Research on Fault-Tolerant Control of Combined Airframe Damage of Electric Aircraft

Abstract

General aviation is an important branch of the aviation field. As a green energy aircraft, the electric aircraft is an important component and development direction of general aviation aircraft, and its safety is crucial. In this paper, the aerodynamic and dynamic characteristics of electric aircraft under collision, lightning strikes, and icing conditions are studied, and the dynamic and kinematics models of the aircraft are established by introducing damage factors. The STAR-CCM+ software is used to simulate the aerodynamic force and aerodynamic moment in the case of combined airframe damage. Based on the estimation ability of the L1 adaptive control algorithm for the parameter uncertainty of the controlled object and the automatic adjustment ability of control output, a fault-tolerant control law for electric aircraft is designed in the case of wing damage, horizontal tail damage after a collision, horizontal tail icing, and wing lightning damage. The results show that the control law has good fault-tolerant control ability for combined airframe damage of electric aircraft, and the control system has adaptability, anti-interference, and robustness, which has a good engineering reference significance for flight safety control of other transport aircraft.

1. Introduction

Although countries around the world attach great importance to aviation safety, ensuring aircraft flight safety still faces various challenges. The aircraft may encounter various problems, such as lightning strikes, icing, wind shear, turbulence, low visibility, etc., especially in complex meteorological conditions [1,2,3]. At the same time, the aircraft may also encounter bird strikes or collisions between two aircraft. As a green and clean energy aircraft that can play a certain role in reducing “carbon emissions”, the electric aircraft is being studied by countries all over the world [4,5,6,7].

Compared with traditional fuel aircraft, the electric aircraft has the following characteristics:

(1) The motor is powered by a battery of clean energy, so the electric aircraft does not emit harmful gases such as carbon dioxide. It is environmentally friendly and has the advantages of a green environment;

(2) The noise of the motor is lower than that of the internal combustion engine during operation, which can reduce the community noise by more than 15 dB, so electric aircraft has the advantage of low flight noise;

(3) Their motors have scale independence; that is, they have a certain power independent of their structural dimensions. A large motor can be divided into several small motors, so that they can be flexibly arranged on the aircraft and then achieve the goal of high lift through the propulsion aerodynamic coupling design. Therefore, electric aircraft has the design potential for unconventional and innovative layouts, and redundant electric propulsion systems can increase flight reliability and safety [8];

(4) The motor power does not decline with the increase in altitude, so the electric aircraft has more advantages than traditional fuel aircraft in plateau areas;

(5) Since the weight of the battery before and after charging and discharging is almost unchanged, the weight, center of gravity, and moment of inertia of this kind of electric aircraft during the whole flight phase are almost unchanged, which is beneficial to the design of the electric aircraft flight control system;

(6) Due to the current low energy density of lithium battery compared with fuel, electric aircraft generally has the disadvantage of insufficient range. In order to make up for this shortcoming, electric aircraft are often designed to have a large aspect ratio and high lift–drag ratio. Although such electric aircraft often have good stability, they may sacrifice some of their controllability, which puts forward higher requirements for the design of electric aircraft flight control systems;

(7) When the motor works, it often has the characteristics of high voltage and large current, so the electromagnetic environment inside the electric aircraft is often poor, and sometimes it will have adverse effects on the flight control system. Therefore, it is necessary to fully consider the anti-electromagnetic interference ability and error tolerance control ability of the electric aircraft flight control system;

(8) Electric aircraft usually install the power battery inside the fuselage rather than inside the wing, so that when the electric aircraft wing is broken to a large extent, it still has the ability to drive the aircraft to fly, thus ensuring flight safety; Traditional fuel powered aircraft usually install fuel tanks and pipelines inside the wings. In the event of a broken wing, damage to the fuel tanks and pipelines may occur, resulting in insufficient or even complete loss of aircraft power, thereby reducing aircraft safety;

(9) Electric aircraft usually use electric actuators to control ailerons, rudders, elevators, flaps, and other control surfaces. In the case of damage to wings and tail, the control ability will not be completely lost to ensure the flight safety of the aircraft; however, traditional fuel-powered aircraft typically use mechanical control systems. Under the same damage situation, the control mechanism also experiences fracture damage, resulting in partial or complete loss of control ability, thereby reducing aircraft safety.

