ML19275A503

Use of Time Series Analysis in Nuclear Reactor Safety Applications

ML19275A503
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Issue date:	08/17/1979
From:	Albrecht R NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES)
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THE USE OF TIME SERIES ANALYSIS IN NUCLEAR REACTOR SAFETY APPLICATIONS A consulting report to U. S. Nuclear Regulatory Commission by Robert W. Albrecht August 17, 1979 requested by William S. Farmer Division of Reactor Safety Research and Chairman, Noise Surveillance and Diagnostics Review Group CONTENTS BACKGROUND.............................................................

1 INTR 000CTION...........................................................

1 REVIEW OF METH0D.......................................................

3 The general model l i ng pro bl em...................................... 3 T i me s e r i e s o p e ra to rs............................................. 3 Moments of stationary time-series model s........................... 5 Es ti ma ti o n o f AR pa rame te rs........................................ 6 E s ti ma ti o n o f MA pa rame te rs........................................ 8 De termi na ti on o f model order....................................... 8 Th e re s i du al va ri a nc e.............................................. 8 APPLICATIONS OF TIME SERIES ANALYSIS...................................

9 Ge n e ra l a s s ump ti o n s................................................ 9 Cl asses of problems and appl ications............................... 9 Fo r e c a s t i n g........................................................ 9 Pl ant and componen t mode 11 i ng..................................... 10 Control system design.............................................

11 I mp rov i n g ro bu s tnes s.............................................. 12 Transfer functions and transit time model s........................ 14 R E C O MME N D AT I O N S.............................................

REFERENCES........................................................... 18 bWYO bb 5910 040 ff p

THE USE OF TIME SERIES ANALYSIS IN NUCLEAR REACTOR SAFETY. APPLICATIONS Robert W. Albrecht August 17, 1979 BACXGROUND During a consulting visit to ORNL (6/6/79) and to NRC (6/7, 6/8/79) the subject of time series analysis by autoregressive and moving average modelling was discussed. 'It was observed that the nuclear industry has made little practical use of these techniques although a limited number of applicaticns have been reported in the literature.

In some cases it was felt that extravagant claims have been made for the potential benefits in noise surveillance and diagnostics of time series analysis.

In other cases it appears that the application of t'me series analysis to the estimation of parameters in noise measure-ments resulted in quite unsatisfactory results. This consultant, ORNL researchers and NRC officials agreed that an appropriate activity in the area of noise surveillance and diagnostics is to inves'.igate the application of time series analysis to nuclear reactor safety problems.

ORNL has been funded to look into this area.

fir. William S. Farmer (NRC/RSR) requested that this consultant prepare a report to assist ORNL in formulating a program to invest-igate time series analysis applications. This report is the response to that request.

INTRODUCTION The subject of this report is the analysis of discrete time series data through the building of stochastic models.

In contrast, the majority of all analyses over the past two decades in the area of reactor noise surveillance and diagnostics has involved spectral density or correlation estimation.

In these methods, characteristics of the moments of the measured process such as mean, variance, periodicities, time delays, break frequencies, etc.

are determined.

To relate measured results to physical models the iUY833[p

2 analysis depends upo.n a theoretical development that predicts the It is often true that such models do not directly measured moments.

For reveal the characteristics of the underlying stochastic process.

example, if one were in posession of a model for the spectral density of a process and a measurement of the spectrum from a finite time series record, this information would give little guidance to forcasting the behavior of the process to future times.

The contrast between auto regressive and moving average (ARMA) methods in time series analysis and traditional spectral density and correlation analysis is basically that traditional methods relate measured averages to averaged models whereas ARMA methods seek to find parameters of a time series model that represent the underlying The determination of correlation functions and spectrum estimates process.

~

ARMA may be performed with no knowledge of the underlying process.

models are of ten constructed using many correlation function and spectrum Each correlation and each spectral ordinate is a parameter to estimates.

Therefore if non-structural approaches alone (correlations be estimated.

and spectra) are used to characterize a time series the procedure may be very prodigal with parameters.

The ARMA model approach has the promise of yielding a representation of the underlying process with very few parameters and therefore being a much more economical (and perhaps physically appealing) description of the process under study, producing a satisfactory ARMA model to represent a physical process generally requires considerably more detailed study of the process by a skilled analyst than does the traditional analysis and interpretation of averaged moments of stochastic processes.

The basic In this report, traditional methods are not reviewed.

principles of ARMA methods are discussed and potential applications in the araa of reactor noise surveillance and diagnostics are suggested.

