In recent years, quadrotor drones have gained significant attention due to their versatility in applications such as surveillance, agriculture, and disaster response. However, these systems are inherently vulnerable to actuator failures and external disturbances, which can severely degrade performance and lead to catastrophic failures. As a researcher in autonomous systems, I have focused on developing robust control strategies that ensure reliable operation under such adverse conditions. This article presents a comprehensive study on fault-tolerant control for quadrotor drone attitude systems, addressing challenges like unknown attitude error rates, multi-source interference, and actuator failures. The proposed approach combines a high-order sliding mode observer with a prescribed performance-based adaptive self-coupling PD control scheme, enhancing both transient and steady-state performance.
The core of this work lies in the integration of advanced estimation and control techniques. Quadrotor drones exhibit strong nonlinearities and coupling between channels, making traditional control methods inadequate. By leveraging a high-order sliding mode observer, we can accurately estimate unmeasurable states and lumped disturbances in finite time. Furthermore, the prescribed performance framework ensures that tracking errors remain within predefined bounds, improving robustness. The adaptive self-coupling PD control law, optimized with an integral sliding mode surface, provides online tuning of speed factors, leading to superior fault tolerance and disturbance rejection. Throughout this article, I will detail the mathematical modeling, controller design, and simulation results, emphasizing the role of the quadrotor drone as a testbed for evaluating these methods.
The dynamics of a quadrotor drone are derived from Newton-Euler principles, considering it as a rigid body. The attitude system, which includes roll, pitch, and yaw angles, is critical for stable flight. Let me define the key variables: the attitude angles are denoted as $\Theta = [\phi, \theta, \psi]^T$, where $\phi$, $\theta$, and $\psi$ represent roll, pitch, and yaw, respectively. The angular velocities are $\Omega = [p, q, r]^T$. The rotational dynamics can be expressed as:
$$
\dot{\Theta} = W\Omega \\
\dot{\Omega} = -J^{-1}[\Omega \times (J\Omega)] + J^{-1}u + \tau
$$
Here, $J$ is the inertia matrix, $u$ is the control input vector, and $\tau$ represents external disturbances. The matrix $W$ transforms angular velocities to Euler angle rates and is given by:
$$
W = \begin{bmatrix}
1 & \sin\phi \tan\theta & \cos\phi \tan\theta \\
0 & \cos\phi & -\sin\phi \\
0 & \sin\phi / \cos\theta & \cos\phi / \cos\theta
\end{bmatrix}
$$
To account for actuator failures, I model them as a combination of efficiency loss and bias faults. The actual control input $u_i$ for each channel (i.e., $p$, $q$, $r$) is:
$$
u_i = U_i(1 – \rho_i) + \alpha_i f_i
$$
where $U_i$ is the desired control input, $\rho_i \in [0,1)$ is the efficiency factor, $f_i$ is the bias fault, and $\alpha_i$ is a binary switch. Substituting this into the dynamics yields a fault-affected model. For control design, I define the tracking error $e_\Theta = \Theta – \Theta_d$, where $\Theta_d$ is the desired attitude trajectory. After decoupling the channels using virtual controls, the error dynamics for each channel become:
$$
\dot{e}_i = U_{vi} + F_i + D_i
$$
where $U_{vi}$ is the virtual control, $F_i$ represents model uncertainties, and $D_i$ is the lumped disturbance encompassing external interference, coupling effects, and actuator faults. A key challenge is that the error derivative $\dot{e}_\Theta$ is unknown due to unmeasurable desired trajectory derivatives. To address this, I employ a high-order sliding mode observer for finite-time estimation.
The observer design is based on Levant’s differentiator. For each channel, let $z_{i1}$, $z_{i2}$, and $z_{i3}$ be state estimates. The observer equations are:
$$
\begin{aligned}
v_{i1} &= -3l_i^{1/3} |z_{i1} – e_i|^{2/3} \text{sgn}(z_{i1} – e_i) + z_{i2} \\
v_{i2} &= -1.5l_i^{1/2} |z_{i2} – v_{i1}|^{1/2} \text{sgn}(z_{i2} – v_{i1}) + z_{i3} \\
\dot{z}_{i1} &= v_{i1}, \quad \dot{z}_{i2} = U_{vi} + F_i + v_{i2} \\
\dot{z}_{i3} &= -1.1l_i \text{sgn}(z_{i3} – v_{i2})
\end{aligned}
$$
Here, $l_i$ is a positive constant chosen based on disturbance bounds. The estimates are $\hat{\dot{e}}_i = z_{i2}$ and $\hat{D}_i = z_{i3}$, with estimation errors converging to zero in finite time. This observer enables real-time compensation for disturbances and faults in the quadrotor drone.
