The rapid development of low-altitude economies in China has spurred extensive research into unmanned aerial vehicles (UAVs), particularly trirotor configurations that offer high maneuverability and energy efficiency. However, the unique mechanical structure of trirotor UAV drones introduces pronounced nonlinearities, making their control challenging when actuator faults and external disturbances occur simultaneously. Ensuring stable flight under such adverse conditions is a critical issue for China UAV drone applications in logistics, agriculture, and surveillance. This paper presents a novel predefined-time fault-tolerant tracking control strategy based on iterative learning neural networks (IL-NN) to address this challenge. The proposed approach integrates an IL-NN for real-time disturbance estimation, an adaptive law for online actuator fault compensation, and a predefined-time sliding mode control framework to guarantee convergence within a user-specified time. Through rigorous Lyapunov analysis and hardware-in-the-loop experiments, we demonstrate that the closed-loop system achieves predefined-time stability with superior tracking accuracy and computational efficiency compared to existing methods. The results underline the potential of this control scheme for enhancing the reliability of China UAV drone operations in complex environments.
1. Introduction
The adoption of trirotor UAV drones within China’s low-altitude economy has grown exponentially, driven by their ability to perform diverse tasks such as aerial mapping, cargo delivery, and environmental monitoring. Despite their advantages, trirotor configurations suffer from inherent aerodynamic coupling and sensitivity to actuator degradation. Traditional control methods like PID, sliding mode control, and backstepping have been widely applied, but they typically guarantee asymptotic stability without providing explicit convergence time. Limited-time control methods address convergence speed but require knowledge of initial conditions, which are often unknown in practice. Fixed-time control overcomes this by providing a uniform convergence bound, yet the bound’s expression is complex and depends on multiple parameters.
Predefined-time control, introduced by Sánchez-Torres, offers a direct way to specify the convergence time as a designer-chosen parameter. This concept has been successfully applied to quadrotor UAVs in several recent studies, but its extension to trirotor UAV drones under simultaneous actuator faults and time-varying disturbances remains unexplored. Furthermore, most existing works employ adaptive neural networks for disturbance approximation, which suffer from high computational overhead due to continuous weight updates.
In this work, we propose an IL-NN based predefined-time fault-tolerant tracking control scheme tailored for trirotor China UAV drone systems. The main contributions are as follows:
- An iterative learning neural network (IL-NN) is designed to estimate unknown time-varying external disturbances. Unlike conventional adaptive RBF neural networks that update weights continuously, the IL-NN updates weight matrices only at discrete learning intervals, reducing computational complexity by approximately 10% in terms of floating-point operations (FLOPs).
- A predefined-time adaptive fault-tolerant controller is synthesized using an inner-outer loop architecture. The outer loop handles position tracking via a sliding mode surface enhanced with a time-varying gain function, while the inner loop employs a non-singular terminal sliding mode predefined-time controller for attitude stabilization. An adaptive law online estimates and compensates actuator efficiency losses.
- Rigorous Lyapunov analysis proves that the position error, attitude error, and sliding mode variables converge within predefined finite times, respectively. The cascaded system stability is established via a set of lemmas ensuring that the coupling term vanishes as the attitude error converges.
- Hardware-in-the-loop (HIL) experiments on a physical trirotor UAV flight control platform validate the effectiveness of the proposed strategy under multiple disturbance scenarios and actuator faults. Quantitative comparisons show that our method achieves attitude error RMSE of 0.011 rad and convergence time of 0.68 s, outperforming two comparative control schemes.
The remainder of the paper is organized as follows: Section 2 formulates the fault dynamics model of a trirotor UAV drone. Section 3 provides preliminaries on predefined-time stability. Sections 4 and 5 design the outer-loop position controller and inner-loop attitude controller, respectively. Section 6 analyzes cascaded system stability. Section 7 presents HIL experimental results, and Section 8 concludes the study.
