A comparative study of two state-of-the-art stochastic model predictive controllers for linear systems with parametric and additive uncertainties is presented. On the one hand, Stochastic Model Predictive Control (SMPC) is based on analytical methods and solves an optimal control problem (OCP) similar to a classic Model Predictive Control (MPC) with constraints. SMPC defines probabilistic constraints on the states, which are transformed into equivalent deterministic ones. On the other hand, Scenario-based Model Predictive Control (SCMPC) solves an OCP for a specified number of random realizations of uncertainties, also called scenarios. In this paper, Classic MPC, SMPC and SCMPC are compared through two numerical examples. Thanks to several Monte-Carlo simulations, performances of classic MPC, SMPC and SCMPC are compared using several criteria, such as number of successful runs, number of times the constraints are violated, integral absolute error and computational cost. Moreover, a Stochastic Model Predictive Control Toolbox was developed by the authors, available on MATLAB Central, in which it is possible to simulate a SMPC or a SCMPC to control multivariable linear systems with additive disturbances. This software was used to carry out part of the simulations of the numerical examples in this article and it can be used for results reproduction.

This paper presents an alternative way to the dynamic modeling of a rotational inverted pendulum using the classic mechanics known as Euler-Lagrange allows to find motion equations that describe our model. It also has a design of the basic model of the system in SolidWorks software, which based on the material and dimensions of the model provides some physical variables necessary for modeling. In order to verify the theoretical results, It was made a contrast between the solutions obtained by simulation SimMechanics-Matlab and the system of equations Euler-Lagrange, solved through ODE23tb method included in Matlab bookstores for solving equations systems of the type and order obtained. This article comprises a pendulum trajectory analysis by a phase space diagram that allows the identification of stable and unstable regions of the system.

The dynamic modeling of a Furuta pendulum using Euler-Lagrange formalism which allows to find the equations of motion described by the model is presented. A basic design of the system mechanical structure was built in SOLIDWORKS software, which provided the required physical variables for modeling based on the defined material and dimensions of the pendulum prototype. Taylor series were used to obtain a linear model of the system and to design the predictive control in state space. This strategy allows control over the Furuta pendulum taking into account restrictions in its manipulated and controlled variables.

In this paper, is used the Lagrangian classical mechanics for modeling the dynamics of an underactuated system, specifically a rotary inverted pendulum that will have two equations of motion. A basic design of the system is proposed in SOLIDWORKS 3D CAD software, which based on the material and dimensions of the model provides some physical variables necessary for modeling. In order to verify the results obtained, a comparison the CAD model simulated in the environment SimMechanics of MATLAB software with the mathematical model who was consisting of Euler-Lagrange's equations implemented in Simulink MATLAB, solved with the ODE23tb method, included in the MATLAB libraries for the solution of systems of equations of the type and order obtained. This article also has a topological analysis of pendulum trajectories through a phase space diagram, which allows the identification of stable and unstable regions of the system.