a combined correlation function-linear regression approach
TRANSCRIPT
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LSU Historical Dissertations and Theses Graduate School
1971
A Combined Correlation Function-LinearRegression Approach to System Identification.James Brian FroisyLouisiana State University and Agricultural & Mechanical College
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Recommended CitationFroisy, James Brian, "A Combined Correlation Function-Linear Regression Approach to System Identification." (1971). LSU HistoricalDissertations and Theses. 2124.https://digitalcommons.lsu.edu/gradschool_disstheses/2124
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FROISY, James Brian, 1945-A COMBINED CORRELATION FUNCTION-LINEAR REGRESSION APPROACH TO SYSTEM IDENTIFICATION.
The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1971 Engineering, chemical
University Microfilms, A XEROX Company, Ann Arbor, Michigan
A COMBINED CORRELATION FUNCTION - LINEAR REGRESSION APPROACH TO SYSTEM IDENTIFICATION
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and
Agricultural and Mechanical College in partial fulfillment of the requirement for the degree of
Doctor of Philosophyin
The Department of Chemical Engineering
byJ. Brian Froisy
B.S., Louisiana State University, 1968 M.S., Louisiana State University, 1969
December 1971
PLEASE NOTE:
Some pages may have
indistinct print.Filmed as received.
University Microfilms, A Xerox Education Company
ACKNOWLEDGEMENT
The research presented in this dissertation was conducted under
the guidance of Dr. Cecil L. Smith and Dr. Armando B. Corripio.A sincere thanks is extended to these gentlemen for giving generously of their time and energy throughout the duration of this work. Particular appreciation is felt for Dr. Smith whose counsel has made
this work both an enjoyable and a valuable experience.I would also like to thank the members of the faculty and staff
under which this work was conducted with particular recognition to Dr. A. J. McPhate of the Department of Mechanical Engineering and Miss Hazel Lacoste of the Department of Chemical Engineering.
The financial support provided by the Kaiser Company Fellowship and by Project THEMIS contract No. F44620-68-C-0021 administered through the Department of Defense by the U.S. Air Force Office of Scientific Research has made this work possible. Appreciation for this support is deeply felt. Thanks are also extended to The Dr. Charles E. Coates Memorial Fund of the L.S.U. Foundation donated by George H. Coates for aid in defraying the expenses associated with
preparation and reproduction of the manuscript.
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TABLE OF CONTENTS
page
ACKNOWIEDGEMENT........................................... ii
LIST OF TABLES......................................... vi
LIST OF FIGURES........................................... viiABSTRACT.................................................. xi
CHAPTERI INTRODUCTION..................................... 1
Literature Cited............................... 4II DISCRETE MODEL IDENTIFICATION BASED ON
CORREIAX ION FUNCTIONS............................ 5Introduction................................... 5Discrete Model Identification.................. 5Some Practical Considerations.................. 8Selecting an Input Signal..................... 9Measurement Noise ......................... 12Implementation of the IdentificationProcedure......................... 13
Results for the Deterministic Case............. 14Addition of Measurement Noise.................. 20Summary ....... 24Nomenclature................................... 25Literature Cited............................... 26
III APPLICATION OF CORREIATION-REGRESSIONIDENTIFICATION TO A SIMULATED PLANT MODEL........ 27
Intr oduc t ion................................... 27Discrete Model Identification.................. 27The Plant - A Plug Flow Reactor................ 29
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pageOpen Loop Identification - Plug Flow
Reactor..................................... 29
Results - Open Loop Identification............ 30Comparison with Another Regression
Scheme...................................... 34Measurement Noise and Parameter Bias.......... 39Closed Loop Identification..................... 43Effect of Measurment Noise on Closed
Loop Identification.......................... 44Results - Closed Loop Identification........... 45Summary........................................ 51Nomenc la ture.................................. 52Literature Cited.............................. 53
IV LINEAR INDEPENDENCE OF INPOT-OUTPUT DATAFOR CLOSED LOOP IDENTIFICATION................... 54
Introduction.................................. 54Regression Analysis of Linear DynamicModels....................................... 54
Mathematical Development of IndependenceCriterion.................................... 57
Linear Independence - Necessary and SufficientConditions ............................... 62
Design of an "Identifying Controller".......... 63Example Problem............................... 64Summary........................................ 67Nomenclature.................................. 69Literature Cited............................... 71
V IDENTIFICATION OF PIANT DYNAMICS IN A CLOSEDLOOP SYSTEM DRIVEN BY EXTERNAL DISTURBANCES 72
Introduction.................................. 72
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page
Development of Correlation-Regression Equations............... .................... 72
Estimation Procedure.......................... 76System Disturbances............................ 79Disturbance Compensation....................... 79Disturbance Transformation Parameter........... 85Proposed Identification Procedure.............. 86Realiability of Parameter Estimates and Model Structure.......................... 88
Results........................................ 91
Summary........................................ 98Nomenclature................................... 105Literature Cited............................... 107
VI APPLICATION OF CLOSED LOOP IDENTIFICATION TO ASIMULATED PHYSICAL SYSTEM........................ 108
Introduction......... 108Process Model.................................. 108Controller Design.............................. IllIdentification Procedure....................... 118
Results........................................ 118Summary........................................ 125Nomenclature................................... 126
VII..... CONCLUSIONS...................................... 127Literature Cited............................... 131
APPENDIXA Dahlin Algorithm............................... 132B Computer Programs.............................. 135
VITA....................................................... 161
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LIST OF TABLES
CHAPTER page
II TABLE 1Comparison of Analytical and Approximate Coefficients................................... 16
III TABLE 1Comparison of Model Errors from Correlation-Regression and Ordinary Regression for300 Data Points................................ 38
VI TABLE 1Mean Square Errors Resulting from IdentificationBased on Three Different Controller ParameterSets........................................... 120
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LIST OF FIGURES
CHAPTER page
II Figure 1Comparison of analytical and approximate autocorrelation functions for a range of time series points................................... 10
Figure 2Comparison of fit error with the input-output record length................................... 17
Figure 3Comparison of system and model stepresponse........................................ 18
Figure 4Comparison of system and model step responses.... 19
Figure 5Comparison of system and model step responses.150 sampled points with measurement noise added.. 21
Figure 6Comparison of system and model step, response500 sampled points with measurement noiseadded................................ 22
... Figure 7Comparison of system and model responses to the actual input signal used for the identification.. 23
III Figure 1Diagram of Data Collection for Plug FlowReactor........................................ 29A
Figure 2Analytical and Approximated Auto-correlation Functions for Filtered White Noise............. 31
Figure 3Plant response and model predictions for openloop identification based on 300 data pointsand discrete interval binary noise input........ 32
Figure 4Plant response and model predictions for openloop identification based on 300 data pointsand discrete Interval binary noise input....... 33
CHAPTER page
Figure 5System Input, output, and model error foropen loop identification based on 300 datapoints with filtered white noise input......... 35
Figure 6System input, output, and model error foropen loop identification based on 300 datapoints with filtered white noise input.......... 36
Figure 7System input, output, and model error for openloop identification based on 300 data pointswith filtered white noise input................ 37
Figure 8Comparison of modeling errors for correlation-regression and ordinary regression withmeasurement noise present....................... 41
Figure 9Effect of Measurement Noise on Modeling Errorfor Open Loop Correlation-RegressionIdentification ........................... 42
Figure 10Manipulated Variable, Plant Response, and Model Errors for a Series of Step Inputs in Setpoint Identification based on 300 data points with no measurement noise.............................. 46
Figure 11Measurement noise auto-correlation function and noise contribution to the cross-correlation function based on 1000 data points............. 47
Figure 12Manipulated variable, plant response, and model error for a series of step inputs in setpoint. Measurement noise standard deviation * 4.0...... 49
Figure 13Effect of measurement noise on modeling errorfor closed loop correlation-regressionidentification................................. 50
IV Figure 1Normalized determinant versus selected PI controller parameters and for Dahlin controller.. 68
-viii-
CHAPTER pageV Figure 1
Configuration of closed loop system driven by unmeasured disturbances........................ 81
Figure 2System configuration for closed loop Identification vith unmeasured disturbances 92
Figure 3Effect of data record length on parameter estimates and von-Neuman ratio. Disturbance:Noise with p ■ 0.1.......................... . 93
Figure 4Effect of data record length on parameter estimates and von-Neuman ratio. Disturbance:Noise with p ** 0.9............................ 94
Figure 5Effect of data record length on parameter estimates and von-Neuman ratio. Disturbance:Load........................................... 95
Figure 6Effect of data record length on parameter estimates and von-Neuman ratio. Disturbance:Combined load and noise with p » 0.9.......... 96
Figure 7Error distribution for feedback control with Dahlin algorithm corresponding t'o the identification shown in Figure 6 with 1000 data points............................... 100
Figure 8Error distribution for feedback control withPI algorithm corresponding to theidentification shown in Figure 6 with1000 points.................................... 101
Figure 9Error distribution for feedback control with Dahlin algorithm corresponding to the identification shown in Figure 7 with 1000 data points............................... 102
Figure 10Error distribution for feedback control with FIalgorithm corresponding to the identificationshown in Figure 7 with 1000 data points....... 102A
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CHAPTER page
Figure 11Error distribution for feedback control with Dahlin algorithm corresponding to the identification shown in Figure 8 with 1000 data points............................... 103
Figure 12Error distribution for feedback control with PI algorithm corresponding to the identification shown in Figure 8 with 1000 data points........ 104
VIFigure 1External disturbance configuration............. 109
Figure 2Comparison of controlled and uncontrolled responses for tubular reactor................... 110
Figure 3Step response of plant and first order lagplug dead time discrete model to a unitincrease in the manipulated variable........... 112
Figure 4Contour map of normalized determinants andthe pole constraint boundaries................. 117
Figure 5Controlled reactor response with external disturbances and a major load change........... 122
Figure 6Comparison of plant and model responses withtwo different reaction rate constants.......... 123
-x-
ABSTRACT
The work presented in this dissertation is concerned with the problem of identifying the parameters describing a dynamic process based on an input-output record of the process. The identification technique is used to approximate the parameters of a pulse transfer function and is structured for utilization in a sampled-data environment. System identification is not ordinarily an end in itself, but is rather means of approaching an overall problem. The identification approach subsequently described is designed for use
in an adaptive control configuration.The basic approach involves the use of discrete correlation
functions approximated from the input-output record as data in a multiple linear regression from which the parameters are estimated. This combined correlation function-linear regression approach retains much of the simplicity of linear regression while benefiting from the filtering capabilities of correlation functions.
The initial presentation is concerned with open loop identification and treats both the deterministic and the stochastic output cases. The same cases are treated for closed loop identification for which setpoint variations supply the necessary dynamic response data. Much of the effort in this work is directed to the problem of closed loop identification of a system driven only by unmeasured, external (i.e., excluding setpoint) disturbances. Guidelines for the design of a feedback controller suitable for use in such an application preceed the presentation of the actual identification procedure. For this application the basic
-xi
Identification approach is retained, but is extended to utilize a form of generalized regression and an iterative solution to obtain
estimates of both the process and disturbance dynamics.Tests of the identification for various system configurations
are implemented using digital simulation. Applications for thesecases are presented with simple linear process models as well asa simulated chemical reactor representative of a complex physicalprocess.
xii-
CHAPTER I
INTRODUCTION
The topic of interest in this work is that of dynamic identi
fication. A great deal of investigation into this general area has been completed within the past two decades. Many different approaches have been taken and even more variations of any given approach are available. Early attempts utilizing correlation functions were followed by approaches based on linear or non-linear regression.The various approaches tend to be general in form but specific in intended application. The assumptions required for a given approach to apply tend to be a measure of the intended range of application.
The wide range of application for identification techniques has resulted in an overwhelming wealth of literature in this case. Several survey papers and sane texts give a good overview of the "state of the art" (1-5). In addition, some recent works related to or with possible application in process control are available in the following references (6-10).
In this work, the intended application is that of control systems design for process systems. This is not to say that the range of application is limited to control systems design, but rather that the various difficulties associated with such an implementation are of primary concern.
The work is based entirely on the estimation of system dynamics in the form of a pulse transfer function, and is directed to systems for which a digital computer forms a primary element in the control loop. In a digital environment, a discrete representation
of the system dynamics Is quite adequate and suited for use with
many of the flexible design methods for discrete controllers. -
The Identification approach Is based on the combined use of correlation functions and linear regression. In Its basic form} the Identification Is Implemented by using the points of discrete correlation functions as data points In a multiple regression. This combined approach relies on the filtering action of correlation functions to overcome difficulties related to measurement noise and unmeasured system disturbances while retaining much of the simplicity of linear regression.
In Chapter II, the correlation-regression Identification pro
cedure Is developed and applied to two simple second-order differential equations. The problem of measurement noise, which Is common to many processes, Is discussed and it Is shown that filtering action of the correlation functions is sufficient to compensate for this noise. The developments of this chapter apply to the open-
loop identification problem.The work presented in Chapter III consists primarily in the ap
plication of the technique to a simulated physical system. This system is non-linear and of high order; however, relatively simple discrete models can be used to describe its dynamics for the purpose of control system design. In this chapter a comparison is made between the results of applying ordinary regression directly to the time series data with the results of the correlation-regression approach. The correlation-regression approach is seen to be superior when measurement noise is present. Some consideration is also given to the case of closed loop identification for which setpoint
3
variations are present, and in addition, the problem of measurement noise in such a configuration is treated.
Chapter IV is primarily a developmental chapter in which the mathematical considerations related to linear independence of input- output records are treated. This development is concerned with the closed loop system identification problem for which no setpoint variations are present. That is, the system is driven only by external disturbances. The chapter concludes with some guidelines for the design of a feedback controller which is adequate for use in an overall closed loop, regression based, identification scheme.
Presentation of a procedure for closed loop identification with only external disturbances driving the system is given in Chapter V. The relationship between correlation-regression and a form of generalized regression is presented and used as a basis for closed loop identification. Results for a simple system indicate the need of using a well designed (with respect to identification) controller for such an application.
In the following chapter, an application of the closed loop correlation-regression approach is presented for a simulated physical system. This chapter represents an application under fairly realistic conditions and points out some areas of further development which may be required for the design of a truly self-optimizing control strategy.
Literature Cited
Cuenod, M., and A. P. Sage, "Comparison of Some Methods Used for Process Identification", IFAC Symposium on Identification in Automatic Control Systems, Prague, Czechoslovakia, June, 1967.Davies, W. D. T., System Identification for Self-Adaptive Control John Wiley & Sons, New York, 1970.Eveleigh, Virgil W., Adaptive Control and Optimization Techniques McGraw-Hill, New York, 1967.Eykhoff, P., "Process Parameter and State Estimation", IFAC Symposium on Identification in Automatic Control Systems,Prague, Czechoslovakia, June, 1967.Oldenburger, Rufus (ed.), Optimal and Self-Optimizing Control.The M.I.T. Press, 1966.Bigelow, Stephen C. and H. Ruge, "An Adaptive System Using Periodic Estimation of the Pulse Transfer Function", IRE Convention Record. Part 4, 1961.Dube, James H., "Application of Identification and Control Methods", Ph.D. Dissertation, Department of Chemical * Engineering, Louisiana State University, Baton Rouge,Louisiana, 1971.Rowe, Ian H., "A Bootstrap Method for the Statistical Estimation of Model Parameters", Int. J. Control. Vol. 12, No. 5, 1970Suh, Samuel, "Design and Application of an Adaptive Control Systems Using Digital Techniques", Proceedings of the Sixth Annual Conference on the Use of DIGITAL COMPUTERS IN PROCESS CONTROL. February, 1971.Wong, Kwan Y. and E. Polack, "Identification of Linear Discrete Time Systems Using the Instrumental Variable Method", IEEE Trans, on Automatic Control. Vol. AC-12, No. 6, 1967.
CHAPTER II
DISCRETE MODEL IDENTIFICATION BASED ON CORRELATION FUNCTIONS
IntroductionUsing most of the techniques currently available, some Informa
tion concerning the dynamics of a system must be known before any meaningful control strategy can be Implemented. This Information can be presented In the form of a plant model which may be obtained In a variety of ways, ranging from a model derived from knowledge of the basic physical phenomena involved to some simple empirical model (e.g., first-order lag with dead time). In this chapter a technique of obtaining a dynamic plant model for a general system
is presented and applied to two specific cases.
Discrete Model IdentificationThe identification technique discussed in this chapter produces
a discrete model, and as such should be useful in a digital control environment. The basic approach of the technique is to apply a straightforward multiple linear regression to points on the discrete auto- and cross-correlation functions calculated from a system's sampled experimental input-output record.
The basic equations are developed in Kuo (2). Consider the following representation of a discrete transfer function.
where$ (z) ■ discrete cross-spectral density
]>C
$rr(z) " discrete auto-spectral density
G(z) ■ discrete transfer functionWriting G(z) in the form of a pulse transfer function gives:
z'M S A zJ
« ■ > - <2 >E B z1 rri=o 1
“Mwhere the z~ term is introduced to represent a possible system dead time. Cross multiplying and expanding gives:
z"M[AN2f)5rr(z) + AN_1zti'1irr(z) + ... Aoz°§rr(z)]
* V D{rc(z) + BD-lzD lirc(z) + ••• V ° {rc(z> (3>
Rearrangement of Equation 3 results in:
z“M[ANz“(D"N)$rr(z) + AN_1z"(D_N+1)$rr(z) + ... Aoz“D$rr(z)]
" BDz°$rc(z) + BD-lz"1$rc(z) + BoZ" \ c (z) (4)
As mentioned, the functions $__(z) and $__(z) represent spectralAA AC
densities of the input-output record. These are simply the z-trans- form (divided by T) of the auto- and cross-correlation functions of the input-output record. That is,
l 005 (z) ■ — E cp (nfl)z”mrcN 7 T __ _ TrcN 'JQB mCO
7
and00
" T £ cPrr(nitr z"m m- -»
where
T - sample timecp « discrete cross-correlation functionrccprr - discrete auto-correlatlon function
-kObserve that in Equation 4 all of the terms are of the form Az Z{cprr(mT)}/T. Taking the Inverse z-transform of each term results in:
ANcprr[(m-M-D4N)T] + + ... Ao«prr[(m-M-D)T]
" BD^rc^m T + + **• B0cPrc^m“D)1 (6>
Without loss of generality, can be set equal to one and Equation6 solved for cp (mT) to. give:TC
N D-lcp (mT) = Z A cp [(m-M-D+j)T] - Z B.cp [(m-IHl)T] (7)rc j»0 J r i-o 1 rc
Equation 7 implies that a given point on the cross-correlation
function, cp (m T), can be predicted if the necessary past values rcof the auto- and cross-correlation functions and the system transfer function are known. However, considered differently, Equation 7 forms the basis of the identification procedure. Knowing several values for the correlation functions at appropriate time shifts, these can be treated as "data points" and a multiple linear regression performed to give a least squares set of values for the constants in the pulse transfer function. Note that the Integer variable M
8
(representing the system dead time) will have to be defined prior to solving the regression equations and that in a general case some
other algorithm must be supplied to determine the "best" value for this parameter.
