y toward a mathematical theory of environmental …
TRANSCRIPT
UCEL-51837
y TOWARD A MATHEMATICAL THEORY OF ENVIRONMENTAL MONITORING THE INFREQUENT SAMPLING PROBLEM
Kenneth D Pimentel (Ph D Thesis)
June 1975
Prepared for US Energy Research amp Development Administration under contract No W-7405-Eng-48
I I I B LAWRENCE I H 3 UVERMORE K s i LABORATORY
bull OWrwijyric^c-
Cy
NOTICE This report was rcpared asan account or work sponsored by the United States Government Neither the United States nor the United States Energy Research laquoV Development Administration nor any of their employees nor any or their contractors subcontractors or their employees makes any warranty express or implied or assumes any legal [lability or responsibility for the accuracy completeness or usefulness of any information apparatus product or process disclosed or represents thst Its use would not infringe privately-owned rights
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LAWRENCE UVEPIORE LABORATORY UnmsityotCaHorr^VmmmCalifarigtW550
UCFSL-51837
TOWARD A MATHEMATICAL THEORY OF ENVIRONMENTAL MONIYOPING
THE INFREQUENT SAMPLING PROBLEM Kenneth D I lcnetitel
(Ph D T h e s i s )
Ms da te June 1975
then cinplorm miklaquo
tubibi oi iltipraquoiuibiLigt fu urriilnnof inraquo ciai-
prooia disdoird tu rrpiri
TOWARD A MATHEMATICAL THEORY OF ENVIRONMENTAL MONITORING
THE INFREQUENT SAMPLING PROBLEM
Kenneth D Pimentel University of California Lawrence Livermore Laboratory
Livermore California
ABSTRACT
An environmental monitor is taken to be a system which generates estimates of environmental pollutant levels throughout an emironmental region for all times within a time interval of interest from measureshyment data taken only at discrete times and only at discrete locations in that region This study addresses the following optimal environshyment monitoring problem determine the optimal monitoring program mdash the numbers and types of measurement devices the locations where they are deployed and the timing of those measurements mdashwhich minimizes the total cost of taking measurements while maintaining the error in the pollutant estimate below some bound throughout the time interval of interest
Diffusive pollutant transport in distributed environmental systems is treated with the method of separation of variables to obtain a set of stochastic first-order ordinary differential state equations for the process Techniques of optimal estimation theory are applied to this set of state equations yielding a set of matrix estimation error co-variance equations whjse solutions are used in accuracy measures for the resulting estimates in the synthesis of optimal monitors
ii
The main results are associated with the infrequent sampling probshylem If the estimation error constraints imposeJ upon the monitor are sufficiently lax the solution for the optimal monitoring program results in relatively long times between required measurements This leads to drastic simplifications in the solutions of the problems of optimally designing and sequencing the measurements where only certain terms in the solutions of the estimation equations are found to effect the reshysponse for large time This dominance of certain asymptotic terms is seen as a potential area for future application in more complex environ-bullintal pollutant transport problems
Owing to the ease in their interpretation numerical applications for one-dimensional diffusive systems are included to illustrate the main results though all the results are shown to generalize to the three-dimensional case Considerable use of graphical computer output is made which clearly exhibits the features of the infrequent sampling problem An extensive list of references in areas relevant to the optishymal monitoring problem completes this report
TABLE OF CONTENTS
Page
TITLE PAGE i ABSTRACT ii ACKNOWLEDGMENTS viii DEDICATION xii LIST OF CONCLUSIONS xiii NOMENCLATURE xiv CHAPTER 1 INTRODUCTION 1
CHAPTER BACKGROUND AND PROBLEM STATEMENT 7 21 Background 7 22 Problem Statement 1
CHAPTER 3 NORMAL MODE MODELS FOR DIFFUSIVE SYSTEMS 19 31 Separation of Variables for the Diffusion
Equation 23 32 One-Dimensional Diffusion 25
321 No-Flow Boundary Conditions 26 322 Fixed Boundary Conditions 33
33 Two-Dimensional Diffusion 35 34 Three-Dimensional Diffusion 40
CHAPTER 4 MODEL DISCRETIZATION AND APPLIED OPTIMAL ESTIshyMATION 42
41 Discretization of the System Model 43 4 1 1 The Systen Model Equations 43
412 The System Disturbance Stat is t ics 46 42 Optimal Estimation - The Kalman F i l t e r 47
421 Optimal Estimation 4 7
2 2 Summary of F i l t e r A l go r i t hm SO
CHAPTER 5 OPTIMAL DESIGN AND MANAGEMENT OF MONITORING
SYSTEMS 52
51 Monitoring and the Kalman F i l t e r 5 2
52 One-Dimensional Piffusion with No-Flow Boundary Conditions 5 6
iv
CHAPTER 5 (Continued) 53 The Design Problem for a Bound on the Error
in the State Estimate 57 531 The Infrequent Sampling Problem 57 532 The Effect of a priori Statistics 66 533 Fixed Number of Samplers at Ech
Heasurment and Fixed Error Limit 70 534 Variable Number of Samplers 73 535 Analytical Measurement Optimization 74 536 Numerical Measurement Position Optishy
mization 77 537 Numerical Measurement Quality Optishy
mization 82 54 The Design Problem for a Bound on the Error
in the Output Estimate 84 541 The Minimax Problem 84 542 Determination of the Position of Maxishy
mum Variance in the Output Estimate 94 55 Diffusive Systems Including Scavenging 98
551 The Infrequent Sampling Problem 100 5 6 One-Dimensional Diffusion with Fixed Boundshy
ary Conditions 105 57 Extension to Monitoring Problems in Three
Dimensions Systems with Emission Boundshyary Conditions 112
58 The Managemeit Problem 122 581 Optimality in the Scalar Case 123 582 Extension to the Vector Case mdashArbishy
trary Sampling Program 132 583 Extension to the Vector Case - Infreshy
quent Sampling Program 133 5E4 Suggestion of a Heuristic Proof for
the Vactor Case 136 59 Extension to Systems in Noncartesian Coordishy
nates General Result for the Infrequent Sampling Problem 138
CHAPTER 6 NUMERICAL EXPERIMENTS 142 61 Problems in One-Dimensional Diffusion with No-
Flow Boundary Conditions 143 62 Problems with Bound on State Estimation Error 157
621 Asymptotic Response of State Estishymation Error 157
v
CHAPTER 6 (Continued) 622 Optintality of Measurement Locations 176 623 Comparison of Performance Criteria 176 624 Effect of Instrument Accuracy 178
63 Problems with Bound on Output Estimation Error 180 631 Asymptotic Responses of Output Estishy
mation Error 188 632 The Effect of a priori Statistics 192 633 Problems with a Fixed Number of Samplers
and Constant Error Bound i99 634 The Effect of Level of Estimation Error
Bound upon the Optimal Monitoring Probshylem 209
635 Examples of Various Levels of Bound upon Output Error 210
636 The Effect of Time-Varying Error Bound upon the Optimal Measurement Design 218
637 The Effect of Time-Varyir^ Disturbance and Measurement Statistics upon the Optishymal Monitoring Design and Management Problems 223
638 Variable ruirher of samplers 227 639 Sensitivity o Results for the Infrequent
Sampling Problem to Model Dimensiorslity 231 6310 Problems Including Pollutant Scavenging 249 6311 Problems with Multiple Sources 257
64 Optimality in the Management Problem 265 CHAPTER 7 SUMMARY AND RECOMMENDED EXTENSIONS OF THE MAIN
RESULTS 268 71 Summary 268 72 Recommended Extensions 270
APPENDIX A DISCRETIZATION OF THE STATE EQUATION 276 APPENDIX B DISCRETIZATION OF THE STATE DISTURBANCE
STATISTICS 278 APPENDIX C STATE AND ERROR COVARIANCE PREDICTION WITHOUT
MEASUREMENTS 285
Vi
APPENDIX D ANALYTICAL MEASUREMENT OPTIMIZATION 289 Dl Minimize Estimate Error 289 D2 Minimize Estimation Error and Estimation
Cost 295 D3 Results 237
APPENDIX E NUMERICAL MEASUREMENT QUALITY OPTIMIZATION 299 APPENDIX F DESCRIPTION AND LISTING OF PROGRAM KALMAN 303 APPENDIX G DESCRIPTIONS AND LISTINGS OF POSTPROCESSOR
PROGRAMS 343 Gl Program CONTOUR 345 G2 Program POFT 348 G3 Program PELEM 35^ G4 Program SIGMAT 356 G5 Program MAXTIME 360 G6 Program POSTPLT 362 G7 Program POSTFP 363 G8 Program POSTSP 364
REFERENCES 365
vii
ACKNOWLEDGMENTS
Many people in a variety of situations have contributed to my doctorial program Academicians colleagues fellow employees and supervisors and members of my family To all of these and more go my gratitude and sincerest good feelings
To John Brewer who started it all for me in automatic controls as an undergrad at Davis this stuff sure beats gear design To the Faculty at Berkeley thank you all Yasundo Takahashi tried to teach me what a state vector was just when I thought I had it he added noise and everything got stochastic To Robert Steidel who helped with my Masters and introduced me to that Lab out there in Livermore To Joseph Frisch who got me the job in the Controls ab and the TAship thanks so much To Dan Mote and Bob Donalu^on out there in eigenspacemdash it finally sank in To Charles Desoer and William Kahan for the clarity which came through their rigor
To the Faculty at the Davis Campus which somehow when I got back was no longer the University Farm my gratitude Dean Karnopp cleaned up my head about systems with one causal stroke Walt Loscutoff not only conveniently graduated from Berkeley so I could have his TAship but he also conveniently went to Davis where I could watch him on TV and have him hulp with my orals
To Charles Beadle and Mont Hubbard who helped with the manuscript thank you for your many hours which might have been more amusingly spent I truly appreciate your help
And then full circle back to John Brewer who has been a continual source of fascination inspiration perspiration frustration and
yiii
resuscitation you are a thesis advisor and friend par exoellenae Your patience understanding and nurturing have not all gone for naught Thank you so very much as I look forward to a long continuing potentially mellower relationship
Howard McCue by far deserves the most thanks of all my colleagues He sat through more baloney poked holes in more theories but learned more about computers from me than anybody else And look where it got you Howard sure do love those computers dont you Thanks too tc Larry Carlson Steve Johnson and Frank Melsheimer for making those days at Berkeley what they were And special thanks to Jerry Alcone for findshying it in his heart to graduate so I could have his office you still owe me a handball it the back too Alcone And at Davis thanks to Steva Moore and Jeff Young who sewed the seeds for a lot of what came from this study
Thanks to the many at Lawrence Livermore Laboratory who have seen fit to employ me while finishing my education Wally Decker and Walt Arnold as Department Heads in Mechanical Engineering have supprted me far beyond what I ever expected I sincerely intend to pay back in my career at the Lab Gene Broadman as Division Leader has helped in ways which mark M m as one of the best in my book John Ruminer and Jerry Goudreau were just the kinds of supervisors we needed great ones
And then there was is and ever shall be Gerry Wright He put up with me put me down got put down and got fed up Hope he forgives Howard and I someday for going back for his Masters Sincerely thank you for all your help Ger all of it for its always been considerable
1x
To Chuck Mi l le r Nort Croft Al Cassell and Gail Dennis did you hear
the one about t h i s Portagee who finished school I knew you hadnt
And f i na l l y to Mildred Rundquist She is no secretary no t yp is t
no c le r ica l type She is a typographical ar t is t - -pure and simple The
i s j s and ks are hers The equations are a l l hers Even some of the
figures are hers And with a l l that my respect appreciation and f r iendshy
ship w i l l always be hers Thanks M i l
To the people of th is country through the United States Energy Research
and Development Administration thank you for your support To the people
of the State of California through the University of Cal i fornia and the
Lawrence Livermore Laboratory my gratitude extends Thank you a l l for
making th is research possible
To Dr Justin Simon a special f r iend in a special way thank you
for your encouragement your kicks i n the mdash your understanding and the
lack of i t Yob now and I know how important a l l this was for me to do
You are the best at what you do and I or we may s t i l l r i p o f f your leaded
glass some day
To my parents who thought i t never could be done i t s done Thank
you for everything you gave me
To ray mother- and father- in- law youve always been there and that s
always counted Your encouragement is ever appreciated I know what f i n i sh shy
ing th is means to you and Im proud that Im able to give i t
The approach of the conclusion of my doctoral studies has prompted a
wide variety of responses from those closest to me From my daughter
Jennifer whos almost f i ve I missed you today From my son John
x
whos almost three Daddy don go wurk anymotmdashstay home now
And from my wife Janet who alone knows how old she rea l ly i s I
dont believe i t Thank you Hunny for always being there and yes
i t is done Now whered you want that pool
DEDICATION
for Jyp PhD
LIST OF CONCLUSIONS
Page
Conclusion I 60 II 64 III 64 IIIA 78 IV 69 V 69 VI 71 VIA 71 VIB 218 VIC 224 VID 224 VII 73 VIII 84 IX 90 X 90 XI 92 XII 94 XIII 105 XIV 112 XV 121 XVI 127 XVII 132 XVIII 1 4 1 XIX 247
Conjecture A 137 B 140 C 230
xU
NOMENCLATURE
Symbol Description
A ( t ) A Continuous-time dynamic system matrix
B ( t ) B Continuous-time deterministic input d is t r ibut ion
matrix
C( t ) C Continuous-time measurement matrix
Cbdquo Discrete-time time-varying measurement matrix at
bullbull time t K
cpound The optimal measurement matrix at time t
C(zK) Measurement matrix as a function of the vector z K of measurement positions at time t bdquo
C Generalized modal capacitance D( t ) D Continuous-time stochastic disturbance d i s t r i shy
bution matrix
pound bull bull Unit matrix with ( i j ) t h element equal to one
~ J and a l l other elements zero
F Pollutant mixing ra t io
G K + Kalman gain matrix at time t R +
I Ident i ty matrix
J Performance cr i te r ion
J(t) First monitor performance criterion estimation error in optimal state estimate at time t
Jbdquo(ct) Second monitor performance criterion value of pollutant concentration estimation error at that point c in the medium where it is a maximum at time t~
K Diffusion coef f ic ient discrete-time index f ina l
value of a discrete-time summation index
L 2L Length of a one-dimensional di f fusive medium
M n Covariance matrix for i n i t i a l state
Symbol Description
N Final value of a discrete-time summation index
P Region in space over which pollutant transport problem is defined
Pbdquo Corrected state estimation error covariance ma-~K t r i x at time t conditioned upon a l l past measureshyments including the measurement at time t
1 P K + 1 Predicted state estimation error covariance matrix
at time t^ +-| conditioned upon a l l past measurements up to ard including the measurement at time t K
v -K+N^-K Predicted state estimation error covariance matrix
at t i ire t K + f j conditioned upon a l l past measurements up to and including the last measurement at time t( and a function of the measurement matrix at timt t bdquo
p ( t ) P Continuous-time state estimation error covariance
matrix
R Generalized modal resistance T Discrete-time integration step-size T r F i rs t monitoring error constraint maximum allow-
able error in the estimate of the monitor state vector
Tr Ppound + N(zj) j Predicted value of the trace of the state estima-l ~ N - t ion error covariance matrix at time t |^ + N condishy
tioned upon a l l past measurements up to and includshying the optimal measurement at zjlt at time t K
V( t ) V Continuous-time measurement error covariance matrix
W(t) W Continuous-time state disturbance covariance matrix
X A matrix used in derivations
Y A matrix used in derivations
c Scalar measurement coefficient used in optimal management problem derivations
c(c) c Readout vector mapping modal states into pollutant concentration at point pound in space
Symbol Description
e Base of natural logarithms (= 271828 ) surshy
face emissivity coeff ic ient
e T Exponential of the matrix [AT]
e Unit vector with i th element equal to one and a l l other elements zero
e (z) Eigenfunction associated with the nth eigenvalue
evaluated at position z
f Stochastic pollutant source term in the transport equations
g Deterministic pol lutant source term in the transshy
port equations
h Emission boundary condit io coeff ic ient
i Vector or matrix element index
j Vector or matrix element index m The dimension of the noise-corrupted measurement
measurement error and measurement position vectors y R y K and z K
j u Mean value of i n i t i a l state
n Discrete-time summation index
n The dimension of the^state and optimal state e s t i shymate vectors x K and x K
p Scalar state estimation variance used in optimal management- iroblem derivations
p The dimension of the deterministic input vector a(t)
r The dimension of the stochastic state disturbance vector w(t)
t Continuous value of time t K The Kth discrete value of time i Convolution of deterministic input vector over the
time interval EtKt|+j
xv 1
Symbol Description u(t) y Continuous-tine deterministic Input vector v K Discrete-time measurement error vector at time tj y(t) v Continuous-time measurement error vector -K+l Convolution of the stochastic disturbance vector
over the time interval [ t K t K + 1 ] w(t) w Continuous-time stochastic disturbance vector x Derivative with respect to time of the state
vector x x K Discrete-time state vector at time t K
xpound Corrected value of the optimal state estimate at time t|lt conditioned upon all past measurements inshycluding the measurement at time t x[ Predicted value of the optimal state estimate at time t K +i conditioned upon all past measurements up to and Including the measurement at time tbdquo x(t) x Continuous-time state vector x(t) x Optimal estimate of continuous-time state vector vbdquo Discrete-time noise-corrupted measurement vector bull at time t K
y(t) y Continuous-time noise-corrupted measurement vecshytor
z Position in a one-dimensional diffusive medium z Position of maximum error (variance) in the estishymate of the pollutant concentration over all values of 7 In a one-cffmenslonal medium zbdquo Discrete-time measurement position vector at time
zj Vector of optimal measurement positions at time t K
z Vector of deterministic input point source loca-~u tlons
xvll
Symbol Description Vector of stochastic disturbance point source loca-
w tions
0 o Zero matrix or vector
a Pollutant scavenging parameter r K + 1 r Time-invariant discrete-time stochastic disturbance distribution convolution matrix for the fixed time step T = (t K + 1 - t K) A K Amount of correction to scalar state estimation varshyiance for a measurement at time t K used in the opshytimal management problem derivations ATr Amount of correction to the trace of the state estishymation error covariance matrix for a measurement at time t|( used in the optimal management problem derishyvations S(t-x) Dirac delta function Kj Kronecker delta function
e A convergence criterion 5 Position coordinate vector for a point in a region
P in a diffusive medium n An intermediate transformation variable 0 A matrix used in certain derivations Eigenvalue or separation constant u Terms involved in determination of eigenvalues for
n emission boundary conditions pound(t) 5 Pollutant concentration at point z in space at
time t (Ct) Optimal estimate of pollutant concentration at
point c In space at time t 4bdquo(z) 5i Discrete-time pollutant concentration at point z
K and time tbdquo
xvlli
Symbol Description
I ( z ) L Optimal estimate of discrete-time pollutant corcen-K t ra t ion at point z and time t
5 (z) I n i t i a l pollutant concentration as a function of bull0
ulim
posit ion z in the medium
= 314159
p A convergenc measure
a 2 ( c t ) Variance in the optimal continuous-time estimate of pol lutant concentration at point z in space at time t
ol(z) Variance in the optimal discrote-time estimate of the pollutant concentration at point z and time h
0 ^ J M ( Z I ^ Z ) Predicted value of the variance at time t K + N in the K N ~ K discrete-time estimate of the pol lut ion concentrashy
t ion at point z conditioned upon measurements up to and including the last measurement with posit ion vector z K at time t K
deg K + N ~ K Z Predicted value of the maximum value over a l l values of z of the variance in the pollutant concentration at time t K + r j conditioned upon a l l past measurements up to and including the optimal measurements at zj at time t K
oK(zJz) Corrected value of the maximum value over all values of z of the variance in the pollutant concentration at time t K conditioned upon all past measurements including the optimal measurements at z at time t K
o Second monitoring error constraint maximum allowshyable error in the estimate of the pollutant concenshytration anywhere in the medium Time used in certain definitions and derivations
An intermediate matrix used in various derivations Scalar measurement error variance used in optimal management problem derivations
xix
Symbol Description
C i gt Time-invariant state t rans i t ion matrix for the ~- ~ f ixed time step T 5 ( t K + 1 - t K )
( t K + t bdquo ) Time-varying state t rans i t ion matrix between times t K and t K + 1
X A matrix used in certain derivations
C + i t I Time-invariant discrete-time deterministic input d is t r ibu t ion convoution matrix for the f ixed time step T = ( t K + t K )
g bdquo + a Discrete-time convolution of the continuous-time state disturbance covariance matrix W(t) over the interval L i t K + - | J
a The discrete-time matrix convolution of the matrix N g K + where N terms in the series are included
a The l i m i t of the discrete-time matrix convolution SS pound2 as N approaches i n f i n i t y with i t s (1 l)-element
to zero
ltD Scalar state disturbance variance used in optimal management problem derivations
- Approximately equals = Identically equals or is defined as gt Greater than raquo Much greater than
lt Less than lt Less than or equal to lt Proport^irtf to or goes like Approaches or goes to - raquo Implies or infers
d [ - ] Total d i f fe ren t ia l operator
g r [ bull ] [ bull ] Derivative with respect to time of the variable in brackets
Symbol Description _3_ 3c
_i 3C
a
diag [bull]
EL-]
min
min max Z K Z
Partial differentiation of a variable with respect to the scalar c Partial differentiation of a variable with respect to the vector c
Partial differentiation of a variable with respect to the matrix C A vector whose elements are the diagonal elements of the matrix enclosed in brackets Expectation operator for a random variable vector or matrix Limiting operation as N approaches infinity Maximum over all scalar values of z Minimum over all vector values of z K
Simultaneous minimum over all vector values z K and maximum over all scalar values z
n=l Tr[-]
bullh
n-l
N r j
Summation from 1 to N over all values of the index n
Trace operator of the matrix enclosed in brackets The 1th_ element of the vector enclosed in bracket [a]^ 1s also denoted a The (ij)th element of the matrix enclosed in brackets [A] 1s also denoted A ^ Transpose operation for a vector or matrix Inverse operation for matrices
A matrix with (ll)-e1ement equal to u and all other elements zero
A matrix with (ll)-element equal to zero and all other elements equal to the elements of the matrix A
xxl
Symbol Description
6 o -cj
A diagonal matrix
p gt 0 The matrix pound i s posit ive def in i t i ve
ltbull I n f i n i t y
CHAPTER 1 INTRODUCTION 1
The problem of the optimal monitoring of pollutants in environshymental systems concerns the minimum cost estimation of pollutant levels throughout a region while maintaining the errors in the estimates within a given bound The optimal monitor synthesis problem considered in this thesis logically separates into the two monitoring subproblems of optimal design and optimal management Optimal monitoring system design includes the specification of a model for the physical system the choice of measured variables measurement devices and their spatial distribution in the medium The optimal management problem concerns finding the best sequencing of measurements in time to result in the minimum cost sampling program The optimal monitor is then defined as that solution of the design and management problems together which results in the minimum cost measurement program necessary to maintain the error in the pollutant estimate below a given bound over the time interval of interest
This is a departure from most studies in the optimization of systems with cost for observation in that use is not made of a comshybiner performance criterion which typically consists of the time integral of a weighted combination of measurement cost and estimation error Insteid in this study advantage is takrn of the separation of the design and management problems whose two solutions separately determine the characteristics of the measurements at the required sample times and the timing of those measurements themselves Thus estimation error is not minimized but rather bounded in a
2
fashion which corresponds with actual applications where legal limits are placed upon allowable errors in the pollutant level estimates in environmental monitors It 1s bounded In such a manner that the minimum total number of samples is necessary over some time Interval resulting in the minimum cost monitoring program
The separation of the monitoring design and maiagement problems was proposed by Brewer and Moore [24] Moore [95] has considered application of such corcspts to the area of aquatic ecosystems where the Extended Kalman Filter 1s applied to the highly nonlinear equashytions of the dynamics of population growth of aquatic constituents This thesis instead concentrates upon strictly linear processes in the hope that the mathematical simplifications possible there may be extendable to the nonlinear case in future studies In the optimal estimation of the state vector of a linear discrete-time stochastic system the Kalman Filter [66] provides a particularly elegant computational solution The two equations for prediction and correction of the associated state estimation error covariance matrix have been conjectured by Brewer and Moore [24j as containing the key to the solution of the management problem it is shown here that they indeed do lead to a problem structure which results In the optimal solution of not only the management problem but to that of the design problem as well
Owing to the anticipated complexities of the optimizations assoshyciated with the various parts of the monitoring problem advantage 1s taken of the simplicity of the separation of variables technique in the theory of linear partial differential equations In obtaining orshydinary differential equation models for distributed systems (see Berg
3
and McGregor I18J) In reducing the resulting state spaces for such normal mode models to spaces of finite dimension the quantitative methods recently developed by Young I131J 1n atmospheric modeling greatly extend the area of applicability of such analytical techniques In particular nonhomogeneous anisotropic media may be handled by the spatial discretization of the medium Into component subregions over which constant average values for system parameters are sufficiently accurate Component coupling by the use of pseudo-sources to make up for differences in the normal mode submodels is the key factor given by Voung which allows for the simple approximation of the dynamic reshysponse of large varied distributed environmental systems The existshyence of these techniques underlies the studies 1n this thesis in their extension to large scale practical problems in environmental monitoring
With the use of a finite-dimensional normal mode state model the resultant continuous-time state equations are discretized in time for use in the Kalman Filter The natura of the Kalman Filter is now well known 1n its applications in the aerospace field Recent applishycations in more diverse areas (see for example the special issue 1n IEEE [62]) have established It as a powerful tool of broad scope 1n the field of system estimation Its numerical advantages over other optimal estimation techniques (well documented 1n Gelb [44]) make it the logical choice for use in environmental monitoring systems where processes of Interest may dictate the use of huge models to obtain desired levels of spatial arid temporal resolution in the results
4
The main results of this thesis concern the special class of monishytor addressed In the infrequent sampling problem This case is charshyacterized by high levels of allowable pollutant estimation error which result in relatively long periods between required sample times These long times between samples allow the transient terms involved in the growth of the uncertainty in the pollutant estimates to reach steady-state values so that only asymptotic solutions of the estimation error covariance equations need be considered in the design and management problems This drastically simplifies the solution of the monitoring problem for the case of infrequent sampling
