In recent years, quadcopters have gained significant attention due to their versatility in applications such as surveillance, delivery, and environmental monitoring. However, the reliability of these systems is often compromised by actuator faults, which can lead to performance degradation or even catastrophic failures. This paper addresses the challenge of designing a fault-tolerant control strategy that ensures the quadcopter’s attitude tracking errors converge within a predefined time and adhere to specified performance bounds. By integrating radial basis function (RBF) neural networks, event-triggered mechanisms, and hysteresis quantizers, an adaptive backstepping control framework is developed to handle unknown actuator faults and system uncertainties. The proposed approach not only guarantees practical predefined-time stability but also reduces communication frequency between the controller and actuators, enhancing the quadcopter’s operational efficiency.
The dynamics of a quadcopter are inherently nonlinear and underactuated, making them susceptible to disturbances and component failures. Actuator faults, including partial loss of effectiveness and bias faults, are common in quadcopter systems due to factors like motor wear or propeller damage. Traditional control methods often fail to account for such faults, leading to inadequate performance. This work focuses on developing a robust control scheme that compensates for these issues while ensuring that the quadcopter’s attitude angles track desired trajectories with high precision. The integration of RBF neural networks allows for accurate approximation of unknown nonlinear functions, while event-triggered control and input quantization mitigate resource constraints. Simulation results demonstrate the effectiveness of the proposed method in achieving predefined-time convergence and prescribed performance under various fault scenarios.
Problem Formulation and Quadcopter Dynamics
The quadcopter is modeled as a rigid body with four rotors, each producing thrust and torque. The attitude dynamics are derived using the Newton-Euler formulation, considering the rotational motion around the body-fixed frame. The state-space representation of the quadcopter’s attitude system is given by:
$$ \dot{x}_1 = x_2 $$
$$ \dot{x}_2 = a_1 x_4 x_6 + b_1 u_1 – c_1 x_2 $$
$$ \dot{x}_3 = x_4 $$
$$ \dot{x}_4 = a_2 x_2 x_6 + b_2 u_2 – c_2 x_4 $$
$$ \dot{x}_5 = x_6 $$
$$ \dot{x}_6 = a_3 x_2 x_4 + b_3 u_3 – c_3 x_6 $$
where \( x_1, x_3, x_5 \) represent the roll, pitch, and yaw angles, respectively, and \( x_2, x_4, x_6 \) are their angular velocities. The parameters \( a_i, b_i, c_i \) are derived from the quadcopter’s inertial and aerodynamic properties. To account for actuator faults, the control input \( u_i \) is modified as:
$$ u_i^F = \lambda_i u_i + \zeta_i $$
where \( \lambda_i \) denotes the effectiveness loss and \( \zeta_i \) represents the bias fault. The faulty dynamics are incorporated into the state-space model, leading to:
$$ \dot{x}_2 = b_1 u_1 + f_1(\underline{x}) + \Delta_1 $$
$$ \dot{x}_4 = b_2 u_2 + f_2(\underline{x}) + \Delta_2 $$
$$ \dot{x}_6 = b_3 u_3 + f_3(\underline{x}) + \Delta_3 $$
Here, \( \Delta_i = (\lambda_i – 1) u_i + \zeta_i \) encapsulates the fault-induced uncertainties. The control objective is to ensure that the attitude tracking errors converge to a predefined bound within a specified time, despite these faults.
Control Design Methodology
The control design involves several key components: error transformation for prescribed performance, RBF neural networks for function approximation, event-triggered mechanisms for reduced communication, and hysteresis quantizers for input processing. The backstepping technique is employed to recursively design virtual control laws and actual control inputs.
Error Transformation for Prescribed Performance
To enforce predefined-time convergence, the tracking errors are transformed using time-varying functions \( \gamma_i(t) \). The transformed error \( \xi_i \) is defined as:
$$ \xi_i = \frac{z_i}{\pi_i} $$
where \( \pi_i = 1 – \gamma_i^2 z_i^2 \). The derivative of \( \xi_i \) is given by:
$$ \dot{\xi}_i = \eta_i \dot{z}_i + \chi_i $$
with \( \eta_i = \frac{2 – \pi_i}{\pi_i^2} \) and \( \chi_i = \frac{2 \dot{\gamma}_i \gamma_i z_i^3}{\pi_i^2} \). This transformation ensures that the original error \( z_i \) remains within the performance bounds \( |z_i| < \frac{1}{\gamma_i} \) for all \( t \geq 0 \).
RBF Neural Network Approximation
RBF neural networks are utilized to approximate unknown nonlinear functions arising from actuator faults and system dynamics. The network output is expressed as:
$$ g(Z) = \vartheta^T \psi(Z) $$
where \( \vartheta \) is the weight vector and \( \psi(Z) \) is the Gaussian basis function vector. For any unknown function \( f(Z) \), the RBF network provides an estimate:
$$ f(Z) = \vartheta^{*T} \psi(Z) + \varepsilon(Z) $$
with \( \varepsilon(Z) \) being the approximation error. The adaptive laws for updating the weight estimates are designed to ensure stability.
Event-Triggered Control and Hysteresis Quantization
To reduce the actuator update frequency, an event-triggered mechanism is incorporated. The control input \( u_i(t) \) is updated only when the triggering condition is satisfied:
$$ t_{k+1} = \inf \{ t \in \mathbb{R} \mid | \Lambda_i(t) | \geq \beta_i |u_i(t)| + \Pi_i \} $$
where \( \Lambda_i(t) \) is the sampling error. The control signal is quantized using a hysteresis quantizer to prevent chattering:
$$ \upsilon[\nu(t)] = \Xi[\nu(t)] \nu(t) + \Omega[\nu(t)] $$
with \( 1 – d \leq \Xi[\nu(t)] \leq 1 + d \) and \( |\Omega[\nu(t)]| \leq \nu_{\min} \). The quantized control input is then applied to the quadcopter system.
