Robust gain scheduled control of spacecraft rendezvous system subject to input saturation

Aerospace Science and Technology 42 (2015) 442–450

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Robust gain scheduled control of spacecraft rendezvous system subject to input saturation ✩ Qian Wang, Bin Zhou ∗ , Guang-Ren Duan Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, 150001, China

a r t i c l e

i n f o

Article history: Received 18 April 2014 Received in revised form 25 October 2014 Accepted 1 February 2015 Available online 7 February 2015 Keywords: Robust gain scheduling Spacecraft rendezvous Parametric Riccati equation Input saturation Invariant sets

a b s t r a c t The problem of robust control of spacecraft circular orbit rendezvous system subject to input saturation is studied in this paper. The relative model with parameter uncertainties caused by linearization error and saturation nonlinearity is established based on C–W equation. By combining the parametric Riccati equation and the existing gain scheduling technique, a new robust gain scheduling controller is proposed to solve the robust control problem. By scheduling the design parameters online, the convergence rate of the state can be improved. With the designed controller, the spacecraft orbit rendezvous is accomplished successfully. Numerical simulations show the effectiveness of the proposed approach. © 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction Spacecraft rendezvous is an important technology for the present and the future space mission. Successful rendezvous is the precondition of many astronautic missions, such as repairing, intercepting, docking, saving, large-scale structure assembling and satellite networking [22]. Considering a target spacecraft in a circular orbit and another chaser spacecraft in its neighborhood, the relative motion between two neighboring spacecrafts can be described by autonomous nonlinear differential equations. If the distance between them is much smaller than the orbit radius, the model is given by C–W equation, derived by Clohessy and Wiltshire in 1960 [4]. During the last few decades, the spacecraft rendezvous has been actively studied and many results in control theory and technologies have been developed. Many advanced methods have been used to solve the rendezvous control problem. For example, model predictive control approach is developed for spacecraft rendezvous [10]; adaption control theory was applied to solve the rendezvous and docking problem [24]; in [27], the problem of rendezvous is

✩ This work was supported in part by the National Natural Science Foundation of China under grant numbers 61273028 and 61322305, by the Fundamental Research Funds for the Central Universities under Grants HIT.BRETIII.201210 and HIT.BRETIV.201305, by the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201343, by the Innovative Team Program of the National Natural Science Foundation of China under Grant number 61321062, and by the National Basic Research Program of China (National 973 Program) under grant number 2012CB821205. Corresponding author. E-mail address: [email protected] (B. Zhou).

*

http://dx.doi.org/10.1016/j.ast.2015.02.002 1270-9638/© 2015 Elsevier Masson SAS. All rights reserved.

cast into a stabilization problem analyzed by Lyapunov theory; the neural-network-based controller is proposed for rendezvous maneuvers [33]; by using sliding mode control in the presence of the Earth’s gravitational perturbation, the problem of spacecraft rendezvous is studied in [21]; a continuous simulated annealing (SA) algorithm is used to design LQ controller for spacecraft rendezvous [19]; a modified adaptive controller is developed in the presence of uncertain orbital parameters and target’s escape trajectory which guarantees the asymptotic stability of closed-loop system [32] and a parametric Lyapunov differential equation approach to the elliptical rendezvous with constrained control is given in [38]. For astronautic missions, the control input constraint is an important issue that we must pay attention to. In practical engineering, the orbital control input force is limited due to the constraints of the thrust equipment, the limited quality of the fuel and the limited power the engine provided and so on. Among these constraints, the one caused by thrust equipment is very important. So far, there are some results on these problems. For instance, [2] and [3] studied the lowest energy controller with fixed time. Because of using the optimal control theory, in order to realize the proposed controller, some differential equations need to be solved online. Gain scheduling methods based on parametric Lyapunov equation are proposed in [28] to solve the stabilization problem for spacecraft rendezvous system with input saturation. In the existing literature, the relative motion of the circular orbit spacecraft rendezvous system is often modeled by C–W equations, which were derived by Clohessy and Wiltshire in 1960 [4]. Generally, the C–W equations are transformed into a state space equation X˙ = A X + BU . Here X is the relative position and velocity

Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450

vector and U is the control input. This description has been used widely to study the spacecraft circular orbit rendezvous problems. However, it should be noted that there exist uncertainties caused by the linearization error in the state matrix A. These uncertainties may degrade the precision, the stability or even the safety of the rendezvous mission. Both adaptive control (see, for example, [34,35,17,18]) and robust control (see, for example, [13,31]) are effective in handling uncertainties in practical systems. In recent years, there have been many studies on the robust control of the spacecraft rendezvous system [24,8]. However, the parameter uncertainties caused by linearization error have not attracted enough attention in the existing studies. This motivates us to pay attention to these uncertainties and find a proper approach to deal with them. The gain-scheduling approach is perhaps one of the most popular nonlinear control design approaches which has been widely applied in fields ranging from aerospace to process control [14,23]. There are basically three kinds of scheduling approaches, namely, the continuous static scheduling approach [25,20,15,11], the continuous dynamic scheduling approach [9,26,37], and the discrete scheduling approach [5,12,30]. In this paper, we intend to develop an effective discrete gain scheduling controller to solve the robust stabilization problem for the uncertain circular orbit rendezvous system subject to input saturation. With independent continuous control accelerations being used as the control signals, the spacecraft rendezvous system is modeled as a linear system with input saturation and parameter uncertainties. By using a class of parametric Riccati equations, a new robust gain scheduling controller is proposed to solve the robust stabilization problem of the spacecraft rendezvous system. The main advantage of the designed controller is that the convergence rate of the state can be improved by scheduling the design parameters. The results in this paper improve that in our recent paper [29]. The rest of this paper is organized as follows. Section 2 presents the dynamic model of the spacecraft rendezvous system and formulates the robust stabilization problem. In Section 3, the robust gain scheduled controller is proposed to solve the problem. Then, the numerical simulation is given to illustrate the effectiveness of the presented approach in Section 4. Finally, Section 5 concludes the paper. Notation. Throughout the paper, the notation used is fairly standard. We use A T to denote the transpose. diag{·} stands for a block-diagonal matrix. I denotes the identity matrix with compatible dimensions. For a real symmetric matrix P , the notation P > 0 ( P < 0) is used to denote its positive (negative) definiteness and the notation P ≥ 0 ( P ≤ 0) denotes its semi-positive (semi-negative) definiteness. I [p , q] denotes the sets of integers [ p , p + 1, . . . , q]. The function sign is defined as sign( y ) = 1 if y ≥ 0 and sign( y ) = −1 if y < 0. The function sat : Rm → Rm is a vectorvalued standard saturation function, i.e.,