Applications of the theory developed here are seen to arise in any environmental or other dispersive system where the dynamics of the disshypersal of the pollutant or variable involved is dominated by diffusion and where convective transport can be ignored This rules out its use in air quality monitoring systems on a regional basis where convection typically dominates diffusion in pollutant transport by a ratio of 301 [76] However as developed by others cited in Young 1131] models of pollutant transport on a global scale are often based upon diffusion as the dominant mechanism of dispersal In fact examples in Young indishycate that the normal mode modeling techniques mentioned earlier can be successfully applied to global atmospheric modeling where only diffusion is included as the dispersion mechanism
An interesting extension of the results of this thesis might be to a study involving assessment of the climatic impact of flying a fleet of SSTs upon the protective ozone layer in the atmosphere (see for exampls Mac Cracken et al [80]) In such an application knowing where and when to best sample atmosphere pollutant levels could greatly
5
facilitate validation of numerical atmospheric models in initial applishycations and greatly reduce long-range monitoring costs upon implementashytion of such a program
Groundwater systems seem to be a probable area of application as indicated in what follows though no experimental verifications have been attempted Systems involving heat transfer by conduction which involve stochastic heat sources could find application for the theory of the infrequent sampling problem For example in nuclear reactor cooling systems a central control computer could be time-shared to consider only the best sites for temperature measurement in the walls of the pressure vessel over time
The need for better environmental monitoring has been described in the literature [4695102] typical measurement costs have been tabulated [14] Propagation of uncertainty in distributed systems has been considered in some detail 15659101] Related studies using other approaches do not address the monitoring problem either as it separates into the design and management problems or with the drastic simplifications which arise in the infrequent sampling problem (see the work of Seinfeld [113] Seinfeld and Chen [114115] Seinfeld and Lapidus [116] Reiquam [104] Bensoussan [17] Soeda and Ishihara [119]) Thus there is a naed for improvement of the synthesis procedures for monitoring systems in large scale environmental problems
The thesis is organized into seven chapters and seven appendices to keep things even Chapter 2 summarizes work by others in germane problem areas and defines the scope of the present study Chapter 3 develops briefly the normal mode modeling technique of the application of the method of separation of variables Chapter 4 deals with the
6
time-discretization of the associated f in i te set of continuous-time
ordinary differential state equations and summarizes the more salient
features of Kalroan Fi l ter Theory Chapter 5 presents the main theory
associated with the infrequent sampling problem punctuated with conshy
clusions as they can be made Application and demonstration of the
analytical results of Chapter 5 are made in the numerical examples of
Chapter 6 in which more conclusions are seen to follow In Chapter 7
the main results for the optimal monitoring problem for the case of inshy
frequent sampling are collected in summary and possible extensions for
future study indicated Some of the more routine analytical developshy
ments as well as al l of the computer program listings are gathered
in the appendices A rather extensive l i s t of references relevant to
the optimal estimation monitoring and measurement system design probshy
lems completes this document
7 CHAPTER 2 BACKGROUND AND PROBLEM STATEMENT
This chapter begins with a suiroary of representative work done by others In fields of Importance to the environmental monitoring problem An attempt Is made to present a reasonably complete survey of pertinent literature in the hope that future researchers may benefit from the sources this author has utilized
The broad area of optimal measurement system design is then narrowed greatly in scope as it applies to problems In certain classes of environshymental pollutant transport The problems of the optimal design and management of environmental quality monitoring systems are finally stated in the contexts of two cases for bound on the allowable error In either the monitor state or the monitor output estimite
21 Background
The major topics of concern in the study of environmental monitorshying systems in this thesis include the following mathematical modeling in dispersive environmental systems the numerical treatment of certain classes of partial differential equations the stability and asymptotic solutions of systems of ordinary differential equations optimization of a function of several variables deterministic dynamical system theory stochastic system theory and optimal estimation optimal measurement sysshytem design in lumped and distributed parameter systems and finally monishytoring system synthesis for environmental applications
Considerable Interest has been turned to problems In the dispersal of pollutants In environmental systems in recent years Some typical contributions 1n the areas of the atmospheric sciences include the modelshying of air pollutant transport on a regional basis [81 J the climatic
8
impact of f l y ing a f lee t of SSTs in the upper atmosphere I80J studies
1n the parameter sens i t iv i ty of models of the planetary boundary layer
[3599J and studies of models of the global transport of pollutants
[36131] In one recent study by Young [131J the classical methods of
applied mathematics were successfully applied to the solution of global
pol lutant transport problems in a unique way that takes advantage of
analytical results available fo r certain classes of part ia l d i f fe ren t ia l
equations By the expansion of solutions for such equations in i n f i n i t e
series form followed by quant i tat ively meaningful truncation of those
serious solut ions approximate solutions for otherwise Targe d i f f i c u l t
problems can be obtained This procedure involves coupling together
solutions for problems in adjacent subregions to e f f i c i en t l y approximate
the response in larger areas The theory for such Fourier-type expanshy
sions is now well established [183482118J but the unique extensions
made by Young possess the potential for applying classical normal-mode
analysis long associated with problems in the mechanics of l inear solids
[9347] to a far braoder class of problems including environmental
pollutant transport in nonhomoqeneous anisotropic media
This author follows Young in the application of normal-mode technishy
ques to problems in the solution of the dynamic equations of environmental
pollutant transport Such methods y ie ld f i n i t e sets of ordinary d i f f e r shy
ent ia l equations whose solutions form time-varying mul t ip l iers for the
spatial mode shapes which comprise the normal mode solut ion bond graphs
are seen to of fer a concise graphical representation of such normal mode
models (see for example Karnopp and Rosenberg [6S]) The study of the
numerical treatment of systems of ordinary d i f fe ren t ia l equations is a
fundamental part of the solution of the monitoring problem when using
9
the normal mode approach recent advances 1n the numerical solution of general nonlinear time-varying possibly stiff ordinary differential equations are typified by the work of Gear [43] Hindmarsh [5758] and Byrne and Hindmarsh [25] Analytical treatments can be found in Coppel [28]
In the case of linear time-Invariant ordinary differential equashytions the class involved in the infrequent sampling problem considered in this study the powerful techniques of linear system theory can be used (see for example Desoer [32] Takahashi et at [121] Brewer [22] Freeman [41] Timothy and Bona [123]and Schultz and Helsa [109]) In the actual implementation of algorithms associated with the solutions of such linear systems certain topics in matrix theory in numerical analysis prove to be useful [3840129] Involved in the optimal design problem in monitoring system synthesis are the problems associated with the optimization of a function of several variables Beveridge and Schechter [20] is found to be an excellent reference in this area while Fleming [37] provides a more firm background in the theory of a function of several variables A gradient routine by Westley [127] was chosen for the constrained minimization of the nonlinear objective functions associshyated with the optimal design problem Such gradient methods are conshytrasted for example with the work of Radcliffe and Comfort [103] in which constrained direct search methods are presented which do not involve the use of derivatives of the objective function gradient methods are found to offer computational advantages over direct search methods in their application to the optimizations involved in the optimal design problem In the particular problems of finding the position of maximum uncertainty in the pollutant estimate for the monitoring problem with
10
bound on error in the output estimate root finding methods for finding zeros in the derivative of the expression for the error were found to be superior to direct search methods for such scalar maximizations (see Hausman [5354])
The field of optimal state estimation in stochastic dynamic system theory is well developed in what it offers for vhe solution of the optishymal monitoring problem Gelb [44122]makes a particularly lucid presenshytation of the more practical topics in applied estimation theory the original work of Kalman [66] and Kalman and Bucy [67] still stand as basic reference material for the concepts involved Sorensen (in Leondes [78]) presents a concise introduction to Kalman Filter techniques Meditch [85] also presents a clear development of the optimal filter Aokr [ 3] contains a considerable amount of material concerned with speshycial topics in stochastic system theory as does Sage [105] Jazwinski [65] is sufficiently complete in its rigor to serve as one single refershyence in the area of stochastic processes and filtering theory for more fundamental material in the theory of stochastic differential equations including a particularly rigorous development of the Kaliran-Bucy Filter see Arnold [ 6]
The Special Issue of IEEE Transactions on Automatic Control Decemshyber 1971 dealing with the Linear-Quadratic Gaussian Problem [62] ofshyfers an extensive collection of topics in optimal estimation theory It Includes a well edited bibliography which should be a basic resource to any researcher 1n this field The proceedings of a special confershyence sponsored by NATO [98] summarizes many military and aerospace apshyplications of estimation theory
11
There are many special topics In estimation theory which could prove of Importance In future extensions of the work in this thesis to practical applications in nonlinear systems Of them adaptive filtershying 1s of particular importance see the work of Mehra [86878889] Jazwinski [64] Berkovec [19] Godbole [45] Nahi and Weiss [97] and Scharf and Alspaeh [108] Extension to nonlinear estimation are conshysidered in Wlshner et aZ[130] Athans et al [9 J Hells [126] Gura [49] and Gura and Hendrikson [52] Moore uses the Extended Kalman Filshyter as cited earlier in his work on the monitoring problem [95] As well as Moore others have examined the effects of using an imprecise model in the optimal filter upon the performance of optimal estimation schemes among them are Jazwinski [65] who considers the area of filter divergence at length Aok1 and Huddle [4 ] Leondes and Novak [77] and Inglehart and Leondes [63]
The area of theory most closely allied to that of the optimal monishytoring problem is known variously as optimal estimation with cost for observation optimal measurement system or subsystem control or the opshytimal timing of measurements Aoki and Li [ 5] were among the first to address such problems along with Meier [909192] Athans uses his Matrix Minimum Principle [ 8 ] along with the work of Schweppe [11] in an application in continuous-time systems this work is strongly based upon direct extensions of optimal control theory (see Bryson and No [26] or Athans and Falb [10]) Schweppe [12110111] has made developments of op timal measurement strategies in radar applications Denham and Speyer [30] did some early work in midcourse guidance Kramer and Athans [73 74] have made recent rigorous contributions to the mathematics associated with the combined optimal control and measurement problems along with PIiska [100]
12
Other studies Involving the optimal timing and use of measurement data include Kushner [75] Breazeale and Jones [21] Sano and Terao [106] Hsia [60] and Dreyfus [70]
Some of the most germane references found in the area of optimal measurement system design include Cooper and Nahi [27] Sauer and Melsa [107] Vande Linde and Lavi [125] Herring and Melsa [55] Shoemaker and Lamont [117] and Soeda and Ishlhara [119]
Studies which concentrate on monitoring and measurement system optishymization in distributed parameter systems include the work of Seinfeld [112113114115116] Draper and Hunter [33] Reiquam [104] Bensoussan [17] Atre and Lamba [13] Murray-Lasso [96] and Prado [10lJ
Bar-Shalom et al [is] consider monitoring systems much like those considered here but for a far more general class of problem Moore [95] and Brewer and Moore [24] serve as the inspirational basis for much of what is developed in this thesis
22 Problem Statement
Consider a region into which pollutants are being injected by a colshylection of deterministic and stochastic point sources Two problems in the monitoring of the pollutant levels in that region over time are conshysidered in this study
First suppose that measurements are required of pollutant levels for the purpose of closed-loop control in which case feedback signals are to be constructed to control seme of the amounts of pollutant being emitted into the medium An example might be thermal pollution near a power station where it is required to optimally monitor temperatures in the surrounding area for the purpose of closed-loop control of the mean
13
power level Assuming that a model can be constructed for the dynamics of the pollutant dispersal in the form of a finite set of first-order orshydinary differential equations whose solution forms the state vector for the model of the process (see Desoer 132]) It is well known that the mean square length of the error between the state vector and the esshytimate of the stochastic state vector fs given by the trace of the estishymation error covariance matrix for such a stochastic process as a funcshytion of time (see Kalman [66]) Thus if it is required to minimize the mean square error 1n the estimate of the stochastic state vector a suitshyable choice for the performance criterion for the optimal monitor with bound on maximum allowable error in the state estimate is
J(t) = Tr[p(t)] (21) where
P(t) = E (x(t) - x(t))(x(t) - x(t)) T ( )
is the estimation error covariance matrix for the optimal estimate S(t) of the state x(t) both of dimension n at time t E[-J denotes the exshypectation operator applied to the random argument and (bull) denotes the transpose operation Here
n
Tr[A] = T [A]^ (23) n=l
is the trace function The notation [ALj means the (ij)Jh_ element of the matrix A
Second suppose legal limits are placed upon the maximum error in the estimate of the pollutant level itself allowable at any time anyshywhere 1n the medium This case represents a problem of practical interest where a monitor might be used on-line to detect infractions of legal pollutant concentration levels in some airshed or watershed
14
Let the concentration of a pollutant of interest as a function of space and time bt denoted by Ut) Define
5(ct) = c(c) T x(t) (24) where x(t) as before is the state vector of dimension n of pollutant dispersal in the region is the coordinate position vector of the point where the concentration pound is being calculated and where c(c) is a vector (typically of eigenfunctions in the spatial coordinates c for the case of normal mode models) which maps the state x into the concentrashytion at the point pound In this application the function of the monitor is to provide an estimate (st) of pound(ct) such that the maximum error between the pollutant concentration and its estimate is maintained below a given constraint or bound for all times of interest and throughout the medium spanned by t Thus a measure of the uncertainty or error in the estimate of the pollutant level at some point c anywhere in the medium is given by the variance in the estimate C(t) denoted by a (ct)
Derive using (22)
o 2(Ct) B E (c(st) - C(t)) Z
= E ^(5) T(x(t) - x(t))c(c) T(x(t) - x(ty
- E[jc)T(x(t) - x(t))(x(t) - x(t) )Tc(s)J
= c ( 5 ) T E[(x(t) - x(t))(x(t - x(t))TJc(c)
= ztflMsty- lt 2 - 5 gt Thus the variance in the estimate of the pol lutant concentration i t s e l f
also termed the monitor output anywhere in the medium can be expressed
d i rec t ly in terms of the monitor state estimation error covariance mashy
t r i x P(t) and the readout vector pound() Hence a logical choice for a
15
performance criterion for the monitoring problem with bound on maximum allowable error in the output estimate is
J 2(ct) = a2(poundt)
= max a (t)
= max c(c)TPCt)c(c) 5 = StffytM) (2-6)
where C is the position of maximum variance in the estimate of uie pol shy
lutant concentration or output at time t
Thus the two estimation error c r i t e r i a to be considered here are
given in (21) and (26) for the optimal monitoring problems with bound
on state and output estimation error Once an error c r i te r ion is seshy
lected in a given problem the requirements of the optimal monitoring
system design problem are to select the optimal choice of monitor model
complexity the optimal number and qual i ty of measurement devices to deshy
ploy and their optimal locations in the environmental medium fo r a l l
measurement times tlaquo over the time interval of interest The added reshy
quirement of the problem of optimal monitoring management is to select
the optimal measurement times t K such that together with the results for
the optimal design problem the minimum cost monitoring program is found
which maintains the chosen estimation error c r i t e r ion within i t s bound
throughout the time interval of interest
This is a somewhat d i f ferent approach from those taken in the o p t i shy
mal design of systems with measurement cost by previous authors Athans
[ 7 ] defines a scalar cost functional which is a l inear combination of
the tota l observation cost and the mean square error in the estimate of the
variables of interest As in a l l problems with such combined performance
16
criteria most of which are direct extension1 of the original concepts of optimal control relative weighting parameters are required amongst the cost and estimation error terms to make the criteria adjustable to the needs of a specific problem (see Bryson and Ho [26] or Athans [10] regarding the concepts of optimal control See Athans [7] Kramer and Athans [73] Athans and Schweppe [12] Meier et al [92] Shoemaker and Lamont [117] Cooper and Nahi [27] Sauer and Melsa [107] Vande Linde end Lavi [125] Kushner [75] Sano and Terao [106] Dreyfus in Karreman [70] and particularly Aoki and Li [5] for examples of work in the area of optimal system design with measurement cost) The choice of such weighting parameters inevitably complicates the measurement system deshysign problem Particularly in applications in the environmental area combining the minimization of costs associated with measuring a process with the minimization of a measure of the errors made in the estimation of the variables in that process does not seem to address the correct problem In any practical implementation legal limits would be placed upon estimation errors allowable in the pollutant estimates On the other hand the use of a combined performance criterion typically admits arbitrarily high estimation error levels at certain points in time since the objective of the optimization is to minimize the time integral of the performance criterion not its instantaneous value Thus the minimization of a performance criterion involving the time integral of a weighted combination of measurement cost and estimation error is not solving the right problem in the context of an environmental monitor
Thus the separation of the optimal monitoring problem into the problems of optimal design and management leads to a problem structure which conforms better to the requirements in actual applications than
17
do those which come from the application of principles of optimal conshytrol with combined quadratic performance indices
If at all measurement times the cost of making a measurement of a given quality is a constant then the total cost of the required monishytoring program over the time interval of interest is directly related to the number of times a measurement of a given quality has to be made scaled by some cost weighting factor which is typically a function of the accuracy of the measurement instrument involved Roughly speaking then the total cost of the whole monitoring program is an increasing
function of the total number of individual samples which must be taken over the time interval of interest in order to maintain the value of the selected estimation error criterion within its bound over that entire time interval With this assignment of measurement cost as a function of measurement instrument accuracy then the two optimal monitoring probshylems to be considered in this study are defined as follows
The Optima] Monitoring Problem of the First Kind -Find the optimal number and quality of measurement deshyvices their optimal locations in the medium and the opshytimal measurement times such that the total cost for the measurements required to maintain the estimation error in the state of system below a given bound over the time interval of interest is minimized (27)
The Optimal Monitoring Problem of the Second Kind -Find the optimal number ana quality of measurement de-vices their optimal locations in the medium and the opshytimal measurement times such that the total cost for the measurements required to maintain the maximum estimation error in the pollutant concentration anywhere in the meshydium below a given bound over the time interval of inshyterest is minimized (28)
Notice that in the above problem definition the choice of model complexity for use in the monitor - the order of the model and perhaps certain aspects of its structure mdash has been excluded It is reintroshyduced later in Chapter 6 in a sensitivity analysis of monitor performance
18
as a function of the number of normal mode states retained in the series solution approximation for the dynamic equations involved
In what follows the problem stated in (27) or (28) are equivashylents referred to as the optimal monitoring problems with bound on error in the state or output estimate respectively
The next chapter considers normal mode models for pollutant transshyport which result in sets of first-order ordinary differential equations of the initial value type these are commonly known in system theory as continuous-time state equations (see Desoer pound32])
In Chapter 4 these continuous-time state equations are discretized in time (see Freeman [41]) for computational implementation and for use in the Kalman Filter in the optimal estimation problem In Chapter 5 attention is finally returned to consideration of the monitoring problems stated above
19
CHAPTER 3 NORMAL MODE MODELS FOR DIFFUSIVE SYSTEMS
The transport and dispersal of a particular pollutant in some reshygion P can be described by the following partial differential equation
K = 5 F + p P $ F laquoF + f + 9 O-1) where
F = mixing ratio of pollutant (grams of pollutant per kilogram of medium)
f = gradient operator y = local velocity of medium
p = mass density K = diffusivity coefficient
a = scavenging rate coefficient
f = stochastic pollutant source term (grams pollutant per unit time per kilogram of medium)
and finally g = deterministic pollutant source term (same units as f)
The terms of the right-hand side of (31) represent respectively (1) forced convection (or advection) (2) Fickian diffusion (3) environmental degradation (or scavenging) of pollutant from the region (4) stochastic and (5) deterministic pollutant production within the region
For some environmental media particularly the atmosphere the propshyerties p and K vary in space and time In some cases (31) will not be an accurate description where K may also vary with direction of diffusion andor the scavenging term may require a far more complicated description The above equation describes the transport of only a single pollutant species F if more than one pollutant is being considered an equation
20
like (31) is required for each one where more terms may be necessary to describe chemical reactions among the various pollutants if they exist Another case where (31) may be an incomplete description is with a meteorologically or hydrologically active pollutant one which can change the energy balance of the medium an example is a pollutant whose presshyence effects optical properties within the region For this latter case the full enevgy and momentum equations of fluid mechanics must be augshymented to (31) to complete the mathematical description of pollutant dispersal [3536] Thus modeling pollutant transport in general is seen to involve a great deal of analytical difficulty
While approaches to the solution of (31) typically evolve from the use of finite difference methods [808199] the extensions of modal analysis techniques proposed by Young [131] to pollutant transport probshylems will be used in this study The powerful results which come from the application of normal mode analysis are felt to extend directly to finite difference models as will be suggested at the end of this report thus use of normal mode models is not a real restriction
In order to gain insight Into the mathematical relationships involved in monitoring the dispersion of pollutants in time and space consider a more tractable simplified version of (31) namely
| | = wh - a + f + g (3)
where 5 - concentration of pollutant (grams of pollutant per
cubic meter of medium) The simplifications adopted in using (32) 1n place of (31) include the following mass density p is assumed to be constant which allows the use of concentration instead of mixing ratio as the dep3ndent variable
21
when the fluid can be assumed incompressible spatial variation of the diffusivity K is negligible and advection is dominated by diffusion as the principle mechanism of transport
Since (32) is linear in pound and since the main emphasis of this study iraquo upon the stochastic nature of its solution the deterministic source term may be eliminated since its effects could be added later to the stochastic solution by the method of superposition The result is
fsect = ltregh - a + f (33) This equation forms the basis for this study It is the stochastic difshyfusion equation including scavenging written in arbitrary coordinates (it should be noted that (33) equally well describes stochastic heat transfer in solids including radiation to the surroundings)
The above assumptions mean that applications of the results which follow to problems in atmospheric pollution are remote at best However (33) is sometimes used for long time scales in global atmospheric studies (see references cited in [131]) In such cases C is interpreted as the pollutant concentration averaged over mixing times sufficiently long that local wind velocities can be viewed as small scale effects of large scale eddies However application of the results to be developed around (33) are thought to be possible in groundwater systems or thgtse surface water systems for which local velocities are small
It should be noted that spatial variation in the density and difshyfusivity can be reintroduced into the problem to extend the results of this work to inhomogeneous anisotropic regions This can be done by dishyviding the region P into component subregions in each of which the asshysumption of constant p and K Is a reasonable approximation Young pound131]
22
has shown that by coupling such component submodels together low order models of relatively high accuracy are able to be formed
For now ignore the inclusion of poll tant scavenging in the transshyport equation It will be introduced later as 1t effects the results for the optimal monitoring problem for diffusive transport alone in Chapshyter 5 Thus with this final simplification the stochastic partial difshyferential equation governing Fickian diffusion results
|| = K7 25 + f (34)
Various methods exist for solving (34) but owing to its simplicity and useful areas of application the method of separation of variables will be used to convert (34) into an infinite expansion of ordinary difshyferential equations ir time whose solutions multiply related eigenfunc-tions in space Study has been made of the number of terms to retain in the expansion for adequate accuracy [131] Determination of this number will not be of concern here though its importance will be demonstrated by example in Chapter 6