Some Practical ConsiderationsEquation 7 provides a straightforward means of determining the
pulse transfer function of a discrete or sample data system if appropriate values are available for the correlation functions. A discrete correlation function T) is defined as:
1 Ncprc(kT) = Lim -^r 2 r(mT)c((m+k)T) (8)N -* ® m=« -N
Disregarding any values for negative time and being content with afinite approximation of the function:
1 Ncprc(kT) ss ~ r 2 r(nT) c((n+k)T) (9)n=o
Although the above notation appears to apply to the cross-correlation function, the results are applicable to an auto-correlation functionif r and c represent the same signal.
Determining a suitable value for N, the number of time domain data points, in Equation 9 for effective identification is a difficult task. First, it can frequently be assumed that the statistics of the process are "reasonably" stationary over the identification interval (0 £ t £ NT). It is highly desirable that the identification interval be as small as possible, particularly if used in a real time
control environment. However, a value for N must be chosen which will give a reasonable approximation of the correlation functions. The net
9
result Is that some trade off must be made between accuracy and time
(both computer execution time and Identification time).
Selecting An Input SignalFor the purpose of verifying the Identification technique and
studying some of Its characteristics on a known system, the Input sequence was chosen as discrete Interval binary noise. Very simply, the signal consists of pulses of magnitude + x q and of a duration T^. For each value of tine corresponding to the end of one pulse (t » nT^, n « 1,2,3,...) one of two actions occur (with equal probability) — the signal either remains at Its current level or switches to the other available level. The discrete auto-correlation
function for such a signal Is (1):(l-|kl|/T.) |kT| s Td
0 |kl| k Td
The simple form of this function makes comparison of the analytical values and the finite approximations very convenient. Figure 1 shows several of the finite approximations and the true auto-correlation function. This is an admittedly small sample, but serves to illustrate the typical behavior of the approximations involved. Note that the functions shown are normalized. It should be noted that
the finite approximations are quite good for small values of the time shift but are erratic once a value of zero is initially reached. Thus, for the actual, identification it was decided to use only the first part of the correlation function such that cp(kT) ^ .lcp(O).
<prr(kT)
Auto
-correlation
10
Figure 1
©
00
number of points used for approximating the auto-correlatlon functit n5.0
Analytical
CM
O
CMN»50■
si-I
0 2.5 5.0 10.07.5Time Shift
Comparison of analytical and approximate auto-correlation for a range of time series points.
Ifunctions
11
The Input pulse duration, T^, was chosen as large as possible with the restriction that the output of the system would not be
expected to be "lined out" for any long periods of time. (The probability that n pulses will all be at the same level Is p(n) =0.5n”*.) In order to supply the regression with enough "data points", It Is necessary that (T^/T > N+D+l) where N+D+l represents the number of terms In the discrete model. From the results of this study It was found that even for the case of slow sampling (large T), a typical system will meet the above set of requirements. That Is, the pulse duration can be chosen small enough so that the system will not be lined out, while being large enough to acquire the necessary set of points on the auto-correlation function whose first (N+D+l) values are greater than 10% of cp(0).
Although the characteristics of the test signal which was used have been presented, this problem is not of primary concern.Eveleigh (1) presents several different types of signals which are
applicable to identification based on correlation techniques.The signals chosen are commonly some approximation of a white noise signal (i.e., their auto-correlation function is an impulse function approximation). The net result is a very restricted signal, but one which has many desirable properties that simplify the calculations required to complete the system identification. In the procedure being presented, the test signal is not restricted as such, rather its characteristics are measured (in the form of its auto-correlation function) and these measurements along with similar measurements of the cross-correlation function are used to complete the identification. The important implication of measuring, rather than specifying, the
12
input signal statistics lies in the ability to use normal operating Inputs of a system for real-time, on-line identification of a process pulse transfer function.
Measurement NoiseThroughout this discussion, no mention has been made of the
possibility of errors in the actual measurement of the input and output signals. It will be assumed that the input signal is measured without error and that the output signal is contaminated with zero- mean gaussian noise and is independent of the input signal. The operations involved are shown below.
As shown below, this noise should not affect the value of the cross
correlation function:
The measured output signal will be of the form:
c(mT) » c*(mT) + cm (mT) (U)where
c*(mT) = true value of the outputc (mT) 113 measurement noise mc(mT) ■ measured value of the output
13
1 Ncj> (kT) - Llm E r(mT) c(mT-HtT) (12)rC N - • 281+1 m— N
1 NLlm E r(mT) c*(n(HkT)N - 09 m — N
+ asr i « r(na:> cn.<nff+kT)N ■* o° m*-N
By virtue of cm having a zero mean and being Independent of r, It follows that
1 N * -LI® vhxr 2 r<mT) c (nB?+kT) » r(mtr) c (nffi-HcT) - 0 (13).. t ^ *WTi M ID mN -* 00 m— NAlthough the finite statistics will not be the same as those assumed, the averaging and the regression tend to remove any serious errors Incurred due to finite signal statistics.
Implementation of the Identification ProcedureThe basic steps Involved In the Identification procedure are
indicated below:1. Compute the normalized (<PrJ.(0) =• 1.0) auto- and cross
correlation functions from the sampled input-output record.2. Arrange the correlation functions in a standard form for
use in the multiple regression. That is, treat the correlation functions as data which will follow the format Indicated by Equation 7.
3. Compute the "normal equations" for the regression. (Sums of squares and cross products.)
4. Solve these algebraic equations to obtain the least squares set of coefficients which represent the pulse transfer function.
14
Note that in the usual context, a linear least squares fit will contain a constant term, but that no such term appears in Equation 7. Thus the "raw data" (correlation points) must be used rather than the deviations from their mean values. This procedure effectively forces the constant term to zero.
Results for the Deterministic CaseTwo linear systems described by the following differential
equations were used for the initial verification and study of the identification procedure.System 1, Overdamped:
4-! + 3 + 2y - 2 f(t)dt2 4t
The equivalent time constants and gain are
t2 - 0.5 K - 1.0
System 2, Underdamped:
Mjr + .8 + y » f(t)d r dt
Describing this system by a damping ratio, natural frequency and gain
gives6 ■ 0.4
u) « 1.0nK - 1.0
A question of considerable initial importance concerns the required number of sampled points from the input-output record which
15
must be used In order to achieve acceptable Identification. For
the present, the criterion for acceptability will be defined as the ability of the discrete model to respond to a sequence of Input signals such that the model response is the same as the system response. More concisely,
1 NFit Error » - E | c(nfr) - (^(mT)! (14)m*l
where c(nff) and c^(mT) are the system response and the model response respectively for the same input signal at the m sampling instant.It should be noted that the (n+l)st value for y^ is calculated using the necessary past values of y , and not the past sampled values of the continuous system, y. This was felt to be a tougher test since
it allows for the accumulation of errors. However, in a control environment, one might prefer to use the past measured output values to predict only the next output value and use this Information in some controller algorithm.
For the overdamped system, Figure 2 indicates that relatively few sampled points can be used to accurately identify the pulse
transfer function. Similar results were obtained in the underdamped case. After 200 sampled points, the error tends to approach a mean value of about 0.002. The fluctuations appear to be due to round-off error.
Although the results displayed in Figure 2 give an overall picture of the identification performance, they may not clearly convey how well the system is identified. Figures 3 and 4 show the step response for the continuous system with the discrete model response to the same step. (500 points were used for this particular
16
TABLE 1
Comparison of Analytical and Approximate Coefficients
System 1
A 1 A0 B1 B0 FitError
Analytical .15482 .09390 -.97441 .22313150 Points .16213 .08700 -.97311 .22209 .00325500 Points .15417 .09457 -.97436 .22320 .00040
System 2A1 Ao B1 B0 Fit
ErrorAnalytical .10767 .09414 -1.46852 .67032150 Points .12000 .09142 -1.46934 .67102 .00282500 Points .10716 .09440 -1.46901 .67068 .00070
Number of points used in the identification
Figure 2. Comparison of fit error with the input-output record length.
Step
Response
18
M
Continuous SystemDiscrete Model
or-i H-lll l-H -Hl+H -H-HHI H I H I l+l- H W -H
o0 Time 25
\
Figure 3. Comparison of system and model step responses.
Step
Response
19
oCM
Continuous System
+ + Discrete Model
o«—4
o0 Time 25
Figure 4. Comparsion of system and model step responses
20
identification.) These results clearly indicate the success of the identification procedure.
Table 1 shows a similar measure of the performance of this procedure by comparing the analytical values with those resulting
from the identification. Note that a second order system can be described exactly by a difference equation of the form:
c((m+l)T) » AQr(mT) + A1(t(m-1)T) + BQc(mT) + B^CCm-ljT)
Addition of Measurement Noise
To be of practical importance, an identification technique must be applicable in a noisy environment. Measurement noise was Simulated in the system by adding digitally generated zero-mean gaussian noise to the deterministic output. Figures 5 and 6 show the step response results for various measurement noise statistics with the underdamped system. The statistics of the measurement noise were specified by its standard deviation. The corresponding signal to noise ratios range from infinite (the deterministic case) to roughly 1/3. As is expected, more identification points are required for accurate results when a significant amount of noise is present.
Figure 7 shows the actual output record which was used to obtain the coefficients for some of the step response comparisons of Figure
5. The discrete model response to the same input record is also shown. In addition, these plots serve to give the reader, who may be unfamiliar with statistics, a feel for the magnitude of the measurement noise which had been simulated. In the three cases shown, the same input signals were used as was the measurement noise with the exception that the noise magnitude was different.
21
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +
+ + + f + + f + + + + + 1 + + + + + + + + + + + + + + + + + + + + + ”1
to<unti0 tx01 0) D4ex0).4Jt/3
+ + + + + f + + + + + + + + + + + + + + + + + + + + + + + + + + + + + *
Continuous SystemDiscrete Model*? Standard deviation
of measurement noise
0 Time 20
Figure 5. Comparsion of system and model step responses. 150sampled points with measurement noise added.
Step
Responses
22
+ + + + + + + + + + + + + + + + + + + + + + + + + * + + + +-
+ + -f + + + 4- + + -f + + 4- + + + + + + + - f - f + + + - H -
Continuous system+ + Discrete modela “ Standard deviation of
measurement noise
0 Time 20
Figure 6. Comparison of system and model step response.500 sampled points with measurement noise added.
System Responses
Continuous system+ + Discrete modela ° Standard deviation of
measurement noise
Time 75
Figure 7. Comparison of system and model responses to the actual inputsignal used for the identification.
24
SummaryA technique £or determining the pulse transfer function of a
dynamic system has been presented. The technique is based on approximating the discrete auto- and cross-correlation functions of the sampled input-output record followed by a multiple linear regression of a suitable arrangement of the points on the correlation functions. Unlike many correlation techniques which require an (approximately) uncorrelated input signal, the one presented here allows a more general input signal.
The results of applying the identification on two specific second-order systems serve to illustrate the potential success of such an approach to the problem of system identification. Admittedly, a large number of questions remain unanswered.
Future work is planned to answer some of these questions. Most notable they include: systems with dead time, approximating higherorder systems with lower order models, approximating non-linear systems, and application to a closed loop adaptive control environment with slowly varying process parameters.
25
Nomenclature
A m coefficient of input term in pulse transfer function
B a coefficient of output term in pulse transfer function
D s number of output terms in discrete model
G BB discrete transfer functionK B gain of example systemsM a delay integer >■ nearest integer to dead time/sample time
N a number of model input terms minus 1 or number of data pointsT » sample time (>.5 in all examples)
Td a duration of a single input pulse
Xo a input pulse magnitude
c m system output or measured output if noise is included
- measurement noise
CM a discrete model output
c* true system output for the case with noise
r = input signal
t a time
y - output of example systems
$rr discrete auto-spectral density
$rc discrete cross-spectral density
- auto-correlation function
^rc - cross-correlation functionT - time constant
6 a damping ratioO)n a natural frequency
a a standard deviation of measurement noise
Literature Cited
Eveleigh, V. W., Adaptive Control and Optimization Techniques. McGraw-Hill, New York, (1967).Kuo,B. C., Analysis and Svntheais of Sample-Data Control Systems. McGraw-Hill, New York, (1967).
27
CHAPTER III
APPLICATION OF CORREIAT1DN-REGRESSION IDENTIFICATION TO A SIMULATED PLANT MODEL
IntroductionThis chapter describes an application of the correlation-
regression technique of identification to a simulated physical system. Four specific topics are considered: 1) A low-order, discrete modelis used to approximate the dynamics of a high-order, non-linear simulated system. 2) Identification is performed using different types of inputs; discrete interval binary noise and filtered white noise.
3) Comparison is made with another regression approach with particular attention to the effects of measurement noise. 4) Identification during closed loop operation is treated.
Discrete Model IdentificationAs the identification scheme was developed in the previous
chapter, the results of the development will be only summarized.The basic equation to which multiple regression can be applied toobtain the parameters of a discrete model if of the form:
NU NCcp (mT) - S A cp [(m-M-j)T] - S B.cp [(m-j)T] (1)rC j-1 3 “ j-1 J rC
wherecp “ discrete input auto-correlation function rrcprc - discrete input-output cross-correlation function
A ■> input coefficient
28
B - output coefficient
T - sample time (hereafter assumed equal to unity)M - integer representing the dead timeNU - input' order of the modelNC - output order of the model
A finite approximation of a cross-correlation function is given by:NP
tPrc(mT) “ NP+1 E r(nT) <2>m=owhere the sum is oyer the available data set. The auto-correlation function is simply represented by the special case of "r" and "c" . representing the same signal. By way of example, for a two input - two output model, and no pure time delay, Equation 1 reduces to:
cprc(m) = A ^ C m - l ) + A2cprr(m-2)-B1cprc(m-l)-B2cprc(m-2) (3)
For time domain regression, the correlation functions can be replacedby the actual input-output record ("c" for cp and "r" for cp ).'rc rr
The heart of the identification lies in performing linear regression using the appropriate correlation functions as "data points". The step-by-step procedure is summarized as:
1) Compute the normalized (cprr(0) = 1.0) auto and cross- correlation functions from the sampled input-output record.
2) Arrange the correlation functions in a standard form for U B e in the multiple regression. That is, treat the correlation functions as data which follow the format indicated by Equation 1.
3) Compute the "normal equations" for regression. (Sums of squares and cross products^)
4) Solve these algebraic equations to obtain a least squares set of coefficients which represent the pulse transfer
29
function*.
The Plant - A Plug Flow Reactor
A simulated jacketed plug flow reactor, illustrated in Figure 1, served as the system whose dynamics were to be identified. The following second-order chemical reaction occurs in the reactor:
K1(T>A + B ------------------> R
Feed to the reactor consists of a recycle stream enriched with3reactant B at a rate of 18 ft /min. and a stream of pure A with
variable flow rate. The identification subsequently described involves storing the sampled input-output record, computing the discrete auto- and cross-correlation functions of that record, and finally applying multiple regression to the functions to obtain the parameters of the approximate discrete model.
Open Loop Identification - Plug Flow Reactor
The correlation-regression identification scheme was applied to the reactor model for two different types of input signals, the input being the flow rate of reactant A. The first was a discrete interval binary noise signal, which results in a series of step inputs. This input is generated by allowing operation at one or two available signal levels for some prescribed period of time after which the signal level may, with equal probability, remain at the same level, or switch to the other available level. The length of time between possible switches is kept short enough to cause the plant to be in a state of dynamic change during most of the data collection period.
A second type of input signal was generated by passing white
Digital Computer Generates upsets and stores data
HoldDevice
■-500/TWater In 55°F
25 ft /min
Non-linear plug flow reactorPure A Product R
0^WA^12 ft /mil10 ft water out
Pure B
Constant recycle
w ■o 18CA o “ 6CBo “ 58% > " 20
t3t3.3
Figure 1. Diagram of Data Collection for Plug Flow Reactor
30
noise through a first order filter. The operation can best be described by means of the block diagram below.
Filtered Noisc^Zero Order Hold
The filter time constant, t^, might be thought of as a parameter which determines the "randomness" of the filtered signal, with small time constants giving the greatest degree of randomness. That is, small time constants result in an auto-correlation function which is more like the impulse correlation function of white noise. The autocorrelation functions for selected time constants are shown in Figure 2.
Results - Open Loop IdentificationThe identification of the simulated reactor was implemented for
various order discrete models for the case of no measurement noise. The results shown in Figures 3 and 4 compare the plant responses with
the responses predicted by the resulting discrete models. In all cases the discrete model responses are the result of "one step ahead predictions". Such predictions are computed by utilizing the past (known) response points for the physical system to predict the response for the next sample point. Four different discrete models were used with the order indicated as an "(ir-jc)" model, where "i" is the number of input terms and "j" is the number of output terms. The predictions are all quite adequate, with only the lowest order (lr-lc) model showing any significant error. Only slight improvement
is realized with the higher order models.
