# Improved Adaptive Fault-Tolerant Control of a Variable Structure Fighter with Multiple Faults Based on an Extended Observer

Authors: Kaiyu Hu; Wenjing Sun; Chunxia Yang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, an improved adaptive fault-tolerant control (FTC) scheme is presented based on an extended state observer (ESO) for a highly maneuverable variable structure fighter (VSF) with simultaneous faults of the rudder surface and sensor. The feedback linearization model of the attitude system with faults and external disturbances is introduced. In order to realize the real-time observation of the rudder and sensor faults, a neural network estimator is used to approximate sensor faults, and the ESO is designed to estimate the rudder surface fault and attitude variables. On this basis, the estimated value is used to replace the measured value for state variables to design the adaptive fault-tolerant controller for the inner and outer loop, combined with the second-order command filter to offset the phenomenon of backstepping differential amplification system noise. Finally, the proposed FTC scheme realizes the variable structure self-repairing of multiple positions, multiple types of faults in VSF using fault estimation. The Lyapunov theory and Barbalat’s lemma prove the stability of the designed observer and controller. The effectiveness of the FTC scheme is verified by a simulation experiment.


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## Body

## 1. Introduction

As the core of next-generation airpower, the variable structure fighter (VSF) has significant advantages such as ultra-maneuverability, ultrahigh speed, and wide flight airspace [1]. However, the strong nonlinearity, uncertainty, and fast time-varying characteristics of VSF during its maneuvering process under variable structure perturbation, high attack angle, and angular velocity may increase the difficulty of controller design [2]. With the increasing complexity of flight environments, more and more physical components such as actuators and sensors are involved, the flight control system is prone to combat damage or fault during flight [3], and the faults of different mechanisms are coupled and transitive, leading to a sharp drop in their flight status and affecting the overall stability of the system. In order to improve the reliability of the VSF’s large maneuvering process and ensure the flight performance of the system when a sudden fault occurs, the design of an effective fault-tolerant control (FTC) scheme is the key and prerequisite for the safe operation of the flight control system [4–6].Rudder is an important executive mechanism of a VSF, which is subjected to multiple constraints such as deflection range and deflection rate. However, the traditional nominal control does not consider the rudder failure systems [7–9]. Due to the exposure of the VSF rudder surface to air flow, it is easy to have partial or complete fault of the rudder surface [10, 11] in a high altitude and high-speed flight environment, especially in the case of multiple rudder faults when the VSF is making a large maneuver. It seriously affects the handling performance of the aircraft and even endangers the safety of the system [12]. In recent years, many scholars at home and abroad have conducted in-depth studies on the FTC methods under different rudder faults of aircraft, and have made a series of progress [13–16]. Singh et al. proposed an FTC scheme based on terminal sliding mode to solve the problem of rudder fault in the super-maneuvering flight of the thrust vector aircraft with parameter perturbation and realized fast compensation for the perturbation term while accurately estimating the rudder fault [17]. Ma Guanfu et al. applied the height fault-tolerant controller based on backstepping and the velocity tracking controller based on dynamic inverse to the longitudinal model of the hypersonic aircraft with a redundant rudder surface and realized the effective FTC for the stuck fault of rudder [18]. Yu Xiang et al. proposed an L1 compensation method with matching/nonmatching uncertainties to ensure the transient and steady-state boundedness of all parameters of the system in view of the unmodeled dynamics, uncertain parameters of the system, unknown input gain, and other problems caused by the aircraft rudder fault [19]. Niu et al. designed an adaptive step-back FTC scheme based on the fuzzy system for the hypersonic vehicle with parameter uncertainty and actuator gain loss faults and realized the stable tracking of speed and height [20].All the schemes above have achieved good control effects, but most of them only considered the rudder fault of the aircraft. However, in the actual flight control system, some state variable is difficult to be directly measured by the sensor, and in the harsh flight environment, the sensor is prone to fault or the measurement accuracy is not high [21], resulting in the system disability of direct usage of the measured data. Complex flight missions of the fighter make the probability of sensor fault greater than that of the rudder surface. Therefore, a hybrid FTC method is of great research value in the case of simultaneous fault of the aircraft rudder and sensor [22–24]. At present, some research achievements have been made in FTC of the aircraft under the condition of multiple faults. Chen et al. designed an adaptive robust fault-tolerant controller based on multidimensional-generalized observers for the longitudinal system of the hypersonic vehicle with sensor cascading faults, which solved the ineffectiveness of ordinary generalized observers in cascading fault estimation due to coupling effects and realized the estimation and compensation of the sensor cascading fault [25]. Zhang Jing et al. introduced an adaptive backstepping fault-tolerant controller based on a dynamic high-gain observer for a multiple input multiple output (MIMO) nonlinear system with actuator and sensor faults, which ensures the global stability of the system tracking error [26]. Chen et al. designed a self-healing control method based on an adaptive and backstepping sliding mode fault estimation observer for the situation of multisensor faults in the height and velocity channels of the hypersonic vehicle longitudinal model, the estimated value of the fault was used to compensate the sliding mode controller to make the original system track the given reference command stably [27].Based on the above discussion, this paper adopts the fighter with variable structure parameters (VSF) to study the variable structure FTC based on the extended state observer (ESO) for the nonlinear VSF system with disturbances, rudder, and sensor faults. First, based on typical fighter nonlinear equations, a feedback linearization model with variable structure parameters, rudder, and sensor multiple faults is established. Then, the radial basis function (RBF) estimator is designed to approximate the sensor fault, an adaptive parameter update law is designed to estimate the rudder fault online, and the stuck fault and the uncertain items formed by the disturbance are estimated through the ESO. Finally, the adaptive backstepping FTC with variable structure harmonic parameters based on the command filter is designed, which compensates for multiple faults while ensuring the robustness and achieves the stable tracking of the high attack angles in VSF. The contributions of the paper can be summarized as follows:(1)
Investigating FTC schemes for the VSF with unknown control parameters. The designed ESO based on the RBF estimator quickly observes the state without knowing the precise control efficiency matrix. The estimation accuracy of the attitude angle and angular velocity can be achieved when the rudder and sensor fail simultaneously.(2)
The neural network approximation of sensor fault and the precise observation of rudder fault are realized simultaneously, and the hybrid variable structure FTC is designed by fusing multiple fault information to realize the compensation of all faults and the stable attitude tracking of the VSF.(3)
For the first time in academia, an FTC scheme is proposed, which solves the actuator failure fault, actuator deviation fault, sensor failure fault, and sensor deviation fault at the same time; it opens the era of the grand unification of the FTC theory.The rest of this paper is organized as follows. Section2 introduces the VSF nonlinear model with the twelve-state variable, rudder fault model, sensor fault model, and multiple fault feedback linearization model. Section 3 shows the design process of an ESO for the nonlinear system with rudder fault, sensor fault, and disturbance. Section 4 presents an improved adaptive backstepping fault-tolerant controller based on the estimation results of the ESO. Section 5 shows the simulation results of the proposed FTC scheme and verifies its effectiveness. Section 6 summarizes the research content and future research prospects.


## 2. Models and Compensation Objective

In this section, based on the typical fighter model, the VSF model with rudder fault, sensor fault, and variable structure parameters is introduced.