Development of a finite set of continuous-time state equations of the form
amp = ampS + B (35) y = Cx + V (36)
from the application of the method of separation of variables to (34) is followed by developments for problems with media of various dimensions in the remainder of this chapter More rigorous theory regarding the separation of variables technique 1s summarized and referenced in [131]
23
31 Separation of Variables for the Diffusion Equation
Here the solution of the inhomogeneous stochastic di f fusion equation
(34) in arbi t rary coordinates is expressed as a f i n i t e set of normal
mode state equations of the form (35) with the use of the method of
variatiOTi trf parameters fcee Berg and fttftrego-r [ I S ] p 152)
Begin by considering the homogeneous counterpart to (3 4) namely
sectsect = KV2C (37)
Assume a solution for of the form
5(Pt) = x(t)e(P) (38)
where P is some point in the medium P Substitute th is into (37) to
obtain
x(t)e(P) = Kx(t)72e(P) (39) or
m=^- raquobullraquogt The left-hand side is a function of t and the right-hand side is a funcshytion of P so that for arbitrary P and t both must equal a constant the so-calle separation constant or eigenvalue Choose this constant to be -X so that the following separated equations result
i(t) + Xx(t) = 0 (311) V 2e(P) + | e(P) = 0 (312)
The equation in time (311) Is already seen to be in the form sought 1n (35) The spatial equation (312] 1s the Helmholtz equation which together with the boundary conditions for the medium forms an eigen-problem over P the region of interest The resultant eigenfunctions e (P) can be used to form bases for solutions of (37) assume a solution of the form
24
C(Pt) = 2 ^ x n(t)e n(P) (313) n=l
Substitute this into the inhomogetieous diffusion equation (34) to obshytain
oo oo
) i n(t)e n(P) = K ^ x n(t)7 2e n(P) + f(Pt) (314) n=l n=l
The eigenfunctions are distinguished by the property of orthogonality which can be stated as
[ 0 n + m ebdquo(P)em(P) dp = (315) rebdquo(P)em(P) dp -
n = m the integration occurring over the whole region P Use th is property in
(314) together with (312) to obtain
E i n ( t ) 1 e nlt P gt e n P gt - - laquo ] [ M ^ e n lt P V P gt d
+ f (P t )e m (P) dp (316) JP
The orthogonality then reduces (316) to the following set of first order ordinary differential equations
+ I f(Pt)ebdquo n(tgt deg -xM + I W^K^ dp (317)
The integral in (317) is the contribution to the nth mode due to the source term f(Pt) If f(Pt) can be expanded in a series of eigenfuncshytions it can be given by
25
f(Pt) = ) f n ^ n ^ - ( 3- 1 8 )
Multiply by e m(P) integrate over the region and apply orthogonality again to obtain
f fn(t) = f(Pt)en(P) dp (319)
Jp
where fbdquo(t) is the modal input for the ntjn_ differential equation Thus wit 19) (317) may be written in the compact form
xbdquo(t) = - y n ( t ) + f n(t) n = 12 (320)
This infinite sequence of ordinary differencial equations is known as the set of normal mode state equations and together with the mode shapes given by the eigenfunctions e n(P) they comprise the normal mode solution in (313) of the inhomogeneous diffusion equation (34)
The remainder of this chapter will concern forms for the eigenfuncshytions e (P) the spatial side of the problem This will involve solving for the eigenfunctions once the coordinate systems are specified and boundary conditions given Thus finding e n(P) the eigenvalues n and solving for the source terms fn(P) will be considered next for a range of different problems Solving for the time response x (t) will be apshyproached in Chapter 4
32 One-Dimensional Diffusion
Here w i l l be considered the problem of di f fusion in a one-dimensional
medium Classical ly th is is the problem of heat conduction between two
i n f i n i t e paral lel f l a t plates The problem also embraces that of po l lu t shy
ant d i f fusion where d i f f u s i v i t y constants dominate in one coordinate
26
direction only Consider then the system described schematically as
follows
bullgt f rtrade w l
^1 Sources f rtrade 1 r 1 t ~ J
Measurements
2 f
2L gt
- i gtJ Measurements
2 f
- 2 laquo^ 2 f
Figure 31
321 No-Flow Boundary Conditions - For the system of length 2L
described 1n Figure 3 1 the following specifies the related i n i t i a l -
boundary value problem
Bpoundjfcjabdquo K 3fpoundi5ja t f ( 2 l t t g ( 2 gt t )
dz-
gjC(0t)=0 5fc(2Lt)s0j
CUO) = bdquo
f^zt) ^ W l ( t ) ^ z - zw y
E[w(t)j = 0
EJytJw^T)] = W6(t - T)
f 2 ( z t ) H bdquo 2 ( t ) laquo ( z - z W z )
E w 2 ( t f = 0
(321)
(322)
(323)
(324)
(324A)
(324B)
(325)
(325A)
27
Erw2(t)w2(T)J = W2 laquo(t - T) (325B)
g i ( z t ) = u^t) oz - z u (326)
Thus the system represents diffusion in a one-dimensional medium of
length 2L and diffusivity K with no influx or efflux of the diffusing
substance at the ends The in i t ia l condition throughout the medium is
chosen as a constant 5 Q There are two stochastic point sources f j at
z = z and f at zbdquo with zero means and constant covariances given by W-l lt- Wn
W and W respectively One determnistic source of strength u^(t) acts
a t z - y Measurements y j ( t ) and y 2 ( t ) are taken at points z 1 and z Expresshy
sions ior these measurements in terns of the resulting system of normal
mode state variables are sought
As in (313) begin the analysis by assuming a solution of (321)
of the form CO
pound(zt) =2__ x n(t) cos ((n - 1) j f z) (327) n=l -
Substitute this into (321) to obtain
xbdquo(t) cos ((n-Dfz) n=i
n=l + f(zt) + g(zt) (328)
Right-multiply by cos Um-1) - z) integrate over the length of the medium and invoke the orthogonality of the eigenfunctions to obtain
28
2 r2L 2Lx n ( t ) = - (n - D 2 i | | x n ( t ) + f ( z t ) cos ( j n - 1) ^ z)dz
+ g (z t ) cos f ( n - 1) g f z ) dz n = l (329) 4=0
2 f 2 L
Lxbdquo( t ) = -(n - D 2 f - x n ( t ) + f ( z t ) c o s N n - 1) j f z ) dz
+ g(z t ) cos ( (n - 1) j f z)dz n = 2 3 ( 3 3 0 gt 4=0
The above may be generalized into one in f i n i te set of f i r s t -o rder ordinary
d i f fe ren t ia l equations in state-space form f i r s t by making the def in i t ions
n = 1 ^L 2L (n-l) zCTr2
n = 2 3 ^mdash (331)
(n-l)2lt7T2
With these definitions the complete normal mode solution for the one-dimensional stochastic diffusion equation equation (321) may be written as the sequence
n ( t ) = bull rr n ( t ) + r I f ( z t ) c o s ( ( n - ^ i f z ) d z
+ ^ - g (z t ) cos f (n - 1) g f z j d z n = l 2 n 4=0 ^ (332)
Thus the concentration pound(zt) is found by solving the modal equations (332) and substituting nto the ssumed solution (327) To do this
29
the solution must fit the initial condition so that
s0
CO
bull ) x n(0) cos((n - 1) ^ - z )
For this case it is easily seen that
x(o) = e 0
x n(0) = 0 n = 23
(333)
(334)
Point sources are the most straightforward types of inputs to represhysent in normal mode form (see Mac Robert I 8 2 ] p 124) The stochastic and deterministic sources are transformed as follows
2L
z=0 f^zt) cos ((n - 1) gf z)dz
-r (t)laquo(2-zH)cos(n-l)fz)dz
i(-raquopound) w(t) n - 12 (335A)
Similarly for f(zt)
-2L J - j f 2 (z t ) cos ((n - 1) 2Tz)dz
n -4=0
c i c o s f t n - l j ^ z ) w ( t ) n 12 (335B)
The deterministic term is
30
J- g(zt) cos((n - 1) z) n -4=0
dz
- | ^ c o s ( ( n - l ) ZL z u J u ^ t ) n = 12 (336
If the infinite series in (313) and (327) are truncated after term ngt the retained modal equation may be written as follows
0 deg Kit
O -lt-D2
1 traquo (ltraquobullgt if s )
(337)
bull with initial condition x^O) x7(0)
xbdquo(0)
(338)
The noise-corrupted measurements
1 c o s ^ z ) cos ((n-1) ^ Z l )
1 c o s ( z 2 ) cos((n-l)jf2 z) (339)
31
In summary the stochastic initial-boundary value problem (321) - (326) las been transformed through the method of separation of variables into a truncated sequence of first order ordinary differential equations (337) with initial conditions (338) Measurements made of the system are exshypressed as in (339) These equations comprise the state and output equations which may be written as
x = Ax + Dw + Bu (340)
y = S + v (36)
As in equation (34) most of the examples of interest here will exclude terms like gu in (340)
Once the truncated sequence of normal mode state equations is deshytermined the resulting pollutant concentration at any point z in the medium for any time t may be found as follows
e(zt) = Y x n(t) cos ((n - 1) |f zj ( 3 4 1 )
Finally insight into the structure of the finite normal mode model of the one-dimensional diffusion process may be gained by portraying relashytionships (337) (338) (339) and (341) in a bond graph [69] see Figure 32 The table at the bottom of the figure defines the functional relationships involved in the coefficients b c and d these are in actuality all modulated transformer elements
32
DETERMINISTIC b SOURCE
1
1 tt
1 -Hyendeg 1 trade NOISV
MEASUREMENTS
A h H yen 0
bdquoltbull
bull laquo ^ 5 ^ 7 l rs ((bull ) f((-gt5f-0 raquoraquo(laquobullI ffr) I ((-I) ^i)
Figure 32 Bond graph of normal mode state measurement and output equations used In the monitoring problem
33
322 Fixed Boundary Conditions - Consider the initial-boundary value problem
M | laquo t i K pound s ^ t i + f ( z gt t ) C 3 i 4 2 )
UOt) = 0 6(2Lt) = 0 (343) S(z0) = 0 (344) f(zt) = w(t)6(z - z w ) (345)
E[w(t)] - 0 (346) E[w(t)w(t)] = WS(t - T ) (347)
The essential difference from lthe problem in Section 321 is in the nature of the boundary conditions The so-called fixed boundary condishytions of (343) are referred to as the Dirichlet conditions by others (see Berg and Mc Gregor [18] Section 36) They represent the physically rare situation where the pollutant concentrations at the ends of the medium are fixed to some specified source levels as functions of time here those levels are arbitrarily chosen to be zero This difference manifests itself in the form for the eigenfunctions e (z) and eigenshyvalues x n
In this case assume a solution of (342) of the form
C(zt) = ) x n(t) sin (n bullpound z Y (348)
Substitute (348) into (342) r ight mult iply by sin ( m ^ f z ) integrate
over the length of the medium and invoke orthogonality to obtain
2 f 2 L
L n t ) = - n 2 bull x n ( t ) + f ( z t ) s i n ( | | pound z) dz (349) Jz=0
34
As before generalized modal resistances and capacitances may be defined n = 12
4L T~ST iTKir
Thus the general modal state equation 1s
(350)
Vgt - bull i bullltgt+ J_ fltzlaquogts1n ( n poundz)dz-(3-51gt The general solution (348) must satisfy the initial condition or
00
e(zo) = o =2_ V 0 ) s i n ( if z C 3 5 2 )
from which n=l
xbdquo(0) = 0 n = 12 (353) The stochastic forcing term 1s treated in a manner similar to (335A) for the case with no-flow boundary conditions
If the Infinite series in (348) is truncated after tern n the fishynite set of normal mode state equations results as follows
lb
o
44 o
laquo bull $ [bullsin (ST)
raquoltt) (354)
Note that the major difference in the dynamics between systems with no-flow at the boundaries (as In Section 321) and systems with fixed boundary concentrations (as in this section) is In the first element of
35
the matrix A In the former it is zero in the latter it is less than zero This implies that the initial condition of the first mode of the problem with no flow at the boundaries will remain unchanged in time whereas that of the fixed boundary concentration problem will vanish for large time This difference is central to the considerations of Chapter 5
33 Two-Dimensional Diffusion
Consider the diffusion of a pollutant in a thin flat three-dimenshysional volume For simplicity consider the region to be of rectangular shape with sides of lengths 21^ 2L 2 and 2L 3 in the C 5 Zraquo a n d 3 c o ordinate directions as shown in Figure 33
Figure 33
If the vertical height 2L 3 is small in comparison to the horizontal dishymensions 2L 2 and 2L 3 the gradient of the pollutant concentration In the C direction can be neglected so that the average concentration In the vertical direction can be assumed for the concentration throughout the vertical dimension for any horizontal location
36
Two dimensional di f fusion applies to such a simpl i f ied model Conshy
sider the case of di f fusion in a homogeneous medium with no-flow boundshy
ary conditions and with r stochastic point sources at various locations
in the medium The init ial-boundary value problem in two dimensions may
be wr i t ten for th is model as fol lows
3 2C(gt) 3 2 5U t ) N
H ( S t ) at
36(Ct)
1
3euro(t)
t) bdquoVg(pound
1 raquolaquo1 + f ( s t ) (355)
0 5 = 0 1 = 2 L r
- g ^ mdash - 0 C2 = 0 5 2 = 2L2i (356)
pound(50) = pound 0 (357)
E[w(t)] = 0
E t y U J w ^ T ) ] = W^t t - T ) 1 = 12 r (358)
The no-flow boundary conditions (356) correspond to the case which has interesting practical applications where many such models may be coupled together to span a larger possibly inhomogeneous region The initial pollutant concentration throughout the medium is chosen to be a constant in the initial condition (357) for simplicity r individual stochastic point sources each located at I = c I are described by the ~ wi [ w i wi^J relationships in (358)
The separation of variables of this two-dimensional initial-boundshyary value problem proceeds much like the one-dimensional case However in this case owing to the inclusion of two spatial dimensions the
37
eigenfunctlons 1n the general case (313) w i l l be products of independent
functions of the two space variables as follows
laquolaquonltSgt E en(laquolgtemltS2gt c o s (J 1 5q-laquo l ) c o s ( ^ h ^ ( 3 - 5 9 )
Thus assume a solution for (355) of the form
5 ( ~ C t ) L L x nm ( t e trade ( pound )
n=l m=l
= Z J Xtradegt(t) cos ( J - gt 217 1 ) ( j 1 1 ^ ^ lt 3 - 6 deggt This is a direct extension of the one-dimensional form in (327)
Applying the same techniques used in the one-dimensional problem leads to the following resultant normal mode problem formulation for the two-dimensional case (for details see Voung [131] p 76 Duff and Nay-lor [34] p 148 Mac Robert [81] sect 13 and particularly Berg and He Gregor [18] Chapter 10)
Define the generalized modal resistances and capacitances v and C as In (331) where v 1s either n or m as in (359) and u 1s either 1 or 2 to correspond with coordinate Ci or cbdquo as follows
R v C v
v = 2 3
2 L U
v = 2 3
(v - 1 ) Z L T I 2 2 L U
v = 2 3 (v - I )2KTT2
2 L U
(361)
As in the one-dimensional case substitute the assumed solution S(jt) given in (360) into the differential equation (355) right-multiply by eigenfunction e U ) integrate over the medium and use orthogonality
38
Transform the i n i t i a l condition (357) in a manner similar to (333) and
(334) and the set of igt stochastic point sources as was done in (335A)
Truncate the double- inf in i te series solution in (360) to include n terms
in each coordinate direct ion in order to obtain the following f i n i t e set 2
of n normal mode state equations
11
21
x n x21
X l bull -feyen7) nl
x l 2 bull(yen7 + yenF) 12
m 0 - ( bull ) xnn
1717)() i ^ - c ) ^ ^ ^ ) -
laquopoundcos ( F S) yenTeos ( )cos (fc S j
^-^)r)-fgt^0
w(t)
w 2(t)
raquobdquo(t)
(362)
with initial condition given by
39
Xbdquo10) x 2 1(0)
Vllt 0 )
x2(0)
x (0) o
(363)
For m noise-corrupted measurements y = Cx + y (36)
as in the one-dimensional case the measurement equation is written as follows
(D(i) raquoraquo(j^raquo2l)ltraquo(5q)
^bull )5frlaquoi) c 0 ( lt ^S)
Lw bull i
gt 2 1it)
bull
2
v
(364)
In the state equation (362) the position of the i t | i point source is
written as
(365)
where the components in each coordinate direction and c are as in
40
Figure 33 Similarly for the jth measurement position in the measureshyment equation (364)
i 5 gt (366)
also as shown in Figure 33 (do not confuse the subscript j with time indices used in later chapters here locally z^ means the vector of the coordinates of the jth measurement position)
The result is that the two-dimensional diffusion problem results in sets of normal-mode state and measurement equations which are directly related to those in the one-dimensional problem The only differences are that here SHOTS of the eigenvalues occur in the diagonal A matrix and products of the eigenfunctions occur in the C and D matrices The order of the system ie the number of states retained goes as the product of the number of modes retained in each coordinate direction Thus for the same number of modes n for each coordinate to obtain accuracy in the solution comparable to that for n modes in the one-dimensional prob-lem a total of (n) modes must be included in the two-dimensional model Dimensionality thus grows as the number of modes in one dimenshysion to a power equal to the number of space coordinates describing the domain of the medium in the problem
34 Three-Oimensional Diffusion
The results for the two-dimensional case can be extended directly to three-dimensional regions In applicable coordinate systems (see refershyences listed in Section 33 for conditions under which this extension is possible) In this case solutions may be assumea to be products of
41
eigenfunctions in the three spatial coordinates and may be written degdeg to traquo
( 5 t = L Z L x i w r ( t ) e n^lgt e bdquoA 2 gtM 3gt- lt 3- 6 7gt n=l m=l r=l
TII details of the development are identical to those in the two-dimenshysional case and lead to the same forms for the A D and C matrices in (362) and (364) except that the diagonal elements of A are sums of eigenvalues for eigenfunctions in three not two coordinate directions and the elements of D and C are triple products of the one-dimensional eigenfunctions Dimensionality of the resultant system of state equations goes as (rc)
Three-dimensional examples are included in the discussion of monishytoring systems in Chapter 5 where the development is carried further
It should be pointed out that the method of separation of variables used in normal mode analysis applies in other coordlante systems as well (eg cylindrical and spherical) See any of the references cited in Section 33 for their development
42
CHAPTER 4 MODEL DISCRETIZATION AND APPLIED OPTIMAL ESTIMATION
The purpose of this chapter 1s two-fold First the continuous-time normal mode state equation models of Chapter 3 are transformed into disshycrete-time recurrence relationships for use in the Aalman Filter The statement of these discretization methods is separated from the continushyous-time model development of the previous chapter since they stand alone and can be applied to a variety of modeling techniques which reshysult in systems of first-order ordinary differential equations In addishytion to the normal mode modeling techniques developed above they would for example apply equally well to uncoupled differential-difference models resulting from applying modal analysis [79] to finite-differshyence models [47] or to models resulting from using collocation methods [94] Thus the discretization methods outlined here are general and form a logical connection between the more familiar theory of continuous-time dynamic processes commonly associated with distributed system modelshying and the theory of discrete-time dynamic systems where the majority of applications have been limited to the fields of control system and aerospace system analysis and synthesis
Second the optimal estimation problem is defined and its solution with the Kalman Filter is stated While details of its development are referenced in the literature a concise summary of an algorithm combinshying the simulation of the response of the model of a physical process with all necessary calculations for the optimal estimation is included at the end of this chapter
43
41 Discretization of the System Model
411 The System Model Equations - The systems under considerashy
t ion are typ ica l ly modeled with sets of continuous-time f i r s t -o rder
ordinary d i f fe rent ia l equations of the form
x = Ax + Bu + Dw (41)
y = Cx + y (42)
where the etatietios of the i n i t i a l state x (0 ) disturbance vi(t) and meashy
surement error v ( t ) are given by
E[x(0j ] = m 0
E[x(0)x(0) T ] = M 0
E[w(t)] = Q
E[w(t)w(x)T] = W(t)6(t - T ) (43)
E[v(t)] = o
E[y(t)v(T)T] = y(t)s(t - x)
E[x(0)w(t)T] = 0
E[x(0)y(t)T] = 0
E[w(t)v(T)] = 0 (43)
The discrete-time counterpart of the above is
~ X K+1 = SW^K + ~ J K+1 + raquoK+1 W-laquo)
K+1 = SK+I^K+1 + X K +1 bull W-Sgt
where the dr iv ing functions are defined by
44
J^+l raquo(t K + 1t)B(t)u(t) dt (46)
~K+1 K+1
j(t K + 1t)D(t)w(t) dt C47)
These two terms are convolutions of the deterministic and stochastic inshyputs and ) the state transition matrix defined by the matrix differshyential equation
I = Araquo (tt) = I (48)
In the above the system matrices A B C and p may be functions of time For the time-invariant case however certain simplifying obsershyvations and approximations may be made Let the time step be fixed ie T = (tv+i (bull) a n d obtain (see Appendix A)
amp1 MlVTV-efiT-I+AT + p - t ^ j mdash (49)
-K+l I)AB
T ( I + 2T CA1) + 57 (AT)2 + )sect (410)
= T(J + 2J-(AT) + 3I (AT)2 + )D (411)
With these definitions i t is possible to discretize the problem which
results in a form necessary for the Kalman Filter The discrete form of
the state equation becomes
K+1 amp1laquoK + amph + poundK+SK- ^ J 2
45
Here it is assumed that the input terms u K and w are sampled at time tbdquo and held constant over the interval ti t lt tv+i t n a t isgt
u(t) = u(t K)
laquo(t) = w(tK) t K lt t lt t K + r (413)
This assumption reduces the calculation of the convolutions for u bdquo + 1 and
w K + in (44) given by pound46) and (47) to the far simpler matrix-vector
mult ipl icat ions in (412) above This is possible since the matrix ser-
ies for K and r pound + in (410) and (411) are analy t ica l ly exact expresshy
sions for the convolutions when the variables are sampled and held as in
(413)
The matrix series in (49) - (411) are c lear ly impossible to evalushy
ate exactly The truncation of those series to a pract ical balance beshy
tween accuracy and computational load has been suggested by H M Paynter
(see Brewer [ 22 ] Ch 8) The number of terms k retained in the series
is found as a function of the maximum size of the elements of the matrix
[AT] A bound on the size of the remainder in the series is used to deshy
termine where the series should be truncated Standard integration
techniques (e g Runge-Kutta or l inear multistep methods) are not used
here under the assumption that i f the time stepsize T = ( t j + - t K ) is
su f f i c ien t ly small smaller than the smallest character ist ic tiroes in
the system response then the accuracy of the truncated series approxishy
mation w i l l be suf f ic ient for the purpose of th is study
46
412 The System Disturbance Stat is t ics - I t can be shown
(Jazwlnski [65 ] p 100) that the convolution w K + 1 of the stochastic
variable w(t) in (47) 1s i t s e l f a zero-mean white Gaussian sequence
with covarlance matrix given by
0 K + 1 1 K+1
= I ( t K + 1 t ) 0 ( t ) W ( t ) D ( t ) T 5 ( t ^ t ) 1 d t (414)
This term represents the increase in uncertainty in the estimate of the system state over the time interval T = (t K + - tbdquo) due to the stochastic disturbance term w(t) as in (41) This term is used in the error co-variance equations in the Kalman Filter in the next section
W(t) is a deterministic quantity so the integral in (414) does not involve a stochastic integrand However its numerical integration in general is still far from trivial For this reason a recursive method for the evaluation of amp + 1 will be used a method which closely follows the truncated series approximations for bdquo + + 1 raquo and I V developed in Appendix A
The development of the algorithm to compute Q+ is detailed in Appendix B The method involves differentiating gbdquo + in (414) with respect to time resulting 1n a matrix Riccati equation Hamiltons equations are then found for the Riccati equation which are then solved as a state transition equation Partitions of its state transition mashytrix are shown to comprise the resultant expression for fi An iterative numerical technique (see DAppolito [29]) is used in the actual implemenshytation
47
Suffice it to say here that a method is used to find state transishytion matrices $ and $bdquo (see Appendix B) such that
OK+1 = 2lt T )$22 ( T ) T- lt 4 - 1 5 )
42 Optimal Estimation -The Kalman Filter 421 Optimal Estimation mdash State estimation in dynamic systems
is covered widely in the literature Various developments of the Kalman Filter for optimal estimation can be found in Kalman [66] Kalman and Bucy [69] Sorensen in Leondes [78] Sage [105] Bryson and Ho [26] Heditch [85] Jazwinski [65] and 1n an extensive Bibliography in IEEE [62]
The reader is referred to any of the above for analytical derivashytions of the Kalman Filter equations The emphasis here is upon their implementation taking advantage of properties peculiar to the models being used in this study
The optimal estimation problem and its solution in the Kalman Filter are now described Given is the discrete-time dynamical system described by the following difference equations
raquoK+1 bull K +1K + amp1laquoK + 4lK C416)
K+1 =poundK + 1K + 1 + X K + T laquobullgt
Here x K is an n-vector u an p-vector w an r-vector and y K and v R
raquoi-vectors The vectors x w and v are white normally distributed ranshy
dom vectors with the following statistics
48
ECs 0] = m Q E Xo So 3 gt pound [ K ^ = 2 E KSj = y^Kj
E t y ^ = 2 E K J = Vty
E o KKJ = Q E _5o raquoK = 2raquo
E raquoK l j bull 9-
(418)
A notational convenience will be that for a normally distributed random vector 5 with mean value p and covariance Z pound is described as follows
K N(uZ) (419) The recursive linear estimation problem for the system above is to
determine an estimate x K of the state x at tj that is a linear combinashytion of an estimate at t| and the measurement y K which minimizes the expected value of the sum of the squares of the errors in the estimate that is that estimate which minimizes
$-$-$bullbull (420)
I t has been shown (see Kalman [66]) that the following comprises a
f i l t e r which generates the best estimate in the mean-square sense of
(420) of the state of the stochastic system (416) - (418)
The predicted error covariance matrix PJ+1 is defined by
K+1 x K
~K+1 K+1 ) (K+1 ~K+lJ (421)
and represents the error in the predicted estimate 3pound + 1
o f X K + 1 a t K+1
based upon measurements up to and inc lud ing y K a t t bdquo and i s given by
~K+1 5K + 1 poundK$K+I + 8 K + r (422)
49
Eg ^ H0- (423)
Note in equation (422) that Q K +i 1s the uncertainty in the estimate due to the stochastic input w(t) acting over the interval tbdquo lt t lt tK+- in the state equation (41) This is discussed 1n Section 412 and at length in Appendix B This is pointed out here since many references for the Kalman Filter assume a discrete form for the stochastic input which 1s sampled and held as in (413) and (416) In those cases the so-called disturbance distribution matrix r+ in (416) comes Into the preshydicted error covariance equation as follows
EK+1 = K+1EK$K+1 + ^ K + l ^ K + T
where Wbdquo is the sampled value of the disturbance covariance matrix W(t) at t = tbdquo in (43) In this thesis since the system being studied is continuous in nature equation (422) will be used instead
The Kalman gain for the optimal filter may be shown to be
K T f K T j 1
-K+1 = EK+l-K+l[K+lEK+l-K+l + -K+lj bull ( 4 2 4 gt
The predicted state estimate at time t K + knowing measurements at times up to and Including t K is
amp1 4l~K + amp1-V lt-25) laquoS = bull (426)
The corrected state estimate at t K + 1 including the measurement at
raquopound bull amp 1 + ~GK+1 ffK+1 fiK+l8K+l] bull ( 4 2 7 gt
time t| + is
50
And finally the corrected error covariance matrix at t bdquo + 1 given statistics of the measurement at t bdquo + 1 is
E pound I bull [l bull - G K + I pound K + I ] E K + I [ I - SK+IpoundK+I ] T + sectK+I~ V K + IsectK + I T - lt 4- 2 8gt
An alternate form of the above can be shown to be
$ 1 - [ l bull e K + ipound K + i ]~ p K + r (4-zraquo)
Each form has Its own advantages as will be shown in the next chapter Note the choices for the initial conditions for the covariance equashy
tion (423) and the state estimate (426) They are precisely those given for the system itself in (418) This 1s the best Information available about the initial state to use 1n the filter It turns out that if knowledge of these initial conditions 1s Imprecise the effect upon the later values of the state estimate diminishes as new measurements are processed
422 Summary of Filter Algorithm - For convenience the system simulation equations and Kalman Filter equations are listed together as in Figure 41
The equations 1n Figure 41 are sufficient to both simulate a physical system((416) and (417)) when the actual system cannot be used and to compute the filter calculations themselves The computational cycle 1s as 1n the figure Time is initialized to zero K = 0 and each equation computed Upon completion of one cycle time 1s Incremented and the recursion 1s carried out again until the final time of interest is reached
SI
K+I = K+I2K + ampISK + TK+ISK- 5O bull N(Sto ftgt (416)
ampi - slampW + 9 m bull E - Ho (422)
^K+1 deg EK+1~K+1 poundK+IEK+IpoundK+I f poundK+IJ (424)
K _ 4K JK VK JO K+1 ~K+1 K + iK+lV 0 3 0
(425)
poundK+I = SK+I^K+I + XK+I (417)