Normalized Auto-Correlation Function
31
1.0 Analytical Finite Approximation Filter Time Constant 1.5 Seconds
0.8.496
0.4
0.2 0.097
0.00.022
- 0.20 102 4 86
Discrete Time Shift - No. of Samples
Figure 2: Analtyical and Approximated Auto-correlationFunctions for Filtered White Noise
Concentration
Response
Concentration
Response
32
00CM
00
Model: 2r - 2c
00
00CM
00
-Plant ResponseDiscrete Model Prediction1.5 seconds
00
Model: lr - lc
Time (Minutes) 7.5
Figure 3: Plant response and model predictions for open loopidentification based on 300 data points and discreteInterval binary noise input.
Concentration
Response
Concentration
Response
33
Model: 4r - 4c
f+tfH+H+4Plant ResponseDiscrete Model Prediction1.5 Seconds
W W W *
WHWW00
Model: 3r - 3c
Time (Minutes) 7.5
Figure 4: Plant response and model predictions for open loopidentification based on 300 data points and discreteinterval binary noise input.
34
Figures 5, 6, and 7 Indicate the performance of the predictor
model identified with filtered white noise inputs for selected filtertime constants. In these, and the following figures, the discretedata are plotted continuously for the sake of clarity. The examplesare limited to (2r-2c) models. It is evident that the fit errorincreases as the filter time constant is reduced. In each case,the errors incurred in one step ahead prediction are based on theactual input-output record upon which the identification was based.A more meaningful comparison can be made by observing the change in
2 2the normalized mean square error which is defined as a / c r ,which is simply the ratio of error variance to signal variance.
The primary observation from these results is that the identification performs satisfactorily for both types of Inputs. The discrete interval binary signal might be representative of the type of input used for a well planned identification experiment whereas the filtered noise signal would be more.representative of the type of data available under normal operating conditions.
Comparison With Another Regression SchemeUsing the approach discussed by Kalman (1) and subsequently
tested on this reactor by Dube (2), the model parameters were obtained by an ordinary regression of the sampled input-output record (rather than regression on the correlation functions). The input signal used was the previously discussed discrete interval binary noise. The errors resulting from one step prediction based on 16 different order models are compared in Table 1. Of particular significance is the fact that the ordinary regression errors are smaller than those of the
Input
Flow
Rate
Concentration
Response
Model
Error
35
%
CO
CM
a1 » 0.047
o^/c^ ■ 0.006c.
2 „ ,,a = 7 . 4 4c
00
“ 0.1 minutes
o
^ Time (Minutes)
Figure 5: System Input, output, and model error for open loopIdentification based on 300 data points with filteredwhite noise input.
4
Input
Flow
Rate
Concentration
Response
Model
Error
36
a - 0.1662 . 2 0.014
i n
ct - 11.47 c
Tj ■ 0.05 minute
Time (Minutes) 7.5
Figure 6: System input, output, and model error for open loopidentification based on 300 data points with filtered white noise input.
Input
Flow
Rate
Concentration
Response
Model
Error
37
T •=» 0.01 minutes
0 Time (Minutes) 7.5
Figure 7: System input, output, an£T model error for open loopIdentification based on 300 data points with filteredwhite noise input.
Number of
Input
Terms
38
TABLE 1Comparison of Model Errors from Correlation-
Regression and Ordinary Regression for 300 Data PointsNumber of Output Terms
1 2 3 42.793 1.187 0.933 0.7361 3.537 2.028 1.039 0.788
0.938 0.215 0.081 0.0382 1.482 0.277 0.182 0.076
0.105 0.022 0.018 0.0163 0.254 0.043 0.040 0.034
0.032 0.021 0.017 0.0154 0.104 0.052 0.052 0.030 I
Top Entries - Ordinary Regression Bottom Entries - Correlation-Regression
1 Np 2Error = ~ S (c(iT) - cm(iT))i*l
c(iT) s true plant response at time (iT)cm(iT) B predicted plant response at time (iT)
39
correlation regression. With these results and the relative ease
of implementing the ordinary regression scheme ( the correlation regression approach apparently offers no advantages for use on deterministic signals.
Measurement Noise and Parameter BiasIn many process systems, the ability to determine the value of
a response variable is hampered by measurement noise. The ramifications of such noise with respect to parameter estimates are treated in this section. The noise is assumed to be white and present only in the output variable.
Formal development of the regression equations for least squares parameter estimation is available in the literature and will not be presented (3). Such developments typically Indicate that the regression matrix can be partitioned into four quadrants, one of which contains only the response variable. A typical term in this quadrantwill have the form E C.C.... The measured state at time "i", C.,
i-1 1 i+J * 1is the sum of the true state, C^, and a measurement error, C^.A general term of this quadrant can be expanded as:
NP
NP NP *- E (c + €.) (c i-1 i
? * * F . * * "p (4)
Assuming that the measurement errors are zero mean white noise and independent of the input signal, Equation 4 can be written as:
40
NP NP * * NPCl C*+J " i-1 °i °1-+J +tfi Ci €i+J <5>
In the limit, the error term takes on the following values:„ 1e«j) i - j
^ * 1 * 1 - J o t * i <6>
The non-zero term for (i"J) causes biased parameter estimated when using ordinary regression for dynamic model identification. No amount of data will remove this bias. Variations of the least squares approach can be developed which will cope with this problem (4).The more complicated maximum likelihood approach can also be used (4).
As shown in the previous chapter, the contribution of measurement noise to the cross-correlation function has a zero expected value. This zero expectation leads to unbiased estimates for this correlation- regress ion approach.
Figure 8 shows the effect of adding gaussian distribution white noise to the plant output on the fit error for both types of identification. The model used in this and the following case was a (2r- 2c) model. In agreement with the errors shown in Table 1, the ordinary regression results are superior when measurement noise is omitted. The degeneration of the ordinary least squares estimates is clearly evident with increasing magnitudes of measurement noise, while the errors shown for the correlation-regression scheme are
relatively insensitive to this noise.The intensity of the noise in the previous case was rather small
(maximum noise standard deviation of 1.5 units versus a concentration range of 20 units). Figure 9 shows the results of adding noise of
Mean
Square Error
0.2
0.4
0.6
0.8
1.0
1.2
41
1.5
0 Ordinary-Regression O Correlation Regression
1.01.51.00.0
0.0
o100 200 300
Number of Data Points
Figure 8: Comparison of modeling errors for correlation-regression and ordinary regression with measurement noise present.
Averaged Mea
n Square Error
42
1.0
. 2
0100 325 550 775 1000
Number of Data Points
Figure 9: Effect of Measurement Noise on Modeling Error forOpen Loop Correlatlon-Regresslon Identification
43
greater intensities for the correlatlon-regreaslon approach only.In this figure, the cummulative effects of the noise Is shown.This is equivalent to an Integrated average of the fit error of the type shown in Figure 8. The curves appear to approach a zero slope, thus indicating that unbiased estimates can be expected from large data samples.
Closed loop IdentificationIn most process control applications where system identification
is to be performed, the ability to do so without unduly interrupting normal operation is highly desirable. For a typical feedback system, this implies performing the identification during closed loop operation. If the identification is to be carried out continuously, it must be done while the loop is closed.
To perform any type of dynamic identification, the system must be in a state of dynamic change. During normal closed loop operation of a process system, several causes of such change may be encountered including unmeasured disturbances, measurement errors, and variations
due to setpoint changes.Of primary consideration is the case for which setpoint changes
offer the impetus for a dynamic response. The frequency of such changes is determined by a system's overall environment and the resultant operating policies. For the purpose of this discussion, the frequency of setpoint.variation will be classified as:
1) Low - such that parameter variations can be expected tooccur during a period of no setpoint activity.
2) Moderate - for example, within a supervisory system which
automatically adjusts the setpoint In real time.
3) High - for example, in a batch process.
With little or no setpoint variation (case 1), the discussion does not apply. Identification under such conditions requires a more detailed investigation. If frequent setpoint variations are to be encountered in normal operation (case 3), a sufficient amount of dynamic data will be available. An intermediate amount of setpoint variations, as in case 2, would require a longer period of time for the data collection. In this situation, the nearly steady state data should be omitted as unmeasured disturbances will typically contribute the greater portion of the near steady state response. Intermittently collected data can then be combined into a single record for identification purposes.
Although setpoint variations supply the necessary dynamics for identification, the input for the identification is the actual input to the plant (i.e., the manipulated variable). By using the actual input-output record, the closed loop identification can be solved
in a manner analogue to the open loop case.
Effect of Measurement Noise on Closed Loop IdentificationAs indicated, uncorrelated (white) measurement noise added to
the true system state presents no difficulties for the correlation- regression scheme when applied to open loop identification. However, when the loop is closed the errors tend to propogate around the loop and thus affect the measured cross-correlation function. The interested reader is referred to an article by Goodman and Res wick for a more complete discussion of this problem (5). Their results
45
show that the noise contribution to the cross-correlation function is restricted to the non-positive time shift part of the function.This implies that the useful correlation data in this situation are restricted to the positive shift axis. Noise which is correlated will have the effect of further reducing the useful portion of the correlation function data, but if it is only slightly correlated this reduction may only amount to a few data points.
Results - Closed Loop IdentificationThe correlation-regression identification scheme was applied to
the simulated plug flow reactor during closed loop operation with a discrete PI controller. The controller was tuned to give a minimum integral of absolute error (IAE) response to a step change in setpoint. All cases discussed in this section involve a 2r - 2c discrete model and a sampling.time of 1.5 seconds.
As a base case, the system was subjected to a series of step setpoint inputs with the response variable being measurable without error. Figure 10 shows the results of a typical run with the identification performed over the interval shown and the error based on a one step prediction model for the given input-output record.These results clearly indicate the success of the identification for this deterministic case.
For identification with measurement noise present, the previously described modifications were implemented. Figure 11 shows a finite approximation (1000 points) of the white noise auto-correlation function and the noise contribution of the measurement error to the cross-correlation function. As predicted, this contribution is
Manipulated
Variable
Concentration
Response
Model
Error
46
tn
•tf-VO
7.50Tine(Minutes)
Figure 10: Manipulated Variable, Plant Response, and ModelErrors for a Series of Step Inputs in Setpoint. Identification based on 300 data points with no measurement noise.
47
40
1.5 seconds30
20
10
Auto-correlation
0
Cros s-correlatIon
-10
-2030030 15 15
Time Shift (Seconds)
Figure 11: Measurement noise auto-correlation functionand noise contribution to the cross- correlation function based on 1000 data points.
48
isrestricted to the non-positive shift portion of the axis. The
small non-zero contribution indicated for the positive shift portion is due to the finite approximation involved. Results of a typical
run with measurement noise present are . shown in Figure 12. The model determined from the noisy case does not fit the deterministic output (Figure 10) as well as the model based on the deterministic input-output
data. However, the resulting error remains within an acceptable level. An additional error comparison was made for the models determined from the data of Figures 10 and 12 comparing the deterministic plant response with the discrete model response based solely on the deterministic input sequence. That is, the discrete model was used to predict the entire plant response rather than predicting one step into the future based on the known plant response points and previous inputs. Using the model in this way resulted in a larger mean square error, but by less than an order of magnitude.
As a final observation concerning the closed loop identification's performance, the modeling errors (one step predictor) were determined for a range of sampled input-output points and a range of standard deviation of the measurement noise. Figure 13 shows the results of this set of runs. The data plotted in the figure were obtained in a manner similar to that of Figure 9. That the errors
shown in Figure 13 are larger than those of Figure 9 is partly due to the reduced nun&er of useful correlation points available for closed loop identification with measurement noise present. However, most of the difference in these errors can be attributed to the fact they are based on different input-output sequences.
Manipulated
Variable
Concentration
Response
Model
Error
49
om
CM
oo
7.50Time (Minutes)
Figure 12: Manipulated variable, plant response, and modelerror for a series of step inputs in setpoint. Measurement noise standard deviation ■ 4.0.
Average
Mean
Square Error
50
0.8
0.6
0.4
0.2
1000100 500Number of Data Points
Figure 13: Effect of measurement noise on modeling error forclosed loop correlation-regression identification.
51
Summary
The ability of the correlation-regression Identification scheme to perform in cases of practical interest has been demonstrated.This technique offers no apparent advantages over identification based on regression of the original input-output data if that data is noise free. The .filtering effect of the correlation functions allows identification of systems with considerable measurement noise. Moreover, the effects of such noise for closed loop identification can be eliminated allowing for a very practical application of the technique. An obvious extension of this work includes on line identification with controller parameter adjustment based on the identified process model. Finding a means of handling unmeasured disturbances to the system will allow the full potential of the
identification technique to be reached.
Nomenclature
coefficient of.Input term In pulse transfer function
coefficient of output term In pulse transfer function
number of output terms In discrete modeldelay Integer ■ nearest Integer to dead time/sample time
number of Input terms In discrete modelnumber of data points used for IdentificationLaplace transform operatorsample timesystem outputdiscrete model outputmean square modeling errorInput signal
/
noise filter time constant auto-correlation function
cross-correlation function standard deviation of measurement noise output signal variance
53
Literature Cited
1. Kalman, R. E., "Design o£ a Self-Optimizing Control System", Transactions of the A.S.M.E.. Vol. 80, Feb. 1958.
2. Dube, J. H., "Applications of Identification and Control Methods", Ph.D. Dissertation, Department of Chemical Engineering,Louisiana State University, 1971.
3. Bigelow, S. C. and Herman Ruge, "An Adaptive System Using Periodic Estimation of the Pulse Transfer Function", I.R.E. Convention Record, Part 5, pp. 25-38, 1961.
4. Johnston, J., Econometric Methods. McGraw-Hill, Hew York, 1963.5. Goodman, T. P. and J. B. Res wick, "Determination of System
Characteristics from Normal Operating Records", Transactions of the A.S.M.E., Vol. 78, No. 2 , pp. 259-271, Feb. 1956.
54
CHAPTER IV
LINEAR INDEPENDENCE OF INPUT-OUTPUT DATA FOR CLOSED LOOP IDENTIFICATION
Introduction
In this chapter, a criterion necessary for the estimation of the parameters of a discrete dynamic model using linear regression is developed. The primary topic concerns the linear independence of input-output data recorded during normal operation of a closed loop control system which is driven solely by external disturbances. The discussion is based on the ordinary least squares approach, but should apply to many variations thereof. Computational results are delayed until the next chapter following the proposal of an overall
closed loop identification scheme.
Regression Analysis of Linear Dynamic ModelsThe premiere paper presented by Kalman discusses a basic method
for the estimation of discrete linear model parameters using multiple linear regression (1). Much of the related literature to follow considered various limitations of this basic approach and methods for overcoming these limitations (2,-6). With few exceptions (7), the development of identification techniques has been limited to the
case of open loop identification.As noted by Young, a vast difference may separate the reliability
of parameter estimates based on realistic data versus similar estimates based on a "well planned experiment" (8). In the present context, realistic data will be defined as the input-output record from
closed loop operation of a system whose dynamics are caused only by external (unmeasured) disturbances.
The purpose of this/ chapter Is twofold: 1) to determine a
criterion necessary for strict linear independence of the model input-output sequences, and 2) to give some guidelines for the selection of controller parameters which will result in closed loop data being more characteristic of that from a well planned experiment .
The development to follow involves considerable matrix notation The following conventions are adopted: 1) scalars - lower caseEnglish and Greek, 2) vectors - upper case English and, 3) matrices upper case Greek. Subscripts, superscripts, etc. will not follow this convention.
A brief review of the application models will be given. More detailed developments are available elsewhere (2,6). A general discrete model has the following representation.
whereu
yd a b
NU NC M
NU NCyl _ £ Vi-M-J Y i - J + di
system input (manipulated variable) system output disturbance input term coefficient output term coefficient number of input terms number of output termsan integer representing the system dead time
(1)
1 ■ time indexEquation 1 can be represented for the entire response by means of •
matrix notation as:Y - 0C + D (2)
where 1
A ‘ ‘Vyk+l•
D - ‘Wi•
c » *2•
•
_yk-HJ• •
®NU-bl-b2
•
•-b
Vm-1’. “k-M-2* •••* V-M-NU’ yk-l* yk-2* *,,yk-NC"k-M* V-M-l* **** Vm-NIH-1’ yk 'k-1
“k-HJ-M-l* uk+N-M-2* “k+n-M-NU* yfcW-l^k-W-l*'*'yk-HJ-NC
y’'-1 * ’yk-NC+l
and k is an arbitrary Initial index.
The development leading from Equation 2 to the least squares estimate of C is widely available; for example see Johnston (9).
The estimate of C can be written as:
C - O V 1 0Ty (3)The validity of Equation 3 is contingent on the applicability of various assumptions, one of which is the linear independence of thevectors represented by the columns of the 9 matrix. The matrix
Trepresented by 6 0 is the familiar "sum of squares and cross-
57
products" matrix, also known as a gram-matrix. Such a matrix has a
inverse i£ and only if the columns of 8 are linearly independent (10). The necessity of this independence is noted in the control literature
for an open loop application, and the author comments that no problems of singularity were experienced (3).
Mathematical Development of Independence CriterionThe possibilities of a singularity occurring in a closed loop
configuration is greatly enhanced by the nature of feedback control. That is, the manipulated variable at time "i" is a function of the past values of the manipulated variable and of the past system outputs. A general representation of a linear control algorithm is given in
Equation 4.NU' NC'
ui ' £ V i - J ,j(8i-j'yi-j) (4)
wherep,q are controller parameters s ■=> setpointNU'jNC' are indices defining the order of the controller.
