Abstract
This paper introduces a fault-tolerant control scheme for the automatic carrier landing of carrier-based aircraft using direct lift control. The scheme combines radial basis function neural network and active disturbance rejection control (RBF-ADRC) to overcome the impact of actuator failures and external disturbances. First, the carrier-based aircraft model, the carrier air-wake model, and the actuator fault model were established. Secondly, ADRC is designed to estimate and compensate for actuator faults and disturbances in real time. RBFNN adjusts the ADRC controller parameters based on the system state. Then, the Lyapunov function is constructed to prove the stability of the closed-loop system. The controller is applied to the direct lift control channel, auxiliary attitude channel, and approach power compensation system. The direct lift control improves the performance of fixed-wing aircraft. Finally, comparative simulations were conducted under various actuator failures. The results demonstrate the remarkable fault tolerance of the RBF-ADRC scheme, enabling precise tracking of the desired glide path by the shipboard aircraft even in the presence of actuator failures.
1. Introduction
Carrier-based aircraft landing technology has received extensive attention as a prerequisite for effective combat applications at sea [1, 2]. The automatic carrier landing system (ACLS) comprises the aircraft flight control system, approach power compensator system, inertial navigation sensors, and precision tracking radar, enabling automated approach control to the carrier deck in all weather conditions. Various intelligent control methods, such as adaptive sliding mode control [3], backstepping and sliding mode [4], adaptive preview control [5], fixed-time backstepping control [6], and inverse optimal control [7], are employed in the ACLS to track the desired glide path during the landing process precisely. Furthermore, the ACLS incorporates adaptive backstepping sliding mode control to mitigate the impact of carrier air-wake turbulence during the landing process [8]. The above studies focus on the conventional maneuvering of shipboard aircraft under ideal conditions without any faults. The conventional aircraft maneuvering method involves utilizing the elevator to alter the pitch attitude of the aircraft and subsequently modify its flight trajectory. However, this approach has inherent limitations, including trajectory-attitude coupling and insufficient landing accuracy. Consequently, the direct lift control is proposed for the automatic carrier landing system (DLC-ACLS) [9, 10]. This technology directly manipulates the force exerted on the aircraft through the flap, eliminating the coupling between trajectory and attitude. Nevertheless, potential actuator failures and other factors during the automatic landing of the shipboard aircraft can considerably impact the landing accuracy, even when utilizing DLC. Hence, it is imperative to incorporate controller fault tolerance performance in the design of DLC-ACLS.
For the design of fault-tolerant control systems for aircraft, many experts have designed different schemes [11–13]. Model predictive control is more widely used in fault-tolerant control, and the literature [14] proposed a fault-tolerant controller using nonlinear model predictive control (NMPC) that can effectively recover a damaged quadrotor. However, MPC requires accurate system models to calculate the control signal. Moreover, failures can lead to sudden changes in model parameters that cannot be predicted in advance. Sliding mode control and backstepping techniques applied to an aircraft fault-tolerant control system allow the dynamic stability of the aircraft to converge [15, 16], but the computational complexity is notably high. The literature [17] uses adaptive controllers that can automatically update parameters to compensate for actuator failures in the system, but adaptive control exhibits limited adaptability to swift system changes, particularly when encountering significant model alterations. A fault-tolerant scheme known as antiwindup incremental nonlinear dynamic inversion (INDI) is proposed for flying wing aircraft afflicted with actuator faults [18]. Nonetheless, the NDI control method heavily relies on the accuracy of the model, and the control performance of the dynamic inverse controller will drop sharply when the model data is imprecise. Consequently, active disturbance rejection control (ADRC) has garnered growing attention. Literature [19, 20] optimizes the controller parameters using an optimization algorithm. Waves cause irregular movements of the deck, making the landing process difficult [21]. Literature [22] improves the accuracy of the landing phase by introducing an algorithm for predicting the motion of the deck of the carrier-based aircraft in the automatic landing system.
