I. Assessment Requirements

In this course work (CW) you may use the concepts you learned during your lectures and your laboratory works together with your ability in performing further research in bringing more details if necessary. In your report, provide the answers to the following questions:

II. Assessment Details

I. Provide your understanding about Proportional (P), Integral (I) and Derivative (D) analogue controllers, their differences, advantages, and disadvantages. Then, clarify why their combinations, such as in PID controllers, can be useful.

II. Based on your understanding from the lecture notes, your laboratory practice, by further research, and with the help of a block diagram, explain how a PID temperature control system works.

III. A digital control system has the following difference equation:

y[n] = 2.0 x[n] - x[n-1] + 1.2 y[n-1] - 0.48 y[n-2]

a. Find the transfer function H(z) for this system.
b. Draw the digital Circuit for H(z)
c. Use MATLAB to draw the pole-zero diagram for the above system.
d. Use MATLAB to derive and plot the impulse response of the system.
e. With the help of MATLAB find and explain the response of the system to the input x[n]=(1-0.5e^-n)u[n], where u[n] is a step function. What is the steady- state solution, i.e. when n becomes very large.

IV. Use MATLAB to design a 4^th order lowpass Butterworth digital filter with a bandwidth of 0.35 (normalised with respect to Πfs). Then,
a. Find the parameters of the equivalent analogue filter through bilinear transformation considering a sampling frequency of 10 Hz.
b. What are the passband and stopband frequencies of both analogue and digital Butterworth filters.
c. Draw the pole-zero diagram of the digital filter.
d. Apply the input x[n] = 2sin(10Πt) + 5cos(24Πt) to the system and comment on the output. Use frequency response to enhance your conclusions.

V. This question is to investigate the stability of analogue and digital systems.
a. Provide the conditions for stability of the systems in both analogue and discrete (digital) domains.
b. Consider the following three analogue systems and comment on their stability:

i. H(s) = s/(s+0.2)
ii. H₂(s) = 1/(s² + 1)
iii. H₃(s) = (s - 1)(s²-0.4s-0.05)

c. Convert all above filters into discrete (digital) domain using sampling frequency of 1 Hz and

i. Impulse invariant transformation
ii. Bilinear transformation

Draw the pole-zero diagrams, and comment on the stability of the equivalent digital systems.

d. Use MATLAB, find the impulse and unit step responses of the digital system, draw the outputs, and comment on the shapes of the responses.

VI. Consider the following analogue plant

H(s) = 2/(s + 0.8)

a. Convert H(s) to the equivalent digital system using bilinear transformation using a sampling frequency of 10 Hz. Draw the pole-zero diagram, impulse response, and frequency response of the system.

b. Control the analogue plant using the following feedback system.

Consider G(s) to be a constant K; Drive Y(s)/R(s) and determine for what range of K values the system remains stable. Y(s) and R(s) are the s-transform of y(t) and r(t) respectively.

c. Using a sampling frequency of fs = 2 Hz and for the two values of K = - 0.3 and K = - 0.5 determine the equivalent digital systems using bilinear transformation.

d. Draw the pole-zero diagram for the two digital systems in Part c and discuss on the overall stability of each system.

e. Find the unit step response of the two systems in part c. and explain about the steady-state response if any.