sat(u ) = sat(u 1 )

sat(u 2 )

· · · sat(um )

T

443

Fig. 1. Circular orbit and coordinate system.

is R and the vector from the target spacecraft to the chase spacecraft is denoted by r. The right-handed coordinate system (rotating coordinate system) O − X Y Z is fixed at the center of mass of the target with X axis along the radial direction, Y axis along the flight direction of the target, and Z axis out of the orbit plane, respectively. Denote the gravitational parameter μ = G M where M is the mass of the center planet and G is the gravitational constant. Then the orbit rate of the target orbit is given by n = μ1/2 / R 3/2 . The relative motion between the target and chaser can be governed by Newton’s equations [1]

⎧ ⎨ x¨ = 2n y˙ + n2 ( R + x) − σ μ( R + x) + satα X (ax ), y¨ = −2nx˙ + n2 y − σ μ y + satαY (a y ), ⎩ z¨ = −σ μ z + satα Z (a z ),

(1)

3

where σ = (( R + x)2 + y 2 + z2 )− 2 ; ax , a y and a z are the accelerations supplied by the thrusts in the three directions; α X , αY and α Z are respectively the maximal accelerations that the thrusts can generate in the three directions. Here, the saturation function is defined as

⎧ ⎨ −α , u < −α , satα (u ) = u , |u | ≤ α , ⎩ α, u > α,

(2)

where α > 0 is the saturation level and sat1 (u ) will be denoted by sat(u ) for short. Notice that, by denoting D = diag{α X , αY , α Z } and a = [ax , a y , a z ]T , we have u  [satα X (ax ), satαY (a y ), satα Z (a z )]T = Dsat( D −1 a). By choosing the state vector and control vector



X= x

y

z

x˙

y˙

z˙

T

, U = D −1 a ,

(3)

the relative motion equation (1) can be rewritten as

X˙ = A X + Bsat(U ) + μ f ( X ),

,

and sat(u i ) = sign(u i ) min {1, |u i |} , i ∈ 1, 2, . . . , m. Here, we have slightly abused the notation by using sat (·) to denote both the scalar- and vector-valued functions. Finally, for a matrix A, we

in which

⎡

and f ( X ) = [0, 0, 0, f 1 ( X ) , f 2 ( X ) , f 3 ( X )]T is a high-order function of X , where

, where ai j is the i-th row and j-th col-



 

umn element of A and  A 2 = max λi A T A

12

.

0 0 0⎥ ⎢0 ⎥ ⎢ 1⎥ 0 ⎥, B = ⎢ ⎢ 1 0⎥ ⎢ ⎥ ⎣0 0⎦ 0 0

⎤

The spacecraft rendezvous system is illustrated in Fig. 1. We assume that the target spacecraft is in a circular orbit whose radius

2

0 1 0 2n 0 0

⎡

2. Dynamic models and the problem formulation

denote  A F =

a2i j

1

1 0 0 0 −2n 0

⎤

0 ⎢ 0 ⎢ ⎢ 0 A=⎢ ⎢ 3n2 ⎢ ⎣ 0 0

 

0 0 0 0 0 0 0 0 0 0 0 −n2

(4)

0 0 0 0⎥ ⎥ 0 0⎥ ⎥ D, 0 0⎥ ⎦ 1 0 0 1

(5)

444

Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450

⎧ 2n2 x n2 R ⎪ ⎪ ⎨ f 1 ( X ) = −2 μ + μ − σ R − σ x, n y f 2 ( X ) = μ − σ y, ⎪ ⎪ ⎩ f ( X ) = n2 z − σ z . 3

√ 

μ

If we drop out the second-order function f ( X ) and the saturation nonlinearity from (4), we obtain the well-known C–W equation X˙ = A X + BU . By the second order Taylor expansion of σ at the origin, we have

σ≈

1 R3

−

3

6

3

R

R

2R 5

x+ 4

x2 − 5

y2 −

3 2R

z2 . 5

(7)

Thus f i ( X ) can be rewritten as

⎧ ⎪ f ≈ − R34 x2 + ⎪ ⎨ 1 f 2 ≈ R34 xy , ⎪ ⎪ ⎩ f ≈ 3 xz. 3

Set

+

(6)

3 2R 4

(8)

μ f ( X ) =  A X , where ⎡ 0 0 0

0 0 0

3μ 2R 4

3μ z R4

y

0 0 0

3μ 2R 4

z

0 0 0 0

0 0 0 0

⎤

0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥.

