In this document a brief review of the real-time systems generalities with emphasis in the state-space use lineal control such as LQR( Linear quadratic regulator) and Full state feedback control. where to use implementation of a pendubot system is used, the device is from the manufacturer quanser which provides different mechatronics models for applications in various control strategies. In this cases, as the control card that comes by default is not compatible with current PCs, it was necessary to change it for a TI LAUCHPAD F28379D, which its manufacturer Texas instru ments has a complement for the MATLAB®suite with which it can be used with its SIMULINK®tool. In addition, simulation was carried out in the same software already mentioned for non-linear controllers using the same plant as the implemen tation in this case the pendubot system, where the performance of sliding planes is compared with linearization feedback, on the other hand, the error was obtained at starting from the methods of least squares of the error and ITAE applied both for the implementation and the simulation.MaestríaMagíster en Ingeniería EléctricaEl proyecto de grado consiste en implementar distintas estrategias de controladores lineal, y comparar su rendimiento durante la ejecución, en una planta subamortiguada multivariable no lineal, en este caso para un sistema PENDUBOT en donde consiste en poseer un par de eslabones anclado uno con respecto a otro cuya entrada es el torque proporcionado por un moto DC, y sus salidas son el ángulo de los eslabones respectivos, además se utilizará simulación para la misma planta con la finalidad de realizar una comparación entre controladores no-lineales, en donde se medirá su desempeño.Contents 0.1 Document structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Objectives 2 1.1 General objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Specific objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Introduction 3 2.1 Generalities of the Real-time systems . . . . . . . . . . . . . . . . . . 3 2.1.1 The real-time systems definition . . . . . . . . . . . . . . . . 3 2.1.2 Pendubot definition . . . . . . . . . . . . . . . . . . . . . . . . 4 3 the pendubot system modeling 5 3.1 Pendubot modeling for linear controller . . . . . . . . . . . . . . . . . 5 3.1.1 Linear approximation . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.2 Top-position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.3 Middle position . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Pendubot modeling for non-linear controller . . . . . . . . . . . . . . 10 4 controller design 13 4.1 linear controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.1 Robust pole assignment in linear state feedback . . . . . . . . 13 4.1.2 Linear Quadratic Regulator (LQR) . . . . . . . . . . . . . . . 13 4.2 non-linear controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2.1 Feedback linearization of mimo systems . . . . . . . . . . . . . 14 4.2.2 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . 16 5 Simulation Results 18 5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.1.1 Linear state feedback control . . . . . . . . . . . . . . . . . . . 19 5.1.2 Linear-Quadratic Regulator Control . . . . . . . . . . . . . . . 20 5.1.3 Non-linear control . . . . . . . . . . . . . . . . . . . . . . . . . 21 vii Contents viii 5.2 Implementation of linear controllers for the Pendubot system . . . . . 21 5.2.1 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2.2 blocks diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2.3 Error comparisons . . . . . . . . . . . . . . . . . . . . . . . . 25 5.3 Simulation of non-linear controllers for the Pendubot system . . . . . 26 5.3.1 Error comparisons . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3.2 blocks diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Conclusions 31