Event-triggered backstepping control for attitude stabilization of spacecraft

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Event-triggered backstepping control for attitude stabilization of spacecraft Feng Wang a, Mingzhe Hou b,∗, Xibin Cao a, Guangren Duan b a Research b Center

Center of Satellite Technology, Harbin Institute of Technology, Harbin, China for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, China

Received 20 October 2018; received in revised form 14 May 2019; accepted 2 September 2019 Available online xxx

Abstract To decrease the communication frequency between the controller and the actuator, this paper addresses the spacecraft attitude control problem by adopting the event-triggered strategy. First of all, a backstepping-based inverse optimal attitude control law is proposed, where both the virtual control law and the actual control law are respectively optimal with respect to certain cost functionals. Then, an event-triggered scheme is proposed to realize the obtained inverse optimal attitude control law. By designing the event triggering mechanism elaborately, it is guaranteed that the trivial solution of the closed-loop system is globally exponentially stable and there is no Zeno phenomenon in the closedloop system. Further, the obtained event-triggered attitude control law is modified and extended to the more general case when the disturbance torque cannot be ignored. It is proved that all states of the closed-loop system are bounded, the attitude error can be made arbitrarily small ultimately by choosing appropriate design parameters and the Zeno phenomenon is excluded in the closed-loop system. In the proposed event-triggered attitude control approaches, the control signal transmitted from the controller to the actuator is only updated at the triggered time instant when the accumulated error exceeds the threshold defined elaborately. Simulation results show that by using the proposed event-triggered attitude control approach, the communication burden can be significantly reduced compared with the traditional spacecraft control schemes realized in the time-triggered way. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author. E-mail address: [email protected] (M. Hou).

https://doi.org/10.1016/j.jfranklin.2019.09.010 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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1. Introduction Attitude control systems which provide spacecraft with attitude stabilization or tracking capabilities are of significant importance for spacecraft to fulfill the predetermined missions. Hence, attitude control of spacecraft is always a hot-spot problem in the past few decades and a variety of control approaches have been developed to solve the spacecraft attitude control problem, such as sliding mode control [1–4], optimal and inverse optimal control [5–9], adaptive control [10–13], finite time control [14–18], and so on. Among these approaches, the nonlinear optimal control is an eye-catching one. But it suffers from the complexity problem of solving the Hamilton–Jacobi–Bellman equation. To circumvent such a problem, the inverse optimal control approach is introduced to design the attitude controller for spacecraft in [7]. In [8], the attitude tracking control problem of rigid spacecraft with disturbances and an uncertain inertia matrix is addressed by using the adaptive control method and the inverse optimality approach. Ref. [9] presents an inverse optimal and robust nonlinear attitude control law by using the attitude quaternion feedback for regulation of rigid spacecraft. More details on the development of the inverse optimal spacecraft attitude control can be found in Refs. [19,20] and references therein. With the development of the spacecraft technology, low-cost plug-and-play spacecraft [21] has received considerable attention in recent years. Plug-and-play spacecraft is a tight integration of a set of independent functional modules, such as sensing module, processing module, actuation module, an so on. To reduce the weight of spacecraft, information transmission among these modules is usually completed through wireless communication networks instead of the cable. Considering that low-cost wireless networks only have limited resources, how to reduce the signal transmission frequency and power consumption is absolutely a problem of great practical value. Although there have been a huge number of results on the spacecraft attitude control in the literature, they are mostly based on the continuous time framework, and implemented on the digital platform in a periodic sampling or time-triggered way because of its simplicity. However, from the resource utilization point of view, the timetriggered way is often not an advisable one. For example, when external disturbances can be ignored, system states will almost keep constant and the variation between two successive control signals will be very little, as a result, it is not necessary to update the control signal. In other words, updating of the control signal is in fact only required when it is necessary rather than periodically without intermission, and the extra operating is a waste of limited resources. To realize updating the control signal only when it is necessary and reduce the communication frequency between the controller and the actuator, the event-triggered control strategy (see for example [22]) was introduced to design the spacecraft attitude control law by some scholars very recently. The pioneering work is given in [23], where a sliding-mode-based event-triggered control approach is proposed for the spacecraft attitude stabilization problem, and two different kinds of event-triggered strategies, i.e., the fixed and the relative threshold strategies, are proposed. In [24], the event-triggered attitude stabilization problem is solved for spacecraft with disturbances also by employing the sliding mode control approach. Another result which utilizes the sliding mode control approach to solve the event-based spacecraft attitude stabilization problem can be found in [25]. In [26], a Lyapunov-based event-triggered control approach is proposed to solve the problem of spacecraft attitude stabilization. In [27,28], the self-triggered attitude stabilization problem is further considered based on the approximately linearized model of spacecraft. In [29], based on the approximately linearized Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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model, an event-triggered control method is proposed to deal with the attitude control problem for spacecraft with external disturbances and actuator faults by designing a learning observer to reconstruct actuator faults and disturbances. And the authors, also based on the approximately linearized model, further solve in [30] the attitude cooperative control problem for multi-spacecraft under undirected information flow. In [31], an event-triggered adaptive neural network control approach is proposed for post-capture flexible spacecraft considering pre-specified tracking performance. Generally speaking, research on the event-triggered control of spacecraft is still in its infancy stage. The number of the literature on the event-triggered attitude control of spacecraft is very limited (only about a dozen) up to now and the related technology is far from mature from the theory perspective. This motivates the presented investigation. In this paper, we consider the event-triggered spacecraft attitude stabilization problem by combining the backstepping technology and the inverse optimal approach. The main feature and contribution of the paper is summarized as follows. Firstly, a backstepping-based inverse optimal attitude control law is proposed. Compared with the existing backstepping-based inverse optimal attitude control law, e.g., Ref. [7], it has the feature that not only the virtual control law but also the actual control law are respectively optimal with respect to certain cost functionals. Secondly, to avoid continuous control signal transmitting and receiving, an event-triggered scheme is proposed to realize the obtained backstepping-based inverse optimal attitude control law. It is proved that when the external disturbance torque can be ignored, the attitude error converges to zero exponentially due to the elaborately constructed event-triggered mechanism, while in [23] where the similar attitude control problem is studied, the attitude error only converges to a residual set around zero with adjustable radius. And thirdly, the proposed event-triggered attitude control law is further modified to deal with the more general case when the external disturbance torque could not be ignored. It is prove that the attitude error ultimately converges to a residual set around zero of which the radius can be adjusted arbitrarily. Simulation results show that by using the proposed event-triggered attitude control approach, the communication burden can be significantly reduced compared with the traditional spacecraft control schemes realized in the time-triggered way. 2. Main results 2.1. Backstepping-based inverse optimal attitude stabilization of spacecraft Under the rigid body assumption and in the case when the disturbance torque can be ignored, the dynamics of spacecraft can be obtained from the Euler’s moment equation as [32] J ω˙ = −Sω J ω + u,

