a) The equations of motion of the inverted pendulum model

Chapter , Problem 11.22

(choose chapter or problem)

a) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about \(\phi=0\), assuming that \(\dot{\phi}\) is very small. b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and \(g=9.81 \mathrm{~m} / \mathrm{s}^2\). c) Use the linearized model developed in part (b) to design a PID controller to stabilize the pendulum angle near \(\phi=0\). 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 \(\phi\) and \(\dot{\phi}\) can be measured.