The setpoint will be treated as a constant and, without loss ofgenerality, be assumed equal to zero. Thus, some simplification of
Equation 4 is possible.NU; NC'
U1 " £ pJui-J +*, Vi-3 (5)Substitution of Equation 1 into Equation 5 for the term y^ gives:
NU' NC' r NU NCHU Hu r- HU nu -i
-i PA - J +£ V w + q° (6>
The following definitions will prove helpful:
NU" - Max.(NU',NlHM)
NC" - Max.(NC',NC)Rearranging Equation 6 results in:
M NU"Ui " S P1Ui-1 + E (pj + V j - M )ui-J J"1 j-M+1 J J 3
NC"+ £ <v«0V yi-J + V i O)
In Equation 7 some of the coefficients (a,b,p,q) may have indices which are greater than the limits indicated in the defining Equations (1 and 4). This situation arises from the possibility of
system models and control algorithms having different orders. Any coefficients not explicitly introduced in Equations 1 or 4 are defined to be zero. Equation 7 can be simplified into the form:
NU" NC"ui ■ A *jVi +sl1 Vi-J+ v*i (8)
Equations 1 and 8 can be combined into a single vector equation.Let
Pir “i rVi-,[ y j Ql ' [ J
ru t,
*•
for k “ 1,2,...K where K - Max.(NU",NC").
With this notation, Equations 1 and 8 are combined to give:
h ' V i - i + V i - 2 + V i - K + Q i (9)A more compact form of Equation 9 can be obtained (11).
zi - r z±ml + Wi (10)where T Is the 2Kx2K matrix
*1 *2 .... $K-1 s®KI 0 .... 0 00• I• 0s 0••_0
•0
•I
•0 .
with I representing a 2x2 Identity matrix and 0 the 2x2 null (or zero) matrix. The vectors and are defined as:
c H*•
M i '
yi di
“i-1 0
V l• Wi "0•• •• •
Ui-K 0
_yi-K 0 .
A general solution of Equation 10 is available under the conditions that T is non-singular. The solution given here is dependent on the terms of r being constant. For the purpose of this development, such a restriction Is of little consequence.It will be seen that the solution Is necessary only for a short time span and thus the assumption of constant parameters Is suitable. The solution, as given by Miller (11), Is:
60
12j-k+1 (11)
In Equation II, k represents the initial time index and is the
vector of initial conditions. The superscript of T is an exponent, with r° defined an identity matrix of suitable order.
Before discussing the implication of Equation 11 on concerning input-output sequence independence, some results due to Miller will be mentioned. First, it can be shown that a necessary condition for the stability of the homogenous system (W^ - 0 for all "i") is that the eigenvalues of T be less than one in modulus. This implies that a finite response will result from any finite set of initial conditions. This requirement should always be fulfilled for a properly tuned control loop. In addition, the same criterion gives assurance that the effect of initial conditions will "die out" for large "1". Thus, after some period of normal operation, Equation 11 represents the response of a general closed loop system whose only inputs are in the form of external disturbances.
Additional simplification of Equation 11 is possible. Arbitrarily letting k « 0 and expanding results in:
- r<1“1)
V i % d2 qodidl d2 di0 + p(i-2) 0 +...r° 00• 0• 0••0 •
0•0_
r V (12)
61
Due to the zeros of the W vector, only the first two columns of the
matrix have an effect on the response, Z^. Let these columns be denoted by the vectors 6. and G* ... That is,let
t* 4\ " Column ' W of for m ■ 1,2
Noting the linear dependence of the non-zero terms of the vector allows further simplification.Let
G(i-j) " qo Gl,(i-j) + G 2,(i-j) (13)
The length of G(i-j) be 2K» Define a 2Kx2K matrix, T , suchthat:
* f 1" LG(2K-1) G(2K-2) •** G(2K-2K)J (14)
Equation 12 can be written for "i" ■ 2K as:
Z,
where“2K r* W* + r2* Z. (15)
W
2KIt is necessary to assume that the individual disturbances (i.e., terms in H ) are linearly Independent. (The statistical properties of this term will be treated in the next chapter.)
Linear Independence - Necessary and Sufficient ConditionsThe development leading to Equation 15 can be used to state
the sufficient conditions for the linear independence of the "independent" variables of Equation 1. The "independent" variables in Equation 1 are a subset of the vector Z^. Thus, if the terms of
are independent, the "independent" variables in the model will also be independent. The sufficient condition for the components of Zgg to be independent is that T be non-singular. It should be noted that these conditions hold for any arbitrary choice of initial conditions, ZQ, and in particular, for the case of ZQ itself being the result of "normal operating conditions". Likewise, it should be noted that a derivation requiring the terms of ZQ to be independent would be equivalent to "assuming the problem away".
The conditions stated here are sufficient for a general model and controller configuration. However, if Z contains more terms than are present in the original model, these conditions are not necessary. If such is the case, the matrix T should be reduced to include only those rows corresponding to the variables appearing
in the system model.Defining T as the reduced (NlHNC)x2K. (where (NIHNC)£2K)
matrix the following condition is both necessary and sufficient to assure the linear Independence of the "independent" variables of the system model. "The rank of T** must be equal to (MIHHC)."
By way of review, the following assumptions have been stated
or implied.1) r is non-singular and has eigenvalues less than 1 in
modulus.
63
2) The external disturbances are linearly independent.
3) The model is an adequate representation of the physical system.
4) Both the model parameters and the control parameters canbe assumed constant for a short length of time (corresponding to 2K units of time).
Design of an “Identifying Controller11The choice of model structure must be based on physical con
siderations. However, both the structure and the parameters of the control algorithm are chosen by the control engineer. The criterion which has been established gives the designer some guidelines by which a controller suitable for closed loop identification can be chosen. However, fulfilling these requirements in a mathematical sense leaves the case of obtaining an ill-conditioned sum of squares and cross-products matrix as definite possibility. Thia is due to the fact that time series data are inherently highly correlated for at least a few time lags. The ideal linear regression situation in which the cross-product terms vanish is clearly unobtainable. However, by proper choice of controller parameters, it may be possible to give some added confidence to the overall
numerical results from a closed loop system identification.Many approaches might be taken in establishing a criterion for
"good" (or at least better) closed loop identification results.It is likely that the criterion for good identification and that for good control will be in conflict. The presence of such a possible conflict is acknowledged, but will not be specifically
64
treated.ftftthe criterion to be established in suited to the case of T
ftbeing of the same order as that of r . An extension to the general case should be reasonably straightforward. For this particular
case (i.e., r of order 2Kx2K), a single determinant must be evaluated in order to establish the singularity or non-singularity of the matrix. To obtain a more meaningful quantity, the determinant will be normalized. A suitable definition, given in Conte (12),
for such a normalized determinant is:
I1**!Norm |r | - 2K a (16)
It i i»l
where2K
ftftand y. . is an element of T .JAn "optimum identifying controller" can be designed based on
ftftmaximizing the absolute value of Norm |r | subject to stability constraints on the control loop.
Example ProblemIn this section, the results of the previous development are
used to analyze a particular control loop configuration, namely a first order system with a proportional plus integral controller. (Equation numbers followed by will be used in this section to indicate correspondence with the previously developed general equations.) The model and control (velocity form of PI controller)
65
equations are:
Model: yt - ajU^j - + dt (1*)
Control: ut - P ^ . j + (5*)
In terms of Equation 8, the control equation is:
ui “ ri"i-i + V i - i + V i (8*>where
“ 1 + q ^ ^ (note that = 1 for PI algorithm)
*1 ■ V o blCombining Equation 1* and 8* results in the vector equation:
u • r_ t. u, . qn d[yp -[vJc^kcoChoosing an initial index of zero (k - 0), the solution as given by Equation 11 is:
„ r, t, « * « , d, r, t, (1> o
u p - 4 r j u ° ]
(ii*)
As indicated in Equation 15, the solution for i « 2K (i.e., i » 2) is necessary. After some manipulation, the result is found to be
c ? . k : : * ; » . < - .
aEvaluation of the determinant (unnormallzed) of the T ( which isftftequivalent to T ) matrix gives
66
|r I - q0 + U * )Or, in normalized form:
* q + q-.N01® lr I - r-,,— (16*>1 2
where
*1 " [qo(1 + V l )2 + 2qo(1 + V l X V ’o V + < V qobl>i
4. 21l/2+ qoJ
[qo 4 - v a + bi + t f 2“2
Further simplification of the quantities and prove fruitless.An immediate result of this example shows that a proportional
only controller (qQ ■ -q^) will result in a singular system. That the sum, qQ + q^, should be large in absolute value is also evident, but the functional relationship given in Equation 16* precludes a more definitive statement. Numerically, the evaluation of Equation 16* is straightforward, however, the dependence of the normalized determinant on the unknown parameters and hinders precise evaluation. It may often occur that some prior knowledge concerning the model parameters is available. Thus, the usefulness of the normalized determinant, although restricted, is evident.
The evaluation of Equation 16* has been carried out for a specific system (a^ - .3, b^ » -.7). The results are presented in Figure 1 for both selected PI controller parameters and for the Dahlln algorithm (13). (See Appendix A for the form of the Dahlin
algorithm used.) The heavy lines indicate PI parameter pairs which result in an unstable closed loop system. The stable portion
does however include a region for which the determinant is quite large and should provide for a set of parameters suitable for both
adequate control action and system identification. The curve for the Dahlln algorithm results, as expected, in a stable system over the entire range of the tuning parameter (excluding the limiting cases of X ■ 0 and X ■ 1). The determinant is smaller than can be obtained by the unconstrained case of the PI algorithm.
SummaryAn approach has been presented whereby a closed loop control
system can be evaluated in terms of the "identifiability" of the system with respect to linear regression. In addition, the requirements for an optimal, in some sense, identifying controller have been defined. Although formally limited to an ordinary least squares approach, the conclusions should be applicable for variations of this approach with little or no modification. In a very broad sense, some of the difficulties associated with the design of a true adaptive control loop have been brought to the foreground.
Normalized Determinant
Tuning Parameter
0.25 0.50 0.75 1.0o
(-5.0,-3.0,-1.0)
o Dahlin Algorithm
o
5 010 5 10
Figure 1: Normalized determinant versus selectedPI controller parameters and for Dahlin controller
Nomenclature
model Input coefficientmodel output coefficient
system disturbancecontroller Input coefficientmodified controller Input coefficientcontroller output coefficientsetpolntmodified controller output coefficientmanipulated variablecontrolled variablemodel coefficient vectorvector of system disturbancesa column of T or TIdentity matrixan Integer representing the system dead timeclosed loop state vectorclosed loop disturbance vectornumber of model output terms (output order)number of controller output termsnumber of modified controller output termsnumber of model Input terms (Input, order)number of controller Input termsnumber of modified controller Input termsclosed loop disturbance vector (single step)closed loop disturbance vector (multiple step)
70
Y SB vector of system outputs ("dependent variables")
Z 8 closed loop state vector
a 8 normalization factor for determinant of T* or T**
Y » an element of T, T , or TX - tuning parameter in Dahlin algorithm (see Appendix I)r - closed loop state transition matrix (single step)*r 8 closed loop state transition matrix (multiple step)r** 8 *a partition of T$ a closed loop transition matrix
e s matrix of "independent" variables
71
Literature Cited
1. Kalman, R. E., "Design of a Self-Optlmlzlng Control System", Trans, of theA.S.M.E.. Vol. 80, Feb. 1958.
2. Bigelow, Stephen C., and H. Ruge, "An Adaptive System Using Periodic Estimation of the Pulse Transfer Function", IRE Convention Record, part 4, pp. 25-28, 1961.
3. Dahlin, E. B., "On-Line Identification of Process Dynamics",IBM J. of Research and Development. Vol. 11, No. 4, July 1967.
4. Rowe, Ian H., "A Boot Strap Method for the Statistical Estimation of Model Parameters", Int. J. Control. Vol. 12, No. 5, pp. 721-738, 1970.
5. Wong, Kwan Y. and E. Polack, "Identification of Linear Discrete Time Systems Using the Instrumental Variable Method", IEEE Trans, on Automatic Control. Vol. AC-12, No. 6, Dec. 1967.
6. Young, P. C., "Process Parameter Estimation and Self Adaptive Control", Proceedings of 2nd IFAC Symposium on Theory of Self- Adaptive Control Systems. Sept. 1965.
7. Goodman, T. P. and J. B. Reswick, "Determination of System Characteristics from Normal Operating Records", Trans, of the A.S.M.E.. Vol. 78, No. 2, pp. 259-271, Feb. 1956.
8. Young, P. C., "Regression Analysis and Process Parameter Estimation: ... a cautionary message", Simulation. Vol. 10,No. 3, March, 1968.
9. Johnston, J., Econometric Methods. McGraw-Hill, New York, 1963.10. Gantmacher, F. R., The Theory of MATRICES. Vol. I, Chelsea,
New York, 1960.11. Miller, Kenneth S., Linear Difference Equations. W. A. Benjamin,
Inc., 1968.........
12. Conte, S. D., Elementary Numerical Analysis. McGraw-Hill, New York, 1965, p. 165
13. Dahlin, E. B., DDC Tuning Reference Book. Instruments and Control Systems, Chilton Co., 1969.
72
CHAPTER V
IDENTIFICATION OF PLANT DYNAMICS IN A CLOSED LOOP SYSTEM DRIVEN BY EXTERNAL DISTURBANCES
IntroductionAn approach for solving the closed loop identification problem
for a dynamic system driven only by external disturbances is presented. A preliminary development relating the correlation-regression approach to ordinary regression is followed by a discussion of the nature of external disturbances in a typical process environment and the effect of such disturbances on the identification scheme.An algorithm based on the combined characteristics of generalized
regression and correlation-regression and a means for estimating the adequacy of the overall model structure is also proposed. Numerical results for a simple closed loop system follow the general development.
Development of Correlation-Regression EquationsIn this section, an approach based on linear regression tech
niques is used to formulate a set of estimating equations which are equivalent to the previously developed correlation-regression technique. Such an approach loosens the bonds of the correlation- regression technique from statistical signals while retaining some desirable properties related to ordinary (or variations thereof)
regression.Except where stated and for subscripts, superscripts, etc., the
following nomenclature rules will be observed: scalars are
73
symbolized by lower case (English or Greek) letters, vectors by
upper case English, and matrices by upper case Greek letters.A general discrete model has the following representation:
y.NU NC
A v i -m-j - a Y i - j + di (i)
wheresystem Input (manipulated variable)
system output (controlled variable) disturbance .Input term coefficient output term coefficient number of Input terms (Input order) number of output terms (output order) an integer representing the system dead time
time IndexEquation 1 can be represented for the entire response by means of
matrix notation as:
u
ydab
NUNCM
1
Y - e C + D (2)where
1£ 'Vyk+l• C - a2• D - \+l•
/k-rti•*NU
•A4**
“bl- h•
1I
74
\-M-l \-M-2 V m-NU yk-l yk-2 ***• yk-NC“k-M “k-M-l *••• “k-M-NU+1 yk yk-l *•** yk-NC+l
\+N-M-l “k+N-M-2* \-HJ-M-NU yk+tf-l yk+N-2* yk+N-NC_
and k is an arbitrary initial index.Consider the following transformation of Equation 2.
a* «*. *0 y « 0 0C + 9 D (3)
where
“k-NC “k-NC+l “k-NC+N
“k-NC-l “k-NC “k-NC+N-10*
ihc-NC-NK “k-NC+-l-NK* * * * “k-NC+N-NK*/cThe (NK+-1 x N+-1) transformation matrix, . 0 , is equivalent to the trans
formation which forms the appropriate correlation functions requiredfor the application of correlation-regression. The number of rows
* * of 0 (**NK+1J is equal to'the number of correlation "data" points which
are generated.Inspection of the components of Equation 3 reveals the equi
valence of this transformation with system auto- and cross-correlationChfunctions. A general term for the i row of the transformed
itindependent variable vector, 0 y, can be written as:* Ny{ • S u yk-i+i ^1 j«i k-NC-i+J k 1+J
The definition for the discrete cross-correlation function for
stationary u and y is:
75
1 NV ”0 " h £ uj-i yj (5>
With the exception of the 1/N term, the transformed independent *variable, y^, and the cross-correlation function, 9Uy(*®)» equiva
lent. The difference of the indices of y and u define the shift (or delay) integer for the correlation function. In the case of Equation 4,
m - (k-l+j) - (k-NC-i+J) - NC-H.-1 (6)The relationship between the correlation function and the transformed variable is therefore:
H yi ■ V * * 1-1) (7)The extension to the transformed disturbance term is direct:
5 d! - (8)A similar analysis of the transformed independent variables, say0^j, of the 6 6 matrix results in the equivalent correlation function
«Dfrepresentation. For j £ NU, the 0 ^ term will be an auto-correlation
function. With this restriction on j, the "i,j" term of 0 0 can be written as:
* N®ij " \-NC-i+n “k+n-l-M-j
Or, in terms of correlation functions,
N 0ij “ <Pu u(NC+1-1-M-J) (10)For j > NU, a cross-correlation function will be formed.
* N0ij " ^ “k-NC-i+n yk+n-l-j-HJU
The relationship with the cross-correlation function Is therefore:
' 'Puy<NlWJC+l-l-J) (12)
An expression for the I***1 row of Equation 3 can be obtained by combining Equations 7, 8, 10, and 12.
* NU * NU4NC * *
- A V ’m b j - N u + d l ( 1 3 )
In terms of correlation functions, Equation 13 can be written as:NU NC
cp (NC+i-1) - S a.cp (NC+l-l-M-j) - E b.cp „(NC+L-l-j)uy jBll j ““ iml JY«y
+ cpud(NC+i-l) (14)
for 1 = 1,2,...NK.Equation 14 represents the discrete model in the correlation
function domain. It is of the same form as the time series model, Equation 1, and is equivalent to the previously developed correlation function approach to model identification. The incentive for developing the correlation-regression approach in this manner is based on a similar development discussed by Johnston (1). With this development, many of the results given by Johnston can be applied to the
correlation-regression approach.
