ENGR202 2017 Q A3 (ENGR5001)

ENGR202 2017 Q A3Past paperOld spec ENGR2022:125 marks30 min

Consider the following model for the heating system in a room,

$X(s) = \frac{b_0}{s^2 + a_1 s + a_2}\,U(s) \qquad \text{(3a)}$

where $X(s)$ represents the temperature and $U(s)$ is the heater input. A Proportional-Velocity (PV) control algorithm takes the following form,

$U(s) = k_p\left(R(s) - X(s)\right) - k_v sX(s) \qquad \text{(3b)}$

where $R(s)$ is the command input and $(k_p, k_v)$ are control gains.

Formulas you may need

Stability shortcuts: all coefficients must exist and have the same sign; $a_n = 0$ gives a pole at the origin; for a second order equation, all coefficients positive is necessary AND sufficient (learn this)
Generalised second order characteristic equation $s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$ (given in the question)
Negative feedback rule $\dfrac{G_1}{1 + G_1 G_2}$ (learn this)
Gain margin and phase margin read from the Bode diagram of the open-loop $C(s)H(s)$ (learn this)

(a)
Write down the open-loop characteristic equation for the heating system model (3a) i.e. before the application of PV control. Without calculating the poles, and without using the Hurwitz determinant, state the stability condition for each of the following sets of coefficients. Give the reason for your answer in each case. Case 1: $a_1 = -1.2$ , $a_2 = 3$ Case 2: $a_1 = -1.2$ , $a_2 = 0$ Case 3: $a_1 = 0$ , $a_2 = 2.5$ Case 4: $a_1 = 1.2$ , $a_2 = -3$
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(b)
Draw a block diagram of the heating control system i.e. including the model (3a) and PV control algorithm (3b). Using either standard algebra or the rules of block diagram manipulation, derive the closed-loop Transfer Function. What is the steady state gain of the closed-loop Transfer Function?
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(c)
Using the room temperature control problem as an example, describe four typical objectives of control. Use the PV control system and associated closed-loop Transfer Function to support your answer where possible e.g. explain how this control system addresses or fails to address the temperature control objectives. For information, the characteristic equation of a generalised 2nd order system is defined as follows: $s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$ .
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(d)
Briefly explain how Bode diagrams can be used to analyse the stability of closed-loop control systems. No numerical or algebraic analysis is required to answer this question.
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