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Nonlinear PID Controller Design for a 6-DOF UAV Quadrotor System
Aws Abdulsalam Najm, Ibraheem Kasim Ibraheem
TL;DR
The paper addresses stabilization and trajectory tracking for an under-actuated, unstable 6-DOF quadrotor while seeking low error and control energy. It uses a complete Euler–Newton nonlinear model with six GA-tuned NLPID controllers and Hurwitz stability analysis. Simulations report faster tracking, lower control energy, and smaller steady-state error than LPID control, while the stability proof uses a linearized model and the design simulations use the complete nonlinear model.
Problem
The central problem is stabilizing an under-actuated unstable 6-DOF quadrotor and making its outputs follow a specified trajectory with optimized time-domain behavior and minimum control energy.
Method
The paper uses an Euler–Newton nonlinear model including velocity and acceleration, six NLPID controllers tuned by GA against a weighted ITAE–USQR OPI, and Hurwitz stability analysis.
Results
The proposed NLPID controller tracked faster than LPID with smaller steady-state errors and overshoot, including LPID position errors of ex = 27%, ey = 18%, and NLPID errors of ex = 0.56%, ey = 4.12%.
Takeaways & Limitations
Within the tested simulations, NLPID control performed better than LPID in speed, control energy, and steady-state error.
Takeaways & Limitations
The stability proof uses a linearized model, while the complete nonlinear model is reserved for controller design and simulations; the model also assumes small φ and θ deviations for x–y control.
Abstract
from arXiv · showhide
A Nonlinear PID (NLPID) controller is proposed to stabilize the translational and rotational motion of a 6-DOF UAV quadrotor system and enforce it to track a given trajectory with minimum energy and error. The complete nonlinear model of the 6-DOF quadrotor system are obtained using Euler-Newton formalism and used in the design process, taking into account the velocity and acceleration vectors resulting in a more accurate 6-DOF quadrotor model and closer to the actual system. Six NLPID controllers are designed, each for Roll, Pitch, Yaw, Altitude, and the Position subsystems, where their parameters are tuned using GA to minimize a multi-objective Output Performance Index (OPI). The stability of the 6-DOF UAV subsystems has been analyzed in the sense of Hurwitz stability theorem under certain conditions on the gains of the NLPID controllers. The simulations have been accomplished under MATLAB/SIMULINK environment and included three different trajectories, i.e., circular, helical, and square. The proposed NLPID controller for each of the six subsystems of the 6-DOF UAV quadrotor system has been compared with the Linear PID (LPID) one and the simulations showed the effectiveness of the proposed NLPID controller in terms of speed, control energy, and steady-state error.
1. Introduction
Quadrotor control is challenging because the vehicle is under-actuated and LPID controllers can amplify noise. This paper proposes six GA-tuned NLPID controllers, verifies stability with Hurwitz analysis, and compares them with LPID control.
- Quadrotors are under-actuated because four rotors control six degrees of freedom, making controller design difficult.Their applications span dangerous environments, disasters, rescue, agriculture, and multi-agent systems.
- LPID control is widely used for its simplicity, but one reported disadvantage is noise amplification.
- The proposed system uses six NLPID controllers: three for translation and three for rotation.Each controller has twelve tuning parameters optimized by a Genetic Algorithm.
- The controllers minimize a multi-objective Output Performance Index combining ITAE with the squared control signal U.
- The paper compares NLPID and LPID performance after Hurwitz-stability verification and evaluates the study structure through modeling, design, simulations, and conclusions.
2. Mathematical Modelling of the 6-DOF UAV Quadrotor
The paper models the quadrotor as a nonlinear six-degree-of-freedom system using Euler–Newton equations, with rotor speeds producing altitude force and attitude torques. Unlike acceleration-only formulations, the model includes both velocity and acceleration vectors.
- Rotor-speed combinations generate force ft for altitude z and torques τx, τy, and τz for angles φ, θ, and ψ.The parameter meanings are listed in Table 1.
- The model represents combined translational and rotational motion using Euler–Newton equations for the rigid body.The four possible quadrotor movements are illustrated in Figure 1.
- Unlike acceleration-only formulations, this model incorporates velocity and acceleration vectors to provide a more accurate nonlinear representation closer to the actual quadrotor.The dynamical relations are shown in Figure 2.
3. Problem statement
The problem is to design six control inputs for an under-actuated, unstable 6-DOF quadrotor so its measured position and attitude outputs track a specified trajectory. The design must satisfy time-domain specifications while minimizing control energy.