### 2.1. Nonlinear Dynamical Model

This paper refers to the six-degree-of-freedom, twelve-state variable model of the traditional fighter published by NASA Langley Research Center in [28], and the equations with variable structure parameters are described as (1).In equation (1), x, y, and z denote the projection of the center of mass of the aircraft onto the Earth shafting; u, v, and w denote the velocity components of aircraft-body coordinate frame; ϕ, θ, and ψ denote the roll angle, pitch angle, and yaw angle, respectively; and p, q, and r denote the roll rate, pitch rate, and yaw rate, respectively; α is the attack angle, β is the sideslip angle, and c1∼c9 denote the rotational inertia coefficients, c1,Δ∼c9,Δ are the inertia increase values generated by changing the structure of the fuselage, that is, the variable structure parameters; L0, M0, and N0 represent the aerodynamic moments for roll, pitch, and yaw, respectively; m is the vehicle mass; FT is the engine thrust, Fy and Fz are the y-axis thrust and z-axis thrust, respectively; MT0 is the pitching moment caused by the longitudinal thrust component Fz, NT0 is the yawing moment generated by the lateral thrust component Fy; g1, g2, and g3 represent the gravitational acceleration components, respectively; D, L, and Y denote the drag, lift, and lateral force, respectively. The specific expressions of above parameters are presented in [28].(1)x˙=ucosψcosθ+vcosψsinθsinϕ−sinψcosϕ+wcosψsinθcosϕ+sinψsinϕ,y˙=usinψcosθ+vsinψsinθsinϕ+cosψcosϕ+wsinψsinθcosϕ−cosψsinϕ,z˙=−usinθ+vcosθsinϕ+wcosθcosϕ,ϕ˙=p+tanθqsinϕ+rcosϕ,θ˙=qcosϕ−rsinϕ,ψ˙=1cosθqsinϕ+rcosϕ,p˙=c1+c1,Δqr+c2+c2,Δpq+c3+c3,ΔL0+c4+c4,ΔN0+NT0,q˙=c5+c5,Δpr−c6+c6,Δp2−r2+c7+c7,ΔM0+MT0,r˙=c8+c8,Δpq−c2+c2,Δqr+c4+c4,ΔL0+c9+c9,ΔN0+NT0,V˙=1mFTcosαcosβ−D+mg1+Fysinβ+Fzcosβsinα,α˙=rsinα−pcosαtanβ+q+1mVTcosβ−L+mg2−FTsinα+Fzcosα,β˙=psinα−rcosα+1mVTmg3−FTcosαsinβ+Y+Fycosβ+Fzsinβsinα.Defining state variables asx1=αβϕT,x2=pqrT, control input variables as u = (δe, δa, δr, δy, δz)T, considering the variable fuselage, the affine nonlinear VSF model of the inner and outer loops of the system without fault is expressed as follows:(2)x˙1=f1x1+g1x1x2+d1x˙2=f2x2+g2x2+gΔu+d2y=x1,where f1 (x1) and f2 (x2) are the nonlinear coupling matrix of outer loop and inner loop, g1 (x1) and g2 (x2) denote the input distribution matrix of outer loop and inner loop, gΔ is the perturbation parameter reflecting the variable structure amplitude of the fuselage, d1 and d2 represent the unknown disturbances, δe, δa, and δr denote elevator, aileron and rudder, respectively, δy is the thrust vector generated by the lateral rudder, and δz is the thrust vector generated by side rudder. The system output y = (α, β, ϕ)T is measured by the attitude angle sensors.Assumption 1.
Matrixg1 (x1) and g2 (x2) are nonsingular.Assumption 2.
Disturbance signalsd1 and d2 are derivable and bounded, satisfied by ǀǀdiǀǀ≤d∗, i = 1, 2, where d∗ is the unknown positive constant.


### 2.2. Rudder Surface Fault

The deflection angle of the fighter rudder surface is driven by the rudder. The second-order rudder system model is considered as follows:(3)δ˙i=δviδ˙vi=−ωni2δi−2ξiωniδvi+ωni2δci,where i = (a, e, r), δvi is rudder deflection rate, δci is the reference command for actual output δi of the rudder surface, ωni is the servo frequency, and ξi is the damping ratio of rudder. Since the deflection angle of the thrust vectoring rudder is mainly affected by the thrust provided by the thrust vectoring engine, the fault of the thrust vectoring rudder is not considered in this work.Assumption 3.
2ξ1ωni≪ωni2,1≪ωni2, i.e., the damping of the rudder is much smaller than the natural frequency, and the steering gear can achieve a faster response speed.
The fault of the rudder can be manifested as an abnormal deflection angle of the rudder surface. Common rudder faults include lock-in-place (LIP) faults, hard-over-fault (HOF), float faults, and loss-of-effectiveness (LOE), where LIP faults, HOF, and float faults are total LOE, and HOF and float faults are special cases of LIP faults. Based on the second-order rudder system (2), rudder fault model is described as follows:(4)δ˙i=liδviδ˙vi=−2ξiωniδvi+ωni2liλiδci−δi,where 0 ≤ λi ≤ 1 represents unknown rudder fault coefficient, li ∈ (0, 1) denotes whether the rudder LIP faults happen, li = 0 represents LIP happens, while li = 1 represents it does not. Substituting li2 = li into (4) yields the following:(5)δ¨i=liδ˙vi=−2ξiωniliδvi+ωni2li2λiδci−δi=−2ξiωniδ˙i−ωni2δi+ωni2liλiδci+1−liδi.
In accordance with Assumption3,(6)2ξiωniωni2≈0,1ωni2≈0,then, by dividing both sides of (5) by ωni2, the rudder fault model (5) can be simplified as follows:(7)δi=liλiδci+1−liδi.
Consideringli ∈ (0, 1), we have the following:(8)1−liδi−δitl=0,where δi (tl) denotes the rudder deflection angle at time tl when LIP on the rudder surface δi occurs, tl denotes the time of fault, substituting (7) into (6) yields as follows:(9)δi=liλiδci+1−liδitl.
Considering that LIP and LOE cannot occur simultaneously on one rudder surface, we rewrite (8) as below:(10)δi=λiδci+δitlλiδitl=0,Then, the equation of the rudder fault is constructed as follows:(11)δ=Γu+Ltl,where u = (δe, δa, δr, δy, δz)T denotes the control signal of rudder deflection angle, the rudder fault coefficient matrix is denoted as Γ = diag (λa, λe, λr, 1, 1), L (tl) = (δa(tl), δe(tl), δr(tl), 0, 0)T denotes rudder LIP coefficient matrix.


### 2.3. Senor Fault

Gain fault and bias fault are two common sensor faults, where bias faults include the drift bias fault and fixed bias fault. The sensor fault of the outer loop attitude angle can be expressed as follows:(12)yf=Gfx1+Bft,where yf denotes the actual measured value of attitude angles, Gf=diaggα,gβ,gϕ. denotes sensor gain coefficient matrix, 0 ≤ gi ≤ 1 denotes unknown gain coefficient, the sensor bias coefficient matrix is described as Bft=bαtbβtbϕtT., bfmin<bi<bfmax, denotes unknown bounded variable, i = (α, β, ϕ).In accordance with (12), by combining different values of the gain matrix and the bias matrix, the sensor fault types can be summarized as shown in Table 1.Table 1
Senor faults.
Faults formulaFault coefficientFault typeyf=x1gi=1,bit=0No sensor faultyf=Gfx10≤gi≤1,bit=0Senor gain faultyf=x1+Bftgi=1,bfmin<bi<bfmaxSenor drift bias faultyf=Bftgi=0,b˙it=0Senor fixed bias faultRemark 1.
The physical manifestation of LOE fault is the decrease of the deflection speed of the rudder surface; the main cause is the increase of the rotational friction force caused by the deformation of the motor shaft. The physical manifestation of the LIP fault is that the deflection angle of the rudder surface does not reach the target angle; the main cause is that the unstable airflow compresses the rudder surface or the deflection angle is limited due to the deformation of the mechanism close to the rudder surface. The physical manifestation of the sensor gain fault is that the signal is sluggish, for which the main reason is the feedback distortion caused by the electromagnetic interference of the enemy. The physical manifestation of sensor bias fault is the nonnegligible error between the feedback amplitude and the real output amplitude, and the main cause is the partial destruction of components due to temperature changes.


### 2.4. Multifault Model and Control Object

Combining equations (2), (11) and (12), the VSF attitude system model with rudder fault and sensor fault can be established as follows:(13)x˙1=x2+F1x+d1x˙2=f2x2+G2+GΔu+d2x2y=x1+sx1,where(14)F1x=f1x1+g1x1−Ix2G2+GΔ=g2x2+gΔΓΓ=g2x2Γ+gΔΓ,denotes unknown input distribution matrix of rudder surface with LOE, and the comprehensive disturbance(15)d2x2=g2x2+gΔLtl+d2=g2x2Ltl+gΔLtl+d2,is composed of rudder LIP fault and the disturbance of the attitude angular velocity channel,(16)sx1=Gf−Ix1+Bf,denotes senor fault, and I is the unit matrix with dimension 3 × 3.The multifault model (12) classifies the processing of different types of faults. For example, the rudder LIP fault is regarded as a part of external disturbance, while the rudder LOE is added because of the structural damage of the input distribution matrix. The sensor fault is reflected in the control output signal, which facilitates the design of subsequent FTC schemes by classifying the fault.Assumption 4.
When no LIP fault occurs, the comprehensive disturbanced2 (x2) is derivable and its derivative is bounded, i.e., d2x2<ρ∗ when t≠t1, where ρ∗ is an unknown positive constant.Assumption 5.
Reference signalx1d of x1 is bounded and derivable, and there exists a compact set(17)Σ=x1d,x˙1d,x¨1d|x1d2+x˙1d2+x¨1d2≤σ2,where σ is the unknown positive constant.Assumption 6.
The nonlinearF1 (x), f2 (x), and s (x) are uniformly bounded, and for any x and its estimation (∀x,x^∈!n) satisfying the Lipschitz condition, namely,(18)F1x−F1x^≤k1x−x^f2x−f2x^≤k2x−x^sx−sx^≤k3x−x^,where k1, k2, k3 > 0 denote the Lipschitz constant corresponding to the nonlinear function.
This paper studies the variable structure FTC of the nonlinear super-maneuvering VSF system. The control objective is as follows: when the VSF has a large maneuver and both a rudder fault and sensor fault, a FTC law should compensate for multiple faults and disturbance to the attitude angle, and the influence of the attitude angular velocity and the rudder deflection channel ensure the closed loop stability of system (13). To achieve the above control objectives, several key issues needed to be resolved:(1)
Due to the increase in the number of controllable rudders, the fault of multiple rudders may cause the control efficiency matrix of the system to be inaccurately known. How to design parameter estimation algorithms and FTC scheme for different types of rudder faults if the system control gain is not fully known.(2)
Sensor faults cause the attitude angle and velocity state variables output by the system to be unobservable. How to design fault estimation and FTC schemes to observe and compensate the system state in case of sensor fault is not known.(3)
The control variables such as the rudder deflection angle may be unconstrained when both rudder fault and sensor fault occur. How to improve the traditional variable structure control method to avoid the differential link amplifying the system noise and achieve stable tracking of reference signals is not known.
The overall control block diagram of the system is shown in Figure1.Figure 1
VSF system overall control block diagram.