(1)

where ω = [ω1 ω2 ω3 ]T ∈ R3 is the angular velocity of the spacecraft with respect to the inertial frame, expressed in the body fixed frame, J ∈ R3×3 , J = J T > 0 is the inertia matrix of the spacecraft, u is the control torque, and for any vector z = [z1 z2 z3 ]T ∈ R3 , the matrix Sx is defined as ⎡ ⎤ 0 −z3 z2 0 −z1 ⎦. Sz = ⎣ z 3 −z2 z1 0 Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Meanwhile, the kinematics of spacecraft can be established based on the modified Rodrigues parameters (MRPs) as [32] σ˙ = G(σ )ω,

(2)

where σ = [σ1 σ2 σ3 ]T ∈ R3 is the modified Rodrigues parameter vector which reflects the attitude of the spacecraft by the body fixed frame relative to the inertia frame, and  1  1 − σ 2 I3 + 2Sσ + 2σ σ T . 4 It is easy to check the following facts G(σ ) =

 1 1 + σ 2 σ T , 4  1 G(σ )σ = 1 + σ 2 σ. 4

σ T G(σ ) =

The objective of this subsection is to design a control law for system (1)–(2) by combining the backstepping method with the inverse optimal control method such that the globally asymptotical stability of the closed-loop system can be ensured, and both of the virtual control law and the actual control law are respectively optimal with respect to certain cost functionals. To facilitate the control law design, we introduce the following Lemma on inverse optimal control. Lemma 1. Consider the nonlinear system x˙ = f (x) + g(x)u, where x ∈ Rn denotes the state, u ∈ Rm denotes the input, f(x) and g(x) are both C 1 functions. Assume that associating with a positive definite, radially unbounded Lyapunov function V(x), the control law

T ∂V 1 u = ξ (x) = − R−1 (x ) g(x ) , 4 ∂x where R(x) = RT (x) and R(x) > 0 for all x ∈ Rn , globally asymptotically stabilizes the system such that ∂V  f (x) + g(x )ξ (x ) = −W (x) V˙ = ∂x with W(x) being a positive definite function. Then the control law

T ∂V 1 −1 ∗ ∗ u = ξ (x) = 2ξ (x) = − R (x ) g(x ) 2 ∂x also globally asymptotically stabilizes the system and is optimal with respect to the cost functional ∞  l (x) + uT R(x)u dt Jc,∞ = 0

with l (x) = W (x) = −

∂V  f (x) + g(x )ξ (x ) . ∂x

Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Moreover, the minimum value of the cost functional is Jc,∗ ∞ = V (0). The above lemma can be easily concluded from Theorem 3.19 in [33]. So the proof is omitted here. When applying Lemma 1 to the inverse optimal control design, a stabilizing feedback control law u = ξ (x) needs to be designed first with the help of a Lyapunov function V(x), then the functions l(x) and R(x) can be determined and the feedback law u∗ = ξ ∗ (x) = 2ξ (x) is just the optimal control law which optimizes the cost functional Jc,∞ . This approach is called “inverse” optimal because the cost functional is determined by the stabilizing feedback law a posteriori, rather than assigned by the designer a priori. The design of the inverse optimal spacecraft attitude control law will be completed by two steps via the backstepping technique. Step 1: Consider subsystem (2) and view ω as the virtual input. Design the virtual control law to be ωd = −Pσ, where P ∈ R3×3 , P = PT > 0 is a design parameter matrix. Define 1 V1 = σ T σ 2 and ωr = ω − ωd ,

(3)

(4)

(5)

then one has σ˙ = G(σ )(ωd + ωr ) = −G(σ )Pσ + G(σ )ωr

(6)

and V˙1 = σ T σ˙ = −σ T G(σ )Pσ + σ T G(σ )ωr   1 1 = − 1 + σ 2 σ T Pσ + 1 + σ 2 σ T ωr . 4 4

(7)

Remark 1. As shown in [34], the trivial solution of subsystem (2) with the control law ω = −Pσ

(8)

is asymptotically stable. There this fact is verified by choosing the Lyapunov function   L(σ ) = 2 ln 1 + σ T σ ,

(9)

of which the time derivative satisfies L˙ = −σ T Pσ.

(10)

However, by using this Lyapunov function, it seems difficult to show that the trivial solution of the closed-loop subsystem is exponentially stable as claimed in [35]. In this paper, a Lyapunov function (4) different from Eq. (9) is chosen. By using this Lyapunov function, it is easy to verify the exponential stability of the closed-loop subsystem. And as we will see, it is more convenient to use this Lyapunov function to analyze the stability of the event-triggered closed-loop system obtained finally. Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Remark 2. It is also shown in [35] that, for subsystem (2), control law (8) is an optimal one with respect to the following cost functional 1 ∞ T σ (t )Pσ (t ) + ωT (t )P−1 ω(t ) dt, Jc1,∞ = (11) 2 0 ∗ and the minimum value of the cost functional is Jc1 ,∞ = L(σ (0)).

