# Fuzzy Adaptive Prescribed Performance Fault-Tolerant Control for HFVs with Fixed-Time Convergence Guarantee

Authors: Zehong Dong; Yinghui Li; Maolong Lv; Zilong Zhao; Binbin Pei
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

A new fixed-time fuzzy adaptive fault-tolerant control methodology is proposed for the longitudinal dynamics of hypersonic flight vehicles (HFVs) in the presence of actuator faults, uncertain dynamics, and external disturbances. In contrast with the conventional fixed-time control schemes that typically contain the fractional powers of errors in their designs, this work develops a low-complexity control structure in the sense of removing the dependence on the need of abovementioned fractional power terms by means of prescribed performance control (PPC) method. Different from the most existing PPC approaches where the initial conditions of tracking errors are required to be known, the newly proposed prescribed performance function (PPF) can relax such restrictions through choosing properly small initial values of PPF. Fuzzy logic systems (FLSs) are employed to handle unknown dynamics, and minimal learning parameter (MLP) technique is incorporated into the design for the purpose of alleviating computation burden. Closed-loop stability is rigorously proved via Lyapunov stability theory, and simulation results are eventually given to validate the effectiveness of the proposed control strategy.


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## Body

## 1. Introduction

Hypersonic flight vehicles have already attracted considerable attention due to their advantages of high flight speed, remarkable penetration ability, and cost-effectiveness [1–4]. Nevertheless, the controller design for HFVs remains an intractable issue due to their peculiar features. For example, the engine-airframe structure results in strong couplings between propulsive and aerodynamic forces, and there exist intricate flexible deformation due to the slender geometry of vehicle structure, which influences the aerodynamic characteristics prominently [5]. In addition, the fast time-varying flight environment and the unknown external disturbances lead to frequent parameter variations and model uncertainties, dramatically increasing the difficulty of controller design. To address these problems, many effective methods have been presented, including robust control [5–7], neural/fuzzy control [8–10], prescribed performance control [11, 12], and disturbance observer-based control [13]. Although these efforts solve the above-mentioned issues to some extent, these results rarely focus on the rate of convergence.To be specific, only the exponential convergence of tracking error is guaranteed in the aforementioned work, which reveals the convergence time tends to be infinite. From a practical perspective, the rate of convergence is of great significance to the transient tracking performance [14]. Recently, the finite-time tracking control for HFVs is investigated in [14–17], which can make the tracking error converge into the predefined compact set within a finite time. Nevertheless, the convergence time, which is commonly achieved in [14–17], depends on the initial states of the system. It inevitably brings up a problem, that is, the convergence time cannot be accurately settled when the initial states of the system are unknown. To solve such problem, the fixed-time control [18–21] is proposed skillfully, by which the tracking error can converge into a predefined impact set within a fixed time and the connection between the convergence time and initial states is eliminated.However, there still exist some shortcomings with the conventional fixed-time tracking control scheme [18–21], where the inequation V̇x≤−τVpx−ςVqx+κ holds. On the one hand, the derivative of virtual and actual control laws will tend to infinite when the tracking error approaches to zero, giving rise to singularity issues in the control design [21]. On the other hand, when the system encounters the unknown external disturbances [22–26], the dynamic uncertainties [27–32], the system faults [33–36], or input nonlinearities [37], it is complicated to both ensure the fixed-time stability of the system and predetermine the convergence accuracy by selecting the design parameters. As is known to us, the actuator faults not only affect handling performance for HVFs, but even cause closed-loop instability [33–36]. The fault-tolerant control is an inevitable issue for HFVs, due to complex and variable flight environment that may lead to actuator faults, such as control effectiveness decline and drifting. Furthermore, it is well known that the transient and steady-state performances are important to the controller design for HFVs [38]. Nonetheless, the existing PPC for HFVs commonly fails to explicitly contain a convergence time T in the performance function. Thus, it is urgent to develop a new low-complexity fixed-time fault-tolerant control (FTFTC) strategy for HFVs with the prescribed performance.Motivated by these observations, we present a fixed-time adaptive fuzzy fault-tolerant control scheme for HFVs by utilizing a new prescribed performance function. The contributions mainly contain the follows:(1)
This paper presents a structurally inexpensive FTFTC framework for HFVs in the sense that no fractional powers are involved in the design. There is no fractional power of tracking error in the controller, and thus the singularity problem caused by the derivative of fraction term is removed(2)
By constructing the intermediate control laws and adaptive laws, the adverse impact of actuator faults of HFVs (e.g., loss of effectiveness and drift) is compensated effectivelyThe remainder of the work is organized as follows. The HFV dynamics and preliminaries are introduced in Section2. In Section 3, the FLSs-approximator-based FTFTC is designed and the closed-loop stability is verified in Section 4. Section 5 provides simulations to demonstrate the effectiveness of the proposed methods and the work is concluded in Section 6.A preprint has previously been published by Zehong Dong et al. [39].


## 2. Problem Formulation and Preliminaries

### 2.1. Hypersonic Flight Vehicle Dynamics

The longitudinal control-oriental model is originally developed by Parker et al. [40, 41], which can be formulated as
(1)V̇=Tcosθ−γ−Dm−gsinγ,ḣ=Vsinγ,(2)γ̇=Tsinθ−γ+LmV−gcosγV,θ̇=Q,(3)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(4)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(5)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~1ψ~2η¨1Iyy,where T, D, L, M, N1, and N2 are expressed as
(6)L=q¯SCLα,δ,η,D=q¯SCDα,δ,η,M=zTT+q¯Sc¯CMα,δ,η,T=q¯SCT,ΦαvΦ+CTα+CTηη,Ni=q¯SNiα2α2+Niαα+Niδδ+Ni0+Niηη,i=1,2,where α=θ−γ. q¯, S, zT, and c¯ denote dynamic press, reference area, thrust moment arm, and reference length. η=η1,η̇1,η2,η̇2T denotes the flexible modes. δ=vδe,δcT, where the deflection of canard δc is set to be ganged with vδe and δc=kecvδe; kec=−CLδe/CLδc. This approach was originally proposed in [41] as a way to remove some non-minimum phase characteristics of the dynamics. Considering the actuator fault, the actual output signals of the fuel equivalence ratio and the elevator angular deflection are denoted as vΦ and vδe, respectively. The aerodynamic model is obtained by curve fitting and can be expressed as
(7)CD·=CDα2α2+CDαα+CDδe2v2δe+CDδevδe+CDδcδc+CDδc2δc2+CD0+CDηη,(8)CM·=CMα2α2+CMαα+CMδevδe+CMδcδc+CM0+CMηη,(9)CL·=CLαα+CLδevδe+CLδcδc+CL0+CLηη,(10)CT,Φ·=CT,Φα3α3+CT,Φα2α2+CT,Φαα+CT,Φ0,(11)CT·=CTα3α3+CTα2α2+CTαα+CT0,(12)Cjη=Cjη1,0,Cjη2,0,j=T,M,L,D,(13)Niη=Niη1,0,Niη2,0,i=1,2,and more detailed definitions can be found in [40–41].


### 2.2. The Actuator-Fault Model

The actuator-fault model is developed by the following formula [34]:
(14)vΦ=ωΦΦ+εΦ,vδe=ωδeδe+εδe,where vΦ and vδe denote the actual output signals of the fuel equivalence ratio and the elevator angular deflection, respectively. ω• and ε• represent the actual control effectiveness and drift distance, respectively.Assumption 1.
There exists an unknown positive constantε¯• such that ε•≤ε¯•, 0<ω•≤1.Remark 2.
Assumption1 is commonly applied in FTC research to ensure the controllability of system when the actuator faults occur [34]. During flight, actuator fault inevitably occurs due to multiple factors such as aging and damage of components or screw shedding, which deteriorates the flight performance and even causes the serious flight accident in severe circumstance. Therefore, it is of great significance to consider the possible actuator faults when designing the control strategy. With different values of ω• and ε•, (14) can be divided into the following four cases:
(1)
ω•=1 and ε•=0, representing the fault-free case(2)
0<ω¯•≤ω•≤ω¯•<1 and ε•=0, where ω¯• and ω¯• are unknown positive constants, denoting partial loss of effectiveness(3)
ω•=1 and ε•≠0, indicating the bias fault(4)
0<ω¯•≤ω•≤ω¯•<1 and ε•≠0, where ω¯• and ω¯• are unknown positive constants, signifying that partial loss of effectiveness and bias fault occur at the same time


