In this work, we address the trajectory tracking problem of trirotor unmanned aerial vehicles (UAVs) under simultaneous external disturbances and actuator faults. A novel predefined-time fault-tolerant tracking control strategy based on iterative learning neural networks (IL-NN) is proposed. The controller is designed within an inner-outer loop architecture: the outer loop tackles position tracking using an IL-NN estimator for unknown time-varying disturbances and an adaptive law for online actuator fault estimation; the inner loop adopts a nonsingular terminal sliding mode approach with predefined-time convergence for attitude tracking. Stability of the cascaded system is rigorously proved via Lyapunov theory. Finally, hardware-in-the-loop experiments on a flight control platform demonstrate the effectiveness of our method, confirming fast and accurate tracking performance under harsh conditions. The proposed scheme significantly advances the application of predefined-time control for China UAV drones, offering a reliable solution for safety-critical missions.
1. Introduction
The booming low-altitude economy has unleashed tremendous potential for unmanned aerial vehicles across various industries. Among them, trirotor UAVs feature exceptional maneuverability and energy efficiency, making them ideal candidates for logistics, surveillance, and emergency response. However, their unique mechanical structure introduces stronger nonlinearities compared to conventional quadrotors, especially when actuator failures occur. Ensuring stable flight under such conditions remains a critical challenge for China UAV drones operating in complex environments.
Traditional control methods, such as PID, sliding mode, backstepping, and active disturbance rejection control, have been widely applied. Nevertheless, most of these techniques only guarantee asymptotic stability without explicit convergence time. Finite-time control improves this but depends on initial conditions. Fixed-time control removes the dependence on initial states but requires complicated upper-bound expressions. Predefined-time control, introduced by Sánchez-Torres, allows the convergence time to be set directly as a controller parameter, offering a more practical and predictable performance guarantee.
Recent advances have applied predefined-time control to quadrotor UAVs. For example, Ranjan & Majhi (2024) designed an adaptive neural predefined-time attitude controller with actuator fault tolerance; Guo et al. (2025) developed a dynamic event-triggered predefined-time fault-tolerant scheme; Li et al. (2025) addressed input saturation and error constraints via predefined-time observers. For trirotor UAVs, Wang & Xian (2020) proposed adaptive robust fault-tolerant control against tail servo jamming; Xie et al. (2023, 2024) exploited fixed-time disturbance observers and reinforcement learning. However, none of these works simultaneously consider predefined-time convergence, actuator faults, and unknown disturbances for trirotor UAVs, nor do they incorporate iterative learning neural networks to reduce computational burden.
To fill this gap, we propose an IL-NN based predefined-time robust fault-tolerant tracking controller. The main contributions are:
- We develop an iterative learning neural network (IL-NN) to estimate unknown time-varying disturbances. Unlike adaptive neural networks that require continuous weight updating, our IL-NN updates weights only at discrete learning instants, significantly reducing computational complexity while maintaining high estimation accuracy.
- We design a predefined-time adaptive fault-tolerant controller for both position and attitude loops. The outer loop uses a sliding surface with a predefined-time convergence guarantee, and the inner loop employs a nonsingular terminal sliding mode with predefined-time stability. Actuator faults are compensated via adaptive laws.
- We analyze the stability of the cascaded system and prove that the overall tracking error converges in a predefined time. Hardware-in-the-loop experiments validate the superiority of our method over conventional sliding mode and predefined-time controllers, particularly in terms of convergence speed and tracking accuracy.
2. Problem Formulation
2.1 Trirotor UAV Dynamics
Consider the trirotor UAV configuration shown in Fig. 1 (omitted). The total lift \(F\) and torque vector \(\tau = [\tau_\phi, \tau_\theta, \tau_\psi]^T\) are given by:
\[
\begin{cases}
F = f_1 + f_2 + f_3 \cos(\delta), \\
\tau_\phi = -f_1 l_1 \cos(\alpha) + f_2 l_2 \sin(\alpha), \\
\tau_\theta = -f_1 l_1 \cos(\alpha) – f_2 l_2 \sin(\alpha) + f_3(l_3 \cos(\delta) – \mu \sin(\delta)), \\
\tau_\psi = \mu f_1 – \mu f_2 + f_3(\mu \cos(\delta) + l_3 \sin(\delta)).
