD1 DT1 : We can see that the inequality condition (37) is equivalent to Eq. (36), which can make Eq. (35) hold. Thus, the stability and the guaranteed cost requirement of the closed-loop system (19) can be ensured by Eq. (37).

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Furthermore, we can obtain the following facts by making some transforms for the inequality conditions (31) and (33), which ensure the control input constraint. By Schur complement, Eq. (31) can be written as " # dI xTe0 r0: ð38Þ  P1 Pre- and post-multiplying (38) by diagfI; X T g and its transpose respectively, we obtain " # dI xTe0 r0: ð39Þ  X Similarly, pre- and post-multiplying (33) by XT and its transpose, respectively, we have d T Y Y X r0: m

ð40Þ

By Schur complement, Eq. (40) can be written as " pffiffiffi T # X dY r0:  mI According to Lemma 3, for any scalar u40, Eq. (41) can be ensured by " # X þ udI 0 r0;  mI þ u1 YY T which can be written as 2 X þ udI 0 6  mI 4  

0

ð41Þ

ð42Þ

3

7 Y 5r0: uI

ð43Þ

Thus, we can see that the inequality conditions (31) and (33) have been transformed into (39) and (43). Therefore, the existence of the proper controller can be ensured by the feasibility of (37), (39) and (43). Obviously, these conditions are all LMIs, and we can solve the feasibility problem by standard software. For a set of feasible solutions ~ the controller can be calculated as ~ QÞ, ðe; d; X ; Y ; R; K1 ¼ YX 1 :

ð44Þ

Also, it should be noticed that the inequalities (37), (39) and (43) are the sufficient conditions of the existence for the proper controllers, and there may be many sets of feasible solutions of the problem. Nevertheless, in practice, it is more significant to find a certain controller with certain optimal performance criteria. Thus, according to the results obtained above and the NCVE property which is discussed in Section 2, we consider the 2-norm index (23) as an optimization criteria. Thus, an optimal controller can be obtained by solving the following optimization problem: min

~ ~ Q e;d;X ;Y ;R;

s:t:

m

LMIs ð37Þ; ð39Þ and ð43Þ

ð45Þ

1876

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~ then u(t)=YX1x(t) is the ~ QÞ, If the problem has a set of optimal solutions ðe; d; X ; Y ; R; optimal state-feedback controller, by which the closed-loop system (19) asymptotic stable with minimal control input satisfying the constraint JuðtÞJ22 rm, and the control cost index (24) has the upper bound. Remark 2. Obviously, the problem (45) is a convex optimization problem subject to LMI constraints. The scalar m has been included as an optimization variable to obtain a bound of the index (23). Then, the minimal bound in terms of the feasibility of admissible controllers can be readily obtained by solving the convex optimization problem (45). 3.2. Initial state of the relative transfer motion According to the inequality condition (39), we can see that in order to apply the proposed controller design method, the initial states of systems (17) and (20) should be ascertained before calculations. For out-plane motion, the initial state of (20) can be easily obtained because it only contains the initial relative position and velocity along z -axis. However, for the in-plane motion, the initial state of Eq. (17) is more complex because it is an error state between the initial orbital and the terminal orbital states and there exists the coupling motion between x- and y-axis. Thus, we specifically propose this subsection to discuss the initial state of the relative transfer for the in-plane motion. The initial orbit and the terminal orbit of the in-plane motion can be shown in Fig. 4, and the semiminor axes of the two orbits are ai and at. The distance between the target’s projection on the orbital plane and the center of the initial relative orbit is di, and the distance between the target’s projection on the orbital plane and the center of the terminal relative orbit is dt. According to different values of ai, at, di and dt, there are many kinds of relative positional relationships for the initial orbit and the terminal orbit. However, by considering the practical situations, we can see that there are some potential rules for these scalars. The terminal orbit, generally, should be regarded as an orbit which is near the target, and the origin of the orbit coincides with the target’s projection on the orbital plane. Thus, it can be readily confirmed that dt=0. Then, the positional relationship of the initial orbit and the terminal orbit can be ascertained according to the scalar di. We can see that if di=0, the two orbits are concentric

Fig. 4. Initial point and terminal point of the orbital transfer.

