# Fixed-Time Fault-Tolerant Tracking Control of Fixed-Wing UAVs with Actuator Fault and Unmatched Disturbances

Authors: Zhonghua Wu; Shuaipeng Zheng
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

This paper presents a dynamic surface fixed-time fault-tolerant control strategy for the longitudinal dynamic model of fixed-wing unmanned aerial vehicles (UAVs). Firstly, a novel disturbance observer is constructed to precisely estimate the lumped disturbance. Secondly, without fractional power terms in the designed fixed-time fault-tolerant controller, the potential singular value problem is tactfully avoided, which often exists in the stability analysis of the traditional fixed-time controller design. Thirdly, a novel fixed-time filter is proposed to overcome the phenomenon of “differential explosion” in the backstepping control scheme. Lyapunov stability analysis guarantees that the tracking errors can converge to the neighborhood of the origin in the fixed time. The simulation results verify the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

Fixed-wing UAVs have played an essential role in high-risk and complex missions due to their flexibility and maneuverability [1]. The complex flight conditions make the UAVs subject to actuator fault during operation, which may degrade the stability and robustness of flight control systems, and even lead to a catastrophic accident, thus requiring to explore the fault-tolerant control (FTC) of UAVs. Many researchers have concentrated on addressing the control problem of fixed-wing UAVs subject to actuator fault, thereby publishing numerous results regarding passive or active FTC. Active FTC identifies faults through fault detection and diagnosis block in real time, while passive FTC uses a single fixed controller through the robust control strategy [2–5].An adaptive control approach consisting of a radial base function neural network (RBFNN) was proposed for coaxial octorotor UAV subject to actuator faults [6]. By introducing a prescribed performance function on the synchronized tracking errors, the decentralized finite-time adaptive fault-tolerant synchronization tracking control scheme is proposed for multi-UAVs in the presence of actuator faults [7]. A distributed adaptive FTC scheme is proposed in virtual of a distributed sliding mode estimator and disturbance observers [8]. In the presence of actuator fault, the distributed fault-tolerant output regulation for heterogeneous linear multiagent systems is proposed using the distributed fixed-time observer and adaptive fault-tolerant controller [9]. A constrained control scheme based on model reference adaptive control is investigated for the longitudinal motion of a commercial aircraft with actuator faults and saturation nonlinearities [10]. By using the extended Kalman filter (EKF) to update the weighting parameters of the neural network (NN), a new online detection strategy is developed to detect faults in sensors and actuators of UAVs [11].The external disturbance may degrade the flight performance of fixed-wing UAVs directly. Therefore, it is essential to construct a disturbance observer against its adverse effect. A disturbance observer combined with a time delay estimation is designed to weaken the influence caused by the unknown dynamic parameters in the actuator of the rehabilitation robot [12]. A novel fixed-time extended state observer is presented to estimate the state errors and the total disturbances in the presence of nonlinear couplings, uncertain parameters, and external disturbances [13]. A disturbance observer is incorporated into the control structure to efficiently estimate the lumped disturbances, including modeling inaccuracies and external perturbations [14]. A fixed-time adaptive fast super-twisting disturbance observer is built to estimate the external disturbance, in which two tunable gains are adjusted in real time by an adaptive law [15]. The disturbance observers, as mentioned above, need to know the upper bound of the external disturbance, which may hinder its practical engineering applications.Backstepping control schemes have been widely used in the control system design of fixed-wing UAVs. However, the repeated derivation of the virtual control law in the backstepping designs will largely increase the computational complexity, called the “differential explosion” phenomenon. By introducing a first-order filter to estimate the derivative of the virtual control laws, the dynamic surface control (DSC) scheme is first proposed in [16]. An observer-based fuzzy neural dynamic surface control is presented for a flexible-joint manipulator system with input saturation and unknown disturbance [17]. Moreover, a composite learning fixed-time DSC scheme is proposed for nonlinear strict feedback systems with parameter uncertainties [18]. A constrained adaptive DSC approach is presented for uncertain nonlinear systems subject to full-state constraints [19].Unlike the traditional finite-time control schemes, the upper bound of convergence time is irrelevant to the initial conditions in the fixed-time control scheme, leading to its extensive applications in UAVs. By introducing a continuously differentiable switching function, an adaptive fixed-time control strategy is proposed for autonomous ship landing operations of UAVs [20]. Moreover, a fixed-time controller combined with the obstacle Lyapunov function is designed for a class of surface ship systems with output constraints [21]. Motivated by the practical requirements of high precision and faster convergence rate for an automatic carrier landing, a fixed-time nonlinear flight controller is presented [22]. Anti-saturation coordinated controller is designed [23], which can guarantee the safe distance constraint of each spacecraft in the process of completing the configuration reconstruction task in a specified time. Using a distributed fixed-time observer to estimate the states of the virtual leader, a fixed-time attitude coordinated control is investigated for multispacecraft systems with unknown external disturbance [24]. However, those mentioned above backstepping fixed-time controllers contain fractional power terms, which may encounter the singular problem caused by the repeated derivation of the virtual control law.Inspired by recent results [25–27], a fixed-time fault-tolerant controller without fractional power terms is proposed for fixed-wing UAVs subject to actuator fault and external disturbances. The main contributions of this work are threefold:
(1)
Different from the conventional fixed-time algorithm, a simple fixed-time fault-tolerant controller scheme without fractional power terms is proposed, which overcomes the potential singularity problem often encountered in fixed-time backstepping designs(2)
Unlike the traditional DSC schemes, this paper proposes a simple and smooth fixed-time filter in the control design, reducing the complexity of the control system and ensuring dynamic surface filter error to satisfy fixed-time convergence(3)
A new type of disturbance observer, with an adaptive term to estimate the upper bound of the lumped disturbance, is proposed, thus leading to an accurate estimation of the lumped disturbanceThe layout of this paper is organized as follows. The problem formulation and preliminaries are introduced in Section2. The construction of the new type of disturbance observer is described in Section 3. The fixed-time fault-tolerant controller and a new fixed-time dynamic surface filter are presented in Section 4. In Section 5, the numerical simulation indicates the effectiveness of the proposed control scheme, followed by the conclusions in Section 6.


