ENGR202 2021 Q1 (ENGR5001)

ENGR202 2021 Q1Past paperOld spec ENGR2022:219 marks23 min

Consider the following Transfer Function:

$X(s) = \frac{3.5s + 1}{(s + 1.2)(s + 2)}\,U(s)$

where $X(s)$ and $U(s)$ are the output and input, respectively.

Formulas you may need

Poles: roots of the characteristic equation (denominator $= 0$ ); zeros: roots of the numerator (learn this)
Stable if ALL poles have negative real parts; dominant pole = the one closest to the imaginary axis (learn this)
Steady state gain: $G(0)$ , i.e. set $s = 0$ in the Transfer Function (learn this)
Second order characteristic equation: $s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$ (learn this)
PI control: $U(s) = \left(k_P + \dfrac{k_I}{s}\right)\left(V(s) - X(s)\right)$ (learn this)
Negative feedback rule: $\dfrac{G_1}{1 + G_1 G_2}$ (learn this)

(a)
What are the pole(s) and zero(s) of this system? Plot the poles on the complex plane and label the dominant pole. Giving the reason for your answer, what is the stability? Determine the steady state gain. For the case that the input is a time-invariant value of 5, what is the value of the output at steady state.
zero =
dominant pole =
other pole =
steady state gain =
x at steady state =
[7]
(b)
Determine the damping ratio of this system. Comment on the physical meaning of the damping ratio and relate your answer to the poles determined in part (a).
omega_n =rad/s
zeta =
[4]
(c)
Draw the block diagram of a standard proportional-integral (PI) control system, as applied to the Transfer Function above. Label all the blocks and signals.
[2]
(d)
Showing your working, derive the closed-loop Transfer Function for the controller in Part (c). What is the order of the control system? What is the significance of this result? Determine the steady state gain and comment on the significance of your answer.
closed-loop order =
closed-loop steady state gain =
[6]