Yusong ZHOU,Wenwu ZHU,Haibo DU?

1.Anhui Vocational College of Press and Publishing,Hefei Anhui 230601,China;

2.School of Electrical Engineering and Automation,Hefei University of Technology,Hefei Anhui 230009,China

Global finite-time attitude regulation using bounded feedback for a rigid spacecraft

Yusong ZHOU1,2,Wenwu ZHU2,Haibo DU2?

1.Anhui Vocational College of Press and Publishing,Hefei Anhui 230601,China;

2.School of Electrical Engineering and Automation,Hefei University of Technology,Hefei Anhui 230009,China

This paper investigates the problem of global attitude regulation control for a rigid spacecraft under input saturation.Based on the technique of finite-time control and the switching control method,a novel global bounded finite-time attitude regulation controller is proposed.Under the proposed controller,it is shown that the spacecraft attitude can reach the desired attitude in a finite time.In addition,the bound of a proposed attitude controller can be adjusted to any small level to accommodate the actuation bound in practical implementation.

Global attitude regulation,finite-time control,bounded feedback

1 Introduction

This paper considers the attitude regulation problem for a rigid spacecraft described by[1]

whereq=[q0q1q2q3]T=[q0qTv]Twithqv=[q1q2q3]Tdenotes the attitude with respect to the inertial frame based on the quaternion,ω ∈R3is the angular velocity,J∈ R3×3and τ ∈ R3are,respectively,the positive definite inertia matrix,and the control torque of spacecraft.Our objective is to designa bounded controllerfor the control torque such that spacecraft attitude will converge to the desired constant attitudeqdin afinite time.

As a classical control problem,the attitude control problem of spacecraft has been studied by many researchers[2–6].Many nonlinear control methods have been employed to solve the attitude control problem,such as optimal control[7],sliding mode control[8,9]adaptive control[10,11],H∞control[12],Fuzzy con-trol[13]and hybrid control[14],and so on.It should be pointed out that the most of the existing attitude control laws only guarantee that the closed-loop system is asymptotically stable,which means that the attitude can be controlled to converge to the equilibrium asymptotically with infinite settling time.

In recent years,as one kind of new developed nonlinear control methods,the finite-time control method has been employed to solve the attitude control problem,which is called finite-time attitude control.The main advantage of finite-time control lies in its faster convergence rates,higher accuracies,better disturbance rejection properties and robustness against uncertainties,see the theoretic analysis[15–17]or experiment test[18–24].

Although there have been some results about finite time attitude control for a rigid spacecraft[16,17,25,26],few results consider the constraint of input saturation.Actually,in practice, saturation nonlinearity has the need to be considered.The main aim of this paper is to provide a solution to design a finite-time attitude controller under input saturation.Based on some structural features,the finite-time control technique is skillfully used to design a bounded attitude controller for a rigid spacecraft.It is shown that the proposed method does not need the knowledge of the inertia matrix.In addition,to obtain a global attitude controller,a switching control method is employed.Finally,an example is given to verify the efficiency of the proposed method.

2 Preliminaries and problem formulation

2.1 Problem formulation

The main goal of this paper is to solve the problem of finite-time attitude regulation for a rigid spacecraft under the constraint condition that the control torque is required to be bounded.Motivated by[27],the definition of finite-time attitude regulation problem is extended from the manipulator system to the spacecraft system.

Definition 1(Finite-time attitude regulation problem)Given a desired constant attitudeqd∈R4,find an attitude control law such that the attitude of system(1)converges to the desired attitude,i.e.,q→qdin a finite time.

To solve the attitude regulation problem,as that in[1,8],defineˉq=[ˉq0ˉq1ˉq3ˉq4]T=[ˉq0ˉqTv]T∈R4as the relative attitude error between the attitudeqand the desired attitudeqd,where

With the help of the notation of relative attitude,the control objective is to design a control law such thatˉq(t)→[±1 0 0 0]Tin a finite time.

2.2 Some useful definitions

First,let us introduce some knowledge about the spacecraft model(1).

Definition 2[1]

.The matrixE(q)is defined as

whereI3denotes the 3×3 identity matrix.

.The symbol(·)×denotes a 3 × 3 skew-symmetric matrix,that is,

where the vectorv=[v1v2v3]T.

Second,in order to design a bounded finite-time controller,some nonlinear functions are introduced.

Definition 3Denote sigα(x)=(sgnx)|x|α,where α ＞ 0,x∈ R and sgn(·)is the standard sign function.In addition,ifx=[x1···xn]Tis a vector,then sigα(x)=[sigα(x1) ···sigα(xn)]T.