Step 2: Consider the dynamic subsystem ω˙ r = ω˙ − ω˙ d = J −1 (−Sω J ω + u ) + PG(σ )ω. Define

 σ y= ωr

(12)

(13)

and uc = u − h(y),

(14)

where h(y) = Sω J ω − J PG(σ )ω − G(σ )σ.

(15)

Then it is easy to check that h(0) = 0, and the system consisting of Eqs. (6) and (12) can be rewritten into the following compact form

 −G(σ )Pσ + G(σ )ωr y˙ = −J −1 G(σ )σ + J −1 uc

 −G(σ )Pσ+ G(σ )ωr  = − 14 1 + σ 2 J −1 σ + J −1 uc = a(y) + Buc , (16) where

 −G(σ )Pσ + G(σ )ωr  , − 41 1 + σ 2 J −1 σ

 0 B = −1 . J

a(y) =

Define the Lyapunov function as 1 T 1 σ σ + ωrT J ωr . 2 2 Then it is easy to check that

  T 0 ∂V2 T ωr J = ωrT , B= σ J −1 ∂y

V2 =

(17)

(18)

and the time derivative of V2 satisfies that V˙2 = −

  1 1 1 + σ 2 σ T Pσ + 1 + σ 2 σ T ωr 4 4

Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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 1 1 + σ 2 ωrT σ + ωrT uc 4  1 = − 1 + σ 2 σ T Pσ + ωrT uc . 4

7

−

(19)

For any Q ∈ R3×3 , Q = QT > 0, denote R = Q−1 , it is easy to obtain that the control law

 1 −1 ∂V2 T 1 uc = ξ (y) = − R B = − Qωr 4 ∂y 4

(20)

globally asymptotically stabilizes system (16) since V˙2 = −

 1 1 1 + σ 2 σ T Pσ − ωrT Qωr = −W (y), 4 4

(21)

where  1 1 1 + σ 2 σ T Pσ + ωrT Qωr 4 4 is a positive definite function. According to Lemma 1, the following control law

 1 −1 ∂V2 T 1 ∗ ∗ uc = ξ (y) = 2ξ (y) = − R B = − Qωr 2 ∂y 2

W (y) =

also globally asymptotically stabilizes system (16). Besides, this control law is optimal with respect to the cost functional ∞  l (y) + ucT Ruc dt, Jc2,∞ = (22) 0

where  1 1 1 + σ 2 σ T Pσ + ωrT Qωr . 4 4 ∗ And the minimum value of the cost functional is Jc2, ∞ = V2 (0). To sum up, we have the following theorem. l (y) = W (y) =

(23)

Theorem 1. The control law 1 u∗ = uc∗ + h(y) = − Qωr + h(y) (24) 2 globally asymptotically stabilize the system consisting of Eqs. (6) and (12), and is optimal with respect to the cost functional   ∞   T 1 1 T T −1 2 1 + σ  σ Pσ + ωr Qωr + (u − h(y) ) Q (u − h(y) ) dt. Jc2,∞ = (25) 4 4 0 ∗ Moreover, the minimum value of the cost functional is Jc2, ∞ = V2 (0).

Remark 3. It is noted that the cost functional (25) will not make no sense since h(0) = 0. In addition, if matrices P and Q are appropriately chosen such that the following matrix inequalities P ≥ ka I3 , Q ≥ kb I3 , Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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where ka >> k1b and kb > > 1, are satisfied, then the cost functional (25) is dominantly determined by the first term. As a result, recalling that the virtual control law is also optimal with respect to the cost functional Jc1,∞ , a shorter path stabilizing is ensured, that is, the closed-loop system could return to the desired attitude through a shorter path in the state space. Hence, the proposed inverse attitude control law provides the potential to overcome the unwinding problem, as discussed in [35]. 2.2. Event-triggered attitude stabilization of spacecraft without disturbances As discussed in Introduction, if the inverse optimal attitude control law (24) is realized in a continuously periodic sampling way, it may lead to unnecessary energy and resource consumption. To avoid this problem, an event-triggered scheme will be proposed to realize the inverse optimal attitude control law such that the frequency of information transmission though the communication channel can be reduced and the globally exponential stability of the resulted closed-loop system can be guaranteed. The following lemma will be used in the subsequent control law design. Lemma 2. For any x, y ∈ Rn and Q ∈ Rn×n , Q = QT > 0, 1 T x Qx + yT Q−1 y. 4 Proof. From the fact that xT y ≤

1 T x Qx + yT Q−1 y − x T y = 4



1 Qx + y 2

T Q

−1



 1 Qx + y ≥ 0, 2

the lemma can be concluded immediately.  Reconsider the system consists of Eqs. (6) and (12). Define  1 1 v = u∗ = Sω J ω − J PG(σ )ω − 1 + σ 2 σ − Qωr , (26) 4 2 and 1 1 V = σ T σ + ωrT J ωr . (27) 2 2 If u(t ) = v(t ), then one has  1 1 V˙ = − 1 + σ 2 σ T Pσ − ωrT Qωr . (28) 4 2 This implies that the trivial solution of the resulted closed-loop system is in fact globally exponentially stable. However, in order to reduce the information transmission frequency though the communication channel, the control law u(t ) = v(t ) should not be utilized directly. Instead, the following event-triggered control law is proposed u(t ) = v(ti ), (t ∈ [ti , ti+1 ), i = 0, 1, 2, . . . ),

(29)

where {ti , i = 0, 1, 2, . . .} is the sequence of time at which the control signal is updated. This means that u keeps constant during the interval [ti , ti+1 ). The updating time ti , i = 0, 1, 2, . . . is determined by the event triggering mechanism as follows   t0 = 0, ti+1 = inf t | t > ti , e(t )2 > α exp(−βt ) , i = 0, 1, 2, . . . , (30) Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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where e(t ) = v(t ) − u(t )

