ISA Transactions 53 (2014) 241–250

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Research Article

Finite-time fault tolerant attitude stabilization control for rigid spacecraft Xing Huo a, Qinglei Hu b,n, Bing Xiao b a b

College of Engineering, Bohai University, Jinzhou 121000, China Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

art ic l e i nf o

a b s t r a c t

Article history: Received 17 June 2013 Received in revised form 27 November 2013 Accepted 28 November 2013 Available online 21 December 2013 This paper was recommended for publication by Dr. Qing-Guo Wang

A sliding mode based finite-time control scheme is presented to address the problem of attitude stabilization for rigid spacecraft in the presence of actuator fault and external disturbances. More specifically, a nonlinear observer is first proposed to reconstruct the amplitude of actuator faults and external disturbances. It is proved that precise reconstruction with zero observer error is achieved in finite time. Then, together with the system states, the reconstructed information is used to synthesize a nonsingular terminal sliding mode attitude controller. The attitude and the angular velocity are asymptotically governed to zero with finite-time convergence. A numerical example is presented to demonstrate the effectiveness of the proposed scheme. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Spacecraft Attitude stabilization Finite-time Sliding mode control Reconstruction Actuator fault

1. Introduction Attitude stabilization is a necessary maneuver for orbital spacecraft. Although several important developments have been witnessed in the design of feedback control for performing maneuvers [1,2], there still remain certain open problems. In particular, from the standpoint of external disturbances rejection, there currently exists no unified framework for design simple control. The stabilization problem is further complicated by the system uncertainties [3]. To achieve attitude stabilization with high-pointing accuracy and stability, a number of approaches have been proposed. Unknown moment of mass inertia for a rigid spacecraft was investigated in [4]. Rest to rest maneuver was accomplished in [5] by developing an optimal controller. In [6], external disturbances were handled by using backstepping technique. In [7], ℋ1 control was applied to guarantee robustness to disturbances. In [8], external disturbances and uncertain inertia parameters were addressed by using adaptive control. Sliding Mode Control (SMC) [9–11] is an effective approach for uncertain systems with highly coupled and nonlinear dynamics.

n Correspondence to: Department of Control Science and Engineering, Harbin Institute of Technology, P.O. box 327, No.92 West Da-Zhi Street, Harbin, Heilongjiang Province 150001, China. Tel.: þ 86 451 86402726. E-mail addresses: [email protected], [email protected] (Q. Hu).

The first attempt to design attitude controller by using SMC was made in [12], and then further pursued in [13]. In [14], a smooth SMC algorithm was derived. The proposed controller accomplished attitude tracking in the presence of external disturbance. In [15], rotational maneuver of a flexible spacecraft was studied. System uncertainties, disturbances, and input saturation were considered. A adaptive SMC-based attitude tracking control was presented in [16] with uncertain inertia matrix and external disturbances addressed. In [17], a disturbance observer was proposed, and a sliding mode controller was then designed to improve attitude control performance. Although most of the previously mentioned SMC ensures robustness to system uncertainties and external disturbances, finite-time convergence of the attitude and the angular velocity are not guaranteed. Such infinite-time settling time is not an option during critical phases of some high real-time mission. Consider a military satellite tasked with providing coverage of a specific high priority area. If the attitude cannot be stabilized in finite time, ground objectives may be lost. In [18], a global set approach was presented to stabilize the attitude in finite time. Attitude tracking problem was investigated in [19]. Finite-time reachability of the desired attitude motion was achieved in the presence of model uncertainties and disturbances. A discontinuous finite-time control law was proposed for a rigid spacecraft in [20] to govern the attitude to a small region of the origin. In [21],

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.017

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X. Huo et al. / ISA Transactions 53 (2014) 241–250

