Qinghu ZHU,Gungfu MA,Xioting WANG,Aiguo WU,*

aDepartment of Control Science and Control Engineering,Harbin Institute of Technology,Harbin 150001,China

bShanghai Aerospace Control Technology Institute,Shanghai 201109,China

cHarbin Institute of Technology Shenzhen Graduate School,Shenzhen 518055,China

dShanghai Key Laboratory of Aerospace Intelligent Control Technology,Shanghai 201109,China

1.Introduction

The attitude control of spacecrafts and satellites has important applications for some space missions such as pointing and formation flying.This topic has attracted much attention from a considerable number of researchers.1–4For attitude control,much investigation is based on the unit quaternion representation.5–8In Ref.5,some attitude controllers with the structure of a Proportional-Derivative(PD)feedback plus feed-forward were designed for a rigid body.In Ref.6,a sliding mode control law was designed and applied to spacecraft attitude tracking maneuvers when the inertia of a spacecraft was not exactly known.In Ref.7,attitude control was considered for a rigid spacecraft with the control signal constrained by a common maximum magnitude in the presence of bounded unknown disturbances.A sliding mode controller was designed for such a type of spacecrafts to achieve global stability.The designed controller was in the form of a proportional feedback plus a smooth switch-like feedback with an auxiliary time-varying attitude gain function.High-order sliding mode controllers were designed in Ref.8for attitude control of a rigid spacecraft.A merit of the designed controller is that the phenomena of chattering can be eliminated.In Ref.9,two Fault-Tolerant Control(FTC)schemes were derived for spacecraft attitude stabilization with external disturbances.In Ref.10,the quaternion model of a rigid spacecraft was firstly transformed into the Lagrange-like,and then two robust sliding mode controllers were proposed to solve attitude tracking problems in the absence of both model uncertainties and external disturbances as well as in their presence.

In the aforementioned attitude control laws,the angular velocity of a rigid spacecraft was used in the construction of an attitude control law.However,in some circumstances,it is not easy to measure the angular velocity.Therefore,it is necessary to design an attitude control law without angular velocity measurement.Such a controller was designed in Ref.11by using a nonlinear filter of the quaternion to replace the angular velocity feedback.In Ref.12,a velocity free attitude stabilization scheme was proposed for a rigid spacecraft.In this control scheme,an angular velocity observer-like system was explicitly designed to construct the stabilizing feedback.In Ref.13,two simple Saturated Proportional-Derivative(SPD)controllers were proposed for asymptotic stabilization of a rigid spacecraft with actuator constraints and without velocity measurement.In Ref.14,a continuous angular velocity observer with fractional power functions was proposed to estimate the angular velocity via quaternion attitude information.

For flexible spacecraft,the effect of the motion of the elastic appendages must be taken into consideration,and thus the attitude control problem is more complicated.In Ref.15,a dynamic controller was proposed for the attitude control of a flexible spacecraft under the assumption that the modal variables describing flexible elements were not available.In Ref.16,an adaptive sliding mode control law with a hybrid sliding surface was proposed for a flexible spacecraft to minimize the effect of uncertainties and disturbances.In Ref.17,an adaptive control law was proposed to solve the attitude tracking problem for flexible spacecrafts subject to a gravity-gradient disturbance under inertia matrix uncertainty.In Ref.18,a nonlinear observer-based state feedback control law was designed to ensure the control objectives for attitude tracking.In Refs.15–18,attitude control laws for flexible spacecraft were designed based on the unit quaternion representation.In Ref.19,the three-axis attitude tracking control problem was investigated in presence of parameter uncertainties and disturbances based on the modified Rodrigues parameterization.An attitude control law was presented in the form of a nonlinear PD term plus a switching function about a sliding variable.

In this paper,we consider the problem of attitude control for flexible satellites based on the unit quaternion representation.It is assumed that the modal variables describing flexible elements are not measurable.For such a class of flexible satellites,a dynamic controller is given to achieve stability for the closed-loop system.The designed controller has two features.One is that it is in the form of an observer-based state feedback.The other is that the angular velocity feedback is not used.