## 2. Problem Formulation and Preliminaries

### 2.1. Model Description

The nonlinear longitudinal dynamic model of fixed-wing UAV is given as follows:(1)γ̇=L+TsinαmV−gVcosγ+Δγ,α̇=q−γ̇+Δα,q̇=MIyy+Δq,where γ,α and q are the flight-path angle, angle of attack, and pitch rate, respectively. m represents the mass of the UAV. g is the gravity acceleration, Iyy denotes the moment of inertia, Δα,Δγ,Δq represent unknown external disturbances, and T=TmaxδT is the engine thrust.The aerodynamic force and moment of the UAV are expressed as(2)L=0.5ρV2SCL,M=0.5ρV2ScCm,where ρ denotes the air density and S represents the wing platform area. CL and Cm are the aerodynamic coefficients for drag force and pitch moment, respectively. The aerodynamic force and moment coefficients can be expressed as
(3)CL=CLαα+CL0,Cm=Cmαα+Cm0+cq2VCmq+Cmδeδe,where CLα,CL0,Cmα,Cm0,Cmq,Cmδe represent the aerodynamic coefficients. δe is the actual elevator deflection.


### 2.2. Actuator Faults Model

To account for the actuator faults, the actual elevator deflectionδe can be modeled as
(4)δe=p1δe0+bf1,where δe0 denotes the designed elevator deflection and p1 is the remaining control effectiveness factor with 0<p1≤1. bf1 is the fault-bias factor.


### 2.3. System Transformation and Preparation

By substituting (2)–(4) into (1) and defining the state vector x1x2x3T=γαqT,u0=δe0, the longitudinal dynamics can be rewritten as
(5)ẋ1=f1+g1x2+d1,ẋ2=f2+g2x3+d2ẋ3=f3+g3u0+d3,,where
(6)f1=0.5ρV2SCL0mV−gVcosγ,f2=−γ̇,g2=1,g1=0.5ρV2SCLα+TmV,g3=0.5ρV2ScCmδeIyy,f3=0.5ρV2ScCmαα+Cm0+cq/2VCmqIyy,d1=Δγ,d2=Δα,d3=Δq+g3p1−1δe0+g3bf1.


#### 2.3.1. Control Objective

The control objective is to design a fixed-time fault-tolerant control scheme for the longitudinal dynamic model of fixed-wing UAVs subject to actuator fault and external disturbances, such that all the signals in the closed-loop system are bounded, and the reference signal tracking error converges to a small neighborhood of the origin in a fixed time.Assumption 1.
All states of the system (1) are measurable. Meanwhile, the reference trajectory γd is smoothly bounded and known.Assumption 2.
There exist unknown positive constantspi such that di≤pi with i=1,2,3.Lemma 3.
[28] For a common dynamical system, ẋt=ft,x with x0=0 and the origin be an equilibrium point, where x∈Rn and f:R+×Rn⟶Rn. Defining a Lyapunov candidate, Vx≥0. If we can get that
(7)V̇x≤−μ1Vxα−μ2Vxβ+C,where μ1, μ2, α, and β are positive real numbers with α∈0,1, β∈1,∞, then the origin x=0 of the system is practical fixed-time stable, and the settling time function Ts can be estimated by
(8)Ts≤1μ11−α+1μ2β−1.Lemma 4.
[29] For any constant ε>0 and any variable x∈R, the following inequality holds
(9)x−x2x2+ε2≤ε.Lemma 5.
[26] For any positive real numbers x1,x2,⋯,xn and α∈0,1, one has
(10)∑i=1nxiα≤∑i=1nxiα.Lemma 6.
[30] If xi≥0, i=1,⋯,n, then, the following inequality holds
(11)∑i=1nxi2≤n∑i=1nxi2.Lemma 7.
[21] For x,y∈R, Young’s inequality holds
(12)xy≤κppxp+1qκqyq,where κ>0, p>1, q>1, and p−1q−1=1.Lemma 8.
[19] For any constant δ>0 and any variable x∈R, the following inequality holds
(13)x−xtanhxδ≤kδ,where k is a positive constant satisfying k=e−k+1.


