Data reliability is of a basic importance for process diagnosis, identification and control. Measurements containing undetected errors mean poor process control.
Computers
Pergamon
FAULT
them.
Engng Vol. Elsevier
18,
No. Science
10, pp. 1001-1004, 1994 Ltd. All rights reserved
0098-1354(94)Eoo28-L
DETECTION OF MULTIPLE LEAKS IN LINEAR STEADY J.-Y.
C.R.A.N.,
KELLER,
L.A.R.A.L.,
(Receiued
C.N.R.S.
BIASES OR PROCESS STATE SYSTEMS
M. DAROUACH and G. KRZAKALA U.A. 821, Universitd de Nancy 1, Route de Romain, 54400 Longwy, France
26 March 1992; final revision received received for publication 24 February
14 February 1994)
1994;
new method for detecting, identifying, and estimating gross errors in steady state processes is described in this oaDer. This method improves the GLR test and can be applied to al1 types of gross errors including lea-ks-and biases. Abstract-A
1.
INTRODUCTION
Data reliability is of a basic importance for process diagnosis, identification and control. Measurements containing undetected errors mean poor process control. Two kinds of errors can be identified in process data. The first is very small errors arising, for example, from normal process fluctuation or random variation in the operation of an instrument. In direct contrast to small errors are the very large errors introduced by the complete malfunction of an instrument or very strong instrument bias. Gross errors may occur occasionally in any of the instruments but the number of such errors is generally small compared with the total number of instruments. The easiest test to improve the consistency of data is to measure the same variable with several sensors, and to use the majority-decision principle. It is more convenient to use the analytical redundancy of systems given by balance equations to detect the bias and correct the measurements. Many papers have appeared in the chemical engineering and mineral processing fields and several statistical tests for detecting, identifying and estimating gross errors in linear systems have been developed: Constraint test (CT) (Mah er al., 1976), Maximum power measurement test (MP) (Mah and Tamhane. 1982) and the Generalized Likelihood Ratio test (GLR) (Narasimhan and Mah, 1987) which gives a method for identifying leaks and biases in a steady state process strategy (Romagnoli serial and by a Stephanopoulos, 1981; Serth and Heenan, 1986, Rosenberg et al., 1987). In this paper, the case of multiple measurement biases or process leaks is treated by improving the serial
compensation
strategy
In the proposed
of the
existing
algor-
strategy, we take into account that the analytical redundancy decreases for ithms.
each treatment of a new gross error. A comparative example is given to illustrate the efficiency of the method. 2. PROBLEMSTATEmNT Faced with one measurement bias and one pure energy or component flow losses, a faulty linear steady state system can be described by: x=x*+e+e,b Ax*
(1)
=mib
(2)
where b is the gross error magnitude. The vector e, have unity at the ith position of measurement bias and zero elsewhere. The vector mj have unity at the jth position of process leak and zero elsewhere. A is an (n, u) full TOW rank constraint matrix describing the linear or linearized mass and energy conservation constraints. X* is a u vector of true state variables, x a u vector of measurements and E a u vector of measurement errors normally distributed with zero mean and known variance matrix Q. As Narasimhan and Mah (1987), the set of fault directions describing all the hypotheses for gross error detection is given by: Z(p)=(A,,mj:i=l..
.o,j=l..
.n}
where Aj is the fault direction for a bias in measurement i and mj the fault direction for a process leak in node j_ p =n + u represent the number of streams and nodes of the process network. Suppose that an unknown subset D(s) of Z(p) describes the failures effectively present on the system. Given the measurement X, the problem is to estimate s (the number of gross errors) and to determine the fault directions of D(s) (the sources of gross errors).
1001
loo2
J.-Y. KELLER
3. MULTIPLE
GROSS ERRORS
DETECTION
Sk= v-‘-
ALGORITHM
To obtain directly the recursive algorithm by induction, iet us suppose that k gross errors are already detected and localized and k
(7)
v-‘FkE;‘F;v-‘.
