In recent years, quadrotor unmanned aerial vehicles (UAVs) have gained significant attention due to their versatility in applications such as surveillance, delivery, and environmental monitoring. However, the performance of quadrotor systems is often compromised by input saturation constraints in the propulsion system and external disturbances, leading to instability and reduced accuracy. Traditional control methods, including nested saturation control (NSC), face challenges such as strict convergence conditions, slow response, and limited robustness. This paper addresses these issues by proposing a robust nested saturation control (RNSC) algorithm that integrates self-coupling proportional-derivative (SC-PD) control and a compensation function observer (CFO) to enhance transient and steady-state performance while ensuring global stability under input constraints. The quadrotor dynamic model is derived using Euler-Lagrange formalism, and the controller design is validated through Lyapunov stability analysis. Simulation results demonstrate the superiority of the proposed approach over conventional NSC and PD controllers in terms of convergence speed, overshoot reduction, and disturbance rejection.
The quadrotor is modeled as a rigid body with six degrees of freedom, comprising translational and rotational dynamics. Define the inertial frame {E} and body-fixed frame {B} as shown in the figure below. The position coordinates are denoted by $\xi = (x, y, z)^T$, and the Euler angles by $\eta = (\psi, \theta, \phi)^T$, representing yaw, pitch, and roll, respectively. The kinetic energy includes translational and rotational components:
$$T_{\text{trans}} = \frac{m}{2} \dot{\xi}^T \dot{\xi}, \quad T_{\text{rot}} = \frac{1}{2} \dot{\eta}^T J \dot{\eta},$$
where $m$ is the mass and $J$ is the inertia matrix. The Lagrangian $L(q, \dot{q})$ accounts for gravitational potential energy:
$$L(q, \dot{q}) = \frac{m}{2} \dot{\xi}^T \dot{\xi} + \frac{1}{2} \dot{\eta}^T J \dot{\eta} – mgz,$$
with generalized coordinates $q = (\xi, \eta)^T$. The equations of motion are derived from the Euler-Lagrange equation:
$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) – \frac{\partial L}{\partial q} = F + F_D,$$
where $F = (F_\xi, \tau)^T$ includes the translational force $F_\xi = R \tilde{F}$ and moments $\tau = (\tau_\psi, \tau_\theta, \tau_\phi)^T$. The rotation matrix $R$ transforms body-frame forces to the inertial frame. The disturbance $F_D = (d_x, d_y, d_z, d_\psi, d_\theta, d_\phi)^T$ encompasses unmodeled dynamics and external effects. The control inputs are related to rotor speeds $\omega_i$ via:
$$\tilde{F} = \begin{pmatrix} 0 \\ 0 \\ \sum_{i=1}^4 k \omega_i^2 \end{pmatrix}, \quad \tau = \begin{pmatrix} -d & d & -d & d \\ -kl & 0 & kl & 0 \\ 0 & kl & 0 & -kl \end{pmatrix} \begin{pmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{pmatrix},$$
where $k > 0$ is the thrust coefficient, $d > 0$ is the drag coefficient, and $l$ is the arm length. The simplified dynamics for control design are:
$$m\ddot{x} = f (\sin\theta \cos\phi \cos\psi + \sin\phi \sin\psi),$$
$$m\ddot{y} = f (\sin\theta \cos\phi \sin\psi – \sin\phi \cos\psi),$$
$$m\ddot{z} = f \cos\theta \cos\phi – mg,$$
$$\ddot{\psi} = \bar{\tau}_\psi, \quad \ddot{\theta} = \bar{\tau}_\theta, \quad \ddot{\phi} = \bar{\tau}_\phi,$$
with $f = \sum_{i=1}^4 k \omega_i^2$ as the total thrust. For controller synthesis, the dynamics are decoupled into subsystems: altitude and yaw control $(z, \psi)$, and position and attitude control $(x, \theta)$ and $(y, \phi)$.