(31)

is the accumulated error of the control signal, α > 0 and β > 0 are two design parameters to be determined, and β is such that c β< , 2   1 1 1 1 where c = min λmin (P ), λmin (J − 2 QJ − 2 ) with J − 2 denoting the inverse of J 2 which is a positive definite symmetric matrix and of which the square is J. Substituting Eqs. (26), (29) and (31) into Eq. (12), one has

  1 1 (32) ω˙ r = J −1 − 1 + σ 2 σ − Qωr − e 4 2 Remark 4. It is easy from Eqs. (29), (30) and (31) to see that, once the event triggering condition in Eq. (30) is satisfied, the time instant will be marked as ti+1 and the control signal v(ti+1 ) will be transmitted to the actuator module. During the time interval [ti , ti+1 ), the control torque u(t) is kept as the constant value v(ti ). That is, no communication is needed to update u(t) during (ti , ti+1 ). Hence, the communication burden can be significantly reduced compared with the traditional spacecraft control approaches realized in the time-triggered way. For the stability of the resulted closed-loop system, one has the following theorem. Theorem 2. The trivial solution of the closed-loop system consisting of Eqs. (6) and (32) with the event triggering time sequence (31) is globally exponentially stable. In addition, there is no Zeno behavior in the event triggering time sequence. Proof. The time derivative of the Lyapunov function V defined in Eq. (27) satisfies that   1 1 1 + σ 2 σ T Pσ + 1 + σ 2 σ T ωr 4 4

  1 1 T −1 2   − 1 + σ σ − Qωr − e +ωr J J 4 2   1 1 ≤ − 1 + σ 2 σ T Pσ − ωrT Qωr − ωrT e 4 2 1 1 1 ≤ − σ T Pσ − ωrT Qωr + ωrT Qωr + eT Q−1 e 4 2 4 1 1 = − σ T Pσ − ωrT Qωr + eT Q−1 e, 4 4 where the following inequality

V˙ = −

1 T ω Qωr + eT Q−1 e 4 r which can be obtained from Lemma (2), is used. Bearing the event triggering condition in mind, one has   1 1 V˙ ≤ − σ T Pσ − ωrT Qωr + Q−1 e2 4 4  1  1   1 T 1 1 1 ≤ − σ Pσ − ωrT J 2 J − 2 QJ − 2 J 2 ωr + α Q−1  exp(−βt ) 4 4 −ωrT e ≤

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λmin (P ) T λmin (J − 2 QJ − 2 ) T ≤− σ σ− ωr J ωr + αλ−1 min (Q) exp (−βt ) 4 4 c ≤ − V + αλ−1 min (Q) exp (−βt ) 2 1

1

since c = min{λmin (P ), λmin (J − 2 QJ − 2 )}. According to the comparison lemma [36], one has 1

1

V (t ) ≤ φ1 (t ), where φ 1 (t) is the solution of the following differential equation c φ˙1 (t ) = − φ(t ) + αλ−1 min (Q) exp (−βt ), φ1 (0) = V (0). 2 Clearly, the solution is



 t ct c(t − s) + αλ−1 − exp(−βs)ds φ1 (t ) = φ(0) exp − (Q) exp min 2 2 0



 2αλ−1 2αλ−1 ct min (Q) min (Q) exp − + = V (0) − exp (−βt ). 2 (c − 2β ) (c − 2β ) Hence,





2αλ−1 2αλ−1 ct min (Q) min (Q) exp − + V (t ) ≤ V (0) − exp (−βt ). 2 (c − 2β ) (c − 2β ) If V (0) −

2αλ−1 min (Q) ≥ 0, (c − 2β )

then



2αλ−1 2αλ−1 min (Q) min (Q) exp (−βt ) + V (t ) ≤ V (0) − exp (−βt ) (c − 2β ) (c − 2β ) = V (0) exp (−βt ). Otherwise, V (t ) ≤

2αλ−1 min (Q) exp (−βt ). (c − 2β )

Hence, the following inequality V (t ) ≤ r 2 exp (−βt ) always holds, where   2αλ−1 2 min (Q) . r = max V (0), (c − 2β ) As a result,  T 1 z  σ J 2 ωr Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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satisfies

 β z ≤ r exp − t . 2 This implies that the trivial solution of the closed-loop system is globally exponentially stable. Next, we prove that there is no Zeno behavior in the event triggering time sequence. Simple computing shows that the closed-loop system can be re-presented in terms of z as follows

 σ˙ z˙ = 1 J 2 ω˙ r

 −G(σ )Pσ + G(σ )ωr   = 1 1 1 − 41 1 + σ 2 J − 2 σ − 21 J − 2 Qωr − J − 2 e = Mz + N e, where 

−G(σ )P   1 M= − 41 1 + σ 2 J − 2

 0 N= 1 . −J − 2

 1 G(σ )J − 2 , 1 1 − 21 J − 2 QJ − 2

On any interval [ti , ti+1 ), one has e˙(t ) = v˙(t ) − u˙ (t ) = v˙(t ) = =

∂v ∂v Mz + N e ∂z ∂z

∂v z˙ ∂z

    since u(t) is constant on [ti , ti+1 ). It is noted that  ∂v M  and  ∂v N  are both continuous ∂z ∂z   function of z which belongs to the compact set Br = {z : z ≤ r }. Therefore, both  ∂v M ∂z   and  ∂v N  have maximum value on Br . Their maximum values are denoted by γ 1 and γ 2 , ∂z respectively. From the event triggering condition, one can obtain that, for t ∈ [ti , ti+1 ),

 β 1 2 e(t ) ≤ α exp − t . 2 Hence, ∂v ∂v 2eT (t )e˙(t ) = 2eT Mz + 2eT N e ∂z ∂z ≤ 2γ1 ez + 2γ2 e2



 β β 1 2 ≤ 2γ1 α exp − t r exp − t + 2γ2 α exp(−βt ) 2 2 ≤ γ exp(−βt ), t ∈ [ti , ti+1 ) where 1

γ = 2γ1 rα 2 + 2γ2 α. That is,  d eT (t )e(t ) ≤ γ exp(−βt ), t ∈ [ti , ti+1 ). dt Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 1. Event triggering time.