Estimation ProcedureThe approach taken by Johnston (1) results in a two-stage least
squares estimator, and is similar in many respects to generalized regression. With respect to Equation 3, the situation of Interest occurs when NK > (NIHNC). That is, more correlation data points are
77
available than the number of parameters to be estimated. This is the usual case for least squares analysis. Within certain restric
tions, the appropriate estimate of C is given as:
c - [ e W e *1)"1 e*e]'1 e V ^ e V V V y (is)The restrictions under which Equation 15 applies can be stated as:
1) E(0*D) - 0 (16)and
2) E(D DT) » ct2I (17)where E is the expected value operator and I is an identity matrix.Stated differently, D must represent a white noise sequence and beuncorrelated with the input (manipulated variable). Attention willbe given to these restrictions in a later section.
The estimator given by Equation 15 can be written in terms ofcorrelation functions. Several of the terms in Equation 15 wererelated to the correlation functions in the previous development.The elements of 6 0 are the appropriate correlation functions definedin Equation 10 and 12 while the vector 0 y contains elements definedin Equation 7. Equation 3 can be rewritten in the form:
X - $C + 0*D (18)*where X - 0 y
ft - 0*0Equation 18 is related to Equation 14 in the same manner that Equation
4k2 relates to Equation 1. The term 0 0 can be expanded as:
78
* * Te e“k-NC “k-NC+1
“k-NC-1 “k-NC
“k-NC+N
“k-NC4N+l
“k-NC-NK “k-NC+1-NK** * * “k-NC+N-NK
uk-NC \-NC-l “k-NC-NK
“k-NC+l “k-NC ***• “k-NC+l-NK(19)
^k-NC+N “k-NC+N-1 “k-NC4N-NK_
If u^ represents a zero mean stationary sequence, the (NK+1 x NK+1) matrix of Equation 19 can be shown to be composed of the points onthe input auto-correlation function (3). Formally, let
■l ^r - i e eThen, a general term of T, g ^ is given by:
Yij cPUu(1’J)
(20)
(21)where cpuu represents the manipulated variable auto-correlationfunction. This matrix is symmetric due to the symmetry of a(stationary) auto-correlation function.
These developments lead to the fundamental least squaresestimation equation upon which this work is based.
Ac - [ i F f H T 1 §Tr-1x (22)Equation 22 gives a form of generalized least squares for which the
cross-correlation function, 9^ 0 0 * represents the dependent variables and the auto-correlation function, <Puu(k), In addition to appropriate (as defined by Equation 14) values of cpyy(k) represent the independent variables. The variance-covariance matrix, 1% is composed of the
terms of the input auto-correlation function, <PUU00« The use of generalized regression is not strictly necessary. Earlier work using the correlation-regression approach was based upon ordinary regression applied to the auto- and cross-correlation functions and is equivalent to the case for which the variance-covariance matrix,T, is assumed to be diagonal.
System DisturbancesThe realistic characterization of external disturbances in a
process environmental is a difficult matter. Such disturbances may be attributed to a complex intermixing of both deterministic and stochastic signals. For very long periods of operation, a purely stochastic approximation might be reasonable. If such an approximation could be justified, the resulting computational burden
(time and storage) would likely be prohibitive. In a similar manner, a purely deterministic representation would not account for the presence of short term process noise. An approach which compensates for the combined deterministic-stochastic nature of most process disturbances is Clearly desirable.
Disturbance CompensationTwo different types of disturbances will be treated. One will
be modeled as resulting from a load change and the other as a result of white noise. Both approaches are clearly approximate. For each
case, the disturbance compensation scheme results in a similar transformation of the time series data.
80
Load changes represent the usual deterministic type of disturbance
which affect a process loop. The block diagram shown In Figure 1
Indicates the configuration being considered. The disturbanceoriginates as q and passes through, the dynamics Indicated by Q(s).Ideally, one would like to perform the Identification based on thetrue plant Input-output sequence (u^,y^). However, the outputsequence Is available only in a form corrupted with the disturbanceterm, and some means of compensation for this term must be available.If the disturbance which originates as q is an abrupt (step) change,a reasonable compensation may be implemented if the dynamics of Q(s)are defined. Physically, Q and G can be expected to be inter-relatedsubsystems of the overall plant and should have similar dynamiccharacteristics. As an approximation, Q(s) will be described by afirst order lag. The highly damped dynamics of very many processsystems is the justification for such an approximation.
*The discrete representation of the term d can be written as:
4 ■ V i - i + p2dI-i (23)
where p^ and Pg are related to the gain and time constant of the transfer function Q(s). For q being a step input, the response will be the familiar first order lag. It should be noted that the
•ftdisturbance terms d^ and d^ of Equations 1 and 23 are related, but not identical. The term d^ is only that portion of the overall
”j|fdisturbance, d^, which enters the system from time "i-1" to time "i". Thus, d* affects the model equation as a difference, with the following relationship being applicable:
DisturbanceDynamics
Controller Plantsetpoint +
Q(s)
G(z)
Figure 1: Configuration of closed loop system driven byunmeasured disturbances.
Equation 23 written for two successive time periods can be combined with Equation 24 to give:
di " pl(qi-l“qi-2) + p2(di-l'di-2) (25)Multiplying the model, Equation 1, by a factor, p, and writing itat time "i-l" gives:
NU NC<*i-l ~ P £ ajui-M-j-l - P £ + P dl-l <M)
And subtracting this result from Equation 1 results in:NU
V ^ i - l " aj(ui-M-j" pui-M-j-l^
NC■ £ V yi-j- + dr pdi-i (27)
Based on Equation 23, the last two terms above can be combined to give:
di-Pdi-l ' Pl(ql-rV2-P(ql-2-‘1i-3))+ p2(di - r di-2'p(di-2'di-3)) (28)
Setting this expression equal to zero, and solving for p gives: Pl(qi-l’qi-2) * p2*dl-l“di-2*
P ■ + p2<di-2-di-3) (29>
Since q is a step function, each of the terms and(qi_2«qi_3) will be non-zero for a single (but different) value of "1". Thus, the following approximation can be made.
83
Substitution for the numerator results in the following relatioship.
Pl(qi-2"qi-3* + p2(di-2"di-3* p - ---------5------5------------ (31)i-2 " i-3
With the restriction on the term as above» Equation 31simplifies to
P ■ P2 (32)Equation 32 defines a constant, p, and is valid for all but the twoinstances at which the restrictions on q do not apply.
The implications of this development are quite significant and as best explained in terms of the transformed model, Equation 27.That is, the transformation implied by Equation 27 accounts for thepresence of external disturbances due to abrupt load changes in sucha way that the transformed disturbance term is approximately zero. Explicitly, the necessary transformations can be stated.as:
u i " pui-l
" P^i-l 03)
di - pdi-i - 0
t t twhere u , y and d are the transformed variables. In terms of these transformed variables, the model equation is:
t NU t NC t tyj - £ v U - j V u + < <*>
The actual transformation for the disturbance is not explicitly made, but rather is accounted for by means of the transformation of the measured system output, y. As this development is related to deterministic signals , no discussion will be given concerning the
84
statistical properties of the residual, d^.
As a second stage of this development, the disturbance term will be treated as a stochastic signal. More explicitly, the signal will be assumed describable as white noise with the dynamics of Q(s) equivalent to a first order lag. Equations 23 through 25 also apply in this case. Equation 25 can be written in the form:
di “ pl(qi-rqi-2* + p2 di-l (35)Defining the same type of transformation as for the deterministic case, and choosing p * p^ gives:
V p di-i - (36)
In this case, the disturbance has been transformed into the difference of two white noise sequences and q^.^* which is also a whitesequence having a variance twice that of the original sequence. The variance-covariance matrix of such a signal is diagonal and fits the assumption under which Equation 15 was stated. Based on results due to Goodman and Reswick (2) and subsequently applied for the correlation-regression technique, the transformed disturbance term will have the following characteristics.
E(ujdJ) = 0 *, i > j (37)4 0 J i s j
Stated differently, Equation 37 indicates that the cross-correlationfunction <p (k) is zero for the positive shift portion of the
udfccorrelation axis. Therefore, the previously applied approach for white noise in a closed loop system should be applicable in this case.
In summary, two equivalent transformations have been proposed
85
which should compensate for the effect of external disturbances
upon system identification using the correlation-regression approach. Due to the assumptions involved, the compensation will be approximate for most actual systems. Higher order transformations for the stochastic case are treated by Johnston (1), and should also be possible for the deterministic case. Utilization of a higher order transformation should be considered for a specific system and only if it is clearly necessary.
Disturbance Transformation ParameterNo indication of a means for determining a numerical value for
p, the disturbance transformation parameter, has been given. In a physical system, the dynamics associated with Q(s) will be, at best, ill-defined. Any attempt to determine a value for p based on physical considerations would be self-defeating to the overall model identification approach. Rather, an iterative approach will be proposed. This approach is equivalent to using the successive approximation solution for simultaneous equations and is described in basic numerical methods texts, for example (3).
In the present context, the iteration proceeds as follows:1) Guess a value for p.
A2) Compute C based on the guess of p.
C - f(y,0,P)The function, f, includes correlation, disturbance transformation,
and regression.3) Compute a new value of p based on most recent
86
estimate of C.
In this case, g is a function defined by the one-step-prediction of y, the formation of the disturbance auto-correlation function, and
the subsequent regression using the disturbance auto-correlation function to compute a new estimate of p.
4) Continue steps 2 and 3 until C has converged to within acceptable limits.
Proposed Identification ProcedureAn algorithm for the identification of the dynamics of a plant
which is part of a feedback control loop is presented below.1) Compute the input auto-correlation function and the
input-output cross-correlation function based on the available input-output sequence (u^»yi).Notes:
The value of NK should be large enough that all of the significant correlation points are included.
2) Guess a value for the disturbance transformation parameter.
Notes:If no basis is available, choose
The appropriate correlation functionsare defined as:
1 N= — E u.u.j, for k = 0,1,...NK N , i i-Hc(38)
87
0 ’ V (1)/t>W(0)-3) Confute the auto- and cross-correlation function of the
transformed input-output sequence.
Notes:This can be accomplished by an equivalent transformation
in the correlation function domain. The definition of the cross-correlation function of the transformed sequence is:
'f' t(k) " 5 .s, V i * : .E, V W i (39)uy i=l N i“lComparing Equations 41 and 42 result:
cp. t(k) - «PW 0 0 - P q>w (k-i) <40>uy * y
The result for the auto-correlation function is similar:
cp t(k) - <PuutfO - p V^Ck-l) (41)uu
Transformation in the correlation domain is numerically more accurate and offers considerable savings with respect to computation time. Thus, this approach should always be taken.
4) Solve Equation 22 for an estimate of C.5) Compute the one step ahead predictions of the response
based on the most recent estimate of C. Subtract the predicted response from the measured response to form the predicted disturbance sequence.
6) Compute the disturbance sequence auto-correlation
function.7) Compute the least-square estimate of p based on the
88
approximate auto-correlation function from step 6.Notes:
The correlation function model is:
" p f o r k " 1 , 2 * * * * N K ^4 2 ^
The subsequent least-squares estimate of p, based on
Equation 45 is:NKE <P 0 0 9a a k”1^A k=l M {ft
* ■ - s r - ^ 2 ------------------ (« >E (?M (k-l»2k=l dd
8) Check for convergence of the estimate of C. The criterion should be based on the vector difference of two subsequent estimates for C being "small". If convergence is obtained, go to step 10.
9) Repeat steps 3 through 8 using the new estimate for p.10) Check for model and parameter adequacy.
Realiabllity of Parameter Estimates and Model StructureApplication of linear regression techniques to time series data
presents many difficulties not common to classical linear regression. One of the significant problems which results concerns the confidence
limits of the estimated parameters. The last step of the proposed Identification procedure is concerned with this general topic amd may prove to be a fruitful area of research for the many diciplines concerned with statistical analysis of time series data. In this section, a rather simplistic approach is taken.
89
It is observed that the iterative scheme presented in theprevious section should terminate when a value of p is obtained
Awhich gives a value of G that is approximately the same as the previous A
value of C. The objective of this approach is to reduce the transformed disturbance into a white noise signal. Once convergence is achieved, a check can be made to determine if the disturbance term has been actually reduced to white noise. Note that the condition stated in Equation 17 implies that the disturbance term should be white in order for Equation 15 to apply.
A given sequence, in this case the transformed disturbance, can be checked for whiteness by a method due to Durbin and Watson (4).This check is based on the von-Neuman ratio, which is defined forthe sequence e^(i = 0,1,2...N) as:
N
V - 1 = L - ------- (44)
S ei i=0If "e" represents a white sequence, the limiting value for large N is v - 2.0. The possible range of values is 0.0 ^ v £ 4.0. A discussion related to the approximate distribution for v is presented in a article by Theil and Nagar (5). If a positive correlation exists for the sequence, the expected value of v is less than 2.0, while negative correlation (i.e., negative p) would give values greater than 2i0. An approximate set of confidence limits for positive
correlation is given in the literature (5) as:
6 . ozw-M/ix-n (45)2/ ^ h T
where6 “ confidence limitN ■ number of data pointsX * number of unknown coefficients
o “ significance points with:o « 635
1 per cent: o ■(46)
5 per cent: a ■ „ / N-l _ 1^64485^m 1
For this application, the relationship between the von-Neuman ratio of the transformed disturbance and the parameter estimates is not clearly defined. Nonetheless, this ratio can be used as some indication of the reliability of the estimates. A von-Neuman ratio significantly outside of the limits shown in Equation 45 Indicates several possible situations, specifically:
1) The plant model is of inadequate order, or 2) the transformation (and hence the noise model dynamics) are of too low an order. In addition, a controller which is poorly designed with respect to the identification aspects of closed loop control may be the cause of such a deviation.
The computation of v for the transformed disturbance can be carried out indirectly from the information available from the estimated disturbance auto-correlation function. Equation 44 can be
91
expanded in terms of the disturbance, d., as: N 1
(47)
i«0S (d±-p di-1)
Expanding the right hand side gives:
(48)
The terms appearing In Equation 48 are similar to the definition of the auto-correlation function of d. For larger N, the following approximation will hold.
where cp^ Is the approximated disturbance auto-correlation function. Little additional effort is required to check for the whiteness of the transformed disturbance sequence based on Equation 49.
The identification scheme was applied to the first order system shown in Figure 2. This system is not without practical interest, as evidenced by the widespread use of first order dynamic approximations in the process industries.
Figures 3 through 6 show the effect of the number of time series data points upon the resulting parameter estimates and upon the von- Neuman ratio of the transformed disturbance sequence. The confidence
v = ~PcPdd(0) + (1 * P2)cPdd(1)~pcpdd(2) (1 + P2>cPdd(0)_2p(Pdd(1)
(49)
Results
'— y
G(Z)
1) Plant: y± - .3 n±ml + .7 + d±
2) Controller: » «i_1 + r ^ + *2yi-ia) PI: ^ » -5.5, r^ ■ 0.5b) Dahlin: r^ « -2.11, r2 ■ 1.48
Note: Both controllers are of the same form. Thedesignations "PI" and "Dahlin" are used for convenience of discussions. PI parameters taken from Lopez (6), Dahlin parameters from Dahlin's article (7), with tuning parameter (time constant) equal to one sample time.
3) Disturbance: dt - pi^i-l”*i-2^ + p2^di-l"di-2^
a) Noise Inputp2 ■ .1 (Fig. 3), p2 - .9 (Figs. 4,6)
b) Load ChangesPj ■ »3, p2 ■ .7 (Figs. 5,6)
Figure 2: System configuration for closed loop identification withunmeasured disturbances.
Parameter
Estimates
von-Neuman
Ratio
True or Expected Values Confidence Limits PI Control
vO
Q>00
vO
m
CO
CM
100 1000500Number of Data Points
Figure 3: Effect of data record length on parameter estimatesand Vbn-Neuman ratio. Disturbance: Noise withp - 0.1
Parameter
Estimates
von-Neuman
Ratio
94
CM
00
A True or Expected Values 0 Confidence Limits Q PI Control M a a Dahlin Controlo\ -
00
CO
100 500Number of Data Points
1000
Figure 4: Effect of data record length on parameterestimates and von-Neuman ratio. Disturbance:Noise with p *» 0.9
Parameter
Estimates
von-Neuman
Ratio cm
CM
00
V0
True or Expected Values Confidence Limits PI Control Dahlin Control
00
500 1000100Number of Data Points
Figure 5: Effect of data record length on parameterestimates and .von-Neuman ratio. Disturbance: Load
Parameter
Estimates
von-Neuman
Ratio
©
00
True or Expected Values Confidence Limits PI Control Dahlin Control
ON
00
CM
100 500 1000Number of Data Points
Figure 6: Effect of data record length on parameterestimates and von-Neuman ratio. Disturbance:Combined Load and Noise with p ■ 0.9
limit8 are the 1% level as defined by Equations 45 and 46.Disregarding strict statistical formality, the limits shown correspond
to the range of expected von-Heuman ratios for white noise sequence with 99% confidence.
The controller parameters and disturbance dynamics shown in these figures are defined in Figure 2. Figure 3 shows estimates for
only the PI controller as those for the Dahlin algorithm were completely erroneous.. This case corresponds to high frequency process noise disturbances. Figures 4 and 5 show the estimates for highly
correlated process disturbances and for load disturbances respectively. The results in Figure 6 correspond to the case for which the disturbances were due to both load changes and highly correlated process
noise.In all cases shown, the parameter estimates tend to improve as
the record length is increased. A notable result indicated in these figures concern the difference between the PI controller and the Dahlin algorithm. The PI algorithm gives estimates which are far superior to those resulting from identification with the Dahlin algorithm. The reason for this difference is explained in the previous chapter which is concerned with the effect of controllers on the linear independence of the input-output records. The normalized determinant for each controller is approximately:
Dahlin: -.25PI : -.80
These results indicate the necessity of considering the controller design when attempting to formulate a closed loop identification procedure.
