Jianzhong QIAO , Xiaofeng LI , Jianwei XU

a School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China

b Beijing Advanced Innovation Center for Big Data-based Precision Medicine, Beihang University, Beijing 100083, China

c Shenzhen Han's Robot Co., Ltd., Shenzhen 518102, China

KEYWORDS Disturbance observer (DO);Flexible spacecraft;H∞control;Input delay;Measurement delay

Abstract In this paper, the attitude control algorithm of f lexible spacecraft with unknown measurement delay and input delay based on disturbance observer is designed. The inf luence of measurement delay and input delay on the attitude control system and disturbance observer is analyzed. The disturbance estimation error equation is transformed into a differential system with a pure delay.Then,the observer gain is chosen based on the 3/2 stability theorem to ensure the stability and disturbance attenuation performance of the pure delay system.Next,the controller gain is designed based on the Linear Matrix Inequality (LMI) approach to guarantee the stability of the composite system and achieve H∞performance with two additive delays. The simulation results show that the proposed method can improve the anti-disturbance ability of the attitude control system.

1. Introduction

With the rapid development of space technology, the requirement of precise attitude control on spacecraft is higher and higher. However, large-scale and low-damping f lexible appendages, such as solar arrays and antenna ref lectors, bring great challenges to the attitude control system. Meanwhile, the spacecraft is affected by multiple sources of disturbances coming from internal system and external environment. Sliding-Mode Control (SMC) has favorable ability of disturbance attenuation, which makes it widely studied in the f ield of spacecraft attitude control.1-3But the chattering phenomenon limits the application of SMC to the engineering.Robust H∞control can attenuate the effect of disturbance under a prescribed level. On-orbit attitude control experiments using Engineering Test Satellite VI (ETS-VI) and VIII(ETS-VIII) launched by Japan Aerospace Exploration Agency were carried out, in which several types of H∞dynamic output feedback controllers were designed and their effectiveness is demonstrated.4,5Input shaping technique and its extensions have been widely applied to the vibration reduction of f lexible spacecraft.6,7Besides, robust optimal control methods are proposed to take uncertainties and disturbances into account.8,9

However,the above mentioned control methods either rely on the model precision of the dynamical system or regard disturbances as a norm bounded one leading to the insuff icient exploitation of available disturbance information. Especially when various disturbances do not have equivalent effect,different disturbances should be classif ied to achieve better anti-disturbance performance. The effect of the main disturbance should be rejected by disturbance rejection methodology, while the effect of other disturbances should be attenuated by disturbance attenuation methodology such as H∞control.10Disturbance Observer Based Control(DOBC) is an effective and potential disturbance rejection methodology which has attracted considerable attention and its extensions have been successfully applied to robots, hard disks and missiles.11-20

Except for the disturbances, time delay is also a crucial issue involving real-time high-precision control problem of spacecraft.21-30There are two main sources of delays in the attitude control system of f lexible spacecraft:(A)the measurement delay,known as sensor-to-controller delay;(B)the input delay, known as controller-to-actuator delay. In Ref.26, state delay is considered and supposed to be constant and known.A known constant delay in state together with nonlinear perturbation and external disturbance is studied in Ref.27Chen M and Chen WH designed a delay-independent algorithm for the system with a time-varying state delay in Ref.28In Ref.29,a composite DOBC and H∞control scheme is applied to solving the problem of spacecraft attitude control with time-varying input delay.

Nevertheless, to the best of the authors' knowledge, few results about the inf luence of measurement delay and input delay considering the disturbances are available in the literature. Inspired by our previous work,15the purpose of this paper is to investigate the problem of the unknown constant measurement and input delay in f lexible spacecraft under the effect of the disturbances.

The main contributions of this paper are summarized as follows:(A) the f lexible spacecraft attitude control model subject to disturbances and measurement and input delays is established. (B) As the disturbance estimation error equation is a differential system with a pure delay, the observer gain is designed based on the existing 3/2 stability theorem to make sure the pure delay system's stability and disturbance attenuation performance. (C) The composite controller is designed to guarantee the overall composite system's stability and achieve the prescribed H∞performance with two additive delays.

