CONTROL THEORY III Mock Exam 3 Mock Exam 3

Consider a continuous-time system described by the transfer function $G(s) = \frac{5}{s^3 + 4s^2 + 7s + 4}$. This system is to be digitally controlled using a sampling period $T = 0.5$ seconds. Assume zero initial conditions for all state variables.

a) (5 marks) Determine the pulse transfer function $G(z)$ of the system. Show all intermediate steps in your derivation, including the application of the z-transform to the transfer function $G(s)$.

b) (10 marks) Assuming a unity feedback configuration, determine the closed-loop pulse transfer function $T(z) = \frac{C(z)}{R(z)}$, where $R(z)$ is the reference input and $C(z)$ is the controlled output.

i) (5 marks) Derive the closed-loop pulse transfer function $T(z)$ in terms of $G(z)$ and the feedback gain (which is unity in this case). Clearly state any assumptions made during the derivation. ii) (5 marks) If the reference input $R(z)$ is a unit step function, i.e., $R(z) = \frac{z}{z-1}$, determine the first three values of the output sequence $c(0)$, $c(1)$, and $c(2)$. Assume initial conditions are zero.

A continuous-time signal $x(t)$ is bandlimited to $f_m = 5 kHz$. Consider sampling this signal to create a discrete-time signal $x[n]$.

a) (5 marks) State the Nyquist-Shannon sampling theorem. Define all terms used in the theorem.

b) (5 marks) Determine the minimum sampling frequency $f_s$ required to avoid aliasing when sampling $x(t)$.

c) (10 marks) Suppose the signal $x(t)$ is sampled at a frequency of $f_s = 8 kHz$.

i) (5 marks) Will aliasing occur? Justify your answer with calculations. ii) (5 marks) If aliasing occurs, what is the resulting aliased frequency? Explain how this aliased frequency would manifest in the sampled signal.

Consider a linear time-invariant system described by the following transfer function: $G(s) = \frac{5}{s^3 + 4s^2 + 7s + 4}$. Determine the state-space representation of this system, assuming a controllable canonical form.

a) (5 marks) State the general form of the state-space equations for a linear time-invariant system. Define each variable in the equations.

b) (10 marks) Derive the state-space representation (A, B, C, and D matrices) for the given transfer function $G(s) = \frac{5}{s^3 + 4s^2 + 7s + 4}$.

i) (5 marks) Determine the state matrix A. Show the steps involved in constructing A from the denominator of the transfer function. ii) (5 marks) Determine the matrices B, C, and D. Explain the reasoning behind your choices for these matrices.