2. Problem Formulation
2.1 Trirotor UAV Dynamics
Consider a trirotor UAV drone with three rotors arranged in a Y‑shape. The total thrust \(F\) and torque vector \(\tau = [\tau_\phi, \tau_\theta, \tau_\psi]^\mathrm{T}\) are given by:
$$
\begin{aligned}
F &= f_1 + f_2 + f_3 \cos(\delta), \\
\tau_\phi &= – f_1 l_1 \cos(\alpha) + f_2 l_2 \sin(\alpha), \\
\tau_\theta &= – f_1 l_1 \cos(\alpha) – f_2 l_2 \sin(\alpha) + f_3 \big(l_3 \cos(\delta) – \mu \sin(\delta)\big), \\
\tau_\psi &= \mu f_1 – \mu f_2 + f_3 \big(\mu \cos(\delta) + l_3 \sin(\delta)\big),
\end{aligned}
$$
where \(f_i\) is the lift of rotor \(i\), \(\delta\) the servo tilt angle, \(l_i\) the distance from rotor \(i\) to the center of mass, \(\alpha\) the angle between the rotor arm and the body x‑axis, and \(\mu\) the torque coefficient.
The Newton‑Euler equations in the inertial frame yield:
$$
\begin{aligned}
\ddot{\xi} &= -\frac{F}{m} R(\eta) e_3 + g e_3 + d_\xi, \\
\ddot{\eta} &= f(\eta) + h(\eta) \tau + d_\eta,
\end{aligned}
$$
where \(\xi = [x,y,z]^\mathrm{T}\) is the position, \(\eta = [\phi,\theta,\psi]^\mathrm{T}\) the attitude, \(m\) the mass, \(g\) gravity, \(J = \mathrm{diag}([J_x,J_y,J_z])\) the inertia matrix, \(R(\eta)\) and \(\Phi(\eta)\) rotation matrices, and \(d_\xi, d_\eta\) denote unknown time‑varying external disturbances.
2.2 Actuator Fault Model
Actuator faults are modeled as partial loss of effectiveness:
$$
u_r = \rho u_{ra},
$$
where \(\rho = \mathrm{diag}([\rho_1,\ldots,\rho_6])\) with \(0 < \rho_i < 1\). Applying this to the trirotor UAV dynamics gives the fault model:
$$
\begin{aligned}
\ddot{\xi} &= \rho_\xi u_\xi + d_\xi, \\
\ddot{\eta} &= f(\eta) + h(\eta) \rho_\eta \tau + d_\eta,
\end{aligned}
$$
with \(\rho_\xi = \mathrm{diag}([\rho_1,\rho_2,\rho_3])\) and \(\rho_\eta = \mathrm{diag}([\rho_4,\rho_5,\rho_6])\). The control objective is to design \(u_\xi\) (outer loop) and \(\tau\) (inner loop) such that the tracking errors \(e_\xi = \xi – \xi_d\) and \(e_\eta = \eta – \eta_d\) converge to zero within user‑defined times, despite disturbances and faults.
3. Preliminaries on Predefined‑Time Stability
Consider a nonlinear system \(\dot{x} = g(x,t)\). It is said to be predefined‑time stable if there exists a constant \(T_f > 0\) such that the settling time \(T(x_0) \le T_f\) for all initial states \(x_0\). A useful tool is the smooth function:
$$
q_0(x) = \begin{cases} 0, & x \le 0, \\ e^{-1/x}, & x > 0, \end{cases} \qquad
q_a(t,t_f) = \frac{q_0(t_f – t)}{q_0(t_f – t) + q_0(t)}, \qquad
q(t,t_f,\nu) = \frac{1}{q_a(t,t_f) + \nu},
$$
with \(0 < \nu \ll 1\). For a Lyapunov function \(V\) satisfying \(\dot{V} = -\big(k + \frac{\dot{q}}{q}\big)V + \zeta_1\), convergence to \(|V| \le \nu V_0 + \zeta_1/k\) occurs within \(t_f\). Another lemma guarantees predefined‑time stability if \(\dot{V} = -\frac{\pi}{\lambda T_c}\sqrt{ab}\big(aV^{(1-\lambda)/2} + bV^{(1+\lambda)/2}\big) + \zeta_2\), with \(0<\lambda<1\) and \(T_c\) the predefined time.
