ZHAO Jing,JIANG Bin,XIE Fei,GAO Zhifeng,and XU Yufei

1.College of Automation Engineering,Nanjing University of Posts and Telecommunications,Nanjing 210023,China;2.College of Automation Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;3.School of Electrical and Automation Engineering,Nanjing Normal University,Nanjing 210046,China;4.Jiangsu Province 3D Printing Equipment and Manufacturing Key Lab,Nanjing 210042,China;5.Shanghai Institute of Satellite Engineering,Shanghai 200240,China;6.Jiangsu Engineering Laboratory for Internet of Things and Intelligent Robots,Nanjing 210023,China

1.Introduction

A great amount of effort has been made on the control scheme for near space vehicles(NSVs)recently,which provides valuable reliability of the flight control system,see for examples,[1–11]and the references there in.Shtessel et al.proposed sliding mode controls for NSVs[1–3],and an uncertainty modeling and fixed order controller were designed in[4].A robust nonlinear control approach was presented in[5–7].To achieve better performance of controllers for NSVs,the advantages of adaptive with robust or sliding mode control are combined[8–14].In a flight process of NSVs,generally there are four stages,such as,the subsonic stage,transonic stage,supersonic stage and hypersonicstage.Consequently,NSV has strong coupling,complexity and serious nonlinearity.The above mentioned factors may give rise to faults,which inevitably affect flight reliability,stability and security,and consequently lead to disaster. Thus, fault tolerance control (FTC)technology for flight control system is crucial in the field of aerospace industry.Generally, fault types are classified into the following three categories[15,16] :actuator, sensor,and component fault.In[17–24],there have been some results aiming at actuator faults and structural faults for NSVs.In the existing literature,the second-order dynamic terminal sliding mode technique have been applied to study FTC approaches of NSVs,such as,Zhao et al.proposed an FTC strategy for NSV attitude system to reduce the bad effects generated by actuator faults,uncertainties and exogenous disturbances[17].In[18],as NSVs suffer from control effector damage and actuator dynamic fault,a fault estimation and fault tolerant control scheme are developed using adaptive sliding mode(ASM).Furthermore,some important results on the active FTC approach for actuator faults were proposed in[19–22].Alwi et al.provided some new research progress on component faults[23,24].The issue of component faults has been investigated for aircraft dynamical systems with partial loss vertical tail in[23].There have been rich researches on sensor faults,see for examples,[25–29]and the references therein.Lee et al.designed FTCs for sensor fault based on structured kernel principal component analysis[25].In[26],an FTC de-sign subject to sensor faults has been given in the active FTC framework,specifically,by introducing a descriptor system approach,the problem of sensor fault reconstruction and compensation have been studied for a class of nonlinear systems [28]. In [30 – 33] different fault tolerant control schemes for structural fault were proposed.An adaptive nonsingular terminal sliding mode controller is proposed for NSVs with parameter uncertainty and external disturbance,what is more,lumped disturbance can be accommodated using an adaptive update law[34].An adaptive sliding mode backstepping controller is designed to cope with external disturbances and parameter variations for NSVs re-entry attitude system[35].A fault accommodation scheme is proposed for NSVs with control effector damage using both adaptive neural observer and backstepping technique[36].An active fault tolerant control strategy is developed for NSVs under actuator fault based on Takagi-Sugeno(T-S)fuzzy model,which is designed by sliding mode observers and two fault accommodation approaches[37].A decentralized fault-tolerant controller is designed for NSV reentry attitude dynamic to enable global stability of NSV with actuator faults and control surface damage by ASM and backstepping control[38].The proposed robust controller of flight system with adaptive gain can guarantee accuracy tracking in the presence of nonlinearities and disturbances[39].An adaptive neural prescribed performance control approach is proposed to enable the NSVs to track the position and attitude of the target under the influence of input nonlinearity,which consists of input saturation and dead-zone[40].

