CHALLENGE: We now give you a problem to test your knowledge of this chapter's objectives. The UFSS vehicle steers via the heading control system shown in Figure $5\mathrm{.}38$ and repeated on the back endpapers. A heading command is the input. The input and feedback from the submersible's heading and yaw rate are used to generate a rudder command that steers the submersible $($Johnson$,1980).$ Let $\backslash ($ K$\_\{1\}=\backslash )\backslash ($ K$\_\{2\}=1\backslash )$ and do the following:

a$.$ Draw the signal$-$flow graph for each subsystem, making sure that heading angle, yaw rate, and rudder deflection are represented as state variables. Then interconnect the subsystems.

b$.$ Use the signal$-$flow graph obtained in a to represent the heading control loop in state space.

c$.$ Use MATLAB to represent the closed$-$loop UFSS heading control system in state space in controller canonical form.