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Table 1 xe0 for different orbit cases. xe0 Case Case Case Case

AA AB AC AD

½0; di dt þ 2ðai at Þ; nðai at Þ; 0T ½at ; di dt þ 2ai ; nai ; 2nat T ½0; di dt þ 2ðai þ at Þ; nðai þ at Þ; 0T ½at ; di dt þ 2ai ; nai ; 2nat T

Case Case Case Case

BA BB BC BD

½ai ; di dt 2at ; nat ; 2nai T ½ai at ; di dt ; 0; 2nðai at ÞT ½ai ; di dt þ 2at ; nat ; 2nai T ½ai þ at ; di dt ; 0; 2nðai þ at ÞT

Case Case Case Case

CA CB CC CD

½0; di dt 2ðai þ at Þ; nðai þ at Þ; 0T ½at ; di dt 2ai ; nai ; 2nat T ½0; di dt 2ðai at Þ; nðai at Þ; 0T ½at ; di dt 2ai ; nai ; 2nat T

Case Case Case Case

DA DB DC DD

½ai ; di dt 2at ; nat ; 2nai T ½ai at ; di dt ; 0; 2nðai þ at ÞT ½ai ; di dt þ 2at ; nat ; 2nai T ½ai þ at ; di dt ; 0; 2nðai at ÞT

ellipses, and di a0 denotes that the two orbits are nonconcentric ellipses. The orbital control performances under the concentric and nonconcentric situations are very different, which will be shown specifically by an example in the next section. To simplify the control and the calculation, we consider four special points Ai, Bi, Ci and Di in the initial orbit as the initial points of the transfer orbit. Simultaneously, along the terminal orbit, assume four virtual terminal points At, Bt, Ct and Dt with the same positions of the initial points. Then, note that xi0A is the initial state when the chaser at the point Ai. The states at other points are notated with the same way. According to the inplane free motion rules, we can obtain the initial states and the virtual terminal states of their four different points, respectively, as follows: xi0A ¼ ½0; di þ 2ai ; nai ; 0T ;

xt0A ¼ ½0; dt þ 2at ; nat ; 0T ;

xi0B ¼ ½ai ; di ; 0; 2nai T ; xi0C ¼ ½0; di 2ai ; nai ; 0T ;

xt0B ¼ ½at ; dt ; 0; 2nat T ; xt0C ¼ ½0; dt 2at ; nat ; 0T ;

xi0D ¼ ½ai ; di ; 0; 2nai T ;

xt0D ¼ ½at ; dt ; 0; 2nat T :

Then, for system (17), there are 16 possible patterns of the initial state xe0, because four points could be selected as the termination for each determined initial point of the transfer orbit. We note that Case AA means the transfer orbit case with the initial point Ai and the terminal point At, and its initial error state can be obtained by xe0AA=xi0Axt0A. Other conditions can be described by similar notations and all the possible initial error states are listed in Table 1.

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With these possible initial states, the controller design method proposed in the above subsection can be utilized, and the corresponding controllers of the different transfer orbital cases can be obtained.

4. Orbit control simulation and analysis In this section, we provide an example to illustrate the usefulness of the controller design method proposed in the above sections. Here, we consider a pair of adjacent spacecraft, and assume that the height of the target spacecraft is 500 km and its angle velocity n ¼ 1:1068  103 rads=s. Next, the in-plane motion and the out-plane motion are analyzed separately, and the concentric and nonconcentric situations for the initial orbit and the terminal orbit are considered, respectively, in the analysis of the in-plane motion. 4.1. In-plane motion 4.1.1. Concentric Assume that the initial orbit and the terminal orbit satisfy (ai, di)=(10 km, 0) and (at, dt)=(1 km, 0), which means that the two orbits are concentric ellipses. By considering the configurations of A1 and B1, assume the uncertain matrices as 2 3 2 3 2 3 0 0 0 0 0 0 1 0 0 0 60 0 0 07 60 0 0 17 60 07 6 7 6 7 6 7 D1 ¼ 6 7; E1 ¼ a1  6 7; G1 ¼ b1  6 7; 40 1 1 05 40 0 0 05 41 05 1 0

0

1

0 0

0

0

0

1

where a1 ¼ 103 and b1 ¼ 102 denote the angle velocity uncertainty and the input uncertainty, respectively. According to the analysis in the above section, there are 16 possible cases for the initial state of the in-plane relative motion, and the different orbits and performance indices can be obtained correspondingly. Here, the notation Case CAA means the transfer orbit case with the initial point Ai and the terminal points At for concentric situation, and the similar notations denote other orbital cases with different initial and terminal points. For Case CAA, by solving the convex optimization problem in (45), we obtain the minimal 2-norm of control input JuðtÞJmin ¼ mmin ¼ 0:0664, the upper bound of control cost 2 ~ and R ~ which are listed as J ¼ d ¼ 5:0878  103 , and the associated matrices X, Y, Q follows: 2 3 1:4478  106 4:2087  105 2164:5 2030:4 6 4:2087  105 5:7044  105 136:06 1088:6 7 6 7 XCAA ¼ 6 7; 4 2164:5 136:06 4:6515 2:4133 5 2030:4 " YCAA ¼

