- Email: [email protected]
State Variable Based Statistical Methods for Auditing Sensors of Multivariable Dynamic Processes
Copyright © IFAC Fault Detection, Supervision and Safety for Technical Processes, Kingston Upon Hull, UK, 1997
STATE VARIABLE BASED STATISTICAL METHODS FOR AUDITING SENSORS OF MULTIVARIABLE DYNAMIC PROCESSES Ali Qinar •. 1 Antoine Negiz •. 2
• Department of Chemical and Environmental Engineering, Illinois Institute of Technology, 10 W. 33rd Street, Chicago, Illinois, 60616 USA
Abstract: A statistical process monitoring method based on a state space model of a dynamic process is introduced for auditing sensor status for bias, drift and excessive noise affecting the sensors of multivariable continuous processes. Changes in the magnitudes of means and variances of residuals between measured and predicted process variables are used to detect and discriminate sensor abnormalities. The statistical model that describes the in-control variability is based on a canonical variate state space (CVSS) model. The CV state variables obtained from the state space model are linear combinations of the past process measurements which explain the variability of the future measurements the most. and they are regarded as the principal dynamic dimensions. The method can detect and discriminate between bias change, drift, and variations in noise levels of process sensors ba-;ed on the analysis of data batches. An experimental application to a high-temperature short-time (HTST) milk pasteurization process illustrates the proposed methodology. Copyright © I998IFAC Keywords: Sensor auditing. data fusion. statistical monitoring, canonical variate analysis. state space models
INTRODUCTION
describes the dynamic behavior of the process and it is developed such that the state variables are statistically independent (orthogonal) at zero lag. The state space models and the statistical characterization of the random components are obtained directly froIll process data collected when the process was operating in-control. This stochastir. Tf~alizatioll algorithm is suitable for handling large number of variables that are autocorrelated, cross correlated. and collinear. Once an accurate statistical description of the in-control variability of a continuous process is available, the next step is the design and implementation of the sensor lIIollitoriug SPM procedure.
A new method is introduced for periodic audit of process sensors in multivariable dynamic processes. The method is ba-;ed on interpreting the magnitude of the mean and variance of the residuals between a data batch and their prediction from a process model. Industrial contill1lOUS processes have a large number of process variables and are usually operated for extellded periods at fixed operating poillts under closed-loop coutrol. yielding process rnea-;urernents which are autocorrelated, cross correlated, and collinear. The statistical model of the in-control variation is Cl. r.anolliral '/Iar'iat~ 8tat(~ spar.e (CVSS) model. The 1lI0del
Misleading process illforlllation call be generated if there is a bias challge. dTift or high leveb of noise ill some of the sensors. Erroneous information oftell causes decisions and actions that are unnecessary. resulting in the deterioration of product
1 Author to whom correspondence should be addressed (cinar@charlie. iit. edu) 2 Present address: COP Inc., 25 East Algonquin Road. P.O. Box 5017. Des Plaines. IL 60017, USA. ([email protected]).
185
quality, safety and profitability. Identifying failures such as a broken thermocouple is relatively easy since the signal received from the sensor has a fixed and unique value. Incipient sensor failures that cause drift, bias change or additional noise are more difficult to identify and may remain unnoticed for extended periods of time. Auditing sensor behavior can warn plant personnel about incipient sensor faults and initiate timely repair and maintenance. A sensor auditing method based on functional redundancy generated by the CVSS space model is presented for detecting abnormal behavior in one or more sensors of multivariable processes.
where RI( = E (Y;-lK Y
CV realization requires that covariances of future and past stacked observations be conditioned against any singularities by taking their square roots. The Hankel matrix is scaled by using RI( and Rj
Yk E RP
denote process outputs and
Xk E
k.
Represent a vector autoregressive moving average (VARMA) process in state space as (Aoki 1990)
T
HJ K
).
The SVD of the scaled
yields
SENSOR AUDITING WITH CV STATE VARIABLES
The random error sequences (innovation vector) ek with covariance matrix a are ek = Yk E(n I Yk"-lK) where E(·) denotes the expectation operator. Define a truncated form of the infinite Hankel matrix from the futnre (Yk"J) and past (Yk"-IK) stacked measurement vectors as HJK
(y.tyt
where UpJxn contains the n left eigenvectors, :E nxn contains the singular values (SV), and V Kpxn contains the n right eigenvectors of the decomposition. The SVD matrices in (4) include only the SV and eigenvectors corresponding to the n state variables retained in the model. The full SV matrix :E is Jp x K p and it contains the SVs in a descending order. If process noise is small, all SVs smaller than the nth SV are effectively zero and the corresponding state variables are excluded froIll the model. The ratio of the specific SV to the sum of all the SVs (Aoki 1990) or an information theoretic approach :mch as the Akaike Information Criterion (AIC) can be used for selecting the value of n . Once Xk is known from Eq. (3), A, B, C, and a can be constructed (Negiz and Cinar 1997b). The covariance matrix of the state vector based on CV decomposition E(XkXn = :E shows that Xk are independent at zero-Iag.
