multivariate statistical signal processing technique for fault detection and diagnostics

MULTIVARIATE STATISTICAL SIGNAL PROCESSING TECHNIQUE FOR FAULT DETECTION A N D DIAGNOSTICS *

B. R. Upadhyaya O. Glockler J. Eklund The University of Tennessee

-7 7

/ /

The normal fluctuation of wideband signals in process indust~'y systems exhibits behavior that is characteristic of process dynamics, sensor dynamics, vibration of components, and product quality. A baseline statistical signature behavior can be established by a systematic processing of multivariate signals and determining the cause and effect relationship among the process variables characterizing a subsystem. Both theoretical and computa- tional basis for processing a set of signals using the multivariate autoregression [MAR) modeling has been developed and applied to establish frequency domain statistical signatures for an aluminum rolling mill. A systerrP atic procedure is developed to interpret the causal relationships for the detection and isolation of process anomalies and sensor malolP eration. This digital signal processing technique and its implementation have clearly demonstrated the applicability of this method

* This research and development was partially sponsored by the University of Tennessee Measurement & Control Engineering Center,

of characterizing and monitoring complex in- dustrial processes.

INTRODUCTION In large and complex process industry systems such

as chemical processes, paper manufacturing, the metals industry, and power-generating systems, the random fluctuation of wide band signals contains information related to process interaction, machinery performance, vibration of components, and product quality. An example of the latter is sheet thickness (gauge) in metal- rolling mills. An extensive and systematic approach for processing multivariate signals may be used to establish baseline signature behavior and for the detection and isolation of plant machinery anomalies and instrument maloperation. Classical digital signal processing (DSP) techniques, such as correlation and Fourier spectral analyses, have limited application in defining and characterizing cause and effect relationships among a set of process variables. Such techniques, however, are very useful in obtaining a general overview of signal behavior and often lead to a proper definition of the problem.

ISSN 0019-0578/90/04/0079/17/$2.50 © ISA 1990 I S A T r a n s a c t i o n s • Vol. 29. No. 4 79

The primary objectives of this work are (1) to de- velop a detailed and implementable methodology for relating a set of random processes (measured from plant sensors) using multivariate autoregression (MAR) models, and (20 to establish spectral domain signatures necessary for describing the baseline information and for fault detection and diagnostics. This technique is applied to the analysis of plant signals from an aluminum rolling mill. The development of an appropriate MAR model requires a systematic approach to the problem as enumerated here:

(1) Data acquisition, choice of sampling frequency, and choice of signal set. The latter requires some experimentation.

(2) Understanding of the limitations of the measurements and location of sensors in relation to the dynamic variables being measured.

(3) Data qualification, data compression, and choice of data length.

(4) Estimation of MAR model parameters. The choice of model order also depends on the pur- pose of the model.

(5) Computation of spectral domain signatures and interpretation of their mutual relationships.

(6) Derivation of diagnostic information.

A good collection of papers on multivariate spectrum analysis may be found in Reference 1. High-resolution spectral analysis for short data records and multi- channel analysis are described by Marple [2]. Several authors have applied this method in the study of pres- surized water reactors and boiling water reactors [3-6]. A systematic information management is lacking in the

past studies. The technique and analysis software developed un-

der this study are appl~able for monitoring rotating machinery vibration and surveillance, sensor fault detection, detection of flow related mechanisms, and oth- ers. The time-series modeling approach has the advan- tage of providing a higher resolution in the spectral estimates. It is possible to obtain a continuous spectrum theoretically, and information can be extracted from short data records. The anomaly detection and diagnostics can be fully automated so that minimum operator input is needed to perform on-line monitoring.

MULTIVARIATE AUTOREGRESSION (MAR) MODELING OF PROCESS SIGNALS

Our primary interests are in establishing cause and effect relationships among the measured signals using frequency-dependent statistical functions and in identi- fying signal contribution ratio functions. The aim is to fit a linear model to the measured set of noise signals, that is, to estimate the parameters of the assumed model using the available data samples, or the correlation matrices estimated from the raw data, and then to use the identified model for estimating statistical functions of measured signals in the frequency domain.

The multivariate autoregressive (MAR) modeling of the digitized set of stationary measurement vectors is described by:

/1

X(t) = ~ A(i)X(t - iAt) + V(t) (1) i= 1

where X(t) = (xl(t), XE(t) . . . . . x ( t ) ) represents the measurement signal vector at time instant t, and At is the Sampling interval. The measured m-dimensional random sequence X(t) has zero mean and finite vari- ance (i.e., they are the noise components of the measured processs signals, digitized after removing the mean values of stationary signals.) According to Eq. (1), the value of a given noise signal in vector X(t) at time instant t depends on its past values and the past values of the other measured signals as well. In the MAR model no instantaneous interactions between signals are assumed. The matrices A(1), A(2) . . . . . A(n) are the (m x m) MAR coefficient matrices to be estimated. They describe the coupling among the signals and the overall memory of the system (i.e., the weights of contributions of past values to the current measurement vector).

