In recent years, quadrotor unmanned aerial vehicles (UAVs) have gained significant attention due to their versatility in applications such as military reconnaissance, agricultural monitoring, and aerial photography. However, the reliability of quadrotor systems is often compromised by actuator faults, which can lead to catastrophic failures during flight. Traditional fault diagnosis and control methods struggle to handle the nonlinearities and uncertainties inherent in quadrotor dynamics. To address this, I propose an enhanced approach leveraging an improved radial basis function (RBF) neural network. This method incorporates adaptive laws for weight vectors and center vectors, along with tuning parameters, to achieve precise fault estimation and robust fault-tolerant control. The quadrotor’s fault model is rigorously derived, and the proposed neural network-based observer ensures rapid and accurate fault diagnosis. Furthermore, a fault-tolerant control law is designed to maintain stability and tracking performance even under actuator failures. Simulation results demonstrate the superiority of this approach over conventional methods, with minimal errors in both fault diagnosis and control tracking. This research significantly enhances the safety and reliability of quadrotor operations in real-world scenarios.
The quadrotor UAV is a highly dynamic system that relies on four rotors for lift and maneuverability. Its mathematical model must account for nonlinear aerodynamics and potential actuator faults. I begin by establishing a comprehensive fault model for the quadrotor. Let the state vector be defined as \( x(t) = [\phi, \dot{\phi}, \theta, \dot{\theta}, \psi, \dot{\psi}]^T \), where \( \phi \), \( \theta \), and \( \psi \) represent the roll, pitch, and yaw angles, respectively, and their derivatives denote angular velocities. The output vector is \( y(t) = [\phi, \theta, \psi]^T \), and the input vector is \( u(t) = [V_f, V_b, V_r, V_l]^T \), corresponding to the voltages applied to the four motors. The fault model incorporates actuator faults \( f(x, u, t) \), external disturbances \( d(t) \), and nonlinear dynamics \( \phi(x, u, t) \). The system is described by the following equations:
$$ \dot{x}(t) = A x(t) + B u(t) + \phi(x, u, t) + D d(t) + E f(x, u, t) $$
$$ y(t) = C x(t) $$
Here, \( A \), \( B \), \( C \), \( D \), and \( E \) are coefficient matrices that capture the quadrotor’s linear dynamics, with \( (A, B) \) being controllable and \( (A, C) \) observable. The nonlinear term \( \phi(x, u, t) \) includes cross-coupling effects and aerodynamic friction, expressed as:
$$ \phi(x, u, t) = \begin{bmatrix} 0 \\ \frac{1}{J_\phi} \left( (J_\theta – J_\psi) \dot{\theta} \dot{\psi} – K_x \dot{\phi}^2 \right) \\ 0 \\ \frac{1}{J_\theta} \left( (J_\psi – J_\phi) \dot{\psi} \dot{\phi} – K_y \dot{\theta}^2 \right) \\ 0 \\ \frac{1}{J_\psi} \left( (J_\phi – J_\theta) \dot{\phi} \dot{\theta} – K_z \dot{\psi}^2 \right) \end{bmatrix} $$
where \( J_\phi \), \( J_\theta \), and \( J_\psi \) are the moments of inertia, and \( K_x \), \( K_y \), and \( K_z \) are aerodynamic coefficients. This model forms the basis for designing the fault diagnosis and control strategies, ensuring that the quadrotor can handle uncertainties and faults effectively.
To improve the accuracy of fault diagnosis, I enhance the traditional RBF neural network by introducing adaptive laws for the weight vectors and center vectors, along with tuning parameters to prevent drift. The RBF neural network approximates nonlinear functions using Gaussian activation functions. For an input vector \( z \), the output of the \( i \)-th hidden node is:
$$ \sigma_i = \exp\left( -\frac{(z – \delta_i)^T (z – \delta_i)}{2s^2} \right) $$
where \( \delta_i \) is the center vector and \( s \) is the width parameter. The overall output of the RBF network is:
$$ g = \sum_{i=1}^{h} w_i \sigma_i + \epsilon = W^T \sigma $$
Here, \( W = [w_1, w_2, \ldots, w_h, -\epsilon]^T \) is the weight vector, \( \sigma = [\sigma_1, \sigma_2, \ldots, \sigma_h, 1]^T \), and \( \epsilon \) is the approximation error. The key innovation lies in the adaptive update laws for \( \hat{W} \) (estimated weight vector) and \( \hat{\delta} \) (estimated center vector), which are designed as follows:
$$ \dot{\hat{W}} = \Gamma_1 \left[ \sigma(\hat{x}, u, \hat{\delta}) e_y^T R^T – \rho_s \hat{W} \right] $$
$$ \dot{\hat{\delta}} = \Gamma_3 \sigma_{\hat{\delta}}’ \hat{W} R e_y $$
where \( \Gamma_1 \) and \( \Gamma_3 \) are positive learning rates, \( R \) is a design matrix, \( e_y \) is the output error, and \( \sigma_{\hat{\delta}}’ = \partial \sigma(\hat{x}, u, \delta) / \partial \hat{\delta} \). The tuning parameter \( \rho_s \) is defined to prevent weight drift:
$$ \rho_s = \begin{cases} 0 & \text{if } \|\hat{W}\|_2 < w_m \\ \Gamma_2 \left(1 – \frac{w_m}{\|\hat{W}\|^2}\right) & \text{if } w_m \leq \|\hat{W}\|_2 \leq 2w_m \\ \Gamma_2 & \text{if } \|\hat{W}\|_2 \geq 2w_m \end{cases} $$
with \( \Gamma_2 > 0 \) and \( w_m > 0 \). These adaptive mechanisms enable the neural network to quickly converge to accurate estimates of actuator faults, enhancing the quadrotor’s fault diagnosis capability.
