Fig$\mathrm{.}1$: Closed loop temperature control of a Heat Exchanger

Tasks:

Section$-$A: Dynamic performance and control system design of the heat exchanger

The transfer functions for the heat exchanger, the control valve and the measurement element are; $G_p(s)=\frac{40}{30s+1},G_v(s)=\frac{2}{4s+1}$ and $H(s)=1.[$Note: Each student should choose thier own value for $$ from Table$-1].$

$($i$)$ Assume that the controller is a proportional controller with a gain, $G_c(s)=K_c=1$

a$.$ Plot the position of closed loop poles on $\text{'}$s$-$plane' and show all parameters $($damped & undamped frequency and the angle $$ of the pole from the horizontal $($real$)$ axis$).$ Show all steps of your calculations.

b$.$ If the same undamped natural frequency ${}_n$ in part $($a$)$ is required, redesign the control system by relocating $($replotting$)$ the poles in splane so that the temperature response of a heat exchanger has the maximum percentage overshot of $y(\%).$ Show all steps of your calculations. $[$Note: Each student should choose his$/$her own value for $y(\%)$ from Table$-1].$

$($ii$)$ If the controller in section$-$ A and Fig. $1$ is now replaced by a Proportional Derivative $($PD$)$ controller whose transfer function is given by;

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$G_c(s)=K_c(0\mathrm{.}2s+1),$ sketch, by hand, the root locus of the control system when $K_c$ changes from $0$ to $.$ Use the same transfer functions of $G_p(s)G_v(s)$ and $H(s)$ as in section$-$ A$.$ Show all steps of your calculations including the rules of root locus plot.

$($iii$)$ Use the same transfer functions in part $A($ii$).$ It is required to design the temperature control system of a heat exchanger using root locus technique so that the temperature response can satisfy all of the following dynamic performance;

A percentage overshoot of less than $15\%$ and,

A settling time of less than $5$ s $($within $10\%$ settling limit$)$ and,

A damped frequency of less than $1\mathrm{.}8.$

Use a MATLAB $($with a help of analytical calculations$)$ to find the value $($or the range$)$ of $K_c$ on root locus which can satisfy all design requirements mentioned above.

$($iv$)$ Verify your answer in part $($iii$)$ either by designing a Simulink control model for the heat exchanger given in Fig. $1$ or by using 'stepinfo' command in MATLAB. Select any value of $K_c$ within the range of $K_c$ obtained from part $($iii$).$ Assume a unit step input.