Recommendations for investigations of ARMA methods are summarized.

lNb 33

. REVIEW 0F METHOD The general modelling problem The process of selecting a time series model is generally iterative.

In the most favorable case, one completely understands the physical mechanisms underlying a time series.

If such were the case, it would b'e possible,.in principle, to write down a mathematical expression that described it~ exactly (perhaps with undetermined parameters).

The other extreme is represented by complete ignorance of the underlying mechanisms.

In this case, a purely empirical model could be attempted.

The usual case lies between these extremes.

Incomplete theoretical knowledge can be used to find a class of mathematical functions to be fitted empirically. The number of terms needed in the model and numerical values of parameters are estimated from experimental data.

It is important, in practice, that the smallest possible number of parameters be employed for adequate representation of the process.

A model possessing this smallest number of parameters is termed parsimonious.

To determir.a the appropriate model an iterative procedure involviag thret steps termed identification, estimation, and diagnostic checking is normally followed.

Identification refers to the activity of selecting an appropriate oarsimonious class of models to be enter-tained based on data and knowledge of the system.

Estimation is performed by fitting the model to data and estimating parameters.

Diagnostic checks are performed to reveal if the chosen model produces an adequate fit and if the number and values of parameters are appropriate to the physical problem under investigation. These procedures are repeated, as necessary, until an adequate model is establishedU.

b3 Time series operators Shift operators are of fundamental importance to time series models.

The backward shift operator, B, is defined by Bzt"Zt-1 where z is the observation of a time series at time t.

Therefore B t

b 0' 0 bb

. operating on z pr duces the value of the previous observation z t

t-l

• Obviously, B*z

=z t

t-m' The forward shift operator F = B-I is given by Fz t " Z +1 '

t The backward difference operator is denoted by 9' and 7z

=z

- z _) = (1-8)z '

t t

g t

The inverse of 7 is the summation operator S given by

-I t " bZt

• t-j * (I-0) Z 7 z t*

f. e lhe autoregressive operator of order p is defined by P

p(B) = 1 - 9') B - 9 0 ' -NB 2

p and the autoregressive model is 9(B)z

=a t

t where 'E is the deviation of z rom s meanft.and a is a white t

t t

noise process with variance The autoregressive model contains p+2 parameters [,Y),'f'***/p'I' 3

2 The moving average operator of order q is defined by 2

9 G( B ) = 1 O) B - 0 B

...GB 2

q and the moving average model is

'z

=G(B)a t

t containing q+2 parameters;[, G, G #q' j

2 The mixed autoregressive - moving average model is

$(B)E "

"t t

with p+q+2 parameters;

u., 4),... (p, &).... G,

l()98 jj q

EI)

Moments of stationary time-series models The mean is defined by

/s.=E[z]=

zp(z)dz t

where p(z) is the probability distribution of z for any arbitrary t

time t.

The variance is

?'

2.

%=E[(z-/03*

(z-f)2(z)dz p

t

-v Estimates of the mean and variance are given by

, 11 1-

_z=772z t

f. ' h l

' 2 = J[J_,(z z)2 7 2 t

t al The autocovariance at lag k is Y = cov [z,z +k] = E[(z i#)(*t+k-#0 k

t t t

and the autocorrelation at lag k is E[(z p)(z +k-?)3 Ik t

t

[k*

a2

• f~o*

z Spectra are the Fourier transforms of correlation functions.

For a general linear stochastic process

$~.

Y=a +2}a _)

t t

t f'

where fj are the weights of the linear operator f(B)=1+9')B+PB2,,,,

2 that transforms a into z called de transfer funcdon.

t t

Ohb

. The autocovariance function of this general linear process is 2

Y " 7 N j Nj+k

• k a

f%

The autocovariance generating function for this process is v.

I(B)=2XB

• T f(B) f(B

)=

f(B) D ).

k g

a Es p When B is treated as a dummy variable in a generating function it may have complex values.

b Estimation of AR parameters To estimate the AR parameters it is useful to formulate a relationship between the autocorrelation function of an AR process and the AR parameters (7,(/2,...

in the autoregressive model

((B)T = a t

t or Y*fZl t-1 +

2 t-2+ +'E zp t-p t*

z

+a t

The autocorrelation functions can be formed by multiplying the above process throughout by Y and averaging (using E[Y t-k t t-k3

• k)*

After dividing through by Y and identifying g

Ek 8*G k

the autocorrelation function is found to satisfy a difference equation of the same form as the AR process above, namely

['k"kl[k-1+ 2[k-2+f[k-p (k)0) or

@(B)[k =0 where the shift operator acts on k (and not t) in this case.