jK+1 _ K - r c Jit -| K+1 K+1 raquoK+1 L~K+1 K+lIC+lJ (427)
Etrade [l - SK+IpoundK+I]EK+I[I - sectK+IpoundK+I] T + S W S K + I sect K + I T (428)
Figure 4 1 System simulation aad Kalman Fi l ter computation
52
CHAPTER 5 OPTIMAL DESIGN AND MANAGEMENT OF MONITORING SYSTEMS
The purpose of this chapter is to propose a method of solution for the monitoring problem as stated in Chapter 2 The models for various processes considered in Chapter 3 are discretized using the methods of Chapter 4 for computation in the Kalman Filter The structure of the filter is studied in the context of the monitoring problem in order to obtain a set of monitoring design and managment equations Properties of these equations are examined in detail to yield the optimal solution for the monitoring problem for the case of time-Invariant systems with constant source and measurement noise statistics and time-invariant estimation accuracy constraint Numerical examples to illustrate the conclusions follow in Chapter 6
51 Monitoring and the Kalman Filter
As stated in Chapter 2 two variations of the monitoring problem arise in practice The first is to maintain the error 1n the estimate of the state of the system beow some bound over the complete time intershyval of interest The emphasis on limiting the error in the estimate of the state arises in the use of that estimate In closed-loop state feedshyback applications where high accuracy in the state estimate is of primary importance The second variation in the monitoring problem is to mainshytain the error in the estimate of the output the system variable itself everywhere in the medium below some bound throughout the time interval of Interest The system variable could be pollutant concentration radiation level temperature etc The thrust behind maintaining high
53
accuracy in the knowledge of the system variable cones with application in the detection problem where it is required to know to some degree of certainty where and when a pollutant concentration exceeds a legal limit
Both of these variants can be approached within the structure of the Kalman Filter As described in Chapter 4 the filter provides an optimal estimate of the state of a linear stochastic prrcess optimal in the sense that the expected mean-square error between the estimate and the state Itself is minimized Thus when taking a measurement of an actual physical system the Kalman Filter uses the information obtained In the measurement 1n the best way 1n order to update the estimate of the state The discrete-time recursive nature of the filter provides a fertile structure from which the solution to the monitoring problem can grow
In either case with a bound on state or output estimate error the basic structure of the problem is the same to take the fewest total number of samples over a given time interval in order to maintain the error in the estimate within some bound This says nothing about the number of samples to be made at each measurement time whether or not that number changes from measurement to measurement whether sample locashytions move from measurement to measurement just that when the time inshyterval is over the least number of samples were necessary to insure the accuracy of the estimate
As summarized 1n Figure 41 the first step 1n the Kalman Filter algorithm 1s to Initialize the estimate of the state vector and state estimate error covarlance matrix (from (426) and (423)) The state esttate and its error covariance matrix are then predicted ahead one
54
step in time 11416) and (422)) Sefore each measurement the Kalman gain 1s computed (424) Next a measurement 1s made of the process Itshyself (417) which starts the correction phase of the algorithm The new information from that measurement 1s used to correct the estimate of the state (427) and the statistics associated with the measurement are used to correct the error covariance matrix (428) Finally the time is incremented and the new corrected values are used to reinitialize the prediction equations at the beginning of the algorithm so that the algoshyrithm may be repeated for the next cycle
This sequence of predicting taking a measurement correcting preshydicting taking another measurement etc was the original calculational form of the Kalman Filter (see Kalntan pound66]) Since then applications to guidance and orbit determination for example have resulted in splitting apart the prediction and correction phase allowing for reshycursive prediction of many cycles before a measurement is taken and its corresponding correction made pound301 [44] [65] Moore [95] has shown how this splitting applies In use of the Extended Kalman Filter in monishytoring system design for nonlinear aquatic ecosystems (see Jazwinski [65] for detailed discussion of the Extended Kalman Filter) Thus separating the prediction and correction of the estimate has been suggested as a beginning for the solution to the optimal monitoring system design and management problems (see Brewer and Moore [24] and Brewer and Hubbard [23])
Suppose then that the Kalman Filter algorithm is initialized as usual but instead of taking measurements at each cycle sampling 1s deshyferred until it 1s absolutely necessary to gain more information about the actual system throufh a measurement in order to mlt- intain the error 1n the estimate within some bound This seems like an approach which
55
would logically lead to the fewest number of samples over a given time interval but in fact the optlmaltty of sampling only at times when the error limit is reached is difficult to prove Since it can be shown that for certain special cases the minimum cost measurement program is to sample only when the estimation error is at its limit assume for now that the optimality of such a sampling schedule extends to all cases in order to proceed in the development of relationships for the optimal deshysign problem defer until later proof of the fact that sampling at the limit is the optimal solution of the management problem
Once the bound is reached it is necessary to take a measurement A major phase 1n the monitoring problem is at hand that referred to as the design problem [24] At a measurement time the design problem seeks to answer the following questions
1) What is the best number of samples to take for this measurement
2) What are the best types of samplers to deshyploy
3) Where are the best sites in the medium at which to locate the samplers
The term bes appears in all these questions but best Is what sense In the context of the monitoring problem here posed best can only mean In the manne- which will lead to the fewest total number of samples being taken over the entire time Interval of interest Thus if the assumption of the previous paragraph is true that is if it 1s optimal to sample at the estimate error limit only then the goal of the design problem should simply be to answer (1) (2) and (3) above such inat the time when the error bound is next reached is maximised Then if at each measurement the time to the next measurement is maximized overall the number of measurement times should be minimized
56
However this doe not take into account changing numbers of samshyplers at various measurements For now ignore this part of the problem in order to establish firm results about the case where the same number of samplers are used at each measurement time deferring until later remarks about the general problem
Thus the result in the solution of the design problem also solves the management problem that of the optimal timing of the measurements With this framework established for solution of the monitoring problem first the case of bound on error in the state estimate is considered then that of bound on error in the estimate of the system variable or
output will be dealt with
52 One-Dimensional Diffusion with No-Flow Boundary Conditions
A most important recent application of normal mode analysis is the bilateral coupling of diffusive elements (see Young [13TJ) Throjgh simshyplifying infinite order normal mode models in a quentitative manner it is possible to approximate the characteristics of an inhomogeneous medium by coupling together homogeneous models This is done by assuming no-flow or Neumann boundary conditions at the junctions and introducing pseudo-sources to account for resultant differences The technique readily extends to multiple space dimensions and is thus very powerful
With the practical importance of this technique established [131J the case of ore-d1mens1onal diffusion with no-flow boundary conditions is a fundamental system to consider 1n optimal monitoring system design and management This case is used as the basis for all the theoretical developments in the following sections For completeness extensions and applications of the results to other diffusive systems are considered in the last sections of this chapter
57
53 The Design Problem for a Bound on the Error in the State Estimate
531 The Infrequent Sampling Problem - In the statement of the recursive linear estimation problem in Chapter 4 the Kalman Filter was stated to be that filter which minimiz 5 the mean-square length of the error vector between the estimate of the state and the state itself of a linear stochastic system That is for all times tbdquo it mirimizes
Notice from (420)and (429) that the covariance matrix is defined by (
EK~K+1 ~K+V~ K+l K+l ltamp]bull lt5-)
that is at time t K + the covariance matrix just after the sample is K+l given by PK+-i- It can be seen from the aDOve that
^K+l bdquo YfcK+l W E ^ x ^ - x K + v ) [ ^ - x R + 1 ) I - T r | p mdash I (52)
Thus in order to minimize the mean-square length of the estimation error vector for a measurement at time t+ that measurement should oe chosen which minimizes the trace of the corrected covariance matrix Thus the choice of a convenient scalar performance index for the probshylem of maintaining the error in the state estimate within some bound is to use the tvaae of the estimation error covariance matrix
Returning then to the requirements of the design strategy of the last section it is necessary to choose a measurernt so that in this case the time when the trace of trie covariance matrix next reaches its
limit will be maximised This might be thought to be the same thing as finding that measurement which minimizes the trace of the covariance matrix at the time of the measurement but as will be seen these are not necessarily equivalent To study the evolution in time of the
58
trace of the covariance matrix repeat the equations for the predicted
and corrected covariance matrices
pK+1 ~K+1
where
[l - sect K + 1 pound K + l ] pound K + 1 [ l - sect K + l S K + l J + 5 K + 1 V K + 1 G K + 1
T (428)
sectK + I - ~ P U K + I [ S K + I amp I S K + I + K + I ] lt 4- 2 4gt Use (424) and (429) to obtain
Note that the two forms for p^Jj (428) and (53) can be shown to be equivalent (see Sorensen [78]) Both are listed since It Is u n shyknown that the former is superior computationally from an accuracy point of view 1n that it tends to preserve the pos1t1ve-def1n1teness of the covariance matrices better (see Aoki [ 3 ] ) but the latter is much simpler to manipulate analytically Thus (53) rill be used 1n all the analysis involved in the solution of the monitoring problem and in any numerical gradient algorithms resulting from that analysis whereshyas (428) vriU be used directly In the filter calculations themselves
To make the problem tractable constrain the range of the problem as follows
Assumption Only systems of the form (340) will be considered tthere the eyetem matrix A aontrol matrix g and disturbance matrix D are all time-invariant and c laquo where the disturbance noise oovarianos matrix W and measurement noise oovarianae matrix V are aonaiant
With this assumption initialize the algorithm at time t Q by setting the
covariance matrix in (422) to tfQ Then predict to time t to get
Pdeg = j H 0 j T + n (55)
59
where the subscripts have been dropped owing to the condition of assumpshytion (54) and $ for a fixed time step Is given 1n (49) Next it is necessary to check to see if the error limit which may be called Tr_ has been reached That 1s 1s
TS lrlim
I f not advance in time to t 2 and predict ahead again
Edeg bull laquoET + 5
Check again
I f not
$ZM$ + 4flraquo + Q (56)
[4 TrIBI gt Tr I i f f l
Edeg - JE 2V bull 0
2 0 2^ T
bull t39(jS3 + S 2S Z + 3 T + 8gt (57) Assume that fter K steps the limit is finally reached From Appendix C (57) can be generalized to the form
bull f sn-VlT eS - raquo bull gt s^V 1 bull (58)
It is now necessary to make a measurement Apply (53) to obtain for the measurement at time t K
Note here that from assumption (54) y 1s a constant thus no subscripts but Q K 1s net Q K 1s what 1s available to change 1n the design of the
60
measurement to be taken It is again to be chosen to maximize the time over which prediction may take place before the limit on the trace of the predicted covariance matrix is reached at the next measurement That is find Q K at time t K such that N is maximized where
DK ANbdquoKN T An-l nn-l T K 1 M
pound K + N EK + gt 4 Si (510)
and (511)
In developing a strategy for the choice of Gi to maximize N the properties of (510) the matrix solution of the linear matrix recurshyrence (422) are now considered Since the recurrence is linear In P its solution may be decomposed into the zero-input response and the zero-state response these terms are more commonly known as the homogeneous or unforced and particular or forced solutions in differential equations or dynamic system theory The first term in (510) is seen to involve only the initial state of the covariance matrix just after the sample at time t K the zero-input response The second term the zero-state response has nothing to do with the covariance at time t K and involves only the strength of the disturbance noise ft An observation can thus already be stated
Conclusion I The selection of C K at time t K to maximize t ^ the time of the next measurement is solely a function of PR and not the forcing function (CI)
This can be seen by rewriting (510) as follows
61
T T pound K + N ( C K ) - J N E pound ( G K ) N + ) n 10raquo B 1 bull (512)
Here it is seen that the predicted value of the covariance matrix at time t K +bdquo is a function of the measurement matrix back at time bdquo However only the first of the two terms in the expression for the predicted co-variance matrix involves that measurement matrix
Thus in order for t bdquo + N to be as large as possible before condition (511) is met it is required that the trace of the covariance matrix at time t K + N be minimized by the appropriate choice of the measurement matrix at time tbdquo This presents a formidable problem in the general case The general solution might be approached through the use of dyshynamic programming or through a direct search algorithm structured as follows
(1) Pick in sone manner Q|q (2) Predict ahead to time t K + N using (512) until (3) Tr[PJlt + N(C K i)] gt T r J i n
(4) Store N in N return to (1) (5) Stop when convergence to largest possible Nj Is assured (513)
Such a procedure could be quite costly to execute since it is a direct search technique rather than a technique for which an analytical expresshysion for the gradient of the objective function cn be found Also each evaluation of the objective function that is the finding of each Nj when (3) 1s satisfied Involves carrying out the solution of the mashytrix equation (422) N ( times (It should be mentioned that since the interest here is only in the trace only the diagonal terms of (422) need be computed each time but this 1s still costly nonetheless)
Since an algorithm of the type In (513) is cumbersome at best seek more concise solutions for the problem in (510) and (511) To do
62
this more information ci the structure of the process Involved Is necesshysary that is more knowledge of the forms of $ and Q Suppose the sysshytem which $ represents is a one-dimensional diffusion process with no-flow boundary conditions see Section 321 for such a system Suppose that the problem 1s formulated in normal modes so that the system matrix from (337) 1s given as
o A =
KIT
o bull lt - I ) 2 F
(514)
Thus for this time-invariant system matrix i ts state transition matrix
for the time step T = ( t K + 1 - t K ) according to (49) is given by
O
pound laquo T
Kn2
T ~~7 4LZ
o -0-1) ^ T
(515)
Notice that with the ordering of the eigenvalues in the system matrix in (514) the diagonal elements of laquo written t^ exhibit the following property
11 raquo 11 1+1 1+1 bull ^ deg l23n-l (5 where n I s t h e number of states retained in the normal mode mode and is thus also the dimension of the square matrices 6 and Choice of
63
a normal mode model has resulted 1n this unique relationship in (516) which allows drastic simplification of the optimization problem in (510) and (511)
Expand equation (510) to obtain
pK
tnlv iwl nraquo1
ML fir1
C517)
From the form of (517) using property (516) shows that for N large
the first term of (510) 1s given by
(518)
1 and j i- 1
64
Thus for N sufficiently ^rge all that 1s left of the homogeneous term 1n (610) at time t K + [ ) U -ie first element of g at time t R This result together with Conclusion I yields
Conclusion II For N large the following are equivalent r bdquo - (1) Find C K which minimizes Tr[EK+N(CK)J i (2) Find CKwh1ch minimizes ^ ( C K ) J CII)
From the discussion just after (512) 1t 1s obvious now that the choice of pound K gt for the optimal measurement matrix at time t K can be stated as
Conclusion III For (Llarge to maximize t|lt+N the time when Tr|E^+H(CK)J gt Tr j i m choose cj at time t K which minimizes ( E R ^ K O H (CIII)
Thus for the asymptotic case of N sufficiently large so that (518) applies within some tolerance level the monitoring problem is solved Such an infrequent sampling program may well apply to many physical sysshytems where the dynamics of the transient response are fast in comparison to the time between samples The above conclusions reduce the monitorshying system design problem to one of minimization of the (ll)-element of P in (59) a procedure for which writing the gradient of the objecshytive function is straightforward
In order to more fully understand the nature of the solution (510) consider the second term the zero-state response in (510) and (517) This term is a matrix convolution of the disturbance covarlance matrix Q and the statf transition matrix 4 As such it possesses qualities of convolutions of other linear processes Write the general element for the second term of (517) as
8 l l 5 l a i j L l W l a n d j ^ l (519) n=l
65
From property (516) 0 gt lt 1 1 + 1 Recognizing the products (ijtj) in the convolution term 1n (517) as conmon ratios in geometric progressions the element of the matrix convolution may be seen to apshyproach the limit
L n d j f 1(5-20)
Thus a l l the elements in the second term of (517) go to steady-state
constants as N gets large except the f i r s t which grows monotonically
as a ramp with slope [ f l j i i
Thus (510) may be wri t ten schematically as
+ pK -K+N
o c a sS
(521)
where the (1l)-elements of the matrices are shown partitioned from all the other elements of those matrices- this 1s a notatlonal convenience used throughout what follows From (521) the simplified relationship for the trace can be written as
[CCeK^^K^NMll^r^J Tr|P^bdquorc^| - |P)(Cbdquo)| + H[BJi + Tr| 8 I- (522)
The meaning of Conclusion II becomes clear In that changing the nature tbdquo by char
only through P K lt G K ) J it at time t K + N Then
(523)
of the measurement at time tbdquo by changing C effects the value of Tr P pound T N ( Q K ) only through P K lt G K ) J f o r N sufficiently large Also say the equality in (511) is just met at time t K + N gt Then
(523) can be used to demonstrate Conclusion III From (520) and with
66
a as defined In ( 5 2 1 ) 1 t Is seen that for various choices o f Cbdquo in SS - K
( 5 2 3 ) T r rn ] remains Invar ian t so long as N remains s u f f i c i e n t l y l a r g e LSSj
Thus In the equality In (523) the f i rs t two terms on the right-hand
side always sum to a constant and as CK 1s chosen to minimize IPKCK)J
N 1n the second term Is maximized Conclusion I I I 1s thus seen to hold
whenever the limit 1n (518) 1s approached
A graphical depiction of the relationships 1n (522) and (523) 1s
shown In Figure 51 In Figure 51A a representation of a typical plot
of the ful l trace of P over tine is shown while 1n Figure 5IB the eleshy
ments of the asymptotic approximation In (522) are drawn Writing the
trace of the matrices In (517) obtain
-W=fe]bdquo+[4^ [44 laquo[laquobdquo bull m2zEfv~) + bullbullbull+ r^yr lt5-24gt
As N grows large (524) t~-t to (522) but during the Initial transient period the last terms of both lines of (524) are going through changes These changes account for the approach to the asymptotic slope near time tu In Figure 51A
Notice how If a different choice of C K results In a smaller value of | P K ( C K ) 1 Figure 5IB that the start of the plot would be transshylated downward with the same offset of Tr[(jJ to result in a longer time
SS interval before the limit Trlim 1s reached again
532 The Effect of a priori Statistics - Choice of H Q and m Q
in the filter equations (416) and (422) has come under considerable study ever since the introduction of the Kalman Filter Much effort has gone Into identifying these terms in actual applications and consider-
67
Tr[ppound+H]
T r [ $
(A) Actual response
Trlpound]
Vim
T-reLj
gt _ T1i
raquo - T 1 M
(B) Asymptotic approximation
Figure 51 Schematic representation of the basic relationships In the Infrequent sampling problem
68
able time spent in assessing the sensitivity of the results to lick of knowledge of the Initial statistics Attention 1s now turned to these topics within the framework of the above results for the case of Infreshyquent sampling
It 1s required to find the effects that various values for M Q the matrix of 1mt1al uncertainties 1n the estimate of the state xX have upon the optimal measurement system design poundbdquo for che first measurement at time tbdquo For the case of bound on (58) It is necessary to sample when at time t For the case of bound on error in the state estimate from
bull [ p 0 K ] c T r [ V T + ^ J n 1 S J n l T gtbull ^ U m - lt 5 - 5 gt
n=l
If K lo sufficiently large at the f i rs t sample so that (518) approxishy
mately applies then (525) may be written as
[]u Mil + T [
s^ l r t i m ( 5 2 6 gt
as 1n (523) Thus only the (lf)-element of matrix H Q 1s of any signishyficance 1n the first sample for K sufficiently large Furthermore sines Tr[ f ] is a constant for various choices of H Q the remaining two
SS terms 1n the left-hand expression of (526) sum to a constant over all choices of M_ To deduce the significance of this write out the mashytrices for (525) in a manner similar to (521)
K Pdeg = $K tyfV 1 (5-27)
n=l for K large (518) allows (527) to be written as
69
]11 K[n ] n 0
pdeg - + +
o O a is
(528)
Note that 1f (520) applies then a par t icu lar ly important result fo l lows
namely that the ( l l ) -element of the predicted covariance matrix at the
f i r s t measurement time is given by
K L K ^ I l laquoSn)= laquowst (529) no natter what HQ may be
For the measurement i t s e l f E K i s used in the following expression
Pdeg - PdegC iyK+v]$- (530)
But from (528) since for K large a is f i xed and since (529) holds is
making the optimum choice C of C^ 1n (530) Is independent of the Inishytial error covariance matrix H Q but directly related toTr which is summarized in the following
Conclusion IV For K large determination of the optimum measurement matrix C K at t K 1s determined by the error limit Trlim and is independent of HQ (CIV)
Conclusion V For K large the only effect (jg has upon the monitoring program is in determining with T r z f m the time of the first measurement t K (CV)
Thus if the constraint T r ^ in (525) Is such that (518) and thus (526) hold choice of the Initial condition for B 0 is of little imporshytance However in practical applications the better approach to the identification of the a priori statistics is to concentrate analytical efforts upon the identification of only the (11)-element of Mg and not ujon identifying the full matrix in cases where the simplifying approxishymations of the infrequent sampling problem apply In this manner a better estimate of the first state should be possible for the same
70
analytical effort leading to a longer time before the first sample is necessary
533 Fixed Number of Samplers at Each Measurement and Fixed Error Limit - Thus far little has been said about the number of sampling devices to be deployed at each measurement time Consider here what happens when the same number of samplers m is to be used at each meashysurement Consider further the case when the error limit placed upon the uncertainty in the state estimate Tr m is the same throughout the problem
Suppose a sample has just been made at time t K In order to study the optimal designs which arise-at different measurement times consider the next two sanples which occur at times t|+N and t K + N + f ) Since T r J i m
1s constant If both N- and N 2 are large in the sense of (518) obtain the following conditions at the two sample times
^ U j ap()] n
+ Wi + T r s f lrnlt r K+N I r K+N lt
gt Tr lim
(531)
(532)
Since Tr[ 8] is the same for both measurements for the case of the
equality in both (531) and (532) I t is seen that
[i$o]n bull W T = p(eK + N l) + NgCfl (533) 11
Now if the full matrices In (532) are written out obtain
r p
K + N l l - PK+N N 2 r s j u
0 1 ^ Jl1 + 1 + N 2
O O ss
(5-34)
71
Substituting N 1 for N in (5211 comparing with C534) and using (533) leads t o
K+N it K + l E K + N = ER+N +N N l a n d N2 s u ^ 1 c 1 e n t 1 y large (535)
Thus the predicted covariance matrices at each sample time must be equal
The corrected coyarJance xoatrices just after both samples magt then he
written from (53) as follows
K+N p -K+N
laquo[c PK C C V T + V T V PK (c (536A) LfK+N^K+N^tyiK+N JJ SK+tl^KtH^K
l + N 2 raquo K+N bdquo K+N bdquo T
l+Nj^K+N+N2 ) EK+NJ+NJ^K+N ) EK+N^NJ^K+N ]poundK+N+N 2
r K+N T 1-1 K+N v [EK+NJ+N^K+NJ+NJI^K+N^K+NJ+NJ + -J ^ K + N + N K + N N J pound K + N )bull
(536B)
By recognizing that the two predicted covariance matrices are equal from (535) equations (536) lead to the most important result for the monishytoring problem
Conclusion VI For the infrequent sampling moni-toring problem with a fixed number of samplers and conshystant error 11mlt the optimal design of the monitoring system - the optimal number of sensors and their placeshyment - need only be done once for the same design is optimal for all other measurement times (CVI)
Also from (535) and (536) can be seen Conclusion VIA In the optim) monitoring probshy
lem measurement times are equally spaced (CVIA) These relationships ara Illustrated in Figures 52A and 52B The firsv curve represents a typical trajectory of the full trace while the second the asymptotic approximation Since P pound + N = E K + N + N bull t h e resulting optimal measurement matrices pound K + N and C K + N + N must be the same
72
r K + N I T l ~ p
r + +
^mdash Time
N [g]
(B) Asymptotic approximation
Figure 52 The infrequent sampling problem with fixed number of samshyplers and constant error bound
73
534 Variable Number of Samplers - The case where the number of samplers to be deployed at each measurement time may vary 1s 1n general quite difficult However in cases where (518) applies the case of infrequent sampling results can be obtained If the error limit Tr is constant over the time interval of interest then the result derives immediately from Conclusion VI
Conclusion VII For the case of infrequent sampling the optimal number of samplers to use may be found by reshypetitively solving the optimal design problem for CJJ at the fi rst measurement over the range of gt=1 tc m-n sam-plers then extending the results over the full time intershyval to find which C^ as a function of m leads to the fewshyest total number of samples The optimal number of samshyples to take at each measurement time is the same for all measurement times (CVII)
Thus for infrequent sampling the optimal number of samplers to use is seen to be constant at each measurement and that optimal number can be found in a computationally straightforward manner at the first measureshyment time
Even though the optimal number of samplers to use at each measureshyment is a constant it is important to note that at any specific sample time the optimal number of samplers to use is independent of the number used in the other samples This can be seen by comparing (531) and (532) as was done in (533) If m samplers had been used at time tbdquo
in the left-hand side of (533) m+ could have been used at time t K + bdquo in the right-hand side Since for the case of the equality the two suras in (533) must be equal if the dimension m K of the measurement on the left-hand side were smaller than u+u on the right-hand side then in general P K would be larger than PixJ a n d simultaneously N smaller than N Thus in the case of infrequent sampling at the sample time t K + N in (531) the value of the covariance matrix Ppound +bdquo for use in (536A) to determine C^ + N at time t R +bdquo is no longer truly a function of CJ nor
74
of mK Its dimension This 1s so since the sumnEjSCcj) + f t g^ - l in