0

0

⎥ 0 0 0⎥ ⎦

0

0

0 0 0

(9)

0 ⎢ 0 ⎢ ⎢ 0 √

E =⎢ ⎢

⎤

0 0 0

0 0⎥ ⎥ 0⎥

⎥

(10)

, 0⎥ ⎥ 0⎦

3

⎢ 2 α 0 ⎣ 0 α 0

α

0

√ ⎡1 0 0 0 0 0⎤ 2λl 3 ⎣ 0 1 0 0 0 0 ⎦, F= 3α

⎡ ⎢ H =⎢ ⎣

0

− 1l x √

1 2l

0 y

1 1 z 2l

0

⎤

0

0 ⎥ ⎦,

3 z 2l

0

0

√

μ

with λ = R 4 and re-expressed as

0

(11)

0

x2 + y 2 + z2

(14)

which completes the proof.

2

Proposition 2. The matrix pair ( A , B ) is controllable, ( A , F ) is observable and all the eigenvalues of A are on the imaginary axis, namely, ( A , B ) is asymptotically null controllable with bounded controls (ANCBC) [36] and

⎤

F ⎢ FA ⎥

⎢ ⎥ . ⎥ = 6. ⎣ .. ⎦

rank ⎢

The whole rendezvous process can be described by the transformation of state vectors X (t ) from nonzero initial states X (t 0 ) to the terminal states X (t f ) = 0, where t f is the rendezvous time [38]. In this paper, we will study the spacecraft rendezvous problem by taking the following issues into consideration:

possible;

• A satisfactory convergence performance. To meet these requirements, we will design a discrete gain scheduled controller which works in the following manner: 1). When the initial condition is far from the origin, the norm of the gain is small so as to guaranteeing convergence; 2). The norm of the gain is increasing during the convergence of the state, which helps to increase the convergence speed of the overall system; 3). The gain is no longer changing when the convergence rate of the closed-loop system is large enough. The following standard lemma will be used in this paper. Lemma 1. (See [7].) Let L , E and N be real matrices of appropriate dimensions with N T N ≤ I . Then for any scalar δ > 0, there holds LN E + E T N T L T ≤ δ −1 LL T + δ E T E. 3. The main results

⎥

y

3 2l

l2

• The actuator saturation nonlinearity; • The parameter uncertainties caused by linearization error; • An estimation of the domain of attraction that is as large as

According to [16], the C–W equation is accurate enough when the relative distance between the target  spacecraft and the chaser spacecraft is less than 50 km, that is x2 + y 2 + z2 ≤ l  50 km. Under this condition, we can derive a bound on the second-order term in (4). Notice that we can write  A = E H F , where

⎡

1

F A5

R4

⎢ ⎢ ⎢ ⎢ 3μ A = ⎢ ⎢ − R4 x ⎢ 3μ ⎢ 4y ⎣ R

2l

2 ⎞ 12 z ⎠

≤ I3,

⎡

3 2 z , 2R 4

y2 +

= I3

3

(12)

α = min{|α X | , |αY | , |α Z |}. So system (4) can be

X˙ = ( A + E H F ) X + Bsat(U ).

(13)



Proposition 1. If x2 + y 2 + z2 ≤ 50 km, ∀t ∈ R, then the timevarying matrix H satisfies H T H ≤ I 3 , ∀t ∈ R.

In this section, we construct a discrete gain scheduled controller that solves the spacecraft rendezvous problem as formulated in the previous section. Our solution is mainly based on a class of parametric algebraic Riccati equations (AREs), which is motivated by the results in [36]. Proposition 3. Let ( A , B ) be given by (5), F be given by (11) and γ be a given scalar. Consider the following ARE

A T P + P A − P B B T P + F T F + γ P = 0,

(15)

and the associated feedback gain K = − B P . T

Proof. By recognizing the special structure of H , we can derive

H T H ≤  H 22 I 3 ≤  H 2F I 3

⎛

2  2  2  √ 2  1 1 1 3 ⎝ − x + = I3 + y z + y l

2l

2l

2l

1. The ARE (15) has a unique positive definite solution P (γ ) for any γ ∈ R. γ 2. There holds λi ( A + B K ) ≤ − 2 , i ∈ I [1, 6]. Hence if γ ≥ 0, then the state of the closed-loop system  x˙ γ(t ) = ( A + B K ) x (t ) converges to the origin no slower than exp − 2 t .

Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450

3. For any γ ∈ R, the unique positive definite solution P (γ ) is continuously differentiable and strictly increasing with respect to γ , i.e. dP (γ ) /dγ > 0. Proof. 1). The ARE (15) can be rearranged as



A+

γ 2

T I6



P (γ ) + P (γ ) A +

γ

= 0.

2



T

T

I 6 − P (γ ) B B P (γ ) + F F (16)

Since ( A , B ) is controllable and ( A , F ) is observable, we know that ( A + γ2 I 6 , B ) is also controllable and ( A + γ2 I 6 , F ) is also observable. By the standard optimal control theory [6], we know that the ARE (16) has a unique positive definite solution P (γ ). 2). The ARE (15) can also be written as

A Te P (

T

! " E ( P γi ) = X ∈ R6 : ρ (γi ) X T P (γi ) X ≤ 1 , i ∈ I [0, N] , where



≤ −F T F ,

(17)

0 D K1

I3 D K2

,

(18)

k=1,2,3

Proposition 4. The γ -parametrized family of ellipsoids E ( P γi ) are nested family sets, that is, E ( P γ2 ) ⊂ E ( P γ1 ), whenever γ1 < γ2 . Proof. According to Item 3 of Proposition 3 we know that dρ (γ ) dP (γ ) /dγ > 0. Hence dγ ≥ 0 and thus

d dγ

Pγ =

dρ (γ ) dγ

P (γ ) + ρ (γ )

√

F

=

F (A + B K )

2λl 3 3α

I6,

(19)

which is nonsingular for arbitrary K . Hence ( A − B B T P (γ ), F ) is observable and thus ( A e , F ) is. Now choose the Lyapunov function V (x) = xT P (γ )x whose time-derivative along the trajectories of the linear system x˙ (t ) = A e x (t ) can be evaluated as

P (γ ) > 0.