2.Motion equations of a flexible satellite and problem formulation

In this section,the mathematical model of a flexible satellite is given.We adopt the unit quaternion to describe the attitude of a satellite.The associated quaternion is given by

with

where ∈ is the unit Euler axis,and φ is the rotation angel about the Euler axis.The quaternion components are not independent on each other,and they satisfy a single constraint as

For the quaternion q in Eq.(1),define the following matrix Ξ:

With the preceding notation,the quaternion kinematics equation is given as6

where ω is the satellite angular velocity.Due to the property of the matrix Ξ,there holds

Under the hypothesis of small deformations,by using the Euler theorem,the dynamic equations of a flexible satellite can be given by

where J is the total inertia matrix which is symmetric,u is the external torque acting on the main body of the satellite,and η is the modal coordinate vector.In the modal Eq.(8),C and K are respectively the damping matrix and the stiffness matrix,which are in the following forms:

δ is the coupling matrix between flexible and rigid dynamics.In this paper,N elastic modes are considered.The corresponding natural frequencies are ωni,(i=1,2,...,N),and the associated dampings are ζi,(i=1,2,...,N).From Eqs.(7)and(8),the following dynamic equations of the flexible satellite can be obtained15:

In dynamic Eq.(9),

is the main body inertia matrix and

is the total velocity of the flexible appendages.By summarizing derivation,a satellite with flexible appendages can be described by the mathematical models in Eqs.(5)and(9).

Compared with attitude sensors,angular velocity sensors are more expensive.Therefore,it is practically important to design a control law without angular velocity measurement.In this paper,we aim to design such a control law so that

3.Partial state feedback control

In this section,it is assumed that the modal variables η and ψ are available.Since the angular velocity is not available,we introduce the following filter:

where Cv（s）is a 4 × 4 linear time-invariant,strictly positive real,strictly proper transfer matrix.With the aid of this filter,we design the following partial state feedback controller for the flexible satellite system described by Eqs.(5)and(9):

where k is the control parameter,in the control gain of Eq.(13),the matrix P will be determined later.The main result of this section is given in the following theorem.

Theorem 1.Consider the satellite system described by Eqs.(5)and(9).Let Cv(s)be a 4×4 linear time-invariant,strictly positive real,strictly proper transfer matrix,and the symmetric matrix P satisfies the following Lyapunov matrix equation

for a prescribed positive definite matrix Q.Then,under the control law in Eqs.(11)–(13),the closed-loop system achieves the performance in Eq.(10).

Proof.Consider an arbitrary minimal realization of Cv（s）as follows:

Since Cv（s）is linear time-invariant,strictly positive real,strictly proper,then according to Kalman-Yakubovich-Popv’s lemma,there exist positive definite matrices P1and Q1such that

For the closed-loop system under the control law in Eqs.(11)–(13),we consider the following Lyapunov function:

Denote

Thus,V=V1+V2+V3+V4.

Next,we give the time derivatives for the four functions V1,V2,V3,and V4along the solution of the closed-loop system.For the function V1,from Eq.(5),we have

For the function V2,by using Eqs.(5),(9),and(12),we have

For the function V3,it follows Eqs.(15),(16),and(9)that

For the function V4,it can be obtained from Eq.(9)that

With the previous time derivatives for the functions Vi,i=1,2,3,4,it can be obtained that

Since V is a continuously differentiable, radially unbounded,and positive definite function,and˙V≤0 over the entire state,then the global asymptotic stability can be stated by using the LaSalle invariant theorem.De fineˉΩ as the largest invariant set,which is contained in

Then we obtain=0 on Ω.With this,it follows from the first expression in Eq.(9)that

which implies that on the set Ω,there holds u=0.By using Eqs.(12),(13),and(15),it can be obtained that

From the preceding expression,it can be derived that qv=0 on the set Ω.Therefore,ˉΩ=｛（q，ω，x，η，ψ）:x=0，ω =0，qv=0，η =0，ψ =0｝，and the global asymptotic stability is proven.□

Remark 1.In the proposed stabilizing control law of Eqs.(12)and(13),the need for direct angular velocity measurement is removed since the filter Cv(s),or equivalently the filter in Eq.(15),is used.

4.Observer-based feedback controllers

In the designed control law in Eqs.(11)–(13),modal measurements are needed.However,in many cases,it is difficult to measure vibration modals.Thus,it is necessary to construct a control law when modal measurements are not available.For this end,we adopt the idea of an observer-based control law.For these vibration modals,we use an observer to give their estimation,and then the overall control law can be obtained by replacing the modals in the controller Eqs.(11)–(13)with their estimates.De fine the estimates of the modal variables η and ψ as^η and^ψ,respectively.We construct the following observer to obtain the estimates of η and ψ:

where

and the positive definite matrices P2and P3are determined later.With the aid of the observer in Eq.(28),now we construct the following dynamic controller for the satellite system in Eqs.(5)and(9):

where F is given by Eq.(13),and z is constructed by Eq.(11).