a faster finite-time convergent attitude controller was reported by using a fast terminal sliding mode. In [22], a robust SMC design was proposed to guarantee a finite-time convergence of the attitude tracking errors. In [23], a SMC scheme was proposed to achieve finite-time attitude stabilization. System uncertainties and disturbances were handled by adaptive method. A global finitetime control strategy was developed in [24]. The controller was synthesized based on a homogeneous method, actuator saturation was handled. When solving the problem of finite-time attitude control, tolerating actuator fault is another key issue. In aerospace engineering, some catastrophic faults or failures may occur due to aging of actuators [25]. Fault Tolerant Control (FTC) [26–28] is a widely used scheme to accommodate component failures with guaranteed attitude control performance ensured. Excellent overviews of FTC are provided in [29]. Passive FTC, one type of the available schemes for FTC, is characterized by designing only a fixed controller to compensate for faults without any on-line fault information. Hence, a variety of passive FTC algorithms have been developed for spacecraft [30]. Automated attitude recovery for rigid and flexible spacecraft was investigated in [31]. In [32], a passive FTC controller was derived to achieve reliable attitude stabilization with actuator outage fault. In [33], attitude tracking control with reaction wheel faults was examined for a rigid satellite. Attitude tracking was investigated in [34], the spacecraft was subject to thrusters failures. An adaptive FTC was presented in [35]. The attitude was stabilized in the presence of slow-varying satellite mass, and several fault scenarios of rotating solar flaps. The authors in [36] looked at a terminal SMC approach for satellite formations flying. However, stability analysis was not provided when the faults occurred. One of the drawbacks of the preceding FTC schemes is that, the attitude stabilization may not be accomplished in finite-time when actuator undergoes faults. Although a terminal SMC scheme was derived in [37] to perform the rest-to-rest maneuver with finitetime convergence, only the degradation of actuation effectiveness fault was tolerated. In view of handling the potential problems of finitetime attitude control and tolerating much more types of actuator faults, this work present a theoretical framework for attitude stabilization of a rigid spacecraft controlled by using reaction wheels. A sliding observer is first designed to precisely reconstruct the sum of external disturbance, actuator fault, and reaction wheel angular momentum. Using the reconstructed information, a nonsingular terminal SMC law is then developed with a compensation effort to handle the faults. The main contribution is that the proposed approach compensates for possible actuator faults with the attitude stabilized to zero in finite time. In comparison with SMC-based attitude control scheme such as [20], the proposed control is able to handle actuator faults. Although both two control schemes can accomplish the attitude stabilization maneuver in finite time, the methodology presented in [20] can only govern the attitude to a small set containing zero, while the developed controller can stabilize the attitude to zero. Furthermore, in contrast to the above-cited works on finite-time attitude control such as [21], the designed controller has capability to tolerate actuator faults, while the scheme presented in [21] does not have such ability. The paper is organized as follows. In Section 2, mathematical model of a rigid spacecraft, reaction wheel faults, and problem formulation are presented. In Section 3, a terminal SMC-based FTC solution is proposed for attitude stabilization with finite-time convergence. Section 4 demonstrates the application of the approach to a rigid spacecraft. Conclusions are given in Section 5.

2. System description and problem formulation Throughout this paper, jjU jj stands for the standard Euclidean norm or its induced norm. I n A ℜnn denotes a n-by-n identity matrix. For a vector x ¼ ½ x1 x2 ⋯ xn T A ℜn , a vector function is

defined as sgnðxÞ ¼ ½ sgnðx1 Þ sgnðx2 Þ ⋯ sgnðxn Þ T , where sgnð UÞ denotes the standard signum function. The pseudo-inverse of any matrix A A ℜmn is represented by A† . 2.1. Mathematical model of spacecraft attitude system Let ℱi denotes the inertial frame, ℱo denotes the orbital reference frame, and ℱb denotes the body-fixed frame of the spacecraft. Let the angular velocity of the spacecraft expressed in ℱi , be denoted by ωbi A ℜ3 . To achieve 3-axis attitude stabilization control with reliability guaranteed, redundant reaction wheels are equipped for the spacecraft considered here. Assume that the number of the reaction wheels fixed is N(N 4 3), the dynamics of a rigid spacecraft attitude motion is described by [38]  _ bi ¼  ω Jω bi Jωbi  ωbi h þ Du þ d 33

ð1Þ 3

where J A ℜ is the positive-definite inertia matrix, h A ℜ is the sum of the angular momentum of those N reaction wheels, D A ℜ3N is the installation matrix of actuators, u A ℜN is the output torque of actuators, d A ℜ3 is the bounded but unknown external disturbance, and the operator ξ for any vector ψ ¼ ½ ψ 1 ψ 2 ψ 3 T denotes the skew-symmetric matrix as 2 3 ψ2 0 ψ3 6 0 ψ 1 7 ψ ¼ 4 ψ3 ð2Þ 5 ψ2 ψ1 0 To represent the spacecraft attitude, the unit-quaternion is used. The vector Q ¼ ½ q0 qT T A ℜ4 subjected to q20 þ qT q ¼ 1, composed of a vector q A ℜ3 , denotes the unit-quaternion mapping from ℱb to ℱo . The rotation matrix that brings ℱb onto ℱo , denoted by RðQ Þ A ℜ33 , is defined as RðQ Þ ¼ ðq20  qT qÞI 3 þ 2qqT  2q0 q . The angular velocity of the spacecraft with respect to ℱo is defined by ωbo ¼ ωbi  RðQ Þωoi . Here, ωoi ¼ ½ 0  n 0 T is the velocity of ℱopwith ffiffiffiffiffiffiffiffiffiffiffiffirespect to ℱi , n is the mean angular velocity and equals μe =a3c (μe is the gravitational parameter of the Earth and ac is the distance from the center of the Earth to the _ Þ ¼  ω RðQ Þ, one has spacecraft’s center of mass). Using RðQ bo  _ bo ¼ ω _ bi þ ωbo RðQ Þωoi . ω From (1), the dynamic model for the described rigid spacecraft can be expressed as [38] q_ 0 ¼  0:5qT ωbo

ð3Þ

q_ ¼ 0:5ðq þ q0 I 3 Þωbo

ð4Þ 

_ bo ¼  ðωbo þ RðQ Þωoi Þ J½ωbo þ RðQ Þωoi  Jω  þ Jω bo RðQ Þωoi  ½ωbo þ RðQ Þωoi  h þ Du þ d

ð5Þ

2.2. Reaction wheel faults In [39], it is discussed that reaction wheel is sensitive device. They are vulnerable to faults such as (F1) Decreased reaction torque, (F2) Increased bias torque, (F3) Continuous generation of reaction torque, and (F4) Failure to respond to control signals. For each reaction wheel, F1–F4 can be mathematically modeled as ui ¼ ei ðtÞuci þ uci ;

i A f1; 2; …; Ng

ð6Þ

where uci is the desired torque, ei ðtÞ is the actuator fault indicator represented by a quantitative value in the range of 0 to 1, and uci is the uncertain input fault.