### 2.3. Model Transformation and Decomposition

According to (1) and (7), V and h are mainly regulated by Φ and δe, respectively. To facilitate the controller design, the HFVs dynamics are decomposed into velocity subsystem and altitude subsystem.Considering the actuator-fault model (14) and inspired by [42, 43], the velocity subsystem is written as
(15)V̇=fV+gVvΦ+dV,where fV=−q¯S/mCDα2α2+CDαα+CDδe2δe2+CDδeδe+CDδcδc+CDδc2δc2+CD0+CDηη−gsinγ+q¯S/mcosαCTα3α3+CTα2α2+CTαα+CT0+CTηη, gV=q¯S/mcosαCT,Φα3α3+CT,Φα2α2+CT,Φαα+CT,Φ0. fV and gV stand for unknown functions due to the time-varying aerodynamic parameters, and dV represents the external disturbance on velocity.Considering the actuator-fault model (14) and taking the assumption sinγ≈γ,cosγ≈1, then the altitude subsystem can be considered as
(16)ḣ=Vγ+dh,γ̇=fγ+gγθ+dγ,θ̇=Q,Q̇=fQ+gQυδe+dQ,Similar to velocity subsystem, the functionsfγ, gγ, fQ, and gQ are unknown functions; dh, dγ, and dQ are the external disturbances of altitude subsystem. Along the standard ideas as [10–11], we assume there exist unknown positive functions gVm, gγm, and gQm such that 0<gVm≤gV, 0<gγm≤gγ, and 0<gQm≤gQ.Remark 3.
In practice, it is rather difficult to know the values of functionsf•,g•,∈V,γ,Q accurately. There are mainly the following reasons: On the one hand, the aerodynamic parameters are constantly changing with the flight environment (i.e., velocity, altitude, and attack of angle), where there inevitably exist measuring errors in the sensors of flight control system [2]. On the other hand, it is impossible to take all flight environment of HFVs into account in a wind tunnel so that we have to rely on curve fitting technology to build the aerodynamic model [7]. Consequently, an exact model for HFVs is difficult to be obtained and in order to facilitate the design of flight control system; we regard f•,∈V,γ,Q as unknown function and regard g• as unknown positive function.Assumption 4. See [44].
The reference trajectoryyreft, together with its i-order derivative yrefit, is continuous and bounded (i=1,2⋯n).


### 2.4. A New Fixed-Time Performance Function

Definition 5. See [44].
A smooth functionρt is called fixed-time performance function (FTPF), if the following conditions are satisfied:
(1)
ρt>0, i.e., ρt is ensured to be a positive function(2)
ρ̇≤0, that is, ρt is monotonically decreasing(3)
limt⟶Tρt=ρT and ρt=ρT for any t>T, where ρT and T denote an arbitrarily small positive constant and settling time, respectively
According to Definition5, we construct an FTPF in the form of
(17)ρt=cothϑtT−t+r−1+ρT,0≤t<TρT,t≥T.Proof.
In view of (17), it can be derived that
(18)ρT−=limt⟶T−cothϑtT−t+r−1+ρT=limt⟶T−eϑt/T−t+r+e−ϑt/T−t−reϑt/T−t+r−e−ϑt/T−t−r−1+ρT=ρT=ρT+,that is, ρt is a continuous function. Furthermore, we can deduce that
(19)ρ̇t=−ϑTTT−t2csch2ϑtT−t+r=−ϑTTT−t22eϑt/T−t+r−e−ϑt/T−t−r2,when t<T, and ρ̇t=0 when t≥T. For the sake of simplification, we denote x=t/T−t and the fact limt⟶T−x=+∞ holds. Then, we can obtain
(20)limt⟶T−ρ̇t=limx⟶+∞−4ϑT1+xeϑx+r−e−ϑx−r2.
With the help of L’Hospital’s rule, we get that(21)limt⟶T−ρ̇t=limx⟶+∞−4Tϑ1eϑx+r−e−ϑx−r2=ρ̇T+.
Next,dρ2t/dt2 can be derived as
(22)dρ2tdt2=−8ϑ21+x2eϑx+r+e−ϑx−rTeϑx+r−e−ϑx−r3−8ϑ1+xTeϑx+r−e−ϑx−r2=T4ρ̇t1+2e−ϑx−reϑx+r−e−ϑx−r+ρ̇t4ϑ/Teϑx+r−e−ϑx−r.
Then, we havelimt⟶T−dρ2t/dt2=dρ2T+/dt2=0. Similarly, it leads to that limt⟶T−dρit/dti=dρiT+/dti=0,i=3,⋯,n. That is to say, ρt is a smooth function. Furthermore, we can also see that ρ̇t≤0 and the function ρt is continuous at T. Thus, we can conclude that the function ρt is a FTPF. This completes the proof. +.Remark 6.
We can easily obtain sufficiently largeρ0 by selecting a sufficiently small r, that is to say, the initial error need not to be known accurately. Consequently, the FTPF without initial error constraint is achieved. Furthermore, we can also conclude that the convergence rate of the error depends on ϑ, which can be seen in Figure 1(a). By setting the steady-state error boundary as ρT=1 and choosing different values of ϑ, we can obtain that a larger ϑ means a faster convergence rate of the error.
Consider the following transformation:(23)qt=etρt,where et represents an error function; the error transformation function is chosen as
(24)zq=q1−q2,
and we abbreviateq=qt there-in-after.Remark 7.
From (24), it can be observed that the inequation −1<q<1 holds if zq is bounded. In view of (23), we conclude that et<ρt holds as long as e0<ρ0. Choosing ϑ=0.5, r=0.1, T=6, and ρT=1, the convergence performance of the proposed FTPF is shown in Figure 1(b). In contrast to traditional PPC [11], where the prescribed performance function is in the form of ρt=cothϑt+r−1+ρ∞, the proposed control scheme explicitly contains a convergence time T in the FTPF. By this means, we can easily preset the convergence time as needed.Figure 1
The convergence performance of the proposed FTPF.
(a)(b)


### 2.5. Fuzzy Logic System

In the process of designing the flight controller, the fuzzy logic system is used to estimate the dynamics uncertainties of HFVs. Define a set of fuzzy IF-THEN rules, where thelth IF-THEN rule is written as follows [22, 23, 45, 46]:
(25)Rl:Ifx1isF1l,and⋯andxnisFnl,thenyisBl.wherex=x1,⋯,xnT∈ℝn, and y∈ℝ are the input and output of the FLSs, respectively, and F1l,⋯,Fnl and Bl are fuzzy sets in ℝ. Let Fx be a continuous function defined on a compact set Ωx. Then, for a given desired level of accuracy ε>0, there exists a FLS WTSx such that supx∈ΩxFx−WTSx≤ε, where W=w1,⋯,wpT is the adaptive fuzzy parameter vector in a compact set ΩW, p is the number of the fuzzy rules, and Sx=S1x,⋯,SpxT is the fuzzy basis function vector with Slx=∏j=1mμFjlxj/∑l=1p∏j=1mμFjlxj where μFjlxj is a fuzzy membership function of the variable xj in IF-THEN rule. Let W∗ be the optimal parameter vector, which is defined as
(26)W∗=argminW∈ΩWsupx∈ΩxFx−WTSx.Then, we can further obtain(27)Fx=W∗TSx+ϕ,where ϕ is the minimum fuzzy approximation error.In order to reduce the computational burden, the functionsℓV=WV2/g~Vm,ℓh=Wh2/V~m,ℓγ=Wγ2/gγm,ℓγ=Wγ2/gγm,ℓQ=WQ2/gQm rather than the ideal weight vectors’ elements are estimated in the controller design and stability analysis.Lemma 8. See [47].
The hyperbolic tangent functiontanh• is continuous and differentiable; for ∀ς∈R and ∀μ>0, it has
(28)0≤ς−ςtanhςμ≤0.2785μ.Lemma 9. See [48].
For any positive constantsω and δ, the following inequation holds
(29)0≤ω−ω2ω2+δ2≤δ.
The control objective of this article is to design an fuzzy adaptive tracking controller such that(1)
The velocity and altitude tracking errors are guaranteed to obey the prescribed performance boundaries at all times and finally converge into the predefined residual sets within preassigned timeT even in the presentence of actuator faults(2)
All signals of the closed-loop system remain bounded


