# Reconfiguration Control Design of UAV against Actuator Faults Based on Control Allocation Method

Authors: Yuwei Cui; Aijun Li; Biao Duan; Shabbir Wasif
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

The paper concentrates on the problem of fault-tolerant control of UAV against actuator faults from the perspective of flight control system architecture. Using backstepping control method and inverse optimization theory, the design of backstepping optimal control law was constructed. Based on the fault monitoring mechanism of vehicle management computer in the distributed flight control and control allocation system, a fault-tolerant control design method was established in the case of multiple failure modes of the actuators, which compensates the influence caused by the failures. Finally, the effectiveness of the proposed strategy was verified by numerical simulation.


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## Body

## 1. Introduction

In recent years, XQ-58A “Valkyrie” developed by the United States has started to verify its combat technology with manned fighters F-15X and F-35. While emphasizing joint operations, it also puts forward low-cost development requirements. This medium-size UAV with low-cost attritable aircraft technology (LCAAT) is more affordable than the X47B stealth UAV. In 2020, Russia demonstrated a high-speed, unmanned loyal wingman, named “Thunder”. Similar to XQ-58A, it is capable of partnering with manned fighters and serving as a wingman in a front-line attack and detecting and destroying enemy and air defense targets, as discussed elsewhere [1–3].With the worldwide research and development of all kinds of unmanned systems, the application of manned/unmanned aerial vehicles and unmanned swarm systems has been promoted. Unmanned systems are no longer synonymous with low cost and high risk but increasingly emphasize the following two aspects of technical requirements:(i)
Low cost and affordable economic cost(ii)
High fault tolerance and affordable security risksFor the fighter or large reconnaissance and combat UAVs, its cost can be increased. Therefore, the architecture of multiredundant hardware design is often used to improve the fault-tolerant ability of the entire aircraft system, especially for the flight control system with high safety and reliability. However, for consumable and low-cost unmanned systems such as loyal wingman, the architecture based on redundant hardware design brings high cost, large volume, large weight, and small load, which is often not acceptable. However, it is still expected to minimize the impact of failure, that is, to have higher fault tolerance.There are many fault diagnosis and fault-tolerant control methods for sensors and actuators of flight control system. As early as 1990, as discussed by Professor Frank [4], international authority on fault diagnosis of control system divided fault diagnosis methods into three categories: analytical model-based methods, signal process-based methods, and knowledge-based methods, which have been accepted by many scholars [5, 6].For the application of UAV flight control system, model-based fault detection, isolation, and adjustment methods can be used to reconstruct low redundancy/no redundancy signals for sensor faults, and the fault tolerance problem of sensor signals can be solved by means of analytical redundancy, as discussed elsewhere [7–11].But when the faults occur in the aircraft surface because of actuators, the detection and isolation method based on model cannot work, and it must be reconstructed through the control compensation of other surfaces to achieve fault tolerance. At present, many scholars have proposed adaptive control algorithms to solve various adaptive control problems in the case of surface failure, so as to achieve flight control, but the real-time performance of these algorithms is generally doubted, and a large number of algorithms need accurate mathematical models [12–15].At present, the fault tolerant control towards actuator’s faults is based on the results of fault detection [16]. Firstly, the faulty actuator should be isolated to avoid its fault spreading. Then, by adjusting the parameters of the controller or changing the structure of the controller, the stability and control performance of the system can be guaranteed. However, this method must be offline calculation of control law parameters required under various faults, and these parameters are stored in the flight control computer in advance. During the flight, according to the fault information obtained by fault detection, the matched control law parameters are selected to obtain the reconstructed flight control law. Since the parameters of the readjusted control law are designed off-line, the reconstructed flight control law can only tolerate the fault modes considered in advance, which limits the scalability of this method.At the same time, we consider the possible failure modes of UAV actuators. At present, for small and medium-sized UAVs, electric actuators are basically used as the driving mechanism; and for large UAVs, with the continuous maturity of high-power electric actuators, the trend of replacing hydraulic actuators is also accelerating, due to the maintenance, pipeline, weight, leakage, and other problems caused by hydraulic actuators. Consider that there are two main failure modes of the electric actuators: nonoutput force and the output shaft stuck. From the perspective of surface control, the influence of the former is relatively small. And the most important thing is to realize fault detection, so that the fault can be found quickly. The second failure mode is trickier because it introduces additional unwanted moments that need to be balanced first in flight control.In view of the multiple failure modes of UAV actuators, the strategy proposed in the paper adopts backstepping optimal control law and reconfiguration design based on a control allocation method to compensate the influence of faulty actuators, so as to achieve the goal of fault-tolerant control. The main contributions of the research are summarized as follows:(1)
Different from the adaptive control method, as referred to References [12–15], based on the optimal control law and the fault diagnosis result, control allocation methods carry out the mode switch. It does not need to carry out complex online calculation through the optimization algorithm, so it will not affect the real-time performance(2)
Compared with constructing off-line databases and adjusting the parameters or structure of controller, the reconfiguration control methods based on control allocation can compensate the influence of fault surfaces without adjusting the control law, thus extending its application scope(3)
More importantly, the paper presents a systematic solution, not just for the control algorithms. Through the modular control structure, the fault detection, control law, and control allocation algorithm are organically integrated, and a relatively comprehensive solution is proposed for the reconfiguration control against actuator faults