## 2.1. Model Description

The nonlinear longitudinal dynamic model of fixed-wing UAV is given as follows:(1)γ̇=L+TsinαmV−gVcosγ+Δγ,α̇=q−γ̇+Δα,q̇=MIyy+Δq,where γ,α and q are the flight-path angle, angle of attack, and pitch rate, respectively. m represents the mass of the UAV. g is the gravity acceleration, Iyy denotes the moment of inertia, Δα,Δγ,Δq represent unknown external disturbances, and T=TmaxδT is the engine thrust.The aerodynamic force and moment of the UAV are expressed as(2)L=0.5ρV2SCL,M=0.5ρV2ScCm,where ρ denotes the air density and S represents the wing platform area. CL and Cm are the aerodynamic coefficients for drag force and pitch moment, respectively. The aerodynamic force and moment coefficients can be expressed as
(3)CL=CLαα+CL0,Cm=Cmαα+Cm0+cq2VCmq+Cmδeδe,where CLα,CL0,Cmα,Cm0,Cmq,Cmδe represent the aerodynamic coefficients. δe is the actual elevator deflection.


## 2.2. Actuator Faults Model

To account for the actuator faults, the actual elevator deflectionδe can be modeled as
(4)δe=p1δe0+bf1,where δe0 denotes the designed elevator deflection and p1 is the remaining control effectiveness factor with 0<p1≤1. bf1 is the fault-bias factor.


## 2.3. System Transformation and Preparation

By substituting (2)–(4) into (1) and defining the state vector x1x2x3T=γαqT,u0=δe0, the longitudinal dynamics can be rewritten as
(5)ẋ1=f1+g1x2+d1,ẋ2=f2+g2x3+d2ẋ3=f3+g3u0+d3,,where
(6)f1=0.5ρV2SCL0mV−gVcosγ,f2=−γ̇,g2=1,g1=0.5ρV2SCLα+TmV,g3=0.5ρV2ScCmδeIyy,f3=0.5ρV2ScCmαα+Cm0+cq/2VCmqIyy,d1=Δγ,d2=Δα,d3=Δq+g3p1−1δe0+g3bf1.


### 2.3.1. Control Objective

The control objective is to design a fixed-time fault-tolerant control scheme for the longitudinal dynamic model of fixed-wing UAVs subject to actuator fault and external disturbances, such that all the signals in the closed-loop system are bounded, and the reference signal tracking error converges to a small neighborhood of the origin in a fixed time.Assumption 1.
All states of the system (1) are measurable. Meanwhile, the reference trajectory γd is smoothly bounded and known.Assumption 2.
There exist unknown positive constantspi such that di≤pi with i=1,2,3.Lemma 3.
[28] For a common dynamical system, ẋt=ft,x with x0=0 and the origin be an equilibrium point, where x∈Rn and f:R+×Rn⟶Rn. Defining a Lyapunov candidate, Vx≥0. If we can get that
(7)V̇x≤−μ1Vxα−μ2Vxβ+C,where μ1, μ2, α, and β are positive real numbers with α∈0,1, β∈1,∞, then the origin x=0 of the system is practical fixed-time stable, and the settling time function Ts can be estimated by
(8)Ts≤1μ11−α+1μ2β−1.Lemma 4.
[29] For any constant ε>0 and any variable x∈R, the following inequality holds
(9)x−x2x2+ε2≤ε.Lemma 5.
[26] For any positive real numbers x1,x2,⋯,xn and α∈0,1, one has
(10)∑i=1nxiα≤∑i=1nxiα.Lemma 6.
[30] If xi≥0, i=1,⋯,n, then, the following inequality holds
(11)∑i=1nxi2≤n∑i=1nxi2.Lemma 7.
[21] For x,y∈R, Young’s inequality holds
(12)xy≤κppxp+1qκqyq,where κ>0, p>1, q>1, and p−1q−1=1.Lemma 8.
[19] For any constant δ>0 and any variable x∈R, the following inequality holds
(13)x−xtanhxδ≤kδ,where k is a positive constant satisfying k=e−k+1.


## 2.3.1. Control Objective

The control objective is to design a fixed-time fault-tolerant control scheme for the longitudinal dynamic model of fixed-wing UAVs subject to actuator fault and external disturbances, such that all the signals in the closed-loop system are bounded, and the reference signal tracking error converges to a small neighborhood of the origin in a fixed time.Assumption 1.
All states of the system (1) are measurable. Meanwhile, the reference trajectory γd is smoothly bounded and known.Assumption 2.
There exist unknown positive constantspi such that di≤pi with i=1,2,3.Lemma 3.
[28] For a common dynamical system, ẋt=ft,x with x0=0 and the origin be an equilibrium point, where x∈Rn and f:R+×Rn⟶Rn. Defining a Lyapunov candidate, Vx≥0. If we can get that
(7)V̇x≤−μ1Vxα−μ2Vxβ+C,where μ1, μ2, α, and β are positive real numbers with α∈0,1, β∈1,∞, then the origin x=0 of the system is practical fixed-time stable, and the settling time function Ts can be estimated by
(8)Ts≤1μ11−α+1μ2β−1.Lemma 4.
[29] For any constant ε>0 and any variable x∈R, the following inequality holds
(9)x−x2x2+ε2≤ε.Lemma 5.
[26] For any positive real numbers x1,x2,⋯,xn and α∈0,1, one has
(10)∑i=1nxiα≤∑i=1nxiα.Lemma 6.
[30] If xi≥0, i=1,⋯,n, then, the following inequality holds
(11)∑i=1nxi2≤n∑i=1nxi2.Lemma 7.
[21] For x,y∈R, Young’s inequality holds
(12)xy≤κppxp+1qκqyq,where κ>0, p>1, q>1, and p−1q−1=1.Lemma 8.
[19] For any constant δ>0 and any variable x∈R, the following inequality holds
(13)x−xtanhxδ≤kδ,where k is a positive constant satisfying k=e−k+1.