When the electric aircraft encounters a collision, lightning strike, and icing, the aerodynamic force, aerodynamic moment, and control surface efficiency will generally change greatly, and its mass, center of mass, and moment of inertia will also change greatly. And the specific impact of each form of airframe damage on the aerodynamic force and moment of electric aircraft is also different, so it is necessary to focus on the analysis of the impact of various types of airframe damage on the aerodynamic characteristics of electric aircraft [9,10,11,12,13]. For the electric aircraft, because its batteries are installed more inside the fuselage than inside the wing, it is especially difficult to completely lose the aircraft power and the aileron effect when the wing of the electric aircraft is damaged. Therefore, compared with traditional fuel aircraft, electric aircraft have better conditions for fault-tolerant flight control against wing damage.

The L1 adaptive control method is achieved by adding a low-pass filter to the control law design stage of the model reference adaptive control algorithm. Therefore, it is possible to separate the design of control law and adaptive law and separate the design of adaptive performance and robust performance [14,15,16,17,18]. The state predictor is used to calculate the estimated values of the controlled object parameters; Then, the two numerical values are input into the adaptive law to obtain stable estimates of uncertain parameters in the sense of Lyapunov; and finally, the output signal is obtained through output compensation and low-pass filtering through the control law.

Liu designed an adaptive dynamic surface flight control, achieving good control of an aircraft with aerodynamic parameter perturbations and external disturbances during high angle of attack post-stall maneuvers [19]. Xue designed an L1 adaptive control system using state feedback, achieving good control of the lateral roll angle of unmanned aerial vehicles with parameter uncertainty [20]. Lu studied a longitudinal control law for unmanned aerial vehicles with modeling uncertainty and input interference based on the L1 adaptive control method [21]. Yang developed an adaptive sliding mode fault-tolerant control scheme based on prescribed performance control and neural networks for an unmanned aerial vehicle [22]. Gao proposed a novel adaptive fault-tolerant controller based on the fuzzy neural network (FNN) and nonsingular fast terminal sliding-mode (NFTSM) control scheme for tracking control and vibration suppression of the flexible wings, and successfully addressed the issues of system uncertainties and actuator failures [23]. Liang designed a fixed-time observer to solve the problem of the back-stepping fault-tolerant control (FTC) based on a fixed-time observer for the morphing aircraft with model uncertainties, external disturbances, and actuator faults [24].

In this paper, the aerodynamic characteristics and control methods of a type of electric aircraft under collision damage, lightning damage, and icing are studied, and the effectiveness and robustness of the control law are verified in the digital simulation environment.

2. Dynamic Model of Electric Aircraft

The dynamic equation of aircraft centroid movement is a mathematical equation that represents how to make the translational motion of an aircraft centroid after the aircraft is subjected to an external force, and is derived based on Newton’s second law.

$${\displaystyle \sum F}=\frac{d}{dt}(m{V}_a)$$

(1)

In the formula, ${\displaystyle \sum F}$ is the external force, $m$ is the mass, and $V_a$ is the velocity vector, and then there is

$${\displaystyle \sum F}=\frac{d}{dt}(m{V}_a)=m\frac{d{V}_a}{dt}=m(\frac{\delta V_a}{\delta t}+\Omega \times V_a)$$

(2)

In this formula,

$$V_a=u{i}_b+v{j}_b+w{k}_b$$

(3)

$$\Omega =p{i}_b+q{j}_b+r{k}_b$$

(4)

In the equation, p is the roll angle rate, q is the pitch angle rate, and r is the yaw angle rate.

The dynamic equation for the linear motion of the damaged aircraft is derived. Assuming that the change in center of mass after aircraft damage is $X_{bh}$ and the mass after the damage is $m_p$, there then is the following:

$${\displaystyle \sum F}=m_p\frac{\delta V_a}{\delta t}+m_p\Omega \times V_a+m_p\Omega \times (\Omega \times X_{bh})$$

(5)

$\frac{\delta V_a}{\delta t}$ is decomposed in the body coordinate system, and then

$$\frac{\delta V_a}{\delta t}=\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array}\right]$$

(6)

And the combined external force ${\displaystyle \sum F}$ can be expressed as the sum of gravity, thrust, and aerodynamic forces, as follows:

$${\displaystyle \sum F}=\left[\begin{array}{c}-m_{p}g\sin \theta +F_{xb}\\ m_{p}g\cos \theta \sin \varphi +F_{yb}\\ m_{p}g\cos \theta \cos \varphi +F_{zb}\end{array}\right]$$

(7)