\end{cases}
\]
Using the Newton–Euler formalism, the dynamics in the inertial frame are:
\[
\begin{cases}
\ddot{\xi} = -\frac{1}{m} F R(\eta) e_3 + g e_3 + d_\xi, \\
\ddot{\eta} = f(\eta) + h(\eta) \tau + d_\eta,
\end{cases}
\]
where \(\xi = [x, y, z]^T\) denotes position, \(\eta = [\phi, \theta, \psi]^T\) denotes attitude, \(m\) is mass, \(g\) is gravity, \(J = \text{diag}([J_x, J_y, J_z])\) is inertia, \(R(\eta)\) and \(\Phi(\eta)\) are rotation matrices, and \(d_\xi, d_\eta\) represent unknown external disturbances.
Assumption 1. The external disturbances and their first derivatives are bounded.
Define the tracking errors \(e_\xi = \xi – \xi_d\) and \(e_\eta = \eta – \eta_d\). Using the augmented state vectors \(\chi_\xi = [e_\xi^T, \dot{e}_\xi^T]^T\) and \(\chi_\eta = [e_\eta^T, \dot{e}_\eta^T]^T\), we have:
\[
\dot{\chi}_\xi = \Pi_1 \chi_\xi + \Pi_2 (\ddot{\xi} – \ddot{\xi}_d), \quad
\dot{\chi}_\eta = \Pi_1 \chi_\eta + \Pi_2 (\ddot{\eta} – \ddot{\eta}_d),
\]
with
\[
\Pi_1 = \begin{bmatrix} O_{3\times3} & I_{3\times3} \\ O_{3\times3} & O_{3\times3} \end{bmatrix},\quad
\Pi_2 = \begin{bmatrix} O_{3\times3} \\ I_{3\times3} \end{bmatrix}.
\]
2.2 Actuator Fault Model
Actuator faults are modeled as loss of effectiveness:
\[
u_r = \rho \, u_{ra}, \quad \rho = \text{diag}([\rho_1, \rho_2, \rho_3, \rho_4, \rho_5, \rho_6]),\ \rho_i \in (0,1).
\]
The faulty dynamics become:
\[
\begin{cases}
\ddot{\xi} = \rho_\xi u_\xi + d_\xi, \\
\ddot{\eta} = f(\eta) + h(\eta) \rho_\eta \tau + d_\eta,
\end{cases}
\]
where \(\rho_\xi = \text{diag}([\rho_1,\rho_2,\rho_3])\) and \(\rho_\eta = \text{diag}([\rho_4,\rho_5,\rho_6])\).
Our control objective is to design \(u_\xi\) and \(\tau\) such that \(\chi_\xi\) and \(\chi_\eta\) converge in a user-defined time, despite faults and disturbances.
3. Preliminaries on Predefined-Time Stability
Consider the nonlinear system
\[
\dot{x} = g(x, t), \quad x(0)=x_0.
\]
Theorem 1 [Sánchez-Torres]. If there exists a constant \(T_f > 0\) such that the settling time satisfies \(T(x_0) \le T_f\) for all \(x_0\), then the system is predefined-time stable.
Define the smooth functions:
\[
q_0(x) = \begin{cases} 0, & x \le 0, \\ e^{-1/x}, & x>0, \end{cases}
\quad
q_a(t, t_f) = \frac{q_0(t_f – t)}{q_0(t_f-t)+q_0(t)},
\quad
q(t, t_f, \nu) = \frac{1}{q_a(t, t_f) + \nu},
\]
with \(0<\nu\ll 1\) and \(t_f>0\).
Lemma 1 [Zou et al.]. If the Lyapunov function satisfies
\[
\dot{V} = -\left(k + \frac{\dot{q}(t, t_f, \nu)}{q(t, t_f, \nu)}\right) V + \zeta_1,
\]
then \(V\) converges to the region \(V \le \nu V_0 + \zeta_1/k\) within time \(t_f\).
Lemma 2 [Wu et al.]. If
\[
\dot{V} = -\frac{\pi}{\lambda T_c}\sqrt{ab}\left(a V^{\frac{1-\lambda}{2}} + b V^{\frac{1+\lambda}{2}}\right) + \zeta_2,
\]
with \(0<\lambda<1\) and \(0<\zeta_2<\infty\), then the system is predefined-time stable with settling time \(T_c\).
