# Index of /skoge/publications/2006/araujo_iecr_large-gains

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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..