98
A controller design approach based on both identification and control considerations should be implemented for this type of application. The actual PI parameters used in this case are based on the work of Lopez (6), with parameters chosen to minimize the integral of absolute error resulting from a step load change. This indicates that both control and identification criteria can be satisfied simultaneously. Figures ;7 through 12 verify that suitable control results from this set of parameters. Moreover, a comparison of the error variance for the same disturbance sequence shows that the PI parameters give better results than the Dahlin algorithm.
The development of the correlation-regression approach to system identification has been presented based on linear regression techniques without recourse to formal correlation domain or spectral density analysis. This approach opens some avenues for the solution of the problems related to closed loop system identification. The same type of compensation approach for both unmeasured load and
noise disturbances was shown to be applicable.The presentation of a simple example problem indicates the initial
success of this identification approach. Perhaps the most significant result concerns the implications of the work presented in the previous chapter indicating the need for utilizing a controller designed for use in a closed loop identification scheme. The results of this chapter also indicate that both good control and good identification can be obtained with the same control algorithm.
/
This work leaves many practical questions unanswered. Notably,
the effects of using a simple model for a high-order system with
complicated disturbance dynamics has not been considered. A successful implementation of the identification scheme on such a system would represent an additional step in the formulation of a general procedure for system identification suitable for use in the process industries.
Setpoint Deviation
- . 3 6 0
-.bOO -.77?.?- . 6 4 0- . b 6 0 - . 4 8 0 - . 4 0 0 - . 3 2 ?-.240- • 16Q - . 0 8 0 -.000.030 • 160 .240 .320 .400 .430 .560.640.720 .800 . 880 .960
10 20 30 40 50 60 70Percentage of Points
Figure 7: Error distribution for feedback control with Dahlin algorithm correspondingto the identification shown in Figure 6 with 1000 data points
XXX* Mean = .00011Av Variance = .169A>.x X x X X XX * XXXxxxxX X a X X X Xx x x x x x xa X / X X X XXXxXXXX\X<XXXXa X X X X X X X\XXXXXXX X X X X.XXXXXx x x xx x x xXXXXXX^________ I______ I_________I_________ I_________ I_________ I______ :-- 1
100
Setpoint Deviation
-.9* d-.880-.bOJ A- . 720 A-.640 A-.560 .<•<-.450 xX-.400 XXx>.- . 320 XXX*/XX-.240 a XXVv v XXX• . 160 aX x XXXXXx X-.080 •<X>CX/.XXXXX-.000 XXXAXXXXXX.050 XXXXXXXXXX.160 XXXXXXXXXX.240 x x x x x x x x• 320 x x x x.400 x x x x.480 :<x<.56) X.640 X.720 X• 803• 8«0.960
Mean = Variance =
.00004
.098
_L _L10 20 30 40
Percentage of Points50 60 70
Figure 8: Error distribution for feedback control with PI algorithm correspondingto the identification shown in Figure 6 with 1000 points
101
Setpoint Deviation
-
-•43.)X
-•*cn- • 360-.320 X
-•340 X— • 200 X— • 160 XX-•120 XXX-•OR 3 XXX-•040 x x x x x x x x.000 x x x x x x x x.0*0 x x x x x x.050 XXX• IPO XX• 160 X.20^ X• 2 4 0 X• PRO X• 320 X.360 tA• 40 0 X• 440 X• 4R.0
MeanVariance
-.00264,034
_L ±10 20 30 40
Percentage of Points50 60 70
Figure 9: Error distribution for feedback control with Dahlin algorithm correspondingto the identification shown in Figure 7 with 1000 data points.
102
102A
• s q t r p o d B q n p 0001 L s j n 8 T , i u f t x w o q s u o f ^ B O f
a q 3 o a 8 u T p u o d s 9 J j o o r a q q g j o S i e q 3 T » x 0 ^ 1 1 0 0 ^ o n q p a e g a o j u o f q n q g j g s T p a o a a a
8 3 U j o a g o a S B a u a o a o a :
:qx aan8xa
0Z.T
09 OS1 ~
0*7r
oeT "
oz 01T 084?*
C***CO**09E*C2E*O b ?.*0*2 *C02*0 91*021*O bO*0*0 *000*0* 0 "f>Orv • i w C» w021"091*'002 *'0*2 "0 8 2 "•J2C"0 9 S • ■00* "»';**• *j8*"
T
X>
XXx x x x x x x
y x x A X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x y x x x x x x x x x x x x x xx x x x x x x
XXX*XX
CIO*eeooo*-
= aouBfJBA= UB3W
Setpoint Deviation
Setpoint Deviation
-.9*0 - . 8 ? J• »oCO- • 720 - .64'j- . b * 0
- . ‘♦HO-.‘♦on - .3 ? o- . 240• . 1*0 - . O H O- . 000• O H O .1*0• 240 .320 .400 .<♦80 . b * 0
.*<♦0
.720
.800
. 580• 9*010 20 30 40 50: 60 70
Percentage of Points
Figure 11: Error distribution for feedback control with Dahlin algorithm correspondingto the identification shown in Figure 8 with 1000 data points
x^ Mean = .00140A Variance = .145X X\ X X X X X X X X X X X X x Xx x x x x x xx x x x x x x x<xxxx xxxX V X X X X X Xx x x x x x xa XXXXXXXx x x x x x x xX X X X XX X a XX X Xa XXX XXX
__________I___________ I___________ l___________ I_______ :___ I__;________ I___________ I
Setpoint Deviation
-.960-.300- • 7?0 A-.640 X— a bfiO X-•*♦80 /-.HOC xxxx-.320 XXXXXX.-.240 /XXXXXXXX-.160 xXXXXXXXXX-.080 XXXXXXXXXXX-.000 XX.xXXXXXXXX.080 < x x x x x x x x x x.160 a XKXXXXXX.240 < x x x x x x x.320 XXXXX. 400 XXxXX• 48.0 XXX• b60 X.640 X.720.800.880• 960 . . 1
MeanVariance
.00020
.079
_L10 20 30/ 40
Percentage of Points50 60 70
Figure 12: Error distribution for feedback control with PI algorithm corresponding to theidentification shown in Figure 8 with 1000 data points.
105
Nomenclature
a ■= Input term coefficient
b = output term coefficientd B disturbance in model equation*d = transformed disturbance (corresponding to a cross-correlation
function)e = white noise sequencep = coefficient of disturbance transfer functionq = disturbance inputs ° Laplace transform operatoru = system input (manipulated variable)y = system output (controlled variable)*y = transformed output (corresponding to a cross-correlation
function)y = true plant output (i.e., excluding disturbance)6 = confidence limits for von-Neuman ratio
V “ input auto-correlation functiono = standard deviation of noise disturbance or significance
points of von-Neuman ratio= disturbance auto-correlation function
9 = input auto-correlation functioncpu(j = input-disturbance cross-correlation function<puy = input-output cross-correlation function•jf6 = transformed "independent" variablep = disturbance transformation parameterv ■ von-Neuman ratioX = number of unknown model coefficientsC * coefficient vector
disturbance vector plant transfer function
an integer representing the system dead time number of points in input-output record
number of output terms (output order)
number of correlation function points number of input terms (input order)
disturbance transfer functionvector of "dependent" cross-correlation functions vector of system outputs
variance-covariance matrix of system inputsmatrix of "independent" auto- and cross-correlation functionsmatrix of system inputs and outputs correlation transformation matrix
107
Literature Cited
1. Johnston, J., Econometric Methods. McGraw-Hill Co., New York,1963.
2. Goodman, T. P. and J. B. Reswick, "Determination of System Characteristics from Normal Operating Records", Trans, of the A.S.M.E.. Vol. 78, No. 2, pp. 259-271, February, 1956.
3. McCracken, D. D. and S. D. Dorn, Numerical Methods and FORTRAN Programming. John Wiley, New York, 1966.
4. Durbin, J. and G. S. Watson, "Testing for Serial Correlation in Least-Squares Regression", pts. 1 and II, Biometrika. 1950 and 1951.
5. Theil, H. and A. L. Nagar, "Testing the Independence of Regression Disturbances", American Statistical Association Journal. Vol.56, pp. 793-806, December, 1961.
6. Lopez, A. M., "Optimization of System Response", Ph.D. Dissertation, Department of Chemical Engineering, Louisiana State ( University, January, 1968.
7. Dahlin, E. B., DDC Tuning Reference Book. Instruments and Control Systems, Chilton Co., 1969.
108
CHAPTER VI
APPLICATION OF CLOSED LOOP IDENTIFICATION TO A SIMULATED PHYSICAL SYSTEM
IntroductionThe Identification procedure formulated In the previous chapter
is applied to a simulated tubular reactor in this chapter. This work is presented with the objective of better defining the steps required to implement the correlation-regression based closed loop identification and to indicate the types of problems which might be encountered in a "real world" application. The tuning of an "identifying controller" for this application represents the bulk of the necessary preparation. The actual implementation of the identification is a straightforward application of the results of
Chapter V.
Process ModelThe process whose dynamics are to be identified is the tubular
reactor described in Chapter III. For this application, several sources of external disturbances were simulated. The plant and disturbance configuration is indicated in Figure 1. The setpoint is assumed constant during the entire data collection period. To give an idea of the effect of these disturbances on the process response, a record of both the controlled and uncontrolled responses is
presented in Figure 2.
109
W.■*BoBAo
Disturbances:
ProductConcentration
W
Gaussian distribution with standard deviation of o. Cooling water temperature
white noise a = 4.0 .26+1
|55°F
T : Feed temperatureK
white noisea = 2.0 .26+1
|70°F
C. : Cone, of AAo6 lb/ft“
white noise a = 1.0 .5s+l
C., : Cone, of B QB0 .58 lb/ft3
white noise 1 .icr = 5.0 .2 8+1 +■
W_: Flow rate of BD18 ft. fain.
Load
Figure 1: External disturbance configuration,
Uncontrolled Response
Controlled Response
o SetpolntCM
O
oa\
oCM
OCM
o\
131.0o.oTime (minutes)
Figure 2: Comparison of controlled and uncontrolled responsesfor tubular reactor.
Controller Design
As Indicated In Chapter IV, special consideration should be
given to the design and tuning of a controller which Is to be used
In an Identification loop. It Is generally necessary to have a' priori estimates of the model structure and parameter values. Such
estimates will often be available, but may require special testing in some cases. For this application, these estimates were obtained based on the open loop response of the controlled variable (product concentration) to a step in the manipulated variable (flow rate of reactant A). Such a:.response is shown in Figure 3. Based on this response, the following model and parameters were chosen as a
reasonable initial approximation:Model: First-order lag with dead timeParameters: Gain ■ 3.76 conc. units/flow unit
Time constant ■ .040 min. ( ■ 2.4 sec.)Dead time * .025 min. ( ■ 1.5 sec.)
Choosing the sample time equal to one dead time, the following discrete transfer function gives an appropriate representation of
the system:- T / t „
K(l-e )aHG(z) - -t /t'-1 (1)1-e z
whereK “ system gain T = sample time T ■ time constant
In the discrete time domain, Equation 1 is equivalent to:
Response
18.0
20.0
22.0
24.0
Plant Response Model Response
0.60.3 0.4 0.50.0 0.1 0.2Time (minutes)
Figure 3: Step response of plant and first order lag plus dead timediscrete model to a unit increase in the manipulated variable.
112
113
(2)
whereCx « K(l-e"T^T)
u = Input (manipulated variable)y ** output (concentration)
Numerical values of and Cg are easily obtained, with the resulting model being:
The step response of this discrete system Is also shown In Figure 3.It appears that a much better fit could be obtained If the dead time were chosen somewhat larger. No attempt to obtain such a fit was made. The Inaccuracies Involved with this model can be expected to be encountered In actual practice with the exception that a larger error In estimating the gain could be anticipated. This model should be adequate for the purpose of Initially tuning the controller.
The guidelines presented in Chapter IV can now be used to tune the controller such that a linearly independent input-output record will result from closed loop operation of the system driven by external disturbances. The algorithm to be implemented is a two- term (FI) controller, which can be represented as:
that y represents the deviation of the controlled variable from the
yi = 1>75 ui-2 + *536 yi-l (3)
Ui ‘ “1-1 + P161 + P2ei-1 (4)
Equation 2 can be substituted into Equation 4, where it is assumed
setpoint, with the result being:
“t ■ ui-rpiciui-2‘(Pic2 + P2)yi-i + Pidi <5>
The term Pjd^ is included to account for the presence of system disturbances. Again including a disturbance term in the model, and combining Equation 2 and 5, the following matrix equation can be written:
1
•H9i i
y i
II
V i
101
-(pl°2 + V -pl°l V i -Pldi
yi-l + di (6)
-Ui-2_ 0
Or, more compactly:
Yt - r Y t-i + Di (7)As shown in Chapter IV, (see Equation 15) the solution of Equation 7 can be written in the form:
* * * r W + r Y (8)where
W
Ld3JThe matrix T is composed of the terms of f, and for this system is:
' - p 1( i - p 1c 1+ c 2+ c 2) - p 2( i * h ; 2 ) - ( p ^ c ^ ) P l ‘
~Pl°l+C2
•<P1’,TlC2+P2) *P,
(9)
It is noted that only 2 terms of Y need be linearly independent, and
thus f can be reduced to a (2x3) matrix whose rank must be checked
115
to assure the necessary Independence. Formally;
P1(1“P1C14C2‘K:2)’P2(1‘W 2) -<Pi*PiC2* 1> P1
-»lCIH4 C2 1(10)
In this case, It seems reasonable to design the controller (within**stability constraints) such that the rank of T Is equal to 2y and
with at least one of the three determinants being large in a
normalized sense.Stability considerations tend to restrict the maximization of
■i-.r.the determinants of r \ The poles of the closed loop system must lie within the unit circle in the z-plane for the system to be stable. For control considerations, it is desirable to further constrain the location of these poles, for example, such that they lie within the ".8 circle". In any case, some reasonable constraint
must be imposed.A closed loop pulse transfer function for a feedback loop can
be represented as:
Gfe> . HG<2>__ (U )' ' 1 + D(z) HG(z)
whereD(z) « controller transfer function
HG(fej ° plant transfer function Taking the z-transform of Equation 4 gives:
U(z) - z-1 U(z) + PjE(z) + p2z-1 E(z) (12)
The resulting expression for D(z) is:
116
Similarly, for the model:
Y(z) - + C ^ V z ) (14)
And the model transfer function is:
H G W ' « * > ‘ i S t 1 <15)
Substitution of Equations 13 and 15 into Equation 11 gives the closed loop transfer function:
(Pi-H?2z"1)(c1z‘2)/(1-z“1)(1-c9z"1)G(z) - — (16) (1“* )(l-C2Z V ( p ^ * )(CX^ z)
The denominator of Equation 16 forms a polynomial in which canbe solved to give the poles of the closed loop system. The polynomial
is:z3 + (-1-C2)z2 + ( p ^ + C2)z + p2Cx - 0 (17)
Because of its low order, the solution of Equation 17 can becarried out with relative ease, with the modulus (absolute value) of the largest pole serving as a constraint boundary in the maximization of the determinants of T . This procedure will apply to a system of general order. However, for this case it is in
formative to actually map the contours of these determinants along with the constraint boundaries. Figure 4 shows such a map in the (PX,P2) parameter space. This map indicates that a suitable degree of linear independence can be obtained while retaining good control of the system.
Normalized DeterminantCM vO O n• • •0 . 0 o
max.(1.0,0.9,0^8)
A Control #1 □ Control #11 ❖ Control #111® Constraint Boundary— Normalized determinant
-0.4 0.0 0.2
Figure 4: Contour map of normalized determinantsand the pole constraint boundaries.
118
Identification Procedure
Having designed a controller to be used in the identification
scheme, the procedure outlined in Chapter V can be applied directly to this reactor system.
Several approximations have been made which are typical of the type of approximations which would be required in a actual application including:
1) Disturbance dynamics are of high order and include both noise and load signals.
2) A very simple model is being used to approximate a high-order, non-linear system.
3) The controller has been designed based on approximate parameters of the simple model.
ResultsThree sets of controller parameters were used in obtaining the
various results to be presented. The location of each parameter set is indicated in Figure 4; the actual parameters are:
Controller: £l £2I 0.10 -0.02II 0.24 -0.12III 0.02 0.08
The identification was carried out under the conditions shown in Figure 1 with a setpoint of 20.0. The step responses of the models resulting from this set of runs indicate the Initial success of the
identification scheme.
119
A more comprehensive set of runs was made under similar conditions, but with a setpolnt of 25.0 The different setpoint results in 8lightly different dynamic characteristics due to the non-linear nature of the system. In particular, the steady state gain is decreased by roughly 15%. A summary of results from this set of runs (10 runs with each controller) is given in Table 1. The errors shown are based on the difference between the model and plant step responses for 15 sampled points. Generally, the mean square error decreases as the data record length is increased. Controllers I and II tend to give similar results and appear to give results inferior to those obtained with controller III. This difference can be explained in terms of the determinants resulting from the different controller parameters. Referring to Figure 4, controller I and II have parameters located near the 0.6 determinant contour, whereas those for controller III are near the 0.9 contour. These results reinforce those of the previous chapter indicating the importance of considering linear independence in the design of a controller for use in a closed loop identification scheme.
The identification discussed thus far has been based on operation of the system with variations about a single steady state point. The disturbances causing these variations are each of relatively short duration and do not contribute significantly to the long term mean values of the manipulated or controlled variables. These disturbances are, in effect, the driving forces upon which the identification is based. In most process systems, the presence of long term disturbances can also be anticipated. Such disturbances do not
120
TABLE 1
Mean Square Errors Resulting from Identification Based on Three Different Controller Parameter Sets
No.500
of Data Points 1000 1500
Best 0.039 0.078 0.125Average 0.452 0.430 0.350 Control: I
Worst 1.293 1.092 0.777
Best 0.032 0.028 0.027Average 0.428 0.350 0.241 Control: II
Worst 1.246 1.476 1.119
Best 0.030 0.029 0.029Average 0.197 0.284 0.089 Control: III
Worst 0.880 1.734 0.231
have a direct, long-range, effect on dynamic response, but will often change the steady state operating conditions. As an example, for the reactor system, the changing of feed-stock might result in
different reaction kinetics. Such an effect was simulated by abruptly changing (a 147. decrease) the reaction rate constant. The results of this change on the product concentration are shown in Figure 5. This response includes dynamics due to the normal operating disturbances indicated in Figure 1. Within approximately one minute of operation, the reactor system is "back to normal". A different steady state value of the manipulated variable is the obvious long term effect upon the system, as this disturbance is essentially a load change.
