A Cart$-$Inverted$-$Pendulum System

The Objectives

$1.$ Briefly research and understand your system. Each system is discussed in the unit textbook.

However, other sources, such as YouTube, may help.

$2.$ Develop the input$-$output equations and transfer functions of your system.

$3.$ Construct a MATLAB program for your system using the input$-$output equations and transfer

functions. That is$,$ two MATLAB scripts that should produce the same output.

$4.$ Construct a Simulink program for your system using the input$-$output equations and transfer

functions. That is$,$ two Simulink diagrams that should produce the same output.

$5.$ Demonstrate that each of your programs produces the same output for a given input and set of

system conditions $($i$.$e$.,$ parameter values$).$

$6.$ Select one of your programs and use this to carry out simulations to analyse your system. Consider

the nature of your system inputs and the values of your parameters when performing your analysis.

$7.$ Document your work and your findings in a formal report.

NOTE: You are not required to develop a control system. Your simulations should be open$-$loop responses.

You will be developing a control system in Project II$.$

The Cart$-$Inverted$-$Pendulum System is shown above. This is a mechanical system equivalent to Figure $5\mathrm{.}79$

in the textbook.

A uniform rod of mass $M$ and length $L$ is pivoted on a cart of mass $m.$ Assume that the pendulum is

constrained to move in a vertical plane, and the cart moves without slipping along a horizontal line.

The external force $f$ applied to the cart is the system input, while the displacement of the cart $x$ and the

angular displacement of the pendulum $$ are the system outputs.

Assume that your system initial conditions are zero.

To help you get started, review Example $5\mathrm{.}14($p$.219)$ and Chapter $10$ Review Problem $10($p$.576)$ in the

textbook.