The ADRC method, a novel model-independent nonlinear controller, enjoys significant popularity in practical engineering applications. It has been successfully applied in many fields due to its simplicity, strong robustness, and immunity to disturbances. In literature [23], the UAV attitude control is designed, and the perturbations and uncertainties can be well estimated and eliminated online using the ADRC and embedded model control methods. The decoupled design of a fighter’s three channels utilizing ADRC treats the coupling between different channels as a comprehensive perturbation, effectively estimating and compensating for this disturbance [24]. For fractional-order systems with uncertainty, a combined fractional tracking controller that integrates backstepping and ADRC is introduced [25].
Although ADRC has the above advantages, when the carrier-based aircraft is affected by external disturbances and ESO estimation errors, a set of fixed feedback rate parameters makes the control efficiency unsatisfactory. RBF neural network can effectively control complex uncertain systems due to its ability of learning and self-adaptation [26–28]. In order to simplify the parameter adjustment process and enhance the fault-tolerant control capability of the controller, the automatic landing control system can quickly and stably fly along the ideal glide path under the conditions of actuator failure and external disturbance. This paper presents a new method for online automatic tuning of ADRC parameters using RBFNN. And the strategy is applied to the control of DLC-ACLS. The stability of the control system is analyzed using the Lyapunov stability theory, and the simulation results are compared with ADRC and PID. Compared with the literature [29], the model in this paper is more accurate and the control method is more practical. The results show that the method can quickly and stably track the ideal glide slope and has good robustness to external disturbances. The main contributions of this work are shown as follows: (1)A novel fault-tolerant control technique based on the combination of RBFNN and ADRC is designed. The proposed control scheme handles actuator failures, internal uncertainties, and external disturbances in real time, enhancing the system’s robustness. The ESO accurately observed and suppressed the disturbance through the state feedback controller, significantly improving the success rate of carrier-based aircraft landing. This control method compensates for actuator failures and air-wake disturbances that seriously affect landing accuracy and safety(2)The automatic control system consisting of a flight trajectory control system, attitude control system, and APCS is designed. The stability of the closed-loop system after introducing the RBF neural network is proved by constructing the Lyapunov function
The structure of this paper is as follows: in Section 2, the nonlinear model of carrier-based aircraft, the carrier air-wake model, and the actuator fault model are described. The control method of RBFNN to adjust the ADRC parameters is designed in Section 3, and the control method is applied to the DLC-ACLS in Section 4; in Section 5, simulation results are used to demonstrate the effectiveness of the proposed control method; the conclusion is presented in Section 6.
2. Landing Model Building
The final landing phase is shown in Figure 1. This section describes the aircraft nonlinear model, the carrier air-wake model, and the actuator failure model. According to the literature [30], the control inputs include elevator rudder (), trailing edge flaps () and thrust (), as shown in Figure 2.
2.1. Nonlinear Modeling of Carrier-Based Aircraft
This section gives the kinematic and dynamical model of the carrier-based aircraft. Define the coordinate system: the geodetic coordinate system (inertial coordinate system) is denoted as with the earth fixed, and any point on the surface is chosen as the origin. The trajectory coordinate system is denoted as , the airframe coordinate system is denoted as with the aircraft fixed, and the center of mass is generally chosen as the origin. The airflow coordinate system is denoted as , and represents the geometric center of the aircraft. where is the ground speed; is the flight trajectory angle; is the heading angle; , , and denote the angle of attack, sliding angle, and the roll angle; , , and denote the angular rates; , , and are roll moment, pitch moment, and yaw moment; is the engine thrust; , , and are lift, drag, and lateral force; , , and are the moments of inertia of the carrier aircraft; and is the product of inertia of the carrier aircraft.
2.2. Carrier Air-Wake Modeling
The carrier air-wake was modeled based on the US military standard MIL-F-8785C [31]. The air-wake consists of horizontal longitudinal component, horizontal lateral component, and vertical component. The corresponding equations are presented as follows: where , , and are free air turbulence components, and are steady components, and are periodic components, and , , and are random components, respectively.