Then,

$$\begin{array}{l}\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array}\right]=\left[\begin{array}{c}-g\sin \theta +\frac{F_{xb}}{m_p}\\ g\cos \theta \sin \varphi +\frac{F_{yb}}{m_p}\\ g\cos \theta \cos \varphi +\frac{F_{zb}}{m_p}\end{array}\right]-\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\left[\begin{array}{c}u\\ v\\ w\end{array}\right]\\ -\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\left(\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\left[\begin{array}{c}\Delta X_g\\ \Delta Y_g\\ \Delta Z_g\end{array}\right]\right)\end{array}$$

(8)

In the equation, $\varphi$ is the roll angle, $\theta$ is the pitch angle, and $g$ represents gravitational acceleration. $F_{xb}$, $F_{yb}$, and $F_{zb}$ are the components of force ${\displaystyle \sum F}$ on the three axes of the body coordinate system. $\Delta X_g$, $\Delta Y_g$, and $\Delta Z_g$ are the components of the change in center of mass after aircraft damage $X_{bh}$ on the three axes of the coordinate system.

The dynamic equation of aircraft rotation around the center of mass is a mathematical equation that describes how an aircraft rotates around the center of mass after being subjected to external moments of force. It is derived based on the theorem of moments of momentum. Usually, the aerodynamic moment coefficient and moment of inertia are described in the body coordinate system, so establishing a moment equation system in the body coordinate system is the easiest.

$${{\displaystyle \sum M}}_{he}=\frac{d}{dt}(L_d)=\frac{d{L}_d}{dt}$$

(9)

In the formula, ${{\displaystyle \sum M}}_{he}$ is the external moment applied to the aircraft, and $L_d$ is the aircraft’s moment of momentum.

$${{\displaystyle \sum M}}_{he}=\frac{d{L}_d}{dt}=\frac{\delta L_d}{\delta t}+\Omega \times L_d$$

(10)

Write the moment of momentum $L_d$ as follows:

$$L_d=J\Omega =\left[\begin{array}{ccc}{J}_x& -J_{xy}& -J_{xz}\\ -J_{yx}& J_y& -J_{yz}\\ -J_{zx}& -J_{zy}& J_z\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]$$

(11)

In the formula, $J$ is the inertia tensor composed of the moment of inertia $J_x$, $J_y$, and $J_z$ and the product of inertia $J_{xy}$, $J_{yx}$, $J_{xz}$, $J_{zx}$, $J_{yz}$, and $J_{zy}$. Specific definitions of other symbols can be found in reference [21].

Then, the dynamic equation of the damaged aircraft rotating around the center of mass is derived, and it can be concluded that

$${{\displaystyle \sum M}}_{he}=\frac{\delta L_d}{\delta t}+\Omega \times L_d+m_p\Omega \times (X_{bh}\times V_a)+m_p{V}_a\times (\Omega \times X_{bh})$$

(12)

Then,

$$\begin{array}{l}\left[\begin{array}{c}\dot{p}\\ \dot{q}\\ \dot{r}\end{array}\right]={\left[\begin{array}{ccc}{J}_x& -J_{xy}& -J_{xz}\\ -J_{yx}& J_y& -J_{yz}\\ -J_{zx}& -J_{zy}& J_z\end{array}\right]}^{-1}\\ \left(\begin{array}{l}\left[\begin{array}{c}\overline{L}\\ M\\ N\end{array}\right]-\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\left[\begin{array}{ccc}{J}_x& -J_{xy}& -J_{xz}\\ -J_{yx}& J_y& -J_{yz}\\ -J_{zx}& -J_{zy}& J_z\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\\ -m_p\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\left(\left[\begin{array}{ccc}0& -\Delta Z_g& \Delta Y_g\\ \Delta Z_g& 0& -\Delta X_g\\ -\Delta Y_g& \Delta X_g& 0\end{array}\right]\left[\begin{array}{c}u\\ v\\ w\end{array}\right]\right)\\ -m_p\left[\begin{array}{ccc}0& -w& v\\ w& 0& -u\\ -v& u& 0\end{array}\right]\left(\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\left[\begin{array}{c}\Delta X_g\\ \Delta Y_g\\ \Delta Z_g\end{array}\right]\right)\end{array}\right)\end{array}$$

(13)

In the equation, $\overline{L}$ is the rolling moment, $M$ is the pitch moment, and $N$ is the yaw moment. They are the components of the external moment ${{\displaystyle \sum M}}_{he}$ on the three axes of the body coordinate system.

In order to describe the changes in aerodynamic and moment parameters of electric aircraft after damage, aerodynamic and moment functions with the damage factor (degree) as independent variables are introduced. These functions are interpolated from the aerodynamic and moment data of the damaged aircraft obtained through aerodynamic simulation analysis later in this paper.

