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.

高水平应用型大学培养人才的目标是具有一定的理论知识水平和较强的实践动手能力的高级专门人才。但从土木工程相关行业企业反馈的信息看，毕业生存在工程素质教育相对缺乏，实践动手能力较弱，团队协作能力不够等方面的问题。

2.Modeling of NSV

2.1 Attitude dynamic description

The attitude angle γ and attitude angular rate ω are described in the following:

where J ∈ R3×3denotes the inertia tensor in the form of

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

where D(·) ∈ 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

Let x1= γ and x2= ω,then system(1)can be transformed as

其中，Rf为f的径向导数，即在范数‖f‖=|f(0)|+‖f‖Β下，Bloch空间Β是Banach空间[1].

From(2),we obtain δ =D?(·)u(t)based on dynamic inverse,where the control allocation matrix D?(·)meets D(·)D?(·)=I as

Key words: public institutions; personnel files; management experience

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

2.2 Engine fault model

Theorem 3 The 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,

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

迄今为止，只有经过严格选择的患者才能在生物制剂治疗中获益，因此，在疾病早期能够全面评估患者对疾病的易感性、临床特征、生物学标记、遗传学特征及疗效影响因素，进一步精准地进行CRSwNP内在型细化分型能有助于提出个性化的特异性治疗措施、最优化疗效，减少反复手术几率及有效预防下呼吸道炎症的进展[10]。另外，在治疗某一种特定发病机制的CRSwNP内在型时，是仅生物制剂就足矣还是需要联合应用其他的治疗手段，业界专家有必要共同探讨并达成相关共识[20]。

where ΔD is 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.

（1）戊申將軍範伯陽，庚申將軍華文陽，戊戌將軍範少卿。（《太上說玄天大聖真武本傳神呪妙經註》卷一，《中华道藏》30/538）1 30/538表示第30册，第538页。

(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 express δc=(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 disturbance d(t),a hypothesis is given in the following.

Assumption 1 d(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(·).

Let x1=γ and x2=ω,(7)can be transformed as

Similarly,

and let G=J-1(D+D)?G0+ΔG,where G0=J-1D,ΔG=J-1ΔD.

Remark 2 It is noted that the research object of unmolded part ΔD is represented by the engine fault item,and it can be estimated by adaptive technology.Let G=J-1(D+ΔD)in(7)represent the influences of engine faults on NSV system,G0denote the healthy item by the engine,and ΔG represent 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 2 It is assumed that both tracking signal ycand its derivative˙ycare piecewise continuous and bounded functions in this paper.

For system(3),define errors e1= x1-yc,and e2=x2-ν,and view x2as a control variable for system˙x1=f1(x1)x2,where ν is the virtual control law to ensure e1→0.We have the following results given in the form of Theorem 1.

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

Proof There will be two steps in the following to prove the theorem.

人力资源是企业内部一笔隐形的财富，如果能够充分利用，对企业的发展和经济效益的提升都会发挥出巨大的帮助。因此，企业应该有意识地为会计电算化部门引进综合性人才，让综合性人才来负责会计电子档案管理工作，在引进人才的时候不仅重视人才的会计知识能力，也要重视会计人员计算机专业知识能力，充分考虑企业自身的经营特点，定期对引进的人才进行再次培养，确保人才能够与时俱进，能够确保单一的“核算人才”朝着“智能人才”方向转变[1]。

Step 1 Here ν is designed to stabilize e1as follows:

Consider a Lyapunov function in the form of

The time derivative of V1and we obtain

An appropriate virtual control ν can be designed as follows:

One obtains

Step 2 The global control law δ will be designed in this step,which makes errors e1→0,e2→0.