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900T0MIWL If the values of k = 1,2,... p are substituted into the above difference equations then a linear. set of equations (called the Yule-Walker equations) for the AR parameters 9),(f ' * *

• Ip are obtained in 2

terms of the autocorrelations[),/>2'

• * ' /' k

+@p['p-l

/'iz =(f)

+ /f ['l

+

2

• Y ['l + 4' 2

'"+7'p['p-2

[2 l

,8 p " @],[p.1+Il'2# p-2+

• * * * + fp or

<p where

,.. - ~ ~ hi 1

I

[

[

{~g~

q,

. f; 1':

f'i I

, -:.I I

e -

I'.

i ff f,'

Y kr

,/h -t,h -7,

' ' ~ ~

f and recalling that for autocorrelation functions

=[-k*

Solving for the AR parameters, one obtains d/ = p -I__p f p,

The parameters If of the AR process can be estimated by replacing the theoretical autocorrelations[k by estimated autocorrelations Y '

k Estimated autocorrelation fundtions may be constructed by a series of partial autocorrelation function estimates. Techniques for accomplishing this by recurrsive methods are at the heart of AR parameter estimation theory and have been the subject of extensive research.

It is not the purpose of this review to discuss this field, only to note that, in 1098 340L

. principle, the AR parameters can be approximately determined from a time series record and these parameters can be used for model identification.

b]

Estimation of the MA parameters Unlike the Yule-Walker equations for an AR process, which are linear, the equations relating autocorrelations to MA parameters are nonlinear.

Except for the simple first order process the equations must be solved iteratively.

Although initial estimates of MA parameters may not have high statistical efficiency, they can provide useful estimates for model identification and starting values for an iterative procedure that converges to efficient maximum liklihood es timates.

Determination of model order [2,3,4]

For AR models the model order for recent publications in the field is determined by Akaike's information criterion. The decision procedure for model order is based on the minimum final prediction error which implies an optimum choice of the order of an AR model in the sense of information theory. The Akaike criterion produces a unique order for AR processes.

The lack of a unique criterion to determine order for ARMA or MA processes is a drawback. The decision on order may be very tedious and involve considerable judgement on the part of the analyst.

In any case the model order is quite critical to the success of time-series analysis in identifying models and estimating parameters.

The required order of the model is related to the form, or structure, of the model. More successful illustrations of the application of time-series analysis to engineering problems make use of physical principles and empirical data to identify the model and estimate parameters.

Careful attention to model structure and order is required for valid interpretations and forecasts to be produced.

3 The residual variance If a model is tentatively identified for a tirr< series and parameters are estimated for the model then a study of the residual variance between the model and the data can often be used to determine if the model is adequate.

Several techniques for performing such diagnostic checking are discussed in the literature.

The variance and distribution 1098 3 @

of the residuals may give immediate clues to the adequacy of the model.

Often correlation functions and spectra of residuals are examined for such tell-tale signs as periodicities.

In general, the residuals are expected to exhibit an approximately white noise characteristic if the model and estimated parameters are adequate.

The distribution of residuals may be used not only to check on the adequacy of a time series model but may also be used constructively to modify the model.

APPLICATIONS OF TIME SERIES ANALYSIS General assumptions In discussing potential applications of the methods of time series analysis to nuclear reactor operating and safety problems several assumptions are made.

Basically it is assumed that sufficient physical information and empirical data is available to construct and identify a madel for the problem of interest, parameters of the model are estimated and diagnostic checks have been performed to give confidence in the performance of the model. That is, it is assumed that all of the foregoing discussion on time series methods has been successfully applied to a particular problem. The question, then, is what classes of problems in the nuclear power plant field appear to be most appropriate for application of these methods and, more particularly, what specific problems are ideal candidates for time series modelling.

Classes of problems and applications In this section, several general classes of problems to which time series analysis may be applied are discussed. After each general class of problems is discussed, potential specific applications in the nuclear power plant field will be mentioned.

Forecasting The use of time series data up to a time t together with an appropriate model to forecast probable values at a future time is a classical application of time series analysis. The objective is to obtain a forecast function for various lead times such that the mean b0 0 b

square of the deviations between the actual and forecast values is as small as possible.

In addition, specifying the accuracy of forecasts is of considerable significance. The accuracy of forecasts is often expressed by calculating probability limits. For example one may find the 95% confidence limit, or the limits such that the realized value of the time series, when it occurs, is expected to be included in these limits 95% of the time.

