Nonlinear Control Sy.. - Free

26 CHAPTER 1. INTRODUCTION
Figure 1.17: Free-body diagrams of the pendulum-on-a-cart system.
22
where we have substituted
g sin x1 - amlx2 sin(2x1) - 2a cos(xl) fx
41/3 - 2aml cos2(xl)
1z
J= 2 , and a
1.9.3 The Ball-and-Beam System
f
=
2(m+M)
The ball-and-beam system is another interesting and very familiar experiment commonly
encountered in control systems laboratories in many universities. Figure 1.18 shows an
schematic of this system. The beam can rotate by applying a torque at the center of
rotation, and the ball can move freely along the beam. Assuming that the ball is always in
contact with the beam and that rolling occurs without slipping, the Lagrange equations of
motion are (see [28] for further details)
0 = (R2 + m)r + mgsin 0 - mrO2
r = (mr2+J+Jb)B+2mr7B+mgrcos0
where J represents the moment of inertia of the beam and R, m and Jb are the radius, mass
and moment of inertia of the ball, respectively. The acceleration of gravity is represented by