Remark 2.When the observer in Eq.(28)for estimating the flexible modals η and ψ is constructed,the basic structure of the sub-equation on η-ψ in Eq.(9)is utilized.With the estimationsη andof η and ψ,the overall controller is then obtained by replacing η and ψ in the control law Eq.(12)withand.

Remark 3.Compared with some existing observers in flexible spacecraft,for example in Ref.15,an important feature of the observer in Eq.(28)is that angular velocity measurement is not used.Thus,the overall dynamic output feedback controller in Eqs.(28),(29)can be implemented without angular velocity measurement.

Theorem 2.Consider the satellite system described by Eqs.(5)and(9).Let Cv(s)be a 4×4 linear time-invariant,strictly positive real,strictly proper transfer matrix,and the symmetric matrices P2and P3satisfy the following Lyapunov matrix equation:

for two prescribed positive definite matrices Q2and Q3.Then under the control law Eqs.(29),(28),and(11),qv，ω，x，η,and ψ tend to zero asymptotically for any initial condition.

Proof.Consider an arbitrary minimal realization of Cv(s)as in Eqs.(15)and(16).For the closed-loop system under the control law Eqs.(29)and(28),introduce the errors

and denote

In addition,we also define functions V1,V2,and V3as in Eqs.(18),(19),and(20),respectively.For the closed-loop system,we consider the following Lyapunov function:

By calculation,it can be found that the time derivatives of V1and V3are the same as those in the proof of Theorem 1.Next,we give the time derivatives of functions V2,V4,and V5along the solution of the closed-loop system.For the function V2,by using Eqs.(5),(9),(29),and(31),we have

For function V4,it can be obtained from Eq.(9)that

Fig.1 Quaternion behavior for case of partial state feedback in Eqs.(11)–(13).

For the function V5,from Eqs.(9),(31),(28),and(6),it can be derived that

With the previous preliminaries,it can be obtained from Eq.(14)that

Next,along a similar line to that in the proof of Theorem 1,the conclusion of this theorem can be obtained according to the LaSalle invariant theorem.□

5.Simulation results

In this section,we apply the control law Eqs.(12),(29),and(28)to a satellite model with four modes to demonstrate the previous theoretical results.The main parameters of the flexible satellite taken from Ref.20are as follows.

The main body inertia matrix(in kg·m2)is

and the coupling matrix between flexible and rigid dynamics(in kg1/2·m)is

For the four natural modes of the flexible satellite,the natural frequencies are as follows:

and the dampings are given as

The maneuver of the considered flexible satellite is a rotation of 160°with an Euler axis,i.e.,

Thus,the initial attitude described by the quaternion is

and the initial angular velocity of the spacecraft is ω（0）= ［0，0，0］T

In addition,the initial values of the four modals of the flexible appendages are

Firstly,we apply the partial state feedback controller Eqs.(11)–(13)to the flexible satellite.The parameters of the controller are as follows:

In addition,the parameter P is determined by solving the Lyapunov matrix Eq.(14).The behavior of the quaternion q=q0+q1i+q2j+q3k is shown in Fig.1 where i，j and k are the fundamental quaternion units.The modal displacements are shown in Fig.2.From Fig.1,it can be seen that the quaternion q of the closed-loop system is asymptotically convergent.

Fig.2 Modal displacements for case of partial state feedback in Eqs.(11)–(13).

Fig.3 Quaternion behavior for case of observer-based state feedback in Eqs.(28)and(29).

Fig.4 Modal displacements for case of partial state feedback in Eqs.(28)and(29).

Secondly,the observer-based feedback controller in Eqs.(28),(29)is also used for the flexible satellite,and the parameters are chosen to be as follows:

In addition,the matrices P2and P3are determined by Eq.(30).In this case, the behavior of the quaternion q=q0+q1i+q2j+q3k is shown in Fig.3,and the modal displacements are shown in Fig.4(where^means estimated value).From Fig.3,it can be seen that the quaternion q is also asymptotically convergent when the observer-based control law is used.In addition,it can be seen from Figs.2 and 4 that the vibration amplitude of the flexible modals under the observer-based control law is bigger than that under the state feedback control law.

