# Adaptive Finite-Time Control for a Flexible Hypersonic Vehicle with Actuator Fault

Authors: Jie Wang; Qun Zong; Xiao He; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The problem of robust fault-tolerant tracking control is investigated. Simulation on the longitudinal model of a flexible air-breathing hypersonic vehicle (FAHV) with actuator faults and uncertainties is conducted. In order to guarantee that the velocity and altitude track their desired commands in finite time with the partial loss of actuator effectiveness, an adaptive fault-tolerant control strategy is presented based on practical finite-time sliding mode method. The adaptive update laws are used to estimate the upper bound of uncertainties and the minimum value of actuator efficiency factor. Finally, simulation results show that the proposed control strategy is effective in rejecting uncertainties even in the presence of actuator faults.


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## Body

## 1. Introduction

Air-breathing hypersonic vehicles (AHVs) are intended to be a reliable and cost-effective technology for access to space. Because the slender geometries and light structures cause significant flexible effects and strong coupling between propulsive and aerodynamic forces resulting from the integration of the scramjet engine, AHVs are confronting many complex problems and challenges, involving many different research areas, such as aerodynamics, thermal protection, and communication, and many problems of these fields have been reported [1–3]. Meanwhile, flight control design for AHVs is a hot topic and a challenging task [4, 5].During the last decades, a kind of flexible hypersonic vehicle model including flexible dynamics has been developed in [6, 7]. Based on this model, there have been several papers discussing the challenges associated with the control of air-breathing hypersonic vehicle (AHV) [8, 9] and many control methods have been employed in the flight control system. In [10], a linear quadratic regulator (LQR) was presented for a linearized FAHV model. In [11–13], sequential loop closure controller was designed for the FAHV based on adaptive dynamic inversion together with backstepping structure. In [14, 15], approximate feedback linearization based on dynamic inversion method was adopted to design controller for the FAHV. In [16, 17], a nonlinear tracking controller was constructed by using a minimax LQR control approach, which provides robust stability and excellent tracking performance with parameter uncertainties.The approaches mentioned above do not specifically consider possible actuator faults, which deteriorate the control performance, affect stability, and security of the AHVs, and sometimes even lead to catastrophic accidents. Consequently, it is essential that the actuator faults must be taken into account in the controller design. In the current papers, some fault-tolerant control schemes for AHVs have attracted more and more research attention and gained fruitful results, which can be reported in [18–21]. In [18–20], the results mainly concentrate on the reentry attitude control of the AHV. Meanwhile, the fault tolerant control strategies for the longitudinal model of the AHVs are studied. In [21], an observer-based fault-tolerant control approach using both robust control and LMI techniques is designed for a linearized longitudinal AHV model in the presence of parameter uncertainties and actuator faults, but this method was effective only in the neighborhood of the operating point. On the other hand, nonlinear fault-tolerant control design methods have been devoted to the longitudinal AHV model. A finite-time integral sliding mode control method was proposed in [22], which could achieve superior velocity and altitude tracking performance with actuator fault. In [23], the longitudinal AHV model with unknown parameters and uncertain actuator faults is formatted into a parametric strict-feedback form, and then an adaptive fault-tolerant control scheme based on a combination of back-stepping control and dynamic surface control techniques is applied to make the velocity and altitude track the desired value.However, the aforesaid methods only consider the rigid body of AHVs without flexible effects. A fault-tolerant control scheme for the FAHV was presented in [24], according to the model obtained by approximate linearization in given flight conditions. So, this scheme may not obtain good control performances when flight dynamics undergo great parameter perturbations. To the best of our knowledge, although considerable effort has been made on the control design for the AHVs, the important issue of fault-tolerant control of the FAHV dynamical system has not been fully investigated yet, which remains challenging and motivates us to do this study.As a typical robust control method, sliding mode control (SMC) scheme is regarded as an effective method to cope with external disturbances and parametric uncertainties [25]. Recently, the SMC method has been widely applied for the fault tolerant control of aircraft system, spacecraft, and so on. In [26], a fault-tolerant sliding mode controller was presented for an aircraft system, which requires the message of the effectiveness factor, while it may be difficult and expensive to obtain the actuator faults online. In [27], a finite-time convergent SMC scheme is developed to solve the problem of fault-tolerant control for a rigid spacecraft. The drawback of this method is that the message of the lower bound of the effectiveness factor and the upper bound of system uncertainties needs to be known in prior.The aforementioned references could achieve desired performance through the SMC methodology affected by actuator faults. Although the traditional SMC can guarantee the stability of the system, it adopts a linear switching function. Then the system states and the errors converge to an equilibrium point asymptotically in infinite time. In other words, it means that finite-time convergence is not ensured. Motivated by the above discussions, we propose a novel adaptive sliding mode control scheme for the longitudinal model of the FAHV with uncertainties and actuator faults in this paper. As compared with the existing results, the main contributions are as follows. Firstly, the design method of sliding mode surface based on homogeneous geometry could assure practical finite-time converged tracking of the desired command. Secondly, the upper bounds of aerodynamic uncertainties and the minimum value of actuator efficiency factor are not required in prior. The adaptive law is designed to adjust the control gains dynamically so as to ensure the establishment of sliding mode motion, and the robustness against uncertainties is ensured at the same time. After the uncertainties and actuator faults are compensated using adaptive sliding mode control scheme, the stability of the closed-loop system can be maintained.The rest of this paper is organized as follows. In Section2 the FAHV model is introduced and control objective is stated. Section 3 designs the sliding mode surface and the corresponding adaptive finite-time fault tolerant controller was proposed with actuator fault. Simulation results are discussed in Section 4 and the conclusions are provided in Section 5.