CVSS mode ling is a s1Lbspace algo1·ithm. for developing empirical dynamic models by using input-output data (Larimore 1983, Overschee and de Moor 1994). Subspace algorithms generate the process model by successive approximation of the memory or the states of the process by determining successively functions of the past that have the most information for predicting the future (Larimore 1983). Only the system order is needed to develop the model from the data, and it is determined by inspecting the dominant singular values of the covariance matrix which are generated by singular value decomposition (SVD).
Rn the state variables at sampling instant
=E
Hankel matrix
CV STOCHASTIC REALIZATION
Let
;:lK). Several realiza-
tion techniques, balanced realization (Aoki 1990), PLS realization (Negiz and Cinar 1997 a), and the canonical variate (CV) realization (Larimore 1983, Negiz and Cinar 1997b) can be used to obtain the terms on the right side of Eq. (3).
Information from various process sensors can be used for assessing the correctness of information generated by a sensor. This approach falls in the category of functional n~d1L'T/.da.ncy. A 7n1Lltipass PLS algorithm was developed for detecting simultaneous multiple sensor abnormalities (Negiz and Cinar 1992). Since the PLS regression algorithm is based on zero lag covariance of the process measurements, the application of the m.nltipass P LS algorithm is based on the assumption of serially independent observations (negligible autocorrelation). This assumption is justified for a continuous process operating at steady state and having only random noise on measurements, but is not valid if the process dynamics are significant and the data are autocorrelated. An alternative tecilllique which handles effectively both autocorrelation alld crosscorrelatioll ill process data can be developed by llSilLg the CVSS model. Sensor
= E (Y:JYk":lK)
J and K are the length of the future and past observation windows. The Hankel matrix can also be expressed in terms of truncated observability anu reachability matrices as (Aoki 1990. Negiz anu Cillar 1997b) HJK = OJOK, yielding (3)
186
audit will be based 011 one-step ahead residuals generated from the CVSS model. A rnfLltipass CVSS technique similar to the rnfLltipas.~ PLS algorithm is developed for detecting IIlultiple sensor failures. This is achieved by eliminating successively the corrupted measurements from both the calibration and test data sets and identifying a different CVSS submodel.
remaullng ones, false alarms might be generated unless the corrupted variable is taken out from both the r:alibT·atioT/. and the test data block. The information loss due to taking the variable out of both the calibration and the test sample set is not significant sillce the testing procedures are based on the iid assumption of the residuals and not on the minimum prediction error criterion by the model. The algorithm discards the variable with the highest corruption level by looking at the ratios of its residual variance and its residual mean to their statistical limits which are based on Eqs.5-6.
Assume that there are p sensors to be monitored and the calibT·atioT/. data set is of length N. The mean and the variance of the residuals for each variable is computed through the N x p residuals block matrix R. Once the CV state space model is identified for the in-control data set, the statistics for the residuals are computed for setting the null hypothesis. Then, a test data block of size N t x p is formed from new process measurements. The residual statistics for the test sample are then generated by using the CV calibmtioT/. model. The statistical test compares the residuals statistics of the test sample with the statistics of the calibmtioT/. set for detecting any significallt departures.
Excluding variables and computing a new CV realization for the remaining variables is the key step of the sensor auditing and fault detection algorithm. The likelihood for all of the process sensors to become simultaneously faulty is extremely small. After several successive steps, if the mean and variance of the remaining residuals still indicate significant variation, then it is more likely that a disturbance is active on the system causing the in-control variability to change.
Denote uy R. i tile ith N x 1 residual vector COlUn1l1 from the N x p residual block matrix R. The statistic for testing the null hypothesis of the equality of means from two normal populations with equal and unknowll variances is
MONITORING OF A HTST PASTEURIZATION PROCESS SENSORS Milk pastenrizatioll illvolves heating milk prodncts in properly designed and operated equipment and keepiug it above 161 0 F for 15 .<;(',., or at other equivalent temperature and residence time combinations yieldiug the same lethality level. In HTST pasteurization, raw milk is first sent to a regenerator heat exchanger to be heated by recovering some of the heat from the pasteurized product (Cinar et al. 1995). Then the raw milk is seut to a heating unit through a timing pump which assures a fixed flow rate required by the pasteurized milk ordinauce. The heating 11Ilit is a plate heat exchallger where raw milk is further heated uy hot water to well above the required pasteurizatioll temperature. The heated milk then goes through a holding tube where it is held for a specific length of time according to the milk ordinance. The holding time is a fnnction of the flow rate and the length of the holdillg tnbe. A flow diversion device is located at the end of the holding tll be. If the temperature of the heated milk exiting the holding tube is lower thall the temperature according to the legal pasteurization temperature, the milk is directed back to the balallce tank uy the How diversioll device for reprocessillg. The state of the process is determilled from five process lIleasurements taken at 1 .H·" illtervals: (1) holding tllbe exit temperatme (prodllct temperature), (2) hot water outlet telllperatme. (3) hot water illlet telllperatme, (4) prodllct flow rate, and (0) differential pressure in J.,[odd D(~lWlopll/'(!1It.
where R.i, •• , aud R.i",o.l., dellote the maximulIl likelihood estimates of the residual means for the variable i ill the test sample and the r:alibmtion set, iT Pi is the pooled stalldard de\"iation of the two residual populatiol1s for the i-th variaule, N alld N t denote the sizes of the r:alibmtion and testillg populations, and tN+N,-2 is the t-distriuution with the N + N t - 2 degrees of freedom (Dudewicz and Mishra Hl88). The statistic for testillg the nllll hypothesis of the equality of variances from two normal populations with unkllown meal1s is (Dudewicz and Mishra 1988) • 2
(T.