However,it is the fact that very limited existing reach results on FTCs for NSVs engine faults,have been reported in academic publications.In[41],in order to remove the effect of engine faults,a fault-tolerant controller is developed by using the indirect robust adaptive technique cooperating with the backstepping procedure and the global asymptotically attitude tracking can be achieved in the case of engine faults.Due to the inherent advance on the application of aeronautical engineering,backstepping control has been applied in flight control system[42].Compared with existing works on faulty system of NSV,the contributions from this paper are in two-fold:(i)the unmodeled engine faults are introduced into the attitude control system of NSV;and(ii)an ASM fault-tolerant controller is proposed for engine fault cases.Section 2 describes engine faulty dynamics of NSV and some necessary assumptions.Problem formulation is presented.In Section3 FTC design is given for the NSV faulty model which contains exogenous disturbances and engine faults.In Section 4,simulation results and some necessary comparisons with existing work demonstrate the validity of FTC design and the superiorities contrast to some existing work.Finally,conclusion is presented in Section 5.

2.Modeling of NSV

2.1 Attitude dynamic description

The attitude angleγand attitude angular rateωare described in the following:

whereJ∈ R3×3denotes the inertia tensor in the form of

whereω=[p,q,r]Tdenotes the angular rate and p,q,r are roll velocity,pitch velocity and yaw velocity.uis the control torque generated by engine and aerodynamic surface,and it is written as

whereD(·) ∈ R3×6denotes the sensitivity matrix.δ∈R6×1is the control input vector.γ=[φ,β,α]Twhere φ,α,β are bank angle,angle of attack and sideslip angle respectively.Ω is a skew symmetric matrix and given by

Letx1=γandx2=ω,then system(1)can be transformed as

From(2),we obtainδ=D?(·)u(t)based on dynamic inverse,where the control allocation matrixD?(·)meetsD(·)D?(·)=Ias

where f1(x1)=R(·),f2(x1,x2)=-J-1ΩJω,andg=J-1D.

In the following,a nominal controller is to be designed for the NSV system(5).

2.2 Engine fault model

Remark 1As the first order actuator dynamics˙u=-Λ(u-uc)is considered,here the actuator dynamic is assumed to be much faster than the dynamics,consequently,the fast actuator dynamic generally can be neglected,that is,u≈uc,which could permit to assumeδ≈δc.

For an engine fault case,the disturbance torque is considered,and system(1)can be updated as

where ΔDis an uncertain component caused by engine faults,and it is described as follows:

where ΔD={dij},i=1,2,3;j=1,2,3,...,6.

(i)No engine damage fault denotes as ΔD/=0,if for∀i,j,s.t.dij≡ 0.

(ii)Engine damage fault denotes as ΔD/=0,if ∃i,j,s.t.dij/=0.

d(t)∈ R3×1represents the external disturbance torque vector,so we can expressδc=(D+ ΔD)?(·)uc.

Considering external disturbances and engine faults,(7)can be rewritten as

Thus it needs to design a controller to compensate the influences exerted by the above factors.In view of disturbanced(t),a hypothesis is given in the following.

Assumption 1d(t)is an unknown nonlinear function,which includes external disturbances and model uncertainties.We assume that there will be a continuous bounded function ς(·),that is,||d(t)||＜ ς(·).

Letx1=γandx2=ω,(7)can be transformed as

Similarly,

and letG=J-1(D+D)?G0+ΔG,whereG0=J-1D,ΔG=J-1ΔD.

Remark 2It is noted that the research object of unmolded part ΔDis represented by the engine fault item,and it can be estimated by adaptive technology.LetG=J-1(D+ΔD)in(7)represent the influences of engine faults on NSV system,G0denote the healthy item by the engine,and ΔGrepresent engine faults which will be estimated later.

3.ASM backstepping control design

3.1 Problem formulation

In this paper,our control objective is to design an FTC scheme which can achieve the following goals:(i)the signals of the closed-loop system are bounded;(ii)the desired signals can be accurately and timely tracked under engine fault cases.First of all,a diagram of FTC system(see Fig.1)and a hypothesis will be given for the desired signals as follows.