 No fault: ui ¼ uci , ei ðtÞ ¼ 1, and uci ¼ 0.  F1: 0 o ei ðtÞ o1 and uci ¼ 0.  F2: ei ðtÞ ¼ 1, and uci goes to a non-zero value of the actuator bias.

X. Huo et al. / ISA Transactions 53 (2014) 241–250

243

 F3: ei ðtÞ ¼ 0 and uci a 0. The actuator has frozen leading to τi ¼ uci

 F4: ei ðtÞ ¼ 0 and uci ¼ 0. Taking fault (6) into consideration, the actual torque u generated by reaction wheels is u ¼ uc þ Pðt  TÞ½ðEðtÞ  I N Þuc þ uc  ¼ uc þ uf ault

ð7Þ

where uc ¼ ½ uc1 uc2 ⋯ ucN  is the torque commanded by the controller, uf ault is the faulty torque, EðtÞ ¼ diagðe1 ðtÞ; e2 ðtÞ; …; eN ðtÞÞ is the actuator effectiveness matrix, and uc ¼ ½ uc1 uc2 ⋯ ucN T . The matrix function Pðt  TÞ A ℜNN with T ¼ ½ t 1 t 2 ⋯ t N T A ℜN denotes the time-profiles of faults. It is a diagonal matrix of the form: T

Pðt  TÞ ¼ diagðp1 ðt  t 1 Þ; p2 ðt  t 2 Þ; …; pN ðt  t N ÞÞ

ð8Þ

where t i , i A f1; 2; …; Ng denotes the unknown fault-occurrence time, pi : ℜ↦ℜ is a function representing the time-profile of a fault affecting the ith reaction wheel. We consider faults with time-profiles modeled by ( 0 if t o t i pi ðt  t i Þ ¼ ð9Þ 1  e  ai ðt  ti Þ if t Z t i where the scalar ai 4 0 denotes the unknown fault evolution rate. It worth mentioning that the fault time-profile (9) denotes only the developing speed of a fault, whereas all its other basic features are defined by ðEðtÞ  I N Þuc þ uc . 2.3. Transformed attitude dynamics Define a new matrix F ¼ 0:5ðq þ q0 I 3 Þ, take time-derivative of (4) and pre-multiplying both sides of the resulting expression by ðF  1 ÞT , the following dynamics can be obtained from (4) and (5) with fault (7) _ q_ þðF  1 ÞT H 1 ¼ ðF  1 ÞT Duc J n ðQ Þq€ þ CðQ ; qÞ þðF  1 ÞT ½Duf ault þ d þ ðF 1 q_ þRðQ Þωoi Þ h

ð10Þ

1 _  JF  1 ; _ ¼ ðF  1 ÞT J F_ CðQ ; qÞ þ ðF  1 ÞT ðF  1 qÞ

and

_  JRðQ Þωoi  JðF  1 qÞ _  RðQ Þωoi : þ ðF  1 qÞ Remark 1. Note that obtaining (10) requires that F is invertible. Hence, it should guarantee that 8t Z0

The control objective to be achieved can be stated as follows: Consider the rigid spacecraft attitude system described by (3)– (5) in the presence of external disturbances and reaction wheel faults (7), design a command control uc to achieve attitude stabilization with the attitude q and the angular velocity ωbo guaranteed to zero in finite time.

3. Main results A SMC-based controller will be developed to achieve asymptotic attitude stabilization. In the approach, reaction wheel faults, angular momentum, and external disturbances are reconstructed. The closed-loop attitude system is shown in Fig. 1. The control input uc contains two term: (a) a normal control uc0 and (b) a compensation control ucom . The effort uc0 is to eliminate the nominal portion of the derivative of (5). The compensation control ucom synthesized from the reconstructed information is to compensate for faults, angular momentum of reaction wheel, and external disturbances. As a result, the controller uc is able to achieve asymptotic stability.

To reconstruct external disturbance, reaction wheel faults, and theirs angular momentum, a new variable is defined as τ lumped ¼ Duf ault þd þ ðF 1 q_ þ RðQ Þωoi Þ h. The problem is then changed into the reconstruction of τ lumped . Define a generalized _ using Property 2, (10) can be rewritten as moment inertia J g ¼ J n ðQ Þq, J_ g ¼ ðF  1 ÞT Duc  τ lumped  H 2 ðQ Þ

H 1 ¼ ðRðQ Þωoi Þ JF  1 q_ þ ðRðQ Þωoi Þ JRðQ Þωoi

detðFÞ ¼ 0:5q0 ðtÞ a0;

2.4. Problem formulation

3.1. Reconstruction of faults, wheel momentum, and disturbances

where J n ðQ Þ ¼ ðF  1 ÞT JF  1 ;

Fig. 1. The proposed attitude control system.