2.3. Actuator Fault Model and Engine Dynamics
The typical actuator and engine dynamics are given as where the subscript refers to the command.
Common actuator failures include jamming, drift, damage, and saturation. This paper focuses on the drift fault, and the fault model is as follows: where represents the drift fault, which is divided into constant value fault and time-varying fault.
3. Controller Design
3.1. RBF Neural Network
RBF neural networks are 3-layer feedforward networks with a single hidden layer [32], as shown in Figure 3. RBF networks simulate the structure of locally tuned neural networks in the human brain, and it has been shown that RBF networks can approximate any continuous function with arbitrary accuracy [33].
The action function in the RBF network is a Gaussian function. where is the input vector for the network; is the center vector of node , , ; is the base width parameter of node , .
Weight vector of the network is
The output of the network is
3.2. ADRC Optimized by RBF Neural Network
Take the example of a second-order system with faults where is the total disturbance of the system; equation (8) is written in the following form.
In order to make the output track the reference signal, it is necessary to design the second-order RBF-ADRC scheme. The controller will be constructed in detail below.
3.2.1. Tracking Differentiator (TD)
The TD arranges the transition process for the input signal, and by selecting the appropriate parameters, it can prevent sudden changes in the input signal and improve the robustness of the control system [34]. The specific implementation of TD is shown in where represents the desired instruction, is the velocity factor, is the filter factor, and specific form reference literature [35].
3.2.2. ESO Design
The extended state observer (ESO) is a new type of disturbance observer, first proposed by Han [36]. The third-order ESO is designed according to equation (9). Using the idea of linearizing the dynamic compensation and treating the sum perturbation in the second-order system as an expanded state variable, equation (9) becomes as follows.
To build a third-order ESO based on the above equation, where is a highly accurate estimate of , is an estimate of the overall perturbation, and is the observer gain. To ensure the estimation performance, the value of can be set to [37] where is the observer bandwidth.
3.2.3. State Error Feedback (SEF)
With ESO accurately estimating the total disturbance, the controller is given
Ignoring the estimation error of reduces the system to a unit gain cascade integrator system
The cascaded integrator system can be easily controlled by the following nonlinear state error feedback law.
In the above equation, the nonlinear function is as follows:
In order to simplify the parameter setting, RBFNN is introduced to adjust and in real time.
The performance index function of the RBFNN is
According to the gradient descent method, the iterative algorithm of output weight vector, node center vector, and node base width parameters is where is for the learning rate and is for the momentum factor.
3.3. Stability Analysis
This section analyzes the stability of the closed-loop system and derives the convergence time.
The Lyapunov function is constructed as follows.
According to equation (20), the change in the Lyapunov function is obtained by
The weights are updated by the amount
Since , for , , and , it reveals that where
The linearized model of the error equation can be represented as
Using equation (18) yields
Using equations (25) and (26) gives
From equation (21) and (27), can be represented as
If is chosen as then in equation (29) will be less than zero. Therefore, the Lyapunov stability of and is guaranteed.
4. Design of the Direct Lift Automatic Carrier Landing System Based on RBF-ADRC
The DLC-ACLS is shown in Figure 4. The main design of the system is based on three channels: the flight path control channel, the attitude control channel, and the APCS. The main flight path control channel receives the longitudinal guidance law command and generates the flap deflection command, the main purpose of which is to adjust the longitudinal altitude. The attitude control channel makes the attitude stable by adjusting the elevator, and the APCS keeps the speed stable by adjusting the throttle lever. The role of the longitudinal guidance law is to convert the altitude command into a trajectory angle command. The detailed design procedure will be explained in the following.
4.1. Attitude Channel Controller Design Scheme
The attitude control channel contains angular control and angular velocity control with the main purpose of maintaining the desired angle of attack, which is set to a constant value.
