The system shown in Figure represents the problem of stabilizing the attitude of a rocket during takeoff or controlling the balance of a personal transporter. The applied force f represents that from the side thrusters of the rocket or the tangential force on the transporter wheels. For small angles, Newton's law for the system reduces to
MLθ - (M + m)gθ = f
Where f is the control variable. Design a control law to maintain θ near zero. The specifications are ζ = 0.707 and a 2% settling time of 8 sec. The parameter values are M = 40 slugs, m = 8 slugs, L = 20 ft, and g = 32.2 ft/sec2.

L.