The control strategy employs SC-PD for the $(z, \psi)$ subsystem and RNSC for the $(x, \theta)$ and $(y, \phi)$ subsystems. The SC-PD control law for altitude is:
$$f = \frac{r + mg}{\cos\theta \cos\phi}, \quad r = \frac{\ddot{z}_d + z_z^2 e_z + 2z_z \dot{e}_z}{b_z},$$
where $e_z = z_d – z$, $z_z > 0$ is a speed factor, and $b_z$ is a gain. Similarly, for yaw:
$$\tau_\psi = \frac{\ddot{\psi}_d + z_\psi^2 e_\psi + 2z_\psi \dot{e}_\psi}{b_\psi},$$
with $e_\psi = \psi – \psi_d$. The RNSC design for $(x, \theta)$ relaxes the traditional NSC conditions. Define saturation functions $S_{\theta_i}(a) = \text{sat}_{\theta_i}(a)$ for $i = 1, \dots, 4$, where $\theta_i > 0$ are saturation limits. The improved control law is:
$$\bar{\tau}_\theta = -S_{\theta_1} \left( \dot{e}_\theta + S_{\theta_2} \left( e_\theta + \dot{e}_\theta + S_{\theta_3} \left( 2e_\theta + \dot{e}_\theta + \frac{\dot{e}_x}{g} + S_{\theta_4} \left( 3e_\theta + \dot{e}_\theta + \frac{e_x}{g} + 3\frac{\dot{e}_x}{g} \right) \right) \right) \right) + \ddot{\theta}_d,$$
where $e_x = x – x_d$, $e_\theta = \theta – \theta_d$, and $\dot{e}_x = \dot{x} – \dot{x}_d$. Global stability is proven using Lyapunov theory by considering the error dynamics and showing convergence to zero. A similar law applies to $(y, \phi)$:
$$\bar{\tau}_\phi = -S_{\phi_1} \left( \dot{e}_\phi + S_{\phi_2} \left( e_\phi + \dot{e}_\phi + S_{\phi_3} \left( 2e_\phi + \dot{e}_\phi – \frac{\dot{e}_y}{g} + S_{\phi_4} \left( 3e_\phi + \dot{e}_\phi – \frac{e_y}{g} – 3\frac{\dot{e}_y}{g} \right) \right) \right) \right) + \ddot{\phi}_d.$$
To enhance robustness, the RNSC incorporates a CFO for disturbance estimation. The state-space representation of the closed-loop system is:
$$\dot{X} = A X + B u + D,$$
with $X = (e_x/g, \dot{e}_x/g, e_\theta, \dot{e}_\theta)^T$, $u = \bar{\tau}_\theta$, and $D$ as the disturbance. The reference model is:
$$\dot{X}_m = A_m X_m + B_m u_m,$$
where $X_m$ is the reference error. The CFO estimates $\hat{D}$, and the control law becomes:
$$u_{\text{RNSC}} = B^* \left( (A_m – A) X + B_m u_m – \hat{D} – K e \right),$$
with $e = X_m – X$, $B^* = (B^T B)^{-1} B^T$, and $K$ as a feedback gain. Lyapunov analysis confirms that $e$ asymptotically converges to zero, ensuring the quadrotor tracks the reference under disturbances.
Simulations evaluate the RNSC against NSC and PD controllers. The quadrotor parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Mass $m$ (kg) | 2.2 |
| Moment of inertia $I_{xx}$ (kg·m²) | 3.904 × 10⁻² |
| Moment of inertia $I_{yy}$ (kg·m²) | 3.904 × 10⁻² |
| Moment of inertia $I_{zz}$ (kg·m²) | 7.113 × 10⁻² |
| Arm length $l$ (m) | 0.275 |
| Thrust coefficient $k$ (N·(rad/s)⁻²) | 1.253 × 10⁻⁵ |
| Drag coefficient $d$ (N·m·(rad/s)⁻²) | 1.852 × 10⁻⁷ |
| Saturation constants $\theta_i$, $\phi_i$ | 2, 1, 0.5, 0.5 |
A wind disturbance model is implemented to test robustness, comprising mean wind, shear, Dryden turbulence, and gusts. The total wind velocity $V_\omega$ is:
$$V_\omega = V_{\text{mean}} + V_{\text{shear}} + V_{\text{Dryden}} + V_{\text{gust}},$$
with components along $x$, $y$, and $z$ axes. The quadrotor starts from $q_0 = (0, 0, 0, 0, 0, 0)^T$ and tracks a square trajectory of 30 m side length at 15 m altitude. Performance metrics for position tracking are summarized in Table 2.
| Metric | PD | NSC | RNSC |
|---|---|---|---|
| Rise time (s) for $x$ | 6.81 | 5.68 | 3.78 |
| Overshoot (%) for $x$ | 8.7 | 13.5 | 0.2 |
| Settling time (s) for $x$ | 26.34 | — | 5.82 |
| Steady-state error (m) for $x$ | 1.05 | 3.23 | 0.07 |
| Rise time (s) for $y$ | 7.20 | 4.80 | 4.09 |
| Overshoot (%) for $y$ | 0.6 | 8.7 | 0.1 |
| Settling time (s) for $y$ | 10.31 | 12.61 | 6.35 |
| Steady-state error (m) for $y$ | 0.18 | 0.63 | 0.04 |
The RNSC controller demonstrates faster response, lower overshoot, and higher accuracy compared to PD and NSC. Attitude responses are smoother, and control inputs remain within saturation limits, ensuring stability. The CFO effectively estimates disturbances, as shown in Figure 1, which plots the estimated forces and moments.
The proposed RNSC algorithm significantly improves quadrotor performance by combining relaxed saturation conditions with robust disturbance rejection. Key contributions include:
- Relaxed convergence conditions for NSC, reducing design complexity.
- Integration of SC-PD and CFO for enhanced transient and steady-state performance.
- Global stability guarantees via Lyapunov analysis.
Future work will focus on experimental validation in outdoor environments to further refine the algorithm for real-world applications. The robustness of the quadrotor system under input constraints makes it suitable for demanding tasks in dynamic conditions.
In summary, this paper presents a comprehensive framework for robust control of quadrotor UAVs, addressing critical challenges in input saturation and disturbance rejection. The RNSC approach offers a practical solution for achieving high-precision tracking and stability in complex operational scenarios.