State-Space Predictive Control Of Inverted Pendulum Model

This paper presents a possible way to control the a very fast nonlinear systems. The system of the inverted pendulum was chosen as an exemplar process. This is an example of the nonlinear single-input multi-output process with a sampling period in order of milliseconds. The state-space predictive control was chosen as a control method and the system is described by CARIMA model. The whole process of the controller design is described in this paper. That includes a description of the inverted pendulum nonlinear mathematical model and its linearization, the inference of the output values prediction and the control signal calculation. The control signal is calculated by predictor-corrector method. The results compare several optimization methods to achieve the fastest calculation of the control signal. All of the simulation was done in Matlab. INTRODUCTION In real life we can come across with many types of processes. Many of them are nonlinear and their mathematical models are very complex. Even the sampling period can be very different. This paper focuses on the very fast processes with a sampling period in the order of milliseconds. The basic control methods may not handle with this situation with required precision so we need a more advanced method. The predictive control is a great example of the modern control method that can be used to solve the complex control problems (Bobál 2008). This method belongs to the model based control methods and the mathematical model is used for the output values prediction. This prediction is determine on the chosen time horizon that should be long enough to cover the step response of the controlled system. The model of the inverted pendulum is described by the state-space CARIMA mathematical model for the single-input multi-output (SIMO) system (Bars et al. 2011; Wang 2009). The control signal calculated by the predictive control ensures the desired output values in the near future time horizon. This is achieved by minimization of the cost function that usually has a quadratic form and it minimize the differences between the reference value and the output value and the control signal increments. If the process require some kind of the process variable constraints, several method such as quadratic programming method, fast-gradient method, predictorcorrector method etc. can be used to minimize the cost function (Camacho and Bordons 2004; Maciejowski 2002; Rossiter 2003). However, the chosen CARIMA mathematical model used to the prediction of the output values works only for the linear models so the nonlinear mathematical model of the inverted pendulum needs to be linearized. This paper is divided into the following sections. The model of the inverted pendulum is described in the first section. The predictive control and the calculation of the control signal are described next. The final sections shows the results of the research and the conclusion (Albertos Peréz and Sala 20014; Hangos et al. 2004). MATHEMATICAL MODEL OF THE CONTROLLED SYSTEM The Amira PS600 inverted pendulum system was used as the exemplar model. The photo of this system is shown at figure 1. The main parts of the system are cart driven by servo amplifier and the pendulum rod attached to the cart (Amira 2000; Chalupa and Bobál 2008). Figure 1 : Amira PS600 Inverted Pendulum system The inverted pendulum system is an example of the single-input two-output system. The force produced by the DC motor that moves with the cart is the input variable and the cart position and the angle of the pendulum rod are the output variables. The figure 2 shows the analysis of the forces acting in the system (Amira 2000; Chalupa and Bobál 2008). Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD) Figure 2 : Analysis of the inverted pendulum The variables in the figure 2 are following. The angle of pendulum rod is φ, M0 and M1 stands for the weight of the cart and pendulum respectively, lS is a distance between centre of gravity of the pendulum and the centre of rotation of the pendulum and g is the gravity acceleration constant. Symbol F represents the force produced by the DC motor. The affect of the pendulum on the cart can be expressed as a horizontal and a vertical forces described by the equations (1) and (2)