## 2.1. Nonlinear Dynamical Model

This paper refers to the six-degree-of-freedom, twelve-state variable model of the traditional fighter published by NASA Langley Research Center in [28], and the equations with variable structure parameters are described as (1).In equation (1), x, y, and z denote the projection of the center of mass of the aircraft onto the Earth shafting; u, v, and w denote the velocity components of aircraft-body coordinate frame; ϕ, θ, and ψ denote the roll angle, pitch angle, and yaw angle, respectively; and p, q, and r denote the roll rate, pitch rate, and yaw rate, respectively; α is the attack angle, β is the sideslip angle, and c1∼c9 denote the rotational inertia coefficients, c1,Δ∼c9,Δ are the inertia increase values generated by changing the structure of the fuselage, that is, the variable structure parameters; L0, M0, and N0 represent the aerodynamic moments for roll, pitch, and yaw, respectively; m is the vehicle mass; FT is the engine thrust, Fy and Fz are the y-axis thrust and z-axis thrust, respectively; MT0 is the pitching moment caused by the longitudinal thrust component Fz, NT0 is the yawing moment generated by the lateral thrust component Fy; g1, g2, and g3 represent the gravitational acceleration components, respectively; D, L, and Y denote the drag, lift, and lateral force, respectively. The specific expressions of above parameters are presented in [28].(1)x˙=ucosψcosθ+vcosψsinθsinϕ−sinψcosϕ+wcosψsinθcosϕ+sinψsinϕ,y˙=usinψcosθ+vsinψsinθsinϕ+cosψcosϕ+wsinψsinθcosϕ−cosψsinϕ,z˙=−usinθ+vcosθsinϕ+wcosθcosϕ,ϕ˙=p+tanθqsinϕ+rcosϕ,θ˙=qcosϕ−rsinϕ,ψ˙=1cosθqsinϕ+rcosϕ,p˙=c1+c1,Δqr+c2+c2,Δpq+c3+c3,ΔL0+c4+c4,ΔN0+NT0,q˙=c5+c5,Δpr−c6+c6,Δp2−r2+c7+c7,ΔM0+MT0,r˙=c8+c8,Δpq−c2+c2,Δqr+c4+c4,ΔL0+c9+c9,ΔN0+NT0,V˙=1mFTcosαcosβ−D+mg1+Fysinβ+Fzcosβsinα,α˙=rsinα−pcosαtanβ+q+1mVTcosβ−L+mg2−FTsinα+Fzcosα,β˙=psinα−rcosα+1mVTmg3−FTcosαsinβ+Y+Fycosβ+Fzsinβsinα.Defining state variables asx1=αβϕT,x2=pqrT, control input variables as u = (δe, δa, δr, δy, δz)T, considering the variable fuselage, the affine nonlinear VSF model of the inner and outer loops of the system without fault is expressed as follows:(2)x˙1=f1x1+g1x1x2+d1x˙2=f2x2+g2x2+gΔu+d2y=x1,where f1 (x1) and f2 (x2) are the nonlinear coupling matrix of outer loop and inner loop, g1 (x1) and g2 (x2) denote the input distribution matrix of outer loop and inner loop, gΔ is the perturbation parameter reflecting the variable structure amplitude of the fuselage, d1 and d2 represent the unknown disturbances, δe, δa, and δr denote elevator, aileron and rudder, respectively, δy is the thrust vector generated by the lateral rudder, and δz is the thrust vector generated by side rudder. The system output y = (α, β, ϕ)T is measured by the attitude angle sensors.Assumption 1.
Matrixg1 (x1) and g2 (x2) are nonsingular.Assumption 2.
Disturbance signalsd1 and d2 are derivable and bounded, satisfied by ǀǀdiǀǀ≤d∗, i = 1, 2, where d∗ is the unknown positive constant.


## 2.2. Rudder Surface Fault

The deflection angle of the fighter rudder surface is driven by the rudder. The second-order rudder system model is considered as follows:(3)δ˙i=δviδ˙vi=−ωni2δi−2ξiωniδvi+ωni2δci,where i = (a, e, r), δvi is rudder deflection rate, δci is the reference command for actual output δi of the rudder surface, ωni is the servo frequency, and ξi is the damping ratio of rudder. Since the deflection angle of the thrust vectoring rudder is mainly affected by the thrust provided by the thrust vectoring engine, the fault of the thrust vectoring rudder is not considered in this work.Assumption 3.
2ξ1ωni≪ωni2,1≪ωni2, i.e., the damping of the rudder is much smaller than the natural frequency, and the steering gear can achieve a faster response speed.
The fault of the rudder can be manifested as an abnormal deflection angle of the rudder surface. Common rudder faults include lock-in-place (LIP) faults, hard-over-fault (HOF), float faults, and loss-of-effectiveness (LOE), where LIP faults, HOF, and float faults are total LOE, and HOF and float faults are special cases of LIP faults. Based on the second-order rudder system (2), rudder fault model is described as follows:(4)δ˙i=liδviδ˙vi=−2ξiωniδvi+ωni2liλiδci−δi,where 0 ≤ λi ≤ 1 represents unknown rudder fault coefficient, li ∈ (0, 1) denotes whether the rudder LIP faults happen, li = 0 represents LIP happens, while li = 1 represents it does not. Substituting li2 = li into (4) yields the following:(5)δ¨i=liδ˙vi=−2ξiωniliδvi+ωni2li2λiδci−δi=−2ξiωniδ˙i−ωni2δi+ωni2liλiδci+1−liδi.
In accordance with Assumption3,(6)2ξiωniωni2≈0,1ωni2≈0,then, by dividing both sides of (5) by ωni2, the rudder fault model (5) can be simplified as follows:(7)δi=liλiδci+1−liδi.
Consideringli ∈ (0, 1), we have the following:(8)1−liδi−δitl=0,where δi (tl) denotes the rudder deflection angle at time tl when LIP on the rudder surface δi occurs, tl denotes the time of fault, substituting (7) into (6) yields as follows:(9)δi=liλiδci+1−liδitl.
Considering that LIP and LOE cannot occur simultaneously on one rudder surface, we rewrite (8) as below:(10)δi=λiδci+δitlλiδitl=0,Then, the equation of the rudder fault is constructed as follows:(11)δ=Γu+Ltl,where u = (δe, δa, δr, δy, δz)T denotes the control signal of rudder deflection angle, the rudder fault coefficient matrix is denoted as Γ = diag (λa, λe, λr, 1, 1), L (tl) = (δa(tl), δe(tl), δr(tl), 0, 0)T denotes rudder LIP coefficient matrix.


## 2.3. Senor Fault

Gain fault and bias fault are two common sensor faults, where bias faults include the drift bias fault and fixed bias fault. The sensor fault of the outer loop attitude angle can be expressed as follows:(12)yf=Gfx1+Bft,where yf denotes the actual measured value of attitude angles, Gf=diaggα,gβ,gϕ. denotes sensor gain coefficient matrix, 0 ≤ gi ≤ 1 denotes unknown gain coefficient, the sensor bias coefficient matrix is described as Bft=bαtbβtbϕtT., bfmin<bi<bfmax, denotes unknown bounded variable, i = (α, β, ϕ).In accordance with (12), by combining different values of the gain matrix and the bias matrix, the sensor fault types can be summarized as shown in Table 1.Table 1
Senor faults.
Faults formulaFault coefficientFault typeyf=x1gi=1,bit=0No sensor faultyf=Gfx10≤gi≤1,bit=0Senor gain faultyf=x1+Bftgi=1,bfmin<bi<bfmaxSenor drift bias faultyf=Bftgi=0,b˙it=0Senor fixed bias faultRemark 1.
The physical manifestation of LOE fault is the decrease of the deflection speed of the rudder surface; the main cause is the increase of the rotational friction force caused by the deformation of the motor shaft. The physical manifestation of the LIP fault is that the deflection angle of the rudder surface does not reach the target angle; the main cause is that the unstable airflow compresses the rudder surface or the deflection angle is limited due to the deformation of the mechanism close to the rudder surface. The physical manifestation of the sensor gain fault is that the signal is sluggish, for which the main reason is the feedback distortion caused by the electromagnetic interference of the enemy. The physical manifestation of sensor bias fault is the nonnegligible error between the feedback amplitude and the real output amplitude, and the main cause is the partial destruction of components due to temperature changes.