2. Problem formulation

Based on the model in Ref.15,we consider the following singleaxis f lexible spacecraft attitude control system with measurement delay and input delay:

where θ is the attitude angle;J is the moment of inertia of f lexible spacecraft; F is the rigid-elastic coupling vector; wd1is the merged disturbance torque including the space environmental torques and unmodelled uncertainties; η is the f lexible modal coordinate; Cdis the modal damping matrix diag(i=1,2,···,N) where ξiis the modal damping ratio; Λ is the matrix of stiffness diag(i=1,2,···,N) where ωiis the modal frequency; u is the control input of spacecraft.

Remark 1. It should be noted that, in this paper, the rotation of f lexible spacecraft attitude control system is considered.The moment of inertia of main rotating axis is bigger than that of the other two axes. On the other hand, the vibration effect of f lexible appendage is most signif icant to the main rotating axis among the three axes because f lexible appendage is located at both sides around the main rotating axis. As compared to the vibration effect caused by the f lexible appendage when spacecraft is rotating around the main axis,the effect of coupled issue can be neglected. Therefore, the vibration problem of main rotation axis is mainly handled by the proposed method. The other two axes are dealt by traditional control methods, which is beyond the scope of this paper.

Suppose r is an unknown constant measurement delay,and thus the measurement output is [θ(t-r) ˙θ(t-r)]T. h is an unknown constant control input delay and the control input torque is supposed to be u(t-h). r and h are assumed to satisfy 0 ≤r ≤τr＜∞and 0 ≤h ≤τh＜∞, respectively.

From Eq. (1), we can obtain

Then we have the state-space form as follows:

where y(t) is the measurement output,

Remark 2. The combination of unmeasurable modal variables, wd0(t)=F(Cd+Λη(t)), is considered as a coupling inf luence of the vibration from the f lexible appendages.We can use a DO to estimate and compensate wd0(t) and design a conventional state feedback controller to guarantee the system's stability.

3. Composite controller

The structure of composite controller is shown in Fig. 1. As there exists input delay h, the composite controller is designed as u(t)=-+Ky(t), while the delayed controller can be obtained as u(t-h)=-(t-h)+Ky(t-h), whereis the estimation of wd0, and K=[KPKD] is the conventional PD controller gain, where KPis proportional controller gain and KDis differential controller gain.

The DO is formulated as

where p(t)is the auxiliary variable as the state of the observer,and L is the gain of the observer. As the presence of input delay h, we can only use ^w0at the time of t-h to estimate wd0at t. So the estimation error of DO is def ined as e(t)=wd0(t)- ^w0(t-h). Then we have

From Eq. (2) and Eq. (4), we have

where d=r+h, and d satisf ies 0 ≤d ≤τ ＜∞, τ=τr+τh.

The reference control output is def ined as

Remark 3. If the composite controller is designed as u(t)=-(t)+Kx(t-2r-h), the term LBKx(t-r-h)-LBKx(t-2r-2h) will be counteracted, which makes the design of the controller much simpler. But the premise is that delay h and r must be known.

Remark 4. As it takes some time to measure the current state x,the control signal will contain measurement delay r.But the input delay is different. The DO receives the control signal as soon as it is calculated,while it takes some time before the control torque receives. Consequently, the control input signal delivered to the actuator includes input delay h while that delivered to the DO does not.

Fig. 1 Block diagram of composite attitude controller.

Remark 5. From Fig.1,it can be seen that,different from traditional control scheme which usually adopts a single control technique,the composite controller implies the idea of Composite Hierarchical Anti-Disturbance Control (CHADC) strategy.13The composite controller consists of two parts: in the inner loop,DO is designed to estimate the effect of disturbance which is compensated in the feed-forward control channel;in the outer loop,the H∞attitude controller is designed to stabilize the overall system and attenuate the estimation error of disturbance.Therefore,the composite controller can effectively control the spacecraft attitude and attenuate disturbances.

4. Design of observer gain and controller gain

4.1. Design of observer gain

Eq.(5b)is a pure delay equation of e(t).To make the design of controller gain K simpler,f irstly we design the observer gain to guarantee Eq. (5b)'s stability.

By writing wd(t)=LBKx(t-d)-LBKx(t-2d)+ ˙wd0(t)-LB wd1(t-d), we count wd(t) as an external disturbance. First of all, we consider Eq. (5b)'s stability in the absence of wd(t).