There is a strong coupling between the longitudinal and lateral motions of the aircraft, and in this case, the lateral coupling is treated as an additional perturbation by ADRC. We design the controller for the inner and outer loops of the attitude separately as in Figure 5. The outer loop is mainly responsible for tracking the desired angle of attack and generating the desired pitch angle velocity command, while the inner loop mainly tracks the desired pitch angle velocity to generate the elevator deflection command.
4.1.1. Attitude Outer Loop
The longitudinal rotational motion model of the aircraft is as follows:
That is,
The attitude outer loop RBF-ADRC controller is as follows:
The parameter is designed according to the above RBF-ADRC design method, and the gradient descent method is used to adjust .
4.1.2. Attitude Inner Loop
The equation for the longitudinal rotational dynamics is as follows:
The expression of pitching moment is as follows: where is the dynamic pressure, is the density of air, is the wing area, is the aerodynamic mean chord, and , , , and are aerodynamic coefficients.
Since the elevator actuator is a first-order inertial link, the equation of the pitch angle velocity with respect to the elevator deflection becomes a second-order form
The RBF-ADRC design for the attitude inner loop is as follows:
The parameters and are designed according to the above RBF-ADRC design method, and the gradient descent method is used to adjust and .
4.2. Approach Power Compensator System Controller Design Scheme
The APCS is used to automatically adjust the throttle to control the approach speed of the aircraft.
The dynamic for ground speed has the expression as
The engine thrust is expressed in equation (44). where is the engine Mach number factor and is the engine height factor.
The equation for the speed relative to the throttle lever deflection command is as follows:
The RBF-ADRC controller of APCS is as follows:
The parameter is designed according to the above RBF-ADRC design method, and the gradient descent method is used to adjust .
4.3. Flight Path Control
Direct lift control was introduced into the trajectory control channel to provide lift directly to the shipboard aircraft through the flaps. Conventional automatic landings change the trajectory and altitude indirectly by changing the pitch angle, and the feedforward of the aircraft dynamics is ; the phase lag and the equivalent time delay are large. When DLC is used, the kinetic feedforward of the aircraft is , and the phase lag is greatly reduced.
4.3.1. The Longitudinal Guidance Law
The carrier-based aircraft was disturbed by air-wake during landing. Therefore, the longitudinal guidance law can be used to effectively suppress the interference.
The altitude deviation amount is calculated by the guidance law to get the trajectory angle command, which is transmitted to the flight control system. The lift force controls the trajectory angle directly and tracks the trajectory angle command quickly.
The guidance of the design uses the PID controller, and the control law is as follows:
The parameters in the control law can be obtained by debugging.
4.3.2. Direct Lift Channel
The dynamic equation describing the trajectory angle relative to the flight path is [38]
The expression for the lift force is where , , , , and are aerodynamic coefficients of the lift.
Since the flap actuator is a first-order inertial link, the equation of the trajectory angle with respect to the flap deflection becomes a second-order form
The direct lift channel RBF-ADRC controller is as follows.
The parameters and are designed according to the above RBF-ADRC design method, and the gradient descent method is used to adjust and .
4.4. Stability Analysis of Closed-Loop Systems
The Lyapunov function is constructed as follows.
can be represented as
If is chosen as then in equation (50) will be less than zero. The Lyapunov stability of and is guaranteed; therefore, the whole closed-loop system is stabilized.
5. Landing Simulation Results
When the carrier-based aircraft enters the entrance of the glide path, the initial condition is set as the flight altitude of the carrier-based aircraft is m, the reference height of the ideal landing point of the carrier-based aircraft is m, and the reference velocity is m/s. The angle is controlled at , and the reference glide path angle is controlled at . In the simulation, RBF-ADRC is applied to DLC-ACLS to verify the automatic landing performance. In addition, the automatic landing performance of the RBF-ADRC scheme is compared with the traditional ADRC and the PID control scheme.
5.1. Constant Faults
Incorporating constant value failures in the actuators into the simulation experiments, the simulation results are shown in Figures 6 and 7 and the maximum deviation of variables is shown in Table 1. The engine fault is introduced at s. The elevator fault is introduced at s. The flap fault is introduced at s.