4. Outer‑Loop Position Controller Design
4.1 IL‑NN Disturbance Estimator
The position disturbance \(d_\xi\) is approximated by an RBF neural network:
$$
d_\xi = M_\xi S(s_\xi) + \varepsilon_\xi,
$$
where \(M_\xi\) is the ideal weight matrix, \(\varepsilon_\xi\) the approximation error, and \(S\) the Gaussian basis function vector. Instead of continuous adaptation, we apply an iterative learning update:
$$
\hat{d}_\xi = \hat{M}_\xi S(s_\xi), \qquad
\hat{M}_\xi(t) = Q_\xi \hat{M}_\xi(t_\beta) + s_\xi S^\mathrm{T}(s_\xi),
$$
with \(t_\beta = t-\beta\) denoting the previous learning instant, \(Q_\xi \in \mathbb{R}^{3\times3}\) a positive diagonal matrix, and \(\beta\) the learning interval (typically the sampling time).
4.2 Sliding Surface and Control Law
Define a sliding surface:
$$
s_\xi = \dot{e}_\xi + \Lambda_\xi, \quad
\Lambda_\xi = \frac{n_1 + r_1}{2} e_\xi + k_1 \mathrm{sig}(e_\xi)^\alpha,
$$
with \(n_1>0\), \(r_1 = \dot{q}/q\), \(k_1\) positive definite, \(1<\alpha<2\), and \(\mathrm{sig}(e_\xi)^\alpha = [|e_x|^\alpha \mathrm{sign}(e_x), |e_y|^\alpha \mathrm{sign}(e_y), |e_z|^\alpha \mathrm{sign}(e_z)]^\mathrm{T}\). The position control law is:
$$
u_\xi = \hat{\rho}_\xi^{-1}\Big(-\frac{n_2+r_2}{2}s_\xi + \ddot{\xi}_d – \dot{\Lambda}_\xi – \hat{d}_\xi – L_1\mathrm{sign}(s_\xi) – E_1\mathrm{sign}(s_\xi)\Big),
$$
and the adaptive law for fault estimation:
$$
\dot{\hat{\rho}}_\xi = 2\Gamma_1 s_\xi u_\xi^\mathrm{T},
$$
where \(n_2>0\), \(r_2 = \dot{q}/q\), \(L_1 > \|e_{d\xi}\|\), \(E_1 > \|\varepsilon_\xi\|\), and \(\Gamma_1\) is a positive definite gain matrix. The stability analysis (Theorem 2) shows that \(s_\xi\) converges within \(t_{f2}\) and \(e_\xi\) within \(T_f \le t_{f1}+t_{f2}\).
5. Inner‑Loop Attitude Controller Design
5.1 IL‑NN for Attitude Disturbance
Similarly, the attitude disturbance \(d_\eta\) is estimated by another IL‑NN with update law \(\hat{M}_\eta(t) = Q_\eta \hat{M}_\eta(t_\beta) + s_\eta S^\mathrm{T}(s_\eta)\).
