The open-loop transfer function G(s) between the input force u(t) and output angle (t) is given by

(For convenience, the sign of u(t) has been reversed.) With a unity feedback of (t), suppose you wish to design a controller C(s) that makes the closed-loop transfer function stable, so that the pendulum does not fall off (or, better yet, gradually becomes upright). Using the Routh stability criterion, show that neither the P controller C(s) = K_P , PI controller C(s) = K_P + K_I/s nor PD controller C(s) = K_P + K_Ds is capable of stabilizing the pendulum. Show that, in contrast, the PID controller C(s) = K_P + K_I/s + K_Ds is able to. For the PID controller, also derive the necessary and sufficient conditions on K_P , K_I , and K_D to ensure stability.

G(s) -= (s) G(s) -= (s)