ð13Þ

_ where τ lumped ðtÞ ¼  ðF  1 ÞT τ lumped , and H 2 ðQ Þ ¼ ðF  1 ÞT H 1 þCðQ ; qÞ n _ q_  J_ ðqÞq. Using (13), the following residual vector is presented: Z t rðtÞ ¼ k ½ðF  1 ÞT Duc H 2 ðQ Þ  rðsÞds  kJ g ð14Þ 0

ð11Þ

To guarantee that (11) remains valid, we will require that the initial conditions be restricted such that q0 ð0Þ a0, and the controller be designed to ensure q0 ðtÞ a0 for t 4 0. Regarding the restriction on the initial conditions, it is clear from the unity constraint of Q that the desired trajectory can always be initialized to guarantee q0 ð0Þ a 0. Therefore, the initial condition restriction is actually a very mild restriction on the desired trajectory.

where k is a positive scalar. The residual signal rðtÞ is then such that r_ ¼  kr þ kτ lumped . It is known from the definition of τ lumped that τ lumped is continuous. Hence, τ lumped can be assumed to be a differentiable function with time-derivative τ A ℜ3 . The reconstruction of τ lumped is then formulated as observing a linear system driven by the residual rðtÞ and by an unknown input. That linear system is described by

Property 1. The matrix J n ðQ Þ is positive-definite and symmetric.

ξ_ 1 ¼  kξ1 þ kξ2

ð15Þ

_ and the time-derivative of J ðQ Þ Property 2. The matrix CðQ ; qÞ are such that [14]

ξ_ 2 ¼ τ

ð16Þ

y ¼ ξ1

ð17Þ

n

n

_ xT ðJ_ ðQ Þ  2CðQ ; qÞÞx ¼ 0;

x A ℜ3

ð12Þ

244

X. Huo et al. / ISA Transactions 53 (2014) 241–250

where ξ1 ¼ r, ξ2 ¼ τ lumped , τ is the unknown input, and y is the measured output. The initial vector ξ2 ð0Þ is chosen such that ξ2 ð0Þ ¼ 0. With the only available output y, a sliding mode observer is proposed: _ ξ^ 1 ¼  kξ^ 1 þ kξ^ 2  ξv  η2 e1

ð18Þ

_ ξ^ 2 ¼  η3 e1  η4 ½xv q=p  η5 sgnðxv Þ

ð19Þ

where

ξv ¼ ½ ξv1

ξv2

ξv3 T ¼ η1 sgnðe1 Þ,½ξv q=p ¼ ½ jξv1 jq=p sgnðξv1 Þ

jxv2 jq=p sgnðξv2 Þjxv3 jq=p sgnðξv3 ÞT , e1 ¼ ξ^ 1  y, ηi , iA f1; 2; 3; 4; 5g are positive gains, p and q are integers such that q o p. Introduce the observer error e ¼ ½ eT1 eT2 T A ℜ6 with e2 ¼ ξ^ 2  ξ2 , the dynamics of e can be obtained from (15) and (16) and (18) and (19) that

The definition of W 1 ðtÞ and (26) results in jje1 ðtÞjj  0 for t ZT F1 ¼ jje1 ð0Þjj=ðk þ η2 Þ. Accordingly, the sliding motion takes place on e1 ¼ e_ 1 ¼ 0 by t ¼ T F1 . When the system states reach the sliding surface e1 ¼ e_ 1 ¼ 0, solving for equivalent output injection yields ðξv Þeq ¼ e2 , where the subscript “eq” denotes the equivalent state on the sliding surface e1 ¼ e_ 1 ¼ 0 in the sense of sliding mode control theory: ðξv Þeq ¼ ðl1 sgnðe1 ÞÞeq ¼ ke2

ð27Þ

(P2) Finite-time convergence of e2 : Once the sliding motion (e1 ¼ e_ 1 ¼ 0) is achieved after T F1 , the error dynamics has the form: e_ 1 ¼ 0

ð28Þ ð29Þ

e_ 1 ¼  ke1 þke2  ξv  η2 e1

ð20Þ

q=p e_ 2 ¼ η4 k ½e2 q=p  η5 sgnðke2 Þ  τ

e_ 2 ¼  η3 e1  η4 ½ξv q=p  η5 sgnðξv Þ  τ

ð21Þ

Choosing another candidate Lyapunov function for the resulted error dynamics (28) and (29)

