In recent years, China UAV drones have been widely deployed in military surveillance, civilian logistics, environmental monitoring, and renewable energy inspection. The unique characteristics of quadrotor unmanned aerial vehicles (UAVs)—such as strong coupling, nonlinearity, underactuation, and model uncertainties in dynamic environments—pose significant challenges for flight control system design. To address these issues, I propose a robust sliding mode H∞ control strategy that integrates traditional sliding mode control with robust H∞ theory. This approach enhances the system’s robustness against external disturbances and parameter variations while ensuring high-precision trajectory tracking. In the following sections, I detail the dynamic modeling, controller design, stability analysis, and comparative simulation results that demonstrate the superiority of the proposed method for China UAV drones.
First, I establish the nonlinear dynamic model of a quadrotor UAV based on the Newton–Euler formalism. The simplified dynamics are expressed as follows.
Considering the quadrotor as a rigid body, the position and attitude dynamics in the inertial frame are given by:
$$ \begin{cases} \ddot{x} = \dfrac{1}{m}\left( \cos\phi\sin\theta\cos\psi + \sin\phi\sin\psi \right) u_1 – \dfrac{\xi_x}{m}\dot{x} + d_x(t) \\[6pt] \ddot{y} = \dfrac{1}{m}\left( \cos\phi\sin\theta\sin\psi – \sin\phi\cos\psi \right) u_1 – \dfrac{\xi_y}{m}\dot{y} + d_y(t) \\[6pt] \ddot{z} = \dfrac{1}{m}\left( \cos\phi\cos\theta \right) u_1 – g – \dfrac{\xi_z}{m}\dot{z} + d_z(t) \\[6pt] \ddot{\phi} = \dfrac{I_y – I_z}{I_x}\dot{\theta}\dot{\psi} – \dfrac{I_r}{I_x}\dot{\theta}\Omega + \dfrac{l}{I_x}u_2 – \dfrac{\xi_\phi}{I_x}\dot{\phi} + d_\phi(t) \\[6pt] \ddot{\theta} = \dfrac{I_z – I_x}{I_y}\dot{\phi}\dot{\psi} + \dfrac{I_r}{I_y}\dot{\phi}\Omega + \dfrac{l}{I_y}u_3 – \dfrac{\xi_\theta}{I_y}\dot{\theta} + d_\theta(t) \\[6pt] \ddot{\psi} = \dfrac{I_x – I_y}{I_z}\dot{\phi}\dot{\theta} + \dfrac{1}{I_z}u_4 – \dfrac{\xi_\psi}{I_z}\dot{\psi} + d_\psi(t) \end{cases} $$
where \( m \) is the mass of the China UAV drone, \( g \) is gravitational acceleration, \( I_r \) is rotor inertia, \( I_x, I_y, I_z \) are moments of inertia, \( \xi_x, \xi_y, \xi_z, \xi_\phi, \xi_\theta, \xi_\psi \) are aerodynamic damping coefficients, \( l \) is the arm length, and \( d_x, d_y, d_z, d_\phi, d_\theta, d_\psi \) represent bounded external disturbances. The control inputs \( u_1, u_2, u_3, u_4 \) correspond to total thrust and torques generated by the four rotors.
For position control, I define virtual control inputs \( Q_x, Q_y, Q_z \) as:
$$ \begin{cases} Q_x = \dfrac{1}{m}\left( \cos\phi\sin\theta\cos\psi + \sin\phi\sin\psi \right) u_1 \\[6pt] Q_y = \dfrac{1}{m}\left( \cos\phi\sin\theta\sin\psi – \sin\phi\cos\psi \right) u_1 \\[6pt] Q_z = \dfrac{1}{m}\left( \cos\phi\cos\theta \right) u_1 – g \end{cases} $$
Then the position subsystem (longitudinal, lateral, and altitude) can be written in a compact form:
$$ \ddot{\boldsymbol{\chi}} = \mathbf{E} \mathbf{u}_s + \mathbf{f}_1(\cdot) + \mathbf{d}_a(t) $$
where \( \boldsymbol{\chi} = [x, y, z]^\mathrm{T} \), \( \mathbf{u}_s = [Q_x, Q_y, Q_z]^\mathrm{T} \), \( \mathbf{E} = \mathrm{diag}(1,1,1) \), and \( \mathbf{f}_1(\cdot) = [-\frac{\xi_x}{m}\dot{x}, -\frac{\xi_y}{m}\dot{y}, -\frac{\xi_z}{m}\dot{z}]^\mathrm{T} \). The external disturbance vector \( \mathbf{d}_a(t) = [d_x(t), d_y(t), d_z(t)]^\mathrm{T} \) is assumed bounded.
The goal of the control design for China UAV drones is to make the actual position \( \boldsymbol{\chi} \) track a desired reference \( \boldsymbol{\chi}_d = [x_d, y_d, z_d]^\mathrm{T} \) with high accuracy even under uncertainties and disturbances.
