In this paper, we address the critical challenge of attitude control for quadrotor unmanned aerial vehicles (UAVs) under conditions of model uncertainty and external disturbances. Quadrotor systems are inherently underactuated, with four inputs and six outputs, making them highly susceptible to environmental factors that can compromise stability and performance. Traditional control methods, such as PID or linear quadratic regulators, often fall short in handling nonlinearities and unknown perturbations, leading to degraded accuracy. To overcome these limitations, we propose an improved linear active disturbance rejection control (ILADRC) strategy that incorporates an error-free disturbance tracking linear extended state observer (EFDT-LESO). This approach enables precise estimation and compensation of disturbances without introducing steady-state errors, while a cascade control structure—comprising an inner loop for angular velocity and an outer loop for angle control—enhances robustness. We validate our method through semi-physical simulations under various disturbance scenarios, demonstrating superior tracking performance and resilience compared to existing techniques. The quadrotor’s dynamics are modeled to capture key aspects of its behavior, and the controller design focuses on achieving rapid convergence and minimal error accumulation in attitude regulation.
The quadrotor UAV, a popular platform for research and applications due to its agility and simplicity, requires advanced control strategies to maintain stability in unpredictable environments. Our work builds upon active disturbance rejection control (ADRC) principles, which estimate and cancel out disturbances in real-time without relying heavily on precise system models. By linearizing the ADRC components and introducing a novel error correction mechanism in the ESO, we achieve a more efficient and tunable controller. The cascade architecture further decouples the control objectives, allowing for independent optimization of inner and outer loops. This paper details the mathematical formulation of the EFDT-LESO, stability analysis using Routh-Hurwitz criteria, and comprehensive simulation results that highlight the controller’s effectiveness. Throughout, we emphasize the applicability of our method to quadrotor systems, ensuring that the term ‘quadrotor’ is frequently referenced to maintain focus on this UAV type.
To model the quadrotor dynamics, we consider a standard “X” configuration, where the system’s pose is described using both geodetic and body-fixed coordinate systems. The transformation matrix between these frames is given by the rotation matrix derived from Euler angles—roll ($\phi$), pitch ($\theta$), and yaw ($\psi$). The lift forces generated by the four rotors are summed to produce the total thrust in the body frame, expressed as:
$$F = \begin{bmatrix} F_x & F_y & F_z \end{bmatrix}^T = \sum_{i=1}^{4} T_i \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}^T = K_t \sum_{i=1}^{4} \omega_i^2 \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}^T$$
where $T_i$ represents the thrust from each rotor, $K_t$ is the thrust coefficient, and $\omega_i$ denotes the propeller angular velocities. The rotational dynamics of the quadrotor can be described by the following equations, which account for moments and disturbances:
$$\dot{p} = \frac{K_t l (q r (I_y – I_z) + f_{D\phi})}{I_x}, \quad \dot{q} = \frac{K_t l (p r (I_z – I_x) + f_{D\theta})}{I_y}, \quad \dot{r} = \frac{K_{\text{Tor}} (p q (I_x – I_y) + f_{D\psi})}{I_z}$$
Here, $p$, $q$, and $r$ are the angular velocities about the body axes; $I_x$, $I_y$, and $I_z$ are the moments of inertia; $l$ is the distance from the center of mass to each rotor; and $f_{D\phi}$, $f_{D\theta}$, $f_{D\psi}$ are drag-related disturbance terms. The virtual control inputs $U_i$ are defined based on the rotor speeds, enabling a simplified input-output relationship for attitude control. For the purpose of this study, we focus on the attitude subsystem, neglecting coupling effects in the translational dynamics. The simplified dynamics are represented as:
$$\ddot{\phi} = u_2 + f_{\phi}, \quad \ddot{\theta} = u_3 + f_{\theta}, \quad \ddot{\psi} = u_4 + f_{\psi}$$
where $u_2$, $u_3$, and $u_4$ are the control inputs derived from the virtual inputs, and $f_{\phi}$, $f_{\theta}$, $f_{\psi}$ encapsulate the total disturbances affecting each axis.