Again, by the comparison lemma, one has that, for t ∈ [ti , ti+1 ), eT (t )e(t ) ≤ φ2 (t ) with φ 2 (t) being the solution of the following differential equation φ˙2 (t ) = γ exp(−βt ), t ∈ [ti , ti+1 ), φ2 (ti ) = eT (ti )e(ti ) = 0. The solution on the interval [ti , ti+1 ) is γ exp(−βti ) − exp(−βt ) . φ2 (t ) = β Hence, eT (t )e(t ) ≤

γ exp(−βti ) − exp(−βt ) , t ∈ [ti , ti+1 ). β

As shown in Fig. 1, ∗ ti+1 ≥ ti+1 , ∗ where ti+1 is such that γ ∗ ∗ ∗ exp(−βti ) − exp(−βti+1 φ2 (ti+1 )= ) = α exp(−βti+1 ). β

This results in that ∗ exp(β(ti+1 − ti )) = 1 +

Therefore, ∗ ti+1 − ti ≥ ti+1 − ti =

αβ . γ

 1 αβ ln 1 + . β γ

That is, the inter-execution time has a constant positive lower bound. Hence there is no Zeno behavior in the event triggering time sequence. This completes the proof.  2.3. Event-triggered attitude control of spacecraft with disturbances When the disturbance torque cannot be ignored, the dynamics of the spacecraft is modified as J ω˙ = −Sω J ω + u + τd ,

(33)

Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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where τd ∈ R3 is the disturbance torque and assumed to be bounded, that is, it satisfies that |τdi | ≤ d, i = 1, 2, 3 with d being a known constant. The kinematics of the spacecraft based on the modified Rodrigues parameters (MRPs) is still given by σ˙ = G(σ )ω,

(34)

Now, the objective is to design an event-triggered control law for systems (33) and (34) such that all states of the closed-loop system are bounded and the attitude error can be made arbitrarily small ultimately by choosing proper design parameters. Step 1: The first step is identical to that of the disturbance-free case. For convenience, it is summarized here. The virtual control law is given by ωd = −Pσ,

(35)

where P ∈ R3×3 , P = PT > 0 is a design parameter matrix. Define V1 =

1 T σ σ 2

(36)

and ωr = ω − ωd ,

(37)

then one has σ˙ = −G(σ )Pσ + G(σ )ωr ,

(38)

and  1 1 V˙1 ≤ − σ T Pσ + 1 + σ 2 σ T ωr . 4 4 Step 2: Consider the dynamic subsystem

(39)

ω˙ r = J −1 [−Sω J ω + u + τd ] + PG(σ )ω.

(40)

Define  dωr 1 1 1 + σ 2 σ − Qωr − d Tanh , (41) 4 2 ε where the last term is used to compensate for the influence of the disturbance torque, and ⎡ ⎤ tanh dωεr1 dωr Tanh = ⎣tanh dωεr2 ⎦ (42) ε tanh dωεr3 v = Sω J ω − J PG(σ )ω −

with ε > 0 being a design parameter to be determined. Construct the event-triggered control law as u(t ) = v(ti ), (t ∈ [ti , ti+1 ), i = 0, 1, 2, . . . ),

(43)

where the event triggering time sequence {ti , i = 0, 1, 2, . . .} is determined by   t0 = 0, ti+1 = inf t | t > ti , e(t )2 > α exp(−βt ) + ρ , i = 0, 1, 2, . . . ),

(44)

where e(t ) = v(t ) − u(t ),

(45)

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is the accumulated error of the control signal, and α > 0, β > 0 and ρ > 0 are three design parameters to be determined. Substituting Eqs. (41), (43) and (45) into Eq. (40), one can obtain that

  1 dωr 1 −1 2 − 1 + σ  σ − Qωr − e − d Tanh (46) ω˙ r = J + τd 4 2 ε For the stability of the resulted closed-loop system, one has the following theorem. Theorem 3. All states of the closed-loop system consisting of Eqs. (38) and (46) with the event triggering time sequence (45) are bounded and the attitude error can be made arbitrarily small ultimately by selecting proper design parameters. In addition, there is no Zeno behavior in the event triggering time sequence. Proof. Define the Lyapunov function as 1 T 1 σ σ + ωrT J ωr . 2 2 Its derivative along the closed-loop system is given by

V =

dωr 1 1 V˙ ≤ − σ T Pσ − ωrT Qωr − ωrT e − dωrT Tanh + ωrT τd 4 2 ε 1 d ωr 1 1 ≤ − σ T Pσ − ωrT Qωr + ωrT Qωr + eT Q−1 e − d ωrT Tanh + ωrT τd 4 2 4 ε d ωr 1 1 T T = − σ T Pσ − ωrT Qωr + λ−1 + ωrT τd . min (Q)e e − d ωr Tanh 4 4 ε Note that the function tanh (·) has the property below

 ξ ≤ 0.2785ε 0 ≤ |ξ | − ξ tanh ε for any ξ ∈ R and ε > 0. Hence, d ωr −d ωrT Tanh ε