(531) is a constant i f CjS changes so wil l N to maintain the sum at
that constant Thus since Trig] in (531) 1s fixed and since the SS
Cher two terms form a constant the trace Tr K 1 ~K+Ni o n t h e l e f t - h a n d
side is determined only by the error limit itself T r ^ Hence P pound + N
for N- large does not directly depend upon C K even though such a funcshytional relationship is implied by writing P pound + N (cpound) Thus various numshybers of samplers could be used at different sample times However it is only in considering the solution over the full time interval of inshyterest that the overall optimum is seen to be the use of the same number of samplers at each measurement This concept is demonstrated at length in the example in Chapter 6
535 Analytical Measurement Optimization - Thus far the optimal monitoring problem posed in Section 52 socialized to the casii of bound on error in the state estimate has been found to be equivalent to the minimization of Pj^(CK) as a function of Q K in Conclusion III Little has been said however about the actual determination of ct the optishymal choice of Cbdquo which minimizes the objective function Pu(Cbdquo)
~K L~ KJn As is well known analytical methods of obtaining extrema are supeshy
rior to numerical methods wherever analytical methods exist (see Beveridge and Schechter [20]) Analytical solutions to extremization problems usually exist however only for very special cases A fortushynate situation arises in the present case since some work has already been done in dealing with extrema and derivatives of the trace functional (see Athans and Schweppe [11] and Athans [8 ])
Pursue an analytical solution of the optimal design problem which with the simplifications of Conclusion III may be stated as follows
75
Find the optimal measurement matrlc C K such that lE^K^n 1S m1n1m1zed- C 5- 3 7)
This Is minimization of the first element of the corrected covariance matrix after a sample at time tbdquo over all choices of possible measureshyment matrices C K Analytical methods exist for approaching an allied problem which may be stated as follows
Find the optimal measurement matrix C K such that Trrj^(CK)] is minimized (538)
As shown in Conclusion II these are not the same problems (538) is minimizing the trace at the time of the eample whereas by Conclusion II (537) is equivalent to minimizing the trace for times far beyond the
aample time However techniques for the solution of (538) could prove to be applicable to (537)
Motivated by the computational efficiency of an analytical solution an attempt is thus made to solve
3 7 TK)]-9- lt 5- 3 9gt The notation in (539) means taking the partial derivative of the trace of P K ( pound K (a scalar) with respect to pound (a matrix) This concept has been developed by Athans and Schweppe [11] and applied to a similar probshylem by Shoemaker [117] In order to find the stationary matrix solution of (539) extensions of concepts of finding extrema in ordinary calshyculus are made to the case of scalar valued functions of a matrix
Consider the system starting at time t Q For a measurement at time t K seek C K such that using (59) in (539)
76
As detailed in Appendix D the result is
C = 0 (541)
This can be seen to correspond with the case of taking no measurements such that the extremum found in (540) is actually a maximum not a minishymum An initial attempt was made at constraining the range of C in such minimizations with the method of Lagrange multipliers with no success
more study is still needed of such analytical techniques One study is currently underway by Shoemaker I117J in which restricted classes of probshylems are treated through the use of analytical techniques such methods were not found to be appropriate for use in this study since they require n measurements at each sample time a severe restriction
Alternate performance indices to that used in (540) yield matrix equations whose solutions are not known so that the analytical approach with the trace function is not found to be fruitful see Appendix D
It can be shown that attempting to solve the more germane problem of finding Cjl in (537) such that
(542) 3CJ [~K(poundK) 11 also results in sets of equations for which solutions are not known An even more appropriate optimization problem might be to maximize the time itself between required measurements For the discrete-time formulation used here however this is equivalent to finding
where N is the number of timesteps between samples Solutions to this problem were pursued but led to less conclusive results since due to the discrete nature of N many choices of C resulted in the same maxishymum value for N Thus the analytical approach though instructive in
77
the erea of matrix calculus is abandoned as a means of solving the monishytoring problem (see Appendix D for details of gradient matrices for the trace function and its calculus)
536 Numerical Measurement Position Optimization - In the last section attempts were made at analytical minimization of TrIP KCbdquo)I or E K ^ K M W 1 t n respect to the matrix Q R itself A fundamental question underlies extremization of measurement functionals directly with respect to the elements of the measurement matric Cbdquo once Q K is found how is it related to the vector of actual optimal sensor locations in the medium z K None of the studies of measurement system optimization found in the literature adequately addresses the optimal measurement design problem from the point of view of optimal placement determination
The normal-mode formulation of the diffusion problem is introduced as a means of tying together Q K and z For the case of one-dimensionai diffusion with the no-flow condition at the boundaries from (339) write Q K as a function of z as follows
1 cos^z) cos(2fz) co((n- 1)^2)
1 cos^Zg) cos(z^-z2y COS((K - 1) 2^2) poundLzK) s
( laquo )
(543)
Thus C K is a continuous function of zK so that all the conclusions deshyveloped thus far apply with pound(z K) substituted for C_K and for minimizashytion with respect to zbdquo Instead of Cbdquo
For example with the use of C(z) as defined in (543) Conclushysion III may be written as follows
78
Conclusion IIIA For N large to maximize t K + N the time when TraquoTE|(+N(C|[ZK)))gtTIpoundWII choose that z K at time t K which minimizes [P^Ctzj^))] (CIIIA)
Consider the problem of the minimization of the scalar-valued objecshytive function pSfc(z K)) of a vector z R Such problems hae received considerable attention (An adequate coverage of the various techniques may be found in Beveridge and Schechter [20]) The monitoring problem where the allowable positions of the samplers are constrained to H e sonewhere within the region of the medium suggests consideration of ton-strained optimization techniques There are various types of constrained minimization methods methods requiring use of only the objective function itself (so called direct methods) methods which require the objective function and its gradient (first-order gradient methods) and those which 1n addition require the Hessian of the objective function (second-order gradient methods) Sscond-order gradient methods are often the fastest of available methods [l03] Thus in the interest of numerical efficiency such second-order methods are considered
Define the objective function of interest to correspond with Conshyclusion IIIA
JltKgt -= [edegK - E K pound T ( laquo K ) ^ K gt $ V + x T ^ e S ] - lt5-44gt As shown by Athans and Schweppe [11J for the case of the trace operator TrlO the total differentia am) trace operators are linear so that
(see Appendix D) d Tr[X] = Tr[dX] (545)
Similarly in (544) what may be called the []^-operator is also linear being a linear part of the trace so that
d [ X ] n = [ d X ] n (546)
79
From Appendix D
Define dX1 = -XHdX) 1 (547)
T 5 |c(z K)PdegC T(z K) + VJ (548 (546) (547) and (548) are used with (544) to find the gradient of the objective function which may be written as follows
^W-LiESfe^r E^
^SEfeOVfer^] (5-49gt
where the unit vector e H [00100] the l in the ith element Thus the gradient of J( K) may be written analytically in a straightshyforward manner Note that the inverse need be cc-mputed only once per evaluation of the gradient and that 1t is an (n x m) matrix not an (w x laquo) matrix Usually the number of measurement sensors m 1s smaller than the number of states in the model n so that this inversion is computationally manageable (As a historical note this quality of Inverting the smaller (m x m) matrix was one of the important features inherent 1n the practical utility of the Kalman Filter see Jazwinski [65])
For the second-order gradient of J ( J K ) known as the Hessian adopt for the time being the following notation
(1) Drop the time subscript K the tildas and the funcshytional relationship so that C = C(j K) P H gdeg
lt2gt c i s S 7 S ( 8 K )
lt3gt c i j E 8 i 7 5 i 7 G ^ - lt 5- 5 0gt
80
With (550) differentiate the ith element of (549) with respect to the jth element of zbdquo to obtain the UraquoJ)th element of the Hessian as follows
ra^ijj bull -[C^VCR - K^fclW+ c K c T ) T l c p
- P C V 1 lt(c1)cT + CP(C|)gtTYCJ)P
+ PCT T 1 ^ ^ ) - P^CJJT^CJ^CVCP
+ P C V 1 (C^PC 1 + C P ^ ^ T V C J J P C V C P
- PCV 1 (C 1 J)PCV 1 CP - PCT1(C)P(CT)T1CP
+ P c V ^ P c V 1 ^(cJPC 1 + Cp(cj)gtT CP
- PCV^CJPCV^CJP - P(CJ)T1CP(C])T1CP
+ P C V V ^ P C 1 + CP^JOT^CP^JJT^P
- PCTT1(c i)p(rI)T1CP - P c V c P ^ c J ^ C P
+ P C V C P ^ T 1 (C^PC 1 + CP^JHT^CP
- PcVcP^TjT^cJpJ (551)
This represents only one term if the m x n Hessian matrix which would be given by
where L is a unit matrix The computational efficiency of second-order gradient methods is seen
to be lost in the horrendous task of defining the Hessian of the objective function and for that reason first-order gradient methods are nought
81
Before going on to first-order gradient methods a word about direct search methods 1s in order While in general less efficient than gradishyent techniques direct search methods possess the distinction of not reshyquiring an analytical expression for the gradient an important practishycal advantage This is of significance first since it permits a user to proceed much more rapidly from his problem statement to its coded form for numerical solution Secondly and more importantly the vast majority of physical problems do not admit the writing of an analytical expression for the gradient so that for those problems direct search methods are all that is available An interesting example of a direct search technique is that due to Radcliffe and Comfort [103] j R w nich Powells unconstrained conjugate directions minimization procedure withshyout derivatives [l03] is extended to the case including nonlinear equality and inequality constraints However in the monitoring problem it is a straightforward process to define a gradient of the form (549) so that first-order gradient methods are preferred over direct methods for their computational efficiency
The algorithm chosen for finding the minimum of J( K) in (514) was written by G W Westley and is named KEELE [127] It is an algorithm to find a loaal minimum of a function of many variables where the variables are subject to linear inequality andor linear equality constraints It represents an extension of a Davidon variable metric procedure reported by Fietcher and Powell [127] using gradient projection methods (see Rosen [54]) to include the case of linear constraints
Note how in the monitoring problem it is necessary to constrain the ranges of the variables so that resultant monitoring positions bear physhysical significance to the problem statement Note also how only linear
82
not nonlinear constraints are required each of the elements of zl must satisfy a constraint of the form
0 lt z lt 2L i = l2m (553)
where the one-dimensional medium 1s of length 2L Note how this algorithm and all gradient algorithms seek only
local not global minima The only way known to approach solution of the global minimization problem is by solving a sequence of local minishymization problems starting from different initial guesses until some meashysure indicates probable convergence to the global minimum (see Beveridge and Schechter L20]raquo p 499 and Radcliffe and Comfort [i03]P- 3) For this reason KEELE has been modified to include random initialization of the starting vector zbdquo This technique has beer found to yield satisshyfactory results provided a sufficient number of random starting points is used 1n each attempt at finding a global minimum in J()
Thus within the probability that the best local minimum found is the global minimum the optimal positioning of the m samplers at any time tbdquo is considered solved
537 Numerical Measurement Quality Optimization - The last quesshytion left to answei at a measurement ltime 1n the design problem of Secshytion 51 is what types of sensors to deploy at a samnle Consider the filter equations of relevance for a measurement at time tbdquo
y K laquo C(z K)x K + y K (554)
Ppound = Pdeg - PdegC(z K) Tfc(z K) PdegC(z K) T+ yj C(z K)Pdeg (555)
83
h PdegCCz K) T|c(z K) P^ (z K ) T + VJ (556)
As presented in Chapter 4 the noise-corrupted measurements 1n (554) are
characterized by mean vector and covariance matrix given as follows
E[vK]i o
M Thus the additive measurement noise forms a sequence of zero-mean white Gaussian random vectors with covariance given by V To conform to this problem structure the only variables lnft to determine in specifying the sensors at a measurement are the strengths of the noise terms in vbdquo as defined by their covariances tha elements [V]^ of the covaHsnce matrix y From the theory of random variables if the measurements in (554) are made with independent sensors the elements of ybdquo the individual random errors among the samples taken will be uncorrelated For this case V is a diagonal matrix which leaves only the specification of the m Elements [JfJlfi i = 1raquo2gt bullbulllaquoampbull The diagonal elements of y may DO interpreted as the mean-square values of the errors in each of the m samples Thus their sizes 4re inversely related to the quality of the measurement inshystrument used so that if a high quality sample is desired for tybdquo] 4 gt then
mdashK 1
OfJii should be small and vice versa Thus if the sole objective In the solution of the monitoring probshy
lem is to minimize the total number of samples necessary over the entire time interval the optimal choice of measurement instruments is clearly that choice which leads to the most accurate measurement - use the highest accuracy sensor available If on the other hand the more meaningful
84
measure of minimizing the total monitoring program cost is to be used in the overall optimization a more complicated problem structure results Contributions to the total cost could include costs associated with every sample that is taken a quantized cost range associated with available measurement instruments of various accuracies etc Tradeoffs result between taking a large number of low accuracy measurements and a small number of high accuracy measurements at a sample time
Though this aspect of the total problem is an important part of the complete optimal design it is left for later study with an outline of the structure of its inclusion within the infrequent sampling problem framework given in Appendix E
What is clear from the conclusions so far is that once the optimal choice of measurement instruments is made for one sample that choice is optimal for all other samples which leads to the final result for the monitoring design problem with bound on error in the state estimate
Conclusion VIII For the case of infrequent samshypling the complete solution of the optimal monitoring design problem with constant bound on error in the state estimate - the determination of the optimal number of samplers to use at each measurement their optimal locashytions and the optmal choice of measurement instrument accuracies -may be obtained at the first measurement time with the same design being optimal for all other measurement times (CVIII)
54 The Design Problem for a Bound on the Error in the Output Estimate
541 The Minircax Problem - The second form of tha monitoring de-siqn problem is considered in this section It is required to make the fewest measurements possible over the time interval of interest while maintaining the error in the estimate of the pollutant concentration itshyself the output within some bound everywhere in the medium This is a
85
more complex situation than that of maintaining the error in the state within some bound the pollutant concentration over the whole region must lie within the error constraint so that the entire region must be conshysidered when testing for violation of the constraint
At time t let the pollutant concentration at a point z in a one-dimensional diffusive medium of length 2L be given by
pound K(z) = c(z) Tx K (558)
where the vector c(z) for the scalar output C K(z) is much like the meashysurement matrix Q(zbdquo) for the veotor measurement ybdquo in (543) and is given by
poundz)T - lcos pound zjcos ^ 2 ^ z J c o s ((n-1) jfj- (559)
Equations (558) and (559) are formalizations of the s2Hes expression in (341) and can be seen schematically in the bond graph in Figure 32 The pollutant concentration at any point is thus simply the sum of the modal concentrations at that point in the medium
Equation (558) applies for the estimated pollutant concentration from the filter as well and may be written as
C K(z) = amp(z) Txdeg (560)
where xbdquo is the value of the state estimate predicted to time tbdquo from time t n (see (C18) in Appendix C) it is required to maintain the error in this estimate to be within some bound Since K(z) is a scalar random variable an expression of the error between the estimate 5 K(z) and the actual value pound K(z) in the mean-square sense is the variance in the estishymate The variance in the estimate of the output in (560) is found to be
86
O 2K C Z ) ^ E [ ( pound K U ) - 5 K U ) ) 2 ]
-=|w Tft-^)(sw TiS-J) T] = E [ e ( z ) T ( s O - x K ^ x K T c ( 2 ) ]
5 S(z)TPdege(z) (561) where the last line follows from the definition of the predicted covari-ance matrix equation (421) Thus at time tbdquo associated with the estimate of the pollutant concentration at any point i given by K(z) is its variance o(z) a measure of the error in that estimate which is merely a function of the predicted state estimate error covariance matrix whose properties are by now well established
Since the monitoring problem with a bound on the error in the outshyput stipulates that everywhere in the medium at all times over the time interval of interest the fewest number of measurements must be made to keep the error in the output below a limit the concern is with checking the maximmi value of the variance ot(z) for all z over the length of the medium as time goes on to find when the error limit is reached The asshysumption is as it was for the problem with bound on error in the state estimate that at the time when the error in the estimate of the output reaches its limit a measurement should be made That measurement should be made so that the time before the error limit is next reached is maxishymized extension of the local optimal design for one measurement period to the overall time interval is assumed possible the proof of which will be considered later in Section58 dealing with the optimal management problem
87
Suppose at time tbdquo the variance In the estimate of the output at some point z in the medium is in violation of the error limit defined as
degUmgt t h a t 1 S gt
a2K(z) gt 4bdquo (562)
It is required to make a measurement at time t K that will result 1n the longest possible time say t K + N when the error limit is reached again This will occur when at some point z in the medium the maximum value of the variance over all other locations in the medium exceeds the limit This suggests the following algorithm for finding the optimal measurement design at time t R that will result in the longest time t K + p | when another measurement is necessary
1) Select in some manner a measurement design at time t K and make a measurement
2) Predict ahead to time t K + 1 31 Find the position z of the maximum variance
max a ( z) z K+l 4) Test for violation of the error limit
max o~ (z) gt c z K+l K m 5) If violated go to (6)
If not violated increment time one step and return to (2)
6) Store the time when the limit was violated 1n N
7) Check for convergence to the global maximum t K + N If not satisfied return to (1) reinitialize time to t K and select a different trial meashysurement If comergemce has accwrved the optimal deshysign is that which resulted in largest N^ the longest time tbdquo N - call it t K + N (563)
Such a direct search technique would be costly to implement The effishyciencies of gradient techniques do not apply since a gradient of the obshyjective function (which would literally be N- the time to the next meashysurement) with respect to the measuremsnt design variables cannot be
expressed analytically Thus more information 1s sought from the strucshyture of the problem to avoid using direct search methods
As in Section 537 exclude for now the choice of measurement instrushyment accuracy from the monitoring design problem Consider only the choice of the number of samplers m to be used in the measurement at time tbdquo and their optimal locations which are the elements of the ra-vector z Then the algorithm (563) may be concisely written as a minimax problem as follows
Find min max abdquobdquo (zbdquoz) gt a bull (564) z z K +N ~K ^m
In general such a minimax problem is quite difficult requiring advanced techniques of mathematical programming for its solution However in the case of infrequent sampling the solution of (564) is virtually complete in the earlier results of this chapter
In order to solve (564) from the definition of crpound(z) in (561) obtain the following
deg K + N M = s( Z) Tepound + N(S | fkltz) bull s( 2 ) T
K ) bullpound nV nl
( ) lt 5 6 5 gt
where
EKSK) bull bull $ ( Z K ) T [ C ( Z K ) P deg C ( Z K 7 bull v] 1 C ( K )Pdeg ( 5 6 6 )
is the corrected error covariance matrix jus- after the first measurement at time t K as a function of C(-) of zbdquo in (543) Expand (565)
T N (z K z) - c(z)TJNp|J(zKgtN c(i) t S ( z ) T V n W 1 pound(z) (567)
n=T
to find the same combination of zero-lnp t response and zero-state response that was found in equation (510)
89
For the physically interesting case of no-flow boundary conditions
in one-dimensional d i f fus ic the eigenvalues of A in the state equation
(41) lead to the ordering of the terms in J given by property (516)
For N sufficiently large conditions (518) and (520) are satisfied so
that (567) may be written as matrices to show
bdquo2 M a[ -(pound0 bullbullbull] M
[l co5(^z) ]
[ raquobull(poundlaquo) bullbullbull]
li[n]
O
o
1
J (ft)
Kir2)
a ss
bull()
(568)
from which the most important result for the monitoring prohlem with bound on output error derives for N sufficiently large
4^KZY [amp)]bdquo + N t 8 ] H + Slaquo 2gt T| Spound^) (5-69) Notice that In the asymptotic case for N sufficiently large even though 2
a +jj at time tbdquo +bdquo is a function of both zbdquo the positions of the measureshyment devices at time tbdquo and z the location in the medium where the varishyance is being tested at time t K + N the functional relationship tepcviateA
90
into Independent functions of each argument The selection of measure ment positions z K Is seen to effect only | E K U K ) exactly as 1t did 1n the problem with bound on state error (see equation (5-22)) The location z In the medium where a^ + N Is being tested effects only the variance associated with the steady-state terra of the matrix convolution of the input disturbance statistics here the matrix 8 was defined 1n (520) and (521) The second term on the right-hand side of (569) N [ g ] 1 1 ( represents the increase in uncertainty in the estimate of the first mode which has a constant value throughout the medium and thus 1s a function of neither zbdquo nor z
This may be summarized as follows Conclusion IX For infrequent sampling the varishy
ance in the estimate of the pollutant concentration the output of the monitor at time t|lt+N separates into indeshypendent functions of the measurement positions at time t|lt and of the pollutant concentration position at time K+N- (CIX)
Returning to the minimax problem stated in (564) application of Conclusion IX leads to the following fortuitous result
Conclusion X For infrequent sampling the followshying problems are equivalent () Find z at time t|lt and z at time t|lt+N such that
(2) Find z at time t K and zat time t K +f| such that m j I - K ^ K U H + N[~-1n + T pound ( z ) T deg e ( z ) - aim- (c-x)
- K gt- SS This result reduces the solution of the monitoring design problem from the oi-|etely unmanageable task of (563) to the relatively simple comshybination of two separate problems in minimization and maximization Solushytion of the former 1s Identical to that treated 1n the monitoring problem with bound on error 1n the state estimate as detailed in the section on
91
numerical measurement position optimization Section 536 Finding zpound
Ni 1s minimized results in the smallest con-at time t such that tribution due to the initial covariance at time t K to the variance in the output at time tj + N
Solution of the latter problem the maximization of the variance due to the steady-state convolution matrix at time t bdquo + N is developed in the following From (517) and (521) an expression for the variance associated with the zero-state or forced response in (567) may be exshypanded as matrices as follows
N
S(z)7Y bull n W - l T c ( z ) = s ( z ) W ) bull lmdash1 I f
[ laquo(i0-raquo] flu poundWbdquoX oX^n -
iPl n i
1
amp) (570)
bull J As before
N
^^ijL^w^^j ( s - 2 deg) n=l s s
so that every element of the matrix convolution in (570) approaches its steady-state value as N becomes Urge except the first which grows as a ramp with slope [nJii- Thus for N large
A T S ( z ) T J11 S(z) H[8]bdquo + c(z) T c (z) (571)
n=l
92
It is to be emphasized that as the limit in (520) is approached the variance associated with the matrix convolution (571) separates into a t1me-vary1ng term and a term which is a constant Thus for N sufficiently
9 large the only term involving z in the expression for oj+N(zz) is not
a function of time and can be precalcylated independently of the actual time that che error limit cC is reached in (564) This separates de-termnization of the maximum over z of a^ + N(zbdquoz) from the actual value of N and thus t|+Nraquo provided only that N is sufficiently large for (520) to apply
The relationships in Conclusion X are portrayed graphically in Fig-ure 53A and B Figure 53A depicts the actual evolution of a with time whereas 53B shows the asymptotic relationships of (569) The important point is that the last term in (569) the term involving z has the same
maximum as a function of z at each sample so iony as the number of time steps between each pair of samples is sufficiently large Thus
Conclusion XI The position of the maximum varishyance in the estimate of pollutant concentration at the time each measurement is required in the monitoring problem with bound on error in the output is independshyent of time provided the time between measurements is sufficiently large and is thus the same position at every measurement (CXI)
The procedure for the solution of the infrequent monitoring problem with bound on error in the output estimate is as follows
(1) At time t|( solve for the optimal measurement posishytions Z|( such that
(2) Compute ffilusing the relationships LSSJ
[4-T^te bull - bull [raquo]bdquo-
93
mjn max o K + N( Kz)
max CT^(Z)
(A) Actual response
Time
min max o^iz^z)
Time (B) Asymptotic approximation Figure 53 The Infrequent sampling problem with bound on error in the
output estimate
94
(3) Find N large enough that the infrequent sampling approximations appiy that is so that
[sL^LW^^^ and j f 1 (4) Find z the position where the variance approaches its steady-state maximum where
ltbull = max c(z) T a c(z) SS z S~S~ (5) For the pair (zpoundz) predict the solution to time
lK+N w n e r e
(6) Reinitialize time tv = t^+Nibull and return to (1) for next measurement t W (572)
All of the results for the monitoring problem with bound on error in the state estimate apply here as well permitting statement of the final result for the monitoring problem with bound on error in the outshyput estimate
Conclusion XII For the case of infrequent sam-pling the complete solution of the optimal monitoring design problem with bound on error in the output estishymate mdash the determination of the optimal number of samshyplers to use at each measurement their optimal locashytions the optimal choice of measurement instrument accuracies and the position of maximum variance in the output estimate at each measurement mdashmay be obtained at the first measurement time with the same design being optimal for all other measurement times (CXII)
542 Determination of the Position of Maximum Variance in the Outshyput Estimate - In the solution procedure (572) steps (3) and (4) must be developed First from the form of
1 bull n gt 22 raquo 22 bull Kn gt deg ( 5 7 3 )
