Chenling WANG,Yun LI,Qinglei HU,Jin HUANG

aSchool of Automation Science and Electrical Engineering,Beihang University,Beijing 100191,China

bBeijing Institute of Automatic Control Equipment,Beijing 100074,China

Abstract Plug-and-play technology is an important direction for future development of spacecraft and how to design controllers with less communication burden and satisfactory performance is of great importance for plug-and-play spacecraft.Considering attitude tracking of such spacecraft with unknown inertial parameters and unknown disturbances,an event-triggered adaptive backstepping controller is designed in this paper.Particularly,a switching threshold strategy is employed to design the event-triggering mechanism.By introducing a new linear time-varying model,a smooth function,an integrable auxiliary signal and a bound estimation approach,the impacts of the network-induced error and the disturbances are effectively compensated for and Zeno phenomenon is successfully avoided.It is shown that all signals of the closed-loop system are globally uniformly bounded and both the attitude tracking error and the angular velocity tracking error converge to zero.Compared with conventional control schemes,the proposed scheme significantly reduces the communication burden while providing stable and accurate response for attitude maneuvers.Simulation results are presented to illustrate the effectiveness of the proposed scheme.

KEYWORDS Adaptive control;Attitude tracking;Event-triggered control;Spacecraft;Uncertainties

1.Introduction

In recent years,the low-cost plug-and-play spacecraft has attracted an increasing amount of attention.Since the investigation of Air Force Research Laboratory in 2004,several types of plug-and-play satellites have been successfully launched,1-5including PnPSat-1,TacSat-2 and MSV.As a new architectural concept for spacecraft,the plug-and-play spacecraft offers more flexibility during spacecraft design and mission operation,and helps to reduce the launch risk and increase the reconfigurability.The functional components of plugand-play spacecraft are connected by low-cost wireless networks,and the wireless technology plays an important role in constructing lighter-weight,faster and cost-effective spacecraft.However,as mentioned in Refs.4,5,the bandwidth of the wireless network is limited.Therefore,how to design control schemes with less communication burden and satisfactory performance is of great importance for plug-and-play spacecraft.

Up to now,considerable achievements have been made for attitude control of spacecraft based on various kinds of approaches such as sliding mode control,6-8optimal control,9,10iterative learning control,11feedback linearization,12and observer-based control.13,14Owing to the advantages in improving transient performance and handling uncertainties,adaptive backstepping control15has also been applied to spacecraft.In Refs.16-19,some effective adaptive back stepping control schemes were proposed for attitude control of spacecraft.Nevertheless,it is noticed that all aforementioned control schemes were developed within the framework of continuous-time control.When these schemes are implemented on digital platforms,the sampling period is required to be sufficiently small and the communication between controllers and actuators is executed at every sampling instant no matter whether it is necessary or not,which would cause overloading of the communication network.As a result,these control schemes may be inappropriate for plug-and-play spacecraft with limited communication capability.

On the other hand,nowadays control systems are often implemented over networks,which has advantages in reducing cost and increasing flexibility.Motivated by the fact that the bandwidth of the communication network is limited,event triggered control has received an increasing amount of attention.In this control strategy,the controller communicates with the actuator and/or the sensor only at some discrete time instants when a predefined triggering condition is satisfied,which significantly reduces the communication burden.In Refs.20-22,some event-triggered control strategies were developed,but the closed-loop system was directly assumed to be input-to-state stable with respect to network-induced errors.This assumption is conservative or even impossible for general nonlinear systems.Under the condition that the system models are exactly known,an event-triggered control scheme was proposed in Ref.23,which was later applied to marine vessel in Ref.24.However,in practice,it is hard to obtain precise system models because of uncertainties.In order to handle system uncertainties,recently an event-triggered adaptive control scheme was presented in Ref.25for a class of nonlinear systems.By co-designing the controller and the event-triggering mechanism,the assumption about input-to-state stability in Refs.20-22was removed in Ref.25.Nevertheless,the scheme in Ref.25is limited to single-input single-output systems and can only ensure that the tracking error converges to a residual set rather than zero.Hence,the same as the schemes in Refs.20-24,it cannot be directly applied to spacecraft which have multiple inputs and multiple outputs and require high control precision.

