# Discrete Integral Optimal Controller for Quadrotor Attitude Stabilization: Experimental Results
## 1 Introduction
In [10], once again an LQR methodology and integral state augmentation were adopted to achieve the desired performance of the control system. The unmeasured state variables were estimated by means of a reduced-order observer. Satisfactory simulation results for the UAV helicopter were presented. [PERSON] et al. [11] conducted similar work in which they used a control scheme based on Integral Backstepping with sliding modes applied to a multi-rotor vehicle. This control methodology was experimentally tested using the vision system Optitrack. Results were reported for the altitude \(z\) and displacement on the \(x\)- and \(y\)-axes. However, no analysis of energy consumption was presented, nor were real-time tests with external disturbances conducted and reported. In [12], [PERSON] published LQR and LQR-PI controllers that are applied to a quadcopter. The high performance and robustness of the Linear Quadratic Regulator controllers ensure the ability to reduce deviations in state trajectories with minimal control effort. In this work, the weighting matrices are automatically adjusted through a novel method using the full state of the flying robot. Feasibility and performance of the closed-loop system is only tested by simulation routines. In the literature, it is possible to find documents in which the LQR technique is combined with other types of controllers, such as fuzzy controllers. For example, in [13], [PERSON] presents the development of a longitudinal controller design for an autonomous unmanned aerial vehicle (UAV). In this work, the researchers proposed a dual-loop (inner-outer loop) control method based on intelligent algorithms. The inner feedback loop of controller uses a Linear Quadratic Regulator (LQR) to ensure adaptive stability. Meanwhile, the outer loop controller employs a Fuzzy-PID algorithm for deal with the trajectory tracking task. Moreover, neuro-fuzzy controllers have also been recently used for the control of UAVs. In [14], [PERSON] et al. developed a position control approach for a quadrotor using a cascade Fuzzy Neural Network (FNN). This approach requires offline neural network training, combining the benefits of fuzzy systems and neural networks. According to authors, this fuzzy control strategy demonstrated its ability to minimize the overshoot and the settling time. This was tested by conducting flight simulations and real-time flights using a DJI Tello quadrotor UAV, showing an acceptable position control performance. In fact, one of advantages of neuro-fuzzy controllers is their ability to handle nonlinear systems, which offers better adaptability. Particularly, the controller architecture proposed in that work allows to optimize the use of the robot's energy resources and also providing robustness to the control loop. Also, this produces a balance between adaptability and efficiency, making it particularly useful for UAV applications where energy efficiency and robust control are crucial. That type of controller can also be used in other fields such as the study of seismic structural control. In this field, the contributions reported works by [PERSON] and his collaborators in [15; 16; 17] should also be mentioned, they use neuro-fuzzy controllers and a PID controllers to tackle both stabilization and trajectory tracking task for aerial robots. In this contribution, UAV stabilization is tackled using a controller that combines an optimal strategy (LQR) with the integral action. A difference from other reported works such as [5; 9; 10; 12] is that this optimal synthesized controller is tested in a real-time setting, providing experimental results for the altitude stabilization problem in a four-rotor helicopter. To control the position and orientation of the vehicle, the system was subdivided into four subsystems, as was proposed in [18], and the integral action is added as an additional state variable [4] in the four subsystems.