Multimodel Approach In State-Space Predictive Control

This paper presents a multimodel approach to control nonlinear systems. The system of the inverted pendulum which has one control input and two measured outputs was chosen as an exemplar process. This system is an example of the nonlinear process with a sampling period in order of milliseconds. The state-space predictive control of the system described by CARIMA model was chosen as a control method. This paper presents a description of the inverted pendulum nonlinear mathematical model, its linearization and the control signal calculation using predictor-corrector method. The results compare three methods of linearized model combination. All of the simulations were done in Matlab. INTRODUCTION Process control area offers a variety of different processes with different level of complexity. Even the sampling period can be very different. This paper focuses on nonlinear processes with fast sampling period. The complex and fast processes need a modern control method to control them effectively such as model predictive control (Bobál 2008). This method predicts the output values on the chosen time interval based on the mathematical model of the controlled process. 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 predictive control method uses a minimization of the cost function, which is usually in quadratic form, to calculate the control signal. The quadratic form of the cost function minimize the differences between the reference value and the output value and the control signal increments. The predictive control method offers a possibility to constrain the process variables then the chosen predictor-corrector minimization method can be used to minimize the cost function (Camacho and Bordons 2004; Maciejowski 2002; Rossiter 2003). However, the chosen state-space predictive control method uses a linear CARIMA mathematical model for prediction of the output values but the chosen process of the inverted pendulum has nonlinear behaviour. That means we have to linearize the nonlinear model first. the nonlinear behaviour of the process can be described with a set of the linear models. The final linear model used to output values prediction is obtained by combination of two or more linear models out of the set of the linearized models according to the current output value (Albertos Peréz and Sala 20014; Hangos et al. 2004). 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. MATHEMATICAL MODEL OF THE CONTROLLED SYSTEM The controlled system in this paper is represented by Amira PS600 inverted pendulum system which is shown at figure 1. The pendulum rod of this system is attached to the cart which is driven by a servo motor (Amira 2000; Chalupa and Bobál 2008). Figure 1 : Amira PS600 Inverted Pendulum system The servo motor produces the input force (input variable) that move with the cart. Position of the cart is the first measured output variable and the pendulum angle is the second measured output variable. The figure 2 shows the analysis of the forces acting in the system (Amira 2000; Chalupa and Bobál 2008). Proceedings 32nd European Conference on Modelling and Simulation ©ECMS Lars Nolle, Alexandra Burger, Christoph Tholen, Jens Werner, Jens Wellhausen (Editors) ISBN: 978-0-9932440-6-3/ ISBN: 978-0-9932440-7-0 (CD) Figure 2 : Analysis of the inverted pendulum The angle of the pendulum rod is expressed as φ, the input force produced by DC motor is symbol F, symbol ls means distance between pendulum gravity centre and rotation centre of the pendulum, weight of the cart is expressed as M0, weight of the pendulum is expressed as M1 and g is gravity acceleration constant. The equations (1) and (2) describe the horizontal and the vertical forces that the pendulum causes on cart.