Lab #5 Feedback Stabilization: Stick Balancing|ALL THAT YOU NEED

Lab #5 Feedback Stabilization: Stick Balancing This lab is based upon the problems in Section 11.1, "Feedback Stabilization: Stick Balancing," in the course text Computer Explorations in SIGNALS AND SYSTEMS by Buck, Daniel, Singer. This exercise considers the problem of stabiiizing the unstable system that governs the balancing of an object like a stick in your hand. Fig. 1 Consider the systems shown in Fig. 1. The cart, whose position is given by , can move forward or backward. The acceleration of the cart follows as A stick of length L is attached to the cart with a hinge so that it can only move in the same direction as the cart. The position of the stick is given by the angle . Assume that all ofthe mass of the stick is concentrated in a ball at the end of the stick. Also shown is an angular accelration imparted on the stick by external forces, such as the wind. In order to balance the stick,the cart must be moved with an appropriate acceleration . Balancing forces on the mass along the direction perpendicular to the rod, the differential equation relating , , and is This is not a linear differential equation. However, we can linearlize it for small to examine the dynamics when the stick is nearly vertical, which is the region of interest when balancing the stick. Using the small angle approximations a