6.Conclusions

In this paper,attitude control is considered without angular velocity measurement for satellites with flexible appendages based on quaternion models.Firstly,a partial state feedback control law is proposed for this kind of satellites to guarantee convergence of the quaternion behaviors of the considered satellites.Secondly,when the modal variables describing flexible elements are not measurable,an observer-based partial state feedback control law is also presented.By applying this dynamic controller to the considered satellites,the convergence of the closed-loop system can also be guaranteed.A common feature of the two control laws proposed in this paper is that angular velocity measurement is not used.In future,we will use the idea in this paper to the problem of attitude tracking for flexible satellites.

Acknowledgements

This work was co-supported by the Major Program of National Natural Science Foundation of China (Nos.61690210,61690212),Shenzhen Municipal Basic Research Project for Discipline Layout(No.JCYJ20170413112722597),and Shenzhen Municipal Project for Basic Research(Nos.JCYJ20170307150952660,JCYJ20170307150227897).

1.Liu H,Guo L,Zhang Y.An anti-disturbance PD control scheme for attitude control and stabilization of flexible spacecrafts.Nonlinear Dyn 2011;67(3):2081–8.

2.Xiao B,Hu QL,Zhang Y.Adaptive sliding mode fault yolerant attitude tracking control for flexible spacecraft under actuator saturation.IEEE Trans Control Syst Technol 2012;20(6):1605–12.

3.Guo Y,Song S.Adaptive finite-time backstepping control for attitude tracking of spacecraft based on rotation matrix.Chin J Aeronaut 2014;27(2):375–82.

4.Sun H,Li S.Composite control method for stabilizing spacecraft attitude in terms of Rodrigues parameters.Chin J Aeronaut 2013;26(3):687–96.

5.Wen TY,Kreutz-Delgado K.The attitude control problem.IEEE Trans Autom Control 1991;36(10):1148–62.

6.Lo SC,Chen YP.Smooth sliding-mode control for spacecraft attitude tracking maneuvers.J Guid Control Dyn1995;18(6):1345–9.

7.Wallsgrove RJ,Akella MR.Globally stabilizing saturated attitude control in the presence of bounded unknown disturbances.J Guid Control Dyn 2005;28(5):957–63.

8.Tiwari PM,Janardhanan S,Nabi MU.Attitude control using higher order sliding mode.Aerosp Sci Technol 2016;54:108–13.

9.Shen Q,Wang D,Zhu S,Poh EK.Integral-type sliding mode fault-tolerant control for attitude stabilization of spacecraft.IEEE Trans Control Syst Technol 2015;23(3):1131–8.

10.Wu S,Radice G,Gao Y,Sun Z.Quaternion-based finite time control for spacecraft attitude tracking.Acta Astronaut 2011;69(1–2):48–58.

11.Lizarralde F,Wen JT.Attitude control without angular velocity measurement:a passivity approach.IEEE Trans Autom Control 1996;41(3):468–72.

12.Benziane L,Benallegue A,Chitour Y,Tayebi A.Velocity-free attitude stabilization with inertial vector measurements.Int J Robust Nonlinear Control 2016;26(11):2478–93.

13.Su Y,Zheng C.Velocity-free saturated PD controller for asymptotic stabilization of spacecraft.Aerosp Sci Technol 2014;39:6–12.

14.Gui H,Vukovich G.Finite-time angular velocity observers for rigid-body attitude tracking with bounded inputs.Int J Robust Nonlinear Control 2017;27(1):15–38.

15.Gennaro SD.Passive attitude control of flexible spacecraft from quaternion measurements.J Optim Theory Appl 2003;116(1):41–60.

16.Shahravi M,Kabganian M,Alasty AM.Adaptive robust attitude control of a flexible spacecraft.Int J Robust Nonlinear Control 2006;16(6):287–302.

17.Gennaro SD.Adaptive robust tracking of flexible spacecraft in presence of disturbances.J Optim Theory Appl 1998;98(3):545–68.

18.Gennaro SD.Output attitude tracking for flexible spacecraft.Automatica 2002;38(10):1719–26.

19.Jin E,Sun Z.Robust attitude tracking control of flexible spacecraft for achieving globally asymptotic stability.Int J Robust Nonlinear Control 2009;19(11):1201–23.

20.Hu QL.Robust adaptive backstepping attitude and vibration control with L2-gain performance for flexible spacecraft under angular velocity constraint.J Sound Vib 2009;327(3–5):285–98.