Definition 5(Homogeneity[28])Consider system

wheref=[f1(x)···fm(x)]T:U0→ Rmis continuous on an open neighborhoodU0of the origin.Let(r1,...,rm)∈Rmwithri＞0,i=1,...,m.f(x)is said to be homogeneous of degreek∈R with respect to(r1,...,rm)if,foranygivenε ＞ 0,fi(εr1x1,...,εrmxm)= εk+rifi(x),i=1,...,m,?x∈ Rm,wherek＞ ?min{ri,i=1,...,m}.System(3)is said to be homogeneous iff(x)is homogeneous.

3 Some lemmas

Lemma 2[27,29]Consider the following system

wheref(x)is a continuous homogeneous vector field of degreek＜0 with respect to(r1,...,rm)and?f(x)satisfiesf(0)=0.Assume thatx=0 is an asymptotically stable equilibrium of system˙x=f(x).Thenx=0 is a locally finite-time stable equilibrium of system(4)if

In addition,if system(4)is globally asymptotically stable and locally finite-time stable,then this system is globally finite-time stable.

Lemma 5[32]For anyxi∈R,i=1,...,n,and a real numberp∈(0,1],(|x1|+...+|xn|)p≤|x1|p+...+|xn|p.

4 Main results

According to the problem statement in Section 2.1,it can be obtained from[1]that the dynamics equation for the relative attitude error is

Due to the quaternion constraint,i.e.,

Under this notation,it can be followed from(6)that

The following property about this model is given in[33].

4.1 Design of a bounded finite-time attitude controller

Theorem 1For the spacecraft attitude control systems(10),if the control torque τ is designed as

Under the control law(11),it follows from system(12)that the closed-loop system is

With the help of the transformed system(13),next,we will prove that system(13)is finite-time stable.

Step 1(Asymptotical stability)

Inspired by[27],the candidate Lyapunov function for system(13)is constructed as follows:

Since the function ψ1(sigα1(s))is an odd function and the matrixM(x1)is positive definite,the Lyapunov functionV(x1,x2)is positive definite.Taking the derivative ofValong system(13)yields

By Property 1,it can be concluded that

which results in

Based on the definition of function ψ2(·),and noting that the function sigα2(·)is an odd function,we have

Noticing that the matrixM(x1)is positive definite,then

which leads to thatx1≡ 0.By LaSalle’s invariant principle[34],it can be concluded that(x1(t),x2(t))→0 ast→∞.As a matter of fact,it can be concluded that the the closed-loop system(13)is globally asymptotically stable.

Step 2(Local finite-time stability)

In this step,we will prove that system(13)is locally finite-time stable.The proof is mainly based on Lemma1.

By the conditions for the odd functions ψ1and ψ2(i.e.,ψi(y)=ciy+o(y)(i=1,2)),rewrite system(13)as follows:

First,we will prove that the nominal system of(21),i.e.,system

is asymptotically stable and homogeneous.Construct the following candidate Lyapunov function for system(23)

whose derivative along system(23)is

By LaSalle’s invariant principle,it can be concluded that the system(23)is asymptotically stable. In addition, note that 0 ＜ α1＜ 1 and α2=2α1/(1+ α1).By Definition 1,it can be verified that system(23)is homogeneous of degreek=(α1? 1)/2 ＜ 0 with respect to the dilation(r1,r1,r1,r2,r2,r2),wherer1=1,r2=(1+α1)/2.

Then,we will show that the nonlinear functionf(x1,x2)of system(21)satisfies the condition of Lemma 1.SinceM?1(x1)is a smooth function,we haveM?1(εr1x1)?M?1(0)=o(εr1)by using the mean value inequality.As a result,for any(x1,x2)≠0,

Then,according to Lemma 1,it can be concluded that system(13)is locally finite-time stable.

Therefore,by the results of Steps 1 and 2,system(13)is finite-time stable,i.e.,x1→0,x2→0 in finite time.Sincex2=˙x1=˙ˉqv=G(ˉqv)ω and matrixG(ˉqv)is nonsingular,we haveˉqv→0,ω→0 in finite time.The proof is completed. □

Since|tanh(·)|≤ 1 and|sat(·)|≤ 1,the proposed finitetime attitude controllers(27)and(28)are bounded by

Hence,the bound of a proposed attitude controller can be adjusted to any small level to accommodate the actuation bound in practical implementation.

Remark 2Compared to the existing finite-time attitude controllers in[16,17,25,26],besides the boundedness of the proposed finite-time attitude controller,it does not need any precise information of the inertia matrixJ.In other words,the proposed finite time control law is model-independent.