## 3. Disturbance Observer Design

In this section, a new type disturbance observer is proposed to eliminate the influence of the lumped disturbances.


### 3.1. Disturbance Observer Design

The disturbance observer is given as follows(14)d^i=λi1πi+λi2πiπ2+εi/λi22+λi3πi3+p^itanhp^iπiδi,where d^i is the estimation of di and λi1,λi2,λi3,δi,εi are positive constants, i=1,2,3. p^i denotes the estimated value of the upper bound of the external unmatched disturbance signal. πi is the auxiliary state of the disturbance observer, which is governed by
(15)πi=xi−si,ṡi=λi1πi+λi2πiπ2+εi/λi22+λi3πi3+p^itanhp^iπiδi+gixi+1+fi.p^i can be calculated as
(16)p^̇i=πi−σ1inip^i−σ2ini2p^i3,where σ1i,σ2i,ni are positive constants, p~i=pi−p^i.Defining the disturbance observer estimation error,d~i=di−d^i. Then, the derivative of πi with respect to time can be written as
(17)π̇i=di−d^i=d~i.Remark 9.
The disturbance observer design presented in [31] requires the upper bound of the external unmatched disturbance in advance, while the proposed disturbance observer releases this restrictive condition. Instead of using the sign function in the disturbance observer design [31], an adaptive term p^itanhp^iπi/δi is introducing in the proposed novel disturbance observer, which successfully eliminates the chattering phenomenon.


### 3.2. Stability Analysis

Theorem 10.
Consider system (5), and assume that Assumptions 1 and 2 hold. If the disturbance observer is designed as (14) and there exist positive constants λi1,λi2,λi3,δi,εi,σ1i,σ2i,ni for i=1,2,3, then the estimated error p~i and auxiliary state πi converge to sets Ωp~i=p~i≤2M/μ2,Ωπi=πi≤2M/μ2 in a fixed time Td≤2/μ1+1/μ2, respectively.Proof.
Consider the Lyapunov function candidate as(18)Vd=∑i=1312πi2+12p~i2.Then, it can be found that(19)V̇d=∑i=13πiπ̇i+p~ip~̇i,=∑i=13πidi−λi1πi2−λi2πi2π2+εi/λi22−λi3πi4−p^iπitanhp^iπiδi+p^iπi−piπi+σ1inip~ip^i+σ2ini2p~ip^i3.According to Lemma4, we have
(20)−λi2πi2π2+εi/λi22≤−λi2πi+εi.Relying on Lemma7, one gets
(21)πidi≤λi1πi2+14λi1di2.From Lemma8, we get
(22)p^iπi−p^iπitanhp^iπiδi≤p^iπi−p^iπitanhp^iπiδi≤kδi.With regard to the termsσ1i/nip~ip^i and σ2i/ni2p~ip^i3 in (19), one can find that
(23)σ1inip~ip^i=σ1inip~ipi−p~i2+σ1iγi2nip~i−σ1iγi2nip~i≤σ1ini12p~i2+12pi2−p~i2+σ1i2nip~i2+σ1iγi24−σ1iγi2nip~i,(24)σ2ini2p~ip^i3=σ2ini2−p~i4−3p~i2pi2+p~ipi3+3pip~i3≤σ2ini2−p~i4−3p~i2pi2+3θi4/34pi34/3+14θi4p~i4+3pi2p~i2+34p~i4,where γiθi are two positive constants.Substituting (20), (21), (22), (23), and (24) into the derivative of Vd results in
(25)V̇d≤∑i=13−2λi212πi21/2−4λi312πi22−σ2iθi4−1θi4ni212p~i22−σ1iγini12p~i21/2+M,where M=∑i=13εi+kδi+3σ2iθi4/3/4ni2pi34/3+σ1iγi2/4+σ1i/2ni+1/4λi1pi2.Then, one has(26)V̇d≤−μ1Vd1/2−μ2Vd2+M≤0.According to the above analysis, ifVd≥M/μ2, then V̇d≤−μ1Vd1/2≤0, we can have that Vd will converge to the set Vd:Vd≤M/μ2 in a fixed-time Td≤2/μ1+1/μ2, and thus, estimated error p~i and auxiliary states πi are uniformly ultimately bounded in a fixed time Td≤2/μ1+1/μ2.


## 3.1. Disturbance Observer Design

The disturbance observer is given as follows(14)d^i=λi1πi+λi2πiπ2+εi/λi22+λi3πi3+p^itanhp^iπiδi,where d^i is the estimation of di and λi1,λi2,λi3,δi,εi are positive constants, i=1,2,3. p^i denotes the estimated value of the upper bound of the external unmatched disturbance signal. πi is the auxiliary state of the disturbance observer, which is governed by
(15)πi=xi−si,ṡi=λi1πi+λi2πiπ2+εi/λi22+λi3πi3+p^itanhp^iπiδi+gixi+1+fi.p^i can be calculated as
(16)p^̇i=πi−σ1inip^i−σ2ini2p^i3,where σ1i,σ2i,ni are positive constants, p~i=pi−p^i.Defining the disturbance observer estimation error,d~i=di−d^i. Then, the derivative of πi with respect to time can be written as
(17)π̇i=di−d^i=d~i.Remark 9.
The disturbance observer design presented in [31] requires the upper bound of the external unmatched disturbance in advance, while the proposed disturbance observer releases this restrictive condition. Instead of using the sign function in the disturbance observer design [31], an adaptive term p^itanhp^iπi/δi is introducing in the proposed novel disturbance observer, which successfully eliminates the chattering phenomenon.