## 2.1. Hypersonic Flight Vehicle Dynamics

The longitudinal control-oriental model is originally developed by Parker et al. [40, 41], which can be formulated as
(1)V̇=Tcosθ−γ−Dm−gsinγ,ḣ=Vsinγ,(2)γ̇=Tsinθ−γ+LmV−gcosγV,θ̇=Q,(3)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(4)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(5)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~1ψ~2η¨1Iyy,where T, D, L, M, N1, and N2 are expressed as
(6)L=q¯SCLα,δ,η,D=q¯SCDα,δ,η,M=zTT+q¯Sc¯CMα,δ,η,T=q¯SCT,ΦαvΦ+CTα+CTηη,Ni=q¯SNiα2α2+Niαα+Niδδ+Ni0+Niηη,i=1,2,where α=θ−γ. q¯, S, zT, and c¯ denote dynamic press, reference area, thrust moment arm, and reference length. η=η1,η̇1,η2,η̇2T denotes the flexible modes. δ=vδe,δcT, where the deflection of canard δc is set to be ganged with vδe and δc=kecvδe; kec=−CLδe/CLδc. This approach was originally proposed in [41] as a way to remove some non-minimum phase characteristics of the dynamics. Considering the actuator fault, the actual output signals of the fuel equivalence ratio and the elevator angular deflection are denoted as vΦ and vδe, respectively. The aerodynamic model is obtained by curve fitting and can be expressed as
(7)CD·=CDα2α2+CDαα+CDδe2v2δe+CDδevδe+CDδcδc+CDδc2δc2+CD0+CDηη,(8)CM·=CMα2α2+CMαα+CMδevδe+CMδcδc+CM0+CMηη,(9)CL·=CLαα+CLδevδe+CLδcδc+CL0+CLηη,(10)CT,Φ·=CT,Φα3α3+CT,Φα2α2+CT,Φαα+CT,Φ0,(11)CT·=CTα3α3+CTα2α2+CTαα+CT0,(12)Cjη=Cjη1,0,Cjη2,0,j=T,M,L,D,(13)Niη=Niη1,0,Niη2,0,i=1,2,and more detailed definitions can be found in [40–41].


## 2.2. The Actuator-Fault Model

The actuator-fault model is developed by the following formula [34]:
(14)vΦ=ωΦΦ+εΦ,vδe=ωδeδe+εδe,where vΦ and vδe denote the actual output signals of the fuel equivalence ratio and the elevator angular deflection, respectively. ω• and ε• represent the actual control effectiveness and drift distance, respectively.Assumption 1.
There exists an unknown positive constantε¯• such that ε•≤ε¯•, 0<ω•≤1.Remark 2.
Assumption1 is commonly applied in FTC research to ensure the controllability of system when the actuator faults occur [34]. During flight, actuator fault inevitably occurs due to multiple factors such as aging and damage of components or screw shedding, which deteriorates the flight performance and even causes the serious flight accident in severe circumstance. Therefore, it is of great significance to consider the possible actuator faults when designing the control strategy. With different values of ω• and ε•, (14) can be divided into the following four cases:
(1)
ω•=1 and ε•=0, representing the fault-free case(2)
0<ω¯•≤ω•≤ω¯•<1 and ε•=0, where ω¯• and ω¯• are unknown positive constants, denoting partial loss of effectiveness(3)
ω•=1 and ε•≠0, indicating the bias fault(4)
0<ω¯•≤ω•≤ω¯•<1 and ε•≠0, where ω¯• and ω¯• are unknown positive constants, signifying that partial loss of effectiveness and bias fault occur at the same time


## 2.3. Model Transformation and Decomposition

According to (1) and (7), V and h are mainly regulated by Φ and δe, respectively. To facilitate the controller design, the HFVs dynamics are decomposed into velocity subsystem and altitude subsystem.Considering the actuator-fault model (14) and inspired by [42, 43], the velocity subsystem is written as
(15)V̇=fV+gVvΦ+dV,where fV=−q¯S/mCDα2α2+CDαα+CDδe2δe2+CDδeδe+CDδcδc+CDδc2δc2+CD0+CDηη−gsinγ+q¯S/mcosαCTα3α3+CTα2α2+CTαα+CT0+CTηη, gV=q¯S/mcosαCT,Φα3α3+CT,Φα2α2+CT,Φαα+CT,Φ0. fV and gV stand for unknown functions due to the time-varying aerodynamic parameters, and dV represents the external disturbance on velocity.Considering the actuator-fault model (14) and taking the assumption sinγ≈γ,cosγ≈1, then the altitude subsystem can be considered as
(16)ḣ=Vγ+dh,γ̇=fγ+gγθ+dγ,θ̇=Q,Q̇=fQ+gQυδe+dQ,Similar to velocity subsystem, the functionsfγ, gγ, fQ, and gQ are unknown functions; dh, dγ, and dQ are the external disturbances of altitude subsystem. Along the standard ideas as [10–11], we assume there exist unknown positive functions gVm, gγm, and gQm such that 0<gVm≤gV, 0<gγm≤gγ, and 0<gQm≤gQ.Remark 3.
In practice, it is rather difficult to know the values of functionsf•,g•,∈V,γ,Q accurately. There are mainly the following reasons: On the one hand, the aerodynamic parameters are constantly changing with the flight environment (i.e., velocity, altitude, and attack of angle), where there inevitably exist measuring errors in the sensors of flight control system [2]. On the other hand, it is impossible to take all flight environment of HFVs into account in a wind tunnel so that we have to rely on curve fitting technology to build the aerodynamic model [7]. Consequently, an exact model for HFVs is difficult to be obtained and in order to facilitate the design of flight control system; we regard f•,∈V,γ,Q as unknown function and regard g• as unknown positive function.Assumption 4. See [44].
The reference trajectoryyreft, together with its i-order derivative yrefit, is continuous and bounded (i=1,2⋯n).


## 2.4. A New Fixed-Time Performance Function

Definition 5. See [44].
A smooth functionρt is called fixed-time performance function (FTPF), if the following conditions are satisfied:
(1)
ρt>0, i.e., ρt is ensured to be a positive function(2)
ρ̇≤0, that is, ρt is monotonically decreasing(3)
limt⟶Tρt=ρT and ρt=ρT for any t>T, where ρT and T denote an arbitrarily small positive constant and settling time, respectively
According to Definition5, we construct an FTPF in the form of
(17)ρt=cothϑtT−t+r−1+ρT,0≤t<TρT,t≥T.Proof.
In view of (17), it can be derived that
(18)ρT−=limt⟶T−cothϑtT−t+r−1+ρT=limt⟶T−eϑt/T−t+r+e−ϑt/T−t−reϑt/T−t+r−e−ϑt/T−t−r−1+ρT=ρT=ρT+,that is, ρt is a continuous function. Furthermore, we can deduce that
(19)ρ̇t=−ϑTTT−t2csch2ϑtT−t+r=−ϑTTT−t22eϑt/T−t+r−e−ϑt/T−t−r2,when t<T, and ρ̇t=0 when t≥T. For the sake of simplification, we denote x=t/T−t and the fact limt⟶T−x=+∞ holds. Then, we can obtain
(20)limt⟶T−ρ̇t=limx⟶+∞−4ϑT1+xeϑx+r−e−ϑx−r2.
With the help of L’Hospital’s rule, we get that(21)limt⟶T−ρ̇t=limx⟶+∞−4Tϑ1eϑx+r−e−ϑx−r2=ρ̇T+.
Next,dρ2t/dt2 can be derived as
(22)dρ2tdt2=−8ϑ21+x2eϑx+r+e−ϑx−rTeϑx+r−e−ϑx−r3−8ϑ1+xTeϑx+r−e−ϑx−r2=T4ρ̇t1+2e−ϑx−reϑx+r−e−ϑx−r+ρ̇t4ϑ/Teϑx+r−e−ϑx−r.
Then, we havelimt⟶T−dρ2t/dt2=dρ2T+/dt2=0. Similarly, it leads to that limt⟶T−dρit/dti=dρiT+/dti=0,i=3,⋯,n. That is to say, ρt is a smooth function. Furthermore, we can also see that ρ̇t≤0 and the function ρt is continuous at T. Thus, we can conclude that the function ρt is a FTPF. This completes the proof. +.Remark 6.
We can easily obtain sufficiently largeρ0 by selecting a sufficiently small r, that is to say, the initial error need not to be known accurately. Consequently, the FTPF without initial error constraint is achieved. Furthermore, we can also conclude that the convergence rate of the error depends on ϑ, which can be seen in Figure 1(a). By setting the steady-state error boundary as ρT=1 and choosing different values of ϑ, we can obtain that a larger ϑ means a faster convergence rate of the error.
Consider the following transformation:(23)qt=etρt,where et represents an error function; the error transformation function is chosen as
(24)zq=q1−q2,
and we abbreviateq=qt there-in-after.Remark 7.
From (24), it can be observed that the inequation −1<q<1 holds if zq is bounded. In view of (23), we conclude that et<ρt holds as long as e0<ρ0. Choosing ϑ=0.5, r=0.1, T=6, and ρT=1, the convergence performance of the proposed FTPF is shown in Figure 1(b). In contrast to traditional PPC [11], where the prescribed performance function is in the form of ρt=cothϑt+r−1+ρ∞, the proposed control scheme explicitly contains a convergence time T in the FTPF. By this means, we can easily preset the convergence time as needed.Figure 1
The convergence performance of the proposed FTPF.
(a)(b)