(25)

2

Assume that the initial condition for system (4) comes from a prescribed bounded set ∈ R6 . Then we define γ0 as

!

If is known, the algorithm of computing γ0 can be found in Remark 4 given later. For any i ∈ [0, N ], consider the set

which is the area in the state space where the actuators with the control U = − B T P (γ ) X , are not saturated. In view of (24), we have, for any X ∈ E ( P γi ),

#2 # 1 # #2 # #2 1 # # # # T # # # B k P (γi ) X # ≤ # B kT P 2 (γi )# # P 2 (γi ) X # = B kT P (γi ) B k X T P (γi ) X = 1, k ∈ I [1, 3] .

(20)

⎤

⎢ ⎥ ⎢ .. ⎥ x = 0. ⎣ . ⎦

(22)

According to Item 2 of this proposition we know that A e is asymptotically stable. So the Lyapunov equation (22) has a unique posidP (γ ) tive definite solution, that is dγ > 0. The proof is finished. 2 Consider a set

 N = γ0 , γ1 , · · · , γ N , γi −1 < γi , i ∈ I [1, N] ,

E ( P γi ) ⊆ L i .

(29)

Hence the actuators are not saturated for any X ∈ E ( P γi ) and  sat B T P (γi ) X can be simplified as B T P (γi ) X , ∀ X ∈ E ( P γi ), namely,



X ∈ E ( P γi ) ⇒ sat B T P (γi ) X = B T P (γi ) X .

(30)

Then we can state the main result of this paper as follows.

As ( A e , F ) is observable, we have x ≡ 0, namely, the linear system x˙ (t ) = A e x (t ) is asymptotically stable. This is further equivalent to λi ( A − B K ) ≤ − γ2 , i ∈ I [1, 6]. 3). Differentiating both sides of (15) with respect to γ gives

γ ) dP (γ ) + Ae = − P . dγ dγ

(28)

Then it follows from (24), (27) and (28) that



(21)

F A 5e

dP ( A Te

(27)

≤ ρ (γi ) X T P (γi ) X

According to the LaSalle invariance principle [13], the state x (t ) converges to the set {x : F x ≡ 0} eventually. It follows that F x(i ) = F A ei x = 0, i = 1, 2, · · · , 5, and hence

F ⎢ F Ae ⎥

(26)

X ∈

≤ −x (t ) F F x (t ) ≤ 0.

"

γ0 = γ0 ( ) = min γ : ρ (γ ) X T P (γ ) X = 1 .

T

= −  F x (t )2

⎡

dγ

So the ellipsoids E ( P γi ) are nested. The proof is finished.

V˙ (x (t )) = xT (t ) ( P (γ ) A e + A Te P (γ ))x (t ) T

d

# ! # " # # Li  X : # B kT P (γi ) X # ≤ 1 , k ∈ I [1, 3] ,

from which it follows that





ρ (γi ) = max B kT P (γi ) B k and P γi = ρ (γi ) P (γi ).

γ

where A e = A + 2 I 6 − B B T P (γ ). We first show that ( A e , F ) is observable. For any constant matrix K = [ K 1 , K 2 ] we can compute

A + BK =

(24)

T

γ ) + P (γ ) A e = − P (γ ) B B P (γ ) − F F



445

(23)

where N is any given positive integer. For any γi ∈  N , define the ellipsoids associated with the quadratic function X T P (γ ) X as

Theorem 1. Let P (γ ) be the unique positive definite solution to the ARE in (15) and η j , j ∈ I [0, N] be a series of nonnegative numbers. Then the discrete gain scheduled controller

 $ ⎧ U 0 = −(1 + η0 ) B T P (γ0 ) X , X ∈ E  P γ0 $ E ⎪ ⎪ ⎪ U = −(1 + η ) B T P (γ ) X , X ∈ E P ⎨ E 1 1 1 γ1 U= .. ⎪ ⎪ . ⎪  ⎩ U N = −(1 + η N ) B T P (γ N ) X , X ∈ E P γN ,

  P γ1 , P γ2 ,

(31)

solves the spacecraft rendezvous problem and the set E ( P γ0 ) = { X : ρ (γ0 ) X T P (γ0 ) X ≤ 1} is within the domain of attraction of the closedloop system. Moreover, the controller U = U i −1 , i ∈ I [1, N] is active less than T i −1 seconds where

T i −1 ≤

1

γ i −1



ln



! " ρ (γi ) λmax P (γi ) P −1 (γi −1 ) . ρ (γi−1 )

(32)

446

Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450

2F T H T E T P (γi −1 ) ≤ F T H T H F + P (γi −1 ) E E T P (γi −1 )

Proof. Define the bounded sets

 $  Si −1 = E P γi−1 E P γi , i ∈ I [1, N] . 

X˙ = ( A + E H F ) X − Bsat (1 + ηi −1 ) B P (γi −1 ) X , ∀ X ∈ Si −1 . T

V˙ i −1 ( X ) ≤ ρ (γi −1 ) X T

Assume that at t = t i −1 , i ∈ I [1, N], the state X (t i −1 ) is on the boundary of the ellipsoid E ( P γi−1 ), that is, X (t i −1 ) ∈ ∂ E P γi−1 , or equivalently,



=⎣

V˙ i −1 ( X ) ≤ ρ (γi −1 ) X T

Let

(37)

μ i −1 I n ≥

= ρ (γi −1 ) X T (−γi −1 P (γi −1 ) − F T F − P (γi −1 ) B B P (γi −1 ) + 2F H E P (γi −1 )) X .