The functions of lift aerodynamic coefficient $C_{Lzy}$, drag aerodynamic coefficient $C_{Dzy}$, lateral force coefficient $C_{Yzy}$, rolling moment aerodynamic coefficient $C_{lzy}$, pitching moment coefficient $C_{mzy}$, yawing moment coefficient $C_{nzy}$ of electric aircraft to damage factor (degree) ${\eta }_{zy}$, and $\alpha$ and $\beta$, is as follows:

$$C_{Lzy}=f_{Lzy}({\eta }_{zy},\alpha ,\beta )$$

(14)

$$C_{Dzy}=f_{Dzy}({\eta }_{zy},\alpha ,\beta )$$

(15)

$$C_{Yzy}=f_{Yzy}({\eta }_{zy},\alpha ,\beta )$$

(16)

$$C_{lzy}=f_{lzy}({\eta }_{zy},\alpha ,\beta )$$

(17)

$$C_{mzy}=f_{mzy}({\eta }_{zy},\alpha ,\beta )$$

(18)

$$C_{nzy}=f_{nzy}({\eta }_{zy},\alpha ,\beta )$$

(19)

3. Tolerant Fault Control Law Design

The L1 adaptive control system includes four parts: the controlled object, state predictor, adaption law, and control law [25,26].

The structural diagram of the L1 adaptive control system is shown in Figure 1.

The controlled object is described in state space form as follows:

$$\left\{\begin{array}{l}\dot{x}(t)=Ax(t)+B(w(t)u(t)+{\theta }_1{}^T(t)x(t)+\sigma (t))\\ y(t)=C^{\mathrm{T}}x(t)\end{array}\right.$$

(20)

Among them, $x(t)$ is the observable system state vector, $u(t)$ is the control vector, $y(t)$ is the output vector, $w(t)$ is the control efficiency change matrix caused by factors such as control rudder surface faults, ${\theta }_1(t)$ is the time-varying unknown parameter matrix, $\sigma (t)$ is the time-varying interference matrix, $A$ is the system parameter matrix, $B$ is the system control efficiency matrix, and $C$ is the system observation matrix, where

$$u(t)=u_1(t)+u_2(t)$$

(21)

and

$$u_2(t)=-K_m^{T}x(t)$$

(22)

Then, Equation (20) can be transformed into

$$\dot{x}(t)=A_{m}x(t)+B(w(t)u_1(t)+{\theta }^T(t)x(t)+\sigma (t))$$

(23)

Among them,

$${\theta }^T(t)={\theta }_1{}^T(t)+(1-w(t))K_m^T$$

(24)

The gain $K_m^T$ is designed so that $A_m$ is the Hurwitz matrix, and its expression is as follows:

$$A_m=A-B{K}_m^T$$

(25)

The state observer is designed as follows:

$$\dot{\widehat{x}}(t)=A_m\widehat{x}(t)+B(\widehat{w}(t)u_1(t)+{\widehat{\theta }}^T(t)x(t)+\widehat{\sigma }(t))$$

(26)

where $\widehat{x}(t)$, $\widehat{w}(t)$, $\widehat{\theta }(t)$, and $\widehat{\sigma }(t)$ are the estimated values of $x(t)$,$w(t)$, $\theta (t)$, and $\sigma (t)$, respectively.

According to the error value between the estimated value and the actual value measured by the state observer, the adaptive law is designed as follows:

$$\left\{\begin{array}{l}\dot{\widehat{w}}(t)={\Gamma }_w\mathrm{Proj}(\widehat{w}(t),-{({\tilde{x}}^{\mathrm{T}}(t)PB)}^{\mathrm{T}}{u}_1{}^{\mathrm{T}}(t))\\ \dot{\widehat{\theta }}(t)={\Gamma }_{\theta }\mathrm{Proj}(\widehat{\theta }(t),-{({\tilde{x}}^{\mathrm{T}}(t)PB)}^{\mathrm{T}}{x}^{\mathrm{T}}(t))\\ \dot{\widehat{\sigma }}(t)={\Gamma }_{\sigma }\mathrm{Proj}(\widehat{\theta }(t),-{({\tilde{x}}^{\mathrm{T}}(t)PB)}^{\mathrm{T}})\end{array}\right.$$

(27)

Among them,

$$\tilde{x}(t)=\widehat{x}(t)-x(t)$$

(28)

P is a positive definite symmetric matrix and satisfies the following equation:

$$A_m^{T}P+P{A}_m=-Q$$

(29)

where $\mathrm{Proj}(*,*)$ is the projection operator, and there is

$$Q=Q^T>0$$

(30)

