In this assignment, you will model, analyze and design a controller for the inverted pendulum shown in the figure. You will apply the principles learned in the model to accomplish the tasks and prepare a report that you will be submitting at the end of the trimester. The parameters of the inverted pendulum system are as follows:

Mass of the cart (M): 0.5 kg

Mass of the pendulum (m): 0.2 kg

Coefficient of friction for cart (b): 0.1 N/m/sec

Length to pendulum center of mass (l): 0.3 m

Mass moment of inertia of the pendulum (I): 0.006 kg.m^2

Tasks

1. Obtain the model of the system in State Space form.
2. Obtain the equilibrium point and linearize the model around the equilibrium point.
3. Design a pole placement controller that can drive the cart to follow a step reference input while keeping the pendulum upright.
4. Design an LQR controller for the task in (c) and compare both approaches.
5. Assuming that the speed of the cart and angular speed of the pendulum is not measured, derive a state observer that can estimate those signals and use the observer in parts (c) and (d).