4.1. Inverted Pendulum Modeling Consider an inverted single link arm of length i carrying a mass m, acceleration due to gravity g and input torque T applied at the base as shown in Figure l Figure 1: Inverted Pendulum Applying Newton's second law of angular acceleration to the link taking the axis of rotation as a reference: The sum of external torques Text acting on the link (with respect to the axis of rotation) is equal to the time derivative of the angular momentum L of the link (with respect to the axis of rotation): d EL = 27-9)"; (1) Two torques act on the link: the applied torque T, and the torque exerted by the force of gravity mgl sin 9. Both are measured positive in the clockwise direction. The pendulum has a rotational inertia of I 2 ml? (modeled as a point mass m, concentrated at the end of the pendulum]. Dening at g :9 as the angular velocity of the pendulum, then the angular momentum L is of the pendulum given by L 2 It) 2 mlz'. In all, (1) becomes mlz = T + mgl sin 6' (2) which results in the nonlinear differential equation gTSHIBZWT (3) 4.1.1. Linearization of the Nonlinear Model. 1) What is vertical equilibrium for this system? Answer". 9 = 0.1:. = 7' = O. 2) Linearize the nonlinear differential equation of the plant (3) about the vertical equilibrium 9 = I], a = T = 0. Answer\". .. g- 1 9 9 = 4 E in?\" i ) 4.1.2. Transfer Function of Plant Model. 1) Using the linear model derived (4), compute the system transfer function POL?) from torque input 1" to pendulum angle 8. Answer\