ENGR202 2019 Q B2 (ENGR5001)

ENGR202 2019 Q B2Past paperOld spec ENGR2022:125 marks30 min

Consider the following Transfer Function model derived for an air conditioning unit:

$X(s) = \frac{6.7}{(s + 2)(s + 3)}\,U(s)$

where $X(s)$ and $U(s)$ represent the outlet and inlet temperature, respectively.

Formulas you may need

Poles, stability, steady state gain $G(0)$ (learn this)
Frequency response: $M = |G(s)|_{s = j\omega}$ , $\varphi = \mathrm{Arg}(G(s))|_{s = j\omega}$ (given in the question)
$|j\omega + a| = \sqrt{\omega^2 + a^2}$ , $\mathrm{Arg}(j\omega + a) = \tan^{-1}(\omega/a)$ (learn this)
Nyquist diagram: $\mathrm{Im}\,G(j\omega)$ against $\mathrm{Re}\,G(j\omega)$ as $\omega$ goes from 0 to $\infty$ ; critical point $-1 + 0j$ (learn this)
Gain margin and phase margin (learn this)

(a)
Write down the characteristic equation of the model. Determine the poles, stability condition and steady state gain of the model. Plot the position of the poles on the complex s-plane. For a unit step input of $20^\circ$ C, what is the steady state outlet temperature? Comment on the physical interpretation of all these results.
dominant pole =
other pole =
steady state gain =
steady state outlet temperature =degC
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(b)
Briefly explain what is meant by the frequency response of a system.
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(c)
Using standard notation from the lectures, and noting that $M = |G(s)|_{s = j\omega}$ and $\varphi = \mathrm{Arg}(G(s))|_{s = j\omega}$ , determine the frequency response of the air conditioning unit Transfer Function model i.e. derive expressions that show how $M$ and $\varphi$ vary with $\omega$ .
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(d)
Using a carefully annotated sketch, including axis labels, explain how a Nyquist Diagram can be used to determine the stability of a system. Your sketch should show appropriate traces for (i) a typical stable and (ii) unstable system. Your written answer should briefly explain what is plotted in the Nyquist Diagram. Hint: the sketch is an approximate graph drawn in your answer book and does not require graph paper nor use of the equations derived in part (c).
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(e)
Explain how the frequency response can be used to analyse and set design criteria for the stability of a closed-loop control system.
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