The m-dimensional Gaussian white noise vector V(t) represents the instantaneous contribution to each individual signal, which cannot be derived from intersignal transmissions. The multiple component white noise sequence has zero mean and frequency-independent spectra by definition. The components of the white noise vector V(t) may be correlated (i.e., the off-diagonal elements of the frequency-independent noise covariance matrix ~ , tr.. are not necessarily equal to zero).

l./

As we will see later, to generate the MAR-based auto power spectral density (APSD), cross power spectral density (CPSD), coherence and phase functions, it is not necessary to have an MAR model with independent noise sources. However, to perform an unambiguous signal decomposition and study cause and effect rela-

8 0 ISA Transactions • Vol. 29, No. 4

tionships using signal transmission path (STP) analysis, the derived MAR model must have independent noise sources. In the above mathematical construction (Eq. (1)), the V(t) term can be interpreted in a physical sense as the external driving noise source (white) of the equivalent multivariate linear system.

The first step is to determine the unknown A and matrices (i.e., to determine the model represented by Eq. (1)) using the estimated correlation matrices of the given set of measurement vectors. The MAR coefficient and noise covariance matrices are estimated by solving the set of matrix Yule-Walker equations [7], derived from the defining equation, Eq. (1). This de- scribes the linear relationship between the MAR model matrices and the measurement-based correlation matrices.

n

C(k) = ~A(i)C(k - i), k = 1, 2 . . . . . n i=1

(2a)

/'1

C(O) = ~A(i)C(-i) + i=l

(2b)

where the biased estimator of the correlation matrices {C(k), k = 0, 1 . . . . . n} of measurement vectors used. The diagonal elements of matrix C(k) are the estimated autocorrelation functions at time lag k, while the off- diagonal elements represent the corresponding cross correlation functions at time lag k. The MAR model, identified by solving the above set of linear equations, is equivalent to the original measured system at the level of the correlation matrices of the first n time lags (because Eqs. (2 a,b) are satisfied).

The biased estimator of the correlation matrix C(k) of a digitized set of stationary measurement vectors is given by:

N - k 1

~., X(i + k)X(i) r (3) C(k) - N i=l

where k = 0, 1, 2 . . . . . n and N is the number of samples. To estimate the correlation matrices with ac- ceptable uncertainties, the number of time lags n must be less than 10% of the number of samples N. Note that all correlation matrices up to time lag n are used in the estimation of MAR model of order n.

Having estimated the correlation matrices, the set of matrix equations in Eqs. (2a,b) can be solved by a corn-

putationally efficient algorithm, which is re.cursive with respect to increasing MAR model order [3]. The calcu- lation of successively higher order MAR matrices pro- ceeds until an optimal model order (n) is defined, (i.e., until the determinant of the covariance matrix of the fictitious driving noise source ]~ is reduced to a desired value). Also, several other criteria [3] were used for selection of the range of optimal MAR model orders, but experienced judgment is still required. Among these, the Akaike information criterion (AIC) performs well for finite sample size problems. The model order is selected such that the joint probability density function is maximized. For Gaussian distribution, this re- duces to:

AIC = N In IEI + 2m2n

where:

N = total number of samples

I~I = determinant of the noise covariance matrix

m = number of signals in the model

n = model order

Choose model order n such that AIC is a minimum. The optimal model order may also depend on the

signature to be estimated by the MAR model. The re- cursive algorithm also provides the MAR matrices for. all the lower order MAR models, so the effect of choos- ing different model orders on the results can be studied in a systematic manner.

ESTIMATION OF FREQUENCY DOMAIN SIG. NATURES FROM THE MAR MODEL

F r e q u e n c y D o m a in E s t i m a t i o n

Once the optimal MAR order is established, that is, the A(1), A(2) . . . . . A(n) and ~ matrices are deter- mined, the model is transformed into the frequency domain, and several signature functions as continuous functions of frequency are calculated for the individual signals or signal pairs. These are:

(1) auto and cross power spectral density functions of the measured signals and those of the derived inherent noise components;

(2) ordinary coherence (COH) and partial coherence (PCOH) and the corresponding phase functions;

ISA Transactions ° Vol. 29, No. 4 81

(3) ordinary and partial noise source contribution ratio functions (NCSR and PNSCR);

(4) cumulative noise source contribution ratio functions (CNSCR); and

(5) transfer functions among the measured signals giving the frequency response from one signal to another.