## 2. Problem Statement

The considered FAHV model is derived from [6, 28], and the longitudinal equations of motion of the FAHV are given by
(1)V˙=(Tcosα-D)m-gsinγ,γ˙=(L+Tsinα)mV-gcosγV,h˙=Vsinγ,α˙=Q-γ˙,Q˙=MyyIyy,η¨i=-2ςmωm,iη˙i-ωm,i2ηi+Ni,i=1,2,3,wherex=[V,γ,h,α,Q]T is a vector of rigid-body state, which includes the vehicle speed, flight path angle, altitude, angel of attack, and pitch rate, respectively; ηi, ωm,i, and ςm are the generalized flexible coordinate, natural frequencies, and damping coefficients of the ith elastic mode. The readers may refer to [7] for a full description of the variables in this model.Because of coupling in aerodynamic forces of the FAHV model (1), some simplifications must be carried out for the purpose of feedback linearization. The simplification of the model is necessary because we want to obtain a linearized model, and the same simplified process can be found in [29]. An input-output linearization model is developed by repeated differentiation of the outputs V and h as follows:
(2)V⃛=fV+b11ϕc+b12δe,(3)h(4)=fh+b21ϕc+b22δe,whereϕc and δe are control inputs and the specific expressions of fV, fh, b11, b12, b21, and b22 are presented in [29, equation (17)].Compared with [29], the main propose of this study is discussing the fault tolerant controller design for the FAHV to follow a given desired output reference signals yd=[Vd,hd]T in the presence of partial loss of actuator effectiveness.


## 3. Adaptive Finite-Time Fault-Tolerant Controller Design

The specific controller design step includes two parts: sliding mode surface design and sliding mode control design, which can be described as follows.


### 3.1. Sliding Mode Surface Design

Define tracking error variable as follows:(4)eV=V-Vd,(5)eh=h-hd.Differentiating (4) and (5) three times, and four times respectively, results in
(6)e⃛V=fV-V⃛d+b11ϕc+b12δe,(7)eh(4)=fh-hd(4)+b21ϕc+b22δe.Equations (6)-(7) can be expressed in matrix form:
(8)[e⃛Veh(4)]=[fV-V⃛dfh-hd(4)]︸F=[F1,F2]T+[b11b12b21b22]︸B[ϕcδe]︸u+[ΔF1ΔF2]︸ΔF=[ΔF1,ΔF2]T.Note that the additional itemΔF is introduced to represent the flexible effects and coupled uncertainties described in [29, equation (14)].Introduce new control variable:(9)U=[U1U2]=[b11b12b21b22][ϕcδe].Then (8)-(9) can be rewritten as
(10)[e⃛Veh(4)]=[F1F2]+[U1U2]+[ΔF1ΔF2].Assumption 1.
The uncertainties discussed in the research are bounded∥ΔF∥≤υ, but the value υ is unknown in advance.Assumption 2.
The matrixB denoted in (8) is nonsingular over the entire flight envelope given in [12], so Assumption 1 is reasonable to be assumed.Now, according to the definition of HOSM [30, 31], our objective is to design controller which makes the eV, eh and their derivatives converge to the neighborhood of origin.Design sliding mode surface as follows:(11)sV=e¨V+∫0tλ1V|eV|a1Vsign(eV)+λ2V|e˙V|a2Vsign(e˙V)+λ3V|e¨V|a3Vsign(e¨V)︸GV,(12)sh=e⃛h+∫0tλ1h|eh|a1hsign(eh)+λ2h|e˙h|a2hsign(e˙h)+λ3h|e¨h|a3hsign(e¨h)+λ4h|e⃛h|a4hsign(e⃛h)︸Ghds.The parametersλiV(i=1,2,3) and λjh(j=1,2,3,4) are some positive constants such that λ3Vs2+λ2Vs+λ1V and λ4hs3+λ3hs2+λ2hs+λ1h are Hurwitz polynomial. The parameters aiV(i=1,2,3) and ajh(j=1,2,3,4) are determined by
(13)a(i-1)V=aiVa(i+1)V2a(i+1)V-aiV,i∈{2,3},a(j-1)h=ajha(j+1)h2a(j+1)h-ajh,j∈{2,3,4}witha4V=a5h=1, a3V∈(1-εV,1), and a4h∈(1-εh,1), where εV∈(0,1),εh∈(0,1).Based on the homogeneity theory provided in [32], it is easily shown that eV, e˙V, e¨V and eh, e˙h, e¨h, e⃛h will converge to the neighborhood of origin in finite time if it is satisfied that sV, sh converge to the neighborhood of origin in finite time.