## 2.4. Multifault Model and Control Object

Combining equations (2), (11) and (12), the VSF attitude system model with rudder fault and sensor fault can be established as follows:(13)x˙1=x2+F1x+d1x˙2=f2x2+G2+GΔu+d2x2y=x1+sx1,where(14)F1x=f1x1+g1x1−Ix2G2+GΔ=g2x2+gΔΓΓ=g2x2Γ+gΔΓ,denotes unknown input distribution matrix of rudder surface with LOE, and the comprehensive disturbance(15)d2x2=g2x2+gΔLtl+d2=g2x2Ltl+gΔLtl+d2,is composed of rudder LIP fault and the disturbance of the attitude angular velocity channel,(16)sx1=Gf−Ix1+Bf,denotes senor fault, and I is the unit matrix with dimension 3 × 3.The multifault model (12) classifies the processing of different types of faults. For example, the rudder LIP fault is regarded as a part of external disturbance, while the rudder LOE is added because of the structural damage of the input distribution matrix. The sensor fault is reflected in the control output signal, which facilitates the design of subsequent FTC schemes by classifying the fault.Assumption 4.
When no LIP fault occurs, the comprehensive disturbanced2 (x2) is derivable and its derivative is bounded, i.e., d2x2<ρ∗ when t≠t1, where ρ∗ is an unknown positive constant.Assumption 5.
Reference signalx1d of x1 is bounded and derivable, and there exists a compact set(17)Σ=x1d,x˙1d,x¨1d|x1d2+x˙1d2+x¨1d2≤σ2,where σ is the unknown positive constant.Assumption 6.
The nonlinearF1 (x), f2 (x), and s (x) are uniformly bounded, and for any x and its estimation (∀x,x^∈!n) satisfying the Lipschitz condition, namely,(18)F1x−F1x^≤k1x−x^f2x−f2x^≤k2x−x^sx−sx^≤k3x−x^,where k1, k2, k3 > 0 denote the Lipschitz constant corresponding to the nonlinear function.
This paper studies the variable structure FTC of the nonlinear super-maneuvering VSF system. The control objective is as follows: when the VSF has a large maneuver and both a rudder fault and sensor fault, a FTC law should compensate for multiple faults and disturbance to the attitude angle, and the influence of the attitude angular velocity and the rudder deflection channel ensure the closed loop stability of system (13). To achieve the above control objectives, several key issues needed to be resolved:(1)
Due to the increase in the number of controllable rudders, the fault of multiple rudders may cause the control efficiency matrix of the system to be inaccurately known. How to design parameter estimation algorithms and FTC scheme for different types of rudder faults if the system control gain is not fully known.(2)
Sensor faults cause the attitude angle and velocity state variables output by the system to be unobservable. How to design fault estimation and FTC schemes to observe and compensate the system state in case of sensor fault is not known.(3)
The control variables such as the rudder deflection angle may be unconstrained when both rudder fault and sensor fault occur. How to improve the traditional variable structure control method to avoid the differential link amplifying the system noise and achieve stable tracking of reference signals is not known.
The overall control block diagram of the system is shown in Figure1.Figure 1
VSF system overall control block diagram.


## 3. Observer Design

In this section, an ESO for the VSF attitude system based on the neural network estimator is designed to estimate the system state and fault signal when rudder fault and sensor fault simultaneously occur. Through dimension expansion, the fault attitude angle signal measured by the sensor is converted into external interference, and the RBF is used to approach the sensor fault, and an adaptive control law is designed to estimate the rudder LOE online. The rudder LIP fault is regarded as a part of the comprehensive disturbance and estimated by the ESO.


### 3.1. Neural Network Estimator Design

The RBF neural network is composed of an input layer, hidden layer, and output layer, when enough neurons exist at the hidden layer, any continuous nonlinear function can be approximated by the RBF neural network [29], the specific equation is described as follows:(19)fx=w∗TΦx+η,where φix=φ1x,φ2x…φmxT denotes the RBF vector, xɛℝP denotes the input vector, ϕix,i=1,2,…m denotes the output function of the ith neuron at the hidden layer, m is the number of hidden layer nodes, η > 0 is the approximation error, and w∗ɛℝm is the optimal weight vector.In general, the RBFφi (x) is a Gaussian function, and the equation is constructed as follows:(20)φix=exp−x−bi22ci2,i=1,2,…,m,where bi is the network center of Gaussian function at ith node and ci is the width of Gaussian function at ith node. The nonlinear sensor fault sx1=f1x1,f2x1,…,fnx1T. can be described as follows:(21)fix1=wi∗TΦx1+ηi,i=1,2,…,nsx1=Ws∗TΦx1+ε,where ε=η1,η2,…,ηnT denotes the approximation error, satisfying ε<ε∗, and ε∗ is an unknown positive constant, Ws∗=w1∗,w2∗,…,wn∗∈ℝm×n denotes the optimal weight matrix, defined by the following:(22)Ws∗=argminW^s∈ℝm×nsupx∈ℝpsx1−W^sTΦx1,where W^s is the estimated value of the optimal weight matrix Ws∗.Then, the estimated value of the sensor faults (x1) using the RBF neural network can be expressed as follows:(23)s^x1=W^sTΦx1.Remark 2.
The nonlinear sensor faultsx1 in the inner loop subsystem is obtained by RBF training. RBF improves the estimation speed of the ESO by accurately estimating the fault terms and ensures the rapidity and accuracy of the estimation module.