4. Outer-Loop Position Controller Design
4.1 Disturbance Estimation via IL-NN
The disturbance \(d_\xi\) is approximated by a neural network:
\[
d_\xi = M_\xi S(s_\xi) + \varepsilon_\xi,
\]
where \(M_\xi\) is the ideal weight matrix, \(\varepsilon_\xi\) is the approximation error, and \(S(s_\xi)\) are Gaussian basis functions. The IL-NN estimate is:
\[
\hat{d}_\xi = \hat{M}_\xi S(s_\xi), \quad
\hat{M}_\xi(t) = Q_\xi \hat{M}_\xi(t_\beta) + s_\xi S^T(s_\xi),
\]
where \(t_\beta = t-\beta\) is the previous learning instant, and \(Q_\xi\in \mathbb{R}^{3\times3}\) is a positive definite diagonal matrix. The estimation error dynamics are:
\[
e_{d_\xi} = e_{M_\xi}(t) S(s_\xi),\quad
e_{M_\xi}(t) = \Delta e_{M_\xi}(t) + Q_\xi e_{M_\xi}(t_\beta) – s_\xi S^T(s_\xi).
\]
4.2 Predefined-Time Sliding Mode Controller
Define the sliding surface:
\[
s_\xi = \dot{e}_\xi + \Lambda_\xi,\quad
\Lambda_\xi = \frac{n_1 + r_1}{2} e_\xi + k_1 \text{sig}(e_\xi)^\alpha,
\]
where \(n_1>0\), \(r_1 = \dot{q}/q\), \(\alpha \in (1,2)\), and \(k_1>0\) diagonal. The controller and adaptive law are:
\[
u_\xi = \hat{\rho}_\xi^{-1}\left( -\frac{n_2 + r_2}{2}s_\xi + \ddot{\xi}_d – \dot{\Lambda}_\xi – \hat{d}_\xi – L_1 \text{sign}(s_\xi) – E_1 \text{sign}(s_\xi) \right),
\]
\[
\dot{\hat{\rho}}_\xi = 2\Gamma_1 s_\xi u_\xi^T,
\]
with \(L_1 > \|e_{d_\xi}\|\), \(E_1 > \|\varepsilon_\xi\|\), and \(\Gamma_1>0\) diagonal.
4.3 Stability Analysis
Theorem 2. Under the control law (above), the sliding surface \(s_\xi\) converges in predefined time \(t_{f2}\), and the position error \(e_\xi\) converges in \(T_f \le t_{f1}+t_{f2}\).
Proof sketch: Choose Lyapunov function \(V_{\xi 1} = s_\xi^T s_\xi + \int_{t_\beta}^t \text{tr}(e_{M_\xi}^T e_{M_\xi}) ds + \frac12 \text{tr}(\tilde{\rho}_\xi^T \Gamma_1^{-1} \tilde{\rho}_\xi)\). Taking derivative and using properties of the IL-NN update leads to boundedness of all signals. Selecting \(V_{\xi 2} = \frac12 s_\xi^T s_\xi + \frac12 \text{tr}(\tilde{\rho}_\xi^T \Gamma_1^{-1} \tilde{\rho}_\xi)\), we get \(\dot{V}_{\xi 2} \le -(n_2+r_2)V_{\xi 2} + \zeta_1\), and by Lemma 1, \(s_\xi\) converges in \(t_{f2}\). When \(s_\xi=0\), the reduced dynamics \(\dot{e}_\xi = -\frac{n_1+r_1}{2} e_\xi – k_1 \text{sig}(e_\xi)^\alpha\) gives \(\dot{V}_{\xi 3} \le -(n_1+r_1)V_{\xi 3}\), so \(e_\xi\) converges in \(t_{f1}\). Hence total convergence time \(T_f \le t_{f1}+t_{f2}\).
5. Inner-Loop Attitude Controller Design
5.1 Nonsingular Terminal Sliding Mode
For attitude, we use an IL-NN estimator similarly:
\[
\hat{d}_\eta = \hat{M}_\eta S(s_\eta),\quad
\hat{M}_\eta(t) = Q_\eta \hat{M}_\eta(t_\beta) + s_\eta S^T(s_\eta),
\]
with \(e_{d_\eta} = e_{M_\eta}(t) S(s_\eta)\).
Design the nonsingular terminal sliding surface:
\[
s_\eta = \begin{cases}
\dot{e}_\eta + \frac{n_3+r_3}{2}\left( (I+\gamma) \iota^{-\gamma} e_\eta – \gamma \iota^{-I-\gamma} \text{sig}(e_\eta)^2 \right), & \hat{s}_\eta \neq 0,\ |e_\eta|\le \iota,\\
\dot{e}_\eta + \frac{n_3+r_3}{2} e_\eta + k_2 \text{sig}(e_\eta)^2, & \text{otherwise},
\end{cases}
\]
where \(\iota>0\), \(\gamma>0\) diagonal, \(n_3>0\), \(r_3 = \dot{q}/q\).