~ (1-:-
f"V
F 1V,-1.tV-1
(6)
f "'n.l~1
where FN,-I,N-l is the F distriuution with respective degrees of freedom. The level of tile test for all the testillg statistics is chosen to ue 5% alld two sided. This part of the procedure is similar to that givell by (Wise f'f nl. 1989). The algorithm takes actiol1 when either the mean or variance of the residuals are out of the statistical limit.s (uased 011 I and F probability distriuutiol1s) for a particular variaulc. Since the (,OITll pted variahle affects the predictions of the
187
the regenerator. The total lethality is computed based on the product temperature and product flow rate measurements. Since it is a critical process variable, it is included in the list of variables monitored a<; the 6th variable. Figure 1 depicts results from an experiment where the product temperature hao.; a l'iet-point value of 161.6° F, and the set-point for the product flow rate is at 300 gph . Ba.o.;ed on the set-points for How rate and product temperature the total lethality is set at 18 se.c which provides a safety margin over the 15 se.c lethality level imposed and corresponds to 20% over-processing. As a result of feedback control actions, the temperatures are varying in a highly positively autocorrelated manner around their set-points (Fig. 1).
inlet temperature, the pressure differential and the total lethality measurements are suitable for the residual mean and variance tests proposed by Eqs. 5·6. The mean test for residuals of all variables is appropriate due to the central limit theorem . The test limits for the variance of flow rate residuals computed from Eq. 6 are extended by multiplying them with the correction factor 1.4. The factor is obtained a.o.; the ratio of the 99th percent point of the flow residual >'D = -0.05 distribution to the 99th percent point of the normal distribution (Negiz 1995). The residual mean and variance tests applied to the residuals based on the mlihmtio1/. model fall within the 95% confidence limits and all the ratios are less than unity (Fig. 2). The numbers on the horizontal axes correspond to the HTST pasteurization process variables ac; numbered in the first paragraph of "Model Development" subsection . The 7l/.1tltipas.~ algorithm is used for detecting multiple sensor faults in a silllilar manner to the 1Tmltipass PLS algorithm. If the residual mean or variallce that belollg to a particular sensor is out of the statistical limits, first a new CV state space model is built bao.;ed on the calihmtion data set by excluding the measurements of the corrupted sellSOI". Then, the new model is used to generate the residuals from the t(~.~t data set by also excludillg the measurements of the corr1lpted sellsor. Testillg of the test data set residuals against the calibration data set residuals is done through Eqs.
The statistical characterization of the in-control process variability is performed with CV stochastic realization. A CV state space model is extracted from the data which yields residuals with insignificant auto allll crol'iS correlations. There are six process variables (p = 6) and the length of the calibrat.ion data is N = 798. Alltoregressive 1Il0dels for each variable indicate the maximlllll significant lag is 20. A robustness margin is added, and the past (K) and the futnre (.I) window lengths for the CV state space model are set at 24 amI 30, respectively. The Hankel matrix is constructed by cOlllpnting the covariance between the 180 x 1 fllture stacked vector and the 144 x 1 past stacked vector according to Equation 2. The CV decomposition is performed acconling to the scaled Hankel matrix (Eq. 4) . The singlllar values of the Hankel matrix obtained from Eq. 4 tend to zero exponeutially and 20 singular values are retained for the state space model. Once the st.ate variables are obt.ained by CV decomposition aud Eq. 3, the least squares solutiou provides the system matrices A, C, B. The zero lag structure of the residual autocorrelations cau be extracted frolll the estimate of the residual covariance matrix (.6.) .
5 6.
P1"()(:~SS . The functioual redulldancy generated by the 20 state CV Illodel of the HTST pastenrization process provides residuals which are essentially iid. (Negiz 1995) . The residuals statistics generated from the 20 state CV lIlodel using the mlihmti(}T/. data set will be utilized to test the hypothesis of 110 change against the hypothesis of change for a test data set. The test data are collected in N t salllpling intervals which is cousistent with the n~sidual testing schellles given iu Eqs. 5 6.