## 2. System Description and Preliminaries

A UAV adopts a conventional layout, and the independently controlled surfaces include the following: left fully moving elevator, right fully moving elevator, aileron, and V-shaped rudder. Because the left and right fully moving elevator can be controlled independently; that is to say, in addition to the pitching moment produced by the same deflection, the rolling moment can also be produced by the differential deflection, which provides additional control moment in the lateral direction. At the same time, the application of V-shaped rudder also provides a supplement for the generation of pitching moment. Therefore, the characteristics of multicontrol surfaces of the UAV provide conditions for the compensation control of actuators faults.Firstly, the aircraft attitude dynamic equation in the body axial coordinate system is as follows:(1)μ̇α̇β̇=cosαcosβ0sinαcosβ−cosαtanβ1−sinαtanβsinα0−cosαpqr+FμXFαXFβX,where α, β, and μ are, respectively, the angle of attack, sideslip angle, and roll angle and p, q, and r are, respectively, the pitch angle rate, yaw angle, rate and roll angle rate.
(2)FμX=sinβcosμmVcosβ−Dsinβsinμ+Lcosμ−Ycosβsinμ−Fγt−mgcosγ+tanγ+tanβsinμmVDsinβcosμ+Lsinμ+Ycosβcosμ+Fχt,FαX=−cosμmVcosβ−Dsinβsinμ+Lcosμ−Ycosβsinμ−Fγt−mgcosγ+sinμmVcosβDsinβcosμ+Lsinμ+Ycosβcosμ+Fχt,FβX=−sinμmV−Dsinβsinμ+Lcosμ−Ycosβsinμ−Fγt−mgcosγ+cosμmVDsinβcosμ+Lsinμ+Ycosβcosμ+Fχt,Fχt=T⋅dT⋅−sinμcosαsinβ−cosμsinα,Fγt=T⋅dT⋅−sinμcosαsinβ−cosμsinα,where m is mass, V is flight speed, γ is flight path angle, T is engine thrust, dT is thrust coefficient, and D, L, and Y are resistance force, lift force, and lateral force, respectively, as described below.
(3)D=q¯⋅S⋅cd,L=q¯⋅S⋅cl,Y=q¯⋅S⋅cy,where S is wing area, q¯ is kinetic pressure, and cd, cl, and cy are drag coefficient, lift coefficient, and lateral force coefficient, respectively.Remark 1.
The flight control system architecture of UAV can support the integrated design of actuator fault detection and flight control law. The following distributed system architecture in Figure1 can be adopted [17–19]. The integrated VMC (vehicle management computer) realizes the fault detection of actuators, the calculation of flight control law, and the calculation of control allocation algorithm and finally forms the control instruction for each surface and transmits them to the SMART actuator located near the surface.Remark 2.
As shown in Figure2, the integrated VMC will perform the fault detection, flight control law, and control allocation algorithm continually during the whole flight mission. The distributed system architecture, depending on the integrated design of VMC, can realize the reconfiguration control based on the control allocation method more conveniently, without affecting the servo control of the back-end actuator control loop in SMART. Meanwhile, it can be seen that after the introduction of control allocation module, the design of control law module in VMC does not need to be changed when some of actuators are faulty, which provides convenience for its application.Remark 3.
The fault detection toward actuators’ fault modes needs to be added in the fault detection module of VMC. That is, the fault detection module can obtain all the data related to actuators fault modes, such as actuator position, motor HALL speed, and clutch drive current. Based on this, VMC can detect the failures of actuators. For example, when the motor HALL speed is high and the position of the actuator does not change, it can be considered that the actuator’s output shaft is stuck at this time. When the clutch drive current is large, then the current control command of the motor is cut off, and at this time, the actuator is in a state of nonoutput force. The paper does not make a comprehensive study of the specific fault detection strategy, only as the input conditions of the research.Figure 1
Distributed flight control system architecture.Figure 2
Reconfiguration control block diagram based on control allocation method.


## 3. Backstepping Optimal Controller Design

As we all know, the design of backstepping control law only considers the stability and convergence of the system and does not consider the optimal performance of the system [20, 21]. For the optimal control of the system, it is usually to find an admissible control with a given performance index, and the target functional takes the minimum value. This kind of problem ultimately comes down to the solution of Hamilton-Jacobi-Bellman partial differential equation. Compared with linear systems, the difficulty of nonlinear systems lies in that the HJB equation to be solved is often nonexistent or nonunique, which becomes the main obstacle to optimal control of nonlinear systems. This problem can be solved by introducing the inverse optimization into the design of uncertain nonlinear control systems through the robust control Lyapunov function [22]. The inverse optimization problem does not minimize the control law of a given target functional, but with the Lyapunov function for the robust stability of the system to make the controller is designed to minimize a backlog of objective functional, that is to say, first of all, get the control law, and then determine its optimal value function, so as to determine the optimal controller.


### 3.1. Derive the Control Law

For second order nonlinear systems,(4)ẋ1=ϕx1+x2,ẋ2=u,y=x1.(Step 1)
For the subsystemx1 in equation (4), virtual control law is designed with x2 as the control input.Choose the following form of control law:(5)x2des=−ψx1.Construct the following control Lyapunov function:(6)Wx1=12x12.Substituting intox2=x2des, its differential is Ẇ=ϕx1−ψx1x1.Whenϕx1−ψx1x1<0 is satisfied, its negative definite can be guaranteed.
(Step 2)
Introduce an error variable:(7)x¯2=x2−x2des=x2+ψx1.So the system equation (4) is equal to
(8)ẋ1=ϕx1−ψx1+x¯2,x¯̇2=u+ψ′x1ϕx1−ψx1+x¯2.Construct the following control Lyapunov function:(9)Vx1,x¯2=Fx1+12x¯22,where Fx1 is any effective control Lyapunov function of subsystem x1, which means that when x2=x2des is satisfied,
(10)Ḟx1=F′x1ϕx1−ψx1=−Ux1,where Ux1 is positive definite.
(11)V̇=F′x1ϕx1−ψx1+x¯2+x¯2u+ψ′x1ϕx1−ψx1+x¯2=−Ux1+x¯2F′x1+u+ψ′x1ϕx1−ψx1+ψ′x1x¯2.ChooseF′x1=−ψ′x1ϕx1−ψx1,F0=0; the item x1 in the second items can be cancelled out.SubstitutingF′x1 into formula (10), the following formula can be obtained:
(12)Ux1=ψ′x1ϕx1−ψx12.Whenψ′x1>0 is satisfied, the positive definite is guaranteed.And then, the Lyapunov function of the system is(13)V̇=−Ux1+x¯2u+ψ′x1x¯2.In order to makeV̇ negative definite, the control law can be selected finally:
(14)u=−k2x¯2=−k2x2+ψx1.Whenk2>ψ′x1 is satisfied, the negative definite of V̇=−Ux1−k2−ψ′x1x¯22 is guaranteed.


### 3.2. Determine Its Optimal Value Function

Lemma 4 (see [23, 24]).
Consider the nonlinear system:(15)ẋ=fx+gxu,where x∈Rn is the state variable and u∈Rm is the control input.
And assumegx≠0. For a given feedback control ux, define the optimal value function:
(16)Jx=∫0∞Qx+uTRxudt,where Qx is positive definite and Rx is symmetric positive definite.
The optimal value functionJx can be selected as the Lyapunov function Vx. Then, the optimal control can be obtained:
(17)u∗x=−12R−1xVxxgxT.
Using Hamilton-Jacobi-Bellman equation(18)minuQx+uTRxu+Vxxfx+gxu=0,it can be obtained inversely:
(19)Rx=−VxxgxT2u∗x,Qx=−Vxxfx−12Vxxgxu∗x.Based on Lemma4, comparing the error system shown in equation (8) with the second order system shown in equation (4), the following can be obtained:
(20)fx=ϕx1−ψx1+x¯2ψ′ϕx1−ψx1+x¯2,gx=01.And(21)Vx=F′x1x¯2=−ψ′x1ϕx1−ψx1x¯2.Therefore, the optimal control performance parameters that can be achieved by the control law (17) are obtained by substituting formulas (20) and (21):
(22)Rx=−VxxgxT2ux=x¯22k2x¯2=12k2,Qx=−Vxxfx−12Vxxgxux=ψ′x1ϕx1−ψx12+12k2−ψ′x1x¯22.If virtual control lawx2des=−ψx1=−k1x1 is selected, then, the linear control law of the system (4) is
(23)u=−k2x2+k1x1In order toQx be positive definite and the cost function to be a meaningful optimal performance, it must be guaranteed that
(24)k2>2k1.Therefore, combining (12), (14), and (23), we can get: when 0<2k1<k2 is satisfied; the origin of the system can be given global asymptotic stability.And minimize the following optimal value function:(25)J=∫0∞ψ′x1ϕx1−ψx12+12k−ψ′x1x2+ψx12+12ku2dt.Similarly, if virtual control lawx2des=−ψx1−r=−k1x1−r is selected, then, the linear control law of the system (4) is
(26)u=−k2ϕr+x2+k1x1−r,0<2k1<k2.Definingx−r=e, the tracking problem of the system at y=r can be transformed into the global asymptotic stability problem of the system at x−r, and the results of equations (23) and (24) above can be applied.Then, the system obtains global asymptotic stability aty=r.And minimize the following optimal value function:(27)J=∫0∞k1ϕx1−ϕr−k1x1−r2+12k2−k1x2+ϕr+k1x1−r2+12k2u2dt.