Consider the following one-dimensional system:

where a(t)≥0 and g(t)≤t are both continuous functions in t ∈[0,∞).

Lemma 1.31,32For system Eq. (6), assume thatand that supt≥0and then the zero solution of Eq.(6)is uniformly asymptotically stable.The symbol ‘‘sup” denotes supremum.

Lemma 1 is the 3/2 stability theorem. Based on Lemma 1,Theorem 1 can be obtained.

Theorem 1. For delay system ˙e(t)=-LB e(t-d),its suff icient condition for the zero solution to be uniformly asymptotically stable iswhere τ is upper bound of d.

When wd(t)≠0, we have the following Theorem 2.Theorem 2. Assume that limt→∞wd(t)=wdsand that observer gain L satisf iesand then disturbance estimation error e(t) converges at stable value es=(LB)-1wds.

Proof. As es=(LB)-1wds, we have

Since limt→∞wd(t)=wds, and observer gain L satisf iesbased on Theorem 1, Eq. (7) is uniformly asymptotically stable, that is, limt→∞e(t)=es. The proof is completed.

Based on Theorem 2,as LB is a one-dimensional scalar,the bigger LB is, the smaller esis, i.e. the better disturbance attenuation ability is.

Remark 6. From Eq.(7),we can see that the stable value of estimation error esis related to wds, that is, esincreases with wds.Therefore,this DO works on condition that wd0is slowly timevarying and that wd1and x(t-d)-x(t-2d)are small enough.

Remark 7. The coeff icient matrix LBK in Eq. (5b) contains unknown L and K, so it is hard to design L and K simultaneously with one linear matrix equation like Ref.15. Consequently, we choose L f irst, and then design K, which simplif ies the problem. This design method is also suitable for Ref.15without time delay.

4.2. Design of controller gain

The next step is to design the controller gain K.Following the work of Refs.33,34, Theorem 3 is obtained.

Theorem 3. To composite system Eq.(5),for given parameters γ1＞0,γ2＞0,γ3＞0, if there exist matrices P1＞0,P2＞0,Q1≥Q2＞0,Q3＞0,M1≥M2＞0,M3＞0,Ni(i=1,2,...,6), K satisfying

where

and L is chosen based on Theorem 2, then the composite system Eq. (5) is robustly asymptotically stable and satisf ies

Proof. See Appendix A.

Theorem 4. To composite system Eq.(5),for given parameters γ1＞0,γ2＞0,γ3＞0, if there exist matricesR1satisfying

where

then the composite system Eq. (5) with controller gain

Proof. Pre-multiplying and post-multiplying diagsimultaneously to the left and right side of inequation (8), and then def ining

where

Noting that M1＞0, we have≥0, which are equivalent to -ˉP1ˉM-11ˉP1≤ ˉM1-2ˉP1.Similarly, we haveˉM3-2ˉP2, respectively. Then inequation (9) is obtained. The proof is completed.

As can be seen from the stability analysis and Eq. (5), by comparing to Refs.33-37, the proposed composite controller reduces the conservativeness.

5. Simulations

In this section, effectiveness of the proposed composite controller will be demonstrated by the numerical simulation.Since vibration energy is concentrated in low-frequency modes in a f lexible structure,in the simulation,the f irst two bending modes are taken into account.We select the upper bound of time delay τ=30 ms.Based on the analysis of different delays in Ref.38,we set max r=2τ/3 and max h=τ/3. Parameters of the f lexible spacecraft model and exogenous disturbances are chosen to be the same as Ref.39:the moment of inertia of the spacecraft J=35.72 kg·m2; the f irst two modal frequency and damping ratio are ω1=3.17 rad/s, ω2=7.38 rad/s, ξ1=0.0001,ξ2=0.00015, respectively; the rigid-elastic coupling vector F=[1.27814 0.91756]; initial attitude angle θ(0)=0.08 rad;initial attitude angular velocity ˙θ(0)=0.001 rad/s;the f lexible spacecraft is designed to move in a circular orbit with the altitude of 500 km and the orbit rate n=0.0011 rad/s where the disturbance torques acted on the satellite are supposed as

The observer gain is set as L=[0 900].

With the given parameters, we have B=[0 0.0301]T.Based on Theorem 2,and hence the delay equation is uniformly asymptotically stable.