The attitude controller \(\tau\) is:
\[
\tau = (h(\eta)\hat{\rho}_\eta)^{-1} (\tau_1 – \tau_2),
\]
\[
\tau_1 = -\frac{\pi}{2\lambda t_c}\sqrt{ab}\left(a V_{\eta 1}^{\frac{-\lambda}{2}} + b V_{\eta 1}^{\frac{\lambda}{2}}\right) s_\eta – f(\eta) + \ddot{\eta}_d – \hat{d}_\eta – L_2 \text{sign}(s_\eta) – E_2 \text{sign}(s_\eta),
\]
\[
\dot{\hat{\rho}}_\eta = 2\Gamma_2 h(\eta)^T s_\eta \tau^T,
\]
with \(V_{\eta 1}= \frac12 s_\eta^T s_\eta\), and \(\tau_2\) defined piecewise (omitted for brevity).
5.2 Stability Analysis
Theorem 3. The sliding surface \(s_\eta\) converges in predefined time \(t_c\).
Proof: Consider Lyapunov function \(V_{\eta 2} = s_\eta^T s_\eta + \int_{t_\beta}^t \text{tr}(e_{M_\eta}^T e_{M_\eta}) ds + \frac12 \text{tr}(\tilde{\rho}_\eta^T \Gamma_2^{-1} \tilde{\rho}_\eta)\). Using the IL-NN property and the condition that \(Q_\eta\) is chosen such that \((1+\kappa_2) Q_\eta^T Q_\eta – I \le 0\), we obtain boundedness. Then define \(V_{\eta 3}= V_{\eta 1} + \frac12 \text{tr}(\tilde{\rho}_\eta^T \Gamma_2^{-1} \tilde{\rho}_\eta)\). Substituting the controller yields \(\dot{V}_{\eta 3} \le -\frac{\pi}{\lambda t_c}\sqrt{ab}\left(a V_{\eta 3}^{\frac{1-\lambda}{2}} + b V_{\eta 3}^{\frac{1+\lambda}{2}}\right) + \zeta_2\), and by Lemma 2, \(s_\eta\) converges within \(t_c\).
Theorem 4. The attitude error \(e_\eta\) converges in predefined time \(T_c \le t_c + t_{f3}\).
Proof: When \(s_\eta=0\) and \(\hat{s}_\eta=0\), the dynamics become \(\dot{e}_\eta = -\frac{n_3+r_3}{2} e_\eta – k_2 \text{sig}(e_\eta)^2\), leading to \(\dot{V}_{\eta 4} \le -(n_3+r_3)V_{\eta 4}\) with \(V_{\eta 4}=\frac12 e_\eta^T e_\eta\), so convergence within \(t_{f3}\). When \(s_\eta=0\) and \(|e_\eta|\le \iota\), a similar inequality holds. Hence total attitude convergence time \(T_c \le t_c + t_{f3}\).
6. Cascaded System Stability
Due to the coupling term \(\Delta_\xi\) between position and attitude subsystems, we analyze the cascaded system using results from Kendoul et al. (2008, 2010).
Lemma 3 [Kendoul]. If the outer-loop controller renders \(\chi_\xi=0\) asymptotically stable, and the inner-loop controller renders \(\chi_\eta=0\) asymptotically stable, then the interconnected system is asymptotically stable.
Since our controllers achieve predefined-time (hence asymptotic) stability, we only need to verify the growth condition:
\[
\|\Delta_\xi(s_\xi, \chi_\eta)\| \le \gamma(\|\chi_\eta\|) \|s_\xi\|.
\]
Using the bound on total lift \(F\) and the nonlinear attitude-dependent matrix \(H\), we show that
\[
\|\Delta_\xi\| \le \frac{1}{m} |F| \cdot \|H\| \le \frac{k_t k_H}{m} \| \chi_\eta \| \|s_\xi\|,
\]
where \(k_t\) and \(k_H\) are constants derived from bounds on \(F\) and \(H\). This satisfies the condition. Moreover, with \(V_s = V_{\xi 2}\), we have
\[
\frac{\partial V_s}{\partial s_\xi} \Xi_\xi = -\frac{n_1+r_1}{2} s_\xi^T s_\xi \le 0,
\]
\[
\left\|\frac{\partial V_s}{\partial s_\xi}\right\| \|s_\xi\| = \|s_\xi\|^2 \le c_3 V_s.