In this paper,within the framework of backstepping design,an event-triggered adaptive control scheme is proposed for attitude tracking of spacecraft with unknown inertial parameters and unknown external disturbances.The proposed scheme has the following features:

(1)An event-triggering mechanism is designed to determine the time instants for communication and continuous communication is avoided.As a result,the communication burden is significantly reduced in comparison with the spacecraft control schemes in Refs.6-14,16-19,which is quite favorable for plug-and-play spacecraft.

(2)A linear time-varying model is constructed to rewrite the event-triggering mechanism.Meanwhile,a bound estimation approach is introduced to tackle the network induced error and external disturbances.With these efforts,the impacts of the network-induced error and external disturbances are successfully compensated for.

(3)With the aid of an integrable auxiliary signal incorporated into the controller design,both the attitude tracking error and the angular velocity tracking error converge to zero asymptotically,regardless of the unknown inertial parameters,external disturbances and the network-induced error.Besides,it is proved that all closed-loop signals are globally uniformly bounded.As a result,the proposed scheme is able to provide stable and accurate response for attitude maneuvers while reducing the communication burden.

The remainder of this paper is organized as follows.In Section 2,the control problem is formulated.Sections 3 and 4 are devoted to controller design and stability analysis,respectively.Simulation results are presented in Section 5 to illustrate the effectiveness of the proposed scheme.Finally,we conclude in Section 6.

2.Problem formulation

The attitude kinematics and dynamics of a rigid spacecraft can be described in terms of quaternion as6

where J ∈ R3×3is the inertial matrix described in the body frame B;I∈ R3×3is the identity matrix;w ∈ R3is the angular velocity of the spacecraft with respect to the inertial frame I and is expressed in the frame B;the unit quaternion（q，q0）∈ R3× R represents the attitude orientation of the spacecraft in the frame B with respect to the frame I and satisfies qTq+=1; τ= ［τ1，τ2，τ3］T∈ R3represents the control torque;d= ［d1，d2，d3］T∈ R3denotes an external disturbance;S（x）is a skew-symmetric matrix acting on x= ［x1，x2，x3］T,which is given by

（qe，qe0）∈ R3× R is denoted as the attitude tracking error from the body frame to the desired frame with orientation（qd，qd0）∈ R3× R, whose motion is governed byWith the definition of quaternion multiplication,11（qe，qe0）can be calculated as

The angular velocity error from the body frame with respect to the desired frame is defined as

Note that the inertial matrix J is symmetric positive definite but may be uncertain during operation.In this paper,we do not need the knowledge of J,i.e.,J is assumed to be unknown.The objective is to design an event-triggered adaptive control scheme so that all closed-loop signals are bounded and the attitude q and the angular velocity w of the spacecraft track the desired attitude qdand the desired angular velocity wd,respectively,where wdand˙wdare bounded.

The following assumption and lemma will be used in our design.

Assumption 1.The disturbancedis bounded.

Lemma 1.For any scalar z∈ Randε＞ 0,the following relationship holds:

Proof:.The first inequality of Eq.(6)is obvious.On the other hand,it can be readily checked that 0 ≤ |z|-z2/（|z|+ ε）=ε|z|/（|z|+ ε）＜ ε, which together withz2/（|z|+ ε）≤

3.Event-triggered adaptive controller design

3.1.Event-triggering mechanism

In this paper,the event-triggering mechanism is designed as

wherek=0，1，2，···,ti，0:=0,‘inf”represents the in fimum of a set(i.e.,the greatest lower bound of a set),ui（t）is theith element of u（t）=［u1（t），u2（t），u3（t）］Twhich will be designed in Subsection 3.2,andm1,m2,m3andm4are positive design parameters withm1∈ （0，1）.