## 2.5. Fuzzy Logic System

In the process of designing the flight controller, the fuzzy logic system is used to estimate the dynamics uncertainties of HFVs. Define a set of fuzzy IF-THEN rules, where thelth IF-THEN rule is written as follows [22, 23, 45, 46]:
(25)Rl:Ifx1isF1l,and⋯andxnisFnl,thenyisBl.wherex=x1,⋯,xnT∈ℝn, and y∈ℝ are the input and output of the FLSs, respectively, and F1l,⋯,Fnl and Bl are fuzzy sets in ℝ. Let Fx be a continuous function defined on a compact set Ωx. Then, for a given desired level of accuracy ε>0, there exists a FLS WTSx such that supx∈ΩxFx−WTSx≤ε, where W=w1,⋯,wpT is the adaptive fuzzy parameter vector in a compact set ΩW, p is the number of the fuzzy rules, and Sx=S1x,⋯,SpxT is the fuzzy basis function vector with Slx=∏j=1mμFjlxj/∑l=1p∏j=1mμFjlxj where μFjlxj is a fuzzy membership function of the variable xj in IF-THEN rule. Let W∗ be the optimal parameter vector, which is defined as
(26)W∗=argminW∈ΩWsupx∈ΩxFx−WTSx.Then, we can further obtain(27)Fx=W∗TSx+ϕ,where ϕ is the minimum fuzzy approximation error.In order to reduce the computational burden, the functionsℓV=WV2/g~Vm,ℓh=Wh2/V~m,ℓγ=Wγ2/gγm,ℓγ=Wγ2/gγm,ℓQ=WQ2/gQm rather than the ideal weight vectors’ elements are estimated in the controller design and stability analysis.Lemma 8. See [47].
The hyperbolic tangent functiontanh• is continuous and differentiable; for ∀ς∈R and ∀μ>0, it has
(28)0≤ς−ςtanhςμ≤0.2785μ.Lemma 9. See [48].
For any positive constantsω and δ, the following inequation holds
(29)0≤ω−ω2ω2+δ2≤δ.
The control objective of this article is to design an fuzzy adaptive tracking controller such that(1)
The velocity and altitude tracking errors are guaranteed to obey the prescribed performance boundaries at all times and finally converge into the predefined residual sets within preassigned timeT even in the presentence of actuator faults(2)
All signals of the closed-loop system remain bounded


## 3. The FTFTC Design

Corresponding to the decomposition in Section2.3, the control design is also decomposed into a velocity control design and an altitude control design.


### 3.1. Velocity Controller Design

We first define the velocity tracking error as(30)eV=V−Vref,where Vref is the velocity reference trajectory. In view of (15), (23), and (24), the time derivative of eV is
(31)ėV=fV+gVvΦ+dV−V̇ref=ρ̇tq+żVρt∂q∂z.Then,żV can be rewritten as
(32)żV=fV+gVvΦ+dV−V̇ref−ρ̇tqρt∂q/∂z=YV+EVfV+gVvΦ+dV−V̇ref,with YV=−ρ̇tqEV, EV=1/ρt∂q/∂z.It can be deduced from (24) that ∂q/∂z=1−q22/1+q2≤1 when −1<q<1, and noticing the fact gV≥gVm>0 leads to
(33)EVgV=gVρt∂q/∂z≥gVmρ0>0.Choose the following quadratic function:(34)LV=12zV2+12μVg~Vmℓ~V2,where g~Vm=gVm/ρ0, ℓ~V=ℓV−ℓ^V, ℓ^V denotes the estimation of adaptive parameter ℓV and μV is the positive user-defined parameter.Utilizing (14) and (31), the time derivative of LV is
(35)L̇V=zVYV+EVfV+EVgVωΦΦ+EVgVεΦ+EVdV−EVV̇ref−1μVg~Vmℓ~Vℓ^V.Define the nonlinear function(36)FV=YV+EVfV+EVdV−EVV̇ref+12zV,where fV and dV are unknown due to the existence of unknown external disturbances and fast time-varying flight environment. An FLSs-approximator is constructed to estimate FV as
(37)FV=WVTSVxV+ϕV,where xV=V,Vref,V̇refT. Substituting (36) and (37) into (35), we have
(38)L̇V=zVWVTSV+ϕV+EVgVωΦΦ+EVgVεΦ−1μVg~Vmℓ~Vℓ^V−12zV2.According to Young’s inequation, we can further have(39)zVWVTSV≤zV2WV2SVTSV4τV+τV,zVϕV≤12zV2+12ϕV2,where τV is a positive constant; then, we can rewrite (38) as
(40)L̇V≤zVEVgVωΦΦ+EVgVεΦ+zV2WV2SVTSV4τV+τV+12ϕV2−1μVg~Vmℓ~Vℓ^V.According to Assumption1, ωΦ and εΦ are unknown. Therefore, we define the upper and lower bounds of fault parameters to achieve robustness, which are expressed as
(41)ω¯Φ=infEVgVωΦ,ϑV=1ω¯Φ,ξV=supEVgVεΦ.Consider the Lyapunov function candidate(42)LV¯=LV+12lVω¯Φϑ~V2+12rVξ~V2,where lV>0 and rV>0 are the parameters to be designed and ϑ~V=ϑV−ϑ^V and ξ~V=ξV−ξ^V represent estimation errors with ϑ^V and ξ^V being the estimations of ϑV as ξV, respectively.The time derivative ofLV¯ gives
(43)L̇V¯=L̇V−1lVω¯Φϑ~Vϑ^V−1rVξ~Vξ^V.DefineℓV=WV2/g~Vm and choose the intermediate control law as
(44)Φ¯=kVzV+zVℓ^VSVTSV4τV+ξ^VtanhzVaV.It can be induced that(45)L̇V¯≤zVEVgVωΦΦ+zV2ℓVg~VmSVTSV4τV+τV+12ϕV2−ω¯ΦlVϑ~Vϑ^V+zVΦ¯−kVzV2−zV2ℓ^Vg~VmSVTSV4τV+ξVzV−zVtanhzVaV−g~VmμVℓ~Vℓ^V+1rVξ~VrVzVtanhzVaV−ξ^V.Choose the adaptive laws as follows:(46)ℓ^V=μVzV2SVTSV4τV−ΓVℓ^V,(47)ξ^V=rVzVtanhzVaV−bVξ^V,(48)ϑ^V=lVzVΦ¯−cVϑ^V,where aV>0, bV>0, and cV>0 are the parameters to be designed.Substituting (46)–(48) into (45) yields
(49)L̇V¯≤zVEVgVωΦΦ+τV+12ϕV2+ΓVg~VmμVℓ~Vℓ^V−ω¯Φϑ~VzVΦ¯+ω¯ΦcVlVϑ~Vϑ^V+zVΦ¯−kVzV2+ξVzV−zVtanhzVaV+bVrVξ~Vξ^V.Now, we design the actual control law as(50)Φ=−zVϑ^V2Φ¯2zV2ϑ^V2Φ¯2+σV2,where σV>0 is a predefined constant, which is designed to avoid the singularity issue. According to Young’s inequation, one has
(51)ℓ~Vℓ^V=ℓV−ℓ~Vℓ~V≤−12ℓ~V2+12ℓV2,(52)ϑ~Vϑ^V=ϑV−ϑ~Vϑ~V≤−12ϑ~V2+12ϑV2,(53)ξ~Vξ^V=ξV−ξ~Vξ~V≤−12ξ~V2+12ξV2.Substituting (50)-(53) into (49) and applying Lemma 8 and Lemma 9, we can rewrite (49) as
(54)L̇V¯≤−kVzV2−12μVΓVg~Vmℓ~V2−12lVω¯ΦcVϑ~V2−12rVbVξ~V2+12μVΓVg~VmℓV2+12lVω¯ΦcVϑV2+12rVbVξV2+τV+12ϕV2+ω¯ΦσV+0.2785aVξV.