+

ωrT τd

=

3

 i=1

≤

3  

dωri ωri τdi − dωri tanh ε

|dωri | − dωri tanh

i=1



ωri  ε

≤ 0.8335ε. As a result, 1 1 T V˙ ≤ − σ T Pσ − ωrT Qωr + λ−1 min (Q)e e + 0.8335ε. 4 4 Keeping the event triggering condition in mind, one has that  1 1 V˙ ≤ − σ T Pσ − ωrT Qωr + λ−1 min (Q) α exp (−βt ) + ρ + 0.8335ε 4 4 c ≤ − V + , 2 Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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where c is defined in the proof of Theorem 2, and  = λ−1 min (Q) (α + ρ) + 0.8335ε. According to the comparison lemma, one has V (t ) ≤ η1 (t ) where η1 (t) is the solution of the following differential equation c η˙1 (t ) = − η1 (t ) +  , η1 (0) = V (0). 2 Obviously, the solution is

 t

 ct c(t − s) +  ds η1 (t ) = η1 (0) exp − exp − 2 2 0



 ct ct 2 + 1 − exp − = V (0) exp − 2 c 2



 ct 2 2 exp − + = V (0) − . c 2 c Hence,



 2 2 ct V (t ) ≤ V (0) − . exp − + c 2 c  T 1 As a result, z = σ J 2 ωr satisfies





 1 4 2 2 ct z ≤ 2 V (0) − ≤ R, exp − + c 2 c where

  21 4 R = max 2V (0), . c This implies that all states system are bounded and ultimately converge to  of the closed-loop    21  the residual set Bres = z : z ≤ 2 c of which the radius can be made arbitrarily small by choosing proper design parameters. Further, we prove that there is no Zeno behavior in the event triggering time sequence. The closed-loop system can be re-presented in terms of z as follows

 σ˙ z˙ = 1 J 2 ω˙ r 

−G(σ )Pσ + G(σ )ωr    = −1 1 J 2 − 4 1 + σ 2 σ − 21 Qωr − e − d Tanh dωε r + τd   ωr 1 = Mz + N e + J − 2 −d Tanh + τd , ε where M and N are defined in the proof of Theorem 2. On any interval [ti , ti+1 ), one has ∂v z˙ ∂z

 ∂v ∂v ∂v − 1 dωr 2 τd − d Tanh . = Mz + N e + J ∂z ∂z ∂z ε

e˙(t ) = v˙(t ) − u˙ (t ) = v˙(t ) =

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       − 21  Obviously,  ∂v M ,  ∂v N  and  ∂v J  are all continuous function of z, which belongs to ∂z ∂z ∂z       1  the compact set BR = {z : z ≤ R}. Hence,  ∂v M ,  ∂v N  and  ∂v J − 2  all have maximum ∂z

∂z

∂z

value on BR . Their maximum values are denoted by μ1 , μ2 and μ3 , respectively. From this and the event triggering condition, one can obtain that, for t ∈ [ti , ti+1 ),

 dωr T T ∂v T ∂v T ∂v − 21 −d Tanh 2e (t )e˙(t ) = 2e Mz + 2e N e + 2e J + τd ∂z ∂z ∂z ε    dωr   ≤ 2μ1 ez + 2μ2 e2 + 2μ3 e τd − d Tanh ε  1 1 √ ≤ 2μ1 (α + ρ) 2 R + 2μ2 (α + ρ) + 2μ3 (α + ρ) 2 2 3d  μ, where the following inequality    3

2  21    dω dω r ri τd − d Tanh = τdi − d tanh  ε  ε i=1  3

 21  dω ri ≤ 2 τdi2 + d 2 tanh2 ε i=1  3  21  ≤ 4d 2 i=1

√ = 2 3d is used. In other words,  d eT (t )e(t ) ≤ μ, t ∈ [ti , ti+1 ). dt By the comparison lemma, one has that, for t ∈ [ti , ti+1 ), eT (t )e(t ) ≤ η2 (t ), where η2 (t) is the solution of the following differential equation η˙2 (t ) = μ, t ∈ [ti , ti+1 ), η2 (ti ) = eT (ti )e(ti ) = 0. The solution on the interval [ti , ti+1 ) is η2 (t ) = μ(t − ti ). Hence,

  eT (t )e(t ) ≤ η2 (t ) = μ t − ti , t ∈ [ti , ti+1 ) Similar to the counterpart of the proof of Theorem 2, one has ∗ ti+1 ≥ ti+1 , ∗ where ti+1 is determined by  ∗ ∗ ∗ η2 (ti+1 ) = μ ti+1 − ti = α exp(−βti+1 ) + ρ.

Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 2. Curves of the attitude errors (Scenario 1).

It can be concluded from this that,  ∗ μ ti+1 − ti > ρ, that is, ∗ ti+1 − ti ≥

ρ . μ

Therefore, ∗ ti+1 − ti ≥ ti+1 − ti =

ρ . μ

In other words, the inter-execution time has a constant positive lower bound, and there is no Zeno behavior in the event triggering time sequence. This completes the proof.  3. Numerical simulation In this section, numerical simulation experiments are conducted to verify the correctness and effectiveness of the obtained results. The nominal inertia matrix of the spacecraft is set as ⎡ ⎤ 900 25 30 J0 = ⎣ 25 980 20 ⎦kg · m2 . 30 20 950 Three scenarios are considered in this section. Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 3. Curves of the attitude angular rates (Scenario 1).

