Derive the discrete-time system transfer function ghpz from

Question

A temperature control system can be modelled by the following transfer function G(s), where e- Τs represents a pure time delay in the system and Τ=1. You are required to implement a digital PID controller to track the temperature setting without error.

G(s) = C(s)/M(s) = (5e^-Τs)/s + 5

1. Analytically find the open-loop system response c(t) to a unit step input and plot the response.

2. Based on sampling theorem, determine a suitable sample interval T for the rest of the assignment. (Hint: Use Bode plots of G(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. Choose k = Τ/T as an integer for calculation convenience).

3. Derive the discrete-time system transfer function G_HP(Z) from G(s).

(Hint: Z [e-ΤsG (s)] = z^k Z [G (s)], where the k = Τ/T, , and keep T as a parameter until final results are available).

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).

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).