Subject: Re: Limit cycles for sticky valves
From: Tore Hagglund <tore@control.lth.se>
Date: Tue, 09 Jan 2007 11:53:03 +0100
To: sigurd.skogestad@chemeng.ntnu.no

Dear Sigurd,

Thank you for a very interesting and well written paper.
Many of the results you present are things that we have
beleived, but not seen proven.

You show that the static process gain is not important,
but the gain at -180 degrees phase. This is true, but
from a practical point of view, a large gain at the
interesting frequency is normally obtained by introducing
a high static gain, e.g. by using too large valves.

I have proposed an alternative to dithering some years
ago. see:
    T. Hagglund, A friction compensator for pneumatic control valves
    Journal of Process Control 12 (2002) 897-904

Best regards
Tore


Sigurd Skogestad wrote:

> Hello,
>
> A.C.B. Araujo and S. Skogestad,
> Limit Cycles with Imperfect Valves: Implications for Controllability of Processes with Large Gains,
> Ind.Eng.Chem.Res., Vol. 45 (26), 9024-9032 (2006).
>
> I think this paper may be of interest to you!
>
> We derive expressions for  the output magnitude (a) of the "natural" limit cycles that result with
> a quantized input (of magnitude q) in combination with a controller with integral action (i.e., with no average steady-state offset) .
> The quantized input may be a simple representation of a sticky valve and certainly represents an on/off valve.
> The main results are given by eq. (10) (general, but approximate since it is based on describing functions)
> and eq. (15) (exact for special case of first-order plus delay process with PI-controller where tauI=tau1):
>
> eq. (10):  a = (4/pi) * q * |G(j wL180)|
>
> eq. (15):  a = ...rather complicated expression which depends on the fraction of time f the input spends at each quantization level (where f  again depends on the magnitude of setpoint changes and disturbances)  ....
> but see plot in Figure 5 where we note that eq.(15) agrees with eq. (10) within 27% (for any f !).
>
>  From eq. (10) we see that output magnitude a is about 1.27 (=4/pi) times the input magnitude q
> multiplied by the gain |G| of the plant at frequency wL180.
>
> Remarks on eq. (10):
> 1. Eq. (10)  is no big surprise for anyone familiar with the Åstrøm-Hägglund relay tuning approach, but it still is very useful!
> 2. Note that. wL180 is the frequency where the phase lag in L=GK is -180 degrees (note: L is the plant PLUS controller).
>    In practice, wL180 is often close to wG180 since the phase of the controller K is often close to 0 at wG180.
> 3. The steady-state plant gain (|G| at w=0) is not important, and the original reason for writing the paper was to show this.
>
> In the paper, we also discuss how to reduce the amplitude a of the limit cycles by using "forced input cycling"
> to make w larger than the "natural" wL180 - this works for the common case where |G| decreases at high frequencies.
> One approach is to use a valve positioner, see the paper for details..
>
> Comments are welcome!
>
> -Sigurd Skogestad
>