(f) (15%%) In order to stabilize the pendulum in the inverted position, the pendulum is attached to a cart as shown below. A force Fi is used to move the cart, displacement z, so that the pendulum remains vertical. (20%) Ms Mc Fi The addition of the motion of the cart changes the equation for the pendulum and adds a differential equation for the motion of the cart, i.e. + Lcose gL 12 +0.1822 - [ 12 +0.1d2] [L 2 sine = 0 --- Lcose 2 + 1 + 1 + Ms. Mc sino + = 0 Ms+Mc For small angles, cose 1 and 0 sine 0. Rewrite the two equations above for small angles. Laplace transform the two equations assuming 0(0) = 0.1 rad and solve for e(s) in terms of F(s). Use L=2 m, d = 0.1 m, Mc=10 kg, Ms=1 kg, and g=10 m/s. You should get the following (s 5.5)0(s) = 0.05F; (s) + 0.1s (g) (30%) A transducer is used to measure the angle 0 which is then the input to a controller which generates the force Fi. Throughout this course, we will be designing control transfer functions to achieve various design specifications. For now, however, assume that the control transfer function for generating Fi is defined below. Fi(s) = -150 61 [S+8] 0 (s) Using this equation and the equation from (c), show that the equation for e(s) is 0(s) = 0.1s(s + 8) (s+7.76)(s + 0.241s + 0.128) What are the poles of e(s)? Note, in this problem, the poles are the same as the eigenvalues since there is no input to the system, only nonzero initial conditions. Is the pendulum stable with this controller? What is the undamped natural frequency? What is the damping ratio? What are the time constants? What is the damped natural frequency? Approximately how long will it take for the pendulum to stabilize at 0 0?