### 3.2. Adaptive Sliding Mode Controller Design

Now, let us consider the situation in which the actuator experiences partial loss of effectiveness fault. Then, differentiating (11) and (12), we obtain
(14)[s˙Vs˙h]=[F1F2]+[GVGh]︸G=[GV,Gh]T+[E100E2]︸E[U1U2]+[ΔF1ΔF2],whereE=diag(E1,E2)∈R2×2 is a matrix characterizing the health condition of the actuators with 0≤Ei≤1(i=1,2). Note that the case Ei=1 means that the ith actuator is totally healthy, the case Ei=0 implies that the ith actuator completely fails, and the case 0<Ei<1 corresponds to the case in which the ith actuator partially loses its effectiveness, but it still has effect all the time. In this sense, the matrix E becomes uncertain and even time varying but remains positive definite. In this study, an assumption 0<Ei≤1 is given.The control objective is to design the control inputs forϕc and δe such that all of the closed-loop signals are bounded and the velocity V and altitude h track desired command trajectories Vd and hd in the presence of flexible uncertainties and loss of effective actuator faults. That is to say, the velocity sliding mode surface sV and altitude sliding mode surface sh converge to an arbitrary small set containing the origin in finite time T0, which is ∥sV∥≤δV and ∥sh∥≤δh for t≥T0, where δV and δh are arbitrary small positive constant numbers.LetEmin=mini=1,2Ei and denote μ=1-Emin and then μ<1. Selecting θ=1/(1-μ), then the main result of the paper is formulated in the following theorem.Theorem 3.
Consider the nonlinear sliding mode dynamic system (14) with Assumptions 1 and 2, if the control U=[U1,U2]T is designed as
(15)U=-F-G-k·sigτ(s)-σ^s∥s∥-υ^s∥s∥,with the adaptive gains(16)σ^=-ψ+θ^ψ,ψ=∥F∥+∥G∥+∥k·sigτ(s)∥+υ^,(17)θ^˙=p0(-ε0θ^+ψ∥s∥),(18)υ^˙=p1(-ε1υ^+∥s∥),wheres=[sV,sh]T, k=[kV,kh]T, and 0<τ<1, and define the function sigτ(·)=sign(·)|·|τ, p0, p1, ε0 and ε1 are positive control constants, and the initial values σ^(0), υ^(0) are chosen as positive constants. Then, the system trajectory will converge to the neighborhood of sV=sh=0 in finite time despite of the uncertainties ΔF and actuator faults E.Proof.
The stability analysis of system (14) is performed via constructing the following Lyapunov function:
(19)W=12(sTs+1-μp0(θ-θ^)2+1p1(υ-υ^)2),
whereσ~=σ-σ^ and υ~=υ-υ^. The derivative of (19) is presented
(20)W˙=sTs˙-1-μp0θ~θ^˙-1p1υ~υ^˙W˙=sT(F+G+EU+ΔF)-1-μp0θ~θ^˙-1p1υ~υ^˙W˙=sT(F+G+U-(I-E)︸ΔEU+ΔF)-1-μp0θ~θ^˙-1p1υ~υ^˙W˙=sT(-k·sigτ(s)-σ^s∥s∥-υ^s∥s∥-(I-Ε)︸ΔEU+ΔF)W˙=-1-μp0θ~θ^˙-1p1υ~υ^˙W˙≤-kV|sV|τ+1-kh|sh|τ+1W˙≤+sT(-σ^s∥s∥-υ^s∥s∥-ΔEU+ΔF)-1-μp0θ~θ^˙-1p1υ~υ^˙W˙≤-kV|sV|τ+1-kh|sh|τ+1+sT(-σ^s∥s∥-ΔEU)W˙≤-1-μp0θ~θ^˙+sT(ΔF-υ^s∥s∥)-1p1υ~υ^˙W˙≤-kV|sV|τ+1-kh|sh|τ+1+sT(-σ^s∥s∥-ΔEU)W˙≤-1-μp0θ~θ^˙+sT(ΔF-υs∥s∥)+υ~∥s∥-1p1υ~υ^˙.
In view of Assumption1 and adaptive update laws (18), inequality (20) can be rewritten as
(21)W˙≤-kV|sV|τ+1-kh|sh|τ+1+sT(-σ^s∥s∥-ΔEU)W˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1-σ^∥s∥-sTΔEUW˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1-σ^∥s∥W˙≤+∥ΔE∥·sT·(∥F∥+∥G∥+∥k·sigτ(s)∥+σ^+υ^)W˙≤-1-μp0θ~θ^˙+ε1υ~υ^.
According to (16), inequality (21) can be rewritten as
(22)W˙≤-kV|sV|τ+1-kh|sh|τ+1+(1-θ^)ψ∥s∥+μ·θ^·ψ∥s∥W˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1+(1-(1-μ)θ^)ψ∥s∥W˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1W˙≤+((1-μ)θ-(1-μ)θ^)ψ∥s∥-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1+(1-μ)θ~ψ∥s∥W˙≤-1-μp0θ~θ^˙+ε1υ~υ^.