Ek = F;V-‘F, and
Sj= (flv-7;)-‘j-jV-‘(r-
F&)_
03)
Substituting (7) into (8) yields 6,=(flv-%_)-‘flv-‘(~-F~(F:V-‘F~)-’ x F:V-‘(r--fi6J) and after some manipulations we obtain the estimation of the new gross error:
(9)
T$ = [var(6j)-“Z6j]* or
Ti _ (fgke *-
ffs,fi .
var(6j) = (flgtifi)-’
(10)
(12)
At step k, the calculation of T& is performed for every vector 6. of Z@ - k) and the test statistic Tk is therefore obtained as: Tk= sup Ti
(13)
Ti has a central chi-square distribution with one degree of freedom if no error is present and a process leak or a bias can be detected by selecting a significant level. Since the Pi are not independent, the distribution of Tk cannot be obtained. However, we can use arguments similar to those used by Mah and Tamhane (1982) and choose as critical value $. ,_+ the upper 1 -p quantile of the chi-square distribution. For a given level of significance CL,the value of /3is computed as /+l-(l-o)‘/P-k
(14)
where p-k is the number of gross errors hypothesized (cardinality of the set Z(p - k)). If T*> xzI. t_B, the gross error j is present in the system and an estimation of D(s) is given by: fi(k+l)=@(k)fi}.
(15)
The fault directions matrix F,,, is obtained from the following recursive expression: (16)
We repeat the algorithm described above until no further gross errors are detected. At the initial step (k =O), the test statistic T, applied with the value p = 1 - (1 - a)“” corresponds to the test statistic used by Narasimhan and Mah (1987). The necessary condition of the no singularity of matrix Ek is given by k Grz. It follows that the maximum number of gross errors which can be treated is equal to the number n of constraints. If matrix &+, = FT+, V-‘Fk+, is partitioned as fo11ows
F:Vfi flV& >
where
(11)
We propose to obtain the variable Pi used in the GLR test from the relation:
Fk+, = [FklAl-
where
and
etal.
(17)
we can avoid the increase of computational requirements by computing EL:, with the help of the partitioned matrix inverse lemma (see Appendix).
Fault detection in linear steady state systems
1003
-the probability of non detection pnd(j) defined as the ratio between the nondetections of j gross errors and the total number of simulations containing j gross errors.
Fig. 1. Process network.
We test the significance only of the new gross error but it is possible that the previously significant gross error could become statistically insignificant speciaIiy when the network of the system is strongly connected. Then, a more correct procedure is to apply the test to ail postulated combinations of gross errors. That is, one at a time, two at a time, and so on. The drawback of this combinatorial approach is the escalated dimensionaiity of the problem which leads to an increasing of computational requirements. 4. PERFORMANCE
EVALUATION
Consider the system described by Fig. 1 formed by n nodes and v streams. We propose to compare our algorithm with the algorithm proposed by Narasimhan and Mah (1987). To do this, several positions of gross errors are generated (first column of Table 1). The two aigorithm are repeated 1000 times with different sequences of random errors. The statistical variables used for the comparison are: -the probability of good localization pbl(j) defined as the ratio between the number of good localizations of j gross errors and the total number of simulations containing j gross errors, -the probability of false alarms pfd(j) defined as the ratio between the number of false detections of j gross errors and the totat number of simuiations containing j gross errors, Table 1. Statistical results of the comparative study Location 1 I,5 1,5,10 2 2.3 2,3.4 11 11.12 11.12.13
j=l j=Z j=3 j=l j=2 j=3 j=l j=2 j=3
pbl(j) 1 0.98 0.73 1 0.97 0.83 1 0.56 0.55
pfd(j) 0 0.0-l 0.03 0 0.002 0.03 0 0.03 0.08
pnd(j)
pbl(j)
0 0.11 0.2 0 0.03 0.09 0 0.09 0.M
1 0.98 0.55 1 0.97 0.53 1 0.5 0.5
pfdtj)
pad(j)
0 0.02 0.06 00.002
0 0.18 0.03 0 0.004 0.06 0 0.1 0.18
0.006 EL3 0.02
We obtain the following results. The first column of Table 1 gives the location and the number of simulated gross errors. Columns 2, 3 and 4 give the statistical results obtained by the new algorithm and the last three columns the results from the algorithm developed by Narasimhan. Our algorithm gives better results especially when the number of gross errors is equal to 2 and 3. The serial compensation strategy given by Narasimhan is improved due to the fact that the statistical test is computed by taking into account that the analytical redundancy decreases with the treatment gross errors (rank(&) = n -k). So, if ail variables are measured directly, the maximum number of gross errors which can be treated is equal to the number of constraints. The proposed algorithm can be applied to general steady state systems, if the unmeasured variables are eliminated by a projection matrix (Crowe et al., 1983) and if the indirectly measured variables are redefined (Narasimhan, 1990).