The spectral matrix of the measurement vector can be derived from the multivariate autoregression assump- tion (see Eq. (1)). It will have the form:

S(f) = H(0-1~ H*(j-')-IAt (4)

where:

H(f) = I - ET=~ A(i)exp (-j27rfiAt)

is an (m x m) complex matrix constructed from the MAR matrices. The spectral matrix S(f) contains the auto power spectral density (APSD) functions of the signals along its diagonal elements, and the off-diagonal elements define the corresponding cross power spectral density (CPSD) functions. Note that the signal spectral matrix S(f)in Eq. (4) contains the full noise covariance matrix •, e.g., the contributions of correlated noise sources are also included. As we see, for calculating the signal spectral matrix S(f), the inde- pendence of the noise sources derived in the MAR model is not required.

The ordinary coherence function between signals i and j is defined as follows:

IS~(t312 C O H ii(f) - S ,( f )S ~jQ') (5)

(similar to a normalized cross power spectral density function). The COHi~Q) function has values between zero and unity and indicates the commonality, as a function of frequency, of the two signals, including the effects of all possible signal transmission paths (STP) connecting the two signals and the effects of correlated noise sources.

The corresponding phase shift between the two signals as a function of frequency is defined a:

82

PHASEo.(1) = atan f/ImlS'/f)l t Re[S#(f)] }

ISA Transactions • Vol. 29, No. 4

(6)

The ordinary noise source contribution ratio from the driving noise source of signal j to the auto power spectral density of signal i is defined as:

NSCRii(f ) = I{H(f)l}ql2~q At (7) s,,0

where ff,i is the jth diagonal element of the matrix ]~. The matrix H(f) -a contains all possible signal transmission path (STP) transfer functions that may exist between any two given signals. The set of NSCR functions represents the global coupling among the measured process variables. The ith signal's auto spectrum S,(f) in the denominator also contains the contributions from the off-diagonal elements of the covariance matrix ]~, while the expression in the numerator gives only the contribution from the noise sources of signal j, (crg.) to the APSD of signal i. In most cases the off- diagonal elements have negligible values compared with the diagonal values (uncorrelated driving noise sources); consequently, the sum of the individual NSCR functions for any given signal is close to one.

CNSCR,(D = ]~ NSCRij(j) (8) j=l

The deviation of this sum from unity, as a function of frequency, indicates the goodness of the selected signal combination and model order for the given frequency range, and it is used for posteriori checking of proper process decomposition. The above frequency functions represent the global contribution of one signal to the other through all possible signal transmission paths between signals i and j. Thus, they do not generally char- acterize the direct relationship or the direct STP of individual signal pairs.

Decomposition of Multivariate Dynamic System and Process Diagnostics

In order to get the individual transfer functions connecting the measured variables by eliminating the effect of the remaining variables, the following multivariate linear system modeling based on the MAR-modeling was introduced [3]:

X ( 0 = G(0X(O + N ( 0 (9)

where G(f) is the transfer matrix with Go.(]) in the (i,/')th element and zeros along the diagonal elements; it can be expressed using the MAR matrices by rearranging Eq. (I):

Go.(/) - - - and G~;(J) = 0 (10) H i i(J:)

The (i, j) element of the spectral matrix of the frequency dependent inherent noise source N(f) is

O'..

Q~i(D = {N,(t)N/(])} = v (11) tt.q)n;)(t)

which defines the auto and cross power spectral density functions of the residual or inherent noise field as a result of the multivariate signal decomposition.

By the system decomposition defined by Eq. (9), the auto spectral density function of each signal can be decomposed into contributions from the other signals' noise sources and from its own inherent noise source. Also, the cross power spectral density function of two given signals can be split into two parts in the model: (1) direct coupling between the two signals due to the contribution from the noise source of one signal to the other signal through the corresponding transfer functions (terms Gij(f)QijQ" ) and Gji(l)QiiO0 ) and (2) conlribu- tions from the correlated inherent noise sources (terms Q;~(]) and Q.;(f)). The spectral properties of the resulting inherent noise field were also calculated (APSD, CPSD, COH, and PHASE functions) and plotted together with the corresponding frequency signatures of measured noise signals. Although the above partition of signal coupling is interpreted correctly in the derived linear MAR model, the physical system in which the signals were measured may have a different structure. This ambiguity is due to the fact that the measured physical system and the fitted linear model are not fully equivalent. Their equivalence is confined only to the first n correlation matrices.