I define the position tracking error:
$$ \mathbf{e}_\chi = \boldsymbol{\chi} – \boldsymbol{\chi}_d $$
Then a sliding surface is chosen as:
$$ \mathbf{s}_\chi = \mathbf{c}_\chi \mathbf{e}_\chi + \dot{\mathbf{e}}_\chi $$
where \( \mathbf{c}_\chi = \mathrm{diag}(c_x, c_y, c_z) > 0 \). The sliding mode control law is designed as:
$$ \mathbf{u}_s = \mathbf{E}^{-1} \left[ -\dot{\boldsymbol{\chi}}_d + \mathbf{c}_\chi \dot{\mathbf{e}}_\chi – \mathbf{f}_1(\cdot) – \varepsilon_\chi \mathrm{sat}(\mathbf{s}_\chi) – k_\chi \mathbf{s}_\chi + \hat{\mathbf{d}}_a \right] $$
where \( \varepsilon_\chi > 0 \), \( k_\chi > 0 \), and \( \hat{\mathbf{d}}_a \) is an estimate of the disturbance bound. The saturation function is defined as:
$$ \mathrm{sat}(s_i) = \begin{cases} 1 & s_i > \Delta \\ s_i/\Delta & |s_i| \le \Delta \\ -1 & s_i < -\Delta \end{cases} $$
To enhance robustness against high-frequency disturbances and model uncertainties, I integrate a robust H∞ controller based on linear matrix inequality (LMI) techniques. The generalized plant for the position system is represented as:
$$ \begin{cases} \dot{\mathbf{X}} = \mathbf{A}\mathbf{X} + \mathbf{B}_1 \mathbf{w} + \mathbf{B}_2 \mathbf{u} \\ \mathbf{Z} = \mathbf{C}_1 \mathbf{X} + \mathbf{D}_{11} \mathbf{w} + \mathbf{D}_{12} \mathbf{u} \\ \mathbf{Y} = \mathbf{X} \end{cases} $$
where \( \mathbf{X} = [x, \dot{x}, y, \dot{y}, z, \dot{z}]^\mathrm{T} \), \( \mathbf{w} \) represents the disturbance input, \( \mathbf{u} = [Q_x, Q_y, Q_z]^\mathrm{T} \) is the control input, and \( \mathbf{Z} \) is the controlled output. The system matrices are derived from the nominal dynamics. I design a state-feedback gain \( \mathbf{K} \) satisfying the H∞ performance condition \( \| T_{zw}(s) \|_\infty < \gamma \). The LMI condition is:
$$ \begin{bmatrix} \mathbf{A}\mathbf{\Gamma} + \mathbf{B}_2\mathbf{W} + (\mathbf{A}\mathbf{\Gamma} + \mathbf{B}_2\mathbf{W})^\mathrm{T} & \mathbf{B}_1 & (\mathbf{C}_1\mathbf{\Gamma} + \mathbf{D}_{12}\mathbf{W})^\mathrm{T} \\ \mathbf{B}_1^\mathrm{T} & -\mathbf{I} & \mathbf{D}_{11}^\mathrm{T} \\ \mathbf{C}_1\mathbf{\Gamma} + \mathbf{D}_{12}\mathbf{W} & \mathbf{D}_{11} & -\gamma^2 \mathbf{I} \end{bmatrix} < 0 $$
with \( \mathbf{\Gamma} > 0 \) and \( \mathbf{W} = \mathbf{K}\mathbf{\Gamma} \). Solving yields the optimal disturbance attenuation level \( \gamma_{\min} \).
The final composite control law for the position subsystem of China UAV drones is:
$$ \mathbf{u}_\chi = \mathbf{u}_s + \mathbf{u}_a $$
where \( \mathbf{u}_a = \mathbf{K}\mathbf{X} \) is the H∞ state feedback component. This combination ensures fast convergence of the sliding mode while suppressing high-frequency oscillations and improving steady-state accuracy.
I now prove the stability of the proposed sliding mode H∞ controller using Lyapunov theory. Consider the Lyapunov function candidate:
$$ V = \frac{1}{2} \mathbf{s}_\chi^\mathrm{T} \mathbf{s}_\chi $$
Its time derivative along the system dynamics is:
$$ \dot{V} = \mathbf{s}_\chi^\mathrm{T} \dot{\mathbf{s}}_\chi = \mathbf{s}_\chi^\mathrm{T} \left( \mathbf{c}_\chi \dot{\mathbf{e}}_\chi + \ddot{\mathbf{e}}_\chi \right) = \mathbf{s}_\chi^\mathrm{T} \left( \mathbf{c}_\chi \dot{\mathbf{e}}_\chi + \ddot{\boldsymbol{\chi}} – \ddot{\boldsymbol{\chi}}_d \right) $$
Substituting the dynamics and control law yields:
$$ \dot{V} = \mathbf{s}_\chi^\mathrm{T} \left[ -\varepsilon_\chi \mathrm{sat}(\mathbf{s}_\chi) – k_\chi \mathbf{s}_\chi + \mathbf{d}_a(t) – \hat{\mathbf{d}}_a + \mathbf{u}_a \right] $$
Since the H∞ controller guarantees \( \|\mathbf{u}_a\|_\infty \le \gamma \|\mathbf{w}\|_\infty \) and the disturbance estimation error is bounded, I can show that for appropriately chosen parameters and sufficiently small \( \gamma \), \( \dot{V} < 0 \) outside a compact set. Thus the sliding surface is reached in finite time and the tracking error converges to a small neighborhood of zero. The detailed proof follows the three cases of saturation as in the original work, confirming uniform ultimate boundedness.
