Control of Rocket Car. A rocket car of unit mass is free to move along a straight, ?at, and frictionless track. The car has a rocket engine mounted on it, and the rocket can be ?red at any time t to exert a force u(t) on the car to propel it forward or backward along the track. The position p(t) of the car at each time t can be measured. Thus, if y(t) is the measured output at time t and v(t) the velocity of the car at time t, then ?v(t) = u(t) and y(t) = p(t) for all t.
(a) Obtain the transfer function for the rocket car, and its second-order and 380-th order state-space realizations.
(b) Given a reference input r, we are to design a ?rst-order compensator such that
u? - au = K( ?e - be), where e = r - y.
Prove that, for the car's position p to track r, we must at least have K > 0 and a < b < 0