CONTROL THEORY III Mock Exam 1 Mock Exam 1

Consider a linear time-invariant system described by the state-space equations: $\dot{x} = Ax + Bu$ and $y = Cx + Du$, where $x$ is the state vector, $u$ is the input vector, and $y$ is the output vector. Given that: $A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}, B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, C = \begin{bmatrix} 1 & 0 \end{bmatrix}, D = 0$ and an initial condition $x(0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, determine the system’s response to a unit step input, $u(t) = 1$ for $t \ge 0$.

a) (5 marks) Determine the eigenvalues of matrix $A$. Show all working.

b) (10 marks) Calculate the state transition matrix, $e^{At}$.

i) (5 marks) Find the eigenvectors corresponding to each eigenvalue calculated in part (a). ii) (5 marks) Using the eigenvectors found in (i), determine the state transition matrix $e^{At}$.

Given the transfer function $G(s) = \frac{2}{s^2 + 5s + 6}$, determine the state-space representation in controllable canonical form. Clearly show all steps and justify your choices for state variable selection.

a) (5 marks) Factorize the given transfer function $G(s) = \frac{2}{s^2 + 5s + 6}$ into its numerator and denominator polynomial forms. Express the denominator as a product of first-order terms. Explain the significance of this factorization in the context of state-space representation.

b) (10 marks) Derive the state-space equations for the system. Specifically, determine the state matrices A, B, C, and D. Clearly define the chosen state variables and justify their selection based on the factored denominator polynomial.

i) (5 marks) Define the state variables $x_1(t)$ and $x_2(t)$ such that $\dot{x_1}(t) = -2x_1(t) + 2x_2(t)$ and $\dot{x_2}(t) = -3x_1(t)$. Express these relationships in matrix form to obtain the A and B matrices. Show the derivation of the A matrix and the input matrix B, assuming a single input u(t) acting on the first state variable. ii) (5 marks) Determine the output matrix C and the direct transmission matrix D such that the output $y(t)$ is equal to the original transfer function's output. Explain how the C and D matrices relate to the numerator and denominator polynomials of the transfer function.

A continuous-time signal $x(t)$ is bandlimited to 4 kHz. It is to be sampled to create a discrete-time signal. Consider the implications of violating the Nyquist-Shannon sampling theorem.

a) (5 marks) State the Nyquist-Shannon sampling theorem. Clearly define all terms used in your statement.

b) (10 marks) Suppose the signal $x(t)$ is sampled at a rate of 6 kHz. Explain what will happen during reconstruction and illustrate your explanation with a sketch showing the original signal spectrum and the resulting spectrum after aliasing.

i) (5 marks) Calculate the aliased frequency. Show your working. ii) (5 marks) Describe how the aliased frequency affects the reconstructed signal. What would an observer perceive? Include a sketch showing the frequency spectra of the original signal and the reconstructed (aliased) signal. Label the axes clearly.