4.2 Design of global bounded finite-time attitude controller

In this section,a global bounded finite-time controller is designed for attitude control system(1).

Theorem 1For the spacecraft attitude control systems(1),if the control torque τ is designed as

wherek1＞ 0,k2＞ 0,0 ＜ α1＜ 1 and α2=2α1/(1+α1),then the attitude of system(1)will reach the desired attitudeqdin a finite time,i.e.,q→qd,in a finite time.

whose derivative along system(6)is

Remark 3According to the proof procedure of Theorem 2,it is easy to get the following conclusion.That is for the spacecraft attitude control system(1),if the control torque τ is designed as

wherek1＞0 andk2＞0,the attitude will converge to the desired attitude be asymptotically,i.e.,q→qd,ast→ ∞.In numerical simulations,we will illustrate the advantages of finite-time control,i.e.,faster convergence rate and better disturbance rejection property.

5 Illustrative examples

Consider the attitude regulation problem for the spacecraft described by system(1).The inertia matrix of the spacecraft is given as in[16]:

The initial attitude and initial angular velocity are also set as in[16]:q(0)=[0.3320 0.4618 0.1915 0.7999]T,ω(0)=[2.2?1.2?3]Trad/s.The desired attitude isqd=[0.8 0?0.6 0]T.Different from[16,25,26],here the control torque is required to be bounded.Therefore,the existing finite-time controller is unavailable.

To show how the fractional powers in the proposed finite-time controller affect the system dynamical performances(e.g.,convergence time),Table 1 gives the convergence time for the closed-loop system under the different fractional powers α1,α2.It can be found that if the fractional power α1→ 0,α2→ 0,the convergent rate is faster.As a result,by regulating the additional parameter,i.e.,the fractional powers α1and α2,the convergent rate can be faster without increasing the control gainsk1,k2.As for the rigorous theoretical analysis why the finite-time control can offer better dynamical performance and how the fractional power affects the system dynamical performances,it can be found that in[15,35].

Fig.1 Response curves of the closed-system under finite-time controller(FC).

Fig.2 Response curves of the closed-system under asymptotically stable controller(ASC).

Table 1 Comparison of convergence time of the closed-system under different fractional powers of finite-time controller(FC).

6 Conclusions

The finite-time attitude stabilization problem for a rigid spacecraft under input saturation has been investigated in this paper.Based on the finite-time control technique,a continuous finite-time attitude regulation controller is proposed.Then,by using a switching control approach,a novel global bounded finite-time attitude regulation controller without inertial matrix information has been developed.Finally,an example is given to verify the effectiveness of the proposed method.Future work includes extending the results in this paper to the cases when the the dynamics of actuator is considered.

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11 April 2016;revised 13 September 2016;accepted 8 October 2016

DOI10.1007/s11768-017-6057-6

?Corresponding author.

E-mail:haibo.du@hfut.edu.cn.Tel.:+86 18256956759.

This work was supported by the National Natural Science Foundation of China(Nos.61304007,61673153),the Ph.D.Programs Foundation of Ministry of Education of China(No.20130111120007)and the China Postdoctoral Science Foundation Funded Project(Nos.2012M521217,2014T70584).

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Yusong ZHOUwas born in Dingyuan,Anhui,in 1972.He received his B.Sc.degree from Beijing Institute Of Graphic Communication, Bejing,China,in 1996 and the M.Sc.degree from Hefei University of Technology,Hefei,China,in 2010.From July 2015 to August 2015,he was a visiting researcher at Hefei University of Technology.Heiscurrently an Associate Professor in Anhui Vocational College of Press and Publishing.His research interests include electrical automation,control theory and applications.E-mail:cdadzhou@126.com.

Wenwu ZHUwas born in Suzhou,Anhui,in 1993.He received his B.Sc.degree in Automatic Control from HeFei University of Technology,Hefei,China,in 2015.He is currently pursuing the M.Sc.degree in the School of Electrical Engineering and Automation,Hefei University Of Technology,Anhui,China.His research interests include nonlinear control,and spacecraft attitude control.E-mail:zhuwenwu003@163.com.

Haibo DUwas born in Tongcheng,Anhui,in 1982.He received his B.Sc.degree in Mathematics from Anhui Normal University,China,in 2004,and the Ph.D.degree in Automatic Control from Southeast University,China,in 2012.He is currently an Associate Professor in the School of Electrical Engineering and Automation,Hefei University of Technology.His research interests include nonlinear system control,cooperative control of distributed multi-agent systems and spacecraft attitude control.E-mail:haibo.du@hfut.edu.cn.