Design adaptive control law input signals the following:

$$u_1(s)=KD(s)(k_{g}r(s)-\widehat{\eta }(s))$$

(31)

Among them, $u_1(s)$, $r(s)$, and $\widehat{\eta }(s)$ are the pull transformations of $u_1(t)$, $r(t)$, and $\widehat{\eta }(t)$, respectively, and there are

$$\widehat{\eta }(t)=(\widehat{w}(t)u_1(t)+{\widehat{\theta }}^T(t)x(t)+\widehat{\sigma }(t))$$

(32)

And there are

$$k_g=\frac{1}{C^{T}H(0)}$$

(33)

and

$$H(s)={(sI-A_m)}^{-1}B$$

(34)

To reduce high-frequency oscillations caused by adaptive control laws, a low-pass filter $C(s)$ is designed, which needs to meet the following conditions:

(1) $C(s)$ is asymptotically stable, strictly regular, and has a low-pass gain of $C(0)=1$;

(2) $C(s){\mathrm{H}}_0^{-1}(s)$ is stable and regular, where $H_0(s)=c^{T}H(s)$;

(3) ${‖G(s)‖}_{L1}L<1$; among them, $G(s)=H(s)(1-C(s))$, and $L=\underset{\theta \in \Theta }{\max }{‖\theta ‖}_{L1}=\underset{i}{\max }\left({\displaystyle \sum _j\left|{\theta }_{ij}\right|}\right)$, where Θ is for a given compact convex set.

The design block diagram of the fault-tolerant control law for electric aircraft with combined airframe damage is shown in Figure 2.

In this article, the controlled object of the L1 adaptive control system is the dynamic and kinematics equation of the electric aircraft with combined airframe damage. The aerodynamic parameters of the dynamic and kinematic equation change with the degree of damage (damage factor). The state predictor is an ideal linearized six-freedom model of aircraft which can predict the electric aircraft L1 adaptive control response after aircraft damage. The adaption law can provide parameter uncertainty calculated from response deviation. The control law can adjust the control output according to parameter uncertainty.

4. Simulation Result and Analysis

4.1. Aerodynamic Characteristics of Electric Aircraft with Airframe Damage

In this paper, lightning strikes and icing, which are prone to occur in meteorological conditions and have a great impact on flight safety, as well as aircraft impact, which may occur and have a great impact on flight safety, are selected to conduct research on the impact of a certain type of electric aircraft airframe damage on its aerodynamic characteristics and fault-tolerant control. As the flight height of the electric aircraft increases, the temperature of the airflow around the aircraft will decrease. With thunderstorm weather, the plane’s horizontal tail may be iced, and the wing tip may be damaged by lightning. In order to analyze the serious combined damage of the electric aircraft airframe, this paper also assumes that when icing and lightning damage occurs, the wing and horizontal tail may also be broken due to the collision of two aircraft, and then compares and analyzes the magnitude of the impact of these three types of damage on aircraft aerodynamics. Since the combined damage condition of the airframe includes four sub conditions, wing tip lightning damage, horizontal tail icing, and wing and horizontal tail impact damage, and each sub condition can have different damage degrees, this paper analyzes the aerodynamic impact of each sub condition on the electric aircraft according to the aerodynamic simulation results of the combined damage of the airframe, and selects the damage degree of each sub condition.

The carbon fiber composite skin of this type of electric aircraft basically adopts 45° and 0° cross laying, bonding, and curing, and its lightning damage can be assumed to be where the carbon fiber composite skin is torn and broken down. Combined with the lightning strike zoning of this type of electric aircraft, it can be seen that the wing tip has a high probability of being struck by lightning, and because the wing is made of carbon fiber composite materials, it is also easy to be penetrated by lightning arcs. Therefore, it is set that the lightning damage occurs near the wing tip of the left wing, and it is assumed that the lightning damage is a round hole with a diameter of 300 mm.

For this type of electric aircraft, 40% of the left wing collision damage, 80% of the right horizontal tail collision damage, the right wing tip puncture hole with a diameter of 300 mm caused by a lightning strike, and the icing at the leading edge of the horizontal tail occur at the same time, as shown in

Table 1. Specific damage details of the combined airframe damage of the electric aircraft.

Damage Location	Type of Damage	Damage Degree
Left wing	Collision damage	40%
Right side of horizontal tail	Collision damage	80%
Right wing tip	Lightning damage	Diameter 300 mm hole
Horizontal tail leading edge	Icing	A-ice type

.