1088:6

2:4133

4:4855

4:1694  106

3:0881  106

2:5136  103

2:7177  104

3:1457  106

2:0684  106

2:2372  104

2:2732  103

# ;

X. Yang et al. / Journal of the Franklin Institute 347 (2010) 1863–1887

2

8:0382  108

6 3:4539  107 6 ~ ¼ Q 6 CAA 4 6:6444  105 3:7281  105 " ~ CAA ¼ R

1:0687  108 6:8902

3:4539  107

6:6444  105

1:9399  108 81044

81044 1:0687  108

1:3922  105

598:23

1879

3:7281  105

3

1:3922  105 7 7 7; 5 598:23 1:0687  108

# 6:8902 : 1:0687  108

Also, the gain matrix for the feedback controller KCAA can be calculated as   5:1597 3:226 2875:6 55:071 1 KCAA ¼ YCAA XCAA ¼ 106  : 3:0866 1:8856 436:1 2126:9 Then, with the obtained controller, the required control input accelerations during the transfer process can be calculated. The maximum input accelerations in x-axis and y-axis 2 max 2 are jumax x j ¼ 0:0294 m=s and juy j ¼ 0:0383 m=s , respectively. For a chaser spacecraft weighting 1000 kg, the maximum input thrusts in x-axis and y-axis are only 29.4 and 38.3 N, which is easy to carry out in practice. Similarly, we can also obtain 16 sets of results for each orbit case. Next, we test the minimal 2-norm of control input JuðtÞJmin 2 , the upper bound of the control cost J and the maximum needed control inputs along the two axes of the 16 orbital cases. The results are listed in Tables 2 and 3. It should be noticed that, as shown in the above two tables, we can obtain eight sets of results, and each set of results can be obtained by two initial state cases, which have similar positional relationship between the initial and terminal points due to the property of concentric ellipses. According to the results, we can select an optimal orbit case for each specific optimization criteria, which are listed in Table 4. Obviously, by considering most of the Table 2 Performance indices if initial point at A or C with concentric situation.

JuðtÞJmin 2 J max ðjumax x j; juy jÞ

Case CAA/CCC

Case CAB/CCD

Case CAC/CCA

Case CAD/CCB

0.0664 5.0878  103 (0.0294, 0.0383)

0.0380 4.0984  103 (0.0128, 0.0234)

0.0525 4.7555  103 (0.0110, 0.0275)

0.0377 4.1307  103 (0.0059, 0.0169)

Table 3 Performance indices if initial point at B or D with concentric situation.

JuðtÞJmin 2 J max ðjumax x j; juy jÞ

Case CBA/CDC

Case CBB/CDD

Case CBC/CDA

Case CBD/CDB

0.0239 3.2604  103 (0.0207, 0.0372)

0.0272 3.1896  103 (0.0481, 0.0148)

0.0300 4.0073  103 (0.0461, 0.0149)

0.0335 4.0406  103 (0.0450, 0.0076)

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Table 4 Optimal transfer orbits cases for different emphasized requirements with concentric situation. Minimal JuðtÞJ2 Optimal case

Minimal J

CBA/CDC

CBB/CDD

Minimal input (x-axis)

Minimal input (y-axis)

CAD/CCB

CBD/CDB

x 104 initial orbit

1

x - axis

0.5 terminal orbit

0 transfer orbit

-0.5 -1 2

1.5

1

0.5

0 -0.5 y - axis

-1

-1.5

-2

Fig. 5. Transfer orbit of Case CBA.