### 3.2. State Observer Design

Considering that the system output (12) has unknown sensor faults and cannot be directly used for observer design, in accordance with the estimation results of the neural network estimator, the following ESO is designed in this section. We define(24)x0=∫0tydτ,and we can take the derivative of that as follows:(25)x˙0=x1+sx1.Combined with the system (12), the augmented system can be obtained as follows:(26)x˙0=x1+sx^1+Δsx˙1=x2+F1x+d1x˙2=x3+f2x2+G2+GΔuy˙=x0,where x^1 is the estimated value of x1, Δs=sx1−sx^1, x3=d2x2 denotes the extended state variable used to estimate the comprehensive disturbance. In accordance with the augmented system (21), the ESO is designed as follows:(27)x^˙0=x^1+s^x^1+a0x0−x^0x^˙1=x^2+F1x^+a1x0−x^0x^˙2=x^3+f2x^2+G^2+G^Δu+a2x0−x^0x^˙3=a3x0−x^0,where x^i is the estimation of the state xi, i = 0,1,2,3, and(28)s^x^1=W^sTΦx^1.(28) denotes the estimated value of sensor fault based on RBF, W^s is the estimated value of the optimal weight Ws, and ai, i = 1,2,3 are positive constants. Defining state estimation error as follows:(29)ζ=ζ0,ζ1,ζ2,ζ3T,ζi=xi−x^i,i=0,1,2,3.The estimation errors of the matrices are defined, namely, as follows:(30)F˜1=F1x−F1x^f˜2=f2x−f2x^G˜2=G2−G^2G˜Δ=GΔ−G^Δ.Then, the observer error dynamic equation is expressed as follows:(31)ζ˙=Aζ+∑i=03Biνi,where(32)A=−a0100−a1010−a2001−a3000ν=W˜sTΦx^1+ε+ΔsF˜1+d1f˜2+G˜2+G˜Δux˙3.Bi denotes a column vector with elements of the i + 1th row being 1, the remaining being 0, νi is the i + 1th row element of ν, νi is a 3 × 1 column vector. In order to facilitate the derivation, based on the idea of block matrix, when the element in νi and one of the matrices ai,i=0,1,2,3, Bi and ai,i=0,1,2,3 appear at the same time, set all elements are ai,i=0,1,2,3.Theorem 1.
Selectai, i = 0, 1, 2, 3 to make matrix A strict Hurwitz matrix, then any positive definite symmetric matrix P and positive definite matrix Q satisfying equation(33)ATP+PA=−Q,and the parameter adaptive law (35) can guarantee the ESO estimation error (29) as uniformly bounded.(34)W^˙s=Λ1Φx^1ζTPB0+μ1W^sG^˙2T+G^˙ΔT=Λ2uζTPB2+μ2G^2T+μ2G^ΔT,where Λ1,Λ2 denote positive definite diagonal gain matrices and μ1,μ2>0 denote the parameters to be designed.Proof.
Considering the following Lyapunov function:(35)V=12ζTPζ+12trW˜sTΛ1−1W˜s+12trG˜2+G˜ΔΛ2−1G˜2T+G˜ΔT.
Differentiating (36), we have the following:(36)V˙=12ζTATP+PAζ+ζTP∑i=03Biνi−trW˜sTΛ1−1W^˙s−trG˜2+G˜ΔΛ2−1G^˙2T+G^˙ΔT.
Substituting adaptive law (35) into (37) and rearranging the yield,(37)V˙=−12ζTQζ+ζTP∑i=03Biνi−trW˜sTΛ1−1W^˙s−trG˜2+G˜ΔΛ2−1G^˙2T+G^˙ΔT=−12ζTQζ+ζTPB0W˜sTΦx^1+ε+Δs+B1F˜1+d1+B2f˜2+G˜2+G˜Δu+B3x˙3−trW˜sTΛ1−1Λ1Φx^1ζTPB0+μ1W^s−trG˜2+G˜ΔΛ2−1Λ2uζTPB2+μ2G^2T+G^ΔT=−12ζTQζ+ζTPB0ε+Δs+B1F˜1+d1+B2f˜2+B3x˙3+B0W˜sTΦx^1+B2G˜2+G˜Δu−trW˜sTΦx^1ζTPB0+μ1W^s−trG˜2+G˜ΔuζTPB2+μ2G^2T+G^ΔT=−12ζTQζ+ζTPB0ε+Δs+B1F˜1+d1+B2f˜2+B3x˙3−trμ1W˜sTW^s−trμ2G˜2G^2T+G^ΔT.
Due toBi=1, the following inequalities can be derived from (21) and Assumptions 4–6 as follows:(38)ζTPB0ε≤ζPε∗ζTPB1d1≤ζPd∗ζTPB3x˙3≤ζPρ∗ζTPB0Δs+B1F˜1+B2f˜2≤ζPk1+k2+k3x−x^≤k1+k2+k3ζ2P.
Inequalities (39) combined with the complete square formula ζPε∗+d∗+ρ∗≤ζ2/2+P2ε∗+d∗+ρ∗2/2 can be obtained as follows:(39)ζPε∗+d∗+ρ∗≤ζ22+P2ε∗+d∗+ρ∗22.
In accordance with Young’s inequality [30] and we have the following:(40)−trμ1W˜sTW^s=−μ1trW^sW˜sT≤−μ12W˜s2+μ12Ws∗2−trμ2G˜2+G˜ΔG^2T+G^2T=−μ2trG^2T+G^2TG˜2+G˜Δ≤−μ22G˜2+G˜Δ2+μ22G2+GΔ2.
Substituting (39)–(41) into (38), we have the following:(41)V˙≤−12λminQζ2−μ12W˜s2+μ12Ws∗2−μ22G˜2+G˜Δ2+μ22G2+GΔ2+ζ22+P2ε∗+d∗+ρ∗22+k1+k2+k3ζ2P=−12λminQ−1−2k1+k2+k3Pζ2−μ12W˜s2−μ22G˜2+G˜Δ2+P2ε∗+d∗+ρ∗22+μ12Ws∗2+μ22G2+GΔ2≤−λminQ−1+2k1+k2+k3PλmaxP12ζTPζ−μ1λmaxΛ1−112trW˜sTΛ1−1W˜−μ2λmaxΛ2−112trG˜2T+G˜ΔTΛ2−1G˜2+G˜Δ+P2ε∗+d∗+ρ∗22+μ12Ws∗2+μ22G2+GΔ2≤−C2V+C1.where λmin and λmax denote the minimum and maximum values of the corresponding matrix eigenvalues, design ai,i=0,1,2,3,μ1,μ2,Λ1−1,Λ2−1 ensuring that C2>0 and the value is big enough, in order that V˙<0, and(42)C1=P2ε∗+d∗+ρ∗22+μ12Ws∗2+μ22G2+GΔ2C2=minλminQ−1+2k1+k2+k3PλmaxP,μ1λmaxΛ1−1μ2λmaxΛ2−1.,
After multiplying both sides of (41), the integral can be obtained as follows:(43)0<Vt<C1C2+V0−C1C2e−C2t,then V ⟶ C1/C2 when t ⟶ ∞, and V is uniformly bounded. In accordance with (35), are uniformly bounded. Therefore, the adaptive law (35) can guarantee the uniform bound of the ESO estimation error (29).Remark 3.
The observer of [25] only estimates the sensor fault, not the actuator fault and disturbance; the observer of [30] can only estimate unmeasured states, the essence of its method is passive compensation. The ESO in this study has been greatly improved, which can simultaneously estimate disturbance, states, sensor fault, and actuator fault, and has more functions.


## 3.1. Neural Network Estimator Design

The RBF neural network is composed of an input layer, hidden layer, and output layer, when enough neurons exist at the hidden layer, any continuous nonlinear function can be approximated by the RBF neural network [29], the specific equation is described as follows:(19)fx=w∗TΦx+η,where φix=φ1x,φ2x…φmxT denotes the RBF vector, xɛℝP denotes the input vector, ϕix,i=1,2,…m denotes the output function of the ith neuron at the hidden layer, m is the number of hidden layer nodes, η > 0 is the approximation error, and w∗ɛℝm is the optimal weight vector.In general, the RBFφi (x) is a Gaussian function, and the equation is constructed as follows:(20)φix=exp−x−bi22ci2,i=1,2,…,m,where bi is the network center of Gaussian function at ith node and ci is the width of Gaussian function at ith node. The nonlinear sensor fault sx1=f1x1,f2x1,…,fnx1T. can be described as follows:(21)fix1=wi∗TΦx1+ηi,i=1,2,…,nsx1=Ws∗TΦx1+ε,where ε=η1,η2,…,ηnT denotes the approximation error, satisfying ε<ε∗, and ε∗ is an unknown positive constant, Ws∗=w1∗,w2∗,…,wn∗∈ℝm×n denotes the optimal weight matrix, defined by the following:(22)Ws∗=argminW^s∈ℝm×nsupx∈ℝpsx1−W^sTΦx1,where W^s is the estimated value of the optimal weight matrix Ws∗.Then, the estimated value of the sensor faults (x1) using the RBF neural network can be expressed as follows:(23)s^x1=W^sTΦx1.Remark 2.
The nonlinear sensor faultsx1 in the inner loop subsystem is obtained by RBF training. RBF improves the estimation speed of the ESO by accurately estimating the fault terms and ensures the rapidity and accuracy of the estimation module.