### 3.3. Design the Optimal Control Law

Considering the dynamics equation above, the following coordinate transformation is introduced:(28)pdqdrd=cosαcosβ0sinαcosβ−cosαsinβcosβ−sinαsinβ−sinα0cosαpqr.The corresponding dynamics equation becomes(29)μ̇α̇β̇=1cos2β0001cosβ000−1pdqdrd+FμXFαXFβX.Therefore, with the angular acceleration after coordinate transformation as the control input, a three-axis decoupled second-order nonlinear system can be obtained, as shown below:(30)α̇=1cosβ⋅qd+FαX,q̇d=u2,(31)β̇=−rd+FβX,ṙd=u3,(32)μ̇=1cos2β⋅pd+FμX,ṗd=u1.It can be found that the structures of equations (30), (31), (32), and (4) are exactly the same.Comparing equations (4) and (30), the following can be found:
(33)x1=α,x2=1cosβ⋅qd,y=x1,ϕx1=FαX,u=1cosβ⋅q̇d=1cosβ⋅u2.Therefore, it is easy to obtain the backstepping optimal controller to track control instructionsr=αcmd:
(34)u2=−kqqd+cosβFααcmd+kαcosβα−αcmd,(35)0<2kα<kq.At the same time, the following optimal value function can be minimized:(36)J=∫0∞kαFαα−Fααcmd−kαα‐αcmd2+kq−kα1cosβ⋅qd+Fααcmd+kαα‐αcmd2dt.Similarly, comparing equations (4) and (31), the following can be found:
(37)x1=β,x2=−rd,y=x1,ϕx1=FβX,u=−ṙd=−u3.Therefore, it is easy to obtain the backstepping optimal controller to track control instructionsr=βcmd:
(38)u3=kr−rd+Fββcmd+kββ−βcmd,(39)0<2kβ<kr.At the same time, the following optimal value function can be minimized:(40)J=∫0∞kβFββ−Fβ−kββ−βcmd2+kr−kβ−rd+Fββcmd+kββ−βcmd2dt.Similarly, comparing equations (4) and (32), the following can be found:
(41)x1=μ,x2=1cos2β⋅pd,y=x1,ϕx1=FμX,u=1cos2β⋅ṗd=1cos2β⋅u1.Therefore, it is easy to obtain the backstepping optimal controller to track control instructionsr=μcmd:
(42)u1=−kppd+cos2βFμμcmd+kμcos2βμ−μcmd,(43)0<2kμ<kp.At the same time, the following optimal value function can be minimized:(44)J=∫0∞kμFμμ−Fμμcmd−kμμ‐μcmd2+kp−kμ1cos2β⋅pd+Fμμcmd+kμμ‐μcmd2dt.Through formulas (34), (38), and (42), the angular acceleration ωd′des=pdcmd′qdcmd′rdcmd′T required to achieve attitude maneuver can be obtained. After integral operation and inverse coordinate transformation, the following formula can be obtained:
(45)ωdes=Tbd1sωd′des.To this end, according to the expected flight attitude requirements, the angular velocity required to complete the maneuver can be obtained by using formula (45), and then, through the following formula, it can be converted into the torque coefficient required by the maneuver, which is also the input of control allocation problem in the next section.
(46)T=ITbdα,βω̇ddes+ω×Iω,v=clcmcnT=1Sq¯diagb,c¯,b−1T,where Tbd is the coordinate transformation matrix converted to the body shafting, I is the rotational inertia matrix of the aircraft, b is the wingspan length, and c¯ is the average aerodynamic chord length.


## 3.1. Derive the Control Law

For second order nonlinear systems,(4)ẋ1=ϕx1+x2,ẋ2=u,y=x1.(Step 1)
For the subsystemx1 in equation (4), virtual control law is designed with x2 as the control input.Choose the following form of control law:(5)x2des=−ψx1.Construct the following control Lyapunov function:(6)Wx1=12x12.Substituting intox2=x2des, its differential is Ẇ=ϕx1−ψx1x1.Whenϕx1−ψx1x1<0 is satisfied, its negative definite can be guaranteed.
(Step 2)
Introduce an error variable:(7)x¯2=x2−x2des=x2+ψx1.So the system equation (4) is equal to
(8)ẋ1=ϕx1−ψx1+x¯2,x¯̇2=u+ψ′x1ϕx1−ψx1+x¯2.Construct the following control Lyapunov function:(9)Vx1,x¯2=Fx1+12x¯22,where Fx1 is any effective control Lyapunov function of subsystem x1, which means that when x2=x2des is satisfied,
(10)Ḟx1=F′x1ϕx1−ψx1=−Ux1,where Ux1 is positive definite.
(11)V̇=F′x1ϕx1−ψx1+x¯2+x¯2u+ψ′x1ϕx1−ψx1+x¯2=−Ux1+x¯2F′x1+u+ψ′x1ϕx1−ψx1+ψ′x1x¯2.ChooseF′x1=−ψ′x1ϕx1−ψx1,F0=0; the item x1 in the second items can be cancelled out.SubstitutingF′x1 into formula (10), the following formula can be obtained:
(12)Ux1=ψ′x1ϕx1−ψx12.Whenψ′x1>0 is satisfied, the positive definite is guaranteed.And then, the Lyapunov function of the system is(13)V̇=−Ux1+x¯2u+ψ′x1x¯2.In order to makeV̇ negative definite, the control law can be selected finally:
(14)u=−k2x¯2=−k2x2+ψx1.Whenk2>ψ′x1 is satisfied, the negative definite of V̇=−Ux1−k2−ψ′x1x¯22 is guaranteed.