## 3.2. Stability Analysis

Theorem 10.
Consider system (5), and assume that Assumptions 1 and 2 hold. If the disturbance observer is designed as (14) and there exist positive constants λi1,λi2,λi3,δi,εi,σ1i,σ2i,ni for i=1,2,3, then the estimated error p~i and auxiliary state πi converge to sets Ωp~i=p~i≤2M/μ2,Ωπi=πi≤2M/μ2 in a fixed time Td≤2/μ1+1/μ2, respectively.Proof.
Consider the Lyapunov function candidate as(18)Vd=∑i=1312πi2+12p~i2.Then, it can be found that(19)V̇d=∑i=13πiπ̇i+p~ip~̇i,=∑i=13πidi−λi1πi2−λi2πi2π2+εi/λi22−λi3πi4−p^iπitanhp^iπiδi+p^iπi−piπi+σ1inip~ip^i+σ2ini2p~ip^i3.According to Lemma4, we have
(20)−λi2πi2π2+εi/λi22≤−λi2πi+εi.Relying on Lemma7, one gets
(21)πidi≤λi1πi2+14λi1di2.From Lemma8, we get
(22)p^iπi−p^iπitanhp^iπiδi≤p^iπi−p^iπitanhp^iπiδi≤kδi.With regard to the termsσ1i/nip~ip^i and σ2i/ni2p~ip^i3 in (19), one can find that
(23)σ1inip~ip^i=σ1inip~ipi−p~i2+σ1iγi2nip~i−σ1iγi2nip~i≤σ1ini12p~i2+12pi2−p~i2+σ1i2nip~i2+σ1iγi24−σ1iγi2nip~i,(24)σ2ini2p~ip^i3=σ2ini2−p~i4−3p~i2pi2+p~ipi3+3pip~i3≤σ2ini2−p~i4−3p~i2pi2+3θi4/34pi34/3+14θi4p~i4+3pi2p~i2+34p~i4,where γiθi are two positive constants.Substituting (20), (21), (22), (23), and (24) into the derivative of Vd results in
(25)V̇d≤∑i=13−2λi212πi21/2−4λi312πi22−σ2iθi4−1θi4ni212p~i22−σ1iγini12p~i21/2+M,where M=∑i=13εi+kδi+3σ2iθi4/3/4ni2pi34/3+σ1iγi2/4+σ1i/2ni+1/4λi1pi2.Then, one has(26)V̇d≤−μ1Vd1/2−μ2Vd2+M≤0.According to the above analysis, ifVd≥M/μ2, then V̇d≤−μ1Vd1/2≤0, we can have that Vd will converge to the set Vd:Vd≤M/μ2 in a fixed-time Td≤2/μ1+1/μ2, and thus, estimated error p~i and auxiliary states πi are uniformly ultimately bounded in a fixed time Td≤2/μ1+1/μ2.


## 4. Controller Design

In this section, a fixed-time fault-tolerant controller without fractional power terms scheme is proposed to eliminate the singular value problem for a fixed-wing UAV against actuator fault. To prevent the phenomenon of “differential explosion,” a new fixed-time dynamic surface filter is added in the backstepping design.


### 4.1. Fixed-Time Controller and Filter Design

The tracking errors are defined as(27)z1=x1−x1c,z2=x2−x2c,z3=x3−x3c,where zi is the tracking error, and xic is the fixed-time filter output signal, i=1,2,3,γd=x1c.The filter errors are defined as(28)y2=x2c−x2d,y3=x3c−x3d,where xid is the virtual control signal and yi is the error of xic, xid with yi0=0,i=2,3.Step 1. According to (27) and (28), we have
(29)ż1=g1z2+y2+x2d+f1+d1−ẋ1c.The virtual control signal is designed as(30)x2d=1g1−f1−d^1−k11z1z12+εz1/k112−k12z13−k132+g122kf23z1+ẋ1c,where k11,k12,k13,kf23,εz1 are positive constants.Remark 11.
Different from the conventional fixed-time algorithm with fractional power terms presented in [22], the proposed fixed-time controllers do not contain fractional power terms, which overcome the potential singularity problem often caused by the repeated derivation of the virtual control law in fixed-time backstepping designs.The Lyapunov function candidate is considered to be(31)V1=12z12.The time derivative ofV1 can be represented as
(32)V̇1=z1d~1−k11z12z12+εz1/k112−k12z14−k132+g122kf23z12+z1z2g1+z1g1y2.By applying Young’s inequality and Lemma4, one has
(33)z1d~1≤k13z122+12k13d~12,−k11z12z12+εz1/k112≤−k11z1+εz1,z1g1y2≤g122kf23z12+kf232y22.Then, one has(34)V̇1≤12k13d~12−k11z1+εz1−k12z14+z1z2g1+kf232y22.Step 2. Using (27) and (28), we have
(35)ż2=g2z3+y3+x3d+f2+d2−ẋ2c.The virtual control signal is designed as(36)x3d=1g2−f2−d^2−k21z2z22+εz2/k212−k22z23−k232+g222kf33z2+ẋ2c−g1z1,where k21,k22,k23,kf33,εz2 are positive constants.The fixed-time filter is designed as(37)ẋ2c=−kf21y2y22+εy2/kf212−kf22y23−kf23y2,where kf21,kf22,kf23,εy2 are positive constants.The Lyapunov function candidate is considered as(38)V2=12z22+12y22.The time derivative ofV2 can be represented as
(39)V̇2=z2d~2−k21z22z22+εz2/k212−k22z24−k232+g222kf33z22−z1z2g1+z2z3g2+z2g2y3−kf21y22y22+εy2/kf212−kf22y24−kf23y22−ẋ2dy2.By applying Young’s inequality and Lemma4, we have
(40)z2d~2≤k232z22+12k23d~22,−k21z22z22+εz2/k212≤−k21z2+εz2,−kf21y22y22+εy2/kf212≤−kf21y2+εy2,z2g2y3≤g222kf33z22+kf332y32.Then, one has(41)V̇2≤12k23d~22−k21z2+εz2−k22z24−z1z2g1+z2z3g2−kf21y2+εy2−kf22y24−kf23y22+kf332y32−ẋ2dy2.Step 3. According to (27) and (28), we have
(42)ż3=g3u0+f3+d3−ẋ3c.The virtual control signal is designed as(43)u0=1g3−f3−d^3−k31z3z32+εz3/k312−k32z33−k33z3+ẋ3c−g2z2,where k31,k32,k33,εz3 are positive constants.The fixed-time filter is designed as(44)ẋ3c=−kf31y3y32+εy3/kf312−kf32y33−kf33y3,where kf31,kf32,kf33,εy3 are positive constants.Consider the following Lyapunov function candidate(45)V3=12z32+12y32.The time derivative ofV3 can be represented as
(46)V̇3=z3d~3−k31z32z32+εz3/k312−k32z34−k33z32−z2z3g2−kf31y32y32+εy3/kf312−kf32y34−kf33y32−ẋ3dy3.Similar to the analysis of (33), (40), one has
(47)V̇3≤14k33d~32−k31z3+εz3−k32z34−z2z3g2−kf31y3+εy3−kf32y34−kf33y32−ẋ3dy3.