Partial Coherence and Partial Noise Source Contribution Ratio Functions

Using the transfer matrix G(f) and the inherent noise source spectral matrix Q from the MAR model, the partial coherence function between signals i and j is defined as follows:

PCOHJ) =

[Qii + GoOd + G].Q,, + G•i,Q,i] z (12)

(Qi, + IGoI2Q~- + 2 Re [GoQ~.](Q~. + IGj2Q,, + 2Re[G..~..~])

where the partial CPSD of the two signals is defined in the numerator, and the partial APSDs were defined in the denominator, excluding the effects of all other signals and allowing only the direct transmission between

the two given signals. The corresponding phase shift between the two signals, excluding the effects of oth- ers, is defined from the real and imaginary parts of the partial CPSD function:

rim[Q# + G#Qj + G,'Q~ + GiiG.**Q~] PPHASE1#(J) = atan /

"Re[Q + C Q,+ a; Q, + C g'Q l

Keeping only the terms connecting the given two signals directly, the partial noise source contribution ratio function from signal j to signal i has the form:

IGiy(])lZQi/f) PNSCR0.(]) - (14)

Qi,(f)+ IGIi(DI2Q~.(/) + 2Re[Gi~(I)Q'o.(])]

for i * j and the self-contribution of signal i from its own noise source:

0.65 PNSCR;j(f) = (15)

Q,,(f) + IGo(f)12Q~.(f) + 2Re[G;i(f)Q:~(f)]

The partial coherence and the partial noise source contribution ration functions relate any two signals directly by excluding the influence of other signals. This procedure is essential in the correct interpretation of the cause and effect relationships. In this context, it must be stated that the selection of a proper signal combination is very important and can be achieved by physical considerations of the system and preliminary experimentation with different signal combinations. The effect of hidden or unmeasured process variables that are not included in the MAR modeling are implic- itly involved in the system decomposition. All the effects of unmeasured signals on the measured signal set are represented by the inherent noise components defined and are specific to the given signal combination. Thus, physically significant process variables, unmeas- urable or accidently excluded from the STP analysis, can cause false STPs among the signals analyzed.

Note that the PNSCRo(]) function becomes zero if there is no direct transfer from signal j to signal i, while the PCOHii function still may have some non- zero value for the case of G0(f) = G/ f ) = 0 due to the contribution of correlated inherent noise components Qii(f). In case of uncorrelated noise sources (Qi~Q) = O) and without feedback from signal i to signal j(G;.(J) = 0), these two functions become identical.

The comparison of the above ordinary and their corresponding partial frequency functions provide information on noise generation and propagation mecha-

ISA Transactions • Vol. 29, No. 4 8 3

nism, and on the cause and effect relationships among the selected signals in a multivariate dynamic system. Besides creating a baseline pattern for normal system operation (containing statistical features sensitive enough to changes in the system), the above STP analysis can also be used for extracting process related tech- nical parameters in an actual application.

Unlike the two-signal Fourier analysis, the MAR- based STP analysis can reveal the inter-signal relationships of multivariate dynamic systems by processing all the recorded signals simultaneously instead of pairwise. The process fluctuation generated by the noise source of one signal can propagate to other signals through the process. This dynamic structure can be characterized by the MAR-based signal flow network of a multivariate stochastic process in a systematic manner. The above statistical features also reflect the possible correlation of the inherent noise components. By monitoring various STP-related signatures, diagnostics of system or sensor anomalies can be performed. The application of this technique in real processes may be automated by using pattern recognition and neural network techniques.

DATA ACQUISITION PROGRAM AT AN ALUMINUM ROLLING MILL

A detailed acquisition program was carried out at an aluminum rolling mill. The signals were recorded directly at the plant patch panel. The recorded signals included forces acting on the backup rolls (load cells), interstand sheet tensions, and exit sheet thickness (gauge) deviation. The sheet thickness is measured using an X-ray gauge. Only the wideband random fluctuation component of the signals was recorded during this measurement. A schematic of the five-stand tandem mill, showing the last two stands, is given in Fig. 1. The data length during the passage of the aluminum sheet through the five-stand mill is 210 seconds for each run. A total of fifty runs were recorded.

