In recent years, the quadcopter has gained significant attention in various fields such as agriculture, mapping, and logistics due to its low cost, high flexibility, and ease of operation. However, the inherent nonlinearity, instability, and multivariable nature of the quadcopter make precise flight control a complex and challenging task. To address this, researchers commonly employ control algorithms like Proportional-Integral-Derivative (PID) and Active Disturbance Rejection Control (ADRC). This study aims to comprehensively compare the performance of PID and ADRC controllers for quadcopter attitude control through MATLAB simulations. By evaluating their stability, response to disturbances, and robustness, we seek to provide theoretical insights for selecting appropriate control strategies in quadcopter applications.
The quadcopter is a type of unmanned aerial vehicle (UAV) characterized by its four rotors, which enable vertical take-off and landing and agile maneuverability. Its dynamics are highly nonlinear and coupled, involving interactions between rotational and translational motions. The equations of motion for a quadcopter can be derived using Newton-Euler formalism. Let the state vector be defined as position and orientation in the inertial frame, and the control inputs be the rotor thrusts. The general dynamic model is given by:
$$ \ddot{x} = \frac{1}{m} \left( \sum F \right) – g $$
$$ \ddot{\phi} = \frac{1}{I_x} \left( \tau_\phi – I_y \dot{\theta} \dot{\psi} + I_z \dot{\theta} \dot{\psi} \right) $$
$$ \ddot{\theta} = \frac{1}{I_y} \left( \tau_\theta – I_z \dot{\phi} \dot{\psi} + I_x \dot{\phi} \dot{\psi} \right) $$
$$ \ddot{\psi} = \frac{1}{I_z} \left( \tau_\psi – I_x \dot{\phi} \dot{\theta} + I_y \dot{\phi} \dot{\theta} \right) $$
where \( x \) is the position, \( \phi, \theta, \psi \) are the roll, pitch, and yaw angles, \( m \) is the mass, \( g \) is gravity, \( I_x, I_y, I_z \) are moments of inertia, and \( \tau \) represents the torques. Controlling the quadcopter requires precise manipulation of these angles to maintain stability.
PID control is a classical feedback algorithm widely used in quadcopter systems due to its simplicity and effectiveness. The PID controller computes the control signal based on the error between the desired setpoint and the actual output. The continuous-time form of the PID controller is:
$$ u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt} $$
where \( e(t) \) is the error, \( K_p \), \( K_i \), and \( K_d \) are the proportional, integral, and derivative gains, respectively. For a quadcopter, separate PID controllers are often designed for each axis (roll, pitch, yaw). The proportional term responds to the current error, the integral term eliminates steady-state error, and the derivative term dampens oscillations. Tuning these parameters is critical for optimal performance; methods like Ziegler-Nichols or manual adjustment are commonly employed. The discrete-time implementation for digital control is:
$$ u[k] = K_p e[k] + K_i T_s \sum_{i=0}^k e[i] + K_d \frac{e[k] – e[k-1]}{T_s} $$
where \( T_s \) is the sampling time. In quadcopter applications, the PID controller must handle coupling effects and external disturbances, which can limit its performance in highly dynamic environments.
| Axis | Proportional Gain (K_p) | Integral Gain (K_i) | Derivative Gain (K_d) |
|---|---|---|---|
| Roll | 1.2 | 0.5 | 0.3 |
| Pitch | 1.1 | 0.4 | 0.25 |
| Yaw | 1.0 | 0.3 | 0.2 |
ADRC is an advanced control technique that enhances robustness by actively estimating and compensating for disturbances. It consists of three main components: Tracking Differentiator (TD), Extended State Observer (ESO), and Nonlinear State Error Feedback (NLSEF). For a quadcopter, ADRC can be applied to each attitude angle. The TD generates a smooth reference signal and its derivative:
$$ \dot{v}_1 = v_2 $$
$$ \dot{v}_2 = fhan(v_1 – v_0, v_2, r, h_0) $$
where \( v_0 \) is the desired value, \( v_1 \) and \( v_2 \) are the tracked signal and its derivative, \( r \) and \( h_0 \) are tuning parameters, and \( fhan \) is a nonlinear function. The ESO estimates the system states and total disturbances:
$$ \dot{z}_1 = z_2 + \beta_1 (y – z_1) $$
$$ \dot{z}_2 = z_3 + \beta_2 (y – z_1) + b_0 u $$
$$ \dot{z}_3 = \beta_3 (y – z_1) $$
where \( z_1, z_2, z_3 \) are estimated states, \( y \) is the output, \( \beta_1, \beta_2, \beta_3 \) are observer gains, and \( b_0 \) is a control gain. The NLSEF computes the control signal based on the error:
$$ u_0 = k_1 fal(e_1, \alpha_1, \delta) + k_2 fal(e_2, \alpha_2, \delta) $$
$$ u = \frac{u_0 – z_3}{b_0} $$
where \( e_1 = v_1 – z_1 \), \( e_2 = v_2 – z_2 \), \( fal \) is a nonlinear function, and \( k_1, k_2, \alpha_1, \alpha_2, \delta \) are parameters. This structure allows ADRC to handle uncertainties and disturbances effectively, making it suitable for quadcopter control in turbulent conditions.
