Consider the following nonlinear system representing a pendu

Consider the following nonlinear system representing a pendulum on a cart: [M + m Ml cos theta Ml cos theta I + Ml^2] [x theta] + [0 - Ml delta sin theta 0 0] [x theta] + [kx - mgl sin theta] = [F 0]. Suppose F* = 0, show that the corresponding equilibria are given by theta* = 0 or theta* = pi, theta* = 0, x* = 0 and x* =0. Let tilde F = F - F* and tilde x = x - x*. Linearize the above set of differential equations about the equilibrium x* = 0, dot x* = 0, theta* = 0, dot theta* = 0 and express them in the state space form. What is the transfer function from tilde F to tilde x? What are the zeros and poles of this system? Fix tilde x identicalto 0, i.e., what should the force F(t) be so that tilde x = 0? Note theta may still be varying with time and the force F(t) may depend on the values of dot theta, theta. What is the resulting governing equation if tilde x identicalto 0? What are the roots of the corresponding characteristic equation?

Solution

wthe forcce f(t) be so that x=0? note 0

may still be varying with time and the force f(t) may bedend on the values of

the resultinggoverning equation if x

the rooting of thecorresponding characteristics eqation