Scenario 1: Simulation for the inverse optimal attitude control First of all, the simulation experiment is conducted to verify the correctness and the effectiveness of the obtained inverse optimal control law, that is, 1 u∗ = − Qωr + Sω J0 ω − J PG(σ )ω − G(σ )σ 2 In this scenario, the real inertia matrix of the spacecraft is set to be the same as the nominal one, the design parameters are set as P = 0.5I3 , Q = 200I3 , and the initial values are set as  T  T σ (0) = 0.15 −0.12 0.1 and ω(0) = −0.02 0.02 −0.02 (rad/s ). Fig. 2 shows the curves of the attitude errors, Fig. 3 shows the curves of the attitude angular rates and Fig. 4 shows the curves of the control torques. From the simulation results, one can see that the proposed the inverse optimal control law can stabilize the spacecraft asymptotically. It is easy to check that both of 1 ua = − Qωr + Sω J0 ω − J PG(σ )ω − G(σ )σ 4 and 3 ub = − Qωr + Sω J0 ω − J PG(σ )ω − G(σ )σ 4 can also stabilize the spacecraft. Define t  c(y(τ )) + (u(τ ) − h(y(τ )) )T Q−1 (u(τ ) − h(y(τ )) ) dτ. Jc2,t = 0

Fig. 5 shows the curves of Jc2,t when u equals to ua , u∗ and ub , respectively. Clearly, all ∗ ∗ of them have steady values, and Jc2, ∞ , that is, the steady value of Jc2,t when u = u , is the Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 4. Curves of the control torques (Scenario 1). ∗ minimum one. In addition, Jc2, ∞ = 2.57 = V (0). These show the correctness of the theoretical results given in Theorem 1. Scenario 2: Simulation for the event-triggered control of spacecraft without disturbances Then, the simulation experiment is conducted to verify the correctness and the effectiveness of the event-triggered control law obtained when the disturbance torques can be ignored. For comparison, simulation results for the event-triggered control law proposed in [23] are also given. The event-triggered control law proposed in this paper is summarized as follows. ⎧ u(t ) = v(ti ), (t ∈ [ti , ti+1 ), i = 0, ⎪ ⎪  1, 2, . . .2), ⎨ 1 v(t ) = Sω J0 ω − J0 PG(σ )ω − 4 1 + σ  σ − 21 Qω  r, t0 = 0, ti+1 = inf t | t > ti , e(t )2 > α exp(−βt ) , i = 0, 1, 2, . . . ), ⎪ ⎪ ⎩ e(t ) = v(t ) − u(t ).

In the simulation, the design parameters are set as P = 0.5I3 , Q = 200I3 , α = 2, β = 0.05. In [23], the dynamics of spacecraft is the same as Eq. (1), but the kinematics of spacecraft is described by  q˙v = 0.5(Sqv + q0 I3 )ω, q˙0 = −0.5qvT ω,  T where q0 qvT ∈ R3 × R is the unit quaternion representing the attitude of the spacecraft by the body fixed frame relative to the inertia frame. The relationship between the modified  T Rodrigues parameter vector σ and the unit quaternion q0 qvT is given by Markley and Crassidis [37] qv σ = , 1 + q0 Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 5. Curves of Jc2,t (Scenario 1).

and # qv = q0 =

2σ , 1+σ T σ 1−σ T σ . T 1+σ σ

The compared event-triggered control law given in [23] is summarized as follows. ⎧ i ui (t ) = vi (tki ), (t ∈ [tki , tk+1 ), i = 1, 2, 3; k = 0, 1, 2, . . . ), ⎪ ⎪ ⎡ ⎡ ⎤ ⎪ s1 ( l1 +ks s1 ) ⎤ ⎪ l + k s ( m1 tanh m1s1 ⎪ 1 s 1 ) tanh  ⎪ ⎪ m s s l + k s ⎪ ⎪v(t ) = −(1 + δ)⎣(l2 + ks s2 ) tanh 2 ( 2  s 2 ) ⎦ − (1 + δ)⎣m2 tanh 2 2 ⎦, ⎪ ⎨ m3 tanh m3s3 (l3 + ks s3 ) tanh s3 (l3 + k$s s3 ) $   i ⎪ t0i = 0, tk+1 = inf t | t > ti , |ei (t )| > δi $ui (tki )$ + d¯i , i = 1, 2, 3; k = 0, 1, 2, . . . ), ⎪ ⎪ ⎪ ⎪ s = ω + kq qv , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩l = kq Sqv J0 ω + 0.5kq J0 (Sqv + q0 I3 )ω, e(t ) = v(t ) − u(t ). In the simulation, the design parameters are set as kq = 0.16, ks = 10000, d¯ = 0.005, δi = δ = 0.1, mi = 0.01,  = 0.01. In this scenario, the real inertia matrix of the spacecraft is set as the same as the  T nominal one, and the initial values are still set as σ (0) = 0.15 −0.12 0.1 , ω(0) =  T −0.02 0.02 −0.02 (rad/s ). Fig. 6 shows the curves of the attitude errors, Fig. 7 shows the curves of the attitude angular rates and Fig. 8 shows the curves of the control torques. From the simulation results, one can see that both of the proposed event-triggered control law and that given in [23] can stabilize the spacecraft well. Fig. 6 shows that, by using the event-triggered control law given in [23], the steady attitude errors eventually vary within a residual set with radius 2 × 10−4 Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 6. Curves of the attitude errors (Scenario 2).

level; and by using the proposed event-triggered control law, attitude errors converge to zero asymptotically. This coincides with the theoretical results. Fig. 9 shows the inter-execution time. For the proposed event-triggered control law, the times of the event triggering, i.e., the times of the signal transmission from the controller to the actuator is 55 in 500 s, and the inter-execution time varies relatively uniformly. For the compared event-triggered control law, the times of the event triggering for three channels are respectively 117, 109 and 125 in 500 s, and very high communication frequency is required in the initial stage. To sum up, in the given condition, compared with the event-triggered control law given in [23], the proposed one could add less communication burden on the spacecraft and results in a higher control precision. This shows the advantage of the proposed method to some extent. Scenario 3: Simulation for the event-triggered control of spacecraft with disturbances Finally, the effectiveness of the proposed event-triggered control law when the disturbance torque cannot be ignored is verified by simulation. To show the robustness of the proposed event-triggered control law, in this scenario, the real inertia matrix of the spacecraft is set as ⎡

920 J = ⎣ 35 50

45 1000 20

⎤ 50 30 ⎦kg · m2 , 975

and the disturbance torque is set as ⎡

⎤ 8 + 7 sin (0.1t ) τd = ⎣−7 − 9 cos(0.1t )⎦ × 10−4 N · m. 9 − 8 cos(0.1t ) Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 7. Curves of the attitude angular rates (Scenario 2).

