For the closed-loop system to control the ship’s heading

a. Find the fourth-order differential equation relating the output θ(t) and input θcom(t).

b. Find a suitable choice for matrices A, B, C, and D in the state variable form x = Ax + Bu, y = Cx where u = θcom and y = θ. Leave your answers in terms of the system parameters KC, KP, KS, τP, and τS. Hint: Draw a simulation diagram.

c. Choose the same values for KP, KS, τP, and τS as in the example. Find the characteristic polynomial Δ(s) as a function of KC by evaluating |sI − A|.

d. Prepare a table with two columns. The first column contains values of KC = 1, 5, 10, 25, 50, 75, …, 200 V/deg heading, and the second column lists the four closed-loop system poles.

e. Use the MATLAB M-file “Ch4_feedback_yaw.m” or write your own to find the value(s) of KC that results in an underdamped quadratic factor of Δ(s) with damping ratio equal to 0.5.