Remark 2. Although the input τ is unknown, the amplitude of τ is bounded by a known scalar ε0 4 0, i.e., jjτjj r ε0 . According to (16), it is known that τ is the time-derivative of the lumped fault τ lumped . That means that, τ denotes the rate of the lumped fault. Furthermore, it is obtained from (8) and (9) that small values of ai characterize slowly developing faults, also known as incipient faults. For large values of ai , the time profile bi approaches a step function that models abrupt faults. Hence, the upper bound of τ mainly depends on the time profiles Pðt  TÞ. The larger ai , i A f1; 2; …; Ng is, the larger τ becomes. Because ai may be unknown in practical engineering, a large value of ε0 should be chosen when implementing the designer controller. Theorem 1. For the linear system (15)–(17), with the application of the sliding mode observer (18) and (19), if k and η5 are chosen such that η5 k  ε0 4 0

Proof. To prove Theorem 1, Lyapunov's direct method is adopted, and it is divided into the two parts. (P1) Finite-time convergence of e1 : Consider the following candidate Lyapunov function for the error e1 : ð23Þ

then, V_ 1 r ðk þ η2 Þjje1 jj2  ðη1  kjje2 jjÞjje1 jj

ð24Þ

Using the same analysis method as given in [40], it follows from (24) that V_ 1 r  ðk þ η2 Þjje1 jj2 if a relatively large η1 is chosen to satisfy η1 4 kjje2 jj þε1 , where ε1 40 is a scalar. This ensures that V_ 1 o 0 for all jje1 jja 0, and leads to finite-time convergence of e1 to the surface e1 ¼ 0. Alternatively, choosing η1 4 maxt A ½0;T 1  jje2 jj where T 1 is greater than the time it takes for e1 to converge to the sliding surface, allows a fixed value for η1 and also guarantees that V_ 1 o0 for all jje1 jj a0. The observation error e1 will converge to zero. To show that pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi finite-time convergence, define a function W 1 ðtÞ ¼ 2V 1 ðtÞ ¼ jje1 jj. With the choice of gain η1 4 kjje2 jjþ ε1 , then _ 1 ðtÞ _ 1 ðtÞ ¼ pVffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r  ðk þ η2 Þ W 2V 1 ðtÞ

ð25Þ

Using the comparison theorem in [41], it is established from (25) that W 1 ðtÞ r W 1 ðe1 ð0ÞÞ  ðk þ η2 Þt

ð26Þ

ð30Þ

Differentiating (30) and inserting (22) result in q=p V_ 2 ¼ eT2 ð  η4 k ½e2 q=p  η5 sgnðke2 Þ  τÞ

jje2 jjðp þ qÞ=p  ðη5 k  ε0 Þjje2 jj q=p ðp þ qÞ=2p o  η4 k 2 ðV 2 Þðp þ qÞ=2p r  η4 k

q=p

ð31Þ

Because p o q ) ðp þ qÞ=ð2qÞ o 1, solving (31) yields V 2 ðtÞ  0 when ðp  qÞ=2p

t Zt F2 ¼

ð22Þ

where ε0 4 0. Then, sliding motion is achieved in finite-time on e1 ¼ 0 and e2 ¼ 0, the amplitude of external disturbances, angular momentum and faults of reaction wheel, can be exactly reconstructed in finite time.

V 1 ðtÞ ¼ 0:5eT1 e1

V 2 ðtÞ ¼ 0:5eT2 e2

2pV 2

ð0Þ

q=p ðp þ qÞ=2p

ðp  qÞ

η4 k

2

ð32Þ

Thus, one finds e2 ðtÞ  0 after finite-time t F2 . Further, the sliding motion is achieved in t F2 on e2 ¼ 0. Consequently, the lumped information τ lumped ¼ ξ2 ðtÞ is precisely estimated by ξ^ 2 ðtÞ within t F2 . With the definition of τ lumped , one has τ lumped ¼  F T τ lumped . Because τ lumped can be exactly estimated by ξ^ 2 ðtÞ, the lumped fault τ lumped is thus precisely reconstructed by τ lumped  F T ðQ Þξ^ 2 ðtÞ

ð33Þ

for t Zt F2 . The amplitude of external disturbances, reaction wheel faults, and theirs angular momentum is thus exactly reconstructed by (33) in finite time. Thereby the proof is completed. ■ As seen in Theorem 1 and its proof, the actuator fault reconstruction error is driven to zero in finite time. Hence, the proposed control approach can exactly reconstruct the fault no matter what type of the fault is. That is to say, the occurring fault can be constant, time-varying or even random signals. 3.2. Finite-time fault tolerant attitude control design To achieve attitude stabilization maneuver in finite-time, inspired by the terminal sliding surface presented in [39], a nonsingular sliding manifold is proposed by using higher-order sliding mode for the transformed attitude system (10): M ¼ ½ M1

M2

M 3 T ¼ q þ γ q_ m=n

ð34Þ

m=n where γ 4 0 is a scalar, q_ ¼ ½ jq1 jm=n sgnðq1 Þ jq2 jm=n sgn ðq2 Þjq3 jm=n sgnðq3 ÞT . The positive odd integers m and n satisfy 1 o m=n o 2.

