We present a nonlinear model predictive control (NMPC) framework integrated with a discrete-time nonlinear disturbance observer (NDOB) for attitude tracking of a fixed-wing China UAV drone. The proposed method addresses actuator constraints, model uncertainties, and unknown external disturbances that commonly degrade the performance of China UAV drone flight control systems. By designing an adaptive discrete-time NDOB that estimates the lumped disturbance bound online, we effectively compensate for modeling errors and external perturbations within the NMPC optimization. This approach ensures robust and precise attitude tracking while respecting physical limits of the China UAV drone actuators. We prove the stability of the closed-loop system via Lyapunov analysis and validate the effectiveness through numerical simulations involving a typical China UAV drone model with aerodynamic parameter uncertainties and actuator biases.
1. Introduction
The control of China UAV drones, especially fixed-wing configurations, requires handling strong nonlinearities, coupling effects, and various operational constraints. Model predictive control (MPC) has become a popular method for such systems due to its ability to explicitly incorporate constraints and optimize future behavior. However, the performance of standard NMPC degrades significantly in the presence of model uncertainties and external disturbances because the prediction relies on an imperfect nominal model. To overcome this limitation, many researchers have combined NMPC with disturbance observers (DOB) to estimate and compensate for unknown perturbations. Nevertheless, most existing DOB designs are formulated in continuous time, while practical implementation on embedded flight controllers of China UAV drones requires discrete-time formulations. Moreover, prior knowledge of the disturbance bound is often needed, which is difficult to obtain in real flight scenarios. Therefore, we propose a novel discrete-time NDOB with an adaptive term to estimate the upper bound of the lumped disturbance, and we integrate this observer seamlessly into the NMPC framework for a China UAV drone attitude system.
The primary contributions of our work are threefold:
- We design a discrete-time nonlinear disturbance observer that adaptively estimates the lumped disturbance bound without requiring prior knowledge of the bound. This is achieved by introducing an auxiliary state and a parameter adaptation law.
- We formulate the NMPC optimization problem using the estimated disturbance to correct the nominal model, thereby improving robustness against model uncertainties and external disturbances for the China UAV drone.
- We rigorously prove the stability of the discrete-time observer and the closed-loop NMPC system using Lyapunov techniques, and we verify the proposed method through comprehensive simulations on a realistic China UAV drone model under severe aerodynamic parameter deviations and actuator biases.
2. China UAV Drone Dynamics and Problem Formulation
The attitude dynamics of a fixed-wing China UAV drone are described by the following nonlinear differential equations:
$$
\begin{aligned}
\dot{\phi} &= p + q \sin\phi \tan\theta + r \cos\phi \tan\theta \\
\dot{\theta} &= q \cos\phi – r \sin\phi \\
\dot{\psi} &= (q \sin\phi + r \cos\phi) \sec\theta \\
\dot{p} &= c_1 p + c_2 r q + c_3 L + c_4 N \\
\dot{q} &= c_5 p r – c_6 (p^2 – r^2) + c_7 M \\
\dot{r} &= c_8 p – c_9 r q + c_4 L + c_9 N
\end{aligned}
$$
where $\phi,\theta,\psi$ are the roll, pitch, and yaw angles; $p,q,r$ are the angular rates; $L,M,N$ are the aerodynamic moments; and $c_i$ are inertia-related coefficients. The aerodynamic moments depend nonlinearly on the control surfaces: aileron ($\delta_a$), elevator ($\delta_e$), and rudder ($\delta_r$). The airspeed $V$, angle of attack $\alpha$, and sideslip angle $\beta$ are governed by additional translational dynamics. In practice, the aerodynamic coefficients are uncertain due to manufacturing tolerances, wear, and varying flight conditions. Moreover, actuator biases and external wind gusts introduce unknown disturbances. Therefore, the actual system can be represented as:
$$
\begin{aligned}
\mathbf{x}_2(k+1) &= \mathbf{f}_2(\mathbf{x}_2(k)) + \mathbf{g}_2 \mathbf{u}_1(k) + \mathbf{d}(k) \\
\mathbf{y}(k) &= \mathbf{H}(\mathbf{x}(k))
\end{aligned}
$$
where $\mathbf{d}(k) = [d_1, d_2, d_3]^T$ is the lumped disturbance, including model uncertainty and external perturbations. The problem is to design a control law that tracks desired attitude angles and airspeed while respecting actuator constraints $|\delta| \leq 25^\circ$ and $|\dot{\delta}| \leq 120^\circ/s$.