(47)

1

1 ρ (γi ) − 1 P 2 (γi −1 ) P (γi ) P − 2 (γi −1 ) , ρ (γi−1 )

(48)

or equivalently

− 2ρ (γi −1 ) X T P (γi −1 ) B B T P (γi −1 ) X T T

(46)

" ! 1 1 ρ (γi ) λmax P − 2 (γi −1 ) P (γi ) P − 2 (γi −1 ) ρ (γi−1 ) " ! ρ (γi ) = λmax P (γi ) P −1 (γi −1 ) , ∀i ∈ I [1, N] . ρ (γi−1 ) 1

T T

T

(45)

Since P − 2 (γi −1 ) P (γi ) P − 2 (γi −1 ) > 0, we can deduce that

+ 2ρ (γi −1 ) X F H E P (γi −1 ) X

T



μ i −1 

T T

 P (γi −1 ) A + A T P (γi −1 ) X T

(43)



V i ( X (t i −1 + T i −1 )) = 1.

 P (γi −1 ) A + A T P (γi −1 ) X

T



which indicates that X will converge to zero as t →∞, namely, the closed-loop system is asymptotically stable with E P γ0 contained in the domain of attraction. Now we give the upper bound estimation of T i −1 . When t i = t i −1 + T i −1 , the state coming from the boundary of E ( P γi−1 ) reaches at the boundary of E ( P γi ). So

which, in view of (30), can be further continued as



(42)

V˙ N ( X ) ≤ −γ N V N ( X ) < 0, ∀ X ∈ E ( P γN )\{0},

j =1

+ 2ρ (γi −1 ) X F H E P (γi −1 ) X   − 2ρ (γi −1 )satT B T P (γi −1 ) X B T P (γi −1 ) X

α −α

2 Z

Similarly to (43), the time-derivative of the Lyapunov function V N ( X ) = ρ (γ N ) X T P (γ N ) X along the trajectories of system (44) can be evaluated as

+ 2ρ (γi −1 ) X T F T H T E T P (γi −1 ) X 3   % − 2ρ (γi −1 ) satT B Ti P (γi −1 ) X B Ti P (γi −1 ) X

T

0

⎦.

2

(44)

 P (γi −1 ) A + A T P (γi −1 ) X

T

α 2 − αY2

⎤

0 0



B Ti P ( i −1 ) X

= ρ (γi −1 ) X T

0 0

0

X˙ = ( A + E H F ) X − Bsat (1 + η N ) B T P (γ N ) X , ∀ X ∈ E ( P γN ).

j =1



α 2 − α 2X

Hence the state will move to E P γi at finite time and thus will eventually move to E ( P γN ) and stay in E ( P γN ) thereafter. When X ∈ E ( P γN ), we have U = −(1 + η N ) B T P (γ N ) X and the closed-loop system becomes

+ 2ρ (γi −1 ) X T F T H T E T P (γi −1 ) X 3   % − 2ρ (γi −1 ) satT (1 + ηi −1 ) B Ti P (γi −1 ) X

≤ ρ (γi −1 ) X T

3 4

< 0, ∀ X ∈ Ci −1 \ {0}.

× B T P (γi −1 ) X   = ρ (γi −1 ) X T P (γi −1 ) A + A T P (γi −1 ) X



(41)

= −γi −1 V i −1 ( X )

+ 2ρ (γi −1 ) X T F T H T E T P (γi −1 ) X   − 2ρ (γi −1 )satT (1 + ηi −1 ) B T P (γi −1 ) X

γ

0 0 , 0

It is clear that ≤ 0 as α = min (|α X | , |αY | , |α Z |). Then we have E E T − B B T ≤ 0 and thus (40) simplifies to

T

× B P (γi −1 ) X )) P (γi −1 ) X   = ρ (γi −1 ) X T P (γi −1 ) A + A T P (γi −1 ) X

×

EE − BB =

V˙ i −1 ( X ) ≤ −γi −1 ρ (γi −1 ) X T P (γi −1 ) X



= 2ρ (γi −1 ) ( A + E H F ) X + Bsat −(1 + ηi −1 ) T

T

⎡

Then, for any X ∈ Si −1 , the time-derivative of V i −1 ( X ) along the trajectories of system (34) can be evaluated as

V˙ i −1 ( X ) = 2ρ (γi −1 ) X˙ T P (γi −1 ) X



T

where

(36)

(40)

In view of structures of the matrices B in (5) and E in (10), we can compute

(35)

Now we choose the following Lyapunov function

V i −1 ( X ) = ρ (γi −1 ) X T P (γi −1 ) X .



− γi −1 P (γi −1 )    + P (γi −1 ) E E T − B B T P (γi −1 ) X .

(34)

V i −1 ( X (t i −1 )) = X T (t i −1 ) P γi−1 X (t i −1 ) = 1, i ∈ I [1, N] .

(39)

Then (38) can be continued as

With the controller (31), we have the closed-loop system



= F T F + P (γi −1 ) E E T P (γi −1 ).

(33)

ρ (γi−1 ) P (γi−1 ) ≥ (38)

On the other hand, it follows from Lemma 1 and Proposition 1 that

1

μ i −1

ρ (γi ) P (γi ) .

(49)

On the other hand, it is easy to see from (43) that, for any t ∈ [t i −1 , t i ), we have

Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450





V i −1 ( X (t )) ≤ V i −1 ( X (t i −1 ))e−γi−1 t −t i−1 .