performance indices for concentric situation of the initial and terminal orbits, the optimal initial point of the transfer orbit is Bi or Di. Next, due to the space constraint, we only give the in-plane transfer orbits of two typical cases Case CBA and Case CAD in Figs. 5 and 7, and the required input accelerations of these two cases are illustrated in Figs. 6 and 8. 4.1.2. Nonconcentric For nonconcentric situation, we assume that the initial orbit and the terminal orbit satisfy (ai, di)=(10 km, 8 km) and (at, dt)=(1 km, 0). The notation Case NAA means the transfer orbit case with initial point Ai and terminal point At, and the similar notations denote other orbital cases with different initial and terminal points. Different from the concentric situation, it can be seen that there is no pair of orbital cases have similar positional relationship, so there are 16 different sets of results for the possible 16 orbital cases. Similarly with the analysis of concentric situation, we also consider the minimal norm of control input JuðtÞJmin 2 , the upper bound of the control cost J and the maximum needed control inputs along the two axes for each orbital case, and the results are listed in Tables 5–8. Then, by considering the results, we can select an optimal orbit case for each specific optimization criteria, which are listed in Table 9. It can be seen that, for most of the performance indices, the point Ci is the optimal initial point for the nonconcentric situation we assumed here.

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0.03 input in x-axis input in y-axis

input acceleration

0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 0

0.5

1 time

1.5

2 x 104

Fig. 6. Control inputacceleration of Case CBA.

x 104 1

x - axis

0.5 0 -0.5 -1 2

1.5

1

0.5

0 -0.5 y - axis

-1

-1.5

-2

Fig. 7. Transfer orbit of Case CAD.

And similarly with the illustrations of concentric situation, we only choose two typical cases here to show their transfer orbit shapes and the required input accelerations. The inplane transfer orbits of Case NCC and Case NBB are shown in Figs. 9 and 11, and the required input accelerations of these two cases are illustrated in Figs. 10 and 12. 4.2. Out-Plane motion Consider the out-plane motion which is described in Eq. (20). Assume x0=[6 km, 0], which means that the maximum distance between chaser and the origin during the free

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input acceleration

0.005 0 -0.005 -0.01 -0.015 -0.02 0

2000 4000 6000 8000 10000 12000 14000 16000 time

Fig. 8. Control input acceleration of Case CAD.

Table 5 Performance indices if initial point at A with nonconcentric situation.

JuðtÞJmin 2 J max ðjumax x j; juy jÞ

Case NAA

Case NAB

Case NAC

Case NAD

0.0945 6.1489  103 (0.0284, 0.0373)

0.1449 7.1556  103 (0.0376, 0.0484)

0.0752 6.0790  103 (0.0296, 0.0394)

0.2076 8.4140  103 (0.0299, 0.0528)

Table 6 Performance indices if initial point at B with nonconcentric situation.

JuðtÞJmin 2 J max ðjumax x j; juy jÞ

Case NBA

Case NBB

Case NBC

Case NBD

0.0172 2.8134  103 (0.0289, 0.0094)

0.0403 4.2132  103 (0.0195, 0.0060)

0.0245 3.0447  103 (0.0285, 0.0087)

0.0308 3.4329  103 (0.0293, 0.0114)

Table 7 Performance indices if initial point at C with nonconcentric situation.

JuðtÞJmin 2 J max ðjumax x j; juy jÞ

Case NCA

Case NCB

Case NCC

Case NCD

0.0420 4.1950  103 (0.0053, 0.0187)

0.0354 3.5526  103 (0.0067, 0.0234)

0.0133 2.5217  103 (0.0049, 0.0170)

0.0313 3.9481  103 (0.0016, 0.0181)

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Table 8 Performance indices if initial point at D with nonconcentric situation.

JuðtÞJmin 2 J max ðjumax x j; juy jÞ

Case NDA

Case NDB

Case NDC

Case NDD

0.0337

0.0235

0.0370

0.0318

4:2904  103 (0.0608, 0.0228)

3:8493  103 (0.0474, 0.0113)

4:4033  103 (0.0496, 0.0130)

4:0517  103 (0.0439, 0.0101)

Table 9 Optimal transfer orbits cases for different emphasized requirements with nonconcentric situation.

Optimal case

Minimal JuðtÞJ2

Minimal J

Minimal input (x-axis)

Minimal input (y-axis)

NCC

NCC

NCD

NBB

x 104 1

x - axis

0.5 initial orbit

terminal orbit

0 -0.5

transfer orbit

-1

-3

-2.5

-2

-1.5

-1 -0.5 y - axis

0

0.5

1

1.5 x 104

Fig. 9. Transfer orbit of Case NCC.

