EE380-1 Design Project
Control of an Inverted Pendulum on a Cart
Dr. Ahmad A. Masoud, Semester 122, 2013
The project is due a week before the last lecture in the semester.
The cart with an inverted pendulum, shown below, is "bumped" with an impulse force, F. The dynamic equations
of motion for the system are:
( )???? ?? ???? cos( ) ?? sin( )
( )???? sin( ) ???? cos( )
M mX bX m l m l F
I m l m g l m l X
+ + + · · · - · · =
+ · + · · · = - · · ·
? ? ? ?
? ? ?
2
2
M mass of the cart 0.5 kg
m mass of the pendulum 0.5 kg
b friction of the cart 0.1 N/m/sec
l length to pendulum center of mass 0.3 m
I inertia of the pendulum 0.006 kg*m^2
F force applied to the cart
x cart position coordinate
theta pendulum angle from vertical
1- Linearize the above dynamic equations about the pendulum's angle, theta = Pi (in other words, assume that
pendulum does not move more than a few degrees away from the vertical, chosen to be at an angle of B) and
obtain the transfer function of the system,
2- Assume that the system starts at equilibrium, and experiences an impulse force of 1N. Design a PID controller
that returns the pendulum to its upright position within 5 seconds, and never move more than 0.05 radians away
from the vertical position.
3- Suggest a control scheme that would keep the pendulum in an upright position and x around zero after the
impulse force is applied. Demonstrate your controller with one working example.
You are expected to hand in an organized structured report
with all the necessary drawings, derivations and simulations.
Document Preview: EE380-1 Design Project
Control of an Inverted Pendulum on a Cart
Dr. Ahmad A. Masoud, Semester 122, 2013
The project is due a week before the last lecture in the semester.
The cart with an inverted pendulum, shown below, is "bumped" with an impulse force, F. The dynamic equations
of motion for the system are:
2
() Mm++XbX+m·l·?? · cos( )-m·l·? sin(? )=F
2
() Im +·l?? +·mg·l·sin( )=-m·l·X· cos(? )
M mass of the cart 0.5 kg
m mass of the pendulum 0.5 kg
b friction of the cart 0.1 N/m/sec
l length to pendulum center of mass 0.3 m
I inertia of the pendulum 0.006 kg*m^2
F force applied to the cart
x cart position coordinate
theta pendulum angle from vertical
1- Linearize the above dynamic equations about the pendulum's angle, theta = Pi (in other words, assume that
pendulum does not move more than a few degrees away from the vertical, chosen to be at an angle of B) and
obtain the transfer function of the system,
2- Assume that the system starts at equilibrium, and experiences an impulse force of 1N. Design a PID controller
that returns the pendulum to its upright position within 5 seconds, and never move more than 0.05 radians away
from the vertical position.
3- Suggest a control scheme that would keep the pendulum in an upright position and x around zero after the
impulse force is applied. Demonstrate your controller with one working example.
You are expected to hand in an organized structured report
with all the necessary drawings, derivations and simulations.