A block diagram, showing the various components of the data acquisition system, is given in Fig. 2. The plant signal patch panel was isolated from the data acquisition equipment by Burr-Brown Model 3456 TM isolation amplifiers. The signals are frequency band- passed using high-pass and low-pass filters. In order to achieve high resolution in the analog-to-digital (A/D) converter, all the signals are amplified using high-gain

S t a n d 5

W o r k R o l l ~, • i ( • ) Y

T e n s i o n ~ ! e 4- 5 ~ Sh t

X - r a y g a u g e

O T a k e - u p S p o o l

Figure 1. Schematic Showing the Last Two Stands of a Five-Stand Tandem Aluminum Rolling Mill

8 4 ISA T r a n s a c t i o n s • Vol. 29, No. 4

amplifiers. The A/D board is an integral part of the PC- based data acquisition. A 12-bit data translation A/D board was used in this application. It is necessary to perform a preliminary study of the signal behavior to set the band-pass filters and the amplifier gains.

A total of twelve signals were recorded during each run. The sampling interval is At = 0.04 sec (sampling frequency, f, = 25 Hz). The Nyquist frequency corresponding to this rate is:

5 - S T A N D

T A N D E M M I L L P R O C E S S S I G N A L S

Illlllll/ll I S O L A T I O N AMPLIFIERS

!!!!!!!!!If IIIGH-PASS FILTERING TO

REMOVE DC COMPONENTS

L O W - P A S S F I L T E R I N G TO

R E M O V E C O M P O N E N T S

A B O V E N Y Q U I S T - FREQUENCY

IIIIii111[ .Wp- .qF . ~ .Wr .V~ . ,V V V v v ,

lilll l .-Brr ANALOG TODIGITAL ] CONVERTEa AND PC-BASEDI

D A T A ACQUISITION [

Figure 2. Mul t ichanne l Data Acquis i t ion at an Alu- m i n u m Rol l ing Mill

L - - = 12.5 Hz 2

Two sets of 25 runs were recorded with different signal combinations. The high-pass and low-pass filter settings were 0.1 Hz and 10 Hz, respectively.

APPLICATION TO DIAGNOSTIC ANALYSIS OF A ROLLING MILL

Several wideband process signals from a five-stand aluminum rolling mill were recorded during more than

\

fifty different runs. These measurements are used to

establish the baseline wideband frequency characterization and to observe typical signature variations. De- viations from the normal behavior indicate either process anomaly or sensor malfunction. The process is not

continuous, and each block of aluminum lakes about 4 minutes to be rolled through the five-stand tandem mill.

Spectral J~'gnatures fop a Typical Run The following four signals out of the

twelve were selected for the MAR analysis: recorded

(1) Stand-4 load cell signal, drive side (STD4.LC.DRS)

(2) Stand-5 load cell signal, drive side (STD5.LC.DRS)

(3) Interstand 4-5 tension, drive side (STD4- 5.TEN.DRS)

(4) Exit sheet thickness (gauge) deviation (THICK- NESS)

An optimal MAR model of order n = 40 was used for the analysis. The model order was selected based on Akaike information criterion (AIC) discussed earlier. The AIC attains a minimum value at this model order. The empirical model order does not represent the order of the system dynamics and is influenced by the signal bandwidth, sampling rate, and the desired model prediction error.

Figures 3a-3d show the auto power spectral density (APSD) functions of the four signals (solid lines) and those of the derived inherent noise components (dotted lines) defined in Eq. (11). The signal bandwidth is 0.1- 10 Hz. The APSD functions of all four signals are characterized by resonances and wideband noise. The dif- ference between the signal APSDs and the ASPDs of the inherent noise components indicates contributions from other signals through signal transfer functions.

ISA Transactions • Vol. 29, No. 4 8 5

\ e

J 0 >

103~ i l l l

F , - - ' SIGNAL APSD: :

. . ------: INHERENT NO~SE APSD -

. . . . . . . . . . . . . . . . . . . . ;

FREQUENCY (HZ)

Figure 3(a). Power Spectral Density Function of Stand-4 Load Cell Drive-Slide Signal (STD4.LC.DRS) and the Inherent Noise Spectrum

N "r \

I- J 0 >

iZ2 ~~'"'i . . . . .

~0~,~j ~. ~! . . . . .

10~ ~ . . . . . . ~ ' ~ ~ £0-j-

I I I P. , , m

. . . , . . , , . . . . , . . . , . , . . . . . , . . ~ m

m

0 FREQUENCY (HZ)

Figure 3(b). Power Spectral Density Function of Stand-5 Load Cell Drive-Slide Signal (STD5.LC.DRS) and the Inherent Noise Spectrum

8 6 ISA T r a n s a c t i o n s ° Vol. 29, No. 4

SIGNAL APS~

N i~ . .i . . . . . . ~ , i . . .'7"7".-.". i. ,IN, H,ER, E.NT ,N.O.']~S]g ,A;PSD,

l]\ !:J ' : L : " A ~ izi ' ~ . .