### 4.2. Stability Analysis

Theorem 12.
Consider system (5), and assume that Assumptions 1 and 2 hold. The disturbance observer is designed as (14). The fixed-time controller is built as (30), (36), and (43), and dynamic filter is constructed as (37) and (44). For a given constant P1>0, if V0≤P1 and there exist the following positive constants ki1,ki2,ki3,εzi,σ1i,σ2i,ni,λi1,λi2,λi3,δi,εi,θi,γi for i=1,2,3 and kfi1,kfi2,kfi3,εyi for i=2,3, then, reference signal tracking errors zi and fixed-time filter errors yi converge to the neighborhood near the origin Ωzi=zi≤2C/μ4,Ωyi=yi≤2C/μ4 in a fixed time Ts≤2/μ3+1/μ4.Remark 13.
By differentiatingx2d,x3d, one can obtain
(48)ẋ2d=−∂x2d∂f1ḟ1−∂x2d∂d^1d^̇1−∂x2d∂z1ż1−∂x2d∂y2ẏ2+∂x2d∂g1ġ1+∂x2d∂ẋ1cx¨1c,ẋ3d=−∂x3d∂f2ḟ2−∂x3d∂d^2d^̇2−∂x3d∂z2ż2−∂x2d∂z1ż1−∂x3d∂y3ẏ3+∂x2d∂g1ġ1+∂x2d∂g2ġ2+∂x3d∂ẋ2cx¨2c.Because each partial ofẋ2d and ẋ3d is a continuous on a compact set ΩV×ΩVd, it has
(49)−ẋidyi≤ẋidyi≤kfi3yi22+M¯id22kfi3, where M¯id denotes a positive constant with M¯id≥ẋid,i=2,3.Proof.
The Lyapunov function candidate is considered to be(50)V=∑i=13Vi.The time derivative ofV can be represented as
(51)V̇=∑i=13ziżi+∑i=23yiẏi≤∑i=13−2ki112zi21/2−4ki212zi22+∑i=23−2kfi112yi21/2−4kfi212yi22+Cwhere C=∑i=131/2ki3d~i2−1/4k33d~32+εzi+∑i=23εyi+M¯id2/2kfi3.Equation (51) can be further transformed as
(52)V̇≤−μ3V1/2−μ4V2+C,where μ3 and μ4 can be expressed as μ3=min2k11,2ki1,2kfi1,μ4=min4k12,4ki2,4kfi2,i=2,3.Note that, from (52), if V≥C/μ4, then V̇≤−μ3V1/2≤0, which validates the boundedness of V. The boundedness of V means the boundedness of zi for i=1,2,3 and yi for i=2,3. Naturally, it can be shown that the derivative of fixed-time filter output signal ẋic is bounded because of the boundedness of yi. Then, we can get that the virtual control signal xid is bounded because yi, z1, and ẋic are both bounded.According to above analysis, one can get thatV will converge to the set V:V≤C/μ4 in fixed-time Ts≤2/μ3+1/μ4. Moreover, we have that the tracking errors zi and the filter errors yii=1,2,3 will converge to the sets
(53)Ωzi=zi≤2Cμ4,Ωyi=yi≤2Cμ4.