(50)

Then it follows from (35), (46) and (50) that

e

 −γi −1 t i −t i −1

=e

 −γi −1 t i −t i −1

P (0) =

⎡ ⎣

V i −1 ( X (t i −1 ))

447

1.4429 × 10−11 −1.0508 × 10−13 3.6609 × 10−25 3.1724 × 10−9 1.0028 × 10−7 −1.0508 × 10−13 1.5901 × 10−15 −5.0208 × 10−27 −4.569 × 10−11 −7.2139 × 10−10 − 25 − 27 − 14 − 20 3.6609 × 10 −5.0208 × 10 5.2566 × 10 2.5306 × 10 1.7725 × 10−21 3.1724 × 10−9 −4.569 × 10−11 2.5306 × 10−20 1.1256 × 10−5 2.1791 × 10−5 − 7 − 10 − 21 − 5 1.0028 × 10 −7.2139 × 10 1.7725 × 10 2.1791 × 10 0.00069938 7.5374 × 10−20 4.6772 × 10−24 4.9399 × 10−13 2.1224 × 10−17 6.912 × 10−16

7.5374 × 10−20 4.6772 × 10−24 4.9399 × 10−13 2.1224 × 10−17 6.912 × 10−16 9.9397 × 10−6

⎤ ⎦.

It is easy to verify that B ⊂E ( P (0)).

≥ V i −1 ( X (t i −1 + T i −1 )) T

= ρ (γi −1 ) X (t i −1 + T i −1 ) P (γi −1 ) X (t i −1 + T i −1 ) ρ (γi ) T X (t i −1 + T i −1 ) P (γi ) X (t i −1 + T i −1 ) ≥

μ i −1

= =

1

V i ( X (t i −1 + T i −1 ))

μ i −1 1

μ i −1

(51)

.

Remark 4. For any initial state X (t 0 ), lution to the nonlinear equation

So the upper bound estimation of T i −1 can be calculated as

T i −1 ≤

=

1

γ i −1 1

γ i −1

ρ (γ0 ) X 0T P (γ0 ) X 0 = 1.

ln (μi −1 )

 ln



" ! ρ (γi ) λmax P (γi ) P −1 (γi −1 ) . ρ (γi−1 )

The proof is finished.

Remark 3. From the proof of Theorem 1, we can see that the controller (31) is applied to system (13) in the order U 0 → U 1 → · · · → U N −1 → U N . According to Proposition 3, γ represents the convergence rate of the closed-loop system. Hence the convergence rate of the closed-loop system becomes faster and faster as the time increases. Hence the designed controller can improve the transient performances of the closed-loop system.

(52)

2

A couple of remarks regarding Theorem 1 are given in order.

⎧ 1  γ ⎨ τ 2 e− i−2 1 t −t i−1  X (t ) , t ∈ [t , t ), i −1 i −1 i i −1  X (t ) ≤ ⎩ 12 − γN (t −t N )  X (t N ) , t ∈ [t N , ∞), τN e 2

Remark 5. The sets  N in (23) can be designed arbitrarily. Two simple methods can be described as follows. The first method is an exponential growth method:

τi =

(53)

(54)

Thus, the state X (t i −1 ) converges to the boundary of the ellipsoid γ i −1

E ( P γi ) from the boundary of E ( P γi−1 ) no slower than e− 2 t . When t ≥ t N , the state converges to the origin no slower than γN e− 2 t . Moreover, after at most T ( N ) seconds, where T (N ) =

N %

T i −1 ,

(55)

i =1

the controller (31) will not switch and the control law becomes a linear one. Remark 2. When

γ → 0, the ARE in (15) becomes

P (0) A + A T P (0) − P (0) B B T P (0) + F T F = 0,

γ i = γ0 +  γ0 +



λmax P (γi ) 

, i ∈ I [1, N] . λmin P (γi )

(58)

where γ > 1 is a given constant. The second method is a linear growth method:

for any i ∈ I [1, N], where



(57)

The unique solution of (57) can be solved by the bisection method in view of the monotonicity of P γ with respect to γ .

γi = γ0 γ i , i ∈ I [1, N] , Remark 1. It follows from (50) that

γ0 can be chosen as the so-

(56)

which has a unique positive definite solution P (0) since ( A , B ) is controllable and ( A , F ) is observable [6]. Hence the ellipsoid E ( P (0)) = ρ (0) X T P (0) X is the maximal domain of attraction for system (13) that the controller (31) can achieve. We use an example  to demonstrate that the ellipsoid E ( P (0)) contains the ball B = X = [x, y , z, 0, 0, 0]T :  X  ≤ 50 km within which the C–W equation is accurate enough [16]. Suppose that the target spacecraft is on a geosynchronous orbit of radius R = 42 241 km thus n = 7.2722 × 10−5 rad/s and μ = 3.986 × 1014 m3 /s2 . Let the maximal accelerations supplied by the thrusts & &in the three directions satisfy, respectively, |ax | ≤ 0.1 N/kg, &a y & ≤ 0.1 N/kg and |a z | ≤ 0.1 N/kg. Then we solve the ARE (56) to get

i N i

(γ∞ − γ0 )

γδ , i ∈ I [1, N] , N where, γ∞ > γ0 is a given constant.

(59)

Remark 6. The controller in Theorem 1 can be implemented as follows. Set a current variable i whose initial value is i = 0 and the corresponding controller is U = U 0 . If i ≤ N − 1, for each X (t ), we calculate

f (X) =

1

ρ (γi )

− X T P (γi ) X .