## 3.2. Determine Its Optimal Value Function

Lemma 4 (see [23, 24]).
Consider the nonlinear system:(15)ẋ=fx+gxu,where x∈Rn is the state variable and u∈Rm is the control input.
And assumegx≠0. For a given feedback control ux, define the optimal value function:
(16)Jx=∫0∞Qx+uTRxudt,where Qx is positive definite and Rx is symmetric positive definite.
The optimal value functionJx can be selected as the Lyapunov function Vx. Then, the optimal control can be obtained:
(17)u∗x=−12R−1xVxxgxT.
Using Hamilton-Jacobi-Bellman equation(18)minuQx+uTRxu+Vxxfx+gxu=0,it can be obtained inversely:
(19)Rx=−VxxgxT2u∗x,Qx=−Vxxfx−12Vxxgxu∗x.Based on Lemma4, comparing the error system shown in equation (8) with the second order system shown in equation (4), the following can be obtained:
(20)fx=ϕx1−ψx1+x¯2ψ′ϕx1−ψx1+x¯2,gx=01.And(21)Vx=F′x1x¯2=−ψ′x1ϕx1−ψx1x¯2.Therefore, the optimal control performance parameters that can be achieved by the control law (17) are obtained by substituting formulas (20) and (21):
(22)Rx=−VxxgxT2ux=x¯22k2x¯2=12k2,Qx=−Vxxfx−12Vxxgxux=ψ′x1ϕx1−ψx12+12k2−ψ′x1x¯22.If virtual control lawx2des=−ψx1=−k1x1 is selected, then, the linear control law of the system (4) is
(23)u=−k2x2+k1x1In order toQx be positive definite and the cost function to be a meaningful optimal performance, it must be guaranteed that
(24)k2>2k1.Therefore, combining (12), (14), and (23), we can get: when 0<2k1<k2 is satisfied; the origin of the system can be given global asymptotic stability.And minimize the following optimal value function:(25)J=∫0∞ψ′x1ϕx1−ψx12+12k−ψ′x1x2+ψx12+12ku2dt.Similarly, if virtual control lawx2des=−ψx1−r=−k1x1−r is selected, then, the linear control law of the system (4) is
(26)u=−k2ϕr+x2+k1x1−r,0<2k1<k2.Definingx−r=e, the tracking problem of the system at y=r can be transformed into the global asymptotic stability problem of the system at x−r, and the results of equations (23) and (24) above can be applied.Then, the system obtains global asymptotic stability aty=r.And minimize the following optimal value function:(27)J=∫0∞k1ϕx1−ϕr−k1x1−r2+12k2−k1x2+ϕr+k1x1−r2+12k2u2dt.


## 3.3. Design the Optimal Control Law

Considering the dynamics equation above, the following coordinate transformation is introduced:(28)pdqdrd=cosαcosβ0sinαcosβ−cosαsinβcosβ−sinαsinβ−sinα0cosαpqr.The corresponding dynamics equation becomes(29)μ̇α̇β̇=1cos2β0001cosβ000−1pdqdrd+FμXFαXFβX.Therefore, with the angular acceleration after coordinate transformation as the control input, a three-axis decoupled second-order nonlinear system can be obtained, as shown below:(30)α̇=1cosβ⋅qd+FαX,q̇d=u2,(31)β̇=−rd+FβX,ṙd=u3,(32)μ̇=1cos2β⋅pd+FμX,ṗd=u1.It can be found that the structures of equations (30), (31), (32), and (4) are exactly the same.Comparing equations (4) and (30), the following can be found:
(33)x1=α,x2=1cosβ⋅qd,y=x1,ϕx1=FαX,u=1cosβ⋅q̇d=1cosβ⋅u2.Therefore, it is easy to obtain the backstepping optimal controller to track control instructionsr=αcmd:
(34)u2=−kqqd+cosβFααcmd+kαcosβα−αcmd,(35)0<2kα<kq.At the same time, the following optimal value function can be minimized:(36)J=∫0∞kαFαα−Fααcmd−kαα‐αcmd2+kq−kα1cosβ⋅qd+Fααcmd+kαα‐αcmd2dt.Similarly, comparing equations (4) and (31), the following can be found:
(37)x1=β,x2=−rd,y=x1,ϕx1=FβX,u=−ṙd=−u3.Therefore, it is easy to obtain the backstepping optimal controller to track control instructionsr=βcmd:
(38)u3=kr−rd+Fββcmd+kββ−βcmd,(39)0<2kβ<kr.At the same time, the following optimal value function can be minimized:(40)J=∫0∞kβFββ−Fβ−kββ−βcmd2+kr−kβ−rd+Fββcmd+kββ−βcmd2dt.Similarly, comparing equations (4) and (32), the following can be found:
(41)x1=μ,x2=1cos2β⋅pd,y=x1,ϕx1=FμX,u=1cos2β⋅ṗd=1cos2β⋅u1.Therefore, it is easy to obtain the backstepping optimal controller to track control instructionsr=μcmd:
(42)u1=−kppd+cos2βFμμcmd+kμcos2βμ−μcmd,(43)0<2kμ<kp.At the same time, the following optimal value function can be minimized:(44)J=∫0∞kμFμμ−Fμμcmd−kμμ‐μcmd2+kp−kμ1cos2β⋅pd+Fμμcmd+kμμ‐μcmd2dt.Through formulas (34), (38), and (42), the angular acceleration ωd′des=pdcmd′qdcmd′rdcmd′T required to achieve attitude maneuver can be obtained. After integral operation and inverse coordinate transformation, the following formula can be obtained:
(45)ωdes=Tbd1sωd′des.To this end, according to the expected flight attitude requirements, the angular velocity required to complete the maneuver can be obtained by using formula (45), and then, through the following formula, it can be converted into the torque coefficient required by the maneuver, which is also the input of control allocation problem in the next section.
(46)T=ITbdα,βω̇ddes+ω×Iω,v=clcmcnT=1Sq¯diagb,c¯,b−1T,where Tbd is the coordinate transformation matrix converted to the body shafting, I is the rotational inertia matrix of the aircraft, b is the wingspan length, and c¯ is the average aerodynamic chord length.


## 4. Fault Tolerant Control Design Based on Control Allocation

As mentioned above, as UAV plays an increasingly important role in the air combat system, the tasks it undertakes become more and more complex, and it needs to bear certain impact of faults. For the flight control system, the UAV has the characteristics of multicontrol surfaces, especially the introduction of multiredundant control surfaces, which provides a prerequisite for the reconfiguration control of UAV against actuator failures.As shown in Figure2, when the fault detection module in VMC finds that there is any fault mode toward actuators, the control allocation module in VMC will switch from the normal state to the specific fault state. Then, the fault tolerant control strategy based on the control allocation works, and the control instructions solved by the control allocation module will reflect the influence of the actuator fault mode.