According to the adaptive update laws defined in (17), the inequality (22) can be rewritten as
(23)W˙≤-k(|sV|τ+1+|sh|τ+1)+(1-μ)ε0θ~θ^+ε1υ~υ^,
wherek=mini=V,hki. In view of Lemma  3.1 in [33], inequality (23) can be written as
(24)W˙≤-k(|sV|2+|sh|2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^W˙≤-k(12|sV|2+|sh|2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
Inspired by [33] for any positive numbers δ0>0.5 and δ1>0.5, inequality (24) can be rewritten as
(25)W˙≤-k(12sTs)(τ+1)/2-((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤-(ε1(2δ1-1)2δ1υ~2)(τ+1)/2W˙≤+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
denote(26)p0=δ0k2/(τ+1)ε0(2δ0-1),p1=δ1k2/(τ+1)ε1(2δ1-1).
Then, inequality (25) can be rewritten as
(27)W˙≤-k[(12sTs)(τ+1)/2+((1-μ)2p0θ~2)(τ+1)/2W˙≤-kD+(12p1υ~2)(τ+1)/2]W˙≤+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
According to Lemma  3.2 in [33], when δ0>0.5, δ1>0.5, and 0.5<0.5(τ+1)<1, the time derivative of the Lyapunov function W˙ becomes
(28)W˙≤-k[(12sTs)+((1-μ)2p0θ~2)+(12p1υ~2)](τ+1)/2W˙≤+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2W˙≤+(1-μ)ε0θ~θ^+ε1υ~υ^W˙≤-kW(τ+1)/2+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
Note that, for any positive constantsδ0>0.5 and δ1>0.5, the following inequality holds:
(29)ε1υ~υ^=ε1(-υ~2+υ~υ)ε1υ~υ^≤ε1(-υ~2+12δ1υ~2+δ12υ2)ε1υ~υ^≤-ε1(2δ1-1)2δ1υ~2+ε1δ12υ2.
Similarly(1-μ)ε0θ~θ^ satisfies the following inequality:
(30)(1-μ)ε0θ~θ^≤-ε0(1-μ)(2δ0-1)2δ0θ~2+ε0(1-μ)δ02θ2.
According to inequality (29), if (ε1(2δ1-1)/2δ1)υ~2>1, we obtain
(31)(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+ε1υ~υ^≤ε1(2δ1-1)2δ1υ~2+ε1υ~υ^≤ε1δ12υ2.
If(ε1(2δ1-1)/2δ)υ~2≤1, we have
(32)(ε1(2δ1-1)2δ1υ~2)(τ+1)/2|(ε1(2δ1-1)/2δ1)υ~2≤1<(ε1(2δ1-1)2δ1υ~2)(τ+1)/2|(ε1(2δ1-1)/2δ1)υ~2>1.
Therefore, combining (31) and (32) yields
(33)(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+ε1υ~υ^≤ε1δ12υ2.
Similar to (33), the following inequality can be obtained:
(34)((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2+(1-μ)ε0θ~θ^≤(1-μ)ε0δ02θ2.
Thus, from (28)–(34), the derivative of the Lyapunov function (28) becomes
(35)W˙≤-kW(τ+1)/2+μ0,
where(36)μ0=ε1δ12υ2+(1-μ)ε0δ02θ2.
According to Lemma  3.6 in [33], the decrease of Wcan drive the sliding mode surfaces sV andsh to converge to a neighborhood of the sliding surface in finite time. Furthermore, selecting 0<β≤1, inequality (35) can be expressed as
(37)W˙≤-βkW(τ+1)/2-(1-β)kW(τ+1)/2+μ0.
If-(1-β)kW(τ+1)/2+μ0<0, then W˙≤-βkW(τ+1)/2. Based on the conclusion from [30], the decrease of Wdrives the trajectories of the closed-loop system into W(τ+1)/2≤μ0/(1-β)k. Therefore, the trajectories of the closed-loop system is bounded in finite time as
(38)limβ→β0s(t)∈(∥s∥≤(μ0(1-β0)k)︸ε1/(τ+1)),
where0<β0<1 and ε is a small set containing the origin of the closed-loop system. And the time needed to reach (38) is bound as
(39)T≤2W(0)(1-τ)/2kβ0(1-τ),
whereW(0) is the initial value of W. After that, the control objective that the eV, e˙V, e¨V and eh, e˙h, e¨h, e⃛h converge to the neighborhood of origin is established.
When the controlU=[U1,U2]T is designed via (9), according to Assumption 2 the actual control variable is calculated as
(40)[ϕcδe]=[b11b12b21b22]-1[U1U2].
It is evident from (40) that the finite-time convergent performance of the proposed adaptive fault tolerant controller can be obtained without the knowledge of the minimum value of actuator effectiveness factor. Meanwhile, the upper bound of uncertainties does not need to be known in advance.