The CATIA three-dimensional digital model of electric aircraft is drawn according to the damage conditions in Table 1. The details of the icing on the leading edge of the horizontal tail (A ice type) and the damage on the right side are shown in Figure 3a, and the combined damage of the airframe is shown in Figure 3b.

The CATIA digital model of the electric aircraft is imported into STAR-CCM+ fluid dynamics simulation software, and its automatic mesh generation function is used to mesh and optimize the electric aircraft and then calculate the aerodynamic coefficient and torque coefficient of the aircraft. Due to the fact that the “Spalart–Allmaras” turbulence model only needs to solve the transport equation of turbulent viscosity and does not need to solve the length scale of the local shear layer thickness, and it is not sensitive to numerical errors caused by grid roughness, it is more suitable for flow problems with small flow scales, especially those involving wall-bounded flow in the aerospace field. Therefore, the “Spalart–Allmaras” turbulence model was selected for this simulation. Because the polyhedral mesh has good orthogonality, its accuracy is similar to the “Hexahedron mesh”, and a tetrahedral mesh is easy to produce automatically by computer software; the polyhedral mesh generator is selected for automatic mesh generation in this simulation. The surface of the electric aircraft is set to “wall” type, the wall specification is “smooth”, the Shear stress is specified as “no slip”, and the tangential speed is specified as “fixed”. The outer space of the aircraft is set to the “Free streaming” type, and the Mach number is 0.1 Ma. The reference area for the overall force and torque coefficients is 12 m². The reference radius for the pitch torque coefficient is 0.868 m, and the reference radius for the roll torque coefficient and yaw torque coefficient is 14.5 m.

The electric aircraft pressure scalar field under the condition of combined damage of the airframe and without damage is obtained, as shown in Figure 4a,b.

The airflow velocity field at the symmetry plane of the electric aircraft with combined airframe damage is shown in Figure 5a. The air velocity field at the lightning strike breakdown of the right wing tip under the condition of combined damage of the airframe is shown in Figure 5b. It can be seen from Figure 5b that the holes punctured by lightning at the wing tip cause dramatic changes in the airflow field, and the wing lift is affected. However, since the hole area is smaller than the wing area, the hole has little effect on the wing lift.

The air velocity fields at 40% damage of the left wing and the undamaged wing on the symmetrical side under the condition of combined damage of the airframe are shown in Figure 6. It can be seen from the figure that the airflow field at the damaged wing is disordered, while the flow field on the upper and lower surfaces of the wing on the undamaged side is smooth, so it can be seen that the lift generated by the wing will be reduced after the wing is damaged.

Under the condition of combined airframe damage, the air velocity fields at the 80% damage position and the symmetrical side of the right horizontal tail with ice type A are shown in Figure 7. It can be seen from the figure that after ice type A is accumulated at the leading edge of the horizontal tail, the airflow around the horizontal tail is affected by the ice type, and the airflow is disordered, which will generate additional drag. Especially when the angle of attack is increased, it can be seen that the airflow gradually separates from the horizontal tail surface. The airflow field at 80% of the damage of the horizontal tail is disordered, which reduces the lift generated by the horizontal tail.

The lift coefficient of the electric aircraft with combined airframe damage is shown in Figure 8a. It can be seen from the figure that the sum of the changes in the lift coefficient of the whole aircraft caused by 40% left wing damage, horizontal tail icing, 80% left horizontal tail damage, and lightning damage is equivalent to the reduction in the lift coefficient of the whole aircraft caused by the combined damage of the airframe, and the wing damage is the factor that has the greatest impact on the lift coefficient in the combined damage. Based on the face symmetry of the wing and horizontal tail of the lossless electric aircraft, it can be approximately considered that the change in the lift coefficient of the whole aircraft caused by the combined damage of the airframe is equal to the sum of the changes in the lift coefficient of the whole aircraft caused by various individual types of damage under the combination.

The drag coefficient of electric aircraft with combined airframe damage is shown in Figure 8b. It can be seen from the figure that 60% of the change in the drag coefficient of the whole aircraft caused by the ice type A at the horizontal tail is equivalent to the increase in the drag coefficient of the whole aircraft caused by the combined damage of the airframe, that is, the ice at the horizontal tail is the factor that has the greatest impact on the drag coefficient in the combined damage, especially due to the 80% loss of the right horizontal tail under the combined damage condition; the impact of the horizontal tail with damage and leading edge icing on the drag coefficient of the whole aircraft can be approximately regarded as 60% of the original case without damage and icing (that is, the impact of the horizontal tail collision damage on the spanwise length of the ice shape at the leading edge of the horizontal tail needs to be superimposed). Based on the symmetry of the wing and the horizontal tail of the non-destructive electric aircraft, the change in the drag coefficient of the whole aircraft caused by the combined damage of the airframe can be obtained.