The core of our approach lies in the linear active disturbance rejection control (LADRC) framework, which simplifies the traditional ADRC by linearizing the extended state observer (ESO) and the control law. For a second-order system, the state-space representation is extended to include disturbances as an additional state. Consider a general second-order system:
$$\dot{x} = A x + B u + E f, \quad y = C x + D u$$
where $A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 0 \\ b \\ 0 \end{bmatrix}$, $E = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$, and $C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$. The ESO is designed to estimate both the system states and the disturbance, formulated as:
$$\dot{z} = A z + B u + L (y – \hat{y}), \quad \hat{y} = C z$$
Here, $z$ is the observer state vector, and $L$ is the gain matrix tuned based on the observer bandwidth. The linear state error feedback (LSEF) control law is given by:
$$u = \frac{u_0 – z_3}{b_0}, \quad u_0 = k_p (r’ – z_1) – k_d z_2$$
where $b_0$ is the control gain, $r’$ is the reference signal, and $k_p$ and $k_d$ are proportional and derivative gains, respectively, determined by the desired controller bandwidth $\omega_c$ as $k_p = \omega_c^2$ and $k_d = 2\omega_c$. This structure allows the LADRC to actively compensate for disturbances by canceling $z_3$, the estimated disturbance.
However, conventional ESOs may exhibit steady-state errors under certain disturbances. To address this, we introduce the error-free disturbance tracking linear ESO (EFDT-LESO), which incorporates an error correction mechanism. For an $n$-th order system, the extended state representation includes the disturbance as an $(n+1)$-th state. The system dynamics become:
$$\dot{\bar{x}} = \bar{A} \bar{x} + \bar{B} u + \bar{E} \dot{f}, \quad y = \bar{C} \bar{x}$$
where $\bar{A} = \begin{bmatrix} A & E \\ 0 & 0 \end{bmatrix}$, $\bar{B} = \begin{bmatrix} B \\ 0 \end{bmatrix}$, and $\bar{C} = \begin{bmatrix} C & 0 \end{bmatrix}$. The EFDT-LESO is designed as:
$$\dot{\bar{z}} = \bar{A} \bar{z} + \bar{B} u + L (y – \hat{y}) + d, \quad \hat{y} = \bar{C} \bar{z}$$
Here, $d$ is a stabilization term that ensures the estimation error converges to zero. The observer gains are selected to satisfy the Routh-Hurwitz stability criterion, guaranteeing that the estimation error dynamics are asymptotically stable. The transfer function analysis shows that the EFDT-LESO achieves zero steady-state error for bounded disturbances, enhancing the overall control accuracy. For the quadrotor attitude control, we apply this to each axis independently, resulting in an improved LADRC (ILADRC) that outperforms standard methods in disturbance rejection.
The cascade control strategy further augments the ILADRC by decomposing the attitude control into two loops: an outer loop for angle control and an inner loop for angular rate control. This separation allows for faster response and better disturbance handling. The outer loop utilizes the ILADRC with EFDT-LESO to generate desired angular rates, which are then tracked by the inner loop using a proportional-derivative (PD) controller. The structure is illustrated in the block diagram, where the Levant differentiator is employed to obtain smooth reference signals for the controller. The differential equations for the Levant differentiator are:
$$\dot{x}_1 = x_2, \quad \dot{x}_2 = -\lambda |x_1 – v(t)|^{1/2} \text{sgn}(x_1 – v(t)) + x_3, \quad \dot{x}_3 = -\alpha \text{sgn}(x_1 – v(t))$$
where $v(t)$ is the input signal, and $\lambda$ and $\alpha$ are parameters chosen based on the Lipschitz constant of the derivative of $v(t)$. This differentiator provides accurate derivatives without introducing excessive noise, crucial for the quadrotor’s high-performance requirements.