### 4.1. Nonlinear Control Allocation Design Considerations

For the nonlinear system, assume its motion equation is(47)ẋ=fx,gx,u,where f:Rn×Rk↦Rn and g:Rn×Rm↦Rk are nonlinear and k<m.It can be written as(48)ẋ=fx+gux,u,gux,u=Bvgx,u,where Bv∈Rn×k, f, and g have the same mapping form as above.Introduce virtual controls:(49)v=gx,u,where v∈Rk; the state equation of the system can be rewritten as
(50)ẋ=fx+Bvv=fx+Bvgx,u.Thus, nonlinear system (48) can be converted into equations (49) and (50), which can also use the standard form of the control allocation problem. It can be seen that with linear control allocation problem the difference is that mapping is a nonlinear form.In the flight control system, the control allocation strategy is required to be solved in real time, but the nonlinear control allocation problem cannot be solved in real time. Therefore, one of the methods to solve this problem is to use the method of local approximation mapping, using linear mapping to approximate nonlinear mapping.Through local Taylor form expansion, linearization at pointu0 can be obtained:
(51)gx,u≈gx,u0+∂g∂ux,u0⋅u−u0.So introduce a linear mappingBx=∂g/∂ux,u0; the nonlinear control allocation problem (49) can be converted to the linear control allocation problem:
(52)v¯=Bxu,(53)v¯=v−gx,u0+Bxu0.When a nonlinear control allocation problem is transformed into a linear problem, it can be solved in many ways. Optimization algorithms based on linear programming, with its lower operation cost and simplex method, have been widely studied in the process of solving control allocation problems [25, 26]. According to the authors’ previous research results [27], the allocation algorithm based on linear programming is directly applied here.Combined with formulas (52) and (53), the control allocation problem is described as follows.For the known control efficiency matrixB and the given virtual control quantity vt, the feasible instruction ut of the control surface is determined by considering the position limit and rate limit u¯t≤ut≤u¯t of the control surfaces, so that But=vt is satisfied.Thus, the linear programming problem can be solved with the following matrix:(54)A=M⋅B,b=−A⋅umin,Aeq=,beq=,fT=−vdT⋅B,lb=0ub=umax−umin.The corresponding optimization objectives are(55)maxρ=B⋅uvd=minJ=−vdT⋅B⋅u.In the three-dimensional control allocation problem, the matrixM has only two rows, i.e.,
(56)M=vd,2‐vd,10vd,30‐vd,1and the coefficient matrix A=M⋅B of constraint conditions also has only two rows, whose rank is 2. Therefore, the solution process is relatively simple.The application of the method in the flight control system is toward tov=clcmcnT=1/Sq¯diagb,c¯,b−1T, and the specific process can be summarized as the following steps:
(Step 1)
According to the theory of nonlinear control allocation, the control efficiency matrix under nonlinear condition should be calculated from the expression of aerodynamic derivativeCM=ClCmCnT:(57)Bx=∂CM∂δα,β,p,q,r,δ0.δ0 can be selected as the control input ut−T of previous sampling time or a fixed point, such as δ0=0.
(Step 2)
Then the virtual control input after Taylor linear expansion is calculated:(58)v¯=v−CMα,β,p,q,r,δ0+Bxδ0.(Step 3)
Using the control allocation method of formula (54) to solve the above formula, the solution of formula (58) can be obtained.


### 4.2. Fault Tolerant Control Design for Actuators

This section describes the fault-tolerant control scheme according to the control allocation design method described in Section4.1 for the two possible fault modes of the electric actuators described in Figure 1. Fault mode 1 is nonoutput force of the actuator, and the other fault mode is output shaft stuck of the actuator.The nonlinear system equations of an aircraft under normal conditions are described as follows:(59)ẋ=fx+Bvv=fx+Bvgx,u,where gx,u represents the control efficiency of surfaces, which will change when the surfaces fails due to actuators. The following equation describes the nonlinear system equation of the aircraft in the case of failure:
(60)ẋ=fx+Bvgfx,uf,where gfx,uf represents the control efficiency matrix in the case of failure and uf∈Rn represents the control input. For different fault modes, equation (60) corresponds to different function forms.According to the requirements of reconfiguration control, control allocation needs to achieve:(61)gx,u=gfx,uf.For the control allocation problem under normal conditions, the allocation objective is(62)vt=gx,u.The nonlinear control allocation problem (62) can be converted to the linear control allocation problem:
(63)v¯=v−gx,u0+Bxu0=Bxu.Similarly, for the control allocation problem under faulty conditions, the allocation objective is(64)vft=gfx,uf.After linearization,(65)gfx,uf≈gfx,u0+∂gf∂ux,u0⋅uf−u0.The control efficiency matrixBfx=∂gf/∂ux,u0 is introduced, and the linear control allocation problem similarly is as follows:
(66)v¯f=vf−gfx,u0+Bfxu0=Bfxu.When the fault modes of the actuators occur, the change of aerodynamic coefficient, weight, and center of gravity of the aircraft can be ignored. On this premise, the change of control capability caused by the faulty actuators only comes from the change of control input.Meanwhile, when the linearized equilibrium points areu0=0, the following formula can be obtained:
(67)gx,u0=gfx,u0,(68)Bxu0=Bfxu0.Thus, compare formulas (64) and (67), and the requirements of reconfiguration control (61) can be converted into
(69)v¯t=v¯ft.


#### 4.2.1. Fault Mode 1: Output Shaft Stuck

The output shaft stuck of actuator refers to the fault mode in which the actuator is in a fixed position due to motor blocking or other mechanical reasons, which can be realized by the fault detection module of VMC. In this fault mode, the control surface connected to the actuator will also be stuck in a certain position. In this case, its deflection will not only fail to produce the desired control effect but also to produce unwanted additional forces and unwanted additional torques. Therefore, the influence of the stuck actuator must be offset in the control allocation.When the shaft of actuator is stuck, the motion equation of the aircraft does not change, as shown in equation (60).According to formula (66), the solution of control allocation for reconstruction control can be obtained:
(70)v¯=Bu=Bfuf=Bf∗⋅uf∗+bjδf,where assuming that the aircraft has m surfaces, uf∗=δ1,δ2,⋯δj⋯δm−1∈Rm−1, Bf∗ is the remaining control efficiency matrix after removing the stuck surface in Bf, bj is the control efficiency coefficient of the stuck surface, and δf is the stuck position of the faulty surface.After further derivation, the following can be obtained:(71)uf∗=Bf∗+Bu−Bf∗+bjδfAmong them, the first item redistributes the torque required by the original surface to the remaining effective surfaces, and the second item is used to offset the additional influence caused by the stuck surface, so that the actual output of the stuck rudder surface is(72)uf=δ1,δ2,⋯,δj−1,δf,δj+1,⋯,δm−1.


#### 4.2.2. Fault Mode 2: Nonoutput Force

Nonoutput force of actuator refers to the failure mode in which the clutch cuts off the current control output due to various reasons, and then, the actuator has no output force. It can be realized by the fault detection module of VMC. In this fault mode, the surface connected with the actuator will also be in a loosely floating state, that is, the control efficiency of the surface is zero. Then, it can be assumed that the loosely floating surface will not produce any aerodynamic and aerodynamic torque; that is, the effect of the control surface on the control input of the aircraft is zero.Similar to fault mode 1, when the actuator has no output force, the state equation of the aircraft does not change. That is, only the control input of the loosely floating surface is zero, and the loss of control capability caused by the loss of control capability needs to be compensated by the remaining effective control surfaces.Similarly, assuminguf∗ is the input of the remaining surface after removing the loosely-floating one, then,
(73)v¯=Bu=Bfuf=Bf∗⋅uf∗+bjδf,where, assuming that the aircraft has m surfaces, uf∗=δ1,δ2,⋯δj⋯δm−1∈Rm−1, Bf∗ is the remaining control efficiency matrix after removing the loosely floating surface in Bf, bj is the control efficiency coefficient of the loosely-floating surface, and δf=0.Further derivation can be obtained:(74)uf∗=Bf∗+Bu−Bf∗+bjδf.Then, the actual output of the surfaces when the actuator has no output force can be obtained:(75)uf=δ1,δ2,⋯δj−1,0,δj+1,⋯δm−1.