Using the improved RBF neural network, I design a fault diagnosis observer to estimate actuator faults in real-time. The observer is constructed as:
$$ \dot{\hat{x}}(t) = A \hat{x}(t) + B u(t) + \phi(\hat{x}, u, t) + E \hat{f}(\hat{x}, u, t) + L (y(t) – \hat{y}(t)) $$
$$ \hat{y}(t) = C \hat{x}(t) $$
where \( \hat{f}(\hat{x}, u, t) = \hat{W}^T \sigma(\hat{x}, u, \hat{\delta}) \) is the neural network approximation of the fault. The estimation errors are defined as \( e_x = x(t) – \hat{x}(t) \), \( e_y = C e_x \), and \( e_f = f(x, u, t) – \hat{f}(\hat{x}, u, t) \). Through Lyapunov stability analysis, I prove that the observer errors converge to zero asymptotically under certain conditions, ensuring reliable fault diagnosis for the quadrotor system. For instance, if there exist positive constants \( \alpha \) and positive definite matrices \( P = P^T \) and \( Y \) satisfying:
$$ L = P^{-1} Y $$
$$ E^T P = R C $$
$$ \begin{bmatrix} A^T P + P A – Y C – C^T Y^T + \alpha \gamma^2 I & \frac{1}{\alpha} P \\ P & -I \end{bmatrix} < 0 $$
then the observer is stable. This guarantees that the fault estimates are accurate, with maximum errors as low as 0.01 in simulations, highlighting the effectiveness of the proposed method for quadrotor applications.
For fault-tolerant control, I develop a control law that compensates for actuator faults and maintains desired tracking performance. The control input is divided into two components: \( u = u_n + u_f \), where \( u_n \) is the nominal control for fault-free operation, and \( u_f \) is the fault compensation term. The nominal control is designed using a PID approach:
$$ u_n = c_1 \tilde{y} + c_2 \dot{\tilde{y}} + c_3 \int \tilde{y} dt $$
where \( \tilde{y} = y – y_d \) is the tracking error, \( y_d = [\phi_d, \theta_d, \psi_d]^T \) is the desired trajectory, and \( c_1 \), \( c_2 \), \( c_3 > 0 \) are tuning parameters. The fault compensation term \( u_f \) leverages the RBF neural network to approximate and cancel out the fault effects:
$$ u_f = -B^* \left[ E \hat{W}_f^T \sigma(y, u, \hat{\delta}_f) + D \mu_1 \right] $$
where \( B^* \) is the pseudo-inverse of \( B \), and \( \mu_1 \) bounds the disturbance \( d(t) \). The adaptive laws for \( \hat{W}_f \) and \( \hat{\delta}_f \) are similar to those in the fault diagnosis observer, ensuring online learning and adaptation. The stability of the closed-loop system is proven using Lyapunov theory, confirming that the tracking errors remain bounded and converge to small values, even in the presence of faults. This approach allows the quadrotor to achieve precise attitude control with maximum tracking errors of only 0.3 degrees, demonstrating robust performance.
To validate the proposed methods, I conduct simulations in MATLAB, comparing the improved RBF neural network approach with traditional techniques. The quadrotor parameters used in the simulation are summarized in the table below:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \( J_\phi \) | 0.124 kg·m² | \( J_\theta \) | 0.124 kg·m² |
| \( J_\psi \) | 0.0635 kg·m² | \( K_x \) | 0.163 N·m·s/rad |
| \( K_y \) | 0.163 N·m·s/rad | \( K_z \) | 0.176 N·m·s/rad |
| \( l \) | 0.21 m | \( K_{lc} \) | 0.124 N·m/V |
| \( K_{vc} \) | 0.0041 N·m/V | \( K_{vn} \) | -0.0041 N·m/V |
The simulation runs for 10 seconds, with an actuator fault injected at \( t = 3 \) seconds. The fault is modeled as \( f(x, u, t) = [-0.5, 0.3 \sin(2t), -0.2t, 0.1t – 1]^T \). For fault diagnosis, the improved RBF observer achieves estimation errors below 0.01, whereas traditional methods exhibit oscillations and errors up to 0.1. In terms of control, the fault-tolerant approach maintains tracking errors within 0.3 degrees for roll, pitch, and yaw angles, outperforming conventional methods that show deviations up to 5 degrees. The following table summarizes the maximum errors observed in the simulations:
| Method | Fault Diagnosis Error | Roll Tracking Error (°) | Pitch Tracking Error (°) | Yaw Tracking Error (°) |
|---|---|---|---|---|
| Traditional | 0.1 | 2.0 | 4.0 | 5.0 |
| Proposed | 0.01 | 0.3 | 0.3 | 0.3 |
These results underscore the superiority of the proposed method in enhancing the reliability and performance of quadrotor systems. The integration of adaptive neural networks ensures rapid adaptation to faults, making it suitable for real-world applications where actuator failures are common.
In conclusion, the improved RBF neural network-based approach for fault diagnosis and fault-tolerant control in quadrotor UAVs offers significant advantages in accuracy and stability. By incorporating adaptive laws and tuning parameters, the method achieves precise fault estimation and robust control, even under challenging conditions. Future work could explore extensions to multi-quadrotor systems or integration with other AI techniques for enhanced autonomy. This research contributes to the ongoing efforts to improve the safety and efficiency of quadrotor operations across various domains.