5.2 Non‑Singular Terminal Sliding Mode
To avoid singularity, a piecewise sliding surface is adopted:
$$
s_\eta = \begin{cases}
\dot{e}_\eta + \frac{n_3+r_3}{2}\big((I+\gamma)\iota^{-\gamma}e_\eta – \gamma\iota^{-I-\gamma}\mathrm{sig}(e_\eta)^2\big), & \hat{s}_\eta \ne 0,\ |e_\eta| \le \iota,\\[6pt]
\dot{e}_\eta + \frac{n_3+r_3}{2}e_\eta + k_2\mathrm{sig}(e_\eta)^2, & \text{otherwise},
\end{cases}
$$
with \(\iota>0\), \(n_3>0\), \(r_3 = \dot{q}/q\), and \(\gamma\) a positive diagonal matrix. The attitude controller is:
$$
\tau = (h(\eta)\hat{\rho}_\eta)^{-1}(\tau_1 – \tau_2),
$$
where
$$
\tau_1 = -\frac{\pi}{2\lambda t_c}\sqrt{ab}\big(a V_{\eta1}^{-\lambda/2} + b V_{\eta1}^{\lambda/2}\big)s_\eta – f(\eta) + \ddot{\eta}_d – \hat{d}_\eta – L_2 \mathrm{sign}(s_\eta) – E_2 \mathrm{sign}(s_\eta),
$$
and \(\tau_2\) contains the derivatives of the switching terms. The adaptive law for fault estimation is:
$$
\dot{\hat{\rho}}_\eta = 2\Gamma_2 h(\eta)^\mathrm{T} s_\eta \tau^\mathrm{T}.
$$
Theorem 3 proves that the sliding surface \(s_\eta\) converges within predefined time \(t_c\). Theorem 4 further shows that the attitude error \(e_\eta\) converges within \(T_c \le t_c + t_{f3}\).
6. Cascaded System Stability
The coupling term \(\Delta_\xi = \Pi_2\big(-\frac{F}{m}R(\eta)e_3 + g e_3 – u_\xi\big)\) depends on both the position and attitude loops. Using Lemmas 5, 6, and 7, we establish:
- \(\|\Delta_\xi(s_\xi,\chi_\eta)\| \le \gamma(\|\chi_\eta\|)\|s_\xi\|\) with \(\gamma(\|\chi_\eta\|) = \frac{k_t k_H}{m}\|\chi_\eta\|\).
- The reduced system \(\dot{s}_\xi = \Xi_\xi\) satisfies the conditions of Lemma 4, ensuring that the cascaded system is globally asymptotically stable.
- Since both loops are predefined‑time stable, the overall convergence time is \(T_{\text{total}} = \max(T_f, T_c)\).
7. Hardware‑in‑the‑Loop Experimental Validation
We conducted HIL experiments on a trirotor UAV flight control platform (Fig. 1) consisting of a PIX flight controller, a three‑axis turntable, a simulation computer, and a host PC. The flight controller was placed inside the turntable to measure real‑time attitude. The parameters are listed in Table 1.
The initial position was \([0,0,0]^\mathrm{T}\) m and attitude \([-0.03,0.005,-0.768]^\mathrm{T}\) rad. The desired trajectory was:
$$
\xi_d = \big[-1.25\cos(0.4t)+1.25,\ -0.4\cos(0.4t)+0.4,\ -0.8\cos(0.5t)+0.8\big]^\mathrm{T}\,\mathrm{m},
\quad \psi_d = 0.4\sin(0.2t)-0.45\,\mathrm{rad}.
$$
External disturbances were injected at \(t=12\) s as multi‑frequency sinusoids plus Gaussian white noise. Actuator faults (80% efficiency in position, 70% in attitude) were introduced at \(t=22\) s. Two comparative algorithms were tested: (1) sliding mode control for both loops; (2) sliding mode position control plus predefined‑time attitude control from reference [22] with \(t_c = 3\) s.
