ENGR202 2023 Q B2 (ENGR5001)

ENGR202 2023 Q B2Past paperOld spec ENGR2022:125 marks30 min

ENGR202 Summer 2023 Q B2[VALID] official answers

In medicine, during an operation, the patient is continually monitored and a combination of anaesthetic drugs are adjusted to ensure normal vital signs are maintained during loss of consciousness. There is some research into automatic anaesthetic drug delivery. However, this is a challenging control problem, due to inter-patient variability, poor understanding of drug mechanisms and difficulties in sensing.

Figure B2-1 shows a Proportional-Integral (PI) control system applied to one aspect of the anaesthetic drug delivery control problem. Here, $X(s)$ represents the measured concentration of a drug in the patient's blood and $U(s)$ is the applied drug dosage. The value of $X(s)$ subsequently impacts on the patient's level of consciousness.

Formulas you may need

Negative feedback rule: $\dfrac{G_1}{1 + G_1 G_2}$ ; series rule $G_1 G_2$ (learn this)
Hurwitz determinant for $s^3 + a_1 s^2 + a_2 s + a_3 = 0$ : $H_3 = \begin{vmatrix} a_1 & a_3 & 0 \\ 1 & a_2 & 0 \\ 0 & a_1 & a_3 \end{vmatrix}$ (given in the question)
Stability needs $\Delta_1 = a_1 > 0$ , $\Delta_2 = a_1 a_2 - a_3 > 0$ , $\Delta_3 = a_3\Delta_2 > 0$ (learn this)
PI controller: $\dfrac{k_1 s + k_2}{s} = k_1 + \dfrac{k_2}{s}$ (learn this)

(a)
In regard to Figure B2-1: (i) What does $V(s)$ represent? (ii) Is Figure B2-1 an example of open-loop or closed-loop control? Explain your answer. (iii) What is $k_2$ usually called?
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(b)
Using the rules of block diagram manipulation, derive the Transfer Function for the relationship between $V(s)$ and $X(s)$ , and hence show that the characteristic equation takes the following form: $s^3 + 2.6s^2 + (1.73 + 4.3k_1)s + 4.3k_2 = 0$
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(c)
Using the characteristic equation stated in part (b), derive and hence sketch the stability relationship between $k_1$ and $k_2$ . Hint: for a third order system, i.e. $s^3 + a_1 s^2 + a_2 s + a_3 = 0$ , use the following Hurwitz determinant: $H_3 = \begin{vmatrix} a_1 & a_3 & 0 \\ 1 & a_2 & 0 \\ 0 & a_1 & a_3 \end{vmatrix}$
k2 limit at k1 = 0 =
slope dk2/dk1 of boundary =
k1 intercept (k2 = 0) =
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(d)
Four standard objectives of control relate to stability, tracking, transient behaviour and disturbance rejection. Carefully define and explain each of these terms, referring to relevant mathematical concepts where possible. In addition, speculate on how these control objectives, and possibly other control objectives that you can think of, could apply in the case of the anaesthetic drug delivery control problem. Finally, comment on the limitations of the proposed PI control system in this context.
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