2.1. INTRODUCTORY EXAMPLES: PENDULUM PROBLEMS 21
position then even the smallest external disturbance on the cart would make
the rod lose balance and hence make the system unstable. The objective is to
overcome these external perturbationswith control action and to keep the rod
in the vertical position. Therefore, in the presence of control actions the force
on the cart is comprised of both external disturbances and the necessary control
actions from a controller to overcome the effects of disturbances.
Figure 2.3. Inverted pendulum on a cartThe task of the controller is to apply an appropriate forceu(t)to the cart to
keep the rod standing upright. We wish to design a controller that can control
both the pendulumís angle and the cartís position.
The following model parameters will be used to develop the mathematical
model of the system.
Mis the mass of the cart.
mis the mass of the pendulum.
bis the friction of the cart resisting motion.
Lis the length of the pendulum to its center of mass.
Iistheinertiaofthependulum.
u(t)is the force applied to the cart.
xrepresents the cart position coordinate.
θis the angle of the pendulum measured from the vertical.
To design a controller for the inverted pendulum from a standard control
viewpoint, it isfirst necessary to determine itsmathematical model. In Figure
2.4, we consider the free-body diagrams of the cart and the pendulum. This
will allow us to write the equations of motion.
Since the cart can only move around in a horizontal line, we are only inter-
ested in obtaining the equation by summing the forces acting on the cart in the
horizontal direction. Summing the forces along the horizontal for the cart, we
obtain the equation of motion for the cart as
Mx®+bx ̇+H=u