3. Discrete-Time Nonlinear Disturbance Observer
To estimate the lumped disturbance $\mathbf{d}(k)$, we propose a discrete-time NDOB with an adaptive term. The observer equations for each component $d_i$ ($i=1,2,3$) are given by:
$$
\begin{aligned}
z_i(k+1) &= z_i(k) + \Delta t \left[ \lambda_{1i} \pi_i(k) + \lambda_{2i} \frac{\pi_i(k)}{|\pi_i(k)| + \mu_i} + \hat{p}_i(k) \tanh\!\left(\frac{\gamma_i \pi_i(k)}{\varepsilon}\right) \right] \\
\hat{d}_i(k) &= \lambda_{1i} \pi_i(k) + \lambda_{2i} \frac{\pi_i(k)}{|\pi_i(k)| + \mu_i} + z_i(k) + \hat{p}_i(k) \tanh\!\left(\frac{\gamma_i \pi_i(k)}{\varepsilon}\right) \\
\pi_i(k) &= x_{2i}(k) – z_i(k) \\
\hat{p}_i(k+1) &= \hat{p}_i(k) + \Delta t \left[ m_i \hat{p}_i(k) – n_i \hat{p}_i^3(k) + \lambda_{2i} \frac{|\pi_i(k)|}{|\pi_i(k)| + \mu_i} + \gamma_i |\pi_i(k)| \right]
\end{aligned}
$$
where $\Delta t$ is the sampling time, $\lambda_{1i}, \lambda_{2i}, \mu_i, \gamma_i, m_i, n_i$ are positive design constants, and $\hat{p}_i$ estimates the unknown upper bound $p_i$ of $|d_i|$. The adaptive term $\hat{p}_i \tanh(\gamma_i \pi_i/\varepsilon)$ provides robustness against bounded disturbances with unknown bounds. The estimation error dynamics are analyzed using a Lyapunov approach. We define the Lyapunov function $V_d(k) = \sum_{i=1}^3 [\pi_i^2(k) + \tilde{p}_i^2(k)]$ where $\tilde{p}_i = p_i – \hat{p}_i$. Under the assumption that the second-order differences of $\pi_i$ and $\hat{p}_i$ are bounded, we can show that $V_d(k+1) – V_d(k) \leq -\delta_1 V_d(k) + \delta_2$, implying that the estimation errors converge to a small neighborhood of zero. The convergence performance is further improved by the adaptive bound estimation.
Table 1 summarizes the design parameters used in our simulations for the China UAV drone.
| Parameter | Value |
|---|---|
| $\lambda_{11},\lambda_{12},\lambda_{13}$ | 45 |
| $\lambda_{21},\lambda_{22},\lambda_{23}$ | 0.5 |
| $\mu_1,\mu_2,\mu_3$ | 0.5 |
| $\gamma_1,\gamma_2,\gamma_3$ | 0.5 |
| $m_1,m_2,m_3$ | 10 |
| $n_1,n_2,n_3$ | 0.01 |
| $\Delta t$ | 10 ms |
4. Nonlinear Model Predictive Control with Disturbance Compensation
The NMPC controller for the China UAV drone solves an optimal control problem over a prediction horizon $N_p$, using the nominal model corrected by the estimated disturbance $\hat{\mathbf{d}}(k)$:
$$
\mathbf{x}(k+1) = \mathbf{F}(\mathbf{x}(k), \mathbf{u}(k)) + \mathbf{B} \hat{\mathbf{d}}(k)
$$
where $\mathbf{B}$ is the disturbance input matrix. The cost function is defined as:
$$
J(k) = \sum_{i=0}^{N_p-1} \left[ \|\mathbf{x}_e(k+i|k)\|_{\mathbf{Q}}^2 + \|\mathbf{u}(k+i|k)\|_{\mathbf{R}}^2 \right] + \|\mathbf{x}(k+N_p|k)\|_{\mathbf{P}}^2
$$
with $\mathbf{x}_e = \mathbf{x} – \mathbf{x}_{ref}$ the tracking error. The terminal penalty matrix $\mathbf{P}$ is obtained from the solution of a discrete Lyapunov equation using a terminal controller $\mathbf{u} = \mathbf{K}\mathbf{x}$ that stabilizes the linearized system within a terminal region $\Omega$. The optimization problem is subject to state constraints $\mathbf{x} \in \mathcal{X}$ and control input constraints $\mathbf{u} \in \mathcal{U}$, including the actuator limits of the China UAV drone. The control sequence $\mathbf{U}(k) = [\mathbf{u}(k), \ldots, \mathbf{u}(k+N_p-1)]$ is optimized and only the first element $\mathbf{u}(k)$ is applied to the drone. At the next time step, the state is updated and the optimization is repeated, forming a receding horizon strategy.