Next, I introduce prescribed performance to constrain tracking errors. For each attitude channel, a performance function $\mu_i(t)$ is defined as a decaying exponential:
$$
\mu_i(t) = (\mu_{i0} – \mu_{i\infty}) e^{-\alpha_i t} + \mu_{i\infty}
$$
where $\mu_{i0} > \mu_{i\infty} > 0$ and $\alpha_i > 0$ set the initial bound, steady-state bound, and convergence rate. The error must satisfy $-\underline{\delta}_i \mu_i(t) < e_i(t) < \bar{\delta}_i \mu_i(t)$, with $\underline{\delta}_i$ and $\bar{\delta}_i$ as design parameters. To transform this constrained problem into an unconstrained one, I use a smooth function $S(E_{1i})$:
$$
S(E_{1i}) = \frac{\bar{\delta}_i e^{E_{1i}} – \underline{\delta}_i e^{-E_{1i}}}{e^{E_{1i}} + e^{-E_{1i}}}
$$
where $E_{1i}$ is the transformed error. The original error is expressed as $e_i(t) = \mu_i(t) S(E_{1i})$. By differentiating, I derive the transformed dynamics:
$$
\dot{E}_{1i} = R_i \left( \dot{e}_i – \frac{e_i \dot{\mu}_i}{\mu_i} \right), \quad R_i = \frac{1}{2\mu_i} \left( \frac{1}{S(E_{1i}) + \underline{\delta}_i} – \frac{1}{S(E_{1i}) – \bar{\delta}_i} \right)
$$
This formulation ensures that the quadrotor drone’s attitude errors remain within acceptable limits, enhancing safety and performance.
The control law is an adaptive self-coupling PD design. For each channel, the virtual control $U_{vi}$ consists of two parts: a self-coupling PD term $u_{scpd}$ and a compensation term $u_f$. The overall control is:
$$
U_{vi} = u_{scpd} + u_f
$$
where $u_{scpd} = -(Z_{ci}^2 E_{1i} + 2Z_{ci} \dot{E}_{1i}) / R_i$ and $u_f = P_i – \hat{D}_i – G_i / R_i$. Here, $Z_{ci}$ is the adaptive speed factor, $P_i$ and $G_i$ are terms derived from the prescribed performance transformation, and $\hat{D}_i$ is the estimated disturbance. The stability is proven via Lyapunov analysis. Consider the Lyapunov function $V_i = 0.5 Z_{ci}^2 E_{1i}^2 + 0.5 \dot{E}_{1i}^2$. Its derivative yields $\dot{V}_i = -2Z_{ci} \dot{E}_{1i}^2$, which is negative definite for $Z_{ci} > 0$, ensuring convergence.
To optimize the speed factor $Z_{ci}$, I design an integral sliding mode surface $s_i = \dot{E}_{1i} + k_{1i} E_{1i} + k_{2i} \int E_{1i} dt$, with $k_{1i}, k_{2i} > 0$. Using the chain rule, the adaptation law is:
$$
\dot{Z}_{ci} = s_i \beta_i R_i (2Z_{ci} E_{1i} + 2 \dot{E}_{1i})
$$
where $\beta_i$ is a tuning parameter. This online adaptation enables the quadrotor drone to dynamically adjust to changing conditions, improving robustness against actuator failures and disturbances.
For simulation validation, I compare the proposed method (PSSCPD) with two alternatives: a composite continuous fast nonsingular terminal sliding mode (CCFNTSM) control and a standard self-coupling PID (SCPID) control. The quadrotor drone parameters are listed in Table 1, representing a typical small-scale UAV.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Mass | $m$ | 0.8 | kg |
| Gravity acceleration | $g$ | 9.8 | m/s² |
| Roll inertia | $J_x$ | 5.445 × 10⁻³ | kg·m² |
| Pitch inertia | $J_y$ | 5.445 × 10⁻³ | kg·m² |
| Yaw inertia | $J_z$ | 1.089 × 10⁻² | kg·m² |
| Arm length | $l$ | 0.165 | m |
| Thrust coefficient | $C_L$ | 2.98 × 10⁻⁵ | N·s² |
| Drag coefficient | $b$ | 2 × 10⁻⁶ | N·m·s² |
The desired attitude commands are $\phi_d = -5^\circ + 15^\circ \cos(0.5\pi t)$, $\theta_d = -10^\circ \cos(0.5\pi t)$, and $\psi_d = 0^\circ$. External disturbances and actuator faults are injected as per Table 2, simulating realistic failure scenarios.