To validate our approach, we conducted semi-physical simulations using a platform that integrates a real-time simulator with a Pixhawk flight controller. The quadrotor parameters used in the simulations are summarized in the table below:
| Parameter | Value | Unit |
|---|---|---|
| Mass (m) | 0.65 | kg |
| Arm Length (l) | 0.32 | m |
| Thrust Coefficient (K_t) | 3.1 × 10^{-5} | N/rad² |
| Torque Coefficient (K_{Tor}) | 7.5 × 10^{-7} | N·m/rad² |
| Moment of Inertia (I_x, I_y) | 7.5 × 10^{-3} | kg·m² |
| Moment of Inertia (I_z) | 1.3 × 10^{-2} | kg·m² |
| Drag Coefficients (f_{Dx}, f_{Dy}) | 0.1 | — |
| Drag Coefficient (f_{Dz}) | 0.15 | — |
The simulation environment includes models for sensor noise and wind disturbances, allowing us to test the controller under realistic conditions. We compare our ILADRC with a conventional ADRC approach from the literature, evaluating performance based on tracking error accumulation defined as:
$$Q_i = \int_0^{t_f} |e_i(t)| dt$$
where $e_i(t)$ is the tracking error for each attitude angle (roll, pitch, yaw), and $t_f$ is the simulation time. Three simulation scenarios are considered: no disturbance, sudden white noise disturbance, and continuous wind disturbance. The reference signals for the angles are set as $u_\phi = \cos(1.5t)$, $u_\theta = \sin(1.5t)$, and a square wave for yaw with period $T=4$ seconds.
In the no-disturbance case, the ILADRC demonstrates faster convergence and lower error compared to ADRC. The roll, pitch, and yaw angles reach steady-state in approximately 0.319 s, 0.532 s, and 0.3 s, respectively, with significantly reduced oscillations. The cumulative errors for ILADRC are only about 16.9% of those for ADRC, as shown in the following table:
| Controller | Q_φ (°) | Q_θ (°) | Q_ψ (°) | Total ∑Q_i (°) |
|---|---|---|---|---|
| ILADRC | 0.105 | 0.052 | 1.861 | 2.018 |
| ADRC | 0.310 | 0.201 | 11.422 | 11.933 |
Under sudden white noise disturbance introduced at 4.5 seconds, the ILADRC shows superior disturbance rejection. The maximum deviations for roll, pitch, and yaw are 0.077°, 0.075°, and 0.292°, respectively, whereas ADRC exhibits larger deviations of 0.242°, 0.196°, and 0.702°. The error accumulation is reduced by approximately 82.95% with ILADRC, highlighting its robustness to abrupt perturbations. The results are summarized below:
| Controller | Q_φ (°) | Q_θ (°) | Q_ψ (°) | Total ∑Q_i (°) |
|---|---|---|---|---|
| ILADRC | 0.114 | 0.085 | 1.876 | 2.075 |
| ADRC | 0.339 | 0.238 | 11.593 | 12.170 |
For continuous wind disturbances modeled using a stochastic wind profile, the ILADRC maintains stable tracking with only a slight increase in error magnitude. The cumulative errors remain low, at about 16.5% of the ADRC values, confirming the controller’s ability to handle persistent disturbances without significant performance degradation. The table illustrates the comparison:
| Controller | Q_φ (°) | Q_θ (°) | Q_ψ (°) | Total ∑Q_i (°) |
|---|---|---|---|---|
| ILADRC | 0.138 | 0.071 | 1.806 | 2.014 |
| ADRC | 0.346 | 0.240 | 11.633 | 12.218 |
The stability of the ILADRC is rigorously analyzed using the Routh-Hurwitz criterion. The characteristic equation of the closed-loop system is derived from the transfer functions, and the controller parameters are tuned to ensure all roots have negative real parts. For the second-order system with EFDT-LESO, the stability condition is satisfied when the determinant of the Routh array is positive, confirming asymptotic stability under bounded disturbances.
In conclusion, our proposed ILADRC with EFDT-LESO and cascade control effectively addresses the attitude control challenges for quadrotor UAVs. The error-free disturbance tracking capability eliminates steady-state errors, while the cascade structure enhances response speed and robustness. Simulation results under various disturbances demonstrate that the ILADRC reduces tracking errors by over 80% compared to conventional ADRC, making it a viable solution for real-world quadrotor applications. Future work will focus on experimental validation and extension to trajectory tracking for complete autonomous flight.
The quadrotor platform continues to be a focal point in UAV research due to its versatility, and our controller contributes to improving its reliability in dynamic environments. By integrating advanced estimation techniques with a modular control architecture, we achieve high-performance attitude regulation that meets the demands of modern quadrotor operations. The mathematical formulations and simulation frameworks provided here serve as a foundation for further innovations in quadrotor control systems.