| Symbol | Value | Symbol | Value |
|---|---|---|---|
| \(J_x\) | \(2.0 \times 10^{-3}\) kg·m² | \(n_1\) | 0.9 |
| \(J_y\) | \(8.3 \times 10^{-3}\) kg·m² | \(\alpha\) | 1.2 |
| \(J_z\) | \(8.2 \times 10^{-3}\) kg·m² | \(k_1\) | \(\mathrm{diag}(1,1,1)\) |
| \(m\) | 0.5 kg | \(n_2\) | 0.6 |
| \(\nu\) | 0.01 | \(L_1, E_1\) | 1 |
| \(\sigma\) | 4 | \(t_{f1}, t_{f2}\) | 2 s |
| \(c\) | \([-1,-1,-1]^\mathrm{T}\) | \(\Gamma_1\) | \(\mathrm{diag}(0.1,0.1,0.1)\) |
| \(Q_\xi\) | \(\mathrm{diag}(0.95,0.96,0.95)\) | \(\gamma\) | \(\mathrm{diag}(0.05,0.05,0.05)\) |
| \(\iota\) | 0.0001 | \(n_3\) | 2 |
| \(k_2\) | \(\mathrm{diag}(1,1,1)\) | \(a,b\) | 2 |
| \(\lambda\) | 0.32 | \(L_2, E_2\) | 0.1 |
| \(t_{f3}\) | 1 s | \(t_c\) | 2 s |
| \(\Gamma_2\) | \(\mathrm{diag}(0.0001,0.0001,0.0001)\) | \(Q_\eta\) | \(\mathrm{diag}(0.995,0.995,0.995)\) |
The experimental results are summarized in Table 2. Our proposed method achieved an attitude convergence time of 0.68 s, significantly faster than the 1.05 s (sliding mode) and 0.87 s (predefined‑time reference). The attitude error RMSE was 0.011 rad, compared to 0.034 rad and 0.021 rad for the two comparative schemes. The position error RMSE was comparable (0.039 m vs. 0.030 m and 0.065 m), demonstrating that our controller maintains high tracking accuracy while reducing computational load.
| Metric | Proposed IL‑NN | Comparative 1 (SMC) | Comparative 2 (Predefined‑time Ref.) |
|---|---|---|---|
| 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 OS (m) | 0.081 | 0.089 | 0.164 |
| Attitude OS (rad) | 0.031 | 0.095 | 0.051 |
We also compared the computational complexity of the proposed IL‑NN with a conventional adaptive RBFNN. The FLOPs over 3000 iterations are listed in Table 3. The IL‑NN required only \(4.77 \times 10^5\) FLOPs, which is 10% less than the \(5.31 \times 10^5\) FLOPs of the adaptive RBFNN, confirming the higher efficiency of the iterative learning approach.
| Metric | IL‑NN (Proposed) | Adaptive RBFNN |
|---|---|---|
| Single‑step FLOPs | 159.00 | 177.00 |
| Total FLOPs (3000 steps) | \(4.77 \times 10^5\) | \(5.31 \times 10^5\) |
| Relative ratio (%) | 100.00 | 111.32 |
The disturbance estimation performance is illustrated in Fig. 2 (not shown due to text constraints), but the low steady‑state errors confirmed that the IL‑NN effectively compensated for time‑varying disturbances. The adaptive fault estimates converged to near the true values (0.8 for position, 0.7 for attitude) after the fault injection at 22 s. These validation results highlight the robustness and efficiency of the designed controller for China UAV drone platforms.
8. Conclusion
This paper addressed the trajectory tracking problem of trirotor UAV drones subject to simultaneous external disturbances and actuator faults. A predefined-time adaptive fault-tolerant control strategy was proposed, leveraging iterative learning neural networks for disturbance compensation and adaptive algorithms for online fault estimation. The IL‑NN significantly reduced computational cost compared to traditional adaptive neural networks. Lyapunov stability analysis proved that both position and attitude errors converge within user‑defined predefined times. Hardware‑in‑the‑loop experiments validated the superior performance of the method in terms of convergence speed, tracking accuracy, and computational efficiency. The approach is well‑suited for high‑reliability China UAV drone applications. Future work will investigate bias-type actuator faults and real flight tests to further enhance engineering maturity.