Theorem 2. For the rigid spacecraft attitude system (3)–(5) with reaction wheel faults and external disturbances, assume that the

X. Huo et al. / ISA Transactions 53 (2014) 241–250

remaining active actuators are able to produce a combined torque sufficient enough to perform given maneuvers. If the nominal control uc0 and the compensation control ucom are designed as   n _ q_ þ ðF  1 ÞT H 1  J n ðQ Þq_ 2  m=n  μJ n ðQ ÞsgnðMÞ uc0 ¼ D† F T CðQ ; qÞ γm ð35Þ ucom ¼ D† F T ξ^ 2 ðtÞ

ð36Þ

Table 1 Main parameters of one rigid spacecraft. Mission

Imaging the earth

Mass (kg) Inertia moments (kg m2) Principal moments of inertia Products of inertia Variation range Orbit Type Altitude (km) The inclination (deg) Attitude control type Reaction wheel Speed range (rpm) Reaction torque (N m) Moment of inertia (kg m2)

65.00

245

where μ is a positive scalar, then the system will reach the nonlinear sliding manifold (34) in finite time. Furthermore, the attitude q and the angular velocity ωbo will converge to zero in finite time. Proof. As shown in Fig. 1, one has uc ¼ uc0 þ ucom . Inserting uc0 and ucom into the time-derivative of (34) yields _ ¼ q_ þ γ mdiagðq_ m=n  1 Þq€ M n m m=n  1 ¼  γ μdiagðq_ ÞsgnðMÞ n n o m m=n  1 n ½J ðQ Þ  1 ξ^ 2 ðtÞ þ ðF  1 ÞT τ lumped þγ q_ n

ð37Þ

Recall from Theorem 1 that τ lumped   F T ðQ Þξ^ 2 ðtÞ for t Zt F2 . For all the time t Z t F2 , it thus yields _ ¼  γ mμdiagðq_ m=n  1 ÞsgnðMÞ M n

55.0, 48.0, 50.0 2,  2.5, 1.5 5% Circular 600 95.4 Three axis control by four reaction wheels

ð38Þ

0.4

q1 q2

3500 0.1 0.0465

q3 0.2

The attitude q

ts + t f =59sec

Z

0

Y O -0.2

X

0

50

100

150

Time (sec)

Fig. 2. Configuration of four reaction wheels for spacecraft.

x 10-4

2

The reconstruction error e2 (Nm)

The reconstruction error e2 (Nm)

5

Fig. 4. The attitude q in the absence of faults.

0

-5

-10

t F2=17sec

e 21

-15

e 22 e 23

-20

0

5

10

15

20

Time (sec)

25

30

x 10-7

1.5 1 0.5 0 -0.5 -1 -1.5 -2 190

192

194

196

Time (sec)

Fig. 3. Reconstruction error e2 in the absence of faults.

198

200

200

246

X. Huo et al. / ISA Transactions 53 (2014) 241–250

Consider another candidate Lyapunov function as V 3 ¼ 0:5M T M, taking its time-derivative and substituting the closed-loop system (38) yield

i ¼ 1; 2; 3. With 1 o m=n o2, it leads to pffiffiffiffiffiffiffiffiffi V_ 3 r  γμλmin jjMjj r  γμλmin 2V 3

ð40Þ m=n  1

m m=n  1 ÞsgnðMÞ V_ 3 ¼  γ μM T diagðq_ n

ð39Þ

Because m and n are positive odd integers, it guarantees that m=n  1 Þ is a positive-definite diagonal matrix if any q_ i a 0, diagðq_

Þ. where λmin is the minimum eigenvalue of the matrix diagðq_ It is thus concluded from (40) that, the states will reach the sliding manifold M in finite-time t s r jjMð0Þjj=ðγμλmin Þ in the case of any q_ i a 0. Note that in (39) q_ i ¼ 0 may hinder the reachability of M. We now prove that q_ i ¼ 0 is not an attractor in the reaching phase. After finite-time t Z t F2 , substituting the control (35) and (36) into (10) yields q€ ¼ 

1 ωbo2

0.5

0 -7

q þ γ q_

m=n

¼0

ð42Þ

-0.5 The finite-time t f taken to travel from qðt s Þ a 0 to one qðt s þt f Þ ¼ 0 can be obtained as the solution to the first-order differential equation in (42). That is given by Z qi ðtsi þ tf Þ i 1 t f i ¼  γ n=m dqi m=n qi ðt si Þ a 0 q i  m  q ðt si Þ1  n=m ¼ γ n=m ð43Þ mn i

3 -1

2 190

195

200

-1.5 t + t =59sec s

f

-2 50

100 Time (sec)

150

200 It thus can be concluded from (4) that the attitude q and the velocity ωbo asymptotically converge to zero in finite time t s þt f . This completes the proof. ■

Fig. 5. The velocity ωbo in the absence of faults.

0.1

0.1

0.05

0.05

uc2 (Nm)

uc1 (Nm)

0

0

-0.05

-0.05

-0.1

-0.1 0

100

200

0

Time (sec) 0.1

0.1

0.05

0.05

0

-0.05

-0.1

-0.1 100

Time (sec)

200

0

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0

100

Time (sec)

uc4 (Nm)

0

uc3 (Nm)

The angular velocity

4

x 10

ð41Þ

In the case of q_ i ¼ 0 and M i a 0, one has q_ i ¼ μsgnðM i Þ a 0. Hence, the finite-time reachability of the sliding manifold M is still guaranteed. When the sliding manifold M ¼ 0 is reached, one has

ωbo3

ωbo

(deg/sec)

ωbo1

n 2  m=n q_  μsgnðMÞ γm

200

0

100

Time (sec)

Fig. 6. The commanded control torque uc in the absence of reaction wheel faults.

