What are the name of the signals $V(s)$, $U(s)$ and $X(s)$? Using Figure A2 as an example, explain the difference between open and closed-loop control. Give a practical example of both open and closed-loop control.

Assume the following Transfer Function for this part of the question: $H(s) = \frac{2.5}{(s^2 + 3s + 2.26)(s + 2)}$ Determine the poles of $H(s)$ and sketch their position on the $s$-plane. What is the stability of $H(s)$. Identify the dominant poles and sketch the likely form of the time response mode. Note: this part of the question concerns $H(s)$ only.

Using either standard algebra or the rules of block diagram manipulation, determine the closed-loop Transfer Function for Figure A2 i.e. the relationship between $V(s)$ and $X(s)$. Next, an engineer implements a simple proportional control system based on $C(s) = K$ and $F(s) = 1$ where $K$ is a scalar control gain. In this case, show how for large values of $K$ the error $E(s) = V(s) - X(s)$ approaches zero. What are the disadvantages of this control system?

Still using Figure A2, now assume the following Transfer Functions: $H(s) = \frac{1}{s + 10} \qquad C(s) = K_1 + \frac{K_2}{s} \qquad F(s) = 1$ What is the name of this controller? Using these definitions of $H(s)$, $C(s)$ and $F(s)$, determine the closed-loop characteristic equation and the associated Hurwitz determinant. Use your answer to find the minimum values of $K_1$ and $K_2$ that ensure stability of the closed-loop system. Note: given $a_0 s^2 + a_1 s + a_2 = 0$ then $H_2 = \begin{vmatrix} a_1 & 0 \\ a_0 & a_2 \end{vmatrix}$