£0-1g iZ

FREQUENCY (HZ)

Figure 3(c). Power Spectral Density Function of Interstand 4-5 Tension Drive-Slide Signal (STD4-5.TEN.DRS) and the Inherent Noise Spectrum

,1~4~ i':: ...... : ~ , : ; " \ ~ i . . . . .

FREQUENCY (HZ)

Figure 3(d). Power Spectral Density Function of Exit Sheet Thickness Deviation Signal (THICKNESS) and the Inherent Noise Spectrum

I S A T r a n s a c t i o n s • Vol. 29, No. 4 87

Figures 4a-4d show the cumulative noise source contribution ratio (CNSCR) functions for each variable (see Eq. (8)). For all the four variables, the CNSCR functions have values close to unity, indicating proper signal and model selection. The unity value for the cumulative noise source contribution ratio indicates that the cross correlations among the noise sources V are close to zero. This ensures the validity of spectral decomposition and thus the developed model.

Figures 5a-5d and 6a-6d show the ordinary and partial coherence functions along with the corresponding phase functions and the ordinary and partial noise source contribution ratio functions for signal pair STD 5.LC.DRS and STD 4-5.TEN.DRS, and signal pair STDS.LC.DRS and THICKNESS, respectively. The inherent noise spectrum for signals i and j is defined as (see Eq. (11)).

{ v Q,ff) = E . H~(D'" H,,(D I-I~(/)"

(11)

This quantity differs from "model prediction" error, which is given by E[Vy 7] = ¢r~r Thus, inherent noise spectrum is a frequency-dependent quantity; when Q.;([) = Su(D, the contribution to the power spectrum of a signal i is due only to its own noise source v:. This analysis provides an indication of the source of driving noise.

The highlights of the main results of this particular signal selection are discussed below. Note the match- ing between ordinary and partial coherence between stand-5 load cell and thickness signals. A similar comparison is seen in the ordinary and partial noise source contribution ratio between these two signals. A large contribution from thickness to load cell at non-resonant frequency band is seen. This indicates a unidirectional effect from one signal to the other.

Summary of Diagnosffc Results The salient features of the multivariate signal proc-

essing and diagnostics are summarized below.

Table 1 Five Stand Tandem Mill Data for a Typical Run

Stand Work Roll Backup Roll Strip Speed, No. Frequency Frequency Actual

(ft./min.) Hz RPM Hz RPM

1 0.295 17.7 0.133 8.0 126.2

2 0.515 30.9 0.234 14.0 224.1

3 0.882 52.9 0.415 24.9 364.7

4 1.280 76.8 0.608 36.5 533.2

5 1.716 103 0.784 47.0 692.8

Reel width = 80 inches. Sheet thickness = 0.12 inch.

(3)

(4)

(5)

delay. The resulting propagation to the backup roll results in the characteristic exhibited by the stand-5 load cell signal spectrum.

The linear phase behavior between these two signals (sometimes corrupted by frequency peaks) characterizes pure propagation dynamics transmitted from the stand-5 location to the X- ray gauge. The power spectrum is a mixture of two types of components in the signals: a wideband noise component, which is part of the physical behavior of metal flow, thickness re- duction, etc., and spectral resonances due to vibration of rotating machinery.

The 0.78 Hz in the spectrum of stand-5 load cell signal peak corresponds to the stand-5 backup roll frequency (see Table 1).

The interstand 4-5 tension signal has a high coherence with stand-5 load cell signal (down- stream stand) compared to the stand-4 load cell signal (upstream stand). This was also observed by Bryant [8] using rolling mill dynamic models.

(1)

(2)

The power spectra of thickness deviation and stand-5 load cell signals have dominant peaks at 3.4 Hz and 6.8 Hz, which are the higher har- monics of the fundamental stand-5 work roll frequency of 1.72 Hz (see Table 1).

The source of these two peaks is the modulation in the work roll separation, which results in the sheet thickness modulation with a constant time

(6) The estimated propagation or transit time from stand-5 load cell to X-ray thickness gauge de- tector is t -- 0.5 sec.

Sheet velocity at stand-5 estimated by using MAR analysis = 659 ft/min. Preset sheet velocity at stand-5 = 660.8 ft/min.

This indicates that a good estimate of the sheet


1 , 5 I

NSCRI4 NSCR13

~ 0 5 . . . . .