The effect of operating with a different rate constant on the dynamic characteristics of the system and on the system identification is not well defined. Indeed, without imposing some type of test on the system , this change might not be noticed. The step responses shown in Figure 6 give an indication of the effect of this change on system characteristics. The dynamic portion of the response is significantly different and the steady state gain is reduced by about 207..
This new set of operating conditions offers a challenge for the identification technique. Several identification.runs were made with the change in rate constant occurring during the initial (2.5% to 10%) part of the data collection period. The discrete model response shown in Figure 6 is based on parameters of a resulting model for which the mean square error was near the average error for the set of runs. The use of a first-order lag with the
Controlled Response
Rate Constant ChangedSetpoint
CMCM
oCM
O00
vO
0.0 0.5 1.0 1.5Time (minutes)
Figure 5: Controlled reactor response with external disturbances anda major load change.
Response
18.0
20.0
22.0
24.0
Plant Responses Model Response (Kq=175)Model Response (K =150)
0.3 0.4 0.60.0 0.50.1 0.2Time (initiates)
Figure 6: Comparison of plant and model responses with two differentreaction rate constants.
N>u
124
given dead time as a model for this response Is clearly a poor approximation. However, the discrete model based on the new operating conditions is better than a similar model based on the original
conditions. The results indicate the feasibility of implementing a
simplified adaptive control scheme based on identification using normal closed loop operating data.
The identification technique proposed in the previous chapter and implemented for this application involves a iterative solution of a complex system of equations. The results presented in this chapter represent 150 separate (but similar) identification cases.Of these, the iterative scheme diverged exactly once. In addition, about 10 runs resulted in an oscillation about two nearly equal sets of parameter values. The usual number of iterations required for convergence ranged from 8 to 12. As for any iterative solution, this approach requires safeguards regarding convergence and acceptability of results.
As indicated in Chapter V, the von-Neuman ratio computed after convergence should give a measure of the acceptability of the overall model structure. For this system, the ratio consistently fell between 1.1 and 1.4. This relatively low value is a clear indication of the higher order plant and disturbance dynamics which were being approximated by simple models. Nonetheless, the simple models do give information useful for controller tuning. A well-identified, higher order, discrete model would be more desirable in many cases.
125
SummaryThe dynamic system identification of a complex process has been
carried out using the correlatlon-regresslon approach for a closed
loop, feedback process. The design of a controller suitable for use in this configuration was based on an initial approximation of
the parameters of a very simple discrete model. Identification based on operation at the same conditions from which the original approximation was made resulted in parameter estimates giving a response much like that of the Initial approximation. The non-linear nature of the process results in different dynamic characteristics for different operating conditions.. Parameter estimates based on data taken at these different conditions give responses which describe the process dynamics more accurately than do the initial approximations. The results of this chapter indicate that an adaptive control strategy based closed loop system identification during normal process operation may be feasible.
Nomenclature
d a system disturbance
e BB error signal
P - controller parameter
u - manipulated variable
y a controlled variable
c m discrete model coefficientD - disturbance vector
D<2) 8 controller transfer function
E(2) - z-transform of error signal
HG(z) - plant and data hold transfer function
K - system gainKo - reaction rate constantT - sample time
U(z) m z-transform of manipulated variable*W - disturbance vectorY a state vectorY(z) 8 z-transform of controlled variablez a z.-transform operatorT 8 time constantX 8 closed loop eignevaluer 8 coefficient matrix*r 8 transition matrixr** . reduced transition matrix
127
CHAPTER VII
CONCLUSIONS
The work presented In this dissertation has been directed to the problem of identifying the parameters decriblng a dynamic process based on input-output data collected during various types of operation of the process. The identification technique is used to estimate the parameters of a pulse transfer function and is structured for utilization in a sampled-data environment. The approach is designed for use in the selecting and tuning of discrete control algorithms and can be applied for both open and closed loop systems
involving deterministic as well as stochastic signals.The initial work shows that a correlation domain model can be
represented in a manner similar to that of the discrete time series model whose parameters are to be estimated. Applications of the identification to open loop systems having noise corrupted outputs indicate that the filtering action of the correlation functions is sufficient to overcome the problem of parameter bias which is associated with applying ordinary regression directly to such time series data. For the purely deterministic case, ordinary regression gives better parameter estimates indicating that correlation-regresslon
has no advantages in this situation.The presence of measurement noise in a feedback configuration
requires special consideration since the noise signal tends to propogate around the loop. The net effect of such noise is that the useful correlation data are limited to one half of the time shift axis.
128
In both the open and closed loop cases, the measurement noise Intensity determines the record length required for suitable identification. In any case, the extremely long data sets usually associated with correlation based identification are not required due to the regression part of the approach.
A means of designing a controller suitable for use when closed loop identification is to be performed based on data taken during normal process operation with only external disturbances driving the system is presented in Chapter IV. This work describes
guidelines by which a given model/controller configuration can be analyzed in terms of the expected linear dependence or independence of the closed loop data. The approach requires a* priori estimates of the model parameters. This limitation should not be too serious in most cases of practical interest since some dynamic response data is usually available or can be obtained.
The presentation of a closed loop identification procedure based on correlation-regression is presented in Chapter V. The approach represents an extension of the basic correlation-regression scheme and involves a form of generalized regression and an iterative solution to estimate the model parameters while simultaneously compensating for disturbance dynamics. The results of this chapter indicate the feasibility of the overall approach and also show the importance of linear independence considerations in the tuning of an "identifying controller".
Throughout this work, applications of the various identification configurations have!.been examined for both low-ofder linear systems
129
and for a simulated chemical reactor. The applications for the
reactor system indicate that the identification is suited for use with a low-order, linear model being applied a high-order, complex system.
An entire chapter is devoted to the identification of the reactor dynamics with data taken during simulated normal operation with only external disturbances driving the system. This represents the most realistic application attempted and involves some rather flagrant violations of the assumptions upon which the closed loop identification is based. The violations are due to the use of extremely simple process and disturbance models to represent a very high order system. The results presented in Chapter VI indicate the success of this application. The use of such a simple model appears to limit the accuracy with which the high-order process dynamics can be described, particularly under the changing dynamics caused by different operating conditions. Nonetheless, the simple model identified under varying load conditions tends to describe the process dynamics better than a similar model with a fixed set of
parameters. This application shows the identification aspects of adaptive control, and could be used in its present form to periodically update the process model based on batch input-output data.
The extension of this work to a continuously updated recursive identification should be possible. Recent work in the area of on-line model identification should provide some guidelines by which the correlation-regression approach could be adapted for continuous
130
on-line use (1*2).The primary objective of this work has been that of system
Identification. The overall objective has been to provide an
Identification approach suitable for automatic controller tuning in systems representative of those found in the process industries..
Literature Cited
Suh, Samuel, "Design and Application of an Adaptive Control System Using Digital Techniques", Proceedings of the Sixth Annual Conference on the Use of DIGITAL COMPUTERS IN PROCESS CONTROL, February, 1971.Dube, James H., "Application of Identification and Control Methods", Ph.D. Dissertation, Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana, 1971.
APPENDIX A
Dahlln Algorithm
133
APPENDIX A
The derivation of the discrete equations for a general Dahllncontrol algorithm are presented here. For additional details, thereader Is referred to an article by Dahlln ( l ).
A general discrete transfer function can be written as:-M-l NU -1+1
HG(Z> - * 1) " -i (A-1)1 + it D . Z
ill 1Cross multiplying results In:
r NC *- 1 M 1 N U - 44- 1Y(z); 1 + S b z I - U(z) z 1 S (A-2)i- i»l 1 -J 1-1 1
With unit sample time, the time domain equivalent Is:NU NC
yk " £ “i V w 'JljVk-i (A'3)
The general Dahlln algorithm can be written as:
R(z> r(l-e'I,Tlz'K"1 -lr 1 I,.,.,D<*> ■ ife) L l-e-^Vl-(l-e-T/T)z-M” j L HGfe) J ( }
—t / t . _ , - •Let \ - e , and combine Equations A-l and A-4.NC -1
R & l » 1 f 1 +1-1 blZ ~j (A-5)E(z) L l-\z"1-(l-X)z"M" *- z-M-l g11 z-i+l -
1-1 1Rearranging Equation A-5 gives:
^ Dahlln, E. B., DDC Tuning Reference Book. Instruments and Control Systems, Chilton Co., 1969.
134
r* 1 - M - i • ' NU .,«R(z). l-Xz -(l-X)z ] 2
J 1-1 1r NC -i'l(1-X) E(z)j 1 + E b.z * i (A-6).U ±ml i J
Inversion Into the discrete time domain results In:NU NU NU
airk-i+l "X±f1 airk-i"(1-X)1f1 airk-M-l
NC(1“X)Lck V k - i j <A '7>
Rearrangement gives the form:
NC NUrk
- f r- MU HU “J— {(l-X b±\-i +Jji airk-M-lJ
NU NU i+ x£ V k - l " ^ 2 Vk - l + l J (A"8)
Equation A-8 represents a form of Dahlln's algorithm which can be easily programmed for any general linear system.
APPENDIX B Computer Programs
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L COLLECT LATA >- N U m Kf-AC TUP S 1 UUL AT i CCTO S ’* VI --it NP i b L = b‘P - ',-t •! = ■' +r i *-fc.AP^*bI
136
El = t • I F ( W
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99 CCNIl'S'UL CC PtKf-5RR liJENT IKICAi I0H fcJJK SELCCiCU SECOND L tNUTHS
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CC C OMPUTE KtSPONSt CK K9K
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\ 33 M X h_ X k a < < ii X or. X X »—< 1— X 23 i— 23 7) 2 X l i X 57 73 < •y or £ nJL. fH X J. a j - r— *—« X —< r. a r X. Tt • I2 1 H* ►—4 S3 X a a a X X a «— <13 rH 2 — a \J 2 j : a —« — z. rH 2 -4 rH Va X *— —• 1— -» •— »- X 7J 1 m + r +
X. n -> ii ii II rH ru »—1 23 X »-h z H Ui i— X r*4 u *—« •a i—i 23 ** X X 73 73 K r a a 73 • ■Si a 23 1 < X-.— i *r 1 I— — — i— jo J3 X 7.3 > <x « 3. X. ?- 'J a .9a /> 2 • 73 2 z Z. •» ♦. X D a a a 1- X 73 7) X. N a •l . — i H x 73 mmf ■—i •0 •C a a II » a X 2 'Ji 2 a rH II n —>7 X «H — z i— X r X V i— ii 73 73 n 73 II < < 2 II #-% X33 2 X. * X 3- X X X a a X 2. >~ X •Si X 70 X 7! X X r'. •H *“4 ra ll 2. •■ a a X —# •—« —< a a X a X 73 X 73 Ii 7) 2 i— li X) w- W4 V.
r-4 II u — — —« — — w < X < — I—X »-4 li UJ 2 X X' UJ z X X XI a. U. < ao u U u.2 1— TT X a \ T; V J.I •—« a 2 >—i < a
a a a a a a a a a a a a a
T)z>
a a 33 a
L LgN^urh L'-AST SL’UAKcLS LST 1 HATE t?K LCALL Uh-'.'LSiMAC^ ACI^LUI /C/V/ 1 L/NU* NL>NAC , Kijr )
-'.'Rl I b ( b r y U ‘fJ ,\'.wAbS.» K H M / I L ( 1 } -» I = 1-» N'hU )LL tST ifATE UiSrUHeANCfc. AU rgLOKKCLAT i 3N hUNCTi i JN
L ALL isw AK { A/ 3 / L / N O ML/ NLJ )103 U = b J S ( A / O y . / N O N L , UT
Ub = u»VjyA J =i/\'KHK-<AC I < 1 )=L*
yA L'CTI1)=U*•jay-3 1 = ij nhjlACI't\:KKKK)=Y( I +1 } - Yb-bUSH ( X ( 1 J-XS)t i = b ::sjL A C r i n k u k k ; >Jb = LK+ALI ( NiKj^K )1 1 = ACT 1 J I F I M ! -NK3K* )b0T01Ub <;ey£J = i ' N<any
yH CCI <J)=CUT ( J ) + H * A C T I N K 0 R K + 1-J?lUb uey y j = i / N A Cy3 ACT U ) = A C n J+i )LL LaKKLCT FOR WIAS IN D I S T U K b A C C C H K t U 1C T 1BN
!;a = !Jb/h La at (n f i ;U U S & I = 1 j \ K 8 K R .
bK C C 1 U 3= L L l (I )-3-*uacL LHtLK. CaNV C W b L N L L AND S'AVt LSr i H A T L S
T 1 = U .jyyi i = 1* \‘t fJ T1 = T l + A » b l C U )-C3t 1 ) )
y i LS I I 1 =LI i )IF( T 1 / X N L K • (i I • TBL«ANU«'lRRINT»LT»HJbOTOlue
LL LHfcLK F9K WHITt K t b I DU ALS .
■* - X+ - n n• (ft t— *\H o h- Z s.
rH 71. —i •»rH M tt .V n tt 1 in
f— X Ha* » z 7. «m tt < UJ X)a *— —4 r. JLw- rH < L> .H •»•_> H tt < t—4 •U • tt X 11* K UJ u. Ll33 J N X Jj — *0 33I • n X 13 4- •—*X tt o « — i—+ a. tt 30 ll <•-4 X 1- Ll -a Xrvj tt D •v u -w H •k tt 3_ . •» (0 ;rf— U V X h- I > tt <J J \ tt D s X /) r~u *a •k X. 23 + 15 7) D* 7) •- tt VJ ■k < uJ
If z - aj 1. ?'M < tt »■ ■v II • 1— • —• ,N VI •a n —+ D f— T) —» - _l JL 1j13 Z < t—« •s aJ •k tt 3'r. < tl tt S X - > - aX • aJ h-< K n ■s x.\ M •k* Ll. J) s W _J -» h- •< na % V »a rH XM 13 V H. 1— X •s i0 UJ t) ul+ X — II z •* - s J z —• 75 X J —t ••- tt n H* TT X X.rl d • ♦H UJ tt 75 I— < <; LJ—* X. *■■«* —* U3 a L— 7. II U. jJI vO 13 a *~4 t— X y. jJ X UJ Xa* tt •t u. rH u jJ tt — < a UJ LJ IJ«—1a* cr* • ... *f H* u u > /T 33 i_> U.I u 7
X •k r-1 uj •a L> * ■— •a X a a u .ii _J•— w* r.i t— - T) tt j. X < tt .0 <j — ~ < «— 5|C -t z *3 3 J n j. —k —4 r ntt tt jj 33 r tt - a' r aJ • JL T) jj tt X uj y.* tt •— I) •H tt •H • » tt •jc. 13 — n J_ —t 33 Jj rH — > —i—* * •—» •— w w w rH — •s. X 73 rH tt wJ X It 7 7>N 2? X _l to 75 t— H- II II Ul 1/3 aJ ;?. UJ #x I/O - 33 vU
r 73 Jj tt u rH tt 7.Ti X 73 > >n •- J — 7) ;j X+ X *-* •— < D J w -r* r 7. D X •k jj D 7) 7)
* f-H r 3: » + + i- X ar o X 03 rH K X 1- < Z)*T • • — jj n *-*. 33 #> 7 3J r*\ •* r 7 Z 7X X r~ ~J ✓r a* z rH 3J < • • 3 Xf rH 73 —i tt+ i tt • «H o < — \ o ■ H aJ i~l ■n — — •k LJ a - -
tt • a* » X. jj < * u ~ *-*■ 9 • 9 tt D s •H »-r «- tt •» •» •>T H r-f jlS — » • z x J X — — X X X r » M »4 X X Kr — Z 7. D D D _> s * < ■t -J al z •% (7 D n rH •s VJ X. 0 n n 0*. { ii ■ •a K X it li .j rH •“ II II *— • « "0 _«* •H 43 73 — *— W> 3-) •a X. — •a 'V23 * tt X X c *"*»» a* * it S) **«• tt 2J D X h> 1— i— a- •k uJ r- f— a* — ;/X * U X r< IT' cH X rH I—» (I rH .\l < X X rH »-4 X a.) oJ <. < < X 33 < < < < 'r*ii: X' * *-a t~ J — — • tt — — II X X r; •—« — ~ sr. •t- X*. >< n >LL L IL rlii II W •a U- II .Tv ^ r- — tt w* — — •/ W« ■v; y. v X f) •* X r y 3' —ru ■J) :xj a. U_ o u r-» tt U- u 1.) :r 'vi_ .73 .73 Li. ll' T 73 rD a.j tt - ID r- 3.3 33 :U ^
tt r- •—« < tt < < > ‘“•f 13 3J •-H It J. a. LL X X j. U- X i: .iiC» C-1 X) .3}
X) rH rH VI ■n rH M 4* ji sf>Cv O i-i O rH n tt C) D tt o tv
-J LJ H 13 -H rH H 7) 5 tt 5 5 13
S U3K3U J I -Mb UtN’LSD ( AC' ACT/ CCT' C/ V/ I C / N U / N C / N C S R K ' K 3 P T )D I M E N S I O N I L l l ' J )D i ^ E N S i UN A C \ N L O K R ) / AC I ( N C D K K ) ; U C n N C O R H > / C C 1 U ) / V ( d U / dU > /X ( dU/ 6 ) /
c v v w 1 1& / c u ; / : v v: cr ( ! ) ; f c ) ; i \ ' i i 5 ) ; r ( d u j / D j r i t u t c o /1 ) / y j M 3 C ( 6 / 1 ) / i i j i v s r ( d u j / klNUExiidu/d)* iNutxcib/d)DA I A W M b NMAXd/dU/ S/
f *
C H R 8 UKAM FOR b t N t K A L i £ E D LEAST SQ U A R E S APPLIED TO TKANS?BKMEDC AIJ i 0 AND C R O S S - C O R R E L A T I O N F U N CTIONSCC SUBROUTINE ARtiUEMCNTS AND D E F I NITIONSC AC = AUr O C S K R E L A H O N FU N C T I O N OH C U K H E L A T I N U VARIABLEC AC! = TRANSFORMED A>JT B CBRREL AT 1 ON I-UNCI I BN OF COWKELA T INu V AK I ABLEC CCI = TWANSF OKHtU CROSS KOKRELAI ION F U NCTIONC C = PAR A M E T E R ESI IflAJE VECTORC NU = A 3T INOUI C O E F F I C I E N T SC NC = U OF OUTPUT C O E F F I C I E N T SC N C B R K = L E N b l H 0 r L U K K t L A U B N F U N C T I O N V E C T O R SC A O F ! = P K O B K A M O F T I O NC LESS THAN £EK3 FOR OKI) i N A K T LEAST SQUARESC EL3UAL ^EKO FOR 'JENE.KAL I £EU LEAST SUUAKES WITH PRE C O M P U T E DC INVERSE OH V A R ] A N C E - C O V A R 1ANCE HA IRIXC U R E A T E R T H A N ^ E R O F O R N O R M A L U F N E R A L I ^ E D L E A S T S W U A R E S -C N M A X l / M H A X d = D I H E N S I O N P A R A M E T E R SC NRBRR = ACTUAL •=? Oh 'DATA* FBINIS USED IN KEliKESS I ON
NEU=NU+NCN'<3RR = N C 0 K R - M A X U ( N U / N C + 1 )\'D = K A X U I NU/N C )IFt AOFI »EU.d)(i0T01Ul IF I lAOF 1 J 1U1/ 1U1/ IUd
CFBKM D I S T U R B A N C E V A K I A N C E - C O V A R 1ANCE HA IRIX
IUd D 8 S S I =1/NKORRD O y y j = i /n k o r r V U / J J ' A C U - 1 + 1 J
i—*-S'
23 v i j j i j s v i i i j )
LL h0Kfi r-iATKiX OF B B S L K V A T I O N ON DtPt^'ULNT AND INUtPtNUfc.Nl VARIABLESIUi UU2SI =1> MK.OKP
Y ( 1 } = LL M .\U+i )If NtN
I - (J«Lt*NU)X1 1/J)=AC1 LND+I-JI 2« i r 1J •U 1 •N U )XIItJ )= L L 1 INO+I+ N U-J)
CALL KLUUCt I X, C r IC/UUNtid/ NHAX I, NMAXd/ N*J}PK.» NELij NtP )I PI K-BP i • L I • U ) CO ! ‘J 1U JIF I K.l-jP I • tU • U ) CO I 0 1 U /
LL C A L C U L A T I O N S F'OK ut NEPAL I £tU LEAST SCJ'JAPESL JNVtltr VAPlAMCt-COVAPIA.NCt NATP 1X
CALL MA ! INV I V/'JUMBl/ iPI VUT* INUtXl/NKBPP/ NM AX 1 / U J iu/ uyy / i = i / N t H
u y y / j = i > n k b k p WWlI I/J )=u.D02/K = l/ NK.OKP
27 a'WI U j J ) =X( I ) *V ( J ) +WW1 II J U)LOiOiUA
LL SEiUP 1-BP 8PUINAPT LEAST SUUARES1 (jJ UOUfrI =X/NEP
u y y t j = i / n k b p p 2 ^ a'WI 11 / j » =x ( j# ijcL COMPLETE UitNEPALlXtU OP CPUINAKY) L t A S ! SUUAPES C A L C U L A T I O N S1U4 U B 2 S I=l/NtP
• i U l ) = U 'U!)2SJ = i^ NKBPP
2b ••T t 1 ) = F * 1 I 1 IJ ) * Y ( J ) + W 1 ( J )’;02*I=i*\’tH U 0 2 2 U=l/NtP p.-rd i I > J J =u.
