Examples from Control Engineering

Abstract

Stabilizing an inverted pendulum (Fig. 9.1) is an example that is widespread in control education. A cart moving on a track without friction carries an inverted pendulum which is attached by a friction-free joint. Suppose that the cart is moved along the track by controlling the input voltage of a DC motor that applies a force on the cart proportional to the voltage. The following variables and constants of the system are used: x = x(t) translation of the cart from the center of the track; a = a(t) angle of the inverted pendulum with the normal of the track; f = f(t) force applied to the cart; M mass of the cart; m mass of the pendulum; l center of mass of the pendulum rod (half of full length) g gravitational acceleration. Let us start by modeling the open-loop system. The force f(t) acting on the cart is chosen as the input of the system. First we shall use Lagrange’s method to obtain the model of the system, then the nonlinear equations will be linearized.