The pitching moment coefficient is shown in Figure 9a. It can be seen from the figure that the total variation of the pitching moment coefficient of the whole aircraft caused by 40% left wing damage, horizontal tail icing, 80% left horizontal tail damage, and lightning damage is equivalent to the variation of the pitching moment coefficient of the whole aircraft caused by the combined damage of the airframe, and the horizontal tail damage is the factor that has the greatest impact on the pitching moment coefficient among the combined damage types. At the same time, the wing damage and horizontal tail icing also have a certain impact on the pitching moment coefficient. Based on the face symmetry of the wing and horizontal tail of the lossless electric aircraft, it can be approximately considered that the change in the whole aircraft pitching moment coefficient caused by the combined damage of the airframe is equal to the sum of the changes in the whole aircraft pitching moment coefficient caused by various individual types of damage under the combination.

The combined damage rolling moment coefficient of the airframe is shown in Figure 9b.

It can be seen from Figure 9 that after summing up the changes in the rolling moment coefficient of the whole aircraft caused by the 40% damage of the left wing and the icing of the horizontal tail, and then calculating the difference with the changes in the rolling moment coefficient caused by the lightning damage and the 80% damage of the left horizontal tail, the obtained value is equivalent to the changes in the rolling moment coefficient of the whole aircraft caused by the combined damage of the airframe, and the wing damage is the most influential factor on the rolling moment coefficient in the combined damage. Then, based on the symmetry of the wing and horizontal tail of the lossless electric aircraft, the variation of the rolling moment coefficient of the whole aircraft caused by the combined damage of the airframe can be obtained.

To sum up, the influence of various individual types of damage conditions on aircraft force and moment coefficients in the combined damage of the airframe can be obtained, as shown in

Table 2. Influence of combined damage on aircraft force and moment coefficient.

Parameter	Primary Impact	Secondary Impact
Lift coefficient	Left wing damage 40%	Others
Drag coefficient	Icing of horizontal tail	Others
Pitching moment coefficient	Right tail damage 80%	Left wing damage 40%, and icing of horizontal tail
Rolling moment coefficient	Left wing damage 40%	Others

.

It can be seen from Table 2 that the 40% damage of the left wing has a greater impact on the lift coefficient and roll moment coefficient than other damage conditions. Ice type A with horizontal tail ice has the greatest influence on the drag coefficient. The 80% damage of the right horizontal tail has the greatest impact on the pitching moment coefficient, while the 40% damage of the left wing and the icing of the horizontal tail have a secondary impact on the pitching moment coefficient.

4.2. Fault-Tolerant Control Effect of Airframe Damage Electric Aircraft

The fault-tolerant control law is designed for the combined damage of the electric aircraft with 43% of the left wing collision damage, 80% of the right horizontal tail collision damage, the right wing tip being struck by lightning and puncturing holes with a diameter of 300 mm, and the front edge of the horizontal tail icing with A-type ice. The combined damage condition of the organism is defined as Condition 43, as detailed in

Table 3. Specific damage details of body combination damage Condition 43.

Damage Location	Type of Damage	Damage Degree
Left wing	Collision damage	43%
Right side of horizontal tail	Collision damage	80%
Right wing tip	Lightning damage	Diameter 300 mm hole
Horizontal tail leading edge	Icing	A-ice type

.

For electric aircraft without airframe damage, given the roll angle command, the roll angle response of L1 adaptive control law and PID control law is shown in Figure 10.

When the left wing is damaged by 43%, the left aileron will fail completely. Only the right aileron can work normally; then, the overall rudder efficiency of the electric aircraft aileron will become 50%. Since the right side of the horizontal tail wing is damaged by 80%, resulting in the failure of the right elevator, only the left elevator can work normally; The overall rudder efficiency of the electric aircraft elevator can be considered to be at 50%. Under Condition 43 of combined damage to the body, the lateral heading control effect is shown in Figure 11. The vertical control effect is shown in Figure 12a. The rudder deflection response is shown in Figure 12b. The three-dimensional track of electric aircraft is shown in Figure 13.

The L1 and PID control effects of aircraft under combined damage Condition 43 is described in

Table 4. L1 and PID control effects of aircraft.