(a) AOA
(b) Flight trajectory angle
(c) Pitch angle
(d) Pitch angle rate
(a) Velocity error
(b) Tracking error
5.1.1. Stability Analysis
It can be seen from Figure 6(a) that when RBFNN is not introduced, the AOA angle deviation of traditional ADRC control is more significant. The AOA under the new design method is more stable through comparative analysis.
The deviation of the trajectory angle between the proposed method and the other two methods is compared in Figure 6(b), where “the proposed method” refers to the RBF-ADRC controller proposed in this paper. The comparative analysis shows that RBF-ADRC can better maintain the stability of the trajectory angle under the influence of faults and perturbations.
It can be seen from Figures 6(c) and 6(d) that after the introduction of RBFNN, compared with the traditional ADRC, the pitch angle deviation and pitch rate deviation fluctuations of the proposed method are significantly reduced, and the maximum deviation is reduced. In contrast, under the RBF-ADRC method, the flight attitude of the carrier-based aircraft can be better maintained. Reducing the deviation of flight attitude can make the carrier aircraft land more smoothly and enhance the robustness of the system.
5.1.2. Tracking Performance Analysis
From Figure 7(a), it can be seen that the velocity disturbance variation of the proposed method is significantly lower than that of PID. In contrast, the speed stability of the new design is better maintained, and the disturbance can be effectively suppressed. Figure 7(b) shows the trajectory tracking comparison of the proposed method and the other two methods. The height deviation represents the deviation between the simulated results and the ideal slope. It can be seen from Figure 7(b) that the tracking error of DLC-ACLS based on RBF-ADRC is within 0.113 m. The results show that the proposed DLC-ACLS based on RBF-ADRC can better track the ideal glide slope in the presence of faults and disturbances.
5.2. Time-Varying Faults
To enhance the assessment of the RBF-ADRC’s robustness, the simulation experiment incorporates time-varying faults in the actuator; the simulation results are shown in Figure 8, and the maximum deviation of variables is shown in Table 2. The engine fault is introduced at s. The elevator fault is introduced at s. The flap fault is introduced at s.
(a) AOA
(b) Flight trajectory angle
(c) Pitch angle
(d) Pitch angle rate
5.2.1. Stability Analysis
It can be seen from Figure 8(a) that when RBFNN is not introduced, the AOA angle deviation of traditional ADRC control is more significant. The AOA under the new design method is more stable through comparative analysis.
5.2.2. Tracking Performance Analysis
As can be seen in Figure 9, the effect of time-varying faults on the aircraft speed is very large, and even with the newly designed method, the speed fluctuations remain large, but the fluctuations are relatively small compared to other methods. Figure 9 shows that the proposed method can track the error of the ideal glide path within a small range.
(a) Velocity error
(b) Tracking error
6. Summary
This paper introduces a novel robust fault-tolerant control strategy based on RBF-ADRC, designed and applied to a DLC-ACLS system suffering from wake disturbances and actuator failures. In this system, the attitude control channel precisely follows the desired angle of attack, the APCS automatically adjusts the throttle to control the aircraft’s approach speed, and the direct lift control channel tracks the desired trajectory angle command. This study considers actuator failures and air-wake disturbances as total disturbances, estimated and compensated by ADRC. The parameters of ADRC are designed in real time using RBFNN.
To verify the effectiveness of the RBF-ADRC scheme, actuator constant faults and time-varying faults are introduced. Comparing the proposed control scheme with PID controller and traditional ADRC, the results show that the proposed method can not only quickly and effectively compensate for actuator faults and enhance fault tolerance but also significantly improve the robustness of the system. Further research will be conducted to investigate how to utilize the robustness and learning capability of the method to deal with complex control problems such as saturation of control inputs of the landing system.
Data Availability
The data that support the findings of this study are available on request from the corresponding author, upon reasonable request and with permission of laboratory confidentiality agency permission.
Conflicts of Interest
The authors declared that they have no conflicts of interest regarding this work.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (Grant Number 52171299).