A test data set of length 200 wao.; generated by adding bias and noise. A bia<; in the amount of one standard deviatioll unit based on the ill-control variability of the flow rate measurements wa.<; added to the original flow rate measnrements, and a IlOise with its stalldard deviatioll being equal to one standan.l tieviation twit of the in-control variability of the product temperature measurements wa.o.; added to the original temperature readings. The residual tests where N = 745 and N t = 200 are depicted ill Figure 3. The bia.o.; of fiow rate measuremellts can be deduced by inspecting the meall ratio chart. Sillce this ratio is the maximuIlI amollg all. the How mea.o.;1lI"elllellt wa.~ tiiagnosed to be faulty. Since its variance is with ill bounds, the flow rate sensor fault wa.o.; deduced to be a i.>iao.; change. Total lethality (variable 6) ha'i residual variance larger than the critical limit. Since the total lethality is computed froll1 fiow rate, both variables were excluded from the r.alihmti()1I. and t.(~sl. data sets.
Aualysis ltas showu that the distributioll of t.he residuals closely follow a Gaussian distribatiou except the How rate residuals (Ncgiz 1995). This means that the CV resid aals for the HTST temperatllre, hot wat!'r outlet temperature, hot water
A CV state space model was compnted for the remainillg process variables and the residllalmean alld variallce charts were developed (Fig. 4) . hi this case, all of the residual means are within bounds, whereas the protillct temperature resid-
Auditing thl' SI' TI.S ()1·.~ ()f HTST Pn.stl~TL1·izati(}T/.
188
ual variance is almost out of bounds. This indicates that the product temperature thermocouple (Variable 1) was noisy. Once variable 1 was also excluded from both the calibT'ation and test data sets, a new 20 state CV realization was obtained for the remaining three process variables. The residual mean and variance statistics were all within bounds, indicating that the changes experienced were due to the measurement system rather than any actual change occurring in the process.
Negiz, A. and A. Ginar (1997a). Pis as a technique in identifying vector autoregressive moving average models in state space. Chemometrics and Intelligent Lab. Systems, Accepted for publication. Negiz, A. and A. Ginar (1997b) . Statistical monitoring of multivariable continuous processes with state space models. AIChE J., accepted for publication. Overschee, P . Van and B. de Moor (1994) . N4sid: Subspace algorithms for the identification of combined determiuistic-stochastic systems. A utomatir.a 30, 75-93. Wise, B. M., N. L. Ricker and D ..1. Veltkamp (1989). Upset and sensor fault detection in multivariable processes. AIChE Annual Meetiug, Paper 164b, Sail FraIlcisco, CA .
CONCLUSIONS A sensor auditillg method is developed for detecting abnormal behavior in one or more sensors of lIlultivariable processes based on functional redundancy generated by the CV state space model. The method can detect and discriminate between sensor drift, bias change or additiollal noise. The method can be implemented to rim repeatedly at frequent intervals and warn plant personnel about illcipient sensor faults to initiate tilllcly repair alltl lIlaintenance. Acknowledgment: The authors gratefully acknowledge the ASA/NSF /NIST Research Associate appointment of A. Negiz in 19D2-1993, and the research assistantship provided to A. Negiz by the National Center for Food Safety and Technology.
REFERENCES Aoki, M. (l!J!J0) . Statl'. Spar.e Moddin.q of Tillw S('.T·il~S. 2nd Ed .. Springer-Verlag. New York. Ginar, A. , .1. E. Schlesser, A. Negiz, P. Ramauauskas, D . .1. Annstrong and W . StrollP (1995) . Automated control of high-temperature short-time pasteurization systems based on lethality limits. In: P1'Or.el'.din.Q$ of tlw Food PT'I)r.l'.s!1in.Q A 1£tomatioll Confl~ T·l'.nr.l~ IV. pp. 394 404 . Dudewicz, E . .1 . and S. N. Mishra (1988). ModPT'11. M(], thl~m.atir.al Stati.~tir.s . .101111 Wiley. New York. Larilllorc, W . E. (I !J83) . Systcnl identification. reduced-order filtering and 1lI0deling via canonical variate analysis. In : PmC'. 198.'1 A 7lt011l.atir COT/.tml Con!.. pp. 445 451. Negiz , A . (1 !J!JG) . Statistical Dynalllic Modeliug aud Mouitoring Methods for 1\1 Illtivariable ContilluollS Processes. PhD thesis. Illinois 111stitllte of Technology. Negiz , A. and A. Ginar (19!J2). On the detectioll of mllltiple sensor abnormalities in IlIl1ltivariable processes. Ill: P1,()r.. l.1J.fJ2 A.",,1'1·il'an (,'on t1'lJ I Con! . pp. 2:361 23GR. 189
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1612t
_ _ IRaIio!
t
,. IW!!J'I
...... J ,...
.
'''2:
i
,
~,
'''0
:00
0100
r_
'f------------n0.8f 0.6
I
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V_ _
2
" -----
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.
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~
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Figure 2. Statistics of Residuals of CV State Space Calibration Model of HTST Pasteurization System. (A) Means of residuals, (B) Ratios of residuals means, (D) Variances of residuals, (D) Ratios of residuals variances.
A_ _ /Rauc!
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.2°. 8
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i ~0.5 Figure 1. The In-Control Variation of the HTST Process. Measurements of (A) Product temperature, (B) Hot water outlet temperature, (C) Hot water inlet temperature, (D) Product flow rate, (E) Pressure differential, and (F) Total lethality.