as seen in (515) Thus in the determination of the number of terms necessary 1n the computation of the matrix convolution [ft] In (3) from N (570) and (520) the critical terms In the matrix those which approach
95
the i r steady-state values slower than a l l the others can be seen to be
[ n ] 1 9 and [pound2 ] 5 1 where from (570) N u N
(574)
As a measure of how rapidly the series in (574) grows as N increases deshyfine
4N-1 4N-1 plj 4A vao
as the ra t io of the contribution to the series for [ f iL- dnp to seep N N 1 J
compared to the contribution from step 1 in the series Thus a meaningful
check for approaching the steady-state value of the convolution is to
f ind N su f f i c ien t ly large that
P^j lt E i j = 12 n i = j f l (576)
where c 1s some practical convergence c r i t e r i on
Since Q I t s e l f is a covariance matrix (see Appendix B) i t is posishy
t i ve -de f in i te hence [8 ] i o = telov T n u K 1 l c a n D e readi ly seen from
(573) (574) and (575) that the series for terms [Q3 and poundpound ] grow N e K i x
more slowly than a l l the others (excluding of course M bdquo ) since N
p12 p21 gt p1j a 1 1 o t h e r ( 1 j ) ( 5 7 7 )
Thus a convenient measure for the convergence
Um [n] = [n] ltdeg 8 SS
is simply to find for just the second element of 2 2 that value of
N such that for some convergence accuracy e
N-1N- 4N-1 N 11 raquo22 22 S-2 c - bdquogt Plraquo - ~mdashZ A mdash 09 e- (578) It n22 22
96
Thus for the infrequent sampling approximations to apply within some
tolerance e at least N time steps must occur between sample times so that
steady-state conditions are adequately approached
In order to f ind the maximum in step (4 ) that i s f ind z such that
c(z) 52 c(z) is maximized an analyt ical approach is f i r s t sought Since SS ~
the problem is a simple extremization of a scalar-valued function of a
single variable elementary calculus techniques apply so that for some
value of z K a necessary condition for an extremum is
From Conclusion IX and (569)
(580)
a f lt amp f l M - 3 F | ^ n
+ ^ S bull poundU)T|s amp(z
i s ( z ) T ) | E ( 2 ) t c ( Z ) T | ( i c ( z ) ) SS SS
Recalling that since U is a covariance matrix
0 = 8 gt
SS SS
so that
al 0 K + N M S 2 ( l l^) )8 e (z )
Thus
S(z) 1 l cos( ^ z j cos^2 ^ z ) |
pound^J = 0 2 f s 1 n ( 5 f z ) - 2 2 f - s i n ( 2 ^ z )
97
M N ( M gt 2 poundpound-ltbull-i [(i - H c o s Ibull 2 taj ( 5 8 )
i-i j - i
2 For an extremum in vt N(zz) set (581) to zero from which it is seen clearly that for finding the solutions of (579) analytical methods are
of little nee
The numerical solution of (579) using (581) and (569) however is straightforward Since the derivative can be so concisely written it is well known that solving for the roots of (579) then checking the value of the function (569) at each root so as to classify each extrema in order to arrive at the global maximum is superior to direct one dimenshysional search methods (such as golden section or Fibonacci search) which do not employ derivatives (see [20] and [53]) Thus any of the widely available root solving methods for nonlinear equations could be suitable for the determinization of z at the maximum cf crK+N(Z|z) (see foi exshyample [61])
55 Diffusive Systems Including Scavenging
Return now to the original problem of monitoring diffusive pollutant dispersal including anvironmental degradation or scavenging of the pollutshyant The relevant transport equation from (33) is given as
| | = KV 2 - a + f (582)
where a is a smaller parameter This equation describes di f fusion in an
arbi t rary homogeneous region P where the small term -a accounts for the
scavenging of the pol lutant from the medium The scavenging term is
typ ica l ly much smaller than either the source or di f fusion terms and
usually leads to a slowly-changing component in the system response
98
Application of separation of variables to the homogeneous form of (582) leads to the following state and Helmholtz equations
x(t) + tt + )x(t) = 0 (583)
7 2e(P) + pounde(P) = 0 (584) Comparison with equations (311) and (312) for the case of simple difshyfusion the case in (34) with a E 0 shows that the only difference in the associated eigenproblem i In the rates of response in the time equashytion The equation regarding the spatial response is identical with that for the case of simple diffusion Thus all the eigenvalues are seen to be shifted by the same amount a the value of the scavenging parameter itself
Notice that nothing has been said that restricts this result to specific coordinate systems boundary conditions etc It 1s a general relationship between the eigensystems of (34) and (582) Thus the modal state equations for the case with scavenging may be written
n(t) = -(Xn + oe)xn(t) + f n(t) n = 12 (585)
where f bdquo ( t ) is the modal input to mode n (see (319)) Comparison of
(585) with (320) for the case of simple di f fusion shows that the probshy
lem with scavenging changes the response of the system with no-flow
boundary conditions to that of a problem which l ies somewhere between
simple di f fusion with no-flow boundary conditions and simple di f fusion
with f ixed boundary conditions I t would seem from what we have seen in
the infrequent sampling problem thus far that for the cases where a
is small in (582) extensions of the ear l ier results of th is chapter to
the problem including scavenging should be possible
99
Another way of seeing how the inclusion of the term -aE in (582 effects the structure of the eigenproblem associated with (582) can be shown by reconsidering the one-dimensional example of Section 32 Conshysider here only the homogeneous response Thus the problem may be stated as follows
bull^tl K 3 ^fi - g(zt) (586)
M|Mi0 ^f^EOi (587) SfzO) = 5 0(z) (588)
Now make the transformation (see Mac Robert [82] p 33) S(zt) = n(zt)eat (589)
Substitute (589) into (586) to obtain
nfzt)^-] + ^ ^ - B a t = K i ^ f L e- a t - an(zt)e-at (5
which reduces to ^1=K^ (691)
3 t 3z 2
But the eigensystem for (591) given boundary conditions (587) is just that for the problem of simple diffusion already discussed in Section 32 from which the homogeneous solution may be written as
^3 - K ( n - l ) 2 ^ nizt) = 2 ^ x
npounddeggt e 4 L cos f(n - 1) J zj (592) n=l ^
where the initial conditions for the modes are given by
100
x n(0) bullr n(z) cos (n - 1) 2L y dz (593)
Sibstitution of (593) into (589) then yields the important result for the case including scavenging
- _K(n-l) 2-Lt S(zt) = e 0 1 ^ xn(0) e 4 L cos Un-1) ^ zj
n=l CO
n=l (0) e
K(n-l) 2 _ C 4L 2 + ltxgtt ((n-l)^z) (594)
Thus the solution to the problem including scavenging has exactly the same eigenfunations as the case without scavenging and a set of shifted eigenvalues each of whose elements is just that of the problem without scavenging shifted by an amount a
551 The Infrequent Sampling Problem - Consider a one-dimensional diffusive system described as follows
Source
Measurements i
1 2
Figure 54
-S(zt)
2Llt - raquo bull
at S z i (595)
101
3z U 32 bull
S(zo) = 5 0
f(zt) = w(t)6(zw bull bull z )
(596)
(597)
HvWh = 0 E[w(t)w(r)] = Wlaquo(t - T ) (598)
After s impl i f icat ion of the series solution of the homogeneous probshy
lem in (594) to a f i n i t e number of terms n i t can be seen from the
form of (337) for the problem without scavenging that the fol lowing set
of modal state equations resul ts
1
- ( $ bull )
o
o
(bdquo-bdquo=pound)
a
w(t) (599)
f COS (lt-gtlaquoraquo) |
102
with in i t ia l condition
x(0) = [ 5 0 0 0 ] T (5100)
The measurement equation is exactly that of (339) for the case with no scavenging
Thus comparison of the dynamic matrix for the case with no scavshyenging in (337) with that in (599) for the inclusion if the a-term shows the one major difference for the Infrequent sampling problem In the former [ A ] ^ = 0 while In the latter [ A ] ^ = -a + 0 Thus the first modal state variable will fn general exhibit a relatively slow reshysponse governed by the term e The effect of the initial condition x(0) will decay at that rate whereas it remained constant in the case with no scavenging This leads to differences In the asymptotic propshyerties of the solutions which are developed in the following
Consider the time discretization of (599) The state-transition matrix laquo given in (48) for the A matrix in (599) is
o m o 4 - ) 2 S + a gt
(5101)
where the integration step T s (t K + - t K ) Assume as before that the problem starts at time t- with initial estimation error covariance mashytrix given by tf0 Assume further that at time tbdquo the estimation error constraint is reached so that a measurement is necessary at time tbdquo It
103
Is required to design the measurement by finding the optimal measurement position vector zt so that the time when the error constraint 1s next reached 1s maximized
Consider the evolution of the predicted estimation error covarlance matrix with time after the sample at t R
nl Expand the above as matrices as was done for the case with no scavenging in (517) to obtain
amp amp ) bull fetoiMi [ilaquo
M
nSl T5t B H
CS3bdquo nraquoi
(5103)
104
Now 1f a in (595) is su f f i c ien t ly small then the diagonal elements of
J cal led ^ i = 1 2 n w i l l be related in (5103) by the fol lowshy
ing ordering property
^N N 1 gt $j| raquo bdquoj2 gt ltjgtN gt 0 (5104)
Using (5104) the matrices in (5103) may be approximated by the follow- ing expression for N large
-K+N(-Kgt
[dtei
o
[Q] v 6 2 ( n- igt u
O 8 ss
(5105) Comparison of (5105) with (521) for the case with no scavenging shows the expected result that here the asymptotic matrix solution approaches that of just the (11)-element of th matrix with time plus the steady-state matrix n due to the forcing function
SS For the monitoring problem with bound on error in the state estimate
from (5105) the trace of the estimation error covariance matrix Is given by
N
Tr[EK-Hl(sK 3 - [ E K ( S K J l + Kill Y l i n 1 gt + T r [ | s J ( 5- 0 6 )
n=l which is similar In form to (522) for the problem without scavenging The only differences H e in the first two terms on the right hand sides of (522) and (5106) Both pairs of terms describe the response of i p K l I with tirno i n the former case the response is that of a fm$]]
w1th time ramp with slope [fl]- starting at efegt] bullvv In the latter case the
11
105
response starts from the same value but then slowly approaches a finite steady-state value in the limit as N + laquo much like all the othar terms do in the matrix The main difference is that the (11)-element of P K + N ( z K ) grows much much slower to its final value than all the other
K elements of P D + N ( z K ) this is the result of requiring the scavenging parameter a to be small leading to property (5104)
A graphical depiction of the trace of (5102) and its asymptotic approximation in (5106) is shown ii Figure 55 Comparison with Figshyure 52 for the case with no scavenging shows the difference in the asshyymptotic responses
For the monitoring problem w h bound on error in the output esti-mate using the form for Ppound+N(poundK) in (5105) in the equation (568) deshyveloped earlier leads to
N
lt 4 N amp gt Z ) a [K(4U + I83bdquo Y bull i i ( n 1 ) + e ( ) T
S V ( Z ) - ( 5 1 0 7 )
n=l Comparison of (5107) with (569) for the case with no scavenging shows the same asymptotic properties as exhibited in the problem with bound on error in the state estimate above which leads to the general result for the problem with scavenging
Conclusion XIII For diffusive systems with scavengshying all the results for the infrequent sampling problem for normal diffusion apply directly so long as the scavengshying parameter is sufficiently small (CXIII)
56 One-Dimensional Diffusion with Fixed Boundary Conditions
Consider the case of a one-dimensional diffusive system with the pollutant concentrations at the ends of the medium fixed at known values throughout the time interval of interest This case was modeled in
106
Tr[P]
Tr[P2]
(A) Actual response
(B) Asymptotic approximation Figure 55 The infrequent sampling problem for systems with scavenging
compare to Figure 52 for systems with no scavenging
107
Section 32 2 Such systems are of much lesser practical Importance than those with ho-flow boundary conditions since It 1s difficult to find many physical situations of any significance where fixed end conditions occur (see Brewer [22] and Young [131])
For such a system the following state and measurement equations apply
x = Ax + Dw y = Q + X
where from (356)
4|Z
A i
o - 4 KiT 5
O -ltraquo)2 K pound 4
D 2
E = raquo(poundl) s 1 n( 22Ti)
Sfff (bullgt)
(5108) (5109)
(5110)
From tne definition of A above and 4 1n (48) and (49) the state transishytion matrix for fixed boundary concentrations is given as follows where the time step T = (t K + - t|A
108
4llt o
raquoST -44 (511)
r 2 Kn T
4L Z
Comparing this transition matrix with that from the case for no-flow boundary conditions (see (515)) shows how the fundamental difference in the two normal mode expansions effects the dynamical responses of such systems In the case with no-flow boundary conditions [] = 1 whereshyas for the case with fixed concentrations at the boundaries 0 lt [Jl lt 1
This difference manifests itself in ways which effect both the monishytoring problems with bound on error in the state and output estimates Consider the predicted covariance matrix equation from time tbdquo to time
S-K+N A M I Pbdquo +
n=l
$ V From (5111) l e t
M = A l l
Then (510) may be expanded as f o i l ows
12
(510)
(5112)
109
[laquo
[lto [4 fll B1
n1 n=1 (5113)
Comparing (5113) with (517) for the case with no-flow boundary condishytions shows that the properties of first elements of both matrices in (517) which proved to be crucial to the simplicity found in the infreshyquent sampling problem do not hold in the case with fixed end concentrashytions
However as in the case with scavenging notice that owing to the ordering of the eigenvalues in the A matrix in (5110) there is a corre sponding ordering in the elements of such that for Pbdquo+ in (5113)
gt A N gt 0 1 gt (f^ gt lttgt22 (5114)
Notice from the matrix A that for the first two terms 4X 1 X 2 (5115)
so that the second mode decays four times faster than the first Thus the two dominant eigenvalues are widely enough separated to proceed with apshyproximations for an infrequent sampling problem
Use (5112) in (5113) to obtain 1 1
amp
o
Braquongt bulli- )
O (5116)
no which is exactly the same result as in (5105) for the case with scavshyenging The trace of (5116) follows the form of (5106) for the scavshyenging problem so that for the monitoring problem with bound on error in the state estimate all the results for the infrequent sampling probshylem apply Trajectories for Tr[ppound + N(zpound)] would appear similar to those for the problem of no-flow boundary conditions including scavenging as shown in figure 55 the rate of approach to steady-state for the (11)-element of P pound + N would be faster if X 1 for this problem is larger than a
in the former problem For the monitoring problem with bound on the error in the output
estimate the case of fixed boundary conditions causes a confusing relashytionship in the minimax problem for finding the location of maximum varishyance in the output estimate From the approximation for P pound + N in (5116)
LEHlt
o [sln(^z) sin ( i r f ) ] ISA
o
sin (tpoundj
sin k plusmn )
[1laquo(poundraquo) m (]pound) - ]
8 ss
sin ( JT )
sin (2 j f ) (5117)
I l l
where c(z) Is derived from the def in i t ion of pound (z t ) in (348) Thus
for N large
V ^ T + sin yz]_ ZJL8J-- ^ bull s~ s~
n=l
which is of the form
0JJ+N(2Kraquo Z) = a ( 2 K z N ) + e^ z N gt + E 2 ) (5119A) = a(z K)|3(zMN) + B(z)6(N) + E( Z ) (5119B)
It is required to find zjj and z such that for N large
4^1) = JjJ T degK+N( ZK Z)- (5-120) From the separation of functions in (5119B) it is clear that finding zt should be done exaotly as before that is
Find zj at t K such that [ t ^ ) ] bdquo = [ ^ Jin ^ ^ It would appear that knowing zpound the optimal measurement positions
for the measurement at time tlaquo one could then substitute its value dishyrectly into (5118) to solve for the position of maximum variance z at time t K + N- However as seen 1n (5119B) the terms (a 8 y) and (B 6) are functions of time t K + N gt such that the relationship between (agy + 86) and (e) in (51198) is always changing A general statement of a separashytion principle like (569) for systems with no-flow boundary conditions cannot be made for the case with fixed boundary conditions However if more knowledge exists about the specific problem under study for example if in (5118) [n] raquo [ Q ] i j i and j f 1 then the term (Blaquo) In (5119B) may dominate the right-hand side of that equation for N large such that
112
for such a special case
T C K+N(K Z ) = trade X s i Z [ t z
What is clear about the general case is that the minimax problem in (5120) simplifies to (1) finding z in the minimization in (521) as before then (2) evaluating the position z for the maximum oy +bdquo(z Kz ) in (5118) iteratively as N increases until for some t R + N o^ + N(z^z) gtcC The latter procedure is greatly simplified using the approximashytions of the infrequent sampling problem as can be seen by comparing the simplicity of the expression for aj + N in (5118) with the complicated
V
expression that would have resulted had the full matrices for P K + [ in (5113) been involved instead
Thus results for the infrequent monitoring problem with no-flow
boundary conditions extend with restrictions to the case with fixed boundary conditions
Conclusion XIV For N large all the results for the infrequent sampling problem with no-flow boundary conditions with bound on error in the state estimate extend to the case with fixed boundary conditions The results for bound on error in the output estimate do not all extend to the case with fixed boundary conditions in general however application of the infrequent samshypling problem approximations does drastically simplify solution of the functionally interdependent minimax problem to the solution of two independent problems in minimization and maximization (CXIV)
57 Extension to Monitoring Problems in Three Dimensions Systems with Liiission Boundary Conditions
As a means of demonstrating the power of the results for the infreshyquent sampling problem consider extensions to diffusive systems in three dimensions examples of applications might include pollutant transshyport in estuaries or bays and radiation level detection in settling basins
113
or in groundwater systems Suppose there is a rectangular three-dimenshysional region into which known stochastic sources are injecting pollutshyant In the case of bay estuary or settling basin systems the upper surface of this region would interface with the earths atmosphere whereshyas in groundwater applications the upper surface of this hypothetical region could coincide with the local level of the water table The reshyquirement of the problem is to place the fewest number of sampling stashytions at the best locations on the surface of the region taking the fewshyest number of samples over a given time interval in order to maintain the error in the estimate of the concentration ttceoughout the three-dimenshysional volume below a given bound This is an interesting variation of the general problem in three dimensions where sources may occur anywhere in the volume but measurements are required to be taken only on one surshyface of the volume
The validity of the description of pollutant transport in such sysshytems by the use of Fickian diffusion has not been thoroughly studied However it seems reasonable to assume that if small enough subregions which may be called components are considered thtn coupling large numbers of such component subregions together each of which is governed by its own diffusion equation could result in a system of submodels which could be used to model a large possibly inhomogeneous anisotripic medium Thus this example is presented for its conceptual interest as a starting point toward a more sophisticated approach to solutions for pollutant monitoring problems of this type
Assume the component subregion is described schematically as in Figure 56 One of the v generalized sources w ^ t ) is shown somewhere in the volume with its position vector defined as
114
Figure 56 Three-dimensional component subregion for a three-dimensional monitoring problem
115
Sw S 1 L M 2 3 Sw S K c w laquosw t 1 = 12 P (5122)
One of the set of m generalized measurements y is shown on the surface with its position given by
2j S [ Z V Z V 2 L 3 ] T J = 12 m (5123)
If the size of the rectangular region 1s sufficiently small the dif-fusivity throughout the medium may be approximated as a constant The boundary conditions of the submerged surfaces are chosen to be of no-flow type so that other such components may be coupled together in order to approximate inhomogeneous material properties over larger regions (see Young [131] Chap 3)
At the upper surface of the component the assumption is made that a no-flow boundary condition adequately models the characteristics of the pollutant exchange across the upper boundary of the region In problems involving transport of a volatile soluble contaminant in water systems (like DDT or disolved radioactive wastes) this assumption could be changed for instance to include emission of the pollutant into the atmosshyphere at the earths surface An approximate model of such emission is Robins boundary condition (see Berg and He Gregor [18] Sections 36 and 49 Mac Robert [82] p 28 and Duff and Naylor [34] Section 73) The only difference such a modification makes in the normal mode analysis is in the eigensystem which results for the coordinate direction which 1s similar in form to that for no-flow boundary conditions but has intershyesting conceptual differences (see 118] Section 49)
Suppose the initial pollutant concentration throughout the medium i given by the function 5 0(c) Thus the initial-boundary value problem for this system is defined as follows
amp bull (
bull bull bull Cj raquo 0 e - ^
K2 2 deg 0raquo 2 = 2L 2
c 3 = 0 3 = 2 L 3
c(co) H e 0 i Ml
116
t)t (5124)
(5125A)
(5125B)
(5125C)
(5126)
iMiltgtlte - s )^ - s 2 ) 6 ( c 3- s 3 gt E^tt)] = 0
E[w(t)w(T)] = W6(t - T ) i = 12 r (5127) The no-flow boundary conditions are specified for all surfaces by
(5125) The initial condition as a function of the spatial coordinate vector 5 is given in (5126) while the stochastic point sources with their statistics are described in (5127)
The essential difference between this problem and the two-dimensional case treated in Section 33 is in the extension to eigensystems in three dimensions and the resultant increase in dimensionality as mentioned in Section 34
Begin the analysis by assuming a solution in separated variables of the form
^ bull ^ L I L L W ) wsgt pound=1 nR n=l
mM e U l gt e m ( 2 en^3gt- ( 5 1 2 8 gt Jt=l m=l nlt
117
From the one- and two-dimensional problems 1n Chapter 3 elgensystems for
the coordinates C 1bull amp 2 and 3 given boundary conditions (5125) can be
w i t ten down Iranedlately as follows
h TT~ 4 = 12 (5129A) 11
(5129B) e l(5 1) = cos U - 1) mdash- c I
= R T m = l 2 (5130A) m m
e m (c 2 ) = cos ( m - l j j j - e j (5130B)
=^4~ bull n = 1 2 (5131A) n n
e n k 3 ) = cos ( n - l ) ^ - 3 (5131B)
The generalized modal resistances and capacitances the Rs and Cs above
are exactly those given for the two-dimensional case in (361) As before
substitution of C(ct) in (5128) into the differential equation (5124)
right-multiplying by eigenfunctions integrating over the volume and apshy
plying orthogonality results in the following generalized normal mode
state equation
fat14 Jf bdquo lt 5 t ) cs ( lt ) a q e 0 c ( - n pound ) C 0 S (ti1gt i ^ W r (5132) The initial conditions for x(t) are found as follows from (5126) and
(5128)
~] ~=LLL x raquo c o ) e U i gt ^ ^J- ( 5 1 3 3 )
xf npl n=l If CQ(C) 1S expandable 1n a triple Fourier series then x J l m n(0) is given
N
118
as Allows (see Mac Robert [82] p 43)f
r Z h r 2 4 r 2 L 3 W deg gt bull r r r o(-5) e i ( igt ^ eM d 3 d t2 d i (5134)
m n -^bullo-tj-o-tj-o
where the eigenfunctions are given in (5129) through (5131)
The stochastic point sources are transformed into modal inputs in a
similar fashion
r c V f (5 t ) efc) ^ ( ^ e n U 3 )d 3 d 2 d i
tradeltXs2H3) where treating the point sources as distributions the eigenfunctions in (5135) are evaluated at the coordinate positions of the ith point source
Truncating the triple Fourier series in (5128) and retaining n terms in each results in a set of state and measurement equations entirely anashylogous to those for the two-dimensional problem in Section 33 The dishyagonal element for A for the (ijk)-th equation is
bull^--jk4i+S ( 5 136 )
so that the eigenvalues of the three-dimensional problem are simply the
sums of those for one-dimensional problems written in each of the three
coordinate directions Similarly (see (362) and (364)) the elements
of the D and C matrices are merely triple products of the eigenfunctions
Thus the similarity with the two-dimensional case is well established
Notice that in the discretization of the elements of A from (5136)
and Table (361) [A] = 0 so that ct^ = 1 thus all the results for
the Infrequent sampling problem with no-flow boundary conditions extend
(laquo i = 12 (5135)
119
directly to multidimensional regions Thus regardless of the dimensionshyality of a region 1f no-flow boundary conditions exist at all boundshyaries the monitoring problem may be treated in a straight orward manner with thp techniques of the infrequent sampling problem
Consider the Inclusion suggested earlier of the emission of pollutshyant into the atmosphere at the surface of the component subregion at C = 2L A model for such emission (see Mac Robert [82] p 28) ibdquo given by the following homogeneous boundary condition
3(Ct) 33
bull+ h[e(poundt) - C 3(c rC 2)] = 0 5 3 = 2L 3 (5137)
where pound is the pollutant concentration in the atmosphere over the surshyface = 2Ltaken to be constant over time Thus the atmosphere acts like a pollutant source with constant concentration pound) h is a constant relating the emlsslvity of the surface e to the diffusivity within the component subregion by
h 5 eK (5138) Berg and Mc Gregor([18] Section 49) show that the eigensystem for a one-uimensional system with a no-flow boundary condition like (5123C) at C = 0 and a boundary condition with emission of the form (5137) at -g = 2U can be described as follows
V ^ = (n - D s r + V n = l2 (5139A)
e n(5 3) = cos (5139B
where J T must be a positive root of the transcendental equatio ^ tan (213^)= h (5139C)
ion
120
A graphical solution of (5139C) shows that there is an ordering of the roots y T 1 such that for u
gt p gt P 2 gt gt p n gt u n + 1 gt gt 0 (5139D)
For example for 2L 3 = 1 and h = 01
n 1 2 3 4 5
03111 31731 62991 94333 125743 (5139E)
Thus it is found that an ordering in this problem exists such that for
V 0 gt A gt Xj gt n = 12 (5139F)
Since the eigenvalues for the three-dimensional problem are the sums of those in eigenproblems written in the three independent coordinate dishyrections 5 c 2 and cbdquo from (5136) it 1s seen that if an emission boundary condition is used at s = 2L 3 the crucial first eigenvalue in the A matrix is given by
Xlll = (deg + 0 + v 2J (5140) 2
where p 1S the first eigenvalue for the modified elgensystem (5139) This leads to an ordering for the matrix elements such that
1 gt n gt 2 2 gt (5141)
so the the concepts developed for the infrequent sampling problems for the cases with fixed boundary conditions and scavenging apply here as well It should be noted that since P 1 gt 0 the first eigenfunction 1n (5139B) will be a function of c 3 so that the minimax problem possesses
121
the modified separation property of (5119) for the case of fixed bound ary conditions Thus the case of practical interest accounting for emisshysion at a boundary is seen to fall within the framework of the infrequent sampling problem
Conclusion XV For N large the results of Conclushysion XIV tor the case with fixed boundary conditions are seen to extend to regions with emission or radiation boundary conditions (CXV)
Another interesting point about the structure of this type of monishytoring problem is that pven though the dynamic response of the process must be computed for the entire region 1n three-space the measurement position optimization is constrained to a two-dimensional subspace that is to the surface
C 3 = 2L 3 (5142)
This reduces the domain of the optimization considerably and emphasizes the power and versatility of constrained optimization techniques In Section 536 a first-order gradient technique with linear constraints was described In the context of the problems of this section the power of such a technique is demonstrated in being able to express the requireshyment (5142) directly as an equality constraint upon the domain of 5 3 in the optimization
In the application to groundwater problems a more practical problem scatement might be to constrain measurements to be taken anywhere down to a depth e below the upper surface of the component subregion that is to a depth E below the water table This form of a constraint is readily placed upon the domain of the optimized variables as follows (see (553))
For the position of the jth measurement device require that z -J3
the element of z^ in the 5 coordinate direction be limited to (2L 3 - e) lt Zj lt 2L3 j = 12m (5143)