(60)

If f ( X ) ≥ 0, we set U = U i +1 and let i = i + 1; otherwise we set U = Ui. 4. Simulation results In this section, a numerical simulation will be carried out to demonstrate the effectiveness and the advantages of the proposed approach to the circular orbital rendezvous. We emphasize that, differently from the controller design, our simulation will be carried out directly on the nonlinear plant described by Eq. (1). Suppose that the target spacecraft is on a geosynchronous orbit of radius R = 42,241 km with an orbital period of 24 h. Thus, the angle velocity n = 7.2722 × 10−5 rad/s and the gravitational parameter μ = 3.986 × 1014 m3 /s2 . For simulation purpose, choose the initial condition in the target orbital coordinate system as

X (0)

T  = 10,000 m 10,000 m 10,000 m 5 m/s 3 m/s −1 m/s ,

Assume that the maximal accelerations supplied by the thrusts in the three directions satisfy, respectively, |ax | ≤ 0.5 N/kg, |a y | ≤ 0.5 N/kg and |a z | ≤ 0.5 N/kg.

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Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450

Fig. 4. Relative distances  and velocities with N = 50 and N = 0, where d =  x2 + y 2 + z2 and v = x˙ 2 + y˙ 2 + z˙ 2 .

Fig. 2. Relative positions in the X -axis, Y -axis and Z -axis with N = 50 and N = 0.

Fig. 5. Control accelerations in the X -axis, Y -axis and Z -axis with N = 50 and N = 0.

Fig. 3. Relative velocities in the X -axis, Y -axis and Z -axis with N = 50 and N = 0.

In view of Remark 4 and the prescribed initial condition, we get

γ0 = 0.00267. We choose the exponential growth method to design  N , where γ = 1.01 and N = 50. Let ηi = 100, here i ∈ I [0, 50]. With these parameters, the unique positive definite solution to the parametric ARE (15) can be computed accordingly. The resulting discrete gain scheduled controller can be constructed according to (31). For comparison purpose, the closed-loop system will also be simulated for N = 0 which corresponds to a static state feedback controller. The state trajectories of the closed-loop system are plotted in Fig. 2, Fig. 3 and Fig. 4 and the control accelerations for the closed-

loop system are recorded in Fig. 5, respectively. From Fig. 2 to Fig. 4, we can see that the rendezvous mission is accomplished by using the discrete gain scheduled controller (31) at about t f = 3000 s with N = 50 which saves about 2000 s compared with the static state feedback controller corresponding to N = 0. In fact, the time used by the gain scheduled controller is about 60% of the time used by the static state feedback controller corresponding to N = 0, which shows the effectiveness of the proposed approach. Fig. 5 shows that the proposed controller not only makes full use of the actuator capacity, but also guarantees that the control inputs don’t exceed the maximal control inputs during the whole rendezvous process. To illustrate that the system dynamic performance becomes better and better as the value of N increases, we carry out the

Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450

449

crete gain scheduled controller based on the parametric Riccati equation is proposed to solve this problem with a large estimation of the domain of attraction. The main advantage of the proposed approach is that the convergence rate of the state can be improved by the well design of the parameters. Simulation results show the effectiveness of the proposed robust gain scheduling method. Conflict of interest statement The authors claim no conflicts of interests. References

Fig.  6. Relative distances and  velocities with N = 50, N = 25 and N = 0, where d = x2 + y 2 + z2 and v = x˙ 2 + y˙ 2 + z˙ 2 .

Fig. 7. Control accelerations in the three directions with N = 50, N = 25 and N = 0.

simulation for N = 50, N = 25 and N = 0, respectively. The relative distances [x, y , z] and the relative velocities [˙x, y˙ , z˙ ] are recorded in Fig. 6. From this figure we can see that the rendezvous mission is accomplished at about t f = 4000 s with N = 25 which saves about 1000 s comparing with N = 0, and expends more 1000 s than with N = 50. Fig. 7 illustrates that the proposed controller not only makes better use of the actuator capacity with the value of N increases, but also guarantees that the control inputs don’t exceed the maximal control inputs provided by the thruster equipment. 5. Conclusion This paper has investigated the robust control problem for circular orbital spacecraft rendezvous system with input saturation and parameter uncertainties caused by linearization error. A dis-