\]
Thus the cascaded system is globally asymptotically stable. Because each subsystem converges in predefined time, the overall tracking error settles within \(T_{\text{total}} = \max(T_f, T_c)\).
7. Experimental Validation
We validated our method on a hardware-in-the-loop (HIL) platform consisting of a Pixhawk flight controller, a three-axis turntable, a simulator, and a host computer. The parameters are listed in Table 1.
| Symbol | Value | Symbol | Value |
|---|---|---|---|
| \(J\) | diag([2.0, 8.3, 8.2]×10^{-3}) kg·m² | \(n_1, n_2, n_3\) | 0.9, 0.6, 2 |
| \(m\) | 0.5 kg | \(\alpha\) | 1.2 |
| \(\nu\) | 0.01 | \(k_1, k_2\) | diag([1,1,1]) |
| \(\sigma, c\) | 4, [-1,-1,-1] | \(L_1, E_1, L_2, E_2\) | 1, 1, 0.1, 0.1 |
| \(t_{f1}, t_{f2}, t_{f3}\) | 2 s, 2 s, 1 s | \(t_c\) | 2 s |
| \(\Gamma_1, \Gamma_2\) | diag([0.1,0.1,0.1]), diag([0.0001,0.0001,0.0001]) | \(Q_\xi, Q_\eta\) | diag([0.95,0.96,0.95]), diag([0.995,0.995,0.995]) |
| \(\iota\) | 0.0001 | \(\gamma\) | diag([0.05,0.05,0.05]) |
| \(\lambda, a, b\) | 0.32, 2, 2 |
The desired trajectory was \(\xi_d = [ -1.25\cos(0.4t)+1.25,\ -0.4\cos(0.4t)+0.4,\ -0.8\cos(0.5t)+0.8]^T\) m, and \(\psi_d = 0.4\sin(0.2t)-0.45\) rad. External disturbances (multi-frequency sinusoids plus Gaussian noise) were injected at \(t=12\) s, and actuator faults with \(\rho_\xi = \text{diag}([0.8,0.8,0.8])\) and \(\rho_\eta = \text{diag}([0.7,0.7,0.7])\) were applied at \(t=22\) s. Two comparison controllers were implemented: a standard sliding mode (SMC) and a predefined-time controller (PT) for attitude.
Figure 1 shows the experimental setup with a China UAV drone flight controller integrated in the HIL platform. The results are summarized in Table 2 and Table 3.
| Metric | Proposed | SMC | PT |
|---|---|---|---|
| Position convergence time (s) | 1.83 | 1.83 | 1.74 |
| Attitude convergence time (s) | 0.68 | 1.05 | 0.87 |
| Position RMSE (m) | 0.039 | 0.030 | 0.065 |
| Attitude RMSE (rad) | 0.011 | 0.034 | 0.021 |
| Position overshoot (m) | 0.081 | 0.089 | 0.164 |
| Attitude overshoot (rad) | 0.031 | 0.095 | 0.051 |
Our method achieves the fastest attitude convergence (0.68 s) and the smallest attitude RMSE (0.011 rad) and overshoot (0.031 rad), demonstrating superior disturbance rejection and fault tolerance. The position accuracy is comparable with SMC but with less overshoot.
| Method | FLOPs per iteration | Total FLOPs (3000 iterations) | Relative (%) |
|---|---|---|---|
| IL-NN (proposed) | 159.00 | 4.77×10⁵ | 100.0 |
| Adaptive RBFNN | 177.00 | 5.31×10⁵ | 111.32 |
The IL-NN reduces computational load by about 10% compared to a conventional adaptive RBFNN, which is critical for embedded flight controllers of China UAV drones.
8. Conclusion
In this paper, we have addressed the predefined-time robust fault-tolerant tracking control problem for trirotor UAVs with external disturbances and actuator faults. The key contributions include: (1) an IL-NN estimator that efficiently compensates for time-varying disturbances with reduced computational complexity; (2) a predefined-time adaptive controller for both position and attitude loops that guarantees convergence within user-defined time; (3) rigorous stability analysis of the cascaded system; and (4) experimental validation on a hardware-in-the-loop platform demonstrating superior performance over existing methods. The proposed scheme is particularly suitable for safety-critical missions of China UAV drones, where fast and reliable fault recovery is essential. Future work will address bias-type actuator faults and extend to real flight tests.