3
_V_(RaIio!
jo...
--I
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J V~Number
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Figure 3. Statistics of Residuals of CV State Space Test Data of HTST Pasteurization System . (A) Means of residuals, (B) Ratios of residuals means. (D) Variances of resiuuals, (D) Ratios of residuals variances. Sensor 4 is biased.
Figure 4. Statistics of Residuals of CV State Space Test Data of HTST Pasteurization System Excluding Variables 4 and 6. (A) Means of residuals, (B) Ratios of residuals means, (D) Variances of residuals, (D) Ratios of residuals variances. Sensor 1 is noisy.
190
STATE VARIABLE BASED STATISTICAL METHODS FOR AUDITING SENSORS OF MULTIVARIABLE DYNAMIC PROCESSES Ali Qinar •. 1 Antoine Negiz •. 2
• Department of Chemical and Environmental Engineering, Illinois Institute of Technology, 10 W. 33rd Street, Chicago, Illinois, 60616 USA
Abstract: A statistical process monitoring method based on a state space model of a dynamic process is introduced for auditing sensor status for bias, drift and excessive noise affecting the sensors of multivariable continuous processes. Changes in the magnitudes of means and variances of residuals between measured and predicted process variables are used to detect and discriminate sensor abnormalities. The statistical model that describes the in-control variability is based on a canonical variate state space (CVSS) model. The CV state variables obtained from the state space model are linear combinations of the past process measurements which explain the variability of the future measurements the most. and they are regarded as the principal dynamic dimensions. The method can detect and discriminate between bias change, drift, and variations in noise levels of process sensors ba-;ed on the analysis of data batches. An experimental application to a high-temperature short-time (HTST) milk pasteurization process illustrates the proposed methodology. Copyright © I998IFAC Keywords: Sensor auditing. data fusion. statistical monitoring, canonical variate analysis. state space models
INTRODUCTION
describes the dynamic behavior of the process and it is developed such that the state variables are statistically independent (orthogonal) at zero lag. The state space models and the statistical characterization of the random components are obtained directly froIll process data collected when the process was operating in-control. This stochastir. Tf~alizatioll algorithm is suitable for handling large number of variables that are autocorrelated, cross correlated. and collinear. Once an accurate statistical description of the in-control variability of a continuous process is available, the next step is the design and implementation of the sensor lIIollitoriug SPM procedure.
A new method is introduced for periodic audit of process sensors in multivariable dynamic processes. The method is ba-;ed on interpreting the magnitude of the mean and variance of the residuals between a data batch and their prediction from a process model. Industrial contill1lOUS processes have a large number of process variables and are usually operated for extellded periods at fixed operating poillts under closed-loop coutrol. yielding process rnea-;urernents which are autocorrelated, cross correlated, and collinear. The statistical model of the in-control variation is Cl. r.anolliral '/Iar'iat~ 8tat(~ spar.e (CVSS) model. The 1lI0del
Misleading process illforlllation call be generated if there is a bias challge. dTift or high leveb of noise ill some of the sensors. Erroneous information oftell causes decisions and actions that are unnecessary. resulting in the deterioration of product
1 Author to whom correspondence should be addressed (cinar@charlie. iit. edu) 2 Present address: COP Inc., 25 East Algonquin Road. P.O. Box 5017. Des Plaines. IL 60017, USA. ([email protected]).
185
quality, safety and profitability. Identifying failures such as a broken thermocouple is relatively easy since the signal received from the sensor has a fixed and unique value. Incipient sensor failures that cause drift, bias change or additional noise are more difficult to identify and may remain unnoticed for extended periods of time. Auditing sensor behavior can warn plant personnel about incipient sensor faults and initiate timely repair and maintenance. A sensor auditing method based on functional redundancy generated by the CVSS space model is presented for detecting abnormal behavior in one or more sensors of multivariable processes.
where RI( = E (Y;-lK Y
CV realization requires that covariances of future and past stacked observations be conditioned against any singularities by taking their square roots. The Hankel matrix is scaled by using RI( and Rj
Yk E RP
denote process outputs and
Xk E
k.
Represent a vector autoregressive moving average (VARMA) process in state space as (Aoki 1990)
T
HJ K
).
The SVD of the scaled
yields
SENSOR AUDITING WITH CV STATE VARIABLES
The random error sequences (innovation vector) ek with covariance matrix a are ek = Yk E(n I Yk"-lK) where E(·) denotes the expectation operator. Define a truncated form of the infinite Hankel matrix from the futnre (Yk"J) and past (Yk"-IK) stacked measurement vectors as HJK
(y.tyt
where UpJxn contains the n left eigenvectors, :E nxn contains the singular values (SV), and V Kpxn contains the n right eigenvectors of the decomposition. The SVD matrices in (4) include only the SV and eigenvectors corresponding to the n state variables retained in the model. The full SV matrix :E is Jp x K p and it contains the SVs in a descending order. If process noise is small, all SVs smaller than the nth SV are effectively zero and the corresponding state variables are excluded froIll the model. The ratio of the specific SV to the sum of all the SVs (Aoki 1990) or an information theoretic approach :mch as the Akaike Information Criterion (AIC) can be used for selecting the value of n . Once Xk is known from Eq. (3), A, B, C, and a can be constructed (Negiz and Cinar 1997b). The covariance matrix of the state vector based on CV decomposition E(XkXn = :E shows that Xk are independent at zero-Iag.
