ENGR202 2017 Q A1 (ENGR5001)

ENGR202 2017 Q A1Past paperOld spec ENGR2022:125 marks30 min

ENGR202 Summer 2017 Q A1[VALID] official answers

The relationship between the pressure $X(s)$ and the flow rate control input $U(s)$ in an oil fractionator is described by the following Transfer Function model:

$X(s) = \frac{b_1}{s + a_1}\,U(s) \qquad \text{(1a)}$

Using standard notation, generalised first and second order differential equations are:

$\tau\frac{dx}{dt} + x = Ku(t) \quad \text{and} \quad \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = Ku(t) \qquad \text{(1b)}$

Formulas you may need

Generalised forms: $\tau\dfrac{dx}{dt} + x = Ku(t)$ and $\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t)$ (given in the question)
Frequency response: $M = |G(s)|_{s = j\omega}$ , $\phi = \mathrm{Arg}(G(s))|_{s = j\omega}$ (given in the question)
Laplace with zero initial conditions: $\dfrac{dx}{dt} \to sX(s)$ , $\dfrac{d^2x}{dt^2} \to s^2X(s)$ (learn this)
Steady state gain: set $s = 0$ (learn this)
Pole placement: match the closed-loop characteristic equation to $s^2 + 2\zeta\omega_n s + \omega_n^2$ (learn this)

(a)
Showing your working and indicating any assumptions made, develop the Transfer Functions forms of the generalized first and second order differential equations (1b), expressed in terms of $\tau$ , $K$ , $\omega_n$ and $\zeta$ .
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(b)
Assume $a_1 = 5$ and $b_1 = 2$ for this part of the question. Determine the pole and stability condition of the oil fractionator model (1a). Determine the time constant and steady state gain. For a flow rate $u(t > 0) = 5$ , what is the steady state pressure.
pole =
time constant =
steady state gain =
steady state pressure =
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(c)
The steady state response of the oil fractionator model (1a) for $u(t) = \sin(\omega t)$ when $t > 0$ , is $M\sin(\omega t + \phi)$ . Determine relationships for $M$ and $\phi$ as a function of frequency $\omega$ and the model parameters $a_1$ and $b_1$ . Note that for a Transfer Function $G(s)$ , then: $M = |G(s)|_{s = j\omega}$ and $\phi = \mathrm{Arg}(G(s))|_{s = j\omega}$
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(d)
An automatic control system is used to regulate the pressure in the oil fractionator model. The closed-loop Transfer Function is shown below, where $D(s)$ is the set point and $(k_1, k_2)$ are gains, $X(s) = \frac{k_1 b_1 s + k_2 b_1}{s^2 + (a_1 + k_1 b_1)s + k_2 b_1}\,D(s)$ What is the stready state gain of the closed-loop Transfer Function? Comment on the significance of your answer. Assuming that $a_1 = 5$ and $b_1 = 2$ , design a control system (i.e. determine values for the control gains) such that the closed-loop system has a damping ratio $\zeta = 1.2$ and natural frequency $\omega_n = 0.1$ .
closed-loop steady state gain =
k_1 =
k_2 =
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