### 3.2. Altitude Controller Design

In the process of altitude controller design, the backstepping methodology is adopted to deal with complex dynamics. The virtual controllers will be designed at first, and then the intermediate control law the actual control law will be constructed to counteract the impact of actuator fault. To initiate the design process, we first define the following tracking errors:(55)eh=h−href,zγ=γ−χγ,zθ=θ−χθ,zQ=Q−χQ,with χγ, χθ, and χQ representing the virtual control laws.Step 10.
Similarly to velocity controller design, one reaches(56)ėh=Vγ+dh−ḣref=ρ̇tq+żhρt∂q∂z.
Then, we have(57)żh=Vγ+dh−ḣref−ρ̇tqρt∂q/∂z=Yh+EhVγ+dh−ḣref,
with(58)Yh=−ρ̇tqEh,Eh=1ρt∂q/∂z.
Noting that∂q/∂z=1−q22/1+q2≤1 when −1<q<1 and the fact that V≥Vm>0 where Vm is minimum permissible flight velocity yields
(59)EhV=Vρt∂q/∂z≥Vmρ0=V~m>0.
Consider the following Lyapunov function candidate:(60)Lh=12zh2+12μhV~mℓ~h2,
withℓ~h=ℓh−ℓ^h, in which ℓ^h denotes the estimation of adaptive parameter ℓh and μh is the positive design parameter.
The time derivative ofLh gives
(61)L̇h=zhWhTSh+ϕh+EhVγ−1μhV~mℓ~hℓ^h−12zh2,with Fh=Yh+Ehdh−Ehḣref+1/2zh being approximated by FLS WhTShxh+ϕh, where xh=h,href,ḣrefT.
Applying Young’s inequation, it gives(62)zhWhTSh≤zh2Wh2ShTSh4τh+τh,zhϕh≤12zh2+12ϕh2,where τh is a positive constant; noting that γ=zγ+χγ we can further rewrite (61) as
(63)L̇h≤EhVzhzγ+zhEhVχγ+zh2ℓhV~mShTSh4τh+τh+12ϕh2−1μhV~mℓ~hℓ^h,with the definition of ℓh=Wh2/V~m. Choose the virtual control law and updating law as follows:
(64)χγ=−khzh−zhℓ^hShTSh4τh,(65)ℓ^h=μhzh2ShTSh4τh−Γhℓ^h.
Substituting (64)–(65) into (63) yields
(66)L̇h≤−khzh2+1μhΓhV~mℓ~hℓ^h+EhVzhzγ+12ϕh2+τh.Step 11.
Take the Lyapunov function candidate as(67)Lγ=Lh+12zγ2+12μγgγmℓ~γ2.
Similar to step10, by defining ℓγ=Wh2/gγm, the time derivative of Lγ can be formulated as
(68)L̇γ=L̇h+zγWγTSγ+ϕγ+gγθ−1μγgγmℓ~γℓ^γ−EhVzhzγ−12zγ2,with Fγ=fγ+EhVzh+dγ−χ̇γ+1/2zγ being approximated by FLS WγTSγxγ+ϕγ, where χ̇γ=∂χγ/∂hḣ+∂χγ/∂ℓ^hℓ^h+∂χγ/∂hrefḣref and xγ=h,γ,href,ḣref,href2,ℓ∧hT.
Design the virtual control law and adaptive law as(69)χθ=−kγzγ−zγℓ^γSγTSγ4τγ,(70)ℓ^γ=μγzγ2SγTSγ4τγ−Γγℓ^γ.
Based on (66)–(70), we obtain
(71)L̇γ≤−khzh2−kγzγ2+1μhΓhV~mℓ~hℓ^h+1μγΓγgγmℓ~γℓ^γ+gγzγzθ+12ϕh2+12ϕγ2+τh+τγ.Step 12.
Consider the following Lyapunov function candidate:(72)Lθ=Lγ+12zθ2+12μθℓ~θ2.
Definingℓθ=Wθ2, the time derivative of Lθ gives
(73)L̇θ=L̇γ+zθWθTSθ+ϕθ+Q−1μθℓ~θℓ^θ−gγzγzθ−12zθ2,with Fθ=gγzγ−χ̇θ+1/2zθ being approximated by FLS WθTSθxθ+ϕθ, where χ̇θ=∑x=h,γ∂χθ/∂xẋ+∑x=h,γ∂χθ/∂ℓ^xℓ^x+∑i=02∂χθ/∂hrefihrefi+1 and xθ=h,γ,θ,href,ḣref,href2,href3,ℓ∧h,ℓ∧γT.
Construct the virtual control law and adaptive law(74)χQ=−kθzθ−zθℓ^θSθTSθ4τθ,(75)ℓ^θ=μθzθ2SθTSθ4τθ−Γθℓ^θ.
From (72)–(75), one has
(76)L̇θ≤−khzh2−kγzγ2−kθzθ2+ΓhV~mμhℓ~hℓ^h+Γγgγmμγℓ~γℓ^γ+zθzQ+Γθμθℓ~θℓ^θ+12ϕh2+12ϕγ2+12ϕθ2+τh+τγ+τθ.Step 13.
Choose the following Lyapunov function candidate:(77)LQ=Lθ+12zQ2+12μQgQmℓ~Q2.
ConstructingℓQ=WQ2/gQm and taking the derivative of LQ yield
(78)L̇Q=L̇θ+zQWQTSQ+ϕQ+gQωδeδe+gQεδe−1μQgQmℓ~Qℓ^Q−zθzQ−12zQ2,with FQ=fQ+zθ+dQ−χ̇θ+1/2zQ being approximated by FLS WQTSQxQ+ϕQ, where χ̇Q=∑x=h,γ,θ∂χQ/∂xẋ+∑x=h,γ,θ∂χQ/∂ℓ^xℓ^x+∑i=03∂χQ/∂hrefihrefi+1 and xQ=h,γ,θ,Q,href,ḣref,href2,href3,href4,ℓ∧h,ℓ∧γ,ℓ∧θT.
The upper and lower bounds of fault parameters are defined as(79)ω¯δe=infgQωδe,ϑQ=1ω¯δe,ξQ=supgQεδe.Construct the Lyapunov function:(80)LQ¯=LQ+12lQω¯δeϑ~Q2+12rQξ~Q2,where lQ>0 and rQ>0 are designed parameters and ϑ~Q=ϑQ−ϑ^Q and ξ~Q=ξQ−ξ^Q represent estimation errors with ϑ^Q and ξ^Q being the estimations of ϑQ and ξQ, respectively. Choose the intermediate control law and adaptive laws as follows:
(81)δ¯e=kQzQ+zQℓ^QSQTSQ4τQ+ξ^QtanhzQaQ,(82)ℓ^Q=μQzQ2SQTSQ4τQ−ΓQℓ^Q,(83)ξ^Q=rQzQtanhzQaQ−bQξ^Q,(84)ϑ^Q=lQzQδ¯e−cQϑ^Q,where kQ, τQ, aQ, bQ, and cQ are designed positive parameters.Finally, we choose the actual control law as(85)δe=−zQϑ^Q2δ¯e2zQ2ϑ^Q2δ¯e2+σQ2,where σQ>0 is a predefined constant. Following similar analysis to velocity subsystem, we can further deduce that
(86)L̇Q¯≤−khzh2−kγzγ2−kθzθ2−kQzQ2−ΓhV~mμhℓ~h2−Γγgγmμγℓ~γ2−Γθμθℓ~θ2−ΓQg~Qm2μQℓ~Q2−ω¯δecQ2lQϑ~Q2−bQ2rQξ~Q2+ΓhV~mμhℓh2+Γγgγmμγℓγ2+Γθμθℓθ2+ΓQg~Qm2μQℓQ2+ω¯δecQ2lQϑQ2+bQ2rQξQ2+12ϕh2+12ϕγ2+12ϕθ2+12ϕQ2+τh+τγ+τθ+τQ+ω¯δeσQ+0.2785aQξQ.The whole FTFTC design for HFVs is shown in Figure2.Figure 2
Overall fixed-time tolerant-control scheme for HFVs.