In the design of the PD controller, we set γ1=2,γ2=γ3=8,C11=[2 0],and others are 0.Based on Theorem 4,with the Linear Matrix Inequality(LMI)toolbox in MATLAB it can be solved that the controller gain is K= [-3.6673 -12.3985].

Fig. 2 shows the actual value, estimated value and estimation error of the disturbance wd0caused by the f lexible appendages. Fig. 3 is obtained by partially amplifying Fig. 2. From both f igures, we can see that the main disturbance can be estimated and rejected accurately by the proposed DO. Through comparisons between the composite H∞+DOBC scheme and the only H∞scheme in Figs. 4-6, the advantage of introducing DOBC is obvious. By using the proposed method, the system states converge to a smaller region as compared to the traditional method,which can be seen from Fig.5.Besides,the vibration of states under traditional method is severer than that under proposed method. Therefore, the proposed composite method shows better overall performance of the attitude and attitude angular velocity, which conf irms that the disturbance inf luence is attenuated by the developed DOBC.

Fig. 2 Time response of vibration and vibration observed.

Fig. 3 Partial amplif ication of time response of vibration and vibration observed.

Fig. 4 Attitude angle of f lexible spacecraft.

Fig. 5 Partial amplif ication of attitude angle.

Fig. 6 Time response of attitude angular velocity.

6. Conclusions

On the basis of Ref.15,this paper concerns the impact of measurement and control input delays on the composite H∞and DOBC attitude control system of the f lexible spacecraft.Firstly, as the disturbance estimation error equation is a pure-delay differential system, based on 3/2 stability theorem in Refs.31,32, the observer gain L is chosen to make sure the pure delay system's stability and disturbance attenuation performance. Secondly, based on the algorithm proposed by Refs.33,34,the controller gain K is designed with LMI toolbox to guarantee the composite system's stability and H∞performance with two additive delays. Finally, simulation results illustrate that with the proposed scheme the main disturbance is estimated accurately and the stability of the system is improved. Certainly, there are some limitations of the proposed scheme, for example, state delay is not taken into account during the process of modeling, which may cause great impacts on the system; computational delays in controller and DO are assumed to be the same.These will be studied in the future research. It should be noted that, in practice,uncertainties are contained because of the inf inite order vibration. The uncertainties will degrade the performance of attitude control system. Further study will be given in the next research work.

Notation:throughout this paper,for a vector s(t),its Euclidean norm is def ined byA real symmetric matrix P ＞0(≥0)denotes P being a positive def inite(positive semi-def inite)matrix.The identity and zero matrix are denoted by I and 0, respectively. Matrices, if not explicitly stated, are supposed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by symbol *. For a square matrix M, we denote sym(M):=M+MT.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61627810, 61320106010,61633003, 61661136007 and 61603021), the Program for Changjiang Scholars and Innovative Research Team, China(No.IRT_16R03),and Innovative Research Team of National Natural Science Foundation of China (No. 61421063).

Appendix A: Proof of Theorem 3

Consider the following Lyapunov function:

Since P1＞0,P2＞0,Q1≥Q2＞0,Q3＞0,M1≥M2＞0,M3＞0, by differentiating V(t) with respect to time, it can be shown that

By Newton-Leibniz formula, we have

First, we prove the system's stability in the absence of ˙wd0(t), wd1(t) and wd1(t-d). Then we obtain

where

We can see that if Φ ＜0, Φ1≤0, Φ2≤0, Φ3≤0, then we have ˙V(t)＜0, which means that system Eq. (5) is asymptotically stable.

If there exist matrices P1＞0, P2＞0, Q1≥Q2＞0, Q3＞0, M1≥M2＞0, M3＞0, Ni(i=1,2,...,6)satisfying inequation (8) which by Schur complement implies

then there must exist matrices P1＞0, P2＞0, Q1≥Q2＞0, Q3＞0, M1≥M2＞0, M3＞0, Ni(i=1,2,..., 6)and matrices

satisfying Φ ＜0, and

The above three inequations are equivalent to Φ1≤0,Φ2≤0,Φ3≤0,respectively, which result in ˙V(t)＜0, namely system Eq. (5) is asymptotically stable.

The next step is to prove the robustness of the system. The following auxiliary function is considered:

which satisf ies the zero initial condition.

Similar to the proof of the stability,from inequation(8)we