## 3.2. State Observer Design

Considering that the system output (12) has unknown sensor faults and cannot be directly used for observer design, in accordance with the estimation results of the neural network estimator, the following ESO is designed in this section. We define(24)x0=∫0tydτ,and we can take the derivative of that as follows:(25)x˙0=x1+sx1.Combined with the system (12), the augmented system can be obtained as follows:(26)x˙0=x1+sx^1+Δsx˙1=x2+F1x+d1x˙2=x3+f2x2+G2+GΔuy˙=x0,where x^1 is the estimated value of x1, Δs=sx1−sx^1, x3=d2x2 denotes the extended state variable used to estimate the comprehensive disturbance. In accordance with the augmented system (21), the ESO is designed as follows:(27)x^˙0=x^1+s^x^1+a0x0−x^0x^˙1=x^2+F1x^+a1x0−x^0x^˙2=x^3+f2x^2+G^2+G^Δu+a2x0−x^0x^˙3=a3x0−x^0,where x^i is the estimation of the state xi, i = 0,1,2,3, and(28)s^x^1=W^sTΦx^1.(28) denotes the estimated value of sensor fault based on RBF, W^s is the estimated value of the optimal weight Ws, and ai, i = 1,2,3 are positive constants. Defining state estimation error as follows:(29)ζ=ζ0,ζ1,ζ2,ζ3T,ζi=xi−x^i,i=0,1,2,3.The estimation errors of the matrices are defined, namely, as follows:(30)F˜1=F1x−F1x^f˜2=f2x−f2x^G˜2=G2−G^2G˜Δ=GΔ−G^Δ.Then, the observer error dynamic equation is expressed as follows:(31)ζ˙=Aζ+∑i=03Biνi,where(32)A=−a0100−a1010−a2001−a3000ν=W˜sTΦx^1+ε+ΔsF˜1+d1f˜2+G˜2+G˜Δux˙3.Bi denotes a column vector with elements of the i + 1th row being 1, the remaining being 0, νi is the i + 1th row element of ν, νi is a 3 × 1 column vector. In order to facilitate the derivation, based on the idea of block matrix, when the element in νi and one of the matrices ai,i=0,1,2,3, Bi and ai,i=0,1,2,3 appear at the same time, set all elements are ai,i=0,1,2,3.Theorem 1.
Selectai, i = 0, 1, 2, 3 to make matrix A strict Hurwitz matrix, then any positive definite symmetric matrix P and positive definite matrix Q satisfying equation(33)ATP+PA=−Q,and the parameter adaptive law (35) can guarantee the ESO estimation error (29) as uniformly bounded.(34)W^˙s=Λ1Φx^1ζTPB0+μ1W^sG^˙2T+G^˙ΔT=Λ2uζTPB2+μ2G^2T+μ2G^ΔT,where Λ1,Λ2 denote positive definite diagonal gain matrices and μ1,μ2>0 denote the parameters to be designed.Proof.
Considering the following Lyapunov function:(35)V=12ζTPζ+12trW˜sTΛ1−1W˜s+12trG˜2+G˜ΔΛ2−1G˜2T+G˜ΔT.
Differentiating (36), we have the following:(36)V˙=12ζTATP+PAζ+ζTP∑i=03Biνi−trW˜sTΛ1−1W^˙s−trG˜2+G˜ΔΛ2−1G^˙2T+G^˙ΔT.
Substituting adaptive law (35) into (37) and rearranging the yield,(37)V˙=−12ζTQζ+ζTP∑i=03Biνi−trW˜sTΛ1−1W^˙s−trG˜2+G˜ΔΛ2−1G^˙2T+G^˙ΔT=−12ζTQζ+ζTPB0W˜sTΦx^1+ε+Δs+B1F˜1+d1+B2f˜2+G˜2+G˜Δu+B3x˙3−trW˜sTΛ1−1Λ1Φx^1ζTPB0+μ1W^s−trG˜2+G˜ΔΛ2−1Λ2uζTPB2+μ2G^2T+G^ΔT=−12ζTQζ+ζTPB0ε+Δs+B1F˜1+d1+B2f˜2+B3x˙3+B0W˜sTΦx^1+B2G˜2+G˜Δu−trW˜sTΦx^1ζTPB0+μ1W^s−trG˜2+G˜ΔuζTPB2+μ2G^2T+G^ΔT=−12ζTQζ+ζTPB0ε+Δs+B1F˜1+d1+B2f˜2+B3x˙3−trμ1W˜sTW^s−trμ2G˜2G^2T+G^ΔT.
Due toBi=1, the following inequalities can be derived from (21) and Assumptions 4–6 as follows:(38)ζTPB0ε≤ζPε∗ζTPB1d1≤ζPd∗ζTPB3x˙3≤ζPρ∗ζTPB0Δs+B1F˜1+B2f˜2≤ζPk1+k2+k3x−x^≤k1+k2+k3ζ2P.
Inequalities (39) combined with the complete square formula ζPε∗+d∗+ρ∗≤ζ2/2+P2ε∗+d∗+ρ∗2/2 can be obtained as follows:(39)ζPε∗+d∗+ρ∗≤ζ22+P2ε∗+d∗+ρ∗22.
In accordance with Young’s inequality [30] and we have the following:(40)−trμ1W˜sTW^s=−μ1trW^sW˜sT≤−μ12W˜s2+μ12Ws∗2−trμ2G˜2+G˜ΔG^2T+G^2T=−μ2trG^2T+G^2TG˜2+G˜Δ≤−μ22G˜2+G˜Δ2+μ22G2+GΔ2.
Substituting (39)–(41) into (38), we have the following:(41)V˙≤−12λminQζ2−μ12W˜s2+μ12Ws∗2−μ22G˜2+G˜Δ2+μ22G2+GΔ2+ζ22+P2ε∗+d∗+ρ∗22+k1+k2+k3ζ2P=−12λminQ−1−2k1+k2+k3Pζ2−μ12W˜s2−μ22G˜2+G˜Δ2+P2ε∗+d∗+ρ∗22+μ12Ws∗2+μ22G2+GΔ2≤−λminQ−1+2k1+k2+k3PλmaxP12ζTPζ−μ1λmaxΛ1−112trW˜sTΛ1−1W˜−μ2λmaxΛ2−112trG˜2T+G˜ΔTΛ2−1G˜2+G˜Δ+P2ε∗+d∗+ρ∗22+μ12Ws∗2+μ22G2+GΔ2≤−C2V+C1.where λmin and λmax denote the minimum and maximum values of the corresponding matrix eigenvalues, design ai,i=0,1,2,3,μ1,μ2,Λ1−1,Λ2−1 ensuring that C2>0 and the value is big enough, in order that V˙<0, and(42)C1=P2ε∗+d∗+ρ∗22+μ12Ws∗2+μ22G2+GΔ2C2=minλminQ−1+2k1+k2+k3PλmaxP,μ1λmaxΛ1−1μ2λmaxΛ2−1.,
After multiplying both sides of (41), the integral can be obtained as follows:(43)0<Vt<C1C2+V0−C1C2e−C2t,then V ⟶ C1/C2 when t ⟶ ∞, and V is uniformly bounded. In accordance with (35), are uniformly bounded. Therefore, the adaptive law (35) can guarantee the uniform bound of the ESO estimation error (29).Remark 3.
The observer of [25] only estimates the sensor fault, not the actuator fault and disturbance; the observer of [30] can only estimate unmeasured states, the essence of its method is passive compensation. The ESO in this study has been greatly improved, which can simultaneously estimate disturbance, states, sensor fault, and actuator fault, and has more functions.


## 4. FTC Scheme Design

In this section, combined with the estimation result of the ESO, the observed value is substituted into the control law, and the estimated states replace the actual states for the faulty VSF system to design an variable structure adaptive backstepping controller based on the command filter to realize the unobservable state tracking.


### 4.1. Command Filter Compensator Design

Considering the constrained state variables of the system and the physical characteristics of the actuator, then according to [31], this section introduces the command filter and designs the compensation control law. The basic structure diagram is shown in Figure 2.where(44)ς∗=ς+ς∗GΔω∗=ω+ω∗GΔ.Figure 2
Command filter structure diagram.The state equation of the command filter is constructed as follows:(45)z˙1z˙2=z22ς+ς∗GΔω+ω∗GΔfavfav=faω+ω∗GΔ22ς+ς∗GΔω+ω∗GΔfvχ0−z1−z2χdχ˙d=z1z2,where χ0 is the input signal of the command filter, χd,χ˙d is the output signal of the command filter, ς is the fix filter damping, ω is the fix filter bandwidth, and xl and xh are amplitude limiting function and rate limiting function, respectively, expressed as follows:(46)fix=xh,x≥xhx,−xl<x<xh−xl,x≤−xl,i=a,v.In order to adapt to the variable structure mode of VSF, the variable structure harmonic functionsς∗ (GΔ) and ω∗ (GΔ) with known expressions are added to the filter, so that it can maintain high performance in both the fixed structure mode and variable structure mode.Lemma 1.
(see [32]). If the input signal χ0 of the command filter is bounded, then the output signals χd,χ˙d are bounded and continuous. For any constant c, there exist(47)ω+ωGΔ>00<ς+ςGΔ<1,that makes the input signal and output signal satisfy χd−χ0≤c.Defining the tracking error of the outer loop attitude angle caused by the command filter asσ1, and then designing the outer loop filter compensation control law as follows:(48)σ˙1=−c1+c1∗GΔσ1+g1x^1x2d−x20,where c1 > 0 denotes the nominal gain parameter of the outer loop filter compensation control law, c1∗ (GΔ) is a known harmonic function designed for automatic adaptation to variable structure parameters, and x20 denotes the outer loop command filter input signal.Defining the tracking error of the inner loop attitude angular velocity due to the command filter asσ2, and then designing the inner loop filter compensation control law as follows:(49)σ˙2=−c2+c2∗GΔσ2+G^2+G^Δu−G^2+G^Δu0.where c2 > 0 denotes the gain coefficient of the inner loop filter compensation control law, c2∗ (GΔ) is a known harmonic function designed for automatic adaptation to variable structure parameters, and u0 denotes the inner loop command filter input signal.Remark 4.
If the attitude anglex1 and rudder deflection angle u of the VSF are saturated during large maneuvering, the tracking error will increase and the system will diverge if the limit is exceeded. The traditional backstepping control method needs to derive the virtual control signal. The differential link amplifies the noise of the system. The improved backstepping control combined with the second-order command filter can effectively offset the “explosion” caused by the differentiation.