Parameter	L1 Adaptive Control Law	PID Control Law
Sideslip angle ($\beta$) variation (°)	−0.5	−9
Roll angle ($\varphi$) variation (°)	0	−8
Yaw angle ($\psi$) variation (°)	−0.4	−1.2
Pitch angle ($\theta$) variation (°)	−0.3	0.7
Angle of attack ($\alpha$) variation (°)	1	1
Aileron deflection angle (da) variation (°)	−20	−16
Rudder deflection angle(dr) variation (°)	−3	−8
Elevator deflection angle (de) variation (°)	3	3

. Under combined damage Condition 43 of the airframe, the electric aircraft has a sideslip angle ($\beta$) of −9°, a roll angle ($\varphi$) of −8°, and a yaw angle ($\psi$) of −1.2° under the action of the PID control law. The aileron deflection angle (da) reaches −16°, the rudder deflection angle (dr) reaches −8°, and the elevator deflection angle (de) increases by 3°. The PID lateral flight attitude control effect is relatively poor. Under the action of L1 adaptive control law, the aircraft roll angle remains unchanged, the sideslip angle changes by −0.5°, the yaw angle changes by −0.4°, the aileron deflection angle reaches −20°, the rudder deflection angle reaches −3°, and the elevator deflection angle increases by 3°; that is, it can control the longitudinal and lateral directions of electric aircraft well. The PID control law in this chapter only uses proportional and differential parts of control, without adding an integral part. Therefore, when the electric aircraft under the action of the PID control law in this chapter suffers from combined airframe damage due to sudden changes in aerodynamic force and moment, as well as changes in control surface effect, the aircraft attitude control has a large steady-state error.

Under the combined damage condition 43 of the airframe, because the left wing is damaged by 43%, the aircraft has a tendency to roll to the left, and because the leading edge of the plane tail is iced (ice type A) and the right wing tip is struck by lightning, resulting in a hole with a diameter of 300 mm, the Lift coefficient of the aircraft decreases, the Drag coefficient increases, and the aircraft has a tendency to rise. As the left aileron and right elevator fail after the wing and horizontal tail are damaged, the efficiency of the aileron and elevator of the electric aircraft becomes 50%. At this point, the traditional PID control law cannot automatically adjust with changes in aircraft aerodynamic parameters, so the control effect of PID control law on aircraft lateral heading, especially roll angle, is relatively poor; the L1 adaptive control law can estimate the changes in aerodynamic parameters of the aircraft at the current moment in real time. So, the aircraft roll angle and pitch angle are basically kept in the original balance state.

In conclusion, in the case of combined airframe damage Condition 43 of this type of electric aircraft, the fault-tolerant control law designed based on L1 adaptive control algorithm can control the longitudinal and lateral flight attitude of the aircraft well. However, due to the limitation of the aileron rudder deflection angle of this type of electric aircraft, when the left wing’s damage exceeds 43% in the airframe combined damage Condition 43, the L1 adaptive control law will also find it difficult to effectively control the lateral flight attitude of the aircraft. Therefore, under the condition of airframe combined damage A, especially when the damage of the left wing is not more than 43%, the fault-tolerant control law designed in this section based on the L1 adaptive control algorithm can realize the fault-tolerant flight control of this type of electric aircraft.

5. Conclusions

In this paper, aerodynamic parameter simulations and the design of the fault-tolerant control law are carried out for electric aircraft in case of collisions, lightning strikes, and icing airframe damage, and the following conclusions are obtained:

(1) The aerodynamic and dynamic characteristics of the electric aircraft under impact, lightning strike, and icing conditions are systematically studied in this paper. By introducing the damage factor, the dynamic and kinematics models of the aircraft are established, which provides a theoretical basis for the aircraft capability and fault-tolerant control in the case of damage;

(2) The influence of left wing damage of 40% on lift coefficient, rolling moment coefficient, and yaw moment coefficient is greater than that of other damage conditions. The horizontal tail icing type A has the greatest impact on the drag coefficient. An amount of 80% of damage to the right horizontal tail has the greatest impact on the pitch moment coefficient, while 40% damage to the left wing and ice formation on the horizontal tail has a secondary impact on the pitch moment coefficient;

(3) Under the condition of airframe combined damage, especially when the damage of the left wing is not more than 43%, the fault-tolerant control law designed in this section based on L1 adaptive control algorithm can achieve fault-tolerant flight control of this type of electric aircraft. The fault-tolerant control law of electric aircraft designed in this paper is adaptive, anti-interference, and robust, and has good engineering reference significance for the flight safety control of other transport aircraft.