## 4.1. Fixed-Time Controller and Filter Design

The tracking errors are defined as(27)z1=x1−x1c,z2=x2−x2c,z3=x3−x3c,where zi is the tracking error, and xic is the fixed-time filter output signal, i=1,2,3,γd=x1c.The filter errors are defined as(28)y2=x2c−x2d,y3=x3c−x3d,where xid is the virtual control signal and yi is the error of xic, xid with yi0=0,i=2,3.Step 1. According to (27) and (28), we have
(29)ż1=g1z2+y2+x2d+f1+d1−ẋ1c.The virtual control signal is designed as(30)x2d=1g1−f1−d^1−k11z1z12+εz1/k112−k12z13−k132+g122kf23z1+ẋ1c,where k11,k12,k13,kf23,εz1 are positive constants.Remark 11.
Different from the conventional fixed-time algorithm with fractional power terms presented in [22], the proposed fixed-time controllers do not contain fractional power terms, which overcome the potential singularity problem often caused by the repeated derivation of the virtual control law in fixed-time backstepping designs.The Lyapunov function candidate is considered to be(31)V1=12z12.The time derivative ofV1 can be represented as
(32)V̇1=z1d~1−k11z12z12+εz1/k112−k12z14−k132+g122kf23z12+z1z2g1+z1g1y2.By applying Young’s inequality and Lemma4, one has
(33)z1d~1≤k13z122+12k13d~12,−k11z12z12+εz1/k112≤−k11z1+εz1,z1g1y2≤g122kf23z12+kf232y22.Then, one has(34)V̇1≤12k13d~12−k11z1+εz1−k12z14+z1z2g1+kf232y22.Step 2. Using (27) and (28), we have
(35)ż2=g2z3+y3+x3d+f2+d2−ẋ2c.The virtual control signal is designed as(36)x3d=1g2−f2−d^2−k21z2z22+εz2/k212−k22z23−k232+g222kf33z2+ẋ2c−g1z1,where k21,k22,k23,kf33,εz2 are positive constants.The fixed-time filter is designed as(37)ẋ2c=−kf21y2y22+εy2/kf212−kf22y23−kf23y2,where kf21,kf22,kf23,εy2 are positive constants.The Lyapunov function candidate is considered as(38)V2=12z22+12y22.The time derivative ofV2 can be represented as
(39)V̇2=z2d~2−k21z22z22+εz2/k212−k22z24−k232+g222kf33z22−z1z2g1+z2z3g2+z2g2y3−kf21y22y22+εy2/kf212−kf22y24−kf23y22−ẋ2dy2.By applying Young’s inequality and Lemma4, we have
(40)z2d~2≤k232z22+12k23d~22,−k21z22z22+εz2/k212≤−k21z2+εz2,−kf21y22y22+εy2/kf212≤−kf21y2+εy2,z2g2y3≤g222kf33z22+kf332y32.Then, one has(41)V̇2≤12k23d~22−k21z2+εz2−k22z24−z1z2g1+z2z3g2−kf21y2+εy2−kf22y24−kf23y22+kf332y32−ẋ2dy2.Step 3. According to (27) and (28), we have
(42)ż3=g3u0+f3+d3−ẋ3c.The virtual control signal is designed as(43)u0=1g3−f3−d^3−k31z3z32+εz3/k312−k32z33−k33z3+ẋ3c−g2z2,where k31,k32,k33,εz3 are positive constants.The fixed-time filter is designed as(44)ẋ3c=−kf31y3y32+εy3/kf312−kf32y33−kf33y3,where kf31,kf32,kf33,εy3 are positive constants.Consider the following Lyapunov function candidate(45)V3=12z32+12y32.The time derivative ofV3 can be represented as
(46)V̇3=z3d~3−k31z32z32+εz3/k312−k32z34−k33z32−z2z3g2−kf31y32y32+εy3/kf312−kf32y34−kf33y32−ẋ3dy3.Similar to the analysis of (33), (40), one has
(47)V̇3≤14k33d~32−k31z3+εz3−k32z34−z2z3g2−kf31y3+εy3−kf32y34−kf33y32−ẋ3dy3.


## 4.2. Stability Analysis

Theorem 12.
Consider system (5), and assume that Assumptions 1 and 2 hold. The disturbance observer is designed as (14). The fixed-time controller is built as (30), (36), and (43), and dynamic filter is constructed as (37) and (44). For a given constant P1>0, if V0≤P1 and there exist the following positive constants ki1,ki2,ki3,εzi,σ1i,σ2i,ni,λi1,λi2,λi3,δi,εi,θi,γi for i=1,2,3 and kfi1,kfi2,kfi3,εyi for i=2,3, then, reference signal tracking errors zi and fixed-time filter errors yi converge to the neighborhood near the origin Ωzi=zi≤2C/μ4,Ωyi=yi≤2C/μ4 in a fixed time Ts≤2/μ3+1/μ4.Remark 13.
By differentiatingx2d,x3d, one can obtain
(48)ẋ2d=−∂x2d∂f1ḟ1−∂x2d∂d^1d^̇1−∂x2d∂z1ż1−∂x2d∂y2ẏ2+∂x2d∂g1ġ1+∂x2d∂ẋ1cx¨1c,ẋ3d=−∂x3d∂f2ḟ2−∂x3d∂d^2d^̇2−∂x3d∂z2ż2−∂x2d∂z1ż1−∂x3d∂y3ẏ3+∂x2d∂g1ġ1+∂x2d∂g2ġ2+∂x3d∂ẋ2cx¨2c.Because each partial ofẋ2d and ẋ3d is a continuous on a compact set ΩV×ΩVd, it has
(49)−ẋidyi≤ẋidyi≤kfi3yi22+M¯id22kfi3, where M¯id denotes a positive constant with M¯id≥ẋid,i=2,3.Proof.
The Lyapunov function candidate is considered to be(50)V=∑i=13Vi.The time derivative ofV can be represented as
(51)V̇=∑i=13ziżi+∑i=23yiẏi≤∑i=13−2ki112zi21/2−4ki212zi22+∑i=23−2kfi112yi21/2−4kfi212yi22+Cwhere C=∑i=131/2ki3d~i2−1/4k33d~32+εzi+∑i=23εyi+M¯id2/2kfi3.Equation (51) can be further transformed as
(52)V̇≤−μ3V1/2−μ4V2+C,where μ3 and μ4 can be expressed as μ3=min2k11,2ki1,2kfi1,μ4=min4k12,4ki2,4kfi2,i=2,3.Note that, from (52), if V≥C/μ4, then V̇≤−μ3V1/2≤0, which validates the boundedness of V. The boundedness of V means the boundedness of zi for i=1,2,3 and yi for i=2,3. Naturally, it can be shown that the derivative of fixed-time filter output signal ẋic is bounded because of the boundedness of yi. Then, we can get that the virtual control signal xid is bounded because yi, z1, and ẋic are both bounded.According to above analysis, one can get thatV will converge to the set V:V≤C/μ4 in fixed-time Ts≤2/μ3+1/μ4. Moreover, we have that the tracking errors zi and the filter errors yii=1,2,3 will converge to the sets
(53)Ωzi=zi≤2Cμ4,Ωyi=yi≤2Cμ4.


