ENGR202 2025 Q2 (ENGR5001)

ENGR202 2025 Q2Past paperOld spec ENGR2022:150 marks60 min

ENGR202 Summer 2025 Q2[VALID]

This question concerns the heating system in a human occupied office building. The relationship between the indoor temperature $X(s)$ and the applied voltage to a heater $U(s)$ is approximated by Equation (2-1):

$X(s) = \frac{s + b_1}{s^2 + a_1 s + a_2}\,U(s) \qquad \text{Equation (2-1)}$

Figure Q2 shows a control system with two negative feedback loops.

Figure Q2: Control system with two negative feedback loops.

Formulas you may need

Steady state gain $G(0)$ (set $s = 0$ ) (learn this)
Poles: roots of the denominator; zeros: roots of the numerator; stable if all poles have negative real parts (learn this)
$s^2 + 2\zeta\omega_n s + \omega_n^2$ : $\omega_n = \sqrt{a_2}$ , $\zeta = a_1/(2\omega_n)$ (learn this)
Negative feedback rule $\dfrac{G_1}{1 + G_1 G_2}$ ; series rule $G_1 G_2$ (learn this)

(a(i))
The following questions are about the model given by Equation (2-1). Equation (2-1) was obtained by considering heat transfer and energy balance equations (details not important). Use this example to briefly explain the difference between data-based and mechanistic models. Suggest potential limitations of Equation (2-1), for example in its ability to represent real world temperature data and/or if it is subsequently used to design a control system.
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(a(ii))
State an algebraic equation for determining the steady state gain of Equation (2-1). Use your answer to determine the steady state indoor temperature for $u(t > 0) = 5$ V, assuming initial conditions $u(0) = 0$ , $x(0) = 20^\circ\mathrm{C}$ , and coefficients $a_1 = 6$ , $a_2 = 8$ and $b_1 = 10$ .
steady state gain =degC/V
steady state temperature =degC
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(a(iii))
Again with $a_1 = 6$ , $a_2 = 8$ and $b_1 = 10$ , what are the pole(s) and zero(s) of Equation (2-1). Plot and label the pole(s) on the complex $s$ -plane. State the stability condition, explaining the reason for your answer.
zero =
dominant pole =
other pole =
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(a(iv))
Still using $a_1 = 6$ , $a_2 = 8$ and $b_1 = 10$ , determine the natural frequency and damping ratio of the model given by Equation (2-1). Comment on what the damping ratio tells us about the dynamics of this model.
omega_n =rad/s
zeta =
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(b(i))
The following questions are about the control system shown in Figure Q2. What is $V(s)$ usually called and what does it represent? Briefly suggest how it might be chosen for the temperature control application.
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(b(ii))
Develop the closed-loop Transfer Function for Figure Q2 and hence show that the closed-loop characteristic equation is as follows: $(1 + K_2)s^2 + (a_1 + K_1 + K_2 b_1)s + a_2 + K_1 b_1 = 0 \qquad \text{Equation (2-2)}$ In addition, calculate the steady state gain of the closed-loop Transfer Function and comment on your answer.
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(b(iii))
Suggest and briefly describe two methods for designing the above controller, i.e., methods to determine suitable values of $K_1$ and $K_2$ in Figure Q2.
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(b(iv))
Figure Q2 represents an initial attempt at designing a controller for this problem. A more experienced control engineer states that Figure Q2 should be revised to include a more suitable control algorithm. Speculate on the reasons for their statement and suggest a more appropriate algorithm. Hints: your answer might refer to the model dynamics from part (a); the closed loop Transfer Function from part (b); and practical aspects of the temperature control application.
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