The potential uses of forecasting in nuclear reactor applications appears to be nearly unbounded.

It is easy to imagine any number of control room actions that may result in the change in a plant variable.

Examples are count rate changes at startup for rod withdrawals, pressure changes in response to throttle variations, secondary level changes in response to feedwater pump failure, power changes associated with coolant conditions, coolant temperature response to pump variations, etc.

In all of these cases a good plant madel based en a combination of analytical methods and empiricism is likely to be capable of producing a reasonable forecast.

An obvious use of forecasting is as an aid to power plant operators.

In many situations the operator could call for a forecast before or during an operational maneuver and then compare actual responses to expected behavior within probabilistic limits. Such comparisons could be automatically made in a plant computer and alerts could be provided if actual behavior deviated significantly from expected behavior for a specific variable or for a set of variables.

Plant and component modellina[1,5,6,7,8,9]

The most common application to date of time series analysis methods to nuclear power plant problems is in the area of plant modelling. As discussed earlier, modelling implies the triad of activities called identification, estimation and diagnostic checking.

In a large power plant the model will be more complex than discussed previously since it wiil be a multivariable model. Modelling can be applied to several types cf problems including:

0]0 b

plant optimizatiori forecasting noise source identification malfunction identification and diagnosis upset analysis stability analysis Application of time series analysis to modelling implies that the activity of modelling involves coordinated use of basic plant physical models and empirical data obtained either in an undisturbed plant or a plant responding to programmed disturbances such as rod bumps or PRBS signals. An objective of the methods of time series analysis in improving power plant modelling is to use these methods not only to identify models and estimate parameters but also to diagnostically test models through study of the distribution of residuals in order to improve both the structure and parametric values of power plant models.

Good models are difficult to argue against in an absolute sense.

On the other hand, empirical testing for the purpose of model identif-ication, parameter estimation, or diagnostic checking can be quite expensive, time consuming, and disruptive.

It is often these latter consequences that significantly limit the amount of empirical data available to the analyst in formulating and evaluating models. Because of costs, it is imperative that model requirements be prioritized and experimental tests be kept within reasonable limits. Time series analysis can be of value in providing a systematic framework to evaluate the needs for more tests and the adequacy of interim models.

D 0]

Control system desian If the purpose of an identification / estimation problem using time series analysis is to design a control system the character of the problem may vary widely depending on the nature of the control problem.

Examples are given below:

design a stable regulator design a control program for optimal transition from one state to another design a regulator which minimizes the variations in process variables d,e to disturbances The following discussion of these examples is excerpted from [10].

To design a stable regulator it may be sufficient to have a fairly b0'Y0 bk crude model of the system (or plant) dynamics. The optimal transition problem may require a more accurate plant model.

In the last problem it is necessary to model the environment as well as the planc.

In most practical problems there is seldom sufficient a priori information about a system and its environment to design a control system from calculated data only. At least certain parameters of the control system are usually left to adjustment. Therefore', an experiment is usually necessary. There are often limitations on the experiments that can be performed, however. The experiments may have to be carried out during normal operation (or small perturbations), other controllers may be in operation during the experiment, and experimental conditions may not represent the range of conditions under which the controller to be designed will be called upon to operate.

When the design of a partially known process is approached using identification / estimation techniques one assumes that the centroller dasign can be divided into two steps. The first is to approximate the system to be controlled and the second is to design the controller.

This procedure is seldom an optimum strategy and it is of ten necessary to iterate between these two activities.

A great deal of research has been directed toward the use of identification methods and time series models in control system design.

It is apparent that the nuclear power plant field can take advantage of this experience to better optimize control systems.

Dl,12]

Imoroving robustness Time series analysis can play a unique role in decreasing the f;>.lse alarm rate associated with a reactor surveillance and diagnostic system. The potential for false alarms is one of the most inhibit.ng problems associated with the wide-spread application of stochastic surveillance and diagnostic systems.

A system that is resistant to a broad class of disturbances represented by a departure from underlying statistics is termed robust. For example, of a few data points having values far from the normal signal levels enter a sample (e. g. from an electrical disturbance) it is desirable that this disturbance not signal a component failure. The algorithm for surveillance md diag-nosis should be capable of distinguishing an insignificant disturbance from a true equipment malfunction. The more resistant the surveillance system is to such disturbances the more robust it is.

