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MMAN3200 Linear Systems and Control
ASSIGNMENT – T1 2022
Overview of the Assignment
This assignment focuses on the control of a pendulum system and is worth 20% of your total mark in
this course.
The due date to submit your assignment report to Moodle is Thursday 6am Week 9 (14 April).
Marks will be returned within 2 weeks after the submission deadline. You will complete this assignment
individually.
In this assignment, you will design a controller for an inverted pendulum system which is a nonlinear
system. We have learned in MMAN3200 how to control different LTI systems, but if we are given a
nonlinear system, we can linearize it around an equilibrium point and then design a controller so that
the closed-loop linearized system is stable. Then we can simulate the controller on the original nonlinear
system and if it works fine,

we can implement it on the real physical system. We will do these steps in
this assignment. For simulations, we will use MATLAB.
1 Background
Consider the pendulum shown in Figure 1, where denotes the length of the rod and m denotes the mass
of the bob. We assume the rod is rigid and has zero mass. The pendulum is actuated by a DC motor
through a gear box. The motor can apply a torque input u that makes the bob move.
Let θ denote the angle subtended by the rod and the vertical axis through the pivot point as shown in
Figure 1. The pendulum can swing in the vertical plane and the bob of the pendulum moves on a circle
of radius . Using Newton’s second law of motion, we can write the equation of motion as
m θ¨(t) +mg sin(θ(t)) + b θ˙(t) = u(t), (1)
where g is the gravity constant and b is the friction constant.
MTRN9400
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