122
the form of a constraint for the optimization algorithm must be z s W lt 5 - 1 4 4 gt
thus decompose the single inequality constraint in (5143) into two of the form (5144) to obtain
zi 2 L 3 -
- Z j lt (2L3 - c) (5145)
Thus the subspace for the measurement posit ion optimization consisting
of a layer of depth e beneath the surface of the region is entered into
the optimization algorithm as two simple inequali ty constraints on the
elements z given in (5145) J 3
Thus formulation of a three-dimensional pol lutant monitoring probshy
lem over a homogeneous region with various boundary conditions amounts
to a straightforward extension of the methods used for one- and two-dishy
mensional problems In addi t ion confining the admissible region for
optimal monitor placement is a natural application of constrained op t i shy
mization techniques
58 The Management Problem
Thus far consideration has been given solely to the problem involved 1n the design of a measurement - the number and quality of measurement sensors and where they should be placed - in order to minimize the total number of samples necessary over some time interval It is the requireshyment on the other hand of the management problem to determine at what times within that time interval the measurements should be made in order to minimize the total number of samples necessary overall
123
It is desired to prove that the optimal management program is to
sample only when the error criterion for the state or output estimate
has reached its limit In general this is a difficult fact to establish
Results are clear for the scalar case however and (algebraically tedishy
ous) constructive proofs for a system with only two normal mode states
and one measurement device indicate that such a sampling program is also
optimal for the vector case However obtaining a comprehensive proof
that sampling only at the limits is optimal for multidimensional normal
mode representations remains an elusive task Heuristically the verishy
fiable resilt for scalar systems still seems to be extendable to the
multivariable case as will be shown
581 Optimality in the Scalar Case - Consider a scalar system whose Kalman Filter covariance equations (see Chapter 4 Figure 41) can be reduced to
(5147)
where ui and v are the disturbance and measurement noise variances p is the variance in x and c is the scalar measurement coefficient
Assume the process starts at time t Q In order to deduce the optishymal sampling program compare the two following monitoring programs which correspond to sampling at the error limit (2) and sampling before-the error limit is reached (1)
(1) Predict to t 1 sample at time t] and predict ahead to tfj (2) Predict to t N then sample (5148)
The optimality of one program over the other will be established after time t K + N by the determination of which of the two has the smaller
bdquo K + 1
= PK+1 v
PK+I = PK+1 PK+I C K+I + v
124
variance p since both wil gtve used the same number of measurements (one each)
a starting point make the assumption that the characteristics of the measurements at the two times (specified by cjL and v in (5147))
2 are the same The more general case where v can vary and c at t in
2 the first measurement program and cf at t N in the second may be differ-
2 2 2 ent is commented upon later Thus for now let ct = cz H c at both samples Case (1)
(A) Predict from t Q to t
0 J- j p1 = Sgt MQ + lto
(B) Sample at t
1 = P V
h = P pdegc 2
+ v
= (ltj2u0 + u) = (ltj2u0 + u)
_ ($ 2 u Q + u ) c 2 + v
(C) Predict to t^ N-l
pj = ( 2 ) N _ 1 P ] + 2 I n=l
-) laquo
(5149)
(5150)
(5151)
Case (2)
(A) Predict to t N
Pbdquo = () bullN Z n-l (bull ) i
n=l
( V bull pound (V
(5152A)
(5152B)
125
(B) Sample at t N
N 0 W+ (5153)
It is required to show that in (5148) program (2) is optimal (which is an analogous case to sampling at the limit in the monitoring problem when pH gt p 7 an error limit) This can be shown by finding conditions under which
(5154)
To illustrate the relationships involved in the optimality of such a monitoring program consider Figure 57
P
P N lt P N
Figure 57 Relationships involved in scalar optimal manageshyment program
126
The optimality of case (Z) is verified if after both programs have included one measurement after time tK+f- the variance for case (2) is below that of case (1)
In order to prove (5154) proceed as follows Consider the amount of correction A to the variance p at a sample as the difference between the predicted and corrected values at the sample time From Figure 57 then define
Al - (P bull P i ) lt 5 1 5 5
A N a (pdeg - p|j) (5156)
t wil be shown in what follows that if pj is a monotonically increasshying function of t K then
(PN gt P) bull (AN gt A l ) - ( G- 1 5 7) Then predict A ahead in time to tbdquo to show
(AN gt A) -ofy gtpjj) (5158)
which proves (5154) Finally it is necessary to show that if sampling at t N is superior to sampling at t then for all times t N + R after t
( P J gt P K ) - ( P J + R gt P NN
+ R (5-159) i
F i r s t consider the evolut ion of p pound + bdquo a f t e r a measurement a t time
bdquoK PK+N ( bull ^ bull ^ ( bull V V
n=l
where if the measurement after tbdquo is the first measurement
P K pK pdegc 2 + v
(5160)
(5161)
127
Since pdeg gt 0 and c Z gt 0 in (5161)
gtl lt Pdeg (5162) that is the variance in the estimate is (expectedly) decreased at a measurement In general the variance or uncertainty will grow beshytween measurements or at least it will under certain conditions upon
K 2 the combination of pj^ lttgt and ltu in (5160) those conditions which are of interest in the monitoring problem Thus restrict the study here to systems which possess monotonically increasing values of predicted varishyance as shown in Figure 57 Hence require that
(5163) Next consider the corrections in (5155) and (5156) To deduce
the inference in (5157) from (5149) through (5153) find
PNdeg gt P-
A - P - P
-5
V -0 2
V L J
V
I 2 + V
(5164)
(5165)
To find conditions under which
A N gt A 1 (5166)
substitute (5164) and (5165) into the above cross multiply by the
denominators aid collect terms to obtain
[(PS)2Plt2 bull ( P ^ ] gt [(-fif bull (ptfv] (5167)
from (5157) and (5167) follows Conclusion XVI For the scalar case of the monishytoring management problem and for problems with increasshying uncertainty 1n the state estimate between sample times the amount of correction made to the predicted variance In the state estimate Is an Increasing funcshytion of the predicted value of the variance at the time of the measurement (CXVI)
128
This concept of the comparison of the amounts of estimation error corshyrection at different measurement times Is suggested in a later section as the basis for a proof in the extension of these results to the vector case
In order to prove (5154) establish now the inference in (5158) Referring to Figure 57 and using (5151) and (51528) obtain
n 0 J PN PN (bullJ-pfL L
n=l V ) m
N-l
bull c 2 ) N - ] P E ^ V
However for a stable system
i i 1
[ P N - P N ] S V Thus by construction from Figure 57
[ gt gt l] [Pi gt P]
7 N-l i V 9 I1
() Pi + gt ( ) ltraquo n=l
bullA]
from which (5158) follows Finally to demonstrate (5159) for case (1) in (5148)
Plaquo+R
ft o R i 9 n-l
= ( ) Pf| + ) ( ) I n=l
(5168)
(5169)
(5170)
(5171)
(5172)
and for case (2)
129
n=l from which (5159) is obviously seen to follow regardless of the value
o o of ltr Hence if pfj gt p ^ m gt some error limit sampling at the limit is seen to be optimal at the sample time and optimal thereafter Thus in the scalar case (2) is the best monitoring program
o Notice how no restrictions were placed upon 4 lto or v except that the system must be stable and to and v as variances must be positive Thus Conclusion XVI includeb both the zero eigenvalue case for $ = 1 and the negative eigenvalue case where 0 lt ltjgt lt 1 Thus it is a general reshysult for scalar models where the asymptotic properties (518) and (520) of the infrequent sampling problem need not necessarily apply
Thus the verification of (5157) through (5159) prove that for a p
fixed measurement position reflected in c and fixed instrument accuracy fixed by v sampling at the estimation error limit is optimal
In the original comparisons for monitoring programs (1) and (2) 2 2 2
the assumption was made that ci = c in (5150) and cjj = c in (5153) The general case is now considered where the characteristics of the meashysurement at time t in program (1) are free to differ from those at time t N in (2) that is c f cjj
The objective of both monitoring programs under the earlier problem definition is to provide a sampling schedule which requires the least
overall number of samples necessary to maintain the estimation error beshylow its limit at all times An important observation for the scalar
case is that for a measurement at time t maximizing the time t K + N beshyfore the error limit is again reached is strictly equivalent to minimizshying the estimation error just after the sample at time t K (this may not
130
be the case in the extension to the vector problem due to the linear combinations of increasingdecreasing responses inherent in theTr[-] and g- [J functions this case is considered later) Thus the Objecshy
ts n tive of sampling schedule (1) is to choose c such that p is minimized and that of sampling schedule (2) is to find that cjj which minimizes pjj The optimality of the two is then established by determining which proshygram after time t N results in the smaller estimation error that is in determining which of Pu(c| ) and pbdquo(cjj ) is the smaller at time t N
for the scalar case it can be shown that the optinal measurement positions reflected in c and oL must be independent of the time each measurement is taken independent of the value of the variance at the times of the measurements and they must strictly be equal to each other To see this compare the first line of (5150) for a sample at time t with the case for a sample at time t N in (5153) Examining the denomishynators of the two expressions leads to the observation that the optimal choice for c in both cases must be the same In order to maximize the time until the estimation error limit is next reached after each measure-
1 N ment p-j and p N must be minimized at the times of those measurements From the forms of the expressions for the corrected variances this is achieved when the denomiators in both cases are maximized Clearly this occurs at the same common value
c 2 = c 2 = c 2 (5174) Thus for the eaalar case the optimal measurement positions as detershymined by c are seen to be independent of the value of the variance p at the times of the measurements and which is actually the same thing independent of time The same Is obviously true of the selection of the best Instrument accuracy as reflected In the measurement error variance
131
v which leads to the general result for the optimal management problem for scalar systems
Conclusion XVH For the scalar case of the tnonl-toring management problem the optimal sampling program is to sample only when the estimation error criterion 1s at its limit (CXVII)
Notice that the results in Conclusions XVI and XVII are general in that no restriction has been made which would limit them to the infreshyquent sampling problem only The infrequent sampling problem is obviously included under them as a special case
582 Extension to the Vector Case mdash Arbitrary Sampling Program mdash Consider the general case with n states retained in the normal mode exshypansion for the model m measurements at r stochastic disturbances for the monitoring management problem with bound on error in the state estishymate As in the scalar case assume the process starts at time tlaquo then compare the following two arbitrary monitoring programs
(1) Predict to t] sample at t and predict to t N (2) Predict to t N then sample
In the problem with bound on error In the state estimate the optimal program will be that which has the smallest value of Tr[P] after t N The relevant equations are for prediction
T r 8- T
ampN
s W +XVV-1 (s-176)
nl
on
nl
and for correction
Assume that the same measurement matrix pound Is used in both sampling programs
132
Ce Q ) (A) Predict from t Q to t
pound = H 0J T + Si (5178)
(B) Sample at t^
Ei bull Si - EdegE T [CPC T + y] _ 1cpO
=(5oJ T + s ) - ryo~ T + s)s T|9(io~ T + ~)~T + xl pound(JHoS T + ) s lt 5 - 1 7 9 )
(C) Predict to tbdquo using (5176) obtain N-l
pound - H Y H - l T +XV _ l T
n=l
^ n=l
- ^ ( J M Q J 1 + Q)pound T fe ( jy 0 T + Q ) E T + y l C ( M 0 T + s ) 1
(5180)
Case (2)
(A) Predict t N
(5181)
n=l
(B) Sample at t N
EM bull eS - E 0
N C T [ Q B deg G T bull y j 1 c E deg
N N
bull (V T + A pound r 1osn _ l TV ( t V T + Z J 1 ^ 1 ^ 7
^ n=l ^ n=l
x U v T + f laquon v- i T V + J V v T + Z jnlsslT (5182)
133
In order to establish the optimal1ty of program (2) it is required to find conditions on J a and MQ such that
Trjjpjj gt Tr[pJjJ (5183)
In general this is difficult to accomplish owing to the complexity inshyvolved in comparing traces of inverses of matrices Since it is so difshyficult to say anything at the symbolic level of (5180) and (5182) an example with n = 2 lt = l and r = 1 was developed algebraically which resulted in the same result as found with the scalar case in Conclusion XVII However an analytical result for the general case has not been found
Thus a general result for the optimal management problem for the vector case has not been obtained analytically though the results for the scalar case do suggest extension to the vector problem Numerical determination of the optimal sampling schedule for specific problems though tedious should be possible with dynamic programming (see Meier et al [92] for a related problem)
583 Extension to the Vector Case - Infrequent Sampling Program -Following the discussion for the scalar case where it was found that the amount of correction to the estimation error criterion was directly proportional to its predicted value at the time of a measurement it is desired to show the following for the vector case of monitoring with a bound on error 1n the state estimate
(A) Predict to time t K sample there and find the correction
poundTrK 5 Trfe - EJ J (5184A)
(B) Predict to time t K + N then sample and find the correction
134
ATr K+N 4degtrade - amp (5184B)
(C) Show that
(5184C)
(D) Finally predict the case in (A) ahead to t K + N and show
(5184D)
Graphically these relationships are shown in Figure 58 which is simply
the vector analog to Figure 57 for the scalar case
the cas
A T rK+N raquo i T r K
I K 1
Figure 58 Asymptotic relationships for Tr[pound] in the vector optishymal management problem
135
It 1s assumed that tines t K and t K + N are sufficiently long to pershymit the approximations in the infrequent sampling problem (518) and (520)) to apply at each sample With these simplifications obtain from (522)
T E H 4 + K deg + T r | j s
~PK = Edeg - efej [ s K $ T
K
+ y ]V p deg -K+N[~K+NEK+N poundK+N + ^J
pK+N -K+N ampamp CK+NEK+N
For consistency as before assume that
~K = poundK+N E ~
a t both measurement t imes Thus in (5 184A)
ATrbdquo = Tr
S i m i l a r l y for (5 184B)
ATr K+N [amp4 pound p K + N E + J
(5185)
(5186)
(5187)
(5183)
(5189)
(5190)
(5191)
I t is required in (5184C) to compare ATrK in (5190) with ATr K + N in
(5191) Making substitutions for pjj and Pdeg+ N for the matrices in (5185)
and (5186) shows that the only difference in pound[ and Ej + N is in the
valua of their (ll)-elements see the second terms in (5185) and (5186)
This results from the infrequent sampling approximations
Even with this simplification analytical comparisons in (5190)
and (5191) could not be found to substantiate (5184C) Approaches used
included use of the following theorem from matrix theory for the inversion
of a partitioned matrix
136
THEOREM I f fln is nonsingular then the inverse of the part i t ioned matrix
6
is given by
where
laquo11 Siz
A21 _ 1
1 5 2 2
A 1 + Xltf^X 1 - sect _ 1
e 1 1 e1
ilaquo x = 6 n f l
1 2
sect = ~22 ^21~
I - A 2 l A i r
(5192)
Attempting to use (5192) in comparing (5190) and (5191) where the
par t i t ion i s taken to ive A include only the ( l l )-elements of those
matrices shows that allowing only the ( l l ) -element of K and P + N to
be d i f ferent effects every element in the inner inverses in (5190) and
(5191) thus use of (5192) does not seem to help
I t was thought that use could be made of the
MATRIX INVERSION LEMMA For pound gt 0 and V gt 0
E - EpoundT[poundpoundST + y]_1poundpound = O f 1 + s V 1 ^ 1 (5193) (see Sorensen in Leondes 1781 p 254)
However though the number of terms in ATr K and ATr K + f | decreases the complexity in their comparison increases Thus the pursuit of an analytical statement for the solution of the optical management problem in the vector case was abandoned
584 Suggestion of a Heuristic Proof for the Vector Case - For the general management problem (of which the infrequent sampling problem is only a special case) the following heuristic proof is offered in substantiation of the optlmality of sampling only at the error limit when the model state is a vector
137
Suppose the problem started at time tQ and now is at time tbdquo The following two sampling programs as before are to be compared
(1) Sample at t|lt and predict to t +f (2) Predict to t K + N and sample (5194)
For consistency assume again that the same measurement matrix C is used in each case Then the optimality of (2) over (1) can be shown by provshying that at t K + N gt
T r ~K+N f o r C a s e ^ lt T r ~K +N f deg r C S S e ^ (5195) The above may be proven with a simple extension of the scalar results of Conclusion XVI to the vector case This extension can be made after making the following
Coniecture A The absolute values of the individual elements of the predicted covariance matrix in the linear recurrence (5175) are monotonically increasing functions of time (CA) Numerical experiments have shown the above to be true but an analytical proof has not been obtained Assume the conjecture to be true in what follows
The optimality of case (2) can be established by reasoning as folshylows at the first measurement time tbdquo
(1) Assume the measurement associated with the matrix C effects allthe modal state variables that is information is gained in the estimate of each state of the filter at a measurement (2) The information obtained in each mode decreases the absolute value of every element of the covariance matrix during a meashysurement
(3) Conjecture A implies that the absolute values of all the eleshyments of the predicted covariance matrix [PR+N3 at time t +tj are larger than those of [pound$] at time t|lt
(4) Then from Conclusion XVI for the scalar case the absolute value of the correction to each element of [J$+N] at t K + N should be greater than that to each element of [E$] at t|lt
(5) By the uniqueness of the solutions of linear recurrences the elements of [P|lt+M] for a sample at time t^+o must thus be smaller in absolute value than those of rPKM] at tiMM for a sample at t R K + N N N (5196)
138
A graphical interpretation of this even for a small number c reshytained modes adds more confusion than clarification to the above Such a pictorial description would follow Figure 57 for the scalar case where such a graph can now be thought to apply to eaah element of the (n x n) covariance matrix
If the above construction has validity 1t applies to both the trace of the state estimate error covariance matrix and to the variance of the system output estimate anywhere in the medium Thus in both the moi toring problem with bound on state estimate error and that with bound on output estimate error the optimal management program would be to sample only when the error criterion reaches its limit
Though a proof has not been found the concepts presented here may prove to form a basis for future solution of the optimal management probshylem for the multidimensional case
59 Extension to Systems in Woncartesian Coordinates General Result for the Infrequent Sampling Problem
Duff and Naylor [34] in Chapter 6 on the general theory of eigenshyvalues and eiaensystems discuss at length conditions under which partial differential equations of applied mathematics are separable Results are given there of conditions under which eigensystems for given coorshydinate systems can be found The results presented in this thesis for the Infrequent sampling problem based upon properties (518) and (520) extend directly to systems 1n any coordinate system for which complete orthogonal eigensystems can be found the requirement Is only that the first eigenvalue must dominate the asymptotic response a condition which has been seen to admit a wide variety of suitable boundary condishytions As developed in Young [131] no-flow boundary conditions can be
139
used in conjunction with pseudo-sources at the boundaries of actual sysshytems in the coupling of component models to one another greatly extendshying the applicati n of infrequent monitoring theory
The results of Conclusion XIV for systems with fixed boundary conshyditions extend as a worst case to systems in any separable coordinate system where a complete set of orthogonal eigenfunctions nay be found In those cases fidegd boundary conditions or emission or radiation boundary conditions lead to the modified separation property in (5119) this results in the necessity of solving for the position of maximum variance in the output estimate in the monitoring problem with bound on output error as a function of time This is not a serious difficulty and does boast the property that as in Conclusion XII for no-flow boundshyary conditions once the position of maximum variance is found at the first sa pie that position will be the position of the maximum varishyance for all subsequent samples Thus the time-varying maximization in (5119) and (51ZC) for one-dimensional diffusion with fixed boundary conditions or for systems with emission or radiation boundary conditions as in Conclusion XV need be solved only at the fivet sample the same result applying for all other samples the result extends directly to all systems of higher dimension in separable coordinates with complete orthogonal eigensystems
The more ideal results of Conclusions VII and XII for systems with no-flow boundary conditions appear to also extend to systems in arbitrary coordinate systems where again complete orthogonal eigensystems exist The requirement in order for the solution of the minimax problem to be Independent of time in Conciusion XI is that the eiaenfunction associated with the dominant eigenvalue in this case the zero eigenvalue be inde-
140
pendent of the spatial coordinates Consistent with this requirement make
Conjecture B For diffusive systems in any coordishynate system where solutions may be expressed in terms of a complete orthogonal eigensystem the case of no-flow boundary conditions leads to a dominant eigenvalue of zero modulus and an associated eigenfunction which is independent of the spatial coordinates (CB)
Examples include diffusive systems in cylindrical coordinates For a system with a no-flow boundary condition at radius r = R the eigenfunc-tions are Bessel functions the eigenvalues are the positive roots of
3 pound J 0 ( A R ) = 0 (519)
the first of which is zero The eigenfunctions are e n(r) = J 0(A nr) (5198)
but since A = 0 the fir-it eigenfunction is independent if r (see Mac Robert [b2] for n extensive treatement of Bessel functions in the area of spherical harmonics)
Another example concerns radial and latitudinal atmospheric pollushytant transport on a global scale (see Young[131] Chapter 4) It can be seen that eigenfunctions in the radial direction are Bessel functions while those in the latitudinal direction are the Legendre polynomials Both eigensystems possess zero first eigenvalues and related eigenfunc-ions which are independent of the spatial variables
In cases such as these the complete separation of the minimax problem as in Conclusion X into two independent problems in minimization and maximization both of which may be solved independently of time leads to in elegantly simple solution of the infrequent monitoring problem with bound on error in the output estimate
141
The following general results for diffusive systems in various dishymensions and coordinate systems summarize the extension of the one-dimensional results of this chapter o the general case in multiple dimensions
Conclusion XVIII The complete solution of the deshysign problem for an infrequent sampling monitor may be determined at the first sample time the results being optimal for all subsequent sample times The optimal sampling management program appears to be to sample only when the estimation error criterion is at its limit These results apply to diffusive systems in separable coordinate systems with homogeneous boundary conditions where complete orthogonal eigensystems exist and to normal mode models of arbitrary finite dimension
(CXVIII)
142
CHAPTER 6 NUMERICAL EXPERIMENTS
Examples are presented in this chapter which serve to numerically substantiate many of the analytical results of Chapter 5 The discrete-time Kalman Filter algorithm of Chapter 4 is programmed as shown in PROGRAM KALMAN (see Appendix F) using the normal mjde problem formulashytion of Chapter 3 and the time-discretization algorithms of Chapter 4 and Appendices A B and C The first-order gradient optimization algoshyrithm with linear constraints described in Section 536 (see Westley [127]) is coded as SUBROUTINE KEELE and included as part ot KALMAN For the case m = 2 for the optimal positioning of two noise-corrupted meashysurement devices and for a one-dimensional diffusive medium it is found to be convenient to generate contour plots of the value of the estimate error criterion (either Tr[Ppound + N(z K)] or [ P J ^ f z J L j ) as a function of the two measurement device position coordinates IKJi and f z K ] 2 at various times t bdquo + bdquo The surfaces shown in these plots with the high level of information they contain were instrumental in arriving at many of the conclusions of Chapter 5
The basic problem to be considered is developed in the following section various examples based upon it to demonstrate the more salient features of the infrequent monitoring problem are included in subsequent section
143
61 Problems in One-Dimensional Diffusion with Ho-Flow Boundary Condishytions Method of Solution
Consider a one-dimensional problem in diffusion including scavenging described as follows
Figure 61 One-dimensional Diffusive system example
For the pollutant concentration pound(T) consider the following initial-boundary value problem
3 5 uraquo 5 = 0 x = U
W e(cO) = V cos ((n - D f E )
(61)
(62)
(63)
The single stochastic point source 1s defined by
144
U U T ) = OI(T)S(C - c j
E[OI(T)] = 0
E[u(T)agt(T2)] = Wlaquo(T - x 2 ) (64)
In the interest of generality transform the problem to dimension-less functions of time and space as follows
t = poundl bull
a fix K
W T raquo (65)
Substitution of (65) into (61) yields the following dimensionless form
for the one-dimensional diffusion initial-boundary value problem 9
| f = S-l - 05 + f(zt)j (66)
bull amp i | f pound U 0 z = o z = 1 (67)
n laquoz0) = cos (n - 1) irzj) (68)
n=l
and where the dimensionless point source is given by
f (z t ) = w(t)lt5(z - z w )
E[w(t)] = 0
ElXt^wttg)] = Wa^ - t ) (69)
With these generalizations the modal resistances capacitances and eigenvalues from Table (331) become the following for the dimenshysionless problem with scavenging
145
n = 1 raquo
n = 23 2 n = 23 (n-l)V
The forcing terms from (335) become
((n-l)V + a)
[ c n cos ((n - 1)TT z w)jw(t)
concentration at any point z CO
pound(z t ) = ) x n ( t ) cos fn - UirzV
12
The pollutant concentration at any point z from (335) becomes
(610)
(611)
(612)
For a sufficient number of modes to be both theoretically interesting and computationally expedient choose n = 5 for the number of terms retained in the expansion in (612) This choice will be studied later as to its effect upon the outcome of the infrequent sampling problem
Thus the modal state equations may be written in dimensionless variables as follows
1 -o
2 bullU2+a) k3 - -(47i 2+a)
4
5
-(9ir z+a)
0
J +
o x l x 2
3
4
+
lt 5
2 cos (IT Z W ) 2 cos (2ir z j 2 cos (3ir z w) 2 cos (4 z )
w(t)
(613) The initial pollutant distribution (z0) is chosen as in (68) so that from (333) the initial modal state variables are written simply as
146
8(0) = m Q (614)
The covariance of the error in the estimate in the Initial state 1s chosen to be
005
Bo s raquoo 001 o
000001
o 000001 (615)
000001 For m = 2 the two noise-corrupted measurements in the vector y are given by
X pound i v
raquo1 1 cos(nz) cos (2irz) 1 cos(nz2) COS (2irz2)
r l1
x 3 x 4
v(t) v 2(t)
(616)
where the mean value of the measurement noise E[y] 5 o (617)
Choose the position of the stochastic source as z w = 03 (618)
For this case scavenging is ignored so that a = 00 (619)
Let the source and measurement noise statistics be defined by the folshylowing covarlance matrices
W = 0125 (620)
147
OOSO 0 (62i)
0 0025 A typical output record of the problem description from KALMAN Is
