The system shown in Figure P10.60 represents the problem

Chapter , Problem 10.60

(choose chapter or problem)

The system shown in Figure P10.60 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

                                                           \(M L \ddot{\theta}-(M+m) g \theta=f\)

where f is the control variable. Design a control law to maintain \(\theta\) near zero. The specifications are \(\zeta=0.707\) and a 2% settling time of 8 sec. The parameter values are M = 40 slugs, m = 8 slugs, L = 20 ft and \(g=32.2 \ \mathrm{ft} / \mathrm{sec}^2\).