- The nonlinear system is expressed with state vector X and measured output Y, both containing linear, angular position, and velocity variables.The model is presented in the nonlinear system equations associated with Figure 3.
- The six-dimensional input vector U contains controls for the quadrotor’s translational and rotational behavior.
- The controller must stabilize the under-actuated unstable system and make it follow a specific trajectory.
- The design objective includes optimum time-domain specifications and minimum control energy.
4. The main results
The paper designs six nonlinear PID controllers for the quadrotor’s translational and rotational subsystems, using nonlinear error gains and virtual controls for under-actuated position motion. Stability is analyzed with Hurwitz conditions under stated assumptions and gain bounds.
- Nonlinear controller design: The NLPID controller replaces each linear PID gain with a nonlinear error-dependent function to improve response for the nonlinear quadrotor system.The nonlinear gain is positive and sector-bounded, while the integral term is modified by adding k31 to increase closed-loop stability.
- Nonlinear controller design: The under-actuated system is divided into inner-loop and outer-loop control, with virtual signals Ux and Uy generated for motion in the x-y plane.The x-y control derivation assumes small roll and pitch deviations so that the trigonometric terms can be simplified.
- Nonlinear controller design: The complete controlled system uses NLPID signals for altitude and attitude motions, with desired and measured values defining the tracking errors.The signals include altitude and roll, pitch, and yaw controls, while the position controls are incorporated through the virtual inputs.
- Stability analysis: The nonlinear model is decomposed into six second-order subsystems in Brunovsky form, and stability of each subsystem implies stability of the overall quadrotor system.The subsystem states and inputs correspond to the six translational and rotational channels, including the virtual position inputs.
- Stability analysis: Under assumptions that the nonlinear exponents are approximately one and all relevant states are observable, the closed-loop system is Hurwitz stable within specified nonlinear-gain intervals.The proof derives Hurwitz conditions from the characteristic equation and requires positive determinant conditions involving k1, k2, and k3.
- Stability analysis: The complete nonlinear model is used for controller design and simulations, whereas the linearized model is used only to prove closed-loop stability.This separates the model used for performance evaluation from the model used in the stability argument.
5. Simulation Results and Case Studies
MATLAB/Simulink simulations tuned the six NLPID and LPID controllers through multi-objective performance indices, then compared their step and trajectory-tracking behavior. Across circular, helical, and square paths, NLPID generally provided faster, smoother, lower-error tracking, while LPID was faster only for altitude step tracking at higher control energy.
- Simulation setup: The controller parameters were tuned using an unconstrained multi-objective optimization based on ITAE and USQR performance indices.The optimization emphasized selected objectives through weighting variables and normalized the objectives for comparable treatment.
- Step reference tracking: NLPID produced faster, less fluctuating responses than LPID for most step outputs, whereas LPID tracked altitude faster but spent more control energy.The higher altitude control energy was identified as undesirable because it can lead to actuator saturation.
- Step reference tracking: NLPID reduced roll and pitch control energy and produced smoother output responses than the linear controller.The comparison used time-domain specifications for position and yaw and peak values for roll and pitch.
- Trajectory tracking: For the circular trajectory, LPID steady-state errors were ex = 27%, ey = 18%, and ez = 0%, versus NLPID errors of ex = 0.56%, ey = 4.12%, and ez = 0%.The NLPID controller followed the circular path in less time and with smaller error.
- Trajectory tracking: For the helical trajectory, LPID maintained a constant offset throughout the simulation, while NLPID improved the tracking performance.The helical case varied altitude over time.
- Trajectory tracking: For the square trajectory, NLPID tracked faster with very small overshoot, whereas LPID overshoot reached approximately 200% of the desired trajectory.The square path tests sudden direction changes at its vertices.
6. Conclusion and Future Work
The paper adopts an exact nonlinear six-DOF UAV model and proposes an NLPID controller for stabilization and reference tracking. Hurwitz-based analysis and simulations indicate better speed, control energy, and steady-state error than LPID; future work addresses wind disturbances with disturbance observation.
- Conclusion: The study uses an exact nonlinear six-DOF UAV model to design a controller for stabilization and reference tracking.The model targets the complex, highly nonlinear quadrotor system.
- Conclusion: Hurwitz stability analysis showed that the proposed NLPID controller stabilized the position and orientation closed-loop systems and achieved the required tracking.The conclusion attributes these findings to the analyzed closed-loop systems.
- Conclusion: Simulations concluded that NLPID outperformed LPID in speed, control energy, and steady-state error.
- Future work: Future work will incorporate wind disturbances and reject them using a disturbance observer based on active disturbance rejection control.