NSCR12 NSCRII

, , . • ° . i , . ° , , . ° ° • i , , , , , . * ~

FREQUENCY (HZ) 1 0

Figure 4(a). Cumulative Fractional Contributions to the Power Spectrum of Stand-4 Load Cell Signal (STD4.LC.DRS) from the Noise Sources of Other Signals

1.5

z.I. 0

o

0 (Z

I

NSCR24 NSCR2 3 NSCR22

4 I~ 8 10 FREQUENCY (I ~'')

Figure 4(b). Cumulative Fractional Contributions to the Power Spectrum of Stand-5 Load Cell Signal (STDS.LC.DRS) from the Noise Sources of Other Signals

ISA Transactions • Vol. 29, No. 4 89

Z 0 I-I

0 ¢

h

1, 5[ NSCR34 NSCR33

i

~ . 5 ~ ~ s ~ 3 . ~ ...: . . . . . . . . . . . . . . . .

FREQUENCY (HZ)

Figure 4(c). Cumulative Fractional Contributions to the Power Spectrum of Interstand 4-5 Tension Signal (STD4-5.TEN.DRS) from the Noise Sources of Other Signals

Z 0 14

I- 0 ¢

1.51

0 . 5

0.~

NSCR44

2 4 S 8 LO FREQUENCY (HZ)

Figure 4(d). Cumulative Fractional Contributions to the Power Spectrum of the Exit Sheet Thickness (Gauge) Deviation Signal (THICKNESS) from the Noise Sources of Other Signals


~' I I I I ~ : ~ , . . . . . . . i . . . . . . . ~: . . . . . . . . . . . . . . . . . . . . . . . . . . ~ :

o . . . . . . . . . . . . . . . . . . . . . . . . .

i l l

' i . . . . . . . : . . . . . . . . : . . . . . . . .

: : . . . . . . : A . . . . . . . . . . .

2 0 0 ~ , , , . ,

~o~:l. ,~ ! ........ ! .A .... !fl =.~/

~"~-I: J ~ " " - ! Y ! ~ ~ "<" : :

FREQUENCY (HZ) 71

Figure 5(a). Coherence (Top) and Phase (Bottom) Relationship between Stand-5 Load Cell Signal (STD5-LC.DRS) and Interstand 4-5 Tension Signal (STD4-5.TEN.DRS)

" - - ' T - " - " - " " ' - ~

. i . . _ i , i . i... °~:

zzz • ~ ' " ' ~ . A ~ , ' . .

~ 1 0 i ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . .

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FREQUENCY (HZ)

Figure 5(b). Coherence (Top) and Phase (Bottom) Relationship Betwen Stand-5 Load Cell Signal (STD5-LC.DRS) and Exit Sheet Thickness (Gauge) Signal (THICKNESS)

ISA Transac t ions • Vol. 29, No. 4 91

0 h

O

Z

0 i -

ra O ¢0 Z

0[ T T "1' T

~,8' . . . . . . . " . . . . . . . . : . . . . . . . . ', . . . . . . . . " . . . . . . . .

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g , , • o i a , , I , * • , a • i o ~ , , * ° . I • , I , • • , ° , , • ° , t g , • , o i i 0 4 , . . . .

o . . . . . . . . . . . . . . . . . . . . . . . . . .

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0, 0 0,

FREQUENCY (HZ)

Figure 5(c). Fractional Contribution to the Power Spectrum of Stand-5 Load Cell Signal (STD5.LC.DRS) from Interstand 4-5 Tension Signal (STD4-5.TEN.DRS) (Top). Fractional Contribution to the Power

Spectrum of STD4-5.TEN.DRS Signal from the STD5.LC.DRS Signal (Bottom)

,4

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FREQUENCY (HZ) 0

Figure 5(d). Fractional Contribution to the Power Spectrum of Stand-5 Load Cell Signal (STD5.LC.DRS) from Exit Sheet Thickness (Gauge) Deviation Signal (THICKNESS) (Top). Fractional

Contribution to the Power Spectrum of THICKNESS Signal from STD5.LC.DRS Signal (Bottom)


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, , , , , , , , , , , , , l , , , , , , , , o , , , , , , , , , , , , , , , , , m

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\ °

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FREQUENCY

I I

[HZ)

Figure 6(a). Partial Coherence (Top) and Phase (Bottom) Relationship Between Stand-5 Load Cell Signal (STD5.LC.DRS) and Interstand 4-5 Tension Signal (STD4-5.TEN.DRS)

T 0 O a.

10 I . . . . . . i . . . . .

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o

FREQUENCY

.d.__

(HZ) Z

Figure 6(b). Partial Coherence (Top) and Phase (Bottom) Relationship between Stand-5 Load Cell Signal (STD5.LC.DRS) and Exit Sheet Thickness (Gauge) Deviation Signal (THICKNESS)

I SA T r a n s a c t i o n s • Vol. 29, No. 4 93

°° i ~0 o0 zO ~ g

0 0 ,

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~0,12

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4 [ - . . . ~ . . . i . . . . . . . . ; . . . . . . . . : . . . . . . . . . : . . . . . . . - J

• i i i i i i i i i i l l . . . . . . . . ! . . . . . . . . . ! . . . . . . . _ F /, . . \ 2 ~L - - J ~ o . 0 ~ T I 8 . . . .