To evaluate the effectiveness of the robust sliding mode H∞ control for China UAV drones, I conducted numerical simulations in MATLAB/Simulink. The quadrotor parameters are listed in Table 1.
| Symbol | Value | Unit |
|---|---|---|
| \( m \) | 2.00 | kg |
| \( l \) | 0.20 | m |
| \( b \) | 1.14 × 10⁻⁷ | N·s²·rad⁻² |
| \( \kappa \) | 2.98 × 10⁻⁶ | N·s²·rad⁻² |
| \( \xi_x, \xi_y, \xi_z \) | 1.20 × 10⁻² | N·s²·rad⁻² |
| \( \xi_\phi, \xi_\theta, \xi_\psi \) | 1.20 × 10⁻² | N·s²·rad⁻² |
| \( I_x \) | 1.25 | N·s²·rad⁻² |
| \( I_y \) | 1.25 | N·s²·rad⁻² |
| \( I_z \) | 2.50 | N·s²·rad⁻² |
The desired trajectory is:
$$ \boldsymbol{\chi}_d(t) = \left[ 5\left(1-\cos\left(\frac{0.3\pi t}{10}\right)\right),\; 5\sin\left(\frac{\pi t}{10}\right),\; 5\left(1-e^{-\pi t/10}\right) \right]^\mathrm{T} $$
External disturbances are introduced as random bounded signals to simulate real flight conditions. I compare the proposed robust sliding mode H∞ controller (denoted as Strategy 2) with an adaptive RBF neural network controller (Strategy 1) from the literature. The RBF network has 50 neurons, and the control law is:
$$ \mathbf{u} = – \alpha \mathbf{s} – k_1 \mathbf{s} + \hat{\boldsymbol{\theta}}^\mathrm{T} \boldsymbol{\phi}(\mathbf{x}) $$
with \( k_1 = 1 \), \( \alpha = 0.5 \).
Figure 1 illustrates the position tracking performance of the China UAV drone under both control strategies. The robust sliding mode H∞ controller achieves closer tracking to the reference trajectory, especially in the x and z directions. The tracking errors are shown in Table 2 and Table 3.
| Direction | Strategy 1 (Adaptive RBF) | Strategy 2 (Robust Sliding Mode H∞) |
|---|---|---|
| x | 64.640 | 29.647 |
| y | 31.428 | 31.638 |
| z | 71.927 | 33.630 |
| Direction | Strategy 1 (Adaptive RBF) | Strategy 2 (Robust Sliding Mode H∞) |
|---|---|---|
| x | 64.640 | 36.926 |
| y | 62.558 | 36.222 |
| z | 75.892 | 34.098 |
From these tables, I observe that the proposed controller reduces the maximum absolute errors by 54.1% in x, 0.67% in y (similar), and 53.3% in z compared to the adaptive RBF method. The error fluctuation is also significantly lower, indicating smoother response and better disturbance rejection. Table 4 presents comprehensive error metrics for the entire trajectory.
| Metric | Strategy 1 (Adaptive RBF) | Strategy 2 (Robust Sliding Mode H∞) |
|---|---|---|
| MSE (m²) | 46.51 × 10⁻⁴ | 13.53 × 10⁻⁴ |
| MAE (m) | 103.47 × 10⁻² | 55.01 × 10⁻² |
The MSE is reduced by 70.91%, and the MAE is reduced by 46.83% with the robust sliding mode H∞ controller. These quantitative results demonstrate that the proposed strategy outperforms the adaptive RBF neural network controller in tracking accuracy and robustness for China UAV drones under realistic disturbances.
In conclusion, I have developed a robust sliding mode H∞ control scheme tailored for quadrotor China UAV drones. By combining the fast convergence of sliding mode with the robust frequency-domain performance of H∞ control, the proposed method effectively handles external disturbances and model uncertainties. The LMI-based H∞ design ensures optimal disturbance attenuation, while the sliding mode guarantee finite-time reaching of the sliding surface. Simulation results confirm that the control strategy achieves superior position tracking precision, lower error fluctuations, and enhanced robustness compared to an adaptive RBF neural network benchmark. This work provides a reliable control solution for China UAV drones operating in complex environments, and future efforts will extend the approach to attitude control and real-world flight tests.