ENGR202 2024 Q B2 (ENGR5001)

ENGR202 2024 Q B2Past paperOld spec ENGR2022:125 marks30 min

The generalized form of a second order model is stated below:

$\frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = Ku(t) \qquad \text{(B2-1)}$

Formulas you may need

Generalised second order form: $\ddot x + 2\zeta\omega_n\dot x + \omega_n^2 x = Ku(t)$ (given in the question)
Laplace with zero initial conditions: $\dot x \to sX$ , $\ddot x \to s^2 X$ (learn this)
Second order steady state gain $K/\omega_n^2$ (learn this)
Hurwitz for $a_0 s^4 + a_1 s^3 + a_2 s^2 + a_3 s + a_4$ : $\Delta_1 = a_1$ , $\Delta_2 = a_1 a_2 - a_0 a_3$ , $\Delta_3 = a_3\Delta_2 - a_1^2 a_4$ , $\Delta_4 = a_4\Delta_3$ (learn this)
Necessary conditions for stability: all coefficients exist and have the same sign (learn this)

(a)
For a second order engineering example of your choice (such as one of the examples from the lectures), develop a linear dynamic model from first principles, defining all the terms and stating all the assumptions made. Convert your answer into generalized second order form, and hence identify the forcing function, steady state gain, damping and natural frequency.
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(b)
Convert the generalised second order model (B2-1) into Transfer Function form, stating any assumptions made. Briefly discuss how Transfer Functions can be useful in the context of control system design.
[4]
(c)
Sketch the general form of a Bode diagram for a second order system, labelling all the axis (including frequency, phase and magnitude) and identifying key features.
[6]
(d)
For the following characteristic equation, what is (i) the inequality defining the necessary condition for stability and (ii) the value of the coefficient $\alpha$ such that the system is marginally stable: $s^4 + 8s^3 + 2s^2 + 3s + \alpha = 0$ .
alpha for marginal stability =
[2]
(e)
In the context of a practical example of your choice, briefly summarise the relative advantages of PI control and its potential limitations.
[6]