fluctuate motion is 6 km. The matrices related with the uncertainties are assumed as       0 1 1 0 1 ; E2 ¼ a2  ; G2 ¼ b2  ; D2 ¼ 1 0 0 0 0 where a2 ¼ 104 and b2 ¼ 102 . Then, By solving the convex optimization problem in Eq. (45), we obtain the minimal 2-norm of control input JuðtÞJmin ¼ mmin ¼ 0:0037, the upper 2 ~ and R ~ bound of control cost J ¼ d ¼ 5:2030  103 , and the associated matrices X, Y, Q which are listed as follows: " # 5:3221  105 256:84 X¼ ; 256:84 0:89958

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input acceleration

0.015 0.01 0.005 0 -0.005 -0.01 0

2000 4000 6000 8000 10000 12000 14000 16000 time

Fig. 10. Control input acceleration of Case NCC.

x 104 1

x - axis

0.5

transfer orbit initial orbit

0 terminal orbit

-0.5 -1

-3

-2.5

-2

-1.5

-1 -0.5 y - axis

0

0.5

1

1.5 x 104

Fig. 11. Transfer orbit of Case NBB.

Y ¼ ½1:8057  107 5:9878  104 ; " # 7:0433  108 1:7438  105 ~ ; Q¼ 1:7438  105 4:1006  108 R~ ¼ 4:1006  108 : And also, the gain matrix for the feedback controller K can be calculated as K ¼ YX 1 ¼ ½3:7255  107 7:7199  104 :

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0.01 input in x-axis input in y-axis

input acceleration

0.005 0 -0.005 -0.01 -0.015 -0.02 0

2000

4000

6000

8000

10000

time

position

Fig. 12. Control input acceleration of Case NBB.

6000 4000 2000 0 -2000 -4000

position in z-axis

input acceleration

0

4

0.5

1 time

1.5

2 x 104

x 10-3 input in z-axis

2 0 -2 -4 0

0.5

1 time

1.5

2 x 104

Fig. 13. The position output and the needed input acceleration of out-plane motion.

With the controller K, the chaser’s position in z-axis and the required input acceleration are shown in Fig. 13. Compared with the in-plane motion, the out-plane motion control is easier to analyze and actualize, and we can also readily obtain the proper out-plane controller by the proposed design method, such that the out-plane orbit transfer can be accomplished with small thrust and the control cost has an upper bound. Finally, to observe the transfer orbit intuitively, we give a whole transfer orbit shape which can be divided into the in-plane motion of Case CBA and the out-plane motion which is analyzed above. The integrative transfer orbit is shown in Fig. 14. The chaser is

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z - axia

4000

chaser

2000 0

initial orbit target

transfer orbit

-2000 -4000 1 x 104

terminal orbit

0

x - axis -1

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

y- axis

Fig. 14. The transfer orbit.

asymptotically transferred into the terminal orbit which is close to the target with small thrusts in x-, y- and z-axis, and the control costs of in-plane motion and out-plane motion all have the upper bounds. 5. Conclusions For the orbital transfer process of spacecraft rendezvous subject to parameter uncertainty, this paper has presented a new robust state-feedback controller design method. By using Lyapunov method and linear matrix inequality technique, the controller design problem has been transformed into a convex optimization problem with LMI constraints. With the designed controller, the orbit transfer is accomplished with small thrust and bounded control cost. Furthermore, by considering the different relative positional relationships between the initial point and the terminal point of the transfer orbit, the different performances are analyzed for concentric and nonconcentric situations, respectively. An illustrative example has shown the effectiveness of the proposed controller design methods. References [1] M. Basin, D. Calderon-Alvarez, Optimal LQG controller for linear stochastic systems with unknown parameters, Journal of the Franklin Institute 345 (3) (2008) 293–302. [2] M. Basin, D. Calderon-Alvarez, Optimal filtering over linear observations with unknown parameters, Journal of the Franklin Institute 347 (6) (2010) 988–1000. [3] W.H. Clohessy, R.S. Wiltshire, Terminal guidance system for satellite rendezvous, Journal of Aerospace Science 27 (9) (1960) 653–658. [4] B. Ebrahimi, M. Bahrami, J. Roshanian, Optimal sliding-mode guidance with terminal velocity constraint for fixed-interval propulsive maneuvers, Acta Astronautica 60 (10) (2008) 556–562. [5] H. Gao, T. Chen, H1 estimation for uncertain systems with limited communication capacity, IEEE Transactions on Automatic Control 52 (11) (2007) 2070–2084. [6] H. Gao, C. Wang, A delay-dependent approach to robust H1 filtering for uncertain discrete-time statedelayed systems, IEEE Transactions on Signal Processing 52 (6) (2004) 1631–1640.

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