FREQUENCY [HZ]

Figure 6(c). Partial Fractional Noise Source Contribution to the Power Spectrum of Stand-5 Load Cell Signal (STDS.LC.DRS) from Interstand 4.S Tension Signal (STD4-S.TEN.DRS)

(Top). Partial Fractional Noise Source Contribution to the Power Spectrum of STD4-S.TEN.DRS Signal from STDS.LC.DRS Signal (Bottom)

o0.8 ~O.B m04 z0.2 ~0.0 NI.O o0.8 ~ g . G u 0 . 4 z 0 . 2

a-O,~

T 1 T I

E a l , l • , I I D I I , , I I I j , , • , I • , , , . ,

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- - - - - - - - r - - - - - - - - - - ~ r

FREQUENCY [HZ]

Figure 6(d). Partial Fractional Noise Source Contribution to the Power Spectrum of Stand-5 Load Cell Signal (STD5.LC.DRS) from Exit Sheet Thickness (Gauge) Deviation Signal

(THICKNESS) (Top). Partial Fractional Noise Source Contribution to the Power Spectrum of THICKNESS Signal from STDS.LC.DRS Signal (Bottom)


(7)

propagation velocity may be obtained by careful signal processing.

The cause and effect analysis shows that the 3.4 Hz and 6.8 Hz resonances are not related to sensor fault but are caused by roll-gap fluctuation. If this is due to the sensor vibration, then coherence between the two signals would be very small, thus indicating an independent noise source.

mated diagnostics may be performed for plant anomaly and sensor maloperation detection. Because of the normal variations in the signatures from one run to another, it is necessary to build a large statistical data base. It is also necessary to incorporate available knowledge about the system behavior and the performance of instrument channels. Sensor redundancy must also be used to improve the reliability of decision making.

(8) Inherent effects, not related to other signals, indicate local sensor or gauge effects only; as an example, the interstand 4-5 tension signal has a resonance at 8.9 Hz and is not related to other signals.

This application shows that the approach may be used to isolate the source of perturbation that causes fluctuations in process variables (e.g., bearing wear, roll eccentricity, or misalignment).

CONCLUDING REMARKS A methodology, based on the multivariate analysis

of process signal fluctuations using time-series modeling, is described in detail. The modeling strategy, estimation of optimal model parameters, and the derivation of spectral-domain signatures are presented. A complete set of software systems for signal processing and diagnostics have been developed and implemented in a PC.

A comprehensive study, including theoretical analysis of fault diagnostics, development of multivariate signal processing algorithms, and application to an op- erating metal rolling mill, has demonstrated the feasi- bility of using this modern digital signal processing approach for process dynamics characterization and possible anomaly detection and diagnostics. After establishing a baseline system characterization, auto-

REFERENCES 1. Kesler, S. B., Modern Spectrum Analysis II, NY:

IEEE Press (1986). 2. Marple, Jr., S. L., Digital Signal Analysis with

Applications, NJ: Prentice-Hall (1987). 3. Upadhyaya, B. R.; Kitamura, M.; and Kerlin, T:

W., "Multivariate Signal Analysis Algorithms for Process Monitoring and Parameter Estimation in Nuclear Reactors," Annals of Nuclear Energy, Vol. 7, pp 1-11 (1980).

4. Glockler, O., and Upadhyaya, B. R., "Results of Interpretation of Multivariate Autoregressive Analysis Applied to the LOFT Reactor Process Noise Data," Progress in Nuclear Energy, SMORN-V (1988).

5. Oguma, R., "Coherence Analysis of Systems with Feedback and Its Application to a BWR Noise Investigation," Progress in Nuclear Energy, SMORN-III, Vol. 9, pp 137-148 (1982).

6. Oguma, R., and Turkcan, E., "Application of an Improved Noise Analysis Method to Investigation of PWR Noise: Signal Transmission Path Analy- sis," Progress in Nuclear Energy, SMORN-IV, Vol. 15, pp 863-873 (1985).

7. Box, G. E. P., Jenkins, G. M., Time Series Analy- sis: Forecasting and Control, CA: Holden-Day (1970).

8. Bryant, G. F., Automation of Tandem Mills, Lon- don: The Iron and Steel Institute (1973).

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multivariate statistical signal processing technique for fault detection and diagnostics

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