## 5. Numerical Simulations

In this section, comparative method [21] is used to verify the effectiveness of the proposed composite control scheme. The detailed parameters of the longitudinal model can refer to [27].The controller parameters are designed as(54)k11=k21=k31=0.1,k12=k22=k32=0.1,εz1=εz2=εz3=0.01,k13=0.2,k23=0.7,k33=0.4,kf21=10,kf22=1,kf23=10,kf31=1,kf32=0.001,kf33=10,εy2=0.1,εy3=0.0001,θ1=θ2=θ3=1.1,λ11=10,λ12=0.5,λ13=0.5,λ21=10,λ22=0.5,λ23=0.5,λ31=10,λ32=0.5,γ1=γ2=γ3=0.1,λ33=0.5,ε1=ε2=ε3=0.5,δ1=δ2=δ3=0.5,σ11n1=σ12n2=σ13n3=10,σ21n12=σ22n22=σ23n32=0.01.The reference signalxd is generated through the following filter:
(55)γds=5s+5yds.The initial state of the system is thatx10=0,x20=0,x30=0 and the disturbance signal including wind disturbance [32] is defined as
(56)Δγ=ω̇h,Δα=ω̇h30−ω̇x30,Δq=ω̇x,where ωx,ωh are wind components.The reference signal and actuator fault model are designed as(57)yd=3,t<205,t≥20,p1=1,bf1=0,t<20p1=0.7,bf1=0.07,t≥20.Case 14.
Against actuator fault and slight external disturbances.In this simulation, external wind disturbances are considered to verify the robustness of the proposed composite control scheme. Figure1 gives the wind disturbances. Figure 2 shows the estimated lumped disturbances. One can find that the compared DO and the proposed DO are both able to estimate lumped disturbances including actuator fault and external unmatched disturbances. However, the compared DO exhibits a larger disturbance estimation error. Curves of elevator deflection are shown as Figure 3. Figures 4 and 5 illustrate the tracking errors of flight-path angle, angle of attack, and pitch rate. It can be found that flight-path angle, angle of attack, and pitch rate react reasonably. The proposed controller exhibits smaller tracking errors than the compared method, when the actuator is fault at t=20s.Case 15.
Against actuator fault and severe external disturbances.Figure 1
The curves of wind disturbance.Figure 2
Estimation ofd1 and d2.Figure 3
Estimation ofd3 and control input.Figure 4
Tracking errore1 and the trajectory of γ.Figure 5
Tracking errorse2 and e3.In this simulation, severe wind disturbances are imposed on the UAV. The other conditions are kept as Case14. Figure 6 gives the curves of the wind disturbances. Figures 7 and 8 illustrate the lumped disturbances estimation. Figures 9 and 10 describe the tracking errors. It can be seen that the disturbance estimation errors of the proposed DO are much smaller than the compared method. It is validated that the proposed controller operates well even though there exist the reinforced wind disturbance and actuator fault.Figure 6
The curves of wind disturbance.Figure 7
Estimation ofd1 and d2.Figure 8
Estimation ofd3 and control input.Figure 9
Tracking errore1 and trajectories of γ.Figure 10
Tracking errorse2 and e3.


## 6. Conclusions

In this paper, the fixed-time fault-tolerant tracking control problem for the fixed-wing UAVs subject to external disturbances and actuator fault has been addressed. To estimate the lumped disturbance with unknown upper bound precisely, a new disturbance observer has been proposed. A new fixed-time fault-tolerant controller without fractional power terms has been proposed in this paper, which overcomes the potential singularity problem often encountered in fixed-time backstepping design procedures. Moreover, a new fixed-time dynamic surface filter has been added in the controller design to prevent the phenomenon of “differential explosion.” Lyapunov stability analysis has proven the stability of the proposed control scheme. The simulation results verify the effectiveness of the proposed method.



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*Source: 6447812-2022-11-21.xml*