Stabilising a cart inverted pendulum system using pole placement control method

A cart inverted pendulum system is one of the most common case to be considered for testing many control algorithms, since it has some challenging problems associated with non linearity, complexity and underactuated system model. In fact, non-linearity behaviour of the inverted pendulum can be observed easily. Different pendulum angle response can be obtained by giving the same velocity in the cart. The cart inverted pendulum can be understood as an under actuated system since the system has a lower number of actuator than the degrees of freedom. One of the most convenient method to model the inverted pendulum system is to use Lagrange's equation. However, at present, many presented inverted pendulum models have been derived using simplified physical model. This simplified model may lead to problems for the implementation of the control algorithm in a real physical inverted pendulum system. In this paper, an inverted pendulum system model is presented, where a mechanical transmission system and a motor models have been included in the derivation of the inverted pendulum model. Hence, the problems for the control implementation in a real system can be minimized. The mathematical model of the inverted pendulum was derived using Lagrange's equation. The determination of the pole zero of the system is discussed. A simple method of pole placement is proposed to stabilise the pendulum at the desired position of the cart. Matlab simulation results show the effectiveness of the proposed method. And yet, this intuitive approach can lead to better understanding of the control behaviours.