An inverted pendulum consists of a pole that acts as the pendulum, which is pivoted at one end and is attached to a cart. The cart can travel backward and forwards on a linear track, and by moving the cart appropriately it is possible to balance the pole and maintain it in its inverted position. In this assignment, you will be using a simple modular 3D-printed design for an inverted pendulum system that is driven using a stepper motor.

More specifically, your coursework task involves designing and implementing a state feedback controller with integral action to control an inverted pendulum that is mounted on a motorized cart.

This system is unstable without control and balancing the pendulum in its inverted configuration can only be achieved by displacing the cart. This assignment consists of two stages:

1. Theory and design

Design a state feedback controller with a setpoint for cart location using a state estimated from a Luenberger observer to stabilize a simulated velocity-controlled pendulum in MATLAB.

2. Implementation and simulation

Implement and run a Luenberger observer-based SFC controller with reference input for cart position in MATLAB using Euler integration so it could later easily be ported to run on a microcontroller.

Task

Develop a transfer function for a given linear system and characterize the linear system in terms of frequency response and transient performance.
Identify the stability of a given linear system.
Design a feedback controller to achieve stability and improve dynamics performance for a given linear system.

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