EL ATED IDENTIF~ STATE

hart, Dept. of Chem. T ty, Lubbock, TX 79 121

mad

Qeveloped herein is statistically-based me periods.

Process conml applicadons for steady-state iden~ficadon indude data validity checks, fault detecdon, data re- con~~l~tion, and adj ent of steadystate models. There are several exlrdng methods for identification. A direct approach is to perform a linear regression over a data window and then to perform a t- test on the regresskm slope. hother method8 uses an F- test type of statistic, the rad0 of variances as measured on the same set of data by two different methods. Both methods have several undesirable features, including necessary user experdse and mmputadonal load.

t

Consider that there is a "me" value of a process variable, but that the measured value contains noise: an ~ndependentidendcally disuibuted (lid) perturbadon. The

a ratio of variances of the estimated by two separate

methods. As illustrated in Figure 1, the numerator variance is estimated from deviadons between iladM$ual data and a filtered value. The de~ominator variance is estimated from de~a~ons between successive data.

If the measured proc value is mly at a steadystate, then the two estimates will be equiv~ent and, i~ea~ly, the

numerator to be large, and the variance greater than unity. However, random even

me steady-state. Therefore, the deviation which triggers a "not at steadystate" have a value which is greater than some crid

gins by ~lcuiadng an average value of the noisy process variable by the conventiona~ first-order Biter. This requires I and is c~mputatlonally fa& In algebraic notadon

(1)

where i is the dme increment counter, and i would represent the current sampling. Then estlmate the mean square deviation, a:, of the current measurement x, from the previous filtered value xr,, . ~ m f ~ e l y this would be

x 4 = x,xi + (1-1,) xkl

The previous filtered value xr,, is used instead of xr, In Equadon (2) to prevent autocorreladon between x and x, This artifice is neither necessary nor detrimental, but it simplifies the subsequent analysis and allows one to estimate a: from the following:

U; = cl; + cl4 (3)

Further, for the exponendally weighted moving average, when (3) are independent

a;f= a, o r / (2-1,) (4)

Then equations (3) and (4) yield

6; = (2-a,) a ; / 2 (5)

However, equation (2) is computadonally expensive; so, use a filtered variance instead of a traditional average.

(a) 6; 4 = q"f-xr,,>2 + (1-a2)e:+1 and then

e:, = (2-q 2 (9

This was method 1 to obtain b:, the estimate of the process variable variance at the I* sampling. Now use method 2. Define

In the limit of large N,

However, equation (8) is computatlonally expensive; so, use a filtered approach.

(10)

Then

U; = p2/2 (9)

Pi = A3(x,-xi-,Y + ("-A.JP;,

er,= O i l 2 (11)

was the second method to estimate the process

the ratio of a:, as determined by equation (7) to n (119

variabl~ variance at the i* sampling.

"/ = (2-1,) I P; (1%

~ u m ~ a ~ ~ ~ n g , use (7) to caiculate ai,. Then use ( 1 ) to calculate x, . m e n use ( 1 I ) to calculate ri:, . Then use

to a h l a t e r,, Each are direct, one-step, low ge, low operation calculations.

ity Fundon and Critical Values

r-statistic is dimensionless and Independent of the Because it is a ratio of estimated

independent of the process variance, However the probability density

the rate with which the value of r, its current value depends on the

Monte Carlo experiments can df(r). Recommended values are

=.02. The shape is characteristic of the The critical value of r for which there is

nce of a larger value when the process is rap5, is 1 .Q for the recommended A values.

Acc~rd~ngly, if a measured value of r k greater than 1.6 en one can claim, at a 95% confidence level, that the

k "not at steady-state."

o ~ ~ i a t i o n in the n o b process behavior at "steady- shifts pdf(r) to the right. At some point an r w o ~ l ~ "see" the autocorrelation and decide that

In the authors' ~ x p e ~ e n c e this visual determination coincides with an r

found that changes in " diameter pilot-scale 20. Consequently a the sritlcal value, and steady-state identifier

accept" or "reject" the y-state hypothesis. Satisfyingly, the identifier agrees

v~sual inspection of the actual Row rate. Note that is^ ~ m p ~ ~ ~ ~ e changes as the Row rate changes.

C o ~ ~ ~ ~ o n

e ~ r o c e ~ k not at steadystate.

Illy-based computationally inexpensive developed for automated identification

The author appreciates discussions with Dr. Karen Yin pt of Ch.E., U. Minn, Duluth) and with Dr. Kamal

anda (Dept of Math, Texas Tech Univ.) and e x p ~ o r a t i o ~ by Dr. Soundar ~mchandran, Venkat

and Songling Cao (Depe of ChE, Texas Tech ~ n ~ . ) .

1 ) Dover Publications, Inc. New York, NY (1955).

Crow, Davis, and Maxfield, Statistics Manual,

4066