To guarantee recursive feasibility, we adopt a terminal constraint $\mathbf{x}(k+N_p|k) \in \Omega$, which is a set invariant under the terminal controller. The feasibility proof follows standard arguments: given a feasible solution at time $k$, we construct a candidate at $k+1$ by shifting the sequence and appending the terminal control, which remains feasible due to the invariance of $\Omega$ under $\mathbf{K}$. Stability is established by choosing the optimal cost as a Lyapunov function. With the disturbance compensation, the closed-loop system becomes asymptotically stable when the disturbance estimation error is zero. In practice, the bounded estimation error results in ultimate boundedness of the tracking error.
We summarize the NMPC design parameters in Table 2.
| Parameter | Value |
|---|---|
| Sampling time $T_s$ | 50 ms |
| Prediction horizon $N_p$ | 2 |
| Control horizon $N_c$ | 1 |
| $\mathbf{Q}$ | diag(3,3,3,0,0,0,0,0,0) |
| $\mathbf{R}$ | 60$\times \mathbf{I}$ |
| $\mathbf{P}$ | diag(3,0.01,0.01,3,3,0.1,0.01,0.01,3) |
5. Simulation Results and Discussion
We evaluate the proposed discrete-time NDOB-based NMPC on a realistic China UAV drone model from the literature. The aerodynamic coefficients are perturbed by 30% scaling, and all actuators are biased by 0.7°. The UAV must track a time-varying reference in roll angle (0° to 30°), pitch angle (0° to 10°), yaw angle (0° to 20°), and airspeed (30 m/s to 25 m/s). Actuator constraints are enforced: $|\delta| \leq 25^\circ$ and $|\dot{\delta}| \leq 120^\circ/s$. We compare three approaches: (i) our method (NDOB + NMPC), (ii) NMPC with the NDOB from [13] (Method 1), and (iii) NMPC with the NDOB from [25] (Method 2). All methods are implemented with the same prediction horizon and cost weights.
Figure 1 shows the lumped disturbance estimation results for $d_1$, $d_2$, and $d_3$ in open-loop (observer outputs not fed back to controller). Our adaptive discrete-time NDOB significantly reduces estimation errors compared to the existing methods, especially during transient phases when the airspeed changes (5–20 s and 40–50 s) and when actuator steps occur at 5 s and 10 s. The root mean square errors (RMSE) for disturbance estimation are listed in Table 3.
| Disturbance | Our method | Method 1 [13] | Method 2 [25] |
|---|---|---|---|
| $d_1$ | 0.0747 | 0.4093 | 0.1653 |
| $d_2$ | 0.0053 | 0.1317 | 0.0770 |
| $d_3$ | 0.1113 | 0.6392 | 0.2344 |
When the estimated disturbances are used to compensate the nominal model inside the NMPC, the attitude tracking performance improves remarkably, as shown in the RMSE results for roll, pitch, and yaw angles in Table 4. Our method achieves the lowest tracking errors: 0.3036° (roll), 0.6025° (pitch), and 0.0373° (yaw). Compared to Method 1, the tracking RMSE is reduced by 16.96%, 1.70%, and 60.82% respectively; compared to Method 2, reductions are 1.81%, 0.43%, and 20.81%. The largest improvement is observed for the yaw angle, which is often the most sensitive to disturbance mismatches. The transient responses at actuator step changes (5 s and 10 s) show that our method suppresses overshoot and settling time more effectively, demonstrating the benefit of accurate disturbance compensation for the China UAV drone system.
| Angle | Our method | Method 1 [13] | Method 2 [25] |
|---|---|---|---|
| Roll | 0.3036 | 0.3656 | 0.3092 |
| Pitch | 0.6025 | 0.6129 | 0.6051 |
| Yaw | 0.0373 | 0.0952 | 0.0471 |
6. Conclusion
We have developed a novel discrete-time nonlinear disturbance observer with an adaptive bound estimation mechanism for the purpose of robust attitude control of a China UAV drone via nonlinear model predictive control. The proposed NDOB accurately estimates the lumped disturbance arising from aerodynamic coefficient deviations and actuator biases, and seamlessly integrates into the NMPC framework to correct the prediction model. Lyapunov-based stability analysis guarantees the convergence of the observer and the ultimate boundedness of the tracking error. Simulation results on a realistic China UAV drone model confirm that our method significantly reduces disturbance estimation errors (up to 95% reduction in RMSE for some channels) and improves attitude tracking accuracy (up to 60% reduction in yaw RMSE) compared to existing NDOB designs. Future work will focus on experimental validation and extension to position control under more severe wind disturbances for China UAV drones.