Modelling and Predictive Control of Inverted Pendulum

The paper is focused on creating a model of Inverted pendulum system and subsequent usage of this model to design a predictive controller of inverted pendulum system. The model is obtained on base of mathematical physical analysis of the system. Unknown parameters of the model are obtained from real-time experiments on the PS600 Inverted pendulum system. The model is designed in MATLAB/Simulink environment. The model was created with respect to most nonlinearities contained in the system. Nonlinearities are caused by fundamental principles of the system and by friction between individual parts of the system. Thus, the model is highly non-linear and therefore linearization around working point was performed and continuous linearized model was calculated as well as its discrete version. The discrete linear model was used to design predictive controller which was also verified by real time experiments. INTRODUCTION Most of current control algorithms are based on a model of a controlled plant (Bobál et al. 2005). It is obvious that some information about controlled plant is required to be able to control output of the plant or to investigate its properties and behaviour. Two basic approaches of obtaining plant model exist: the black box approach and the mathematical-physical analysis of the plant. The black box approach to the modelling (Liu, 2001) is based on analysis of input and output signals of the plant. The main advantages of this approach lies in the possibility of usage the same identification algorithm for wide set of different controlled plants. Also, the knowledge of physical principle of controlled plant and solution of possibly complicated set of mathematical equation is not required. On the other hand, model obtained by black box approach is generally valid only for signals it was calculated from. For example, if step response was used to obtain the model, this model should not be used for analysis of the system behaviour when high frequency changes of input signals are applies. The mathematical-physical analysis of the plant and following derivation of the relations between plant inputs and outputs provides general model which can be valid for whole range of plant inputs and states. Contrary, there is usually a lot of unknown constants and relations in the model description when performing mathematic-physical analysis. The second method, mathematical-physical analysis, is used in this paper. The goal of the work was to obtain a mathematical model of the PS600 Inverted Pendulum System (Amira, 2000), to design the model in MATLAB-Simulink environment and use this model for a design of model predictive controller. The PS600 laboratory equipment was developed by Amira Gmbh, Duisburg, Germany and serves as a real-time model of unstable, highly nonlinear system. When the model of the controlled system is known the problem of selecting an appropriate control synthesis arises. Many successful control techniques have been developed in past decades. One of them is model predictive control (MPC) (Camacho and Bordons, 2004). Contrary to most other approaches, MPC uses not only current and previous values of control circuit signals but also future values of reference signal. Future course of reference signal is known in many applications and thus can be used in controller synthesis. The scheme of a simple control circuit with self-tuning predictive controller is shown in Figure. 1. Note that the reference signal is marked as w(t), which means that the course of reference signal is sent to the controller, not only the current value w(k). model predictive controller u(k) w(t) y(k) controlled system Figure 1: Control circuit with model predictive controller INVERTED PENDULUM SYSTEM The PS600 inverted pendulum system is shown in figure 2. The main parts of the system are cart driven by Proceedings 22nd European Conference on Modelling and Simulation ©ECMS Loucas S. Louca, Yiorgos Chrysanthou, Zuzana Oplatková, Khalid Al-Begain (Editors) ISBN: 978-0-9553018-5-8 / ISBN: 978-0-9553018-6-5 (CD) servo amplifier and the pendulum rod attached to the cart. Figure 2: Photo of PS600 Inverted Pendulum system A simplified scheme of the inverted pendulum system is shown in figure 3. This scheme was used for mathematical physical analysis of the system. Figure 3: Scheme of inverted pendulum The PS600 Inverted Pendulum is designed as a system with one input and two measured outputs – SITO (single input two outputs). The input of the system is a control voltage of the servo amplifier (U) and the outputs are cart position and angle of pendulum rod. Both outputs are measured by incremental encoders (Amira, 2000). Systems allows different control objectives with various difficultness of control design. The most common cases are: • control of cart position with pendulum acting as a disturbance • control of cart position with stabilization of the pendulum in the stable equilibrium position (ie. pendulum underneath the guiding bar) • stabilization of the pendulum in the upright position • control of the cart position with pendulum in upright position. MATHEMATICAL MODEL Forces and moments acting in the system were analysed using figure 4 where φ represents the angle of pendulum rod, 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.