Design a digital proportional controller

Question

Assignment Problem: Ships at sea undergo motion about their roll axis. Fins called stabilizers are used to reduce this roll motion. The stabilizers can be positioned by a closed-loop roll stabilizing system shown below

where the transducer works like a proportional (P) controller. If given the open-loop transfer function

G(s)H(s) = (22.5/((s+4)(s²+0.9s+9))x1

Problem 1: Analytically find the open-loop response c(t) to a unit impulse input, show your mathematical working and plot this open-loop system response.

Problem 2: Based on sampling theorem, determine a suitable sample interval T for the following parts of the assignment and show your reasoning. (Hint: Use Bode plots of G(s)H(s) to determine the system's cut-off frequency. At cut-off frequency the magnitude plot is about 3dB below the magnitude of the low frequency. The cut-off frequency can be considered as the highest frequency component).

Problem 3: If the G(s) can be simplified into G1(s) 22.5/4(s²+0.9s+9), derive the discrete-time system transfer function GHP(Z) from G1(s) and show your mathematical working.

Problem 4: Design a digital Proportional (P) controller to form a unit feedback control system, and optimise its parameter P with respect to the performance criterion IAE using the steepest descent minimisation process. Simulate the P controller system and plot its response for a unit step input (The performance criterion IAE = ∑_k=0^M| e_k|).

(Please provide the plots that show the initial and the final/optimal responses).

Problem 5: If the P controller is replaced with a PID controller, using the steepest descent minimisation process again to optimise the PID controller with respect to the performance criterion IAE = ∑_k=0^M| e_k | for a unit step input. (Please also provide the plots that show both the initial and the final/optimal responses).

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