Consider the automatic ship-steering system discussed in Problem P8.11. The frequency response of the open-loop portion of the ship steering control system is shown in Figure P8.11. The deviation of the tanker from the straight track is measured by radar and is used to generate the error signal, as shown in Figure P9.15. This error signal is used to control the rudder angle δ(s).

(a) Is this system stable? Discuss what an unstable ship-steering system indicates in terms of the transient response of the system. Recall that the system under consideration is a ship attempting to follow a straight track.

(b) Is it possible to stabilize this system by lowering the gain of the transfer function G(s)?

(c) Is it possible to stabilize this system? Suggest a suitable feedback compensator.

(d) Repeat parts (a), (b), and (c) when switch S is closed.

Problem P8.11

Automatic steering of a ship would be a particularly useful application of feedback control theory [20]. In the case of heavily traveled seas, it is important to maintain the motion of the ship along an accurate track. An automatic system would be more likely to maintain a smaller error from the desired heading than a helmsman who re corrects at infrequent intervals. A mathematical model of the steering system has been developed for a ship moving at a constant velocity and for small deviations from the desired track. For a large tanker, the transfer function of the ship is

where E(s) is the Laplace transform of the deviation of the ship from the desired heading and δ(s) is the Laplace transform of the angle of deflection of the steering rudder. Verify that the frequency response of the ship, E(jω)/δ(jω). is that shown in Figure P8.11.