CVSS mode ling is a s1Lbspace algo1·ithm. for developing empirical dynamic models by using input-output data (Larimore 1983, Overschee and de Moor 1994). Subspace algorithms generate the process model by successive approximation of the memory or the states of the process by determining successively functions of the past that have the most information for predicting the future (Larimore 1983). Only the system order is needed to develop the model from the data, and it is determined by inspecting the dominant singular values of the covariance matrix which are generated by singular value decomposition (SVD).
Rn the state variables at sampling instant
=E
Hankel matrix
CV STOCHASTIC REALIZATION
Let
;:lK). Several realiza-
tion techniques, balanced realization (Aoki 1990), PLS realization (Negiz and Cinar 1997 a), and the canonical variate (CV) realization (Larimore 1983, Negiz and Cinar 1997b) can be used to obtain the terms on the right side of Eq. (3).
Information from various process sensors can be used for assessing the correctness of information generated by a sensor. This approach falls in the category of functional n~d1L'T/.da.ncy. A 7n1Lltipass PLS algorithm was developed for detecting simultaneous multiple sensor abnormalities (Negiz and Cinar 1992). Since the PLS regression algorithm is based on zero lag covariance of the process measurements, the application of the m.nltipass P LS algorithm is based on the assumption of serially independent observations (negligible autocorrelation). This assumption is justified for a continuous process operating at steady state and having only random noise on measurements, but is not valid if the process dynamics are significant and the data are autocorrelated. An alternative tecilllique which handles effectively both autocorrelation alld crosscorrelatioll ill process data can be developed by llSilLg the CVSS model. Sensor
= E (Y:JYk":lK)
J and K are the length of the future and past observation windows. The Hankel matrix can also be expressed in terms of truncated observability anu reachability matrices as (Aoki 1990. Negiz anu Cillar 1997b) HJK = OJOK, yielding (3)
186
audit will be based 011 one-step ahead residuals generated from the CVSS model. A rnfLltipass CVSS technique similar to the rnfLltipas.~ PLS algorithm is developed for detecting IIlultiple sensor failures. This is achieved by eliminating successively the corrupted measurements from both the calibration and test data sets and identifying a different CVSS submodel.
remaullng ones, false alarms might be generated unless the corrupted variable is taken out from both the r:alibT·atioT/. and the test data block. The information loss due to taking the variable out of both the calibration and the test sample set is not significant sillce the testing procedures are based on the iid assumption of the residuals and not on the minimum prediction error criterion by the model. The algorithm discards the variable with the highest corruption level by looking at the ratios of its residual variance and its residual mean to their statistical limits which are based on Eqs.5-6.
Assume that there are p sensors to be monitored and the calibT·atioT/. data set is of length N. The mean and the variance of the residuals for each variable is computed through the N x p residuals block matrix R. Once the CV state space model is identified for the in-control data set, the statistics for the residuals are computed for setting the null hypothesis. Then, a test data block of size N t x p is formed from new process measurements. The residual statistics for the test sample are then generated by using the CV calibmtioT/. model. The statistical test compares the residuals statistics of the test sample with the statistics of the calibmtioT/. set for detecting any significallt departures.
Excluding variables and computing a new CV realization for the remaining variables is the key step of the sensor auditing and fault detection algorithm. The likelihood for all of the process sensors to become simultaneously faulty is extremely small. After several successive steps, if the mean and variance of the remaining residuals still indicate significant variation, then it is more likely that a disturbance is active on the system causing the in-control variability to change.
Denote uy R. i tile ith N x 1 residual vector COlUn1l1 from the N x p residual block matrix R. The statistic for testing the null hypothesis of the equality of means from two normal populations with equal and unknowll variances is
MONITORING OF A HTST PASTEURIZATION PROCESS SENSORS Milk pastenrizatioll illvolves heating milk prodncts in properly designed and operated equipment and keepiug it above 161 0 F for 15 .<;(',., or at other equivalent temperature and residence time combinations yieldiug the same lethality level. In HTST pasteurization, raw milk is first sent to a regenerator heat exchanger to be heated by recovering some of the heat from the pasteurized product (Cinar et al. 1995). Then the raw milk is seut to a heating unit through a timing pump which assures a fixed flow rate required by the pasteurized milk ordinauce. The heating 11Ilit is a plate heat exchallger where raw milk is further heated uy hot water to well above the required pasteurizatioll temperature. The heated milk then goes through a holding tube where it is held for a specific length of time according to the milk ordinance. The holding time is a fnnction of the flow rate and the length of the holdillg tnbe. A flow diversion device is located at the end of the holding tll be. If the temperature of the heated milk exiting the holding tube is lower thall the temperature according to the legal pasteurization temperature, the milk is directed back to the balallce tank uy the How diversioll device for reprocessillg. The state of the process is determilled from five process lIleasurements taken at 1 .H·" illtervals: (1) holding tllbe exit temperatme (prodllct temperature), (2) hot water outlet telllperatme. (3) hot water illlet telllperatme, (4) prodllct flow rate, and (0) differential pressure in J.,[odd D(~lWlopll/'(!1It.