## 4.1. Nonlinear Control Allocation Design Considerations

For the nonlinear system, assume its motion equation is(47)ẋ=fx,gx,u,where f:Rn×Rk↦Rn and g:Rn×Rm↦Rk are nonlinear and k<m.It can be written as(48)ẋ=fx+gux,u,gux,u=Bvgx,u,where Bv∈Rn×k, f, and g have the same mapping form as above.Introduce virtual controls:(49)v=gx,u,where v∈Rk; the state equation of the system can be rewritten as
(50)ẋ=fx+Bvv=fx+Bvgx,u.Thus, nonlinear system (48) can be converted into equations (49) and (50), which can also use the standard form of the control allocation problem. It can be seen that with linear control allocation problem the difference is that mapping is a nonlinear form.In the flight control system, the control allocation strategy is required to be solved in real time, but the nonlinear control allocation problem cannot be solved in real time. Therefore, one of the methods to solve this problem is to use the method of local approximation mapping, using linear mapping to approximate nonlinear mapping.Through local Taylor form expansion, linearization at pointu0 can be obtained:
(51)gx,u≈gx,u0+∂g∂ux,u0⋅u−u0.So introduce a linear mappingBx=∂g/∂ux,u0; the nonlinear control allocation problem (49) can be converted to the linear control allocation problem:
(52)v¯=Bxu,(53)v¯=v−gx,u0+Bxu0.When a nonlinear control allocation problem is transformed into a linear problem, it can be solved in many ways. Optimization algorithms based on linear programming, with its lower operation cost and simplex method, have been widely studied in the process of solving control allocation problems [25, 26]. According to the authors’ previous research results [27], the allocation algorithm based on linear programming is directly applied here.Combined with formulas (52) and (53), the control allocation problem is described as follows.For the known control efficiency matrixB and the given virtual control quantity vt, the feasible instruction ut of the control surface is determined by considering the position limit and rate limit u¯t≤ut≤u¯t of the control surfaces, so that But=vt is satisfied.Thus, the linear programming problem can be solved with the following matrix:(54)A=M⋅B,b=−A⋅umin,Aeq=,beq=,fT=−vdT⋅B,lb=0ub=umax−umin.The corresponding optimization objectives are(55)maxρ=B⋅uvd=minJ=−vdT⋅B⋅u.In the three-dimensional control allocation problem, the matrixM has only two rows, i.e.,
(56)M=vd,2‐vd,10vd,30‐vd,1and the coefficient matrix A=M⋅B of constraint conditions also has only two rows, whose rank is 2. Therefore, the solution process is relatively simple.The application of the method in the flight control system is toward tov=clcmcnT=1/Sq¯diagb,c¯,b−1T, and the specific process can be summarized as the following steps:
(Step 1)
According to the theory of nonlinear control allocation, the control efficiency matrix under nonlinear condition should be calculated from the expression of aerodynamic derivativeCM=ClCmCnT:(57)Bx=∂CM∂δα,β,p,q,r,δ0.δ0 can be selected as the control input ut−T of previous sampling time or a fixed point, such as δ0=0.
(Step 2)
Then the virtual control input after Taylor linear expansion is calculated:(58)v¯=v−CMα,β,p,q,r,δ0+Bxδ0.(Step 3)
Using the control allocation method of formula (54) to solve the above formula, the solution of formula (58) can be obtained.


## 4.2. Fault Tolerant Control Design for Actuators

This section describes the fault-tolerant control scheme according to the control allocation design method described in Section4.1 for the two possible fault modes of the electric actuators described in Figure 1. Fault mode 1 is nonoutput force of the actuator, and the other fault mode is output shaft stuck of the actuator.The nonlinear system equations of an aircraft under normal conditions are described as follows:(59)ẋ=fx+Bvv=fx+Bvgx,u,where gx,u represents the control efficiency of surfaces, which will change when the surfaces fails due to actuators. The following equation describes the nonlinear system equation of the aircraft in the case of failure:
(60)ẋ=fx+Bvgfx,uf,where gfx,uf represents the control efficiency matrix in the case of failure and uf∈Rn represents the control input. For different fault modes, equation (60) corresponds to different function forms.According to the requirements of reconfiguration control, control allocation needs to achieve:(61)gx,u=gfx,uf.For the control allocation problem under normal conditions, the allocation objective is(62)vt=gx,u.The nonlinear control allocation problem (62) can be converted to the linear control allocation problem:
(63)v¯=v−gx,u0+Bxu0=Bxu.Similarly, for the control allocation problem under faulty conditions, the allocation objective is(64)vft=gfx,uf.After linearization,(65)gfx,uf≈gfx,u0+∂gf∂ux,u0⋅uf−u0.The control efficiency matrixBfx=∂gf/∂ux,u0 is introduced, and the linear control allocation problem similarly is as follows:
(66)v¯f=vf−gfx,u0+Bfxu0=Bfxu.When the fault modes of the actuators occur, the change of aerodynamic coefficient, weight, and center of gravity of the aircraft can be ignored. On this premise, the change of control capability caused by the faulty actuators only comes from the change of control input.Meanwhile, when the linearized equilibrium points areu0=0, the following formula can be obtained:
(67)gx,u0=gfx,u0,(68)Bxu0=Bfxu0.Thus, compare formulas (64) and (67), and the requirements of reconfiguration control (61) can be converted into
(69)v¯t=v¯ft.


### 4.2.1. Fault Mode 1: Output Shaft Stuck

The output shaft stuck of actuator refers to the fault mode in which the actuator is in a fixed position due to motor blocking or other mechanical reasons, which can be realized by the fault detection module of VMC. In this fault mode, the control surface connected to the actuator will also be stuck in a certain position. In this case, its deflection will not only fail to produce the desired control effect but also to produce unwanted additional forces and unwanted additional torques. Therefore, the influence of the stuck actuator must be offset in the control allocation.When the shaft of actuator is stuck, the motion equation of the aircraft does not change, as shown in equation (60).According to formula (66), the solution of control allocation for reconstruction control can be obtained:
(70)v¯=Bu=Bfuf=Bf∗⋅uf∗+bjδf,where assuming that the aircraft has m surfaces, uf∗=δ1,δ2,⋯δj⋯δm−1∈Rm−1, Bf∗ is the remaining control efficiency matrix after removing the stuck surface in Bf, bj is the control efficiency coefficient of the stuck surface, and δf is the stuck position of the faulty surface.After further derivation, the following can be obtained:(71)uf∗=Bf∗+Bu−Bf∗+bjδfAmong them, the first item redistributes the torque required by the original surface to the remaining effective surfaces, and the second item is used to offset the additional influence caused by the stuck surface, so that the actual output of the stuck rudder surface is(72)uf=δ1,δ2,⋯,δj−1,δf,δj+1,⋯,δm−1.