## 3.1. Sliding Mode Surface Design

Define tracking error variable as follows:(4)eV=V-Vd,(5)eh=h-hd.Differentiating (4) and (5) three times, and four times respectively, results in
(6)e⃛V=fV-V⃛d+b11ϕc+b12δe,(7)eh(4)=fh-hd(4)+b21ϕc+b22δe.Equations (6)-(7) can be expressed in matrix form:
(8)[e⃛Veh(4)]=[fV-V⃛dfh-hd(4)]︸F=[F1,F2]T+[b11b12b21b22]︸B[ϕcδe]︸u+[ΔF1ΔF2]︸ΔF=[ΔF1,ΔF2]T.Note that the additional itemΔF is introduced to represent the flexible effects and coupled uncertainties described in [29, equation (14)].Introduce new control variable:(9)U=[U1U2]=[b11b12b21b22][ϕcδe].Then (8)-(9) can be rewritten as
(10)[e⃛Veh(4)]=[F1F2]+[U1U2]+[ΔF1ΔF2].Assumption 1.
The uncertainties discussed in the research are bounded∥ΔF∥≤υ, but the value υ is unknown in advance.Assumption 2.
The matrixB denoted in (8) is nonsingular over the entire flight envelope given in [12], so Assumption 1 is reasonable to be assumed.Now, according to the definition of HOSM [30, 31], our objective is to design controller which makes the eV, eh and their derivatives converge to the neighborhood of origin.Design sliding mode surface as follows:(11)sV=e¨V+∫0tλ1V|eV|a1Vsign(eV)+λ2V|e˙V|a2Vsign(e˙V)+λ3V|e¨V|a3Vsign(e¨V)︸GV,(12)sh=e⃛h+∫0tλ1h|eh|a1hsign(eh)+λ2h|e˙h|a2hsign(e˙h)+λ3h|e¨h|a3hsign(e¨h)+λ4h|e⃛h|a4hsign(e⃛h)︸Ghds.The parametersλiV(i=1,2,3) and λjh(j=1,2,3,4) are some positive constants such that λ3Vs2+λ2Vs+λ1V and λ4hs3+λ3hs2+λ2hs+λ1h are Hurwitz polynomial. The parameters aiV(i=1,2,3) and ajh(j=1,2,3,4) are determined by
(13)a(i-1)V=aiVa(i+1)V2a(i+1)V-aiV,i∈{2,3},a(j-1)h=ajha(j+1)h2a(j+1)h-ajh,j∈{2,3,4}witha4V=a5h=1, a3V∈(1-εV,1), and a4h∈(1-εh,1), where εV∈(0,1),εh∈(0,1).Based on the homogeneity theory provided in [32], it is easily shown that eV, e˙V, e¨V and eh, e˙h, e¨h, e⃛h will converge to the neighborhood of origin in finite time if it is satisfied that sV, sh converge to the neighborhood of origin in finite time.


