Hot and cold liquids are mixed at the junction of two pipes. The temperature of the resulting mixture is to be controlled using a control valve on the hot stream. The dynamics of the mixing process, control valve, and temperature sensor/transmitter are negligible and the sensor-transmitter gain is

Question

Hot and cold liquids are mixed at the junction of two pipes. The temperature of the resulting mixture is to be controlled using a control valve on the hot stream. The dynamics of the mixing process, control valve, and temperature sensor/transmitter are negligible and the sensortransmitter gain is [latex]6 mA / mA[/latex]. Because the temperature sensor is located well downstream of the junction, an [latex]8 s[/latex] time delay occurs. There are no heat losses/gains for the downstream pipe.

(a) Draw a block diagram for the closed-loop system.

(b) Determine the Ziegler-Nichols settings (continuous cycling method) for both PI and PID controllers.

(c) For each controller, simulate the closed-loop responses for a unit step change in set point.

(d) Does the addition of derivative control action provide a significant improvement? Justify your answer.

Accepted Answer

a) Schematic diagram:

Block diagram:

b) [latex]\quad G_{v} G_{p} G_{m}=K_{m}=6 ma / ma[/latex]

[latex]\begin{aligned}G_{T L} &= e ^{-8 s} \G_{O L} &=G_{v} G_{p} G_{m} G_{T L}=6 e ^{-8 s}\end{aligned}[/latex]

If [latex]G_{O L}=6 e ^{-8 s}[/latex]

[latex]\begin{aligned}&\left|G_{O L}(j \omega) ight|=6 \&\angle G_{O L}(j \omega)=-8 \omega[ rad ]\end{aligned}[/latex]

Find [latex]\omega_{c}:[/latex] The critical frequency corresponds to an open-loop phase angle of [latex]-180^{\circ}[/latex] phase angle [latex]=-\pi[/latex] radians

[latex]-8 \omega_{c}=-\pi \quad ext { or } \quad \omega_{c}=\pi / 8 rad / s[/latex]

Find [latex]P_{u}: \quad P_{u}=\frac{2 \pi}{\omega_{c}}=\frac{2 \pi}{\pi / 8}=16 s[/latex]

Find [latex]K_{c u}: K_{c u}=\frac{1}{\left|G_{p}\left(j \omega_{c} ight) ight|}=\frac{1}{6}=0.167[/latex]

[ Note that for this unusual process, the process AR is independent of frequency]

Ziegler-Nichols [latex]1 / 4[/latex] decay ratio settings:

PI controller:

[latex]\begin{aligned}&K_{c}=0.45 K_{c u}=(0.45)(0.167)=0.075 \& au_{I}=P_{u} / 1.2=16 s / 1.2=13.33 s\end{aligned}[/latex]

PID controller:

[latex]\begin{aligned}&K_{c}=0.6 K_{c u}=(0.6)(0.167)=0.100 \& au_{I}=P_{u} / 2=16 / 2=8 s\end{aligned}[/latex]

[latex] au_{D}=P_{u} / 8=16 / 8=2 s[/latex]

c)

Note: The MATLAB version of PID control uses the following controller settings: [latex]k i=K_{c} / au_{I}[/latex] and [latex]k d=K_{c} au_{D}[/latex]

d) Derivative control action improves the closed-loop response by reducing the settling time, at the expense of a more oscillatory response.

Question 14.25: Hot and cold liquids are mixed at the junction of two pipes....

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