## 3.1. Velocity Controller Design

We first define the velocity tracking error as(30)eV=V−Vref,where Vref is the velocity reference trajectory. In view of (15), (23), and (24), the time derivative of eV is
(31)ėV=fV+gVvΦ+dV−V̇ref=ρ̇tq+żVρt∂q∂z.Then,żV can be rewritten as
(32)żV=fV+gVvΦ+dV−V̇ref−ρ̇tqρt∂q/∂z=YV+EVfV+gVvΦ+dV−V̇ref,with YV=−ρ̇tqEV, EV=1/ρt∂q/∂z.It can be deduced from (24) that ∂q/∂z=1−q22/1+q2≤1 when −1<q<1, and noticing the fact gV≥gVm>0 leads to
(33)EVgV=gVρt∂q/∂z≥gVmρ0>0.Choose the following quadratic function:(34)LV=12zV2+12μVg~Vmℓ~V2,where g~Vm=gVm/ρ0, ℓ~V=ℓV−ℓ^V, ℓ^V denotes the estimation of adaptive parameter ℓV and μV is the positive user-defined parameter.Utilizing (14) and (31), the time derivative of LV is
(35)L̇V=zVYV+EVfV+EVgVωΦΦ+EVgVεΦ+EVdV−EVV̇ref−1μVg~Vmℓ~Vℓ^V.Define the nonlinear function(36)FV=YV+EVfV+EVdV−EVV̇ref+12zV,where fV and dV are unknown due to the existence of unknown external disturbances and fast time-varying flight environment. An FLSs-approximator is constructed to estimate FV as
(37)FV=WVTSVxV+ϕV,where xV=V,Vref,V̇refT. Substituting (36) and (37) into (35), we have
(38)L̇V=zVWVTSV+ϕV+EVgVωΦΦ+EVgVεΦ−1μVg~Vmℓ~Vℓ^V−12zV2.According to Young’s inequation, we can further have(39)zVWVTSV≤zV2WV2SVTSV4τV+τV,zVϕV≤12zV2+12ϕV2,where τV is a positive constant; then, we can rewrite (38) as
(40)L̇V≤zVEVgVωΦΦ+EVgVεΦ+zV2WV2SVTSV4τV+τV+12ϕV2−1μVg~Vmℓ~Vℓ^V.According to Assumption1, ωΦ and εΦ are unknown. Therefore, we define the upper and lower bounds of fault parameters to achieve robustness, which are expressed as
(41)ω¯Φ=infEVgVωΦ,ϑV=1ω¯Φ,ξV=supEVgVεΦ.Consider the Lyapunov function candidate(42)LV¯=LV+12lVω¯Φϑ~V2+12rVξ~V2,where lV>0 and rV>0 are the parameters to be designed and ϑ~V=ϑV−ϑ^V and ξ~V=ξV−ξ^V represent estimation errors with ϑ^V and ξ^V being the estimations of ϑV as ξV, respectively.The time derivative ofLV¯ gives
(43)L̇V¯=L̇V−1lVω¯Φϑ~Vϑ^V−1rVξ~Vξ^V.DefineℓV=WV2/g~Vm and choose the intermediate control law as
(44)Φ¯=kVzV+zVℓ^VSVTSV4τV+ξ^VtanhzVaV.It can be induced that(45)L̇V¯≤zVEVgVωΦΦ+zV2ℓVg~VmSVTSV4τV+τV+12ϕV2−ω¯ΦlVϑ~Vϑ^V+zVΦ¯−kVzV2−zV2ℓ^Vg~VmSVTSV4τV+ξVzV−zVtanhzVaV−g~VmμVℓ~Vℓ^V+1rVξ~VrVzVtanhzVaV−ξ^V.Choose the adaptive laws as follows:(46)ℓ^V=μVzV2SVTSV4τV−ΓVℓ^V,(47)ξ^V=rVzVtanhzVaV−bVξ^V,(48)ϑ^V=lVzVΦ¯−cVϑ^V,where aV>0, bV>0, and cV>0 are the parameters to be designed.Substituting (46)–(48) into (45) yields
(49)L̇V¯≤zVEVgVωΦΦ+τV+12ϕV2+ΓVg~VmμVℓ~Vℓ^V−ω¯Φϑ~VzVΦ¯+ω¯ΦcVlVϑ~Vϑ^V+zVΦ¯−kVzV2+ξVzV−zVtanhzVaV+bVrVξ~Vξ^V.Now, we design the actual control law as(50)Φ=−zVϑ^V2Φ¯2zV2ϑ^V2Φ¯2+σV2,where σV>0 is a predefined constant, which is designed to avoid the singularity issue. According to Young’s inequation, one has
(51)ℓ~Vℓ^V=ℓV−ℓ~Vℓ~V≤−12ℓ~V2+12ℓV2,(52)ϑ~Vϑ^V=ϑV−ϑ~Vϑ~V≤−12ϑ~V2+12ϑV2,(53)ξ~Vξ^V=ξV−ξ~Vξ~V≤−12ξ~V2+12ξV2.Substituting (50)-(53) into (49) and applying Lemma 8 and Lemma 9, we can rewrite (49) as
(54)L̇V¯≤−kVzV2−12μVΓVg~Vmℓ~V2−12lVω¯ΦcVϑ~V2−12rVbVξ~V2+12μVΓVg~VmℓV2+12lVω¯ΦcVϑV2+12rVbVξV2+τV+12ϕV2+ω¯ΦσV+0.2785aVξV.