[1] T.E. Carter, State transition matrices for terminal rendezvous studies: brief survey and new example, J. Guid. Control Dyn. 21 (1) (1998) 148–155. [2] T.E. Carter, J. Brientt, Fuel-optimal rendezvous for linearized equations of motion, J. Guid. Control Dyn. 15 (6) (1992) 1411–1416. [3] T.E. Carter, M. Humi, Fuel-optimal rendezvous near a point in general keplerian orbit, J. Guid. Control Dyn. 10 (6) (1987) 567–573. [4] W.H. Clohessy, R.S. Wiltshire, Terminal guidance system for satellite rendezvous, J. Aeronaut. Sci. 27 (9) (1960) 653–658. [5] J.A. De Dona, S.O. Reza Moheimani, G.C. Goodwin, A. Feuer, Robust hybrid control incorporating over-saturation, Syst. Control Lett. 38 (3) (1999) 179–185. [6] J.C. Doyle, K. Glover, P.P. Khargonekar, B.A. Francis, State-space solutions to standard H 2 and H ∞ control problems, IEEE Trans. Autom. Control 34 (8) (1989) 831–847. [7] G.R. Duan, H.H. Yu, LMIs in Control Systems: Analysis, Design and Applications, CRC Press, 2013. [8] X.Y. Gao, K.L. Teo, G.R. Duan, Robust H ∞ control of spacecraft rendezvous on elliptical orbit, J. Franklin Inst. Eng. Appl. Math. 349 (8) (2012) 2515–2529. [9] F. Grognard, R. Sepulchre, G. Bastin, Improving the performance of low-gain designs for bounded control of linear systems, Automatica 38 (10) (2002) 1777–1782. [10] E. Hartley, P. Trodden, A. Richards, J. Maciejowski, Model predictive control system design and implementation for spacecraft rendezvous, Control Eng. Pract. 20 (7) (2012) 695–713. [11] T. Hu, Z. Lin, On improving the performance with bounded continuous feedback laws, IEEE Trans. Autom. Control 47 (9) (2002) 1570–1575. [12] T. Hu, Z. Lin, Y. Shamash, Semi-global stabilization with guaranteed regional performance of linear systems subject to actuator saturation, Syst. Control Lett. 43 (3) (2001) 203–210. [13] H.K. Khalil, Nonlinear Systems, third edition, Prentice Hall, Upper Saddle River, 2002. [14] D.J. Leith, W.E. Leithead, Survey of gain-scheduling analysis and design, Int. J. Control 73 (11) (2000) 1001–1025. [15] Z. Lin, Global control of linear systems with saturating actuators, Automatica 34 (7) (1998) 897–905. [16] D. Liu, Y. Zhao, The Dynamics of Space Vehicle, Harbin Institute of Technology Press, 2003. [17] Y.J. Liu, S.C. Tong, Adaptive NN tracking control of uncertain nonlinear discretetime systems with nonaffine dead-zone input, IEEE Trans. Cybern. 45 (3) (2015) 497–505. [18] Y.J. Liu, C.L.P. Chen, G.X. Wen, S.C. Tong, Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems, IEEE Trans. Neural Netw. 22 (7) (July 2011) 1162–1167. [19] Y. Luo, G. Tang, Spacecraft optimal rendezvous controller design using simulated annealing, Aerosp. Sci. Technol. 9 (8) (2005) 732–737. [20] A. Megretski, Output feedback stabilization with saturated control, in: Proceedings of the 13th IFAC World Congress, San Francisco, CA, 1996, pp. 435–440. [21] J. Park, K. Choi, S. Lee, Orbital rendezvous using two-step sliding mode control, Aerosp. Sci. Technol. 3 (4) (1999) 239–245. [22] M.E. Polites, Technology of automated rendezvous and capture in space, J. Spacecr. Rockets 36 (2) (1999) 280–291. [23] W.J. Rugh, J.S. Shamma, Research on gain scheduling, Automatica 36 (10) (2000) 1401–1425. [24] P. Singla, K. Subbarao, J.L. Junkins, Adaptive output feedback control for spacecraft rendezvous and docking under measurement uncertainty, J. Guid. Control Dyn. 29 (4) (2006) 892–902. [25] R. Suarez, J. Alvarez-Ramirez, J. Solis-Daun, Linear systems with bounded inputs: global stabilization with eigenvalue placement, Int. J. Robust Nonlinear Control 7 (9) (1997) 835–845. [26] A.R. Teel, Linear systems with input nonlinearities: global stabilization by scheduling a family of H∞ -type controllers, Int. J. Robust Nonlinear Control 5 (5) (1995) 399–411.

450

Q. Wang et al. / Aerospace Science and Technology 42 (2015) 442–450

[27] A. Tiwari, J. Fung, J.M. Carson, et al., A framework of Lyapunov certificates for multi-vehicle rendezvous problems, in: Proceedings of the 2004 American Control Conference, Boston, Massachusetts, 2004, pp. 5582–5587. [28] Q. Wang, B. Zhou, G. Duan, Global stabilization of spacecraft rendezvous system subject to input saturation via gain scheduling, in: The 32nd Chinese Control Conference, Xi’an, China, 2013, pp. 1436–1441. [29] Q. Wang, B. Zhou, G. Duan, Discrete gain scheduled control of input saturated systems with applications in on-orbit rendezvous, Acta Autom. Sin. 40 (2) (2014) 208–218. [30] G.F. Wredenhagen, P.R. Belanger, Piecewise-linear LQ control for systems with input constraints, Automatica 30 (3) (1994) 403–416. [31] Z. Wu, P. Shi, H. Su, J. Chu, Passivity analysis for discrete-time stochastic Markovian jump neural networks with mixed time-delays, IEEE Trans. Neural Netw. 22 (10) (2011) 1566–1575. [32] S. Wu, Z. Wu, G. Radice, Adaptive control for spacecraft relative translation with parametric uncertainty, Aerosp. Sci. Technol. 31 (1) (2013) 53–58.

[33] E.A. Youmans, F.H. Lutze, Neural network control of space vehicle intercept and rendezvous maneuvers, J. Guid. Control Dyn. 21 (1) (1998) 116–121. [34] W. Zhao, H. Chen, Adaptive tracking and recursive identification for Hammerstein systems, Automatica 45 (2009) 2773–2783. [35] W. Zhao, T. Zhou, Weighted least squares based recursive parametric identification of the submodels of a PWARX system, Automatica 48 (2012) 1190–1196. [36] B. Zhou, Parametric Lyapunov approach to the design of control systems with saturation nonlinearity and its applications, Ph.D. Dissertation, Harbin Institute of Technology, China, 2010. [37] B. Zhou, Z. Lin, G. Duan, Robust global stabilization of linear systems with input saturation via gain scheduling, Int. J. Robust Nonlinear Control 20 (4) (2010) 424–447. [38] B. Zhou, Z. Lin, G.R. Duan, Lyapunov differential equation approach to elliptical orbital rendezvous with constrained controls, J. Guid. Control Dyn. 34 (2) (2011) 345–358.