Current methods of setting alarm thresholds for portions of a sampled PSD (for example) can be improved by employ.ing estimation methods that are specially designed to be more robust. This can both lower the false alarm rate and allow a more sensitive threshold for true failures.

Non-parametric spectral estimation is relatively robust for spectra consisting of a narrow band or low frequency component and little else. However non-parametric spectral estimation techniques are lacking in robustness for the case of comparatively low amplitude peaks in a wide range spectrum.

An attractive method.for improving robustness is to make spectral estimates by different methods that have differing sensitivity to disturbances.

For example, the two estimates can effectively have different approaches to pre whitening.

The smoothed periodogram and autoregressive spectral densit) estimates are limiting versions of the pre whitening approach. The forme amounts to no pre whitening and the latter corresponds to total pre whitening if the process is truly a finite order auto-regression with known order p.

In practice the autoregressive spectral estimate is only an approximation to total pre whitening.

From the two spectral estimates just discussed, it is possible to compute a hybrid parametric / nonparametric spectral estimate.

This hybrid estimate can be computed by robustifying both the parametric and non-parametric procedures.

The cost of robust estimation is more computation.

In addition, robustifying the estimation procedure can introduce some additional variability and bias when the stochastic process is gaussian.

Nevertheless, it appears probable that the use of robustifying algorithms in conjunction with reactor surveillance and diagnostics systems may have a considerable

~

payoff.

. An example method for applying nbust estimators to reactor surveillance problems is to use a two-step proces ;.

Assume that alarms are based on the threshold of a PSD estimate being exceeded in a pre-specified frequency range (e. g. for vibration monitoring).

If a threshold is exceeded using standard nonparametric estimation, this could auto-matically call a robustifying algorithm that would re-calculate the spectral estimate from setored data using appropriate robust techniques. The result of this second estimate can be used in pre-programmed decision logic to decide if the cause of the potential alarm is likely to be from an external disturbance or from an actual equipment fault. This technique allows more time consuming robust estimates to be made only when there is cause to believe that a fault or a false alarm may exist. Therefore, this application can be considered as a first step in the clagnosis phase of a surveillance and diagnostics system.

b' Transfer functions and transit time models ORNL is heavily engaged in a program to optimize data analysis of impedance probes to deduce transit time. The data is a non-stationary stochastic process at two probes and it is required to estimate the delay of the cnss correlation function or the phase of the cross power spectral density function. ORNL's methods are based on non-parametric estimation techniques involving no model relating the probe signals. The phase is simply related to the delay time and thus the flow velocity.

It is suggested that the accuracy and reliability of time-delay estimation could be improved by adopting parametric methods.

Non parametric athods for transfer function estimation are often unsatisfactory because they involve the estimation of too many parameters.

For example, to determine the gain and phase characteristics between two sensors it is necessary to estimate two parameters at each frequency.

The parametric approach changes the problem to the estimation of parameters in parsimonious difference equation models. The number of parameters can easily be greatly reduced and, in addition, the precision of the estimates can be more systematically assessed. Another potential advantage of parametric techniques is that non-stationaries enter the estimation problem in a definite way and they can be essentially removed by data manipulation.

!Olb

. In the case of stationary stochastic processes the autocorrelation function will die out for moderate to large time delays. A tendency for the autocorrelation function not to die out quickly suggests that the underlying stochastic process be treated as non-stationary. This non-stationary characteristic can be removed by differencing, that is, applying the operator V to the original time series data. The success of differencing in removing non-stationarity can be tested by examining the rate of decay of the autocorrelation function.

After differencing, the resulting time series can be identified and estimated by parametric methods in either the time or frequency domain.

The transfer function model may be identified not only by a time delay (or phase shift in the frequency domain) but also by estimating other parameters such as decay constants, etc. The estimatien of parameters in addition to the time delay gives added information upon which to judge the quality of the time delay estimate (by examining how well the estimates agree with model expectations).

In the time domain estimation methods considerable simplification in identification and parameter estimation can be achieved by pre whitening the input spectrum using the parametric model for this purpose.

If frequency domain methods are used, pre whitening may not be required.

Applications of transfer function identification and estimation have apparent value to NRC's program in the area of impedance probe data reduction for the 3D experiments and this technology should be investigated.

RECOMMENDATIONS The principle agent of NRC to provide technical assistance in the grea of noise surveillance and diagnostics is ORNL.

Recent developments ia the field of time series analysis and parametric estimation ass 7ciated with identification, estimation, and diagnostic checking show pronise of having valuable applications to surveillance and diagnostics pro'al ems.