### 4.2.2. Fault Mode 2: Nonoutput Force

Nonoutput force of actuator refers to the failure mode in which the clutch cuts off the current control output due to various reasons, and then, the actuator has no output force. It can be realized by the fault detection module of VMC. In this fault mode, the surface connected with the actuator will also be in a loosely floating state, that is, the control efficiency of the surface is zero. Then, it can be assumed that the loosely floating surface will not produce any aerodynamic and aerodynamic torque; that is, the effect of the control surface on the control input of the aircraft is zero.Similar to fault mode 1, when the actuator has no output force, the state equation of the aircraft does not change. That is, only the control input of the loosely floating surface is zero, and the loss of control capability caused by the loss of control capability needs to be compensated by the remaining effective control surfaces.Similarly, assuminguf∗ is the input of the remaining surface after removing the loosely-floating one, then,
(73)v¯=Bu=Bfuf=Bf∗⋅uf∗+bjδf,where, assuming that the aircraft has m surfaces, uf∗=δ1,δ2,⋯δj⋯δm−1∈Rm−1, Bf∗ is the remaining control efficiency matrix after removing the loosely floating surface in Bf, bj is the control efficiency coefficient of the loosely-floating surface, and δf=0.Further derivation can be obtained:(74)uf∗=Bf∗+Bu−Bf∗+bjδf.Then, the actual output of the surfaces when the actuator has no output force can be obtained:(75)uf=δ1,δ2,⋯δj−1,0,δj+1,⋯δm−1.


## 4.2.1. Fault Mode 1: Output Shaft Stuck

The output shaft stuck of actuator refers to the fault mode in which the actuator is in a fixed position due to motor blocking or other mechanical reasons, which can be realized by the fault detection module of VMC. In this fault mode, the control surface connected to the actuator will also be stuck in a certain position. In this case, its deflection will not only fail to produce the desired control effect but also to produce unwanted additional forces and unwanted additional torques. Therefore, the influence of the stuck actuator must be offset in the control allocation.When the shaft of actuator is stuck, the motion equation of the aircraft does not change, as shown in equation (60).According to formula (66), the solution of control allocation for reconstruction control can be obtained:
(70)v¯=Bu=Bfuf=Bf∗⋅uf∗+bjδf,where assuming that the aircraft has m surfaces, uf∗=δ1,δ2,⋯δj⋯δm−1∈Rm−1, Bf∗ is the remaining control efficiency matrix after removing the stuck surface in Bf, bj is the control efficiency coefficient of the stuck surface, and δf is the stuck position of the faulty surface.After further derivation, the following can be obtained:(71)uf∗=Bf∗+Bu−Bf∗+bjδfAmong them, the first item redistributes the torque required by the original surface to the remaining effective surfaces, and the second item is used to offset the additional influence caused by the stuck surface, so that the actual output of the stuck rudder surface is(72)uf=δ1,δ2,⋯,δj−1,δf,δj+1,⋯,δm−1.


## 4.2.2. Fault Mode 2: Nonoutput Force

Nonoutput force of actuator refers to the failure mode in which the clutch cuts off the current control output due to various reasons, and then, the actuator has no output force. It can be realized by the fault detection module of VMC. In this fault mode, the surface connected with the actuator will also be in a loosely floating state, that is, the control efficiency of the surface is zero. Then, it can be assumed that the loosely floating surface will not produce any aerodynamic and aerodynamic torque; that is, the effect of the control surface on the control input of the aircraft is zero.Similar to fault mode 1, when the actuator has no output force, the state equation of the aircraft does not change. That is, only the control input of the loosely floating surface is zero, and the loss of control capability caused by the loss of control capability needs to be compensated by the remaining effective control surfaces.Similarly, assuminguf∗ is the input of the remaining surface after removing the loosely-floating one, then,
(73)v¯=Bu=Bfuf=Bf∗⋅uf∗+bjδf,where, assuming that the aircraft has m surfaces, uf∗=δ1,δ2,⋯δj⋯δm−1∈Rm−1, Bf∗ is the remaining control efficiency matrix after removing the loosely floating surface in Bf, bj is the control efficiency coefficient of the loosely-floating surface, and δf=0.Further derivation can be obtained:(74)uf∗=Bf∗+Bu−Bf∗+bjδf.Then, the actual output of the surfaces when the actuator has no output force can be obtained:(75)uf=δ1,δ2,⋯δj−1,0,δj+1,⋯δm−1.


## 5. Numerical Simulation

### 5.1. Control Objective

Based on the design of backstepping optimal controller in Section3 and fault-tolerant control based on control allocation in Section 4, numerical simulations are conducted for the UAV.In order to prove the correctness of the control law, the numerical simulation is carried out under the comparison of the two cases. The first set is three-axis maneuver without actuator failures, and the control objectives areαcmd=25∘,βcmd=0∘,μcmd=25∘.The fault tolerant control of three-axis maneuver under the condition of actuator faults is the second set. As above, the same simulated UAV object is used, and the initial trim conditions are the same. In the simulation, the parameters of backstepping optimal control law do not change, but the allocation algorithm switches from normal modes to the fault modes in control allocation. Also consider the three-axis maneuver; the control objectives areαcmd=25°,βcmd=0°,μcmd=25°.Firstly, the deflection characteristics of the four surfaces of the UAV, namely, left elevator, right elevator, aileron, and rudder, are limited as follows:(76)Umax=30°,30°,30°,30°,Umin=−30°,−30°,−30°,−30°.It should be added here that the surface is stuck beyond the deflection limit due to mechanical reasons, and VMC cannot give the correct diagnosis through the fault detection module. The control allocation method may invalidate the input saturation strategy in these fault modes. And this failure condition is not covered in the paper because it cannot be correctly diagnosed.Then, the initial trim conditions of the simulation are shown below: the trimming velocity isV=350ft/s, the trimming angle of attack is α0=6.76°, the remaining trimming flight statuses are β0=μ0=p0=q0=r0=0, and the trimming angles of deflection are, respectively δel,δel,δa,δr=−0.4449°,−0.4449°,−1.6193°,0.3226°.Finally, the parameters of the optimal controller described in Section3 are kα=2.53, kq=5.76, kβ=1.36, kr=5.14, kμ=1.64, and kp=6.23.


