 
Summary: 1 Stabilization of a Double Pendulum
The purpose of this exercise is to illustrate the full extent of the theory developed in Chapters 9
and 10. The exercise uses many of the concepts introduced in this book (modeling, controllability,
observability, stability, pole placement, observers, feedback compensation). This exercise requires
extensive use of computer aids: Maple for formula manipulation, and Matlab and Simulink
for control system design and numerical simulation.
1.1 Modeling
We study the stabilization of a double pendulum mounted on a movable cart. The relevant
geometry is shown in Figure 1. It is assumed that the motion takes place in a vertical plane. The
M1
M
q
1
L1
M2
L2
u
2
Figure 1: A double pendulum on a cart.
significance of the system parameters is as follows:
