13Inverted Pendulum, Magnetic Levitation, and Cart on a Track
13.1 Inverted Pendulum
Figure 13.1 shows a rod on a cart that is free to rotate about a pivot. The objective is to keep the rod upright (
) by pushing/pulling on the cart using the input force 
. We will refer to this as the inverted pendulum control problem. The first thing to do is to derive the mathematical model of this system.
Figure 13.1. (a) Inverted pendulum. (b) Free body diagram. (c) 
and 
.
Mathematical Model of the Inverted Pendulum
Let 
denote an external force on the cart in the horizontal direction. The cart has mass 
and the angle between the vertical and the pendulum rod of length 
is denoted by . The center ...
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