The event-triggered control law proposed in this paper is summarized as follows. ⎧ u(t ) = v(ti ), (t ∈ [ti , ti+1 ), i = 0, 1, 2, . . . ), ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ tanh dωεr1 ⎪ ⎪   ⎨ v(t ) = Sω J0 ω − J0 PG(σ )ω − 41 1 + σ 2 σ − 21 Qωr − d ⎣tanh dωεr2 ⎦, dωr3 ⎪ ⎪   tanh ε ⎪ 2 ⎪ ⎪ t = 0, ti+1 = inf t | t > ti , e(t ) > α exp(−βt ) + ρ , i = 0, 1, 2, . . . ), ⎪ ⎩0 e(t ) = v(t ) − u(t ). In the simulation, the design parameters are set as P = 0.5I3 , Q = 200I3 , α = 2, β = 0.05, ε = 1 × 10−5 , ρ = 1 × 10−4 , d = 2 × 10−3 . The initial values are still set as σ (0) =  T  T 0.15 −0.12 0.1 , ω(0) = −0.02 0.02 −0.02 (rad/s ). Fig. 10 shows the curves of the attitude errors, Fig. 11 shows the curves of the attitude angular rates and Fig. 12 shows the curves of the control torques. From the simulation results, one can see that the proposed event-triggered control law can stabilize the spacecraft well. Fig. 13 shows the inter-execution time. The times of the signal transmission from the controller to the actuator is 51 in 500 s. These illustrate the correctness and the effectiveness of the obtained results. If the attitude control law is realized in the periodic sampling or time-triggered way, the sampling period is usually set as 0.5 s, as a result, the times of the signal transmission from the controller to the actuator is 1000 in 500 s. Therefore, communication resources utilization is reduced by nearly 95% when adopting the proposed event-triggered approach. This shows the significant advantage of proposed event-triggered attitude control approach compared with the traditional ones realized in the time-triggered way. Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 8. Curves of the control torques (Scenario 2).

Fig. 9. The inter-execution time (IET) (Scenario 2). Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 10. Curves of the attitude errors (Scenario 3).

Fig. 11. Curves of the attitude angular rates (Scenario 3).

Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Fig. 12. Curves of the control torques (Scenario 3).

Fig. 13. The inter-execution time (IET) (Scenario 3).

Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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4. Conclusion This paper considers the spacecraft attitude stabilization problem based on the eventtriggered strategy. The basic control law is obtained via backstepping together with inverse optimal, and both the virtual and the actual control laws are respectively optimal with respect to certain cost functionals. Then, the obtained backstepping-based inverse optimal attitude control law is realized in an event-triggered way to decrease the communication frequency between the controller and the actuator. In the proposed event-triggered control approach, the control signal is only transmitted when the accumulated error exceeds the pre-specified threshold, and the unwanted influence caused by the error is completely compensated for by designing the event triggering mechanism elaborately. It is proved by using Lyapunov theory that the trivial solution of the closed-loop system is globally exponentially stable when the disturbance torque can be ignored, and there is no Zeno phenomenon in the closed-loop system. When the disturbance torque cannot be ignored, to deal with it, the afore-obtained attitude control law and the event triggering mechanism are further modified. It is proved that all states of the closed-loop system are bounded, the attitude error can be made arbitrarily small ultimately by choosing proper design parameters, and the Zeno behavior is excluded in the closed-loop system. Finally, the correctness, the effectiveness and the advantage of the proposed approaches are verified by numerical simulation. It is easy to see that no complex computation is needed in the proposed backsteppingbased inverse optimal attitude control law. However, to realize it in the proposed eventtriggered scheme, a specific software is required to compute and monitor the control signal in real time. So how to further reduce the computing and monitoring burden is an interesting problem to be solved in the future. Another interesting problem to be solved in the future is how to estimate the disturbance torque and compensate for its influence in an active way as done in [38,39], instead of the passive way presented in this paper. Acknowledgement The authors would like to thank the anonymous reviewers for their insightful comments and constructive suggestions. This work was supported by the Major Program of National Natural Science Foundation of China (Grant Nos. 61690210, 61690211, and 61690212), the National Natural Science Foundation of China (Grant No. 11972130), and the Open Fund of National Defense Key Discipline Laboratory of Micro-Spacecraft Technology (Grant No. HIT.KLOF.MST.201702). References [1] Q. Hu, Robust adaptive sliding mode attitude maneuvering and vibration damping of three-axis-stabilized flexible spacecraft with actuator saturation limits, Nonlinear Dyn. 55 (4) (2009) 301–321. [2] B. Xiao, Q. Hu, Y. Zhang, Adaptive sliding mode fault tolerant attitude tracking control for flexible spacecraft under actuator saturation, IEEE Trans. Control Syst. Technol. 20 (6) (2012) 1605–1612. [3] Z. Song, C. Duan, H. Su, Full-order sliding mode control for finite-time attitude tracking of rigid spacecraft, IET Control Theory Appl. 12 (8) (2018) 1086–1094. [4] Z. Song, C. Duan, J. Wang, Chattering-free full-order recursive sliding mode control for finite-time attitude synchronization of rigid spacecraft, J. Frankl. Inst. doi:10.1016/j.jfranklin.2018.02.013. [5] R. Sharma, A. Tewari, Optimal nonlinear tracking of spacecraft attitude maneuvers, IEEE Trans. Control Syst. Technol. 12 (5) (2004) 677–682. Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010

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Please cite this article as: F. Wang, M. Hou and X. Cao et al., Event-triggered backstepping control for attitude stabilization of spacecraft, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.09.010