a) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that ( is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use the linearized model developed in part (b) to design a series compensator to stabilize the pendulum angle near ( = 0. It is required that the 2% settling time be no greater that 4 s and that the response be nonoscillatory. This means that the dominant root should be real and no greater than -1. No restriction is placed on the motion of the base. Assume that only ( can be measured.