Remark 1.As can be seen from Eqs.(7)and(8),whenever the triggering condition in Eq.(8)is satisfied,the time instant will be marked as ti，k+1and the control value ui（ti，k+1）will be transmitted to the actuator module of the spacecraft.During the time interval［ti，k，ti，k+1）,the control torqueτiis kept as the value of uiat the time instant ti，k.Since no communication is needed to updateτiduring（ti，k，ti，k+1）,the communication burden is significantly reduced in comparison with the spacecraft control schemes proposed inRefs.6-14,16-19.

Remark 2.The above event-triggering mechanism is inspired by the switching threshold strategy proposed inRef.25.When the control torque is not very large,i.e.,when|τi|≤m4,the magnitude of the threshold is proportional to that of the control torque and the relative threshold strategy is activated so that precise control can be obtained.Once the control torque becomes large,i.e.,when|τi|＞m4,the magnitude of the threshold is a constant and the fixed threshold strategy is applied to prevent the control torque from large impulses.

Remark 3.By constructing the linear time-varying model in Eqs.(10)and(11),we unify the description of the relationship betweenτ（t）andu（t）,no matter the relative threshold strategy or the fixed threshold strategy is activated.As can be seen from what follows,this model will significantly facilitate the construction ofu（t）and the disposal of the network-induced errorΔ（t）:=u（t）- τ（t）.

3.2.Backstepping design procedure

Based on the above event-triggering mechanism,in this subsection,u（t）in Eqs.(7)and(8)will be designed within the framework of backstepping design,where the design procedure consists of two steps.To simplify the presentation,we first define

where α1is a stabilizing function to be designed.Besides,we will employ positive scalarsc1,c2,γ1and γ2and a symmetric positive definite matrix Γ ∈ R6×6as design parameters in the design procedure without restating.

Let ? := ［J11，J22，J33，J23，J13，J12］TwithJijbeing the （i，j）th element of J.Then,it can be easily proved that Jb=L（b）?,which implies that Eq.(18)can be rewritten as

The block diagram of the proposed control scheme is shown in Fig.1.

Remark 4.It is worth pointing out that,in the above design,the network-induced error and the external disturbance are dealtwith by the combination of the bound estimation approach,Lemma1and the integrable auxiliary signalε（t）.The smooth functionin Lemma1and the bound estimation approach play a key role to tackle the time-varying uncertainties brought by the network-induced error and the disturbance(see Eqs.(21),(22)and(28)),whileε（t）makes the corresponding residual terms integrable and lays a foundation for asymptotic tracking.Besides,the construction ofuand the design of the event-triggering mechanism are completely decoupled and the former does not depend on the parameters of the latter,which gives the user more freedom to choose the design parameters in order to seek less communication burden and better tracking performance.

Fig.1 Block diagram of proposed scheme.

4.Stability analysis

The main theorem of this paper is established as follows.

Theorem 1.Consider the closed-loop system consisting of the spacecraft in Eq.(1),the adaptive laws in Eqs.(25)and(27)and the event-triggered control law in Eqs.(7),(8)and(26).Suppose that Assumption1holds.Then,all signals of the closed-loop system are globally uniformly bounded and the attitude tracking errorqeand the angular velocity tracking errorωeconverge to zero asymptotically.Besides,Zeno phenomenon is avoided.

Proof:.Define a Lyapunov function candidate of the closed loop system as

Remark 5.Compared with the event-triggered adaptive control scheme proposed inRef.25,where the tracking error only converges to a residual set rather than zero,Theorem1indicates that our control scheme can ensure that all closed-loop signals are globally uniformly bounded and both the attitude tracking error and the angular velocity tracking error converge to zero,regardless of the unknown inertial parameters,external disturbances and network-induced error.As a result,the proposed scheme provides not only an effective way to reduce the communication burden but also a control method with global stability and high precision,which is quite favorable for low-cost plug-and-play spacecraft.Besides,it should be pointed out thatthe linear time-varying model in Eqs.(10)and(11),the smooth functionin Lemma1,the integrable auxiliary signalε（t）and the bound estimation approach used in our paper are not involved inRef.25,and the scheme inRef.25is limited to singleinput single-output systems and cannot be directly applied to spacecraft.

