# Discrete Integral Optimal Controller for Quadrotor Attitude Stabilization: Experimental Results
## 1 Introduction  
In recent years, several works have been reported in the literature addressing the application of optimal control of quadrotors [1; 2]. However, this issue remains interesting to the scientific community because it represents a current challenge from the point of view of control theory and engineering (mainly its real-time applications). Many techniques related to optimal control philosophy have been applied to UAVs, but this paper proposes to add an integral action to Linear Quadratic Regulator, in order to improve the performance of the closed-loop system. Moreover, both the mathematical model and synthesized control strategy are obtained in discrete time domain, ensuring a better representation to be programmed in a microcontroller such that Rabbit 4300. In this sens, recent reported works have demonstrated that it is possible to ensure the stability in discrete-time domain for multi-input-multi-output (MIMO) systems. This has been achieved through a generalization of the Letov formula, as mentioned in [3]. A typical control strategy with integral action is the Proportional-Integral-Derivative (PID) controller, which is widely used in industrial processes and devices because it can be heuristically tuned independently of the mathematical model of the system and can be used in both linear and nonlinear systems. However, there also exist many graphical and theoretical methods to tune it considering the dynamical behavior or the mathematical model of system. It is well known that the integral action of a PID controller provides a correction for steady-state tracking error even in the presence of uncertainties [4], providing some degreeof robustness to the control loop. Although this three-action controller is a powerful idea, sometimes it is not efficient or appropriate for a specific tasks. Then, it can be improved by modifying the control parameters with an adaptive scheme or by adding a nonlinear part. This contribution uses the advantages of the integral action of the PID controller combined with modern control techniques through regulation via optimal control in order to improve the performance of a quadrotor when it executes an attitude and altitude stabilization task. Integral control applied to Unmanned Aerial Vehicles has been explored in some reported works. An integral predictive and nonlinear robust control strategy was synthesized in [5], wherein the authors solve the path-following problem for a quadrotor helicopter. The proposed control structure has a hierarchical scheme consisting of a model predictive controller to track the reference trajectory together with a nonlinear \(H_{\infty}\) controller to stabilize the rotational movements of the considered vehicle; satisfactory simulation results were presented. In [6], an Integral Backstepping controller and motion planning are combined to stabilize the helicopter using point-to-point steering stabilization. Simulation results were presented to test the performance of the closed-loop system and its robustness. The same technique was presented in [7]: the goal of this work was to stabilize the attitude, altitude, and position of the vehicle. Satisfactory results for autonomous take-off, hover, landing and collision-avoidance tasks were presented, and all were validated on the OS4 simulation platform. In addition, a comparison between PID, Linear Quadratic Regulator (LQR) and nonlinear controllers (Adaptive Integral Backstepping Controller) was exposed in [8]. A nonlinear control approach was proposed based on a recursive Lyapunov methodology using the Backstepping technique and an adaptive scheme. Satisfactory simulations and real-time experiments were conducted. In [9], LQR continuous control was used to stabilize the attitude and altitude of an Octocopter. Numerical simulations demonstrated the effectiveness of the control strategy under nominal conditions, and the authors improved the LQR by adding integral action to the altitude controller.