shown in Figuure 62 The data corresponds to a problem with a bound on the error 1n the state estimate where the error limit Tr = 0075 At each measurement time NSEARCH pound 5 random starting vectors are to be used In the measurement position optimizations The Initial guess for the measurement positions Is chosen as zbdquo = pound015015] (called Z) The computed values for A and D are shown For a steps1ze of OT 5 001 the so-called Paynter number raquo 35 that is the number of terms used in the series approximation for e- In (49) for an error criterion of EPS = 000001 The state transition matrix pound + 1 (called AK) and the discrete disturbance distribution matrix lpound + 1 (called OK) from (412) are computed along with the Incremental disturbance noise covariance matrix g K + 1 from (414) and Appendix B (called WKP1) The steady-state disturbance covariance matrix n from (519) and (520) including the
r - SS term | ft I ) Is listed as WSS along with the number of tlmesteps NSS
Nn necessary for the Infrequent sampling approximations to be valid see (578) for the value e - 100EPS (same EPS given above)
For the monitoring problem with bound on error in the state estishymate a measurement is necessary whenever at time t bdquo + N Tr[gpound+N(zpound)T gt Tr At each sample an attempt 1s made to locate the global optimum of the measurement position vector jJ + N such that
For the initial guess of z K + H = [015015]1 and for NSEARCH S 5 other randu^ starting vectors the constrained first-order gradient algorithm
DISCft i Te KALHAN F I L T E R SIMULATION PROGRAM V E R S 2 7 5 ftOV f
PFJ03LE1 INPUT JS AS FOLLOWS
EXAMPLE TO SHOW GROWTH OF T R A C E I P ( K K + N ) J Slf l lFACE WITH T I 1 E T ( K N ) ITS SHAPE APFRCACHES THAT OF I P l K K h l l SURFACE ASYHPTOTI ALLY FOR LARGE H
WO VECTOR I S
1OODE00 1OCOEOO
CAPMO MATRIX IS 500DE-O2 -DElaquo00
-OCraquoOC 1000E-O2 OE+CO -OE+OD CE+O0 -CE+OO -CE+OO -OE+OP CAPW MATRIX IS 1250E-01
CAPV MATRIX IS
10D0EO0 tOOOE00 IOODE+OO
-OEDO -OEraquo00 000E-05 -OE+OO -OE+OO
-OE+OO -OE+OD OE00 OOOE-03 -OE+OO
-OE+OO -OE+OO -OE+OO -OE+OO 1OOOE-03
2W VECTOR IS 3000E-01
Z VECTOR IS 1500E-0 1500E-01
NUWSEft OF POINTS FOR RANDOM SEARCH INITIALIZATION IN5EARCH) bull
THIS IS A MONITORINS PROBLEM OF TKE FIRST KIND WITH A CONSTRAINT ON THE ALLOWABLE ERROR IN THE STATE EST MATE THE ESTIMATION LRROR CRITERION IS THE TRACEIPltKK+N)3 THE CONSTRAINT ON THE ERROR IU THE STATE ESTIMATE IS FIXE) AT
Figure 62A Problem description from PROGRAM KALMAN
PARAMETERS FOR SYSTEM DESCRIPTION ARE
D IFFUSION CONSTANT K 1O00E+O0 LENOT OF MTPUW L = 1 OO0E-00 SCAVCKSINO RATE ALPHA = OE+OO
MATRIX I S - O E D D OE+OO
017+00 - 9 8 7 C E + O 0
OEOO OE+OO
OF+03 OEOD
OE00 OE+OO
MATRIX 1 5
1 O0JE+O0 1 1 7 6 E + 0 D
- 6 1 0 0 E - 0 1 - 1 9 0 2 E + 0 0 - 1 6 1 8 E + O D
OE+OO
OEDD bull3 94BE+01
OE+OO
CEOO
OE+OO CE+OD
OE+OO CE+DO
CE+OO OE+OO -eee3Eoi OEraquoOO
OE-00 -1S79E+02
1OOOE+00 bullOE+OD DEC0 OE+OO OE+CO
DK MAT)
10DDE-02 1119E-02 -5106E-03 -126CE-C2 -a134E-C3
OE+DO OOeOE-01 CE+OO OE+CD OE+DO
OE+00
OE+OO 673BE-01
OE+OO
OE+OO
OE+OO OE+OO OE+OO 1ME-D1 CE+OO
OE+OO OE+OO OE+OO OE+OO 2062E-0T
WKPt MATRIX I S
1 2 S 0 E - D 3 1 3 9 9 E - C S - S 3 B 3 E - 0 raquo l Q 7 6 E - 0 - 1 0 1 7 E - 0 3 1 3 3 B E - P 3 1 S 6 B E - 0 3 - 7 1 6 0 E - 0 4 - 1 7 7 6 E - 0 3 - I 1 5 2 E - 0 3
- 6 3 B 3 E - 0 4 - i e 6 E - C 4 3 3 0 1 E - 0 4 6 2 7 pound E 0 4 0 4 5 3 E - 0 4 - 1 3 7 0 E - 0 3 - I 7 7 6 E - D 3 8 2 7 0 E - 0 4 2 1 1 5 E - 0 3 I 4 2 7 E - 0 3 - 1 0 1 7 E - 0 3 - 1 1 S 2 E - 0 3 5 4 D 3 L - 0 4 1 4 2 7 E - 0 3 9 9 2 I E - 0 4
WSS MATRIX I S
9000E-02 143BE-02 - I 9 5 7 E - 0 3 -2 C77E-03 14d6E-02 A7MG-03 - 1 M 0 E O 3 - 2 0 3 2 E - 0 3
-1 957E-03 -I e^OE-03 6047E-04 1 I45E-03 -2 677E-03 -20(2E-Q3 1145E-03 254GE-03 - I 231E-C3 -I 4 1 E - 0 3 6333E-04 1 559E-03
bull1281E-03 bull l 417pound 03 6333E-04 1559E-03 1-036E-O3
THE NUMBER OK TEftK$ I N THE TRUNCATED MATRIX CCMVOLUTION SERIES FOR THE STEAOT-STATE VALUE OF tUSS) NSS 71
Figure 62B Problem description from PROGRAM KALMAN
150
KEELE produced the results for the first measurement partially shown in Figure 63 The global minimum is chosen as the best minimum found after the NSEARCH + 1 attempts
Figure 64 is a time history of Trlppound+N(zJ)] that is a plot of the performance criterion with the optimal measurement positions from time t K used in its evaluation between measurement times t K and t K N Three sample times are shown at t = 009 048 and 088 At each samshyple the optimal positions of the m = 2 measurement devices with covari-ances given in (621) are found such that the time to the next sample is maximized Examples of actual state and optimal state estimates are shown 1n Figure 65 In the plots those labeled X() are plots of states with time those labeled XH() (mnemonic for ( or x-hat) are the corresponding state estimates
In assessing the globil optimality of zpound and P found at time t K
(as in (62)) contour plots are constructed for the objective function [P^(j K)] 1 1 plotted against [z K] horizontally and Is K] vertically The minimum plotted value is noted with a the maximum with a 0 In between are nineteen equally spaced levels denoted with the symbols ()( )(D( )(2)( )(9)( )(U) The actual evolution of the optimizashytion calculations can be followed with such contour plots in order to understand the procedures of the algorithm More importantly study of the contours serves as an important method of understanding the nature of the design problem since the plots convey a level cf information otherwise not available through tabular listings or other means
At each sample time say t K + N the predicted covsriance matrix IK+N is written out for post processing and after the entire time intershyval in the monitoring problem is covered contour plots of the
THE NUMBER OF CALLS TO FVAL IG 1 1309346B3E-02 1ODO00000E+00 213471279E-01
THE KUKBER OF CALLS TO FVAL IS 7 127494646E-02 1OOOOOOOOElaquo00 1C3265064E-01
THE NUriBHR OF CALLS TO FVAL IS t 1 367C4440E-02 437O71939E-01 601669468E-OI
THE KUM3CR OF CALLS TO FVAL IS 19 12644I4E9E-02 21 J255890poundgt01 515S4B271E-01
THE NUMBER CF CALLS TO FVAL IS 1 146922GD4E-02 374187311E-01 B92S8163eE-01
THE NUMBER OF CALLS TO FVAL IS IS 1264J1463E-02 211254872C-01 S15347999E-01
THE NUMBER OF CALLS TO FVAL S 1 162042943E-02 5O7662490E-01 laquo00351916E-01
THE KUKBER OF CALLS TO FVAL 13 13 12B441469E-02 2t126264SE-01 3155529S3E-01
THE NUMBER OF CALLS TO FVAL I S 1 1SB617996E-02 3a5314991tgt01 27e840503E-01
THE NUMBER OF CALLS TO FVAL IK 11 126982870E-02 6621772E5H-01 1 67144930E-01
THE NUMBER OF CALLS TO FVAL IS 1 132010362E-02 2273t1246E-01 663S29703E-01
THE NUMBCR OF CALLS TO FVAL 16 442 1 E6441469E-02 2 U235SC4r-01 6I3540379E-O1
BEST LOCAL MINIMUM FOUND AFTER B TRTS I S 126441469E-02 211234672E-01 315347999E-01
Flpure 63 Sunmary of results of minimization of F P ^ Z ^ ] at the f i r s t sample time from SUBshyROUTINE KEELE r K ^ K ltJ l l
eooooE-o2
B3000E-02
42500E-OZ
X X X x ) x x x x x bull 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 X X X X X X X X X A X X X
x x x x x X X X X X X X X X X X
x x
Figure 64 Time response of TrJpK+MfZ|)Jraquo the performance criterion for the optimal monitorshying problem with bound on error in the state estimate samples occur at t K = 00D 048 and 088
B6900E-01
S5BOOE-01
947O0E-01
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
XX XX
X
x X X X
X X X
XX
X X X X
X X X K X
X X
X
X
X
X
XX X X X
XRXX
XX XX
X X
X XXX
X
X
K
Figure 65A Trajectory of the f i r s t modal state [ K + N ] raquo versus time t K + f J
1OO3Opound00
xxxxxxxxxxxxxxxxxwoooutxxxwuwxxxxxxxxxxx
xxwoouooc
XXWOWKXXXXX) OIMXXXXXXXXXXXXXXXXW ucwxxxx
Figure 65B Trajectory of the optimal estimate of the f i rs t modal state time t K + s bull [ -K+NJ T versus
1000OElaquoO0 X
XXXJUUM WWXXX
-IOOOOE-01
Figure 65C Trajectory of the second modal state [SR+H] 2 versus tine t K + N
6000GE-01
JOOCIE-01
ZOOOOE-Ot
IX
1 X
I X
1 X
1 X
I X I X
I X I XX I X 1 X 1 X I X I X I X 1 X
i V I X 1 ft K XX XX XXX xxxx xxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Figure 65D Trajectory of the optimal estimate of the second modal state I E K + N ] versus time t K + N- L J 2
157
[ppound +J)(z + N)] surfaces are made for each sample time Much use of
these plots is made in what follows
62 Problems with Bound on State Estimation Error
621 ftsyaptotic (tesporso of Stats Estimation Error mdash Fov the
monitoring problem with bound on allowable error in the estimate of the
modal state vector i t is necessary to make a measurement whenever for
a time tK bullK+N
T BK + N(SK) i T r l t a (623)
that i s whenever the trace of the error covariance matrix predicted
from the last measurement at positions z bdquo at time t bdquo to time t K + N reaches
the estimation error l i m i t T r
In order to numerically substantiate the fundamental results for
the Infrequent sampling problem contained in conclusions I I I I I and
I I IA the relationship between T l lpoundJ( + N( K)J and [pound()] is now conshy
sidered Suppose the monitoring problem is started at time t Q with
PS 5 Hbdquo as the i n i t i a l value of the error covariance matrix le t i t s -0 -0 value then be predicted ahead to lime t bdquo when
Tr[pdeg]= Tr i V nV l T gt T r z i r a (624)
at which point a measurement must be made The monitoring design probshy
lem is to choose pound at time t K so that the maximum time t K + N results when
For a measurement at 2 K the corrected estimation error covanance mashy
t r i x 1ltmed1ately af ter the measurement is given by
158
$(h) - PKdeg - $ ( [5(2K)EK-C(K)T + secthgt ampbullgt where
^ K )
1 cos (TTZ) cos (2TTZ)
1 cos (irz2) cos (2TTZ 2) (627)
In order to generate a contour plot of Tr[ppound(jK)] from (626) plot values of Tr[Pj(zK)] for all values of the elements of zraquo over the full length of tne medium (0 lt z lt 1 and 0 lt z lt 1 in (627)) The surface for the first sample at t R = 009 1s shown 1n Figure 66
To study the evolution of the trace of the predicted error covari-ance with time as a function of the measurements at time tbdquo let
-PK+I(SK) bull lt(SK)S T +
~PK+2(K) lti(Kgt T + 8
n=l (628)
Contours of the traces of the above predicted covariance matrices at tines t K + t K + 5 t K + 1 0 t|+11 and t K + 5 as functions of jo are shown in Figure 6-7 Notice how tht global minfmum moves with time ote also how the error 1n the estimate In the region near the stochastic source (z w = 03 along both coordinates z 1 and z 2) Increases in v ^e as time grows relative to the rest of the surface due to greater uncershytainty in the estimate in that area
CONTOUR PLOT OF TRACE[P(KK+Ngt(2(Kgt11 A3 FUNCTICI CF [Z(K)31 HORIZ C2(KM2 VERT EXAMPLE TO SHOW CROWTH OF TRACECPIfcK+Hll SURFACi UlTH TIME TIK+N) ITS SHAPE APPROACHES THAT CF [P(KK1J11 SURFACE ASVMPTOTICALLT FOR LAROE H
10 393 44 3 222 599 44 3 222 555 44 3 222 39 44 33 222 3 44 33 222
OS bull 44 33 222 44 33 222
444 33 222 444 33 222
444 33 222 06 444 33 222
4444 33 22 4444 33 22Z 4444 33 222
4444 33 222 07 bull 4444 33 22
4444 33 22 4444 33 22
4444 33 22
444444 33 22 0 6 M4444 33 2 1
4444 33 222
44 333 22 1 U f K l l Z 333 22 1
3333333 222 1 0 3 333 222 I
Z2222 2222222222
2222 33 4 S 6 77 2222 33 44 S laquo 77 2222 33 44 3 6 77 2222 33 44 3 66 777 2222 33 44 3 3 BB mdash 2222 33 44 S3 06 2222 33 44 S3 i-2322 33 4 S 61 2222 33 4 S 6L
222 33 44 33 66 222 3 44 S3 66 222 33 44 S BBS
222 33 ~
8(138 99999999 BUSS 99999999 S03B 999999B9 1)388 99903399 03986 93999999
777 888886 S9 J9999999 7 6883886 9999939999999 777 8838888 9S99999999
7 77 68868688 95J99 777 eeeeasses
77777 6888888688 77777 866886868666888
777777 6086666868886 666888686
04 -111111 111111 1111111 1111111 03 +1111111
1111111 i u m i m m 111111
oa + i i n
22222222222222222 22222222222222222222222
22222222 2222222 22222 22222 2222 333333 2222 222 3333333333 222 222 333333333333 222 222 33333333333 222 222 333333333 2222 2222 2222 222222 22222
2222222222 222222222 22222222222222222222
222 33 4 33 6G6 777777777 22 33 44 S 3 66S 7777777777 22Z 3 44 33 6663 7777777777777
22 3 3 4 4 3 3 GGXC 77777777777777 22 33 4 33 BE5636 77777777777777 222 33 44 33 3pound66S6GS6 777777
22 3 44 353 -36666666666666666 22 39 44 533S 6666666666666666686 22 33 444 355553553 222 39 4444 33333333553533353355333353333
22 33 44444444 222 333 4444444444444444444444444444 2222 3333333333333333333333333333 lt
22222 222222322222222 2222222222222222222222222pound222222
2222Z222222 11111111111111111111111111 t i l 111 m i l -
1111111111111111111111111111111111111 11111111 111111 111111111 111111 1 i u u i n 11111 11111 11111 11111 11111 11111 _ 11111 0
22222222222222 222222 01 +333333 2222 3933 2222 4444 333 222 44444 333 222 444 33 222 OO + 444 33 222 HH
urn i n i t i n
1 l 1 l l I 1 t l 1 l 1 1 1 l 1 1 1 1 U 1 l m i n i u m t m m t i i n n n i i
2222222222222 2222222 22222222222222222222222222222222 2222 3333333 222 333333 3333333333333333333333399333333 2222 3333 222 333 4444444444 222 33 44444444444444
TtK+N)raquo 90000E-02 T(K bull SO000E-O2 N bull O STEPS AFTER FIRST MEASUREMENT
T K S S S (S) (91
d616pound-02 3369TE-02
i e i (6)
33166E-02 32440E-02
(7) C7) 31713E-02
3O690E-O2 16) (6) 30265E-02
29540E-Q2 (3) (31
26814E-02 26089E-02
(4) (4)
27364E-02 26539E-02
(3 ) (3 )
23914E-02 23163E-0Z
(2) (2gt
244E3E-02 23738E-02
(1) lt1gt
23013E-02 22268E-02
(8jraquo 21363E-02 ESTIMATION ERROR CRITERION CONSTRAINT bull
78000E-02
figure 66 Contour plot of TnP|[(Sv)| a t f i r s t measurement time t K = 009
ITS SHti-e APPROACHES THAT CF lPtKKgt311 EbRFAC ~ IAYMPTOTICALLV FOR LAROE N
tZ(K)32 0 5
laquo 4 444
44444 ltgt444 4144 4444
444 3 444 3 444 33
222 222 222
2222 2222 2222 2222
i 222 pound2
2222 222 272 2ZZ
-1J4 444 44 444 44
33 222 03 222 33 222 33 22 333 333 2 3133 22 33333 222 2222 22222 222222
11111 1111111 111111 Mill 1111 111
111 1 111
22222 33 4 5 66 7 8338 9999999 0-22222 33 4 S 66 7 8e88 S999939 22222 33 4 S 6 7 BBC3B 933999S9 22222 33 4 5 6 7 7 8B380 99992939 22222 33 4 S 63 7 eSBGQ 93939993 22222 33 44 S3 66 77 6C8E68 933^9999999 222222 33 44 S3 60 777 8386888 99999S93999 22222 33 44 0 66 777 683B(8d S9999939 22222 33 4 55 6S6 7777 CSBBBSBB 2222 33 44 53 66 77777 088888888 22I-2 33 44 5 66 777777 08866886888 laquobull 2222 33 44 55 St 6 777777 8833668888880 222 33 A S3 664 77777777 88888888 222 33 44 55 tB-1 7777777777 111 222 3 44 55 6iC6 77777777777 11111 222 33 44 55 60566 7777777777777 1111111 222 33 44 55 UE66666 7777777777777 II 111 111 22 33 44 555 666666S666 777777 11111111 222 3 44 551 66666666666666 111111111 222 33 44 6-5 66666666G666666666
III 111111 2 2 3 3 44 pound5^5533553 66666 1111111111 222 33 444 5355355555533555555555 1111111111 222 33 4444ltM 55555 11111111111 222 3333 444444444444444444444444444444 11111111111 2 2 2 333333333 111111111111 22222 33333333333333333 111111111111 22222222pound22222222222222222222222 111111
222222 22222222222222222
222222 22222 2222 33=3 22222
2222 333333333333 222 222 33333333333333 222
2222 33333 333333 2 2 2 2222 33333 33333 222 222 33333333333333 2 2 2 2222 3333333333 2222 22222 pound222 1
222222 222222 11 2222222222222222
1111111111 111111111111 111111111 II 1111111 II II 111111 1111 1111111111 n i m i i i i 111111 H I m m 11111 i n i i
pound22222222 22222
333333 2222 3333 222
444lt] 333 222 441444 333 2pound22 4M444 33 222
1111111 1111111 1111111 111111 111111 1111111 1111111 1111111 111111
111111111111111111111
11111111 111111111111 11111111111 1111111111
111111 222222222
2222222 2222222222222222222222222222 2222 333333
2222 333333333333333333333333333333333333333 2222 3333 333333333333333333333333 222 3333 3333333333333
TCKNgt 10000E-01 T(K) bull 90000E-02 N bull 1 STEPS AFTER F r RST HEASUHEHENT
SYPcopy LEVEL RANGE
-s-srapoundsi m 35902E-02
35248E-02
i 34594E-02 33940E-02
33265E-02 32631E-02
i 31977E-02 3 1323E-02
30668E-02 30014E-02
s 29360E-O2 26706E-02
26051E-02 27397E-02
i 26743E-02 26CB9E-02
3434E-02 247eOE-02
(copy) 24T26E-02 ESTIMATION ERROR CRITERION CONSTRAINT =
750D0E-D2
12SJCE-013
Figure 67A Contour plot of measurement
T rfei(0] U K+1 010 one timestep af ter f i r s t
161
r w S z
m m n_ lnM bull MM ampnm J 5 8
pound8 SS8
totacopy t^f
I WW
laquo5S N K Jill timctmo B O O
ltoia mio v mm vn hi
ogtn M O W --
- w o n mdash ni Bin bull bull- w o n - w o n
a-o w - raquo - bdquo bdquo _ _
_ _ n n (M mdashmdashraquo- ~mdash^mdashlaquo-mdashmdash~raquo m r t r t o V T I V laquo o w - - ^ - - _ - - - - -
- n n m o n m ltrwMM nn w w - raquo - - - bull - - - -mU)D M T H J ^ M laquo r n w ^ ^ raquo - mdash mdash mdash bull mdash
M lt T M laquo n n n t i i ajpi raquo - - - bull bull nnnnnei laquo laquo - ^raquo - r - r - r -
n n n n ftiNw ^ - bull w w w m i i i - i n n o gtWNlaquo mdash _ bdquo raquo - _ CMVWMIM
n d n o n n n wcyNWh) mdashmdashmdash_- - ^ NNMt twNN laquo OjttOjCVWN bdquo - ^ raquo filtM laquoM
- - bull bull - bull - mdash -bull MU OO laquo
W
- N nnn bull
bullmdashgt- w w
III NiMdiuW
(MCMNfcrw
Bio
F-uu cvw lt laquo(jftlfCVJ
U S O -WMWtVWhJ
raquo-raquo- w
N mdash bull- mdash mdash
si WAituww n o n W N
WMW mdashZZ
CONTOUR PLOT OF TRACpoundtPfKK+HgtltZOOU AS FUNCTION OP t2ltK131 HOR1Z EZltK))2 VERT EXAMPLE TO SHOW GROWTH OF TRACEIPIKK+Hll SURFACE WITH TIME T(ftN) T6 SHAPE APPROACHES THAT OF (F(KK)311 SURFACE ASYMPTOTICALLY FOR LAROE N
0 2
Z S 2 2
aa 2g2 933 2222
333 22g 3333 222
333 33 222 333333 Z22
33333 222 bull33333 22
33J33 pound2 33333 22 35333 222
3333 22 33J 222
444 3 444 3
Aft W 44444 33 44444 33 +4444 333 444 33
333
22ZZZ222222222222 Z22222222222222222 22222222222225^222 Z22ZZ222Z222222222 22ZJ22222222222222
22222 22222222222 ~ 222222Z222
22Z22Z2222 222222222
2222222
333 44 H S 333 4 9 6 933 4 B TO
33 4 S3 66 33 44 S3 61 333 44 9 61
mdash 44 33
999399 O 999999
999999 9939999 99999399993 9999SS9
Mil
7 0B0BB 7 88088 n eases 7 BBSBS 77 BBSBBB T 7 BBBBBB
bdquo - ^ r 77 B680CB 33 44 9 3 68 77777 098888
33 4 9 668 77777 B6BBBBB3 33 44 raquo3 BB1 777777 BBBBBBBBBSBB
- - - 6raquor 7777777 60086886 - - laquo16G6 77771
33 4 S3 66666 77777777 -~ 353 66666B 777777777777
333 66666B66 777777777 SSJ 666666666 777
I SU5S 6066686666 bullJ353333 666666B666666B C666666B
222222 33 22222 33 44 33
222 mdash -222
222 i 22 222
11111111 11111111111111
11111111111111111 111111111111111111 111111 111111111
11111 111111 bdquo - -m i m i l 22^
1111 222 1111 222
111 222 3333133 111 2222
11111 22222122 2222 11111
1111111
2222 111 22222 1111
1111 111111
111111 l l i m
i m i m i i m m 11111 111111 11111 222222222222 11111 111111 222 2222 11111
111111m J g z 2 M 3 3 3 M 2 L - L - 1
444 pound33333333333 1 4444 9S5S35555S3B 13 44lt14ltM4 033333533353
44444444444
bull bull 1 1 U I U 1 1
liliSHn
3333 333
333 444444 333 444444
3333 222 33 222 333 22 333 22 333 222
333 222
11111111111111 111111111111111
m j u i m 01 +222222 111111 222 1111 33333 222 11 333 ---
_ 333 _ 222 33333333333 -raquo2 11111 2222 2222 11111 22222222222222 11111
1111111111111111111 bull 1111111111111 1111111111 1111111111111111111111111111 111111111111111111111111111111111111111 11111111111111111111111 1111111111 1 1111111111111111111111111111111111 m u 1 1 m m 111111111111111111111111111111
111111111111111111111111
0 0 bull144 33 333
sect22 22 111111
111 (11 11111111 11111 222222222 12pound2222222222g22a222222222222 111 2222 1 222 1 2222
T(KN1 19000E-01 TOO bull bull -0000E-02 N bull mdash 0 i E 3 AFTE FIRT BEASUREHENT bull bull bull ^ bull bull bull bull bull bull bull bull bull laquo B
COHTOUR LEVELS laquo0 SYPBaLS bull i i ^ i i i m i i i i i V1Vamp LEVEL RANGE i g g f i e e a s a t i i i i i
(0 47GG7E-02 19 (raquo m 171
47143E-02 46623E-02 46102E-02 40380E-02 4S059E-02 44S37E-02
( 6 ) CB)
44015E-02 43494E-02
IB) (31
42972E-02 42431E-02
11 t4J 41929E-02 41407E-02
13) 40Be6E-02 4 0984E-OS
(2J C2)
3SB43E-02 39321E-02 38799E-02 3S27SE-0Z
CM 37756E-02 I H ^ t H I I I I I I I I I ESTIMATION ERRdR CRITERION CONS^-AINT
7S000E-02 COVARIANCE tWJ
Figure 57C Contour plot of Tr ElLinfe) a l t 1 m e t ^ m - 0-19 ten timesteps after first measurement
CONTOUR PLOT OP TRACEIP(KKraquoN)lt2(KgtJ3 A3 FUraquoeTteM Of [ZCKI31 HORIZ CZIK12 VERT EXAMPLE TO SHOW OROWTM OF TRACEIPtk KN)3 itftACE WITH TIKE TltK+N) ITS SHAPE APPROACHES THAT CF [PIKfltJ311 SURFACE ASYMPTOTICALLY FOB LARUE l
bull 444 444 444 44144
333 33
06 bull 333 2222
333 pound22 3333 222 333333 222 33333 pound22 33333 22 0 7 raquo33333 22 33333 22 33333 22 33333 222 1 3333 22 1 OC 333 222 11 222 111 2222 1111 EZltKgt)2 22222 111 1111 0 9 bullU111111 11111
3 22222222222222222 3 22222222222222222 3 222222222pound2222222 2 22 222222222222222 222222222222222222 22222 222222222 HZ 2222 2222222222 2222 222
SBBflS eoeos 63886 eeeee 777 695808
laquo99999 0 939999 999999 999S939 99999999 9999909999 9999999
333 46 3 0 7T7 333 4 fl 66 -7 333 4 a ee -7 33 44 55 66 ~~ 33 44 55 61 333 44 S 6B 777 608689 _ 33 44 S3 63 7777 6BSofl8a 2222222222 33 44 59 CF 77777 aaSOBd 2222222222 33 44 53 6(6 77777 6638668 2222222 33 44 53 pound68 777777 680308888888 222222 33 44 33 ecEB 7777777 66380888 22222 33 44 S3 Gamp666 7777777 222 33 44 35 66666 77777777 1111111 Z22 33 44 35 6665666 7777777777777 111111111111 22 33 44 3L5 66666669 777777777 11111111111111 222 33 44 ESr3 66666866 77 111111111111111 22 33 444 311555 666666666 11 11111111 7 33 444 i353S533 6666666B66666 11111 22 333 444 55553535553 6666666 11111 22 33 444 55555535333 1111 22 333 44444444 33353533335lt 1111 22 333 4444444444444444 111 222 33533333 4444444444 1111 2222 333333333333333333 1111 222222222222222 111111 222222222222222222 1111111111111111111 1111111111111111111111111111 111111111111111 1111111111111111 11111 11111 bull 11111 222222222222 11 111111 222 2222 111111111 222 3333333333 222 1111111 11111 03 raquo111 111 111 11111 bullIU111 02 - 1111 11 -111111 11111111111111 1111111111111111 1111111111 bull222222 u n t i l 2222 1111 33333 222
333 222 33 222 00 +44 333 222
222 3333 333 22 333 333 pound22 222 333 4444444 33 22 222 333 4444444 33 22 222 33 444 333 2ZZ 222 333 333 222 2222 33333333333 222 11111 2222 2222 11111
1111 1111111111111111111 11111 11111l1111l1llt1l1111 111111 11111111111111111111111 11111111111111111111111111111111111 111111111111 111111111 nil 111111111)111111111111111111111111 11111111111111111111111111111111111 11111 11111111111111111111111
22222222222Z2 1111
01 11111111 11111111 11111111 11111111 11111111 11111111 11111111
1111 111 1 111 I 111111111111111111111111111 II111111111111111 111111111 111111 2222222222222222222222222222 1111 2222 22 111 222 333933333 3333333 I I 2222 333393333333 3333333333
T(KN)raquo 20000E-D T(K) bull 9C003E-Q2 N raquo I I STEPS AFTER FIRST MEASUREMENT
SYlaquoe LEVEL RAN3E (01 4B911E-02 (9) (9)
483g4E-02 47677E-02
(61 (8 ) 4735SE-02 46B42E-02
(71 (7 )
46323E-02 4S807E-02 (6) 16)
43200E-02 4 4773E-D2 IS) (5 )
44255E-02 43738E-02
(4 ) (4gt
43221E-02 42703E-02
C3) (3)
42166E-D2 4I6SSE-02
(2 ) C2)
41I31E-02 40634E-02
( 1 ) 40117E-02 33539E-C2
(6Jgt 390B2E-02 ESTIHAT ION E ROR CRITERION CONSTRAINT bull
75000E-02 souacEINPUT COVARIANCE [U]gt t I2530E-011
Figure 67D Contour plot of T H P I M I U K M moaciirAmont
at time t measurement
K+ll 020 eleven timesteps after first
CONTOUR PLOT OF T R A C E I P 1 K K N H Z ( K J ) J A S FUNCT13K OF t Z ( K ) 1 1 H C S l Z ( Z ( K ) 3 2 VEftT EXAMPLE Tfl s w a y cRCWTH CF TftACElPCRKH) 1 S U R F E WITH TIHE T(Kraquofl ITS SHAPE APPRCACML3 THAT OF I P t K K j - SURFACE A-irKPTOTICALLY FOR UtfWE K
444 33 444 33 4244 33 44a44 333 44444 33 09 +4444 333
22222Z2222222222 2222222222222222 2222222 2X22222 22222^222222222 222222 222lt222222 222^2iVLaJi222222Z 444 03 222222 2222222222222 33 22222 222222222222 333 2222 22^2^22222 333 2222 222222222 33 222 22222 222 pound2222 222 222 222 11111 2 22 m n i n n m 2i 22 11111111111111111 22 1111111111111111111 122 1111111 1111 111 till
333 44 S 68 333 4 3 65 333 4 5 66 33 44 S3 66 33 44 55 66 333 4 53 66 mdash 44 33 61 77
6BB0C 8003 esses csoese esses
99P999 339333 993999 9939339 99393399
3333 333333 33333 33333 33333 3i333 33333 33333 3333 222 Z22 2222 111 2222 111 1111 1111111 1111
1111 111 111 11
1111 1111 Mil 111 11U 111
777 eOOSSfi 9999999399 __ 7777 688888 9999999 33 44 35 6S 7777 6088898 33 44 03 5pound 777777 eceaeceo 33 44 53 euro6S 777777 608828833038 mdash - -s r66 7777777 60888008 bull55 6EG6 7777777 3raquo5 665666 7777777 _ 33 fifl 3 5- 66C65B6 777777777777 gt22 33 44 5 5ES 66366666 77777777 22 33 44 3Si3 65B6SSB6 222 33 44 SJSSSS 6666666BB 3 444 53353333 66666G656666 33 444 3U55S35S333 666866 bull 333 elaquo4 533S353353 2 333 c444444444 5355S35333 22 3333 44444444444444 222 3 3^13333333 444444444 2222 33333333333333
22 222
222222222222222 1111 1111111111111111 11111 11111 11111 22222222222 1111 111111 2222 222 1)111 111111111 222 33333333333 222 111111 III 1111 22 333 3333 222 1111111
1111 pound 2f2222222222222 111U1 111111-1111111111111 m i i n H i m t u i m m m i i i i t i i m i i i t
11111 222 33 44444 333 222 11 22 33 44444444 33 222 222 32 4444444 33 222 1 222 333 44444 333 222 1111 222 233 333 222 111111 111111 222 333333333333 22 1111 11111111 pound222 222 1111 11111111111 22222222222222 11111 1111111111 1111 n m m i m m i i i i m i t m i i i i i 222222 1111111111111 2222 1111111111 33333 222 11111111 333 222 I1M111 33 Z22 111111 44 333 222 11111
11111111111111111111 i i m n m i t r i m m u r n 1 1 1 1 1 1 m m m m m m n l i m i t 111111111111111 IU1111 m m i m i i i m m u r n l i m u m m u i i m m i m m i n i i i i i i m i i n i i i i i i m i m i m m i m i m m m m m m m m m m m
i m m m i m m m I m m 11
m i m i n i u r n m i m m i m i m t m m i m 1111111111 222222 raquo222222222222222222Praquo222222Z22222 11111111 22222 2222 1111111 2222 3131333 3333333 111111 222 33133333333 3333333333
TCKNgtraquo 240C0E-O1 TIKI bull 9000CE-OZ N bull 13 STEPS AFTEB FIRST MEASUREMENT
SYR3 LEVEL RANGE (0) 338S9E-02 (9) 19) 3 3389E-02 32asOE-02
(6) 3237IE-02 51862E-02 17) 17) 3 13S3E-02 30B43E-02 (6) (6) 39S34E-02 49S25E-02 (5) t5) 4931CE-G2 46607E-C2 (4) 14) 48297E-02 477Q0E-O2 (3) (3)
47279E-02 46770E-02 (2) 12) 462G1E-02 45751E-02 11) (1) 4S242E-02 44733E-02 (copy) 44224E-02
ESTIHATTON11
ERROR CRITERI0M CONSTRAINT =
7SO00E-02
IS500E-011
Figure 67E Contour plot of Tr p pound 1 ( z bdquo M at time t bdquo 1 i 024 f i f teen timesteps after f i r s t measurement L K + 1 5 ^ K J K + 1 5
CONTOUR PLCT OF TftftCEIFlKKN) (ZtK) J 7 AS FUNCTION OF IZ tKI I I KeRIZ IZtK)32 Vf=T OIAMPJS O SHOW GfCiWTH CF TRACEtP(KKlaquoN)l SUKFCr WITH TIKE TltKNgt ITS CH-PE APPROACHES THAT OF |P(HK)]11 CURFACt laquoSYKPTCTfCALLY FOR LARGE N
1 0 544 33
OG
EZCJOJS
533laquo3
+1313J J3H33 33333 3333
laquo
3 3 3
2K
l l | S l l l | | 2 J 3 CC d 53 poundCgt
0OCB3 Epound-008 pound3088
poundbull)
Z2 111 1 1 1 222 111 2222 111 2222 111 1111 bull1111111
111111111 1111111111111 111111111111111 1111111111111111 1111111 1111 111
90J099 99909ltJS9 55 6CG 7777 8B00CG 9990993959 44 23 GC 7777 688086 99999D9 333 44 C5 C6 77777 pound00386 33 aa 55 t5 77777 eooeraee 333 44 S3 (1pound6 777777 8380C8e0923 33 44 53 e6Dr 7 777777 noc8309 333 -14 Sf 56tgtDS 7777777 33 44 515 GG666 777777777 333 444 fji 0065656 777777777777
111111 11117 11 11111 1111111111111111 11111 11111 11111 2222222222pound2 1111 111111 222 222 11111 111111111 2 333333333333 22 11111 111111 2 333 333 222 11 24 333 4444444 33 22 2laquoipound 33 444444444 333 222 232 353 44414441 333 222 22 33 4444444 33 22 11 222 333 333 22 1111111 11111 222 33333 3^3333 222 11 HI 11111 ill 222 333 2222 1111 llllllllli 2222222 222222 11111
222 33 44 pound55 egt6igtEEGG6 77777777 22 33 444 Ii3i5 GG36CG666 222 33 44 35SS5 6(gtGGG66GG 22 33 44 SS5amp55555 C360GGDC5G3 22 33 4V4 55555555535 6G6GS 22 33 Mfl4 555S555555 22 333 44444444444 555555555 1 322 331 444444444444 1111 222 333333333333 44144444 1111 22227 33333333333333 1111 E222222r2222poundZ2222
m i l l t u i m i i M u 111111 i i i i i i n i t i i i m u
1111111111111111 1111111111111H1I
11111T11 1111111111111
2222222222222 111111111111 1111111111111111 llllltl 111111111 11 ill 1111111 111111111111111 1 111111111 11111111111 11111111111111111111111111 1111111111
11111111 111111111111111111111 11 11111111111111111111
111 1111111111 111111111 11111 111111111 +222222 111111 2222 11 1 33333 22 1 323 222 33 222 333 222
2222 i t t u m m bull1111 i n m i i i i i i i m m i i i i i m i m m i i i i m i i n i i i i i i -n i i m i u rn 111111111 22222222222222222222222222222222222pound 1 til 11 2222 2222 111111 2222 33C3333 3333333 111111 222 3333333333 3333333333
TIHE = 90000E-O2 F1R3T MEASUREMENT ELEMENT( 1 1)
CCNTO h LEVELS AND 5YKEULS SYMB LEVEL RANGE (0) pound 2200E 02 (91 2 1697C 1 1S4E Q2 02 ltegt 2 (6) 2
0C91F 01 OLE 02 02 (7) 1 (71 1 9680E 3103E 02 02 (5) 1 16 1 eampeoF 8177S
02 02 (5) 1 iSgt 1 7G74E 71gt1E 02 02 (4) 1 (4) 1 65^ TIE 6165E 02 02 (31 1 (31 1 5663E 5160E -02 -02 (21 1 (2) 1 4fr57E 4154pound bull02 -02 (1 ) 1 lt1gt 1 365 IE 314DL -02 -02 lQ)_t 2645E-02
ESTIMATION ERROR CRITERION CONSTRAINT =gt 75000E-02
SOURCE INJUT COVAKIANGE IU1 = C 1 2500E -on MEASIttCMEHT ERROR C0VAR rvj = t 050 I -0 -01 0251
Figure 68 Contour plot of E K ^ I I I asymptotic response of
at f i r s t measurement time t R = 009 compare with T r [~W M surface at t K+15 024 in Figure 67E
166
68 shows that for all values of z R
4 - bull bull bull
As N increases so does the convergence to the result
Finally to demonstrate the result in Conclusion II a contour plot of [Ppound(Zbdquo)] is shown in Figure 68 Comparing the traae of P at time
-f -N I] Vt-15 1 n F i 9 u r e 6- 7 E w i t h t n e OU-efceman of P at time t K in Figure
r all values of zbdquo
[EWB^K)]-^)]- lt 6- 2 9) o does the convergence to the result
^ T K + N ( K ) ] = [ ~ P f e ) ] n - (630)
Another way of seeing these relationships is as follows Write the trace of both sides of