Fig.1 Diagram for FTC system

Assumption 2It is assumed that both tracking signalycand its derivative˙ycare piecewise continuous and bounded functions in this paper.

For system(3),define errorse1=x1-yc,ande2=x2-ν,and viewx2as a control variable for system˙x1=f1(x1)x2,whereνis the virtual control law to ensuree1→0.We have the following results given in the form of Theorem 1.

Theorem 1Consider NSV attitude dynamics(3)with engine fault free cases,a nominal control scheme(11)and(12)based on backstepping is designed,such that the asymptotic output tracking of NSV attitude control system can be guaranteed,that is

ProofThere will be two steps in the following to prove the theorem.

Step 1Hereνis designed to stabilizee1as follows:

Consider a Lyapunov function in the form of

The time derivative of V1and we obtain

An appropriate virtual controlνcan be designed as follows:

One obtains

Step 2The global control lawδwill be designed in this step,which makes errorse1→0,e2→0.

Another Lyapunov function is chosen as

Then,we can obtain the derivative of V2as

In the following,we will design an appropriate controllerδ,which can remove some terms withe1,e2,x1,x2,

where k2＞0,so one obtains

The proof of Theorem 1 has been completed. □

Remark 3Note thatg-1in(12)cannot be always ensured,sog-1is replaced by the pseudo-inverse ofg#=gT[ggT]-1.The controller(12)can be updated based on Remark 1 and Assumption 2,

3.2 Fault tolerant control design

The control objective is to design the actuator deflection commandδc,which can make the output vectorytracks a commandsignalycasymptotically under external disturbances and engine faults.Similarly,an ideal controller is designed for faulty system(7)as follows:

whereδcdenotes the FTC input for engine fault cases.Since ΔG,ηare unknown in(12),so in the next,an adaptive algorithm will be designed to obtain estimated values of ΔG,andη,respectively.In the following,a dynamic sliding mode observer will be designed in order to obtain the estimation values

3.2.1 Adaptive sliding observer design

In order to estimate some parameters of controller(14),we need to design an adaptive observer to estimate both of the information for external disturbance and engine faultrespectively.The adaptive observer has the form of

Then the error system is obtained from(10)subtracting(15)

The corresponding results are given in the form of Theorem 2.

Theorem 2Under the dynamic sliding mode observer design(15),the adaptive laws(18),(19)for estimating ΔGandηcan make system stable and meanwhile such thatz1→0.

ProofA dynamic sliding mode surface is designed as follows:

An appropriate Lyapunov function is chosen as in the form of

where Γ1and Γ2are positive definite adaptive gain matrices,andand letbe the estimated errors,whereare the estimations ofη,ΔGrespectively.Then the derivative of Lyapnunov of V3is

Choose appropriate estimated algorithms for ΔGandηas follows:

where Proj[ζi,1]denotes the projection operator,which projects the estimateto the interval[ζi,1],and 0 ≪ζi＜ 1,sign(·)denotes the sign function.

Remark 4The estimation algorithm of engine fault(18)owns the following property referred to[14]as follows:

On the basis of Remark 4,the derivative of Lyapunov of V3further is rewritten as follows:

So far,the proof of Theorem 2 has been completed. □

3.2.2 Adaptive backstepping FTC design

From the above analysis,some results can be obtained in the form of Theorem 3.

Theorem 3The backstepping controllers(11)and(12)updated by adaptive algorithms(18)and(19),as applied to the engine faulty model system subject to(9),could guarantee the asymptotic output tracking performance and also the boundedness of all the closed-loop signals,that is,

ProofConsider an appropriate Lyapunov function as follows:

Then the derivate of V4is as follows:

Based on(11)and adaptive laws(18)and(19),one obtains

One obtains via further calculating

where k1＞0,and k2＞0,one obtains

So far,the proof of theorem 3 has been completed. □

Remark 5Due toin(21)cannot be always ensured,letbe replaced by its pseudo-inverseThen,(21)can be rewritten as

Remark 6The effect of engine faults for NSV flight control system is slowly changing,and the main effect objects are height and velocity during the cruise phase.The effect of attitude control system can be compensated by the rudder control allocation.Thus the proposed method can guarantee attitude system’s stability.