## 3.2. Altitude Controller Design

In the process of altitude controller design, the backstepping methodology is adopted to deal with complex dynamics. The virtual controllers will be designed at first, and then the intermediate control law the actual control law will be constructed to counteract the impact of actuator fault. To initiate the design process, we first define the following tracking errors:(55)eh=h−href,zγ=γ−χγ,zθ=θ−χθ,zQ=Q−χQ,with χγ, χθ, and χQ representing the virtual control laws.Step 10.
Similarly to velocity controller design, one reaches(56)ėh=Vγ+dh−ḣref=ρ̇tq+żhρt∂q∂z.
Then, we have(57)żh=Vγ+dh−ḣref−ρ̇tqρt∂q/∂z=Yh+EhVγ+dh−ḣref,
with(58)Yh=−ρ̇tqEh,Eh=1ρt∂q/∂z.
Noting that∂q/∂z=1−q22/1+q2≤1 when −1<q<1 and the fact that V≥Vm>0 where Vm is minimum permissible flight velocity yields
(59)EhV=Vρt∂q/∂z≥Vmρ0=V~m>0.
Consider the following Lyapunov function candidate:(60)Lh=12zh2+12μhV~mℓ~h2,
withℓ~h=ℓh−ℓ^h, in which ℓ^h denotes the estimation of adaptive parameter ℓh and μh is the positive design parameter.
The time derivative ofLh gives
(61)L̇h=zhWhTSh+ϕh+EhVγ−1μhV~mℓ~hℓ^h−12zh2,with Fh=Yh+Ehdh−Ehḣref+1/2zh being approximated by FLS WhTShxh+ϕh, where xh=h,href,ḣrefT.
Applying Young’s inequation, it gives(62)zhWhTSh≤zh2Wh2ShTSh4τh+τh,zhϕh≤12zh2+12ϕh2,where τh is a positive constant; noting that γ=zγ+χγ we can further rewrite (61) as
(63)L̇h≤EhVzhzγ+zhEhVχγ+zh2ℓhV~mShTSh4τh+τh+12ϕh2−1μhV~mℓ~hℓ^h,with the definition of ℓh=Wh2/V~m. Choose the virtual control law and updating law as follows:
(64)χγ=−khzh−zhℓ^hShTSh4τh,(65)ℓ^h=μhzh2ShTSh4τh−Γhℓ^h.
Substituting (64)–(65) into (63) yields
(66)L̇h≤−khzh2+1μhΓhV~mℓ~hℓ^h+EhVzhzγ+12ϕh2+τh.Step 11.
Take the Lyapunov function candidate as(67)Lγ=Lh+12zγ2+12μγgγmℓ~γ2.
Similar to step10, by defining ℓγ=Wh2/gγm, the time derivative of Lγ can be formulated as
(68)L̇γ=L̇h+zγWγTSγ+ϕγ+gγθ−1μγgγmℓ~γℓ^γ−EhVzhzγ−12zγ2,with Fγ=fγ+EhVzh+dγ−χ̇γ+1/2zγ being approximated by FLS WγTSγxγ+ϕγ, where χ̇γ=∂χγ/∂hḣ+∂χγ/∂ℓ^hℓ^h+∂χγ/∂hrefḣref and xγ=h,γ,href,ḣref,href2,ℓ∧hT.
Design the virtual control law and adaptive law as(69)χθ=−kγzγ−zγℓ^γSγTSγ4τγ,(70)ℓ^γ=μγzγ2SγTSγ4τγ−Γγℓ^γ.
Based on (66)–(70), we obtain
(71)L̇γ≤−khzh2−kγzγ2+1μhΓhV~mℓ~hℓ^h+1μγΓγgγmℓ~γℓ^γ+gγzγzθ+12ϕh2+12ϕγ2+τh+τγ.Step 12.
Consider the following Lyapunov function candidate:(72)Lθ=Lγ+12zθ2+12μθℓ~θ2.
Definingℓθ=Wθ2, the time derivative of Lθ gives
(73)L̇θ=L̇γ+zθWθTSθ+ϕθ+Q−1μθℓ~θℓ^θ−gγzγzθ−12zθ2,with Fθ=gγzγ−χ̇θ+1/2zθ being approximated by FLS WθTSθxθ+ϕθ, where χ̇θ=∑x=h,γ∂χθ/∂xẋ+∑x=h,γ∂χθ/∂ℓ^xℓ^x+∑i=02∂χθ/∂hrefihrefi+1 and xθ=h,γ,θ,href,ḣref,href2,href3,ℓ∧h,ℓ∧γT.
Construct the virtual control law and adaptive law(74)χQ=−kθzθ−zθℓ^θSθTSθ4τθ,(75)ℓ^θ=μθzθ2SθTSθ4τθ−Γθℓ^θ.
From (72)–(75), one has
(76)L̇θ≤−khzh2−kγzγ2−kθzθ2+ΓhV~mμhℓ~hℓ^h+Γγgγmμγℓ~γℓ^γ+zθzQ+Γθμθℓ~θℓ^θ+12ϕh2+12ϕγ2+12ϕθ2+τh+τγ+τθ.Step 13.
Choose the following Lyapunov function candidate:(77)LQ=Lθ+12zQ2+12μQgQmℓ~Q2.
ConstructingℓQ=WQ2/gQm and taking the derivative of LQ yield
(78)L̇Q=L̇θ+zQWQTSQ+ϕQ+gQωδeδe+gQεδe−1μQgQmℓ~Qℓ^Q−zθzQ−12zQ2,with FQ=fQ+zθ+dQ−χ̇θ+1/2zQ being approximated by FLS WQTSQxQ+ϕQ, where χ̇Q=∑x=h,γ,θ∂χQ/∂xẋ+∑x=h,γ,θ∂χQ/∂ℓ^xℓ^x+∑i=03∂χQ/∂hrefihrefi+1 and xQ=h,γ,θ,Q,href,ḣref,href2,href3,href4,ℓ∧h,ℓ∧γ,ℓ∧θT.
The upper and lower bounds of fault parameters are defined as(79)ω¯δe=infgQωδe,ϑQ=1ω¯δe,ξQ=supgQεδe.Construct the Lyapunov function:(80)LQ¯=LQ+12lQω¯δeϑ~Q2+12rQξ~Q2,where lQ>0 and rQ>0 are designed parameters and ϑ~Q=ϑQ−ϑ^Q and ξ~Q=ξQ−ξ^Q represent estimation errors with ϑ^Q and ξ^Q being the estimations of ϑQ and ξQ, respectively. Choose the intermediate control law and adaptive laws as follows:
(81)δ¯e=kQzQ+zQℓ^QSQTSQ4τQ+ξ^QtanhzQaQ,(82)ℓ^Q=μQzQ2SQTSQ4τQ−ΓQℓ^Q,(83)ξ^Q=rQzQtanhzQaQ−bQξ^Q,(84)ϑ^Q=lQzQδ¯e−cQϑ^Q,where kQ, τQ, aQ, bQ, and cQ are designed positive parameters.Finally, we choose the actual control law as(85)δe=−zQϑ^Q2δ¯e2zQ2ϑ^Q2δ¯e2+σQ2,where σQ>0 is a predefined constant. Following similar analysis to velocity subsystem, we can further deduce that
(86)L̇Q¯≤−khzh2−kγzγ2−kθzθ2−kQzQ2−ΓhV~mμhℓ~h2−Γγgγmμγℓ~γ2−Γθμθℓ~θ2−ΓQg~Qm2μQℓ~Q2−ω¯δecQ2lQϑ~Q2−bQ2rQξ~Q2+ΓhV~mμhℓh2+Γγgγmμγℓγ2+Γθμθℓθ2+ΓQg~Qm2μQℓQ2+ω¯δecQ2lQϑQ2+bQ2rQξQ2+12ϕh2+12ϕγ2+12ϕθ2+12ϕQ2+τh+τγ+τθ+τQ+ω¯δeσQ+0.2785aQξQ.The whole FTFTC design for HFVs is shown in Figure2.Figure 2
Overall fixed-time tolerant-control scheme for HFVs.


## 4. Closed-Loop Stability Analysis

Theorem 14.
Despite the occurrence of unknown actuator fault (14), consider the closed-loop system composed by (15) and (16); the virtual control laws (64), (69), and (74); the intermediate control laws (44) and (81); the actual control laws (50) and (85); and the parameter adaptation laws (46)-(48), (65), (70), (75), and (82)–(84). Let Assumptions 1–4 hold. By designing the parameters properly, it therefore holds the following.
The tracking errorseV and eh can converge into a predefined residual set within an user-defined time T.
(1)
The overshoot and convergence rate are guaranteed by FTPF, and all signals of the closed-loop system are SGPFSProof.
Take the Lyapunov function candidate as(87)L=LV¯+LQ¯.
Applying (54) and (86), the derivative of L gives
(88)L̇≤−kVzV2−khzh2−kγzγ2−kθzθ2−kQzQ2−ΓVg~Vm2μVℓ~V2−ΓhV~mμhℓ~h2−Γγgγmμγℓ~γ2−Γθμθℓ~θ2−ΓQg~Qm2μQℓ~Q2−ω¯ΦcV2lVϑ~V2−ω¯δecQ2lQϑ~Q2−bV2rVξ~V2−bQ2rQξ~Q2+ΓhV~mμhℓh2+Γγgγmμγℓγ2+Γθμθℓθ2+ΓVg~Vm2μVℓV2+ΓQg~Qm2μQℓQ2+ω¯ΦcV2lVϑV2+ω¯δecQ2lQϑQ2+bV2rVξV2+bQ2rQξQ2+12ϕV2+12ϕh2+12ϕγ2+12ϕθ2+12ϕQ2+τV+τh+τγ+τθ+τQ+ω¯ΦσV+ω¯δeσQ+0.2785aVξV+0.2785aQξQ.
Recalling the definitions ofLV¯ and LQ¯, the following inequation holds
(89)L̇≤−kL+C,where k=min2kV,2kh,2kγ,2kθ,2kQ,ΓV,Γh,Γγ,Γθ,ΓQ,bV,bQ,cV,cQ, C=ΓhV~m/μhℓh2+Γγgγm/μγℓγ2+Γθ/μθℓθ2+ΓVg~Vm/2μVℓV2+ΓQg~Qm/2μQℓQ2+ω¯ΦcV/2lVϑV2+ω¯δecQ/2lQϑQ2+bV/2rVξV2+bQ/2rQξQ2+1/2ϕV2+1/2ϕh2+1/2ϕγ2+1/2ϕθ2+1/2ϕQ2+τV+τh+τγ+τθ+τQ+ω¯ΦσV+ω¯δeσQ+0.2785aVξV+0.2785aQξQ.
Integrating both sides of (89) yields
(90)L≤L0+Ck.
In accordance with (34) and (60), we have
(91)zV≤2L0+Ck,zh≤2L0+Ck,
In view of (42), (60), (67), (72), (77), (80), (87), and (90), the estimation errors of the adaptive parameters will converge to the following compact sets:
(92)ℓ~V≤2μVL0+Ckℓ~h≤2μhL0+Ckℓ~γ≤2μγL0+Ckℓ~θ≤2μθL0+Ckℓ~Q≤2μQL0+Ck,ϑ~V≤2lVω¯ΦL0+Ckϑ~Q≤2lQω¯δeL0+Ckξ~V≤2rVL0+Ckξ~Q≤2rQL0+Ck.
Therefore, the transformational errorszV and zh are bounded. By reviewing (17)–(24), we can conclude that the velocity and altitude tracking errors converge to a residual set within a fixed time T and the prescribed performances are guaranteed. Besides, all signals of closed-loop system are SGPFS. According to the design of velocity controller (44), altitude controller (64), and the FTPF (17), the overshoots of velocity and altitude do not exceed their preset threshold. This completes the proof. +Remark 15.
The existing fixed-time control strategies for HFVs [18–20] fail to take system transient and steady-state performances into account, and it is fairly complicated to make tracking error convergence into a preassigned compact set within the fixed time by selecting design parameters. It is worth noting that the fixed-time tracking control is achieved as long as the bounded condition is satisfied in the proposed design. Consequently, the complexity of the control structure is reduced and the initial states need not to be known accurately via the proposed control approach.Remark 16.
It is worth mentioning that the control performance is depended closely on the designed parameters of prescribed function. In (17), large initial errors are allowed by choosing a small enough r; larger ϑ and smaller T will increase the convergence rate. However, too large ϑ or too small T will give rise to actuator input saturations. In (44), (64), (69), (74), and (81), designed parameters kV, τV, aV, kh, τh, kγ, τγ, kθ, τθ, kQ, τQ, and aQ determine the convergence rate and convergence accuracy. In (46)-(48), (65), (70), (75), and (82)–(84), designed parameters μV, ΓV, rV, bV, lV, cV, μh, Γh, μγ, Γγ, μθ, Γθ, μQ, ΓQ, rQ, bQ, lQ, and cQ effect convergence rate of the adaptive parameters. In the controller design, we need to design the parameters properly to improve the tracking performance and avoid the saturation phenomenon.Remark 17.
When the actuator failure occurs, the upper and lower bounds of fault parameters are estimated by the adaptive laws (47), (48), (83), and (84), then the intermediate control laws (44) and (81) and the actual control laws (50) and (85) are executed. By this way, the flight control system is robust to actuator failure. Compared with the state-of-the-art FTC methods [30–33], the fixed-time stability is guaranteed, and the prescribed performance is ensured.