### 5.2. Simulation Result

It can be seen from the simulation results that the backstepping optimal controller can achieve a good control effect. In the control channels of the three axes, the controlled quantity can meet the requirements of the control instruction within 5 s, and the response has no overshoot and no steady-state error, refer to Figure3. The black dotted lines in Figure 4 are the upper and lower limits of surface deflection. This proves that it is feasible and correct to design flight control system using backstepping optimal controller. At the same time, it also shows that the control allocation method can realize the correct distribution of control instructions on each controllable surface.Figure 3
The state response of three-axis maneuver.Figure 4
The deflection angle of surfaces. The forces and torques created by the four independent surfaces contribute to the aircraft’s three-axis maneuver.


### 5.3. Simulation of Fault Mode 1

In the simulation, the electric actuator attached to the rudder is set to be stuck at 5°. Then, the rudder will also be stuck at 5°, and its deflection will fail not only to produce the desired control effect but also to produce unwanted control effect.


### 5.4. Simulation of Fault Mode 2

In the simulation, the electric actuator attached to the rudder is set to nonoutput force. Then, the rudder will also be in a loosely floating state. Then, it can be assumed that the loosely floating rudder will not produce any aerodynamic and aerodynamic torque.In Figures5 and 6, the red curve is the deflection angle of surfaces without the actuator failure, and the blue curve is the deflection angle of surfaces in the corresponding fault mode, and the black dotted lines are the upper and lower limits of surface deflection. It can be seen that when actuator failures occur, the reconfiguration based on the control allocation method ensures remaining surfaces can compensate the influence caused by the faulty actuator. The angle of attack and the angle of roll track the control objectives quickly with stable performance; refer to Figures 7 and 8. Therefore, the design of control allocation completes the task of reconfiguration control well, obtains good control effect, and achieves the goal of fault-tolerant control.Figure 5
The deflection angle of surfaces against fault mode 1. The remaining surfaces compensate for the abnormal forces and torques caused by the stuck actuator attached to rudder.Figure 6
The deflection angle of surfaces against fault mode 2. The remaining surfaces also compensate for the loss forces and torques caused by the loosely floating actuator attached to rudder.Figure 7
The state response of three-axis maneuver against fault mode 1.Figure 8
The state response of three-axis maneuver against fault mode 2.


## 5.1. Control Objective

Based on the design of backstepping optimal controller in Section3 and fault-tolerant control based on control allocation in Section 4, numerical simulations are conducted for the UAV.In order to prove the correctness of the control law, the numerical simulation is carried out under the comparison of the two cases. The first set is three-axis maneuver without actuator failures, and the control objectives areαcmd=25∘,βcmd=0∘,μcmd=25∘.The fault tolerant control of three-axis maneuver under the condition of actuator faults is the second set. As above, the same simulated UAV object is used, and the initial trim conditions are the same. In the simulation, the parameters of backstepping optimal control law do not change, but the allocation algorithm switches from normal modes to the fault modes in control allocation. Also consider the three-axis maneuver; the control objectives areαcmd=25°,βcmd=0°,μcmd=25°.Firstly, the deflection characteristics of the four surfaces of the UAV, namely, left elevator, right elevator, aileron, and rudder, are limited as follows:(76)Umax=30°,30°,30°,30°,Umin=−30°,−30°,−30°,−30°.It should be added here that the surface is stuck beyond the deflection limit due to mechanical reasons, and VMC cannot give the correct diagnosis through the fault detection module. The control allocation method may invalidate the input saturation strategy in these fault modes. And this failure condition is not covered in the paper because it cannot be correctly diagnosed.Then, the initial trim conditions of the simulation are shown below: the trimming velocity isV=350ft/s, the trimming angle of attack is α0=6.76°, the remaining trimming flight statuses are β0=μ0=p0=q0=r0=0, and the trimming angles of deflection are, respectively δel,δel,δa,δr=−0.4449°,−0.4449°,−1.6193°,0.3226°.Finally, the parameters of the optimal controller described in Section3 are kα=2.53, kq=5.76, kβ=1.36, kr=5.14, kμ=1.64, and kp=6.23.


## 5.2. Simulation Result

It can be seen from the simulation results that the backstepping optimal controller can achieve a good control effect. In the control channels of the three axes, the controlled quantity can meet the requirements of the control instruction within 5 s, and the response has no overshoot and no steady-state error, refer to Figure3. The black dotted lines in Figure 4 are the upper and lower limits of surface deflection. This proves that it is feasible and correct to design flight control system using backstepping optimal controller. At the same time, it also shows that the control allocation method can realize the correct distribution of control instructions on each controllable surface.Figure 3
The state response of three-axis maneuver.Figure 4
The deflection angle of surfaces. The forces and torques created by the four independent surfaces contribute to the aircraft’s three-axis maneuver.


## 5.3. Simulation of Fault Mode 1

In the simulation, the electric actuator attached to the rudder is set to be stuck at 5°. Then, the rudder will also be stuck at 5°, and its deflection will fail not only to produce the desired control effect but also to produce unwanted control effect.


## 5.4. Simulation of Fault Mode 2

In the simulation, the electric actuator attached to the rudder is set to nonoutput force. Then, the rudder will also be in a loosely floating state. Then, it can be assumed that the loosely floating rudder will not produce any aerodynamic and aerodynamic torque.In Figures5 and 6, the red curve is the deflection angle of surfaces without the actuator failure, and the blue curve is the deflection angle of surfaces in the corresponding fault mode, and the black dotted lines are the upper and lower limits of surface deflection. It can be seen that when actuator failures occur, the reconfiguration based on the control allocation method ensures remaining surfaces can compensate the influence caused by the faulty actuator. The angle of attack and the angle of roll track the control objectives quickly with stable performance; refer to Figures 7 and 8. Therefore, the design of control allocation completes the task of reconfiguration control well, obtains good control effect, and achieves the goal of fault-tolerant control.Figure 5
The deflection angle of surfaces against fault mode 1. The remaining surfaces compensate for the abnormal forces and torques caused by the stuck actuator attached to rudder.Figure 6
The deflection angle of surfaces against fault mode 2. The remaining surfaces also compensate for the loss forces and torques caused by the loosely floating actuator attached to rudder.Figure 7
The state response of three-axis maneuver against fault mode 1.Figure 8
The state response of three-axis maneuver against fault mode 2.


## 6. Conclusion

The fault-tolerant control of UAV in the case of actuator failure in the flight control system is studied. Based on the backstepping control method derived from Lyapunov function, the nonlinear optimal control law was constructed by introducing inverse optimization strategy. And reconfiguration design based on control allocation is proposed. Through linearized nonlinear mapping, a fault-tolerant control scheme is designed for the actuators in two typical fault modes: stuck and nonoutput force. Through rigorous mathematical analysis and numerical simulation, the effectiveness of the control strategy is verified.The integrated VMC in the FCS can realize online monitor of limited fault modes of actuators, which ensures the scheme proposed achieves better fault-tolerant control after the occurrence of known fault modes. However, mechanical transmission from the actuators to the surfaces, or the surfaces itself, cannot be directly dealt with by using the strategy proposed in the paper due to the lack of monitor results, such as the loss of part of the surface. It is worth further study.



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*Source: 4481553-2022-05-28.xml*