4.Example

In the following,the proposed method in this paper will be verified by simulation experiments,we choose the initial state:γ(0)=[0,0,0]T,ω(0)=[0.2,0,-0.4]T,and tracking signals,γd=[1,0,2]T,ωd=[0,0,0]T,respectively.

The disturbanced(t)with upper bound ‖d(t)‖ =102is denoted by

Engine damage factor ΔDis given as follows:

and k1=k2=0.3,Γ1= Γ2=diag{0.5}.

To verify the effectiveness of the proposed FTC approach,we consider the following different cases during simulation experiments to observe the responses of control torquesu=[u1,u2,u3]T,attitude ratesw=[w1,w2,w3]Tand attitude anglesx=[x11,x12,x13]T.

Case A No engine fault

In this case,ΔD=0,that is,there is no engine fault.The corresponding simulation results using the method proposed nominal control law(12)in this paper are depicted in Fig.2(a)–Fig.8(a).From Fig.2(a)–Fig.4(a),it can be seen that,control torques u1,u2,u3can be stable quickly in engine fault free cases.Similarly,the responses of ω1,ω2,ω3and x11,x12,x13can be seen respectively from Fig.5(a)–Fig.7(a),which show that tracking signal is tracked accurately and quickly.

Case BEngine fault without FTC

As the engine fault occurs,that is,ΔD/=0,in order to compare the proposed method,we experiment without fault tolerance first,and the corresponding simulation results of the nominal control law(12)are demonstrated in Fig.2(b)–Fig.8(b),which show that control torques responses of u1,u2begin to wave after 15 s.While the influence of the engine fault on u3is not so obvious from Fig.4(b).Fig.5(b)–Fig.8(b)show the responses of attitude rates ω1,ω2,ω3and the states of x11,x12,x13.It can be seen that system performance requirements cannot be met with the nominal controller law,due to the angle velocities ω1,ω2,and x11cannot track the desired signals once the engine fault occurs.

Fig.2 Responses of u1under cases A,B,C

Fig.3 Responses of u2under cases A,B,C

Fig.4 Responses of u3under cases A,B,C

Fig.5 Responses of ω1under cases A,B,C

Fig.6 Responses of ω2under cases A,B,C

Fig.7 Responses of ω3under cases A,B,C

Fig.8 Responses of x under cases A,B,C

Case CEngine fault with FTC

Similarly,as the engine fault occurs,that is,ΔD/=0,and the corresponding simulation results are expressed as Fig.2(c)–Fig.8(c).Compared to the designed control law(12),it can be easily seen that the tracking precision of control torqueucan be markedly improved,especially for u1,u2from Fig.6(b)– Fig.7(b).ω2and ω3can also track the desired signals within 20 s while ω1has a slight fluctuation near the tracking target.Fig.8(b),shows attitude angles’responses of NSV’s nonlinear dynamics under engine faults but without FTC technique.Obviously, the responses of x11begin to show a big fluctuation near the tracking signal.While from Fig.8(c),we can see that the proposed fault tolerant control method in this paper could make the system accurately track the desired signals.As the engine fault occurs, the proposed FTC scheme can compensate the engine fault within effective time,and consequently keep the flight control system stable.In summary,the simulation results can prove that the proposed scheme has better tolerance to engine faults from the simulation results.

5.Conclusions

In this paper,an FTC technique based on ASM backstepping is proposed for the attitude control systems of NSV considering engine faults and external disturbances.The proposed scheme combining adaptive backstepping with the sliding mode control strategy could guarantee the stability of the system and track the desired signals under external disturbances and enginefaults. Firstly,attitude mode description and the engine faulty model are given.Secondly,a nominal control law is designed.Thirdly,a sliding mode observer is given to estimate both the information of engine faults and external disturbances.An ASM technology based previous nominal control law is developed via updating faulty parameters.Finally,the effectiveness of the developed scheme in this paper can be verified by some simulation results.

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