A process that can be modeled as a time delay $($ gain $=1)$ is controlled using a proportional feedback controller. The control valve and measurement device have negligible dynamics and steady-state gains of $K_{v}=0.5$ and $K_{m}=1$ , respectively. After a small set-point change is made, a sustained oscillation occurs, which has a period of $10 min$ .

(a) What controller gain is being used? Explain.

(b) How large is the time delay?

Question

A process that can be modeled as a time delay $($ gain $=1)$ is controlled using a proportional feedback controller. The control valve and measurement device have negligible dynamics and steady-state gains of $K_{v}=0.5$ and $K_{m}=1$ , respectively. After a small set-point change is made, a sustained oscillation occurs, which has a period of $10 min$ .

(a) What controller gain is being used? Explain.

(b) How large is the time delay?

(a) What controller gain is being used? Explain.

(b) How large is the time delay?

Accepted Answer

Using the Bode plot, at a phase angle of [latex]-180^{\circ}[/latex], we require that [latex]K_{c} K_{V} K_{p} K_{m}<1[/latex]

[latex]G_{p}(s)=e^{- heta_{s}} \quad G_{V}=0.5 \quad G_{m}=1.0[/latex]

The gain of [latex]G_{p}=1.0[/latex] for all [latex]\omega[/latex].

At the critical frequency [latex]\left(\omega_{c} ight)[/latex], a sine wave is formed with period

[latex]P_{m}=10 \min s=\frac{2 \pi}{\omega_{c}}, ext { so } \quad \omega_{c}=\frac{2 \pi}{10} \frac{ rad }{ min }=0.628 \frac{ rad }{ min }[/latex]

(a) The critical gain is easily found from

[latex]K_{c} K_{V} K_{p} K_{m}=1 ext { at } \omega=\omega_{c}[/latex]

[latex]K_{c u}(0.5)(1)(1)=1, ext { or } K_{c u}=2.0[/latex]

(b) The phase angle of [latex]G_{c} G=G_{c u}=[/latex] phase angle of [latex]e^{- heta_{s}}[/latex], or [latex]\Phi=-\omega heta( rad )[/latex] (Eq. 14-33)

[latex]\phi=-\omega heta[/latex] (14-33)

when [latex]\Phi=-180^{\circ}=-\pi=-\omega_{c} heta[/latex]

Because [latex]\omega_{c}=\frac{2 \pi}{10} \frac{ rad }{ min }[/latex] then [latex] heta=\frac{10}{2}=5 min[/latex].

Question 14.20: A process that can be modeled as a time delay (gain = 1) is ...

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