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247

Fig. 7. Reconstruction of τ lumped . The solid line denotes the reconstructed value, while the dashed-line denotes the real value.

x 10-5 e 21

0.05

The reconstruction error e2 (Nm)

The reconstruction error e2 (Nm)

3

e 22 e 23

0

tF2=95sec

-0.05

-0.1 0

50

100

Time (sec)

150

200

2 1 0 -1 -2 -3 -4 -5 300

2000

4000

Time (sec)

Fig. 8. Reconstruction error e2 in the presence of faults.

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X. Huo et al. / ISA Transactions 53 (2014) 241–250

Remark 3. Due to the usage of sign function in nominal control (35), chattering effect may be caused. This chattering is practically undesirable since it may excite the neglected high frequency dynamics. This work will apply a commonly used approach to reduce the chattering. In this approach [42], the discontinuous function sgnðMÞ is approximated by a continuous M=ðsgnðMÞ þ πÞ, where π is a positivel scalar. As a result, the control (35) can be written as  n _ q_  J n ðQ Þq_ 2  m=n uc0 ¼ D† F T CðQ ; qÞ γm  M ð44Þ þ ðF  1 ÞT H 1  μJ n ðQ Þ sgnðMÞ þπ

4. Simulation results To verify the effectiveness of the proposed control, a rigid spacecraft is numerically simulated. The physical parameters of the spacecraft are listed in Table 1. The configuration of those four reactions wheels is illustrated in Fig. 2. Three wheels are mounted orthogonally, aligned with the body-fixed axes, i.e., þX, þY and þZ, respectively. A fourth, redundant, wheel is mounted skewed at equal angles (54.71) to each of the body axes, aligned diagonally in the þ X, þY, þZ quadrant. This “skew” wheel could be used to provide control about any of the other axes if one of the orthogonal wheels were to fail. With such configuration, the installation matrix D can be obtained as 2 pffiffiffi 3 1 0 0 1= 3 pffiffiffi 7 6 7 D¼6 4 0 1 0 1=p3 ffiffiffi 5 0 0 1 1= 3 The external disturbances for d in (1) are calculated as in [38]. To accomplish the planned image taking mission, it requires the attitude system to provide 0.0051 attitude pointing accuracy with 0.00121/s stability. The initial attitude of the spacecraft is Q ð0Þ ¼ ½ 0:89 0:350:25  0:12T with an initial body angular velocity of ωbo ð0Þ ¼ ½ 2  1 0:75 T 1/s. To implement the developed control, we should first choose m, n, p, q, η5 , and k such that 1 o m=n o 2, 0 o q o p, and (22), respectively; the other control gains are chosen by trial-and-error until a good stabilization performance is obtained. The control gains for the controller are finally chosen as m ¼ 11, n ¼ 9, γ ¼ 0:7, π ¼ 0:001, and μ ¼ 0:05. The gains for the observer (18) and (19) are chosen as q ¼ 9; p ¼ 20; η1 ¼ 0:015;η2 ¼ 0:075;η3 ¼ 0:05;η4 ¼ η5 ¼ 0:001; and k ¼ 10:5.

uc can completely compensate for the external disturbances d and wheel angular momentum ω bi h. Consequently, the closed-loop system is guaranteed to be asymptotically stable. The angular velocity is illustrated in Fig. 5. It results in an attitude stability of 4.0e  71/s. The associated commanded torque uc is shown in Fig. 6. 4.2. Performance in the presence of actuator faults To investigate the controller performance, the reaction wheels mounted on the spacecraft are assumed to be faulty: ● The reaction wheel mounted in line with the X-axis of ℱb experiences F1 with e1 ðtÞ ¼ 0:3, and uc1 ¼ 0 after 20 s. ● The actuator mounted in line with the Y-axes of ℱb undergoes F2 after 10 s, and the bias torque is uc2 ¼  0:005 N m. ● The reaction wheel fixed in line with the Z-axes of ℱb is in normal operation in the first 10 s. It then experiences F3 with uc3 ¼ 0:01 and e3 ¼ 0 between the 10th second and the 40th second. Finally, F4 occurs. ● The redundant reaction wheel is always healthy. When the controller is implemented, Fig. 7 shows the successful fault reconstruction using the incorporated sliding mode 0.4 q1 q2

0.3

The attitude q

248

q3

0.2

ts + t f = 95 sec

0.1

0

-0.1

-0.2

0

50

100

150

200

Time (sec) Fig. 9. The attitude q in the presence of faults.