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

Another Lyapunov function is chosen as

Then,we can obtain the derivative of V2as

本系统是借助LabVIEW平台搭建的在线监测系统。LabVIEW是由美国国家仪器公司(NI)开发的一款适用于多操作系统的基于图形程序的虚拟仪器仪表开发平台，应用范围覆盖了工业自动化、测试测量、嵌入式应用、计算机仿真等众多领域[5]。LabVIEW的程序由前面板和程序框图组成。前面板集合了用户输入和显示控件，相当于传统仪器的面板；程序框图中用功能模块和适当的连线构成控制流程图，其中每个功能模块均为封装好的子程序(子VI)，即图形化代码。程序的执行顺序依据数据流走向决定，从而控制前面板中的对象[6]。

In the following,we will design an appropriate controller δ,which can remove some terms with e1,e2,x1,x2,

where k2＞0,so one obtains

The proof of Theorem 1 has been completed. □

Remark 3 Note that g-1in(12)cannot be always ensured,so g-1is replaced by the pseudo-inverse of g#=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 command δc,which can make the output vector y tracks a commandsignal ycasymptotically under external disturbances and engine faults.Similarly,an ideal controller is designed for faulty system(7)as follows:

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非公募体育基金会不可以直接面向公众开展慈善公益募捐活动募集资金，主要是通过向特定的自然人、法人和组织募集资金及确保基金会资金保值增值的前提下，开展投资活动获得资金，以此从事体育慈善公益事业。非公募体育基金会登记部门为省、自治区、直辖市及以上的民政部门，因不能面向公众公开募集资金，所以资金来源无地域性限制。近年来，我国非公募体育基金会发展迅猛，截至目前，我国非公募体育基金会已有41家，占体育基金会的57.7%，已经超过公募体育基金会。其中在民政部注册的有2家，分别是桃源居公益事业发展基金会和萨马兰奇体育发展基金会，其余39家为地方民政部门注册的公募体育基金会。

where δcdenotes the FTC input for engine fault cases.Since ΔG,η are unknown in(12),so in the next,an adaptive algorithm will be designed to obtain estimated values of ΔG,and η,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 fault respectively.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 2 Under the dynamic sliding mode observer design(15),the adaptive laws(18),(19)for estimating ΔG and η can make system stable and meanwhile such that z1→0.

ProofA dynamic sliding mode surface is designed as follows:

An appropriate Lyapunov function is chosen as in the form of

根据物理检测结果，其配制的水泥需水量、凝结时间等重要性能均达到要求，水泥3d、28d强度开始偏低，后经微调熟料、石膏和混合材料的配比，配制水泥的SO3和MgO含量、需水量、凝结时间、3d、28d强度等物理化学性能，符合国家标准要求。说明该石膏矿石经加工后可作为生产水泥用缓凝剂使用。

where Γ 1and Γ 2are positive definite adaptive gain matrices,andand let be the estimated errors,where are the estimations of η,ΔG respectively.Then the derivative of Lyapnunov of V3is

Choose appropriate estimated algorithms for ΔG and η as follows:

One obtains via further calculating

Remark 4 The 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:

小型农田水利工程单体工程量较小、分布较为分散，增加了建设管理难度，特别是工程建设机械都要从农村农田中经过，工程进场道路不具备，农村河网密集，很多建设机械、建筑材料都要通过水路翻运，大大增加了施工成本。田间灌溉闸、防渗渠均建在农户庄稼中，与农民存在较大的矛盾协调工作。

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.

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 assume δ ≈ δc.

Proof Consider an appropriate Lyapunov function as follows:

Case A No engine fault

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

根据回归结果显示，主要解释变量GDP增长率对碳排放量的影响呈现正向，解释了67%的部分，且在1%的显著水平上显著，说明湖北省在近20年的时间里，经济增长在很大程度上影响了碳排放。虽然代表经济发展和产业结构的第二产业占比也对其显示正向影响，但结果并不显著，人口规模和对外贸易规模也是同样的情况，说明除了经济增长，在武汉这样的非暖气供应的内陆城市，其他因素对于碳排放的影响程度较小。

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.

where k1＞0,and k2＞0,one obtains

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

Theorem 1 Consider 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

Remark 6 The 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.

（1）本项目处理对象为浊漳河南源的受污河水，设计处理规模为12万m3/d，受污河水原水质属于地表水环境质量劣V类水质，经本工程处理后达到地表水环境质量Ⅴ类水标准。

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

Engine damage factor ΔD is 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 torques u=[u1,u2,u3]T,attitude rates w=[w1,w2,w3]Tand attitude angles x=[x11,x12,x13]T.

Then the derivate of V4is as follows:

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 B Engine 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 C Engine 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 torque u can 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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