where R.i, •• , aud R.i",o.l., dellote the maximulIl likelihood estimates of the residual means for the variable i ill the test sample and the r:alibmtion set, iT Pi is the pooled stalldard de\"iation of the two residual populatiol1s for the i-th variaule, N alld N t denote the sizes of the r:alibmtion and testillg populations, and tN+N,-2 is the t-distriuution with the N + N t - 2 degrees of freedom (Dudewicz and Mishra Hl88). The statistic for testillg the nllll hypothesis of the equality of variances from two normal populations with unkllown meal1s is (Dudewicz and Mishra 1988) • 2
(T.
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where FN,-I,N-l is the F distriuution with respective degrees of freedom. The level of tile test for all the testillg statistics is chosen to ue 5% alld two sided. This part of the procedure is similar to that givell by (Wise f'f nl. 1989). The algorithm takes actiol1 when either the mean or variance of the residuals are out of the statistical limit.s (uased 011 I and F probability distriuutiol1s) for a particular variaulc. Since the (,OITll pted variahle affects the predictions of the
187
the regenerator. The total lethality is computed based on the product temperature and product flow rate measurements. Since it is a critical process variable, it is included in the list of variables monitored a<; the 6th variable. Figure 1 depicts results from an experiment where the product temperature hao.; a l'iet-point value of 161.6° F, and the set-point for the product flow rate is at 300 gph . Ba.o.;ed on the set-points for How rate and product temperature the total lethality is set at 18 se.c which provides a safety margin over the 15 se.c lethality level imposed and corresponds to 20% over-processing. As a result of feedback control actions, the temperatures are varying in a highly positively autocorrelated manner around their set-points (Fig. 1).
inlet temperature, the pressure differential and the total lethality measurements are suitable for the residual mean and variance tests proposed by Eqs. 5·6. The mean test for residuals of all variables is appropriate due to the central limit theorem . The test limits for the variance of flow rate residuals computed from Eq. 6 are extended by multiplying them with the correction factor 1.4. The factor is obtained a.o.; the ratio of the 99th percent point of the flow residual >'D = -0.05 distribution to the 99th percent point of the normal distribution (Negiz 1995). The residual mean and variance tests applied to the residuals based on the mlihmtio1/. model fall within the 95% confidence limits and all the ratios are less than unity (Fig. 2). The numbers on the horizontal axes correspond to the HTST pasteurization process variables ac; numbered in the first paragraph of "Model Development" subsection . The 7l/.1tltipas.~ algorithm is used for detecting multiple sensor faults in a silllilar manner to the 1Tmltipass PLS algorithm. If the residual mean or variallce that belollg to a particular sensor is out of the statistical limits, first a new CV state space model is built bao.;ed on the calihmtion data set by excluding the measurements of the corrupted sellSOI". Then, the new model is used to generate the residuals from the t(~.~t data set by also excludillg the measurements of the corr1lpted sellsor. Testillg of the test data set residuals against the calibration data set residuals is done through Eqs.
The statistical characterization of the in-control process variability is performed with CV stochastic realization. A CV state space model is extracted from the data which yields residuals with insignificant auto allll crol'iS correlations. There are six process variables (p = 6) and the length of the calibrat.ion data is N = 798. Alltoregressive 1Il0dels for each variable indicate the maximlllll significant lag is 20. A robustness margin is added, and the past (K) and the futnre (.I) window lengths for the CV state space model are set at 24 amI 30, respectively. The Hankel matrix is constructed by cOlllpnting the covariance between the 180 x 1 fllture stacked vector and the 144 x 1 past stacked vector according to Equation 2. The CV decomposition is performed acconling to the scaled Hankel matrix (Eq. 4) . The singlllar values of the Hankel matrix obtained from Eq. 4 tend to zero exponeutially and 20 singular values are retained for the state space model. Once the st.ate variables are obt.ained by CV decomposition aud Eq. 3, the least squares solutiou provides the system matrices A, C, B. The zero lag structure of the residual autocorrelations cau be extracted frolll the estimate of the residual covariance matrix (.6.) .
5 6.
P1"()(:~SS . The functioual redulldancy generated by the 20 state CV Illodel of the HTST pastenrization process provides residuals which are essentially iid. (Negiz 1995) . The residuals statistics generated from the 20 state CV lIlodel using the mlihmti(}T/. data set will be utilized to test the hypothesis of 110 change against the hypothesis of change for a test data set. The test data are collected in N t salllpling intervals which is cousistent with the n~sidual testing schellles given iu Eqs. 5 6.