### 4.2. Adaptive Backstepping FTC Design

The state tracking errors of attitude angle and attitude angular velocity are defined as follows:(50)e1=x^1−x1de2=x^2−x2d,where x1d denotes outer loop attitude angle reference signal and x2d denotes the inner loop angular speed reference signal. Provided by the outer loop output signal, the smoothness of the reference command signal is guaranteed by the command filter. Combined with (26), the derivative of e1 can be obtained as follows:(51)e˙1=g1x^1x^2+f1x^1+a1ζ0−x˙1d,then, the tracking error of the outer loop attitude angle state correction with the command filter is expressed as follows:(52)γ1=e1−σ1.The outer loop backstepping adaptive law is designed as follows:(53)x20=−g1−1x^1f1x^1+a1ξ0−x˙1d+κ1∫0tγ1dτ+c1+c1∗GΔe1−σ2,where κ1 > 0 is the parameter to be designed, the actual control law x2d is obtained by the adaptive law of virtual control of outer loop x20 through the command filter (47).In accordance with (26), we can obtain the derivative of e2 as follows:(54)e˙2=x^3+f2x^2+G^2+G^Δu+a2ξ0−x˙2d.Then, the attitude angle state correction tracking error of the outer loop with a command filter can be expressed as follows:(55)γ2=e2−σ2.The inner loop backstepping adaptive law is designed as follows:(56)u0=−G^2+G^Δ−1f2x^2+a2ξ0−x˙2d+x^3+κ2∫0tγ2dτ+c2+c2∗GΔe2+g1x^1γ1,where κ2 > 0 is the parameter to be designed, the actual inner control law u is obtained by the adaptive law of virtual control of inner loop u0 through the command filter (48).Theorem 2.
For nonlinear systems (12), if there exist command filter compensation control laws (50) and (51) and variable structure adaptive FTC laws (55) and (58), the closed-loop system is stable and the system state tracking error is uniformly bounded.Proof.
Considering the following Lyapunov function for the outer loop subsystem:(57)V1=12γ1Tγ1+12κ1∫0tγ1dτ2.
Taking the derivative of (59) and considering (50), and (53), we can obtain the following:(58)V˙1=γ1Tγ˙1+κ1γ1T∫0tγ1dτ=γ1Tg1x^1x20+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2.
Define the following Lyapunov function for the inner loop subsystem:(59)V2=12γ2Tγ2+12κ2∫0tγ2dτ2.
Taking the derivative of (61) and considering (51), and (56), we can obtain as follows:(60)V˙2=γ2Tγ˙2+κ2γ2T∫0tγ2dτ=γ2TG^2+G^Δu0+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ.
The Lyapunov function for the FTC system is designed as follows:(61)V=V1+V2.
Taking the derivative of (61), and we have the following:(62)V˙=V˙1+V˙2=γ1Tg1x^1x20+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2+γ2TG^2+G^Δu0+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ.
Substituting (55), (58) into (64) and we can obtain as follows:(63)V˙=γ1Tg1x^1x20+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2+γ2TG^2+G^Δu0+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ=γ1Tg1x^1−g1−1x^1f1x^1+a1ξ0−x˙1d+κ1∫0tγ1dτ+c1+c1∗GΔe1−σ2+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2+γ2TG^2+G^Δ−G^+G^Δ−1f2x^2+a2ξ0−x˙2d+x^3+κ2∫0tγ2dτ+c2+c2∗GΔe2+g1x^1γ1+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ=γ1T−c1+c1∗GΔγ1+g1x^1γ2+γ2T−c2+c2∗GΔγ2−g1x^1γ1=−c1+c1∗GΔγ1Tγ1−c2+c2∗GΔγ2Tγ2≤−c0γ1Tγ1+γ2Tγ2,where(64)c0=minc1+c1∗GΔ,c2+c2∗GΔ.
In accordance with (63), V˙ is negative semi-definite, so V (∞) is bounded. From(65)V˙≤−c0γ1Tγ1+γ2Tγ2.
Lett ⟶ ∞, we have as follows:(66)∫0∞γ1Tγ1+γ2Tγ2dτ≤−1c0∫0∞V˙dτ=−1c0V∞−V0<∞.
Therefore,γ1,γ2∈L2. From Barbalat’s Lemma [33], we can obtain as follows:(67)limt⟶∞γ1=0limt⟶∞γ2=0.
From Lemma1, we can obtain that there exists ω > 0, making the tracking error of the command filter bounded and arbitrarily small, while the corrected tracking errors γ1, γ2, converge to the state tracking errors e1, e2. In accordance with (62), the variable structure backstepping FTC law (55) and (58) based on the command filter can ensure that the state tracking error is bounded, and the VSF system state can track the reference signal.


## 4.1. Command Filter Compensator Design

Considering the constrained state variables of the system and the physical characteristics of the actuator, then according to [31], this section introduces the command filter and designs the compensation control law. The basic structure diagram is shown in Figure 2.where(44)ς∗=ς+ς∗GΔω∗=ω+ω∗GΔ.Figure 2
Command filter structure diagram.The state equation of the command filter is constructed as follows:(45)z˙1z˙2=z22ς+ς∗GΔω+ω∗GΔfavfav=faω+ω∗GΔ22ς+ς∗GΔω+ω∗GΔfvχ0−z1−z2χdχ˙d=z1z2,where χ0 is the input signal of the command filter, χd,χ˙d is the output signal of the command filter, ς is the fix filter damping, ω is the fix filter bandwidth, and xl and xh are amplitude limiting function and rate limiting function, respectively, expressed as follows:(46)fix=xh,x≥xhx,−xl<x<xh−xl,x≤−xl,i=a,v.In order to adapt to the variable structure mode of VSF, the variable structure harmonic functionsς∗ (GΔ) and ω∗ (GΔ) with known expressions are added to the filter, so that it can maintain high performance in both the fixed structure mode and variable structure mode.Lemma 1.
(see [32]). If the input signal χ0 of the command filter is bounded, then the output signals χd,χ˙d are bounded and continuous. For any constant c, there exist(47)ω+ωGΔ>00<ς+ςGΔ<1,that makes the input signal and output signal satisfy χd−χ0≤c.Defining the tracking error of the outer loop attitude angle caused by the command filter asσ1, and then designing the outer loop filter compensation control law as follows:(48)σ˙1=−c1+c1∗GΔσ1+g1x^1x2d−x20,where c1 > 0 denotes the nominal gain parameter of the outer loop filter compensation control law, c1∗ (GΔ) is a known harmonic function designed for automatic adaptation to variable structure parameters, and x20 denotes the outer loop command filter input signal.Defining the tracking error of the inner loop attitude angular velocity due to the command filter asσ2, and then designing the inner loop filter compensation control law as follows:(49)σ˙2=−c2+c2∗GΔσ2+G^2+G^Δu−G^2+G^Δu0.where c2 > 0 denotes the gain coefficient of the inner loop filter compensation control law, c2∗ (GΔ) is a known harmonic function designed for automatic adaptation to variable structure parameters, and u0 denotes the inner loop command filter input signal.Remark 4.
If the attitude anglex1 and rudder deflection angle u of the VSF are saturated during large maneuvering, the tracking error will increase and the system will diverge if the limit is exceeded. The traditional backstepping control method needs to derive the virtual control signal. The differential link amplifies the noise of the system. The improved backstepping control combined with the second-order command filter can effectively offset the “explosion” caused by the differentiation.