ORNL has had little experience with these methods but in order tc properly fulfill their mandate to remain abreast of technical developments it is advisable that either ORNL improve their capability in this area 1098 f$

. or that NRC request another contractor to provide this service.

In view of the above, it is generally recommended that NRC provide support to a qualified contractor (ORNL or other) to develop the capability to advise NRC on the state of the art and most promising applications in this field. As was mentioned in the " background" section, claims for time series analysis have ranged from extravagent to disappointing.

It is important that NRC's agent be competent and objective so that this technology can be evaluated in a balanced manner. As mentioned in the " introduction" section, producing a satisfactory result using time series methods generally requires more skill and persistence from the salyst. This means that NRC's agent should have exoerience with real problems in significant depth in order to adequately evaluate the feasibility and level of effort required to apply these methods.

To accomplish this general recommendation it is suggested that the following actions be taken by the designated contractor.

(1) Algorithm development to generate time series based on AR, MA, and ARMA models, uni and multi variate.

(2) Familiarization with relevant techniques and applications for identification / estimation reported in current literature including an extensive literature review.

(3) Algorithm development to identify models, estimate parameters, and diagnostically check a large class of sample time series data using the methods discussed in

..s report plus many of the ancillary techniques reported in the literature.

A special suggestion is that NRC might consider the participation of two laboratories who would trade data and analyses.

That is one labor-atory would generate (contrive) time series data and send it with a description of some (but not all) of the rules and parameters used in the generation to a second laboratory in a suitable digital format.

The second laboratory would apply their methods to the analysis of the data in a best effort to uncover the underlying model and its parameters.

This procedure would also operate in reverse.

This suggested procedur would greatly help in evaluating the capability of laboratories to efficiently analyze data and draw reasonable inferences. Naturally the classes of data and the analysis methods can become increasingly refined as this procedure continues.

Of course this suggested procedure could be an intrala'aoratory instead of interlaboratory exercize.

However, the competition may not be so keen in this case.

199835f

. (4) Application of time series analysis methods to forecasting, modelling, control system design, robustifying, transfer function estimation or other areas. The most promising applications to be decided upon by NRC and the designated contractor (s).

(5) Assistance to NRC in evaluating and testing proposed applications of time series analysis in the area of reactor surveillance and diagnostics.

e

. REFERENCES

[1]

G. E. P. Box, G. M. Jenkins, " Time Series Analysis:

Forecasting and Control", Holden-Day, San Francisco (1976).

[2]

H. Akaike, " Fitting Autoregressive Models for Predictions",

Annals of the Institute of Statistical Mathematics, 21, 243-247 (1969).

[3]

H. Akaike, "A New Look at the Statistical Model Identification",

IEEE Transactions on Automatic Control, AC-19, 716-722 (1971).

[4]

E. Parzen,"Some Recent Advances in Time Series Modelling", IEEE Transactions on Automatic Control, AC-19, 723-730 (1974).

[5]

K. Fukunishi," Diagnostic Analysis of a Nuclear Power Plant Using Multivariate Autoregressive Processes", Nuclear Science and Engineering, 62, 215-225 (1977).

[6]

K. Fukunishi, " Noise Source Estimation of Boiling Water Reactor Power Fluctuation by Autoregression", Nuclear Science and Engineering, 67, 296-308 (1978).

[7]

S. I. Chang, T. W. Kerlin, " Identification of Nuclear Plant Parameters Using Experimental Data and High-Order Dynamic Models",

Nuclear Science and Engineering, 64, 673-683 (1977).

[8]

K. Matsubara, R. Oguma, M. Kitamura, "A Multivariable Autoregressive Model of the Dynamics of a Boiling Water Reactor", Nuclear Science and Engineering, 65, 1-16 (1978).

[9]

M. Kitamura, K. Matsubara, R. Oguma, " Identifiability of Dynamics of a Boiling Water Reactor Using Autoregressive Modeling", Nuclear Science and Engineering, 66, 106-110 (1978)

[10] K. J. Astrom, P. Eykhoff, " System Identification - A Survey" Automa tica, 7, 123-162 (1971).

[11] L. D. Denby, R. D. Martin, " Robust Estimation of the First Order Autoregressive Parameter", JASA, March (1979).

[12] B. Kleiner, R. D. Martin, " Robust Estimation of Power Spectra", JRSS, May (1979).

[13] G. M. Jenkins, D. G. Watts, " Spectral Analysis and its Applications",

Holden-Day, San Francisco (1968).

1U98 M3