A test data set of length 200 wao.; generated by adding bias and noise. A bia<; in the amount of one standard deviatioll unit based on the ill-control variability of the flow rate measurements wa.<; added to the original flow rate measnrements, and a IlOise with its stalldard deviatioll being equal to one standan.l tieviation twit of the in-control variability of the product temperature measurements wa.o.; added to the original temperature readings. The residual tests where N = 745 and N t = 200 are depicted ill Figure 3. The bia.o.; of fiow rate measuremellts can be deduced by inspecting the meall ratio chart. Sillce this ratio is the maximuIlI amollg all. the How mea.o.;1lI"elllellt wa.~ tiiagnosed to be faulty. Since its variance is with ill bounds, the flow rate sensor fault wa.o.; deduced to be a i.>iao.; change. Total lethality (variable 6) ha'i residual variance larger than the critical limit. Since the total lethality is computed froll1 fiow rate, both variables were excluded from the r.alihmti()1I. and t.(~sl. data sets.
Aualysis ltas showu that the distributioll of t.he residuals closely follow a Gaussian distribatiou except the How rate residuals (Ncgiz 1995). This means that the CV resid aals for the HTST temperatllre, hot wat!'r outlet temperature, hot water
A CV state space model was compnted for the remainillg process variables and the residllalmean alld variallce charts were developed (Fig. 4) . hi this case, all of the residual means are within bounds, whereas the protillct temperature resid-
Auditing thl' SI' TI.S ()1·.~ ()f HTST Pn.stl~TL1·izati(}T/.
188
ual variance is almost out of bounds. This indicates that the product temperature thermocouple (Variable 1) was noisy. Once variable 1 was also excluded from both the calibT'ation and test data sets, a new 20 state CV realization was obtained for the remaining three process variables. The residual mean and variance statistics were all within bounds, indicating that the changes experienced were due to the measurement system rather than any actual change occurring in the process.
Negiz, A. and A. Ginar (1997a). Pis as a technique in identifying vector autoregressive moving average models in state space. Chemometrics and Intelligent Lab. Systems, Accepted for publication. Negiz, A. and A. Ginar (1997b) . Statistical monitoring of multivariable continuous processes with state space models. AIChE J., accepted for publication. Overschee, P . Van and B. de Moor (1994) . N4sid: Subspace algorithms for the identification of combined determiuistic-stochastic systems. A utomatir.a 30, 75-93. Wise, B. M., N. L. Ricker and D ..1. Veltkamp (1989). Upset and sensor fault detection in multivariable processes. AIChE Annual Meetiug, Paper 164b, Sail FraIlcisco, CA .
CONCLUSIONS A sensor auditillg method is developed for detecting abnormal behavior in one or more sensors of lIlultivariable processes based on functional redundancy generated by the CV state space model. The method can detect and discriminate between sensor drift, bias change or additiollal noise. The method can be implemented to rim repeatedly at frequent intervals and warn plant personnel about illcipient sensor faults to initiate tilllcly repair alltl lIlaintenance. Acknowledgment: The authors gratefully acknowledge the ASA/NSF /NIST Research Associate appointment of A. Negiz in 19D2-1993, and the research assistantship provided to A. Negiz by the National Center for Food Safety and Technology.
REFERENCES Aoki, M. (l!J!J0) . Statl'. Spar.e Moddin.q of Tillw S('.T·il~S. 2nd Ed .. Springer-Verlag. New York. Ginar, A. , .1. E. Schlesser, A. Negiz, P. Ramauauskas, D . .1. Annstrong and W . StrollP (1995) . Automated control of high-temperature short-time pasteurization systems based on lethality limits. In: P1'Or.el'.din.Q$ of tlw Food PT'I)r.l'.s!1in.Q A 1£tomatioll Confl~ T·l'.nr.l~ IV. pp. 394 404 . Dudewicz, E . .1 . and S. N. Mishra (1988). ModPT'11. M(], thl~m.atir.al Stati.~tir.s . .101111 Wiley. New York. Larilllorc, W . E. (I !J83) . Systcnl identification. reduced-order filtering and 1lI0deling via canonical variate analysis. In : PmC'. 198.'1 A 7lt011l.atir COT/.tml Con!.. pp. 445 451. Negiz , A . (1 !J!JG) . Statistical Dynalllic Modeliug aud Mouitoring Methods for 1\1 Illtivariable ContilluollS Processes. PhD thesis. Illinois 111stitllte of Technology. Negiz , A. and A. Ginar (19!J2). On the detectioll of mllltiple sensor abnormalities in IlIl1ltivariable processes. Ill: P1,()r.. l.1J.fJ2 A.",,1'1·il'an (,'on t1'lJ I Con! . pp. 2:361 23GR. 189
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Figure 4. Statistics of Residuals of CV State Space Test Data of HTST Pasteurization System Excluding Variables 4 and 6. (A) Means of residuals, (B) Ratios of residuals means, (D) Variances of residuals, (D) Ratios of residuals variances. Sensor 1 is noisy.
190