5.Simulation results

In this section,simulation results are presented to illustrate the effectiveness of the proposed control scheme.In the simulation,the inertia matrix is set as

be zero,and the design parameters and the auxiliary signal are chosen asm1=0.3,m2=0.01,m3=1.5,m4=4,c1=0.9,c2=1.7, γ1=7, γ2=3, Γ =diag｛15，15，15，15，15，15｝and ε=0.2e-0.01t.The simulation results are given in Figs.2-7.As shown in Figs.2 and 3,both the attitude tracking error qeand the angular velocity tracking error weconverge to zero.The boundedness of the parameter estimations is illustrated in Fig.4,while the control torque τ= ［τ1，τ2，τ3］Tis plotted in Fig.5,from which it can be seen that the control torque is updated only when the triggering condition is satisfied.

Fig.2 Error quaternion （qe，qe0）.

Fig.3 Angular velocity error we.

Fig.4Parameter estimation of^l,^p and||^?||.

Fig.5 Control torque.

Moreover,in order to make a comparison,a representative adaptive control scheme proposed in Subsection 3.A of Ref.6is applied to the spacecraft with the same condition and the same simulation step size as before.This scheme does not use event triggered control strategy and its control signal is updated at every sampling instant.The design parameters are set ask0=0.5, δ1=0.01, δ2=1.5 and μ =1.5(their definitions can be found in Ref.6)to have the same energy consumption as the proposed control scheme,where the energy consumption is defined asE=∫300‖ τ（t）‖2dt.The corresponding simulation results are shown in Figs.6-11,where Schemes 1 and 2 denote the proposed control scheme and the control scheme in Ref.6,respectively.From Fig.6,we can see that the two control schemes consume the same amount of energy.As shown in Fig.7,for the control scheme in Ref.6,the control signal τi(i=1,2,3)is updated 3000 times in the first 30 seconds.By comparison,for our event-triggered control scheme,τ1,τ2and τ3are updated only 204,236 and 251 times,respectively.Figs.8 and 9 give the energy comparison ofqeiandwei,where the energy is defined asEφ=φ2（t）dtwith φ denotingqeiandwei(i=1,2,3).The responses of the attitude tracking error in terms of Euler angles and the responses of the angular velocity tracking error are shown in Figs.10 and 11.Figs.6-11 indicate that the proposed control scheme not only significantly reduces the communication burden but also provides better tracking performance.

Fig.6 Energy consumption.

Fig.7 Number of times for communication.

Fig.8 Energy comparison about qe1,qe2and qe3.

Fig.9 Energy comparison about we1,we2and we3.

Fig.10 Attitude tracking error of yaw.

6.Conclusions

Fig.11 Angular velocity tracking error.

In this paper,an event-triggered adaptive control scheme has been proposed for attitude tracking of spacecraft with unknown inertial parameters and unknown external disturbances.A linear time-varying model has been introduced to unify the description of the relative threshold strategy and the fixed threshold strategy used to construct the eventtriggering mechanism.With the aid of a bound estimation approach,a smooth function and an integrable auxiliary signal,the effects of the network-induced error and the disturbances are successfully compensated for.We have shown that all closed-loop signals are globally uniformly bounded and both the attitude tracking error and the angular velocity tracking error converge to zero.Simulation results have been presented to illustrate the effectiveness of the proposed scheme.Future investigation should include the disposal of input constraints.

Acknowledgements

This study was supported by the National Natural Science Foundation of China(Nos.61673036,61661136007 and 51777013)and the Beijing Natural Science Foundation of China(No.4182036).