## 5. Simulations

In this section, simulation results are used to demonstrate the effectiveness and superiority of the proposed methodology. The model parameters of HFVs can be consulted from [40]. HFVs are expected to climb a maneuver from the initial trim conditions, depicted in Table 1, to the final values V=8700ft/s and h=88000ft. The external disturbances in velocity subsystem and altitude subsystem are set as dV=sin0.1πtft/s,dγ=0.001sin0.01πtdeg,dQ=0.01sin0.01πtdeg/s. The reference trajectories of velocity and altitude are generated via the following filters [11]:
(93)VrefsVcs=0.032s2+2×0.95×0.03×s+0.032,hrefshcs=0.032s2+2×0.95×0.03×s+0.032,where Vrefs and hrefs represent the inputs of filters and Vcs and hcs represent the outputs of filters, respectively. It is assumed that HFVs actuators failed at 100 s and the details of failure are formulated in the form of
(94)vΦ=0.8Φ−0.1,vδe=0.8δe+0.0349.Table 1
The initial states.
StatesValueUnitsV7700ft/sh85000ftγ0degθ1.6325degQ0deg/sη10.97ft slugs0.5/ftη̇10ft/s slugs0.5/ftη20.7967ft slugs0.5/ftη̇20ft/s slugs0.5/ftThe FTPFs are selected as(95)ρVt=cotht50−t+0.4−0.9,0≤t<500.1,t≥50,ρht=cotht50−t+0.4−0.5,0≤t<500.5,t≥50.The fuzzy rules inWV∗TSV are listed asRl: If V is FVi, then y is Bl, where i=1,2,3 and l=1,2,3.The fuzzy rules inWγ∗TSγ are listed asRl: If h is Fhi, and γ is Fγj, then y is Bl, where i=1,2,3; j=1,2,3; and l=1,2,⋯,9.Then, the fuzzy rules inWθ∗TSθ are listed asRl: If h is Fhi, and γ is Fγj, and θ is Fθk, then y is Bl, where i=1,2,3; j=1,2,3; k=1,2,3; and l=1,2,⋯,27.The fuzzy rules inWQ∗TSQ are listed asRl: If h is Fhi, and γ is Fγj, and θ is Fθk, and Q is FQp, then y is Bl, where i=1,2,3; j=1,2,3; k=1,2,3; p=1,2,3; and l=1,2,⋯,81.The fuzzy membership function is given as follows:(96)1μFV1=exp−V−75002/300,μFV2=exp−V−81002/300,μFV3=exp−V−87002/300,μFh1=exp−h−850002/1000,μFh2=exp−h−870002/1000,μFh3=exp−h−890002/1000,μFγ1=exp−γ−02/0.0005,μFγ2=exp−γ−0.0052/0.0005,μFγ3=exp−γ−0.012/0.0005,1μFθ1=exp−θ−02/0.002,μFθ2=exp−θ−0.022/0.002,μFθ3=exp−θ−0.042/0.002,μFQ1=exp−Q+0.032/0.002,μFQ2=exp−Q−02/0.002,μFQ3=exp−Q−0.032/0.002.The controller parameters are selected askV=μV=rV=lV=μh=μγ=μθ=μQ=rQ=lQ=1, g~Vm=0.00001, g~Qm=1, τV=ΓV=cV=bV=τh=Γh=τγ=Γγ=τθ=Γθ=τQ=ΓQ=cQ=bQ=0.1, aV=0.2, aQ=0.25, σV=σQ=1/t2+0.01, kh=10, kγ=2, kθ=10, and kQ=50. Taking the engineering practice into account, the limitations of the actuators are set as Φ∈0,1, δe∈−20deg,20deg [40]. To prove that the proposed FTFTC is realizable and independent of the initial states, the first example is taken, while the superiority of the proposed FTFTC over the conventional fixed-time control (CFTC) in [49] is explained in the second example.Example 18.
In this example, two different initial states cases, i.e., case 1 (eV0=eh0=0) and case 2 (eV0=eh0=1), are considered, respectively.
The obtained simulation results, depicted in Figures3–5, reveal that the proposed FTFTC has full capabilities to deal with the condition that aerodynamic parameters are perturbed and initial states are uncertain. It can be observed from Figures 3(a)–3(b) that the velocity and altitude tracking errors are guaranteed not to exceed the prescribed bounds. Furthermore, the velocity and altitude can rapidly track their reference trajectories even if the actuator fails. Figures 4(a) and 5(a) depict that there is no high-frequency chattering in the attitude angles as well as the flexible states, and they can converge to their steady values rapidly. Figure 5(b) shows that the control inputs are smooth and within realistic limits. The estimated values of adaptive parameters are bounded, which is presented in Figure 4(b).Example 19.
In this example, simulations via the CFTC [49] and the proposed FTFTC are demonstrated, where the initial states are set as eV0=eh0=1. In order to expound the advantages of tracking performances of the proposed FTFTC, the performance index of tracking error Ee=∫0te2dt is introduced where e denotes the tracking error. In order to compare the energy consumption between the FTFTC and CFTC, the performance index of control input Eu=∫0tu2dt is defined where u denotes the control input.Figure 3
The velocity and altitude tracking performance under different initial states.
(a)
Velocity tracking performance(b)
Altitude tracking performanceFigure 4
The attitude angles and adaptive parameters under different initial states.
(a)
Attitude angles(b)
Adaptive parametersFigure 5
The flexible states and control inputs under different initial states.
(a)
Flexible states(b)
Control inputsSimulation results are depicted in Figures6–9. Figures 6(a)–6(b) shows the velocity and altitude tracking performance, in which the velocity and altitude tracking errors are limited in the preset bounds by the proposed FTFTC and the proposed FTFTC can provide higher rate of convergence compared with the CFTC. Besides, the attitude angles and flexible states are shown in Figures 7 and 8(a), indicating that smaller oscillation amplitudes of attitude angles and flexible states are achieved in the presence of actuator failures by means of the proposed FTFTC. Figure 8(b) shows that the control inputs are smooth and within realistic limits by means of the proposed method. In addition, Figure 9 gives that less error energy is produced via the proposed FTFTC in contrast with the CFTC. Table 2 shows that the energy consumption of actuator with FTFTC is almost equal to that with CFTC. That is to say, the proposed FTFTC can achieve more accurate tracking.Figure 6
The velocity and altitude tracking performance via different methodologies.
(a)
Velocity tracking performance(b)
Altitude tracking performanceFigure 7
The attitude angles via different methodologies.Figure 8
The flexible states and control inputs via different methodologies.
(a)
Flexible states(b)
Control inputsFigure 9
The performance indexes via different methodologies.Table 2
The performance index of control inputs.
Control methodPerformance index ofΦPerformance index ofδeFTFTC99.468315.2615CFTC99.457415.2620


## 6. Conclusions

A novel fixed-time fuzzy adaptive fault-tolerant control methodology based on performance function is developed for hypersonic flight vehicles in this work. In contrast with the conventional fixed-time control, the proposed approach not only guarantees that the velocity and altitude tracking errors converge into a preassigned compact set, but also satisfies both the prescribed transient and steady performance. In addition, the proposed scheme can avoid the singularity problem caused by the differential of fractional order tracking error and remain valid in spite of actuator faults. Comparative simulation results confirm the validity and superiority of the presented control strategy. Note that the distributed adaptive containment fault-tolerant control of multi-HFVs formation is an important research region for the future [50–53]; thus, the extension of our control scheme to the case of multi-HFVs formation will be an interesting topic for further investigation.



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*Source: 2438657-2022-09-20.xml*