## 4.2. Adaptive Backstepping FTC Design

The state tracking errors of attitude angle and attitude angular velocity are defined as follows:(50)e1=x^1−x1de2=x^2−x2d,where x1d denotes outer loop attitude angle reference signal and x2d denotes the inner loop angular speed reference signal. Provided by the outer loop output signal, the smoothness of the reference command signal is guaranteed by the command filter. Combined with (26), the derivative of e1 can be obtained as follows:(51)e˙1=g1x^1x^2+f1x^1+a1ζ0−x˙1d,then, the tracking error of the outer loop attitude angle state correction with the command filter is expressed as follows:(52)γ1=e1−σ1.The outer loop backstepping adaptive law is designed as follows:(53)x20=−g1−1x^1f1x^1+a1ξ0−x˙1d+κ1∫0tγ1dτ+c1+c1∗GΔe1−σ2,where κ1 > 0 is the parameter to be designed, the actual control law x2d is obtained by the adaptive law of virtual control of outer loop x20 through the command filter (47).In accordance with (26), we can obtain the derivative of e2 as follows:(54)e˙2=x^3+f2x^2+G^2+G^Δu+a2ξ0−x˙2d.Then, the attitude angle state correction tracking error of the outer loop with a command filter can be expressed as follows:(55)γ2=e2−σ2.The inner loop backstepping adaptive law is designed as follows:(56)u0=−G^2+G^Δ−1f2x^2+a2ξ0−x˙2d+x^3+κ2∫0tγ2dτ+c2+c2∗GΔe2+g1x^1γ1,where κ2 > 0 is the parameter to be designed, the actual inner control law u is obtained by the adaptive law of virtual control of inner loop u0 through the command filter (48).Theorem 2.
For nonlinear systems (12), if there exist command filter compensation control laws (50) and (51) and variable structure adaptive FTC laws (55) and (58), the closed-loop system is stable and the system state tracking error is uniformly bounded.Proof.
Considering the following Lyapunov function for the outer loop subsystem:(57)V1=12γ1Tγ1+12κ1∫0tγ1dτ2.
Taking the derivative of (59) and considering (50), and (53), we can obtain the following:(58)V˙1=γ1Tγ˙1+κ1γ1T∫0tγ1dτ=γ1Tg1x^1x20+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2.
Define the following Lyapunov function for the inner loop subsystem:(59)V2=12γ2Tγ2+12κ2∫0tγ2dτ2.
Taking the derivative of (61) and considering (51), and (56), we can obtain as follows:(60)V˙2=γ2Tγ˙2+κ2γ2T∫0tγ2dτ=γ2TG^2+G^Δu0+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ.
The Lyapunov function for the FTC system is designed as follows:(61)V=V1+V2.
Taking the derivative of (61), and we have the following:(62)V˙=V˙1+V˙2=γ1Tg1x^1x20+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2+γ2TG^2+G^Δu0+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ.
Substituting (55), (58) into (64) and we can obtain as follows:(63)V˙=γ1Tg1x^1x20+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2+γ2TG^2+G^Δu0+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ=γ1Tg1x^1−g1−1x^1f1x^1+a1ξ0−x˙1d+κ1∫0tγ1dτ+c1+c1∗GΔe1−σ2+f1x^1+a1ζ0−x˙1d+c1+c1∗GΔσ1+κ1∫0tγ1dτ+g1x^1e2+γ2TG^2+G^Δ−G^+G^Δ−1f2x^2+a2ξ0−x˙2d+x^3+κ2∫0tγ2dτ+c2+c2∗GΔe2+g1x^1γ1+x^3+f2x^2+a2ξ0−x˙2d+c2+c2∗GΔσ2+κ2∫0tγ2dτ=γ1T−c1+c1∗GΔγ1+g1x^1γ2+γ2T−c2+c2∗GΔγ2−g1x^1γ1=−c1+c1∗GΔγ1Tγ1−c2+c2∗GΔγ2Tγ2≤−c0γ1Tγ1+γ2Tγ2,where(64)c0=minc1+c1∗GΔ,c2+c2∗GΔ.
In accordance with (63), V˙ is negative semi-definite, so V (∞) is bounded. From(65)V˙≤−c0γ1Tγ1+γ2Tγ2.
Lett ⟶ ∞, we have as follows:(66)∫0∞γ1Tγ1+γ2Tγ2dτ≤−1c0∫0∞V˙dτ=−1c0V∞−V0<∞.
Therefore,γ1,γ2∈L2. From Barbalat’s Lemma [33], we can obtain as follows:(67)limt⟶∞γ1=0limt⟶∞γ2=0.
From Lemma1, we can obtain that there exists ω > 0, making the tracking error of the command filter bounded and arbitrarily small, while the corrected tracking errors γ1, γ2, converge to the state tracking errors e1, e2. In accordance with (62), the variable structure backstepping FTC law (55) and (58) based on the command filter can ensure that the state tracking error is bounded, and the VSF system state can track the reference signal.


## 5. Simulation

In this section, the simulation results are presented to demonstrate the effectiveness of the proposed variable structure FTC scheme. We use the Links-Box semiphysical simulator (Figure3) to verify the effectiveness. Links-Box automatically converts MATLAB models to the embedded control prototype and supports the engineering hardware used to test the models.Figure 3
Simulation platform.In order to show the influence of rudder fault and sensor fault in the VSF system more intuitively, simulation analysis includes the following: (1) FTC simulation for three rudder LIP faults; (2) Aileron LIP fault, angle of attack sensor gain fault and disturbance FTC simulation where (1) and (2) are in the same simulation environment.Set the initial flight state:α0 = 0deg, β0 = 0deg, ϕ0 = 0 deg, p0=q0=r0=0deg/s. Set the reference signal as: αd, ϕd produced by command generator, βd = 0deg. Based on the reference signal, the fixed-roll-angle post-stall pitch maneuver is completed, the simulation time is 50 s, and the simulation step is 0.1 s. The system parameters and command filter parameters are shown in Table 2.Table 2
VSF system parameters.
SignalAmplitude rangeRate rangeFilter dampingςFilter bandwidthωAngleα±4.3°—0.85Angleβ±5.7°—0.82Angleϕ±3.6°—0.84Roll ratep±60° (s)—0.812Pitch rateq±60° (s)—0.812Yaw rater±60° (s)—0.812δa±21.5°±80° (s)0.825δe±25°±60° (s)0.825δr±30°±120° (s)0.825δy±15°±45° (s)0.825δz±15°±45° (s)0.825Each element in the original variable structure functiongΔ obeys the hyperbolic tangent function, as shown below.(68)gΔ=10tanh10t−4+105tanh10t−4+107tanh10t−4+103tanh10t−4+105tanh10t−4+1kg⋅m3,t≥4s0,0≤t<4stanh10t−4=e10t−4−e−10t−4e10t−4+e−10t−4.Therefore, whent = 4 s, gΔ is approximately 0 in engineering, which meets the simulation requirements of continuously variable structure motion.Since no sensor fault occurs to the system, the neural network estimator in ESO has fewer nodes. Set the number of nodesm = 4, the network width of the Gaussian function ci = 2, the network center bi is uniformly distributed in the interval (−4, 4) × (−5, 5), the initial value of the estimated state of the remaining observers is the same as the initial value of the system state. The parameters of the observer and controller are shown in Table 3. Both Tables 2 and 3 only show a part of the main parameters because there are too many parameters and showing them all will cause confusion and reduce the readability of the paper.Table 3
Control parameters.
ParameterValuea05a18a24a32μ10.05μ20.08Λ1−14I4×4Λ2−16I5×5c12c25κ10.2κ20.1The disturbance of the attitude angle channeld1 and the angular velocity channel d2 are added, respectively; the disturbance value is set as (69), and the stochastic white noise is set as whi(0.2), 0.2 is the standard deviation learn from [31–33].(69)d1=dαdβdϕ,dα=2sin2πt+whi0.2,t>10s,whi0.2,others,dβ=0.2sin2πt+whi0.2,t>10s,whi0.2,others,dϕ=sin2πt+whi0.2,t>10s,whi0.2,others,d2=dpdqdr,dp=2sin2πt,t>10s,0,others,dq=4sin2πt+whi0.4,t>10s,whi0.4,others,dr=sin2πt,t>10s,0,others..The parameters of the neural network estimator in ESO are set as follows: the number of nodesm = 32, the network width of the Gaussian function ci = 2, and the network center bi is uniformly distributed in the interval of (−4, 4) × (−5, 5) × (−5, 5) × (−2.5, 2.5) × (−4, 4). The remaining initial values of the estimated states of the observer are the same as the initial values of the system states, Λ1−1=3.2I32×32, Λ2−1=5.6I5×5, and the remaining control parameters are the same as in the first simulation.The estimation results of disturbance are substituted into the system model, whilst the effective results of LIP fault estimation are shown in Figure4. The estimation curves of the three attitude angles are slightly different. However, the rudder LIP faults are well estimated.Figure 4
Fault estimation of rudder LIP fault. (a) Superimposed deviation fault of aileronδa. (b) Superimposed deviation fault of yaw rudder δr. (c) Superimposed deviation fault of elevator δe.
(a)(b)(c)In the experiment, the sensor gain fault set as the output signal response speed is slowed down by −0.07∼0.07, a sinusoidal time-varying fault in attack angle rate sensor. Moreover, the fault estimation result is shown in Figure5. Like rudder fault, sensor gain fault can also be accurately estimated. These estimates provide the basis for an effective FTC.Figure 5
Fault estimation of sensor gain fault.Figure6 shows the mean response curves of the three attitude angles. The simulation system can generate the attitude angle mean curves. The fast and stable attitude response satisfies the technical requirements of VSF for attitude control.Figure 6
Mean values of attitude angles. (a) Mean of bank roll angleφ. (b) Mean of sideslip angle β. (c) Mean of attack angle α.
(a)(b)(c)Figure6 shows that the fault-tolerant control scheme can ensure that the VSF system stably tracks the desired attitude angle under multiple fault conditions. Although the fault-tolerant control process takes time to return to equilibrium, it does not affect the stability.


## 6. Conclusion

This paper proposes an variable structure adaptive FTC scheme based on the ESO for the next generation fighter system with rudder fault, sensor fault, and external disturbance. The designed ESO based on the RBF estimator can accurately estimate the system’s attitude variables, sensor fault, and rudder fault in real time. Based on the fault observation information, the designed adaptive backstepping control scheme based on the second-order command filter can avoid the differential explosion of traditional backstepping control and compensate for the drift and jitter of the attitude system state variables caused by multiple faults and external disturbance. The FTC scheme makes up for the defect that the nominal variable structure controller cannot repair multiple faults and realizes the robust tracking control of the large maneuver reference signal. Future work will focus on the study of finite-time convergence, high-frequency stochastic attitude control, FTC for incipient, intermittent faults, etc.



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