Adaptive Backstepping Control Design
The backstepping control design proceeds in steps for each attitude channel. For the roll angle subsystem, the virtual control law \( \alpha_1 \) is designed as:
$$ \alpha_1 = \dot{\phi}_d – \frac{\chi_1}{\eta_1} – B_1 \frac{\xi_1}{\eta_1} $$
where \( B_1 > 0 \) is a design parameter. The actual control input \( u_1 \) is derived using the event-triggered quantizer:
$$ u_1(t) = \upsilon_1[\nu_1(t_k)], \quad \forall t \in [t_k, t_{k+1}) $$
The adaptive law for updating the RBF network weights is given by:
$$ \dot{\hat{\Theta}}_1 = \frac{\sigma_1 \xi_2^2 \eta_2^2 \Psi_1^T \Psi_1}{2 \delta_1^2} – \rho_1 \hat{\Theta}_1 $$
Similar procedures are applied to the pitch and yaw angle subsystems. The control parameters are summarized in Table 1.
| Parameter | Description | Value |
|---|---|---|
| \( B_1, B_2, B_3 \) | Virtual control gains | 5, 1, 6 |
| \( \sigma_1, \sigma_2, \sigma_3 \) | Adaptive gains | 30, 15, 25 |
| \( \delta_1, \delta_2, \delta_3 \) | Approximation bounds | 1 |
| \( \beta_i \) | Event-triggering thresholds | 0.5 |
| \( \Pi_i \) | Triggering constants | 1 |
Stability Analysis
The stability of the closed-loop system is analyzed using Lyapunov theory. Consider the Lyapunov function candidate for the roll subsystem:
$$ V_2 = \frac{1}{2} \xi_1^2 + \frac{1}{2} \xi_2^2 + \frac{1}{2\sigma_1} \tilde{\Theta}_1^2 $$
where \( \tilde{\Theta}_1 = \Theta_1 – \hat{\Theta}_1 \). The time derivative of \( V_2 \) is derived as:
$$ \dot{V}_2 \leq -B_1 \xi_1^2 – B_2 \xi_2^2 – \frac{\rho_1}{2\sigma_1} \tilde{\Theta}_1^2 + D_1 $$
with \( D_1 = \frac{\rho_1}{2\sigma_1} \Theta_1^2 + \frac{\delta_1^2}{2} + \frac{2\beta_2}{1-\beta_1} + \frac{m_1 \nu_{\min}^2}{2(1-\beta_1)^2} \). This implies that \( V_2 \) is uniformly ultimately bounded, ensuring that the tracking errors converge to the prescribed performance bounds within the predefined time.
Simulation Results and Discussion
To validate the proposed control scheme, simulations are conducted using MATLAB/Simulink. The quadcopter parameters are listed in Table 2.
| Parameter | Value | Unit |
|---|---|---|
| Rotor arm length \( l \) | 0.23 | m |
| Moment of inertia \( I_x \) | 0.075 | kg·m² |
| Moment of inertia \( I_y \) | 0.075 | kg·m² |
| Moment of inertia \( I_z \) | 0.013 | kg·m² |
| Drag coefficient \( K_i \) | 0.01 | N·s/m |
The desired attitude trajectories are set as \( \phi_d = 2.5 \sin(t) \), \( \theta_d = 5 \sin(t) \), and \( \psi_d = 2.5 \sin(t) \). Actuator faults are introduced with \( \lambda_1 = 0.8 \), \( \lambda_2 = 0.95 \), \( \lambda_3 = 0.7 \), and bias faults \( \zeta_i = 0.5 \sin(3t) \) for \( t < 3 \) s, and changed to \( \lambda_1 = 0.9 \), \( \lambda_2 = 1 \), \( \lambda_3 = 1 \), \( \zeta_1 = 0.8 \sin(3t) \), \( \zeta_2 = 2 \), \( \zeta_3 = 0.8 \sin(3t) \) for \( t \geq 3 \) s. The predefined time is set to \( T^* = 1 \) s, and the performance bounds are \( \kappa_1 = 0.2 \), \( \kappa_2 = 0.8 \).
The simulation results demonstrate that the attitude tracking errors converge within the specified bounds within 1 second, as shown in Figure 1. The event-triggered quantized control inputs significantly reduce the actuator update frequency compared to traditional time-triggered control. For instance, the roll subsystem requires only 184 updates over 20 seconds, representing a 90.8% reduction. The comparison with a standard event-triggered controller further highlights the efficiency of the proposed method, as summarized in Table 3.
| Subsystem | Proposed Controller | Standard Event-Triggered | Reduction |
|---|---|---|---|
| Roll | 184 | 213 | 13.6% |
| Pitch | 182 | 192 | 5.2% |
| Yaw | 132 | 170 | 22.4% |
The effectiveness of the RBF neural network in approximating unknown dynamics is evident from the accurate tracking performance under fault conditions. The event-triggering mechanism ensures that the control inputs are updated only when necessary, reducing communication overhead without compromising stability.
Conclusion
This paper has presented a novel fault-tolerant control strategy for quadcopter attitude systems subject to actuator faults. By combining RBF neural networks, event-triggered control, and hysteresis quantization, the proposed approach ensures predefined-time convergence with prescribed performance bounds. The Lyapunov-based stability analysis guarantees practical stability, while simulations validate the method’s effectiveness in reducing communication frequency and handling uncertainties. Future work will extend this framework to multi-quadcopter formations and real-world experimental validation.