| Component | Parameter | Value |
|---|---|---|
| TD | r | 10 |
| TD | h_0 | 0.01 |
| ESO | β_1 | 100 |
| ESO | β_2 | 300 |
| ESO | β_3 | 1000 |
| NLSEF | k_1 | 1.5 |
| NLSEF | k_2 | 0.5 |
To evaluate the performance of PID and ADRC controllers, we developed a simulation environment in MATLAB/Simulink. The quadcopter model was implemented with the following physical parameters: mass \( m = 1.2 \, \text{kg} \), moments of inertia \( I_x = 0.01 \, \text{kg·m}^2 \), \( I_y = 0.01 \, \text{kg·m}^2 \), \( I_z = 0.02 \, \text{kg·m}^2 \), and gravitational acceleration \( g = 9.81 \, \text{m/s}^2 \). The control inputs were converted to rotor thrusts using a mixing matrix. The simulation time was set to 20 seconds, with a fixed-step solver and sampling time of 0.01 seconds. The desired setpoints for roll, pitch, and yaw angles were square waves with amplitude of 3 degrees and frequency of 0.1 Hz, simulating typical maneuver commands.
For PID control, the controllers were tuned empirically to achieve stable response. The Simulink model included blocks for error calculation, PID controllers, control allocation, and the quadcopter dynamics. Sensor noise was modeled as Gaussian white noise with zero mean and variance of 0.001 to simulate real-world conditions. The performance metrics included rise time, settling time, overshoot, and steady-state error. Similarly, for ADRC, the TD, ESO, and NLSEF were implemented as S-functions, with parameters adjusted to optimize disturbance rejection.
The simulation results demonstrated that both controllers could stabilize the quadcopter, but ADRC exhibited superior performance in terms of response speed and disturbance rejection. For instance, in the roll axis, the PID controller had a rise time of approximately 0.5 seconds and a settling time of 1.2 seconds, while ADRC achieved a rise time of 0.3 seconds and settling time of 0.8 seconds. The overshoot was reduced from 15% with PID to 5% with ADRC. External disturbances, such as wind gusts, were introduced at 10 seconds, and ADRC showed faster recovery, with the error converging to zero within 0.5 seconds compared to 1.5 seconds for PID.
| Metric | PID Control | ADRC Control |
|---|---|---|
| Rise Time (s) | 0.5 | 0.3 |
| Settling Time (s) | 1.2 | 0.8 |
| Overshoot (%) | 15 | 5 |
| Steady-State Error (deg) | 0.1 | 0.02 |
| Disturbance Rejection Time (s) | 1.5 | 0.5 |
Mathematical analysis further supports these findings. The transfer function for a quadcopter attitude loop can be approximated as a second-order system. For PID control, the closed-loop characteristic equation is:
$$ s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 $$
where \( \zeta \) is the damping ratio and \( \omega_n \) is the natural frequency. With ADRC, the disturbance estimation allows the system to behave like a nominal plant, reducing the effects of nonlinearities. The stability of ADRC can be analyzed using Lyapunov methods, ensuring bounded errors under disturbances. The control law for ADRC ensures that the quadcopter tracks the reference accurately even in the presence of model uncertainties.
In terms of implementation complexity, PID control is straightforward and requires fewer computational resources, making it suitable for simple quadcopter systems with limited processing power. However, ADRC, though more complex, offers better adaptability to changing environments. For example, in a quadcopter performing aggressive maneuvers, ADRC maintained stability with minimal tuning, whereas PID needed frequent adjustments. This highlights the trade-off between simplicity and performance in quadcopter control systems.
In conclusion, both PID and ADRC controllers are viable for quadcopter attitude control, but ADRC provides enhanced robustness and faster response in the face of disturbances. The choice of controller should depend on the specific application requirements: PID for simpler, linear quadcopter systems, and ADRC for complex, nonlinear scenarios where disturbance rejection is critical. Future work could explore hybrid approaches or adaptive tuning to further improve quadcopter performance in diverse operational conditions.