148
IDS
23CU106
w-j 'i it j) =wv:i i it«.)*xr\t jj+wwgi it J)! F ( N L K * 6 r . l ) 130 I 6 1 U 3 ou^ti'd lit 1) = W I ( 1 ) / WWe i lit 1)09 16106 C 9 N !INUt J923J = 1.»NE^OU tt'dl i-* 1)=W1 ( I )
39LVL LINtAK A L 6 L6KAIC LU U A T 1 B N S FOK AN tST I NATE: OF U CALL MAI INVlrtWifiDUnbiSj IPIVOT* I N D E X Z / N t K / N M A X 2 J 1)C0N1INUtCALL KtUWC(C>DUNBE/NbU,Nr'iAX2 JKLIUKNEN'J
-p-VO
r. r
vx r
r r
SUcJKSUTINt. CURKT X/T/ AC/CC/SI / St / NP FS I rK I NT >PIP-tNSlONXtNPTS) /YtNPf SO / ACTnnAT/CCtnriT)
CC SUbKOU? i Nfc. |tf COMPUTE C O K K t L A T I O N PUNCT IONSC X = INPUT StUUtNCtL T = OUTPUI StUUt.NCtC AC = INPUT AUTtt-CUKKELA FI thN F UNCTIONC CC = iNPUl-OUfP’J i CKeS S - C O K K E L A I IKN F U NCTIONC SI = INPUT Fit ANC b'H = OUTPUT N£ ANC NH1S = NCHbtN 01- UATA POINTS IN I/O PtCOKDC ' N M A = NtJPafclN OF CORRELATION FUNCTION POINTS T O H E C A L C U L A T E DC IPK1NT = OUTPUT OPTIONC = U FOR NO OUTPUTL = 1 FOR OUTPUT/ WITH NOKNAL'I «£ED C O KKELAT ION FUN C T I O N S
= t FUR OUTPUT/ WITH U N N O K H A L U t U CBKRtLA TI ON FUNCTIONS
L u H P U T t INPUT AND OUTPUT MEANS A X = 1 •X \ ' P 1 = N P I S51 =U«SH = U.U O U H I = 1 / N P T S S1=S1+A T 1 )SH = S'ef+T TIT s i = s i / x N p r52 = S*d/ANPT
COMPUTE CORR E L A T I O N FUN C T I O N S •■•1!C = NP ! S-HTT4 \ X f. = ” X UO^/K = i/HM'f ACTXJ=U*C C ( N > =U*:n}UbN = I/MK.
150
N M O 1( Q * / T J P . ' o j f Y O J V i ^ f i . H P O P
( i N f i l i v l - . ^ M C D S S f l > n f c o l I V l 1 H N C 1 « i n V H 3 9 1 I W T J . . - J T H S I ' / / / ) I V U M P 4 PCiq{ q . q - [ . j ' . = N i V l u i i r M i n a » ' / ' c * c i Af t = M V i U l O r i N T » '//) 1 v u i m C " 0 9
{ n m * ( i ) n v r y f r n o ' q m t q qT - T = X
fruiy *1- iCSSCi C H P , a ' 9 ) 3 i T fc.v
2 S * - I P ( i n q ' 9 ) 3 i . t * m N ^ m n * io • n i • i >vt wh t ) a i
x v * { y . m s ( v n m L &x v * { x) q v = ( X) ."3V
{ I 3 n v / * I = XV ( P. • I N • i M T Hrl I • f i N Y • I » n l ’ X ) d IV U i V / ( V \ 1 V = ( X 1 3 Y
X w Y / ( X ) T J = ( N ) 3 ^I I P - ( WN ) X ) * ( I P - ( . \ ) X ) + { X ) 9 V = ( X 1 n v 9 A( ? . $ - I N N ) A } * / t . q - ( , \ ) X ) + ( V i 1*1= { >• 5 D J
T - X + N = i \ : \
S U S K H U I 1 N L K t D ' J C t I X * T / C / I Z f D U f l d j N P 1 A X X f IS!* A X d / N « 3 K K i N t U / N t R )
i J i r i t N S l S n X ( N i ^ A X l i N' hUJ / Y I N M J K K } . C I N t U ) j IJU^IO [ N M A X d t 1 ) > I H N L 9 )
SUbKBUriNli fo KhUULh THL OKUEK OF iULNI 1FICAI 1 ON*** UOtS I KANSF'OKMATI gNS NtCLSSAKT F0K H i L N L S U ’ TO KKeLEfcOAS wjlH A FULL I'JtN T I F 1 LA I 1 ONLNTKT KEUOC INSLKiS THE IULNT I F i LU C8LF F 1CI fc.NTS 8F L CBNI'AIMEU iN UUriB INTO THE FULL VtCTOK CI
NL."< = NLU K'J = NLU-1 *SUNT=U uosfy i = i/ Nty <!JU*T = * 8 U N 7 + I C U )I - C NOUN I • t U ’U J K t T ’JKN N E K = K t U - ^ O U \ ' t ijrjy»j = ii Ntyi f ti c i j j •tu»u)uoroy>suyy/l =i /n k o k kr ( D = r u ) - C ( J ) * x ( i v J jCONTIN'Jt K. S; u 7 = Uuoybj=i^ m u ^OUNr=^OUNT-flC(U)I r ti'-jU-Ni ! • LU • u } ua Ttjyfeusyoi = 'U NK0WKX ( 1 t J + l - M J U N ! 3 = X C \ > J + l )LON I INUL Kr i UKNL\;! WY KL U8C (L,DUr1B>NtU/NYiAX2?^yuN r=u
J = N E K *UUNT=*.BUNT + ICt J)C (U+KBUN r ) =L5'JM»t J/ I )~’L i UKN
bUOKfJU! INt n'XTlNVt IPIVtJf / INptX'NjNMAXjn)L M A i KI X INVtKblON WITH ACXOMFANT I Nb SOL U T I O N OF LINtAK L'JUAT I £NS
PI M t N b l O N A IN n A X * N 11 3 I N r-iA X / I ; / 1PI VO f Ifxb I NUfc.X ( NMAX> 'd ) hOL 1 VALt.NCt I I KOW/OKOW } , 11 CoLUM/ULOLUfl ) * \ AMAX/ T j SWAP )
C ].Vl j IALi^AT ION0 islall=ufa K1 = 1U* V * *-leS/ \<’<i = 1 • U/ K I
iu uei tKr»=i • u13 DO d'O U = 1 > Nd'O lvJ)=Udo ;;o ot>u I-1jN
C bLAKLH tOK PIVOT tLtntNT<tU A -1 A X = U 'Ua s d o m s j=ii\'3D IF II H i V 3 T (JJ- 1 )6Uj1U3>6UfcU DO 1UU X = 1 / N/O I F t I P i V O T ( K, ) - 1 idUf IU U j 7*0yu If ( AfcSS I AP1AX ) -AOS I A ( Jj K } J ) JStjJ IUUj l’OUbD lPOWsJyo lC0LUr!=*03
1UU CONTINUE IUD ‘CUN ! INUtl
IF IANAX;11 tV 1U6>11U I'JG !.)r } tHT1 = U« U
IbUALt=U 00 I 0 /“ U
11V ] P] vy T I J UOLIH ; =1 FI VO I U C O L U M j+1 C IN!tPCPANUE K3Wb 10 PUI PI VO! fc.Lh.MtNl ON UIAbONAL
lb'J IF II Ktivi- ICO'-UM J 1*0* ZbU, l^U JL'+O UL ! tPM130 DO POU L = l/N 1fcU SWAF=At 1 rCO.•••/!_ )1 / 0 A I 1 K 0 L J = A I I C O L U M N L >
<r
o p c * n i n s ' n x n ? ( ?.h -I I i G a T p \ i - i it - 9 i y n 9 T = 9 ~ y n R T
t o * T I. 0 A I H = I J . S U I r ir.P.c 'non?. 'nnn? (?.h- ( I I Ga Trnqqy > JI
opr fii fin I + q “l V " l S T = m v n s i
I M / I I G A T ^ = T 1 G A I ri nant 'nant 'n?.c c im- ( t i Ga Th ) srw * a I
I + T I V n R l ^ . l y n S Tt H / 1 I H A T r ) = T . l G A l r !
n / n t ' n / n t ' n ^ n i ( I * - { i i G a t a \ s h v ) A IT - T i v n q i r
i i "3 f tnant 'ncot 'nonr (2^- ( ljx ai *ri)s;w j AJ
t - a i v n q T = ' - n y n s i
I K u L H l i T H = u ;> i^ I 3 f i
G a n t ' n * n t ' n - f > n t ( ( u m n n i ) A l d q o i n ! SP.
t + ivn.qT = iH/u.>n i in=.u!,sq i ~.r:
n ? n t ‘■ n p n i ' n a n t { t * - ( u M i . i 9 n \ s s y i j i
t + a n v n R T s T i y ^ s i
t x / u H 3 n n = uiM'-5 n r :
ntnt ' n t m 'nrni ( im- c u-m * n m sr.v > .1 fJ G A t r l = T iHAlrJ
I M V N; I U!?-’ 9 I 9 C! I "i l V 7 i ?f u m f m ‘urnGni w= lga ir
um«ni = ( P' t i v qriN rMf i V- . T : ( 1 » ! ) X ^ n i \ !
Gyi-'q= (“i *u;men f) c c i ' w n i f i D 1 5 9 = c n '■.v k.h r ) q
( iflsrtvwqu / 1 = '1 fin? fir:
ntS 'DSP. 'OQP. (ui) dl MV.MS=(“i'wmGni Jv
nr.o?.osnt
nant
n/.nir«9ntrqnl
nfrnl nan I
r.Pot
Pint cnnl nnn I
p i pn / pr.q?r.Q?r.ppCP.Pnip on? on p.
vuiv vsr I=ri V(J M *K1 Ib>LALL= IbLALt-i
d'd'J 'Jfc.lt KN = Ut f LK;1 * K 1VUI iL JiViJt MVfjT KUW 3T K 1VH1 fc.l_tnfc.Nr
J d ' j M U . e L U ^ / iCULUnj = 1 - U J** J jj dSU L = 1 / NJbU A { JLJLUr^Li =* 1 ItSL'J'l/LJ/HIVJ ! b b b l r U J Jr>U/ b b Uj f o j o y J / u L = i / . ' it / J b \ i L b L U n j L ) = b ( i t J L ' J i J / L ) / H 1 Vt J I
L ^LUU Ct NrjN-Kl VJ I' WCJWSJS'J Jt? L 1 = 1,»Nj y j IF t L l - l L « L U n J <fUU/ 3 D U ; <+UU auu i =a ili »iuunr.)4-efU A ( L 1 / I LC/LU.1J = U • U a j j j y A b u L = i / N'10U , M L 1 / L > = M L 1 ' L > - A U L U L U M / L > * !A-yo ifctnj bbU/bbU/A6U * b ‘j j j b u u L = i ^ r ,3U'J b l L l / L J = 3 l L l * L J “ r U ICCJLUM/L I * I bb'J CJN1INUL
L I N I t K L H A N U t L S L U r i N b5 U J J J / 1 U 1 = 1 / N b l ' J I J \ + 1 - 1£>d:J IF I 1NULX I L/ 1 ) - 1 NL/tX I L J d)) €>JCJ.r /1U/ fc>JU fcd'J .JKU W = I NUt X { L » 1 1 bFU .JCJLU !'I=i^L )fc .X lL^cI)fcb'J jy /ris :<=l/\:bb'J bWAK = A 1 < / JKU'A' )b / U A ( !A i J K J a- ) ■- A I !A / J L 9 L'JM J/ U U A ( X / J L B L U n ) =bWAH/ U O L L ' N i l - N U t/1U UUNT INUt/A-U ‘T'r ! U:-;.\!
fc'.N'U
$
Uiw
r r. r
S ’JOKOU! I Mb. S ’-'iAKl Ai»jrL/NU>NC^NLU)
S U b K C J U l l N t . TO S E K E K A T E ' H E P A R T I T I O N S O f T H E V E C T O R C I N T O T H E V E C i O K SA ANU b
i) I H E N S 1 tJ M A ( N U ) / y I N L )t C ( N E U } irjWi =1 * nu A ! ! ) = U I )onvts i = i/nc
2 5 i J = - H i+NU)K E 1 URN ENU
Ulcr>
157
t..jti♦—m>-IT)
Ull—LiY 3 10 ■—tCi
3<xUJ">•uj.0
73 3JL
[Jj7)2r>a .'XI
r.UJI—to>-•j)
UJI -UJX_Jtort>
UJ L-> T>in b7 I/) r> D XZi CO I— 2 X 2X L- 7* UJ < UJ »—ito H) Ll! T. 7) 1— aUJ —♦ lU —» x.. 7) i— 3 UJ l .X
J—«w)
J-L 2
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VITA
J. Brian Froisy is the son of Mr. and Mrs. J. Froisy of Plaquemine, Louisiana. He was born in Vancouver, Washington on July 20, 1945. He graduated from St. John High School, Plaquemine, Louisiana in May of 1963.
In the summer of 1963 he enrolled in Louisiana State University and received his Bachelor of Science Degree in Chemical Engineering in January of 1968. In August of 1969 he was awarded a Master of Science Degree in Chemical Engineering from Louisiana State University.
He is presently a candidate for the degree of Doctor of Philosophy in Chemical Engineering.
EXAMINATION AND THESIS REPORT
Candidate: J . Brian Froisy
Major Field: Chemical Engineering
Title of Thesis: A Combined C orrelation Function - Linear RegressionApproach to System Id e n tif ic a t io n
Approved:
Major Professor and Chairman
Dean of the Graduate School
EXAMINING COMMITTEE:
Date of Examination:
November 29, 1971