ENGR202 2018 Q A3 (ENGR5001)

ENGR202 2018 Q A3Past paperOld spec ENGR2022:225 marks30 min

The following information is provided for this question:

Time Function	Laplace Transform
Unit step $u(t)$	$\dfrac{1}{s}$
$\exp(-\alpha t)$	$\dfrac{1}{s + \alpha}$

The frequency response of a Transfer Function $G(s)$ is given by:

$M = |G(s)|_{s = j\omega} \quad \text{and} \quad \phi = \mathrm{Arg}(G(s))|_{s = j\omega}$

Formulas you may need

Laplace pairs: unit step $u(t) \leftrightarrow \dfrac{1}{s}$ ; $e^{-\alpha t} \leftrightarrow \dfrac{1}{s + \alpha}$ (given in the question)
Frequency response: $M = |G(s)|_{s = j\omega}$ , $\phi = \mathrm{Arg}(G(s))|_{s = j\omega}$ (given in the question)
Closed loop with feedback element: $\dfrac{X}{V} = \dfrac{CH}{1 + CHF}$ ; characteristic equation $1 + CHF = 0$ (learn this)
Partial fractions: $\dfrac{A}{s(s + a)} = \dfrac{A/a}{s} - \dfrac{A/a}{s + a}$ (learn this)

(a)
Draw a block diagram of a generalized control system, in which $H(s)$ is the model, $C(s)$ is the control algorithm and $F(s)$ is a feedback element. On your diagram, label the set point $V(s)$ , control input $U(s)$ and output $X(s)$ .
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(b)
Using either standard algebra or the rules of block diagram manipulation, determine the Closed Loop Transfer Function and write down the associated characteristic equation.
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(c)
List and explain, with examples of your own choice, the four common objectives of control system design as considered in the course. Where relevant, support your answer with the block diagram from part (a) and the Transfer Function from part (b).
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(d(i))
Consider the model: $H(s) = \dfrac{5}{2s + 1}$ Using the Laplace transforms stated above, determine the unit step response of the model.
final value x(t -> infinity) =
exponent coefficient (x = 5(1 - e^(-a t)), a =) =
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(d(ii))
Determine the frequency response of the model i.e. the magnitude $M$ and phase $\phi$ .
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