5. CONCLUSION
We have presented a new algorithm for the treatment of several measurement biases or process leaks in linear steady state systems. The proposed recursive algorithm improves the GLR test in the case of several gross errors appearing simultaneously.
NOMENCLATURE
A = constraint matrix Ai = ith column of matrix A B = vector of gross error estimation at step k l$= optimal gross error estimation of an hypothetical gross error j D(S) = set of s fault directions describing the failures effectively present on the system e, = unit vector with unity in row i n = number of constraints Q = covariance matrix of measurement errors rank (C) = rank of matrix C r = constraint residuals Pi = GLR test statistic for gross error j knowing that k gross errors are already detected var(r) = covariance matrix of r V= covariance matrix of constraint residuals o = number of variables x = vector of measurements x’ = vector of true measurements Z(p) = set of p hypothetical fault directions
J.-Y. KELLER etal.
1004 Greek characters a = level of significance j3 = adjusted level of significance E = vector of measurement errors q5= least square criterion
Romagnoli J. A. and G. Stephanopoulos, Rectification of process measurement data in the presence of gross errors. Chem. Engng Sci. 36, 1849 (1981). Rosenberg J., R. S. H. Mah and C. Iordache, An evaluation of schemes for detecting and estimating gross errors in process data. ind. Engng Chem. Process Des. Dev.
Other~rnn
upper 1 - @ quantile of chi-square distribution with one degree of freedom
26, 555 (1987).
Serth R. W. and W. A. Heenan, Gross error detection and data reconciliation in stream-metering systems. AlChE J. 32, 733,
(1986).
REFERENCES
Crowe C. M., Y. A. Garcia Campos and A. Hrymak, Reconciliation of process flow rates by matrix projection 1. The linear case. AIChE J. 29, 881 (1983). Mah R. S. H. and A. C. Tamhane, Detection of gross errors in process data. AZChE J. 28, 828 (1982). Mah R. S. H., M. Stanley and D. M. Dowing, Reconciliation and rectification of process flow and inventory data. had. Engng Chem. Process Des. Dev. 15, 175 (19761. Narasihhan’ S., Maximum power tests for gross error detection using likelihood ratios. AIChE J. 36, 1589 (1990).
Narasimhan S. and R. S. H. Mah, Generalized-likelihood ratio method for gross error identification. AIChE J. 33, 1514
(1987).
APPENDIX Partitioned
LetE=
A (
c
B
D
)
Matrix
Inverse
Lemma
a non singular matrix where A and D
are square matrices. If det(A)fO and det(D - CA-‘B)P 0, the inverse of matrix E is given by: A-‘+A-‘B(D-CA-‘B)-‘CA-’ -(D-
CA-‘B)-’ -A-‘B(D(D-
CA-‘B)-’ CA-‘B)-’
>