## 3.2. Adaptive Sliding Mode Controller Design

Now, let us consider the situation in which the actuator experiences partial loss of effectiveness fault. Then, differentiating (11) and (12), we obtain
(14)[s˙Vs˙h]=[F1F2]+[GVGh]︸G=[GV,Gh]T+[E100E2]︸E[U1U2]+[ΔF1ΔF2],whereE=diag(E1,E2)∈R2×2 is a matrix characterizing the health condition of the actuators with 0≤Ei≤1(i=1,2). Note that the case Ei=1 means that the ith actuator is totally healthy, the case Ei=0 implies that the ith actuator completely fails, and the case 0<Ei<1 corresponds to the case in which the ith actuator partially loses its effectiveness, but it still has effect all the time. In this sense, the matrix E becomes uncertain and even time varying but remains positive definite. In this study, an assumption 0<Ei≤1 is given.The control objective is to design the control inputs forϕc and δe such that all of the closed-loop signals are bounded and the velocity V and altitude h track desired command trajectories Vd and hd in the presence of flexible uncertainties and loss of effective actuator faults. That is to say, the velocity sliding mode surface sV and altitude sliding mode surface sh converge to an arbitrary small set containing the origin in finite time T0, which is ∥sV∥≤δV and ∥sh∥≤δh for t≥T0, where δV and δh are arbitrary small positive constant numbers.LetEmin=mini=1,2Ei and denote μ=1-Emin and then μ<1. Selecting θ=1/(1-μ), then the main result of the paper is formulated in the following theorem.Theorem 3.
Consider the nonlinear sliding mode dynamic system (14) with Assumptions 1 and 2, if the control U=[U1,U2]T is designed as
(15)U=-F-G-k·sigτ(s)-σ^s∥s∥-υ^s∥s∥,with the adaptive gains(16)σ^=-ψ+θ^ψ,ψ=∥F∥+∥G∥+∥k·sigτ(s)∥+υ^,(17)θ^˙=p0(-ε0θ^+ψ∥s∥),(18)υ^˙=p1(-ε1υ^+∥s∥),wheres=[sV,sh]T, k=[kV,kh]T, and 0<τ<1, and define the function sigτ(·)=sign(·)|·|τ, p0, p1, ε0 and ε1 are positive control constants, and the initial values σ^(0), υ^(0) are chosen as positive constants. Then, the system trajectory will converge to the neighborhood of sV=sh=0 in finite time despite of the uncertainties ΔF and actuator faults E.Proof.
The stability analysis of system (14) is performed via constructing the following Lyapunov function:
(19)W=12(sTs+1-μp0(θ-θ^)2+1p1(υ-υ^)2),
whereσ~=σ-σ^ and υ~=υ-υ^. The derivative of (19) is presented
(20)W˙=sTs˙-1-μp0θ~θ^˙-1p1υ~υ^˙W˙=sT(F+G+EU+ΔF)-1-μp0θ~θ^˙-1p1υ~υ^˙W˙=sT(F+G+U-(I-E)︸ΔEU+ΔF)-1-μp0θ~θ^˙-1p1υ~υ^˙W˙=sT(-k·sigτ(s)-σ^s∥s∥-υ^s∥s∥-(I-Ε)︸ΔEU+ΔF)W˙=-1-μp0θ~θ^˙-1p1υ~υ^˙W˙≤-kV|sV|τ+1-kh|sh|τ+1W˙≤+sT(-σ^s∥s∥-υ^s∥s∥-ΔEU+ΔF)-1-μp0θ~θ^˙-1p1υ~υ^˙W˙≤-kV|sV|τ+1-kh|sh|τ+1+sT(-σ^s∥s∥-ΔEU)W˙≤-1-μp0θ~θ^˙+sT(ΔF-υ^s∥s∥)-1p1υ~υ^˙W˙≤-kV|sV|τ+1-kh|sh|τ+1+sT(-σ^s∥s∥-ΔEU)W˙≤-1-μp0θ~θ^˙+sT(ΔF-υs∥s∥)+υ~∥s∥-1p1υ~υ^˙.
In view of Assumption1 and adaptive update laws (18), inequality (20) can be rewritten as
(21)W˙≤-kV|sV|τ+1-kh|sh|τ+1+sT(-σ^s∥s∥-ΔEU)W˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1-σ^∥s∥-sTΔEUW˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1-σ^∥s∥W˙≤+∥ΔE∥·sT·(∥F∥+∥G∥+∥k·sigτ(s)∥+σ^+υ^)W˙≤-1-μp0θ~θ^˙+ε1υ~υ^.
According to (16), inequality (21) can be rewritten as
(22)W˙≤-kV|sV|τ+1-kh|sh|τ+1+(1-θ^)ψ∥s∥+μ·θ^·ψ∥s∥W˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1+(1-(1-μ)θ^)ψ∥s∥W˙≤-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1W˙≤+((1-μ)θ-(1-μ)θ^)ψ∥s∥-1-μp0θ~θ^˙+ε1υ~υ^W˙≤-kV|sV|τ+1-kh|sh|τ+1+(1-μ)θ~ψ∥s∥W˙≤-1-μp0θ~θ^˙+ε1υ~υ^.
According to the adaptive update laws defined in (17), the inequality (22) can be rewritten as