An ideal case is simulated. The attitude stabilization maneuver is performed in the absence of reaction wheel faults. It thus leads to τ lumped ¼ d þ ðF 1 q_ þ RðQ Þωoi Þ h. Fig. 3 shows the error between τ lumped and its reconstruction value in (33). The nonlinear observer (18) and (19) achieve finite-time convergence of the reconstruction error e2 . That finite-time is roughly 17 s. More specifically, from the steady-state behavior of e2 as shown in the right plot of Fig. 3, high reconstruction accuracy ( o2.0e 7) for e2 is guaranteed. From the results, it is known that the observer can precisely reconstruct the external disturbances and reaction wheel angular momentum, when all actuators are fault-free. The nonsingular terminal sliding mode controller uc results in the attitude q shown in Fig. 4. The attitude stabilization maneuver is successfully accomplished in a finite-time of 59 s. That is because the lumped fault τ lumped can be exactly estimated by the observer (18) and (19). The compensation effort ucom included in

The angular velocity ωbo (deg/sec)

1

4.1. Performance in the absence of reaction wheel faults

ω bo1 ω bo2 ω bo3

0.5

0 ts + t f =95 sec

-0.5 2

-1

x 10-4

0

-1.5 -2 190

195

200

-2

0

50

100

150

Time (sec) Fig. 10. The velocity ωbo in the presence of faults.

200

0.1

0.1

0.05

0.05

uc2 (Nm)

uc1 (Nm)

X. Huo et al. / ISA Transactions 53 (2014) 241–250

0

0

-0.05

-0.05

-0.1

-0.1 0

100

200

0

0.1

0.1

0.05

0.05

0

200

0

-0.05

-0.05

-0.1

-0.1 100

100

Time (sec)

uc4 (Nm)

uc3 (Nm)

Time (sec)

0

249

200

0

Time (sec)

100

200

Time (sec)

Fig. 11. The commanded control torque uc in the presence of reaction wheel faults.

observer. It can be observed from Fig. 7 that, the error between τlumped i , i ¼ 1; 2; 3 and its reconstruction is less than 2.0e 4 N m, respectively. Hence, a good reconstruction of actuator faults is obtained. Moreover, as shown by the steady-state behavior of the fault reconstruction error e2 in Fig. 8, the accuracy of the reconstruction are smaller than 2e  5 N m. Due to the precise reconstruction information of the reaction wheel faults, reaction angular of momentum, and external disturbances supplied by the reconstruction scheme (18) and (19), the controller uc can completely compensate for actuator faults and reject external disturbances. As a result, the controller produces an asymptotical convergence of the velocity tracking error and the attitude tracking error. The resulted attitude is shown in Fig. 9, and the angular velocity is illustrated in Fig. 10. As the steady-state behavior shows in Fig. 10, the attitude stability is within 2e  41/s. On the other hand, by transforming the unit quaternion into attitude Euler angle [38], the resulted attitude control performance in Fig. 9 corresponding to the attitude pointing accuracy is o 0.0011. The obtained attitude stability and pointing accuracy satisfy a set of stringent pointing requirements to accomplish the planned mission, even in the face of faults and external disturbances. The corresponding commanded torque is illustrated in Fig. 11. It is interesting to observe that uc2 and uc3 is never zero, even when the attitude is asymptotically stabilized. That is due to the fact that, reaction wheel F2 and F3 needs to be compensated. Summarizing above two cases, it can be concluded that the proposed approach can stabilize the attitude in finite time whether actuator fault occurs or not; moreover, extensive simulations were also performed in the presence of time-varying faults. It was also shown that the error between the lumped fault τlumped i , i ¼ 1; 2; 3 and its reconstruction is always guaranteed to be less than 2.0e  4 N m, respectively. The control synthesized by using the precisely reconstructed fault information, stabilized the attitude and the angular velocity to zero in finite time. The results

verified that the proposed controller performed very well and accomplished the attitude stabilization maneuver despite this type of severe fault.

5. Conclusions A finite-time control approach was developed for rigid spacecraft attitude stabilization. To deal with the external disturbances and reaction wheel faults simultaneously, a nonlinear observer was first synthesized to precisely reconstruct those amplitudes and zero observer errors were achieved in finite-time. Then, a nonsingular terminal sliding mode controller is investigated and guaranteed that the attitude stabilization was accomplished in finite time. Contrary to the existing fault tolerant attitude control schemes, the attitude and angular velocity converged to zero in finite-time even in the presence of faults and external disturbance. However, the drawback of the scheme remains its dependence on the availability of the upper bound ε0 on the rate of the lumped fault τ lumped . To implement the control, a relatively large ε0 was usually chosen. As a result, conservativeness may be resulted. For future work, this issue should be investigated. This may be done using adaptive control technique to estimate the upper bound ε0 online.

Acknowledgments This present work was supported by the National Natural Science Foundation of China (61004072, 61273175), Program for New Century Excellent Talents in University (NCET-11-0801), Heilongjiang Province Science Foundation for Youths (QC2012C024), Research Fund for Doctoral Program of Higher Education of China (20132302110028), Scientific Research Fund of Hunan Provincial Education Department of Liaoning Province (L2010009) and National Key Basic Research Program of China (2013CB035605). The authors would also like to

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