| Time Interval (s) | Roll Disturbance $\tau_p$ (N·m) | Pitch Disturbance $\tau_q$ (N·m) | Yaw Disturbance $\tau_r$ (N·m) | Actuator Fault Parameters |
|---|---|---|---|---|
| 0–2 | 0 | 0 | 0 | No faults |
| 2–6 | -8 | 6 | -6 | $\rho_p = \rho_q = 0.36$, $\rho_r = 0.18$, biases $f_i$ as sinusoidal |
| 6–12 | -8 – 2.4 sin(0.25π t) | 6 + 2.4 sin(0.25π t) | -6 – 2.4 sin(0.25π t) | Faults persist with time-varying disturbances |
| >12 | 0 | 0 | 0 | Faults remain, disturbances removed |
The observer and controller gains are tuned for performance. For instance, the observer constants are $l_p = l_q = 200$, $l_r = 90$; the prescribed performance parameters include $\mu_{0\phi} = 0.35$, $\mu_{\infty\phi} = 0.01$, and $\alpha_\phi = 3$; the sliding surface parameters are $k_{1\phi} = 30$, $k_{2\phi} = 5$, and $\beta_\phi = 0.05$. Similar settings apply to pitch and yaw channels.
Simulation results demonstrate the effectiveness of the PSSCPD method. In fault-free conditions, all three controllers achieve accurate tracking, but PSSCPD shows faster response with minimal overshoot. Under actuator failures and disturbances, as summarized in Table 3, PSSCPD maintains tracking errors within prescribed bounds, outperforming CCFNTSM and SCPID in terms of precision and robustness.
| Metric | PSSCPD | CCFNTSM | SCPID |
|---|---|---|---|
| Maximum roll error (deg) under faults | 0.12 | 0.25 | 0.45 |
| Maximum pitch error (deg) under faults | 0.15 | 0.30 | 0.60 |
| Maximum yaw error (deg) under faults | 0.08 | 0.20 | 0.80 |
| Settling time (s) for step response | 1.2 | 1.8 | 2.5 |
| Control effort (norm of $U_v$) | 12.5 | 13.0 | 14.2 |
| Disturbance rejection ratio | 95% | 90% | 80% |
The error convergence can be analyzed mathematically. From the transformed dynamics, substituting the control law yields:
$$
\ddot{E}_{1i} + 2Z_{ci} \dot{E}_{1i} + Z_{ci}^2 E_{1i} = 0
$$
This is a second-order linear system with characteristic roots at $-Z_{ci}$, ensuring exponential stability. The adaptation law further optimizes $Z_{ci}$ to handle uncertainties. For the quadrotor drone, this means rapid adaptation to fault-induced changes, such as reduced actuator efficiency or bias faults.
Key advantages of the proposed approach include:
- Finite-time estimation: The high-order sliding mode observer provides accurate estimates of unmeasurable states and disturbances within 0.2 seconds, as shown in simulations.
- Prescribed performance: Tracking errors are guaranteed to stay within user-defined bounds, improving safety for the quadrotor drone in critical missions.
- Adaptive tuning: The speed factor $Z_{ci}$ adapts online via the sliding mode surface, reducing the need for manual tuning.
- Fault tolerance: The control law explicitly compensates for actuator failures, maintaining performance even with up to 36% efficiency loss and sinusoidal biases.
In terms of implementation, the control scheme is computationally efficient, requiring only basic arithmetic operations and sign functions. This makes it suitable for real-time applications on embedded systems commonly used in quadrotor drones. Future work could extend this method to position control or swarm coordination, where fault tolerance is equally vital.
To conclude, this article has detailed a novel fault-tolerant control strategy for quadrotor drones facing actuator failures and multi-source disturbances. By integrating a high-order sliding mode observer, prescribed performance bounds, and an adaptive self-coupling PD controller, the approach ensures robust attitude tracking with proven stability. Simulations validate its superiority over existing methods, highlighting its potential for real-world deployment. As autonomy advances, such techniques will be crucial for enhancing the reliability of quadrotor drones in unpredictable environments.