(23)W˙≤-k(|sV|τ+1+|sh|τ+1)+(1-μ)ε0θ~θ^+ε1υ~υ^,
wherek=mini=V,hki. In view of Lemma  3.1 in [33], inequality (23) can be written as
(24)W˙≤-k(|sV|2+|sh|2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^W˙≤-k(12|sV|2+|sh|2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
Inspired by [33] for any positive numbers δ0>0.5 and δ1>0.5, inequality (24) can be rewritten as
(25)W˙≤-k(12sTs)(τ+1)/2-((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤-(ε1(2δ1-1)2δ1υ~2)(τ+1)/2W˙≤+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
denote(26)p0=δ0k2/(τ+1)ε0(2δ0-1),p1=δ1k2/(τ+1)ε1(2δ1-1).
Then, inequality (25) can be rewritten as
(27)W˙≤-k[(12sTs)(τ+1)/2+((1-μ)2p0θ~2)(τ+1)/2W˙≤-kD+(12p1υ~2)(τ+1)/2]W˙≤+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
According to Lemma  3.2 in [33], when δ0>0.5, δ1>0.5, and 0.5<0.5(τ+1)<1, the time derivative of the Lyapunov function W˙ becomes
(28)W˙≤-k[(12sTs)+((1-μ)2p0θ~2)+(12p1υ~2)](τ+1)/2W˙≤+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2W˙≤+(1-μ)ε0θ~θ^+ε1υ~υ^W˙≤-kW(τ+1)/2+((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2W˙≤+(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+(1-μ)ε0θ~θ^+ε1υ~υ^.
Note that, for any positive constantsδ0>0.5 and δ1>0.5, the following inequality holds:
(29)ε1υ~υ^=ε1(-υ~2+υ~υ)ε1υ~υ^≤ε1(-υ~2+12δ1υ~2+δ12υ2)ε1υ~υ^≤-ε1(2δ1-1)2δ1υ~2+ε1δ12υ2.
Similarly(1-μ)ε0θ~θ^ satisfies the following inequality:
(30)(1-μ)ε0θ~θ^≤-ε0(1-μ)(2δ0-1)2δ0θ~2+ε0(1-μ)δ02θ2.
According to inequality (29), if (ε1(2δ1-1)/2δ1)υ~2>1, we obtain
(31)(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+ε1υ~υ^≤ε1(2δ1-1)2δ1υ~2+ε1υ~υ^≤ε1δ12υ2.
If(ε1(2δ1-1)/2δ)υ~2≤1, we have
(32)(ε1(2δ1-1)2δ1υ~2)(τ+1)/2|(ε1(2δ1-1)/2δ1)υ~2≤1<(ε1(2δ1-1)2δ1υ~2)(τ+1)/2|(ε1(2δ1-1)/2δ1)υ~2>1.
Therefore, combining (31) and (32) yields
(33)(ε1(2δ1-1)2δ1υ~2)(τ+1)/2+ε1υ~υ^≤ε1δ12υ2.
Similar to (33), the following inequality can be obtained:
(34)((1-μ)ε0(2δ0-1)2δ0θ~2)(τ+1)/2+(1-μ)ε0θ~θ^≤(1-μ)ε0δ02θ2.
Thus, from (28)–(34), the derivative of the Lyapunov function (28) becomes
(35)W˙≤-kW(τ+1)/2+μ0,
where(36)μ0=ε1δ12υ2+(1-μ)ε0δ02θ2.
According to Lemma  3.6 in [33], the decrease of Wcan drive the sliding mode surfaces sV andsh to converge to a neighborhood of the sliding surface in finite time. Furthermore, selecting 0<β≤1, inequality (35) can be expressed as
(37)W˙≤-βkW(τ+1)/2-(1-β)kW(τ+1)/2+μ0.
If-(1-β)kW(τ+1)/2+μ0<0, then W˙≤-βkW(τ+1)/2. Based on the conclusion from [30], the decrease of Wdrives the trajectories of the closed-loop system into W(τ+1)/2≤μ0/(1-β)k. Therefore, the trajectories of the closed-loop system is bounded in finite time as
(38)limβ→β0s(t)∈(∥s∥≤(μ0(1-β0)k)︸ε1/(τ+1)),
where0<β0<1 and ε is a small set containing the origin of the closed-loop system. And the time needed to reach (38) is bound as
(39)T≤2W(0)(1-τ)/2kβ0(1-τ),
whereW(0) is the initial value of W. After that, the control objective that the eV, e˙V, e¨V and eh, e˙h, e¨h, e⃛h converge to the neighborhood of origin is established.
When the controlU=[U1,U2]T is designed via (9), according to Assumption 2 the actual control variable is calculated as
(40)[ϕcδe]=[b11b12b21b22]-1[U1U2].
It is evident from (40) that the finite-time convergent performance of the proposed adaptive fault tolerant controller can be obtained without the knowledge of the minimum value of actuator effectiveness factor. Meanwhile, the upper bound of uncertainties does not need to be known in advance.


## 4. Simulation

To illustrate the efficiency of controller designed previously, a climbing maneuver with longitudinal acceleration for a 100 ft/s velocity change and a 1000 ft altitude change is considered. Simulation studies have been done on the full nonlinear flexible hypersonic vehicle defined in (1). The reference commands have been generated by filtering step reference commands by a second-order prefilter with natural frequency ωf=0.06 rad/s and damping ratio ζf=0.95.The initial trim condition is selected asV=7710 ft/s and h=85000 ft. Simulation parameters are provided in Table 1.Table 1
Simulation parameters setting.
Items
Values
Items
Values
Items
Values
λ
1
V
10
λ
4
h
10
a
3
h
3/5
λ
2
V
15
a
1
V
1/2
a
4
h
3/4
λ
3
V
15
a
2
V
3/5
τ
0.7
λ
1
h
15
a
3
V
3/4
k
V
10
λ
2
h
25
a
1
h
3/7
k
h
10
λ
3
h
20
a
2
h
1/2It is assumed that actuator faults are chosen as(41)E1=0.7,t≥100,E2=0.7,t≥100.The simulation results are provided in Figures1–4. Figure 1 denotes the response to the 100 ft/s step velocity and 1000 ft step altitude. It has been observed that the velocity and altitude converge to the desired value. The control inputs of ϕc and δe could be seen in bottom plots of Figure 1.Regulated outputs and control inputs with actuator faults.
(a)
(b)
(c)
(d)Other flight states with actuator faults.
(a)
(b)
(c)
(d)Sliding mode surface and adaptive parametersυ^ and θ^ with actuator faults.
(a)
(b)
(c)
(d)The dynamic response curves of flexile modes with actuator faults.
(a)
(b)
(c)Figure2 shows the performance of the angle of attack α and the pitch rate Q at the top, as well as the canard deflection δc and the flight path angle γ at the bottom.The velocity and altitude sliding mode surfacessV, sh are shown in Figure 3, which are oscillation with small magnitudes when actuator fault occurred. The convergent performance verifies the effectiveness of the proposed control strategy. The adaptive parameters υ^ and θ^ in control laws of (15)–(18) could be seen in bottom plots of Figure 3, where the convergence of υ^ is confirmed. From the simulation results in Figure 3, the approximate equation θ^≈3 can be obtained. According to the relationship based on equation θ=1/(1-μ), we can solve that μ^=0.67 and denote the estimated error as
(42)eμ=μ-μ^≈0.7-0.67=0.03.The value ofeμ in our research is in tolerance. Meanwhile, the stability of flexible states is depicted by Figure 4. And it can be seen that the flexible states η1, η2, and η3 converge to constant values, respectively.In summary, the simulation results demonstrate that, although there are actuator faults and uncertainties in the system, the good tracking performance and satisfactory system responses can be guaranteed.


## 5. Conclusions and Future Work

In this paper, an effective method has been proposed for linearizing the nonlinear model of the FAHV via feedback, which simplifies the complexity of the controller design process. Furthermore, an adaptive fault-tolerant control scheme based on finite-time sliding mode control technique has been brought forward for the FAHV without any information about the upper bound of uncertainties or the minimum value of actuator effectiveness. Simulation results have been presented to evaluate the validity of the proposed control scheme and to show its robustness to uncertainties and the loss of actuator effectiveness.Further research work includes two aspects. Firstly, only the loss-of-effectiveness fault has been investigated in this paper; other types of actuator faults such as float failure and actuator faults in FAHV with unknown structure are worth being dealt with. Furthermore, the FAHV model considered in this paper is highly nonlinear and strongly coupled, and a more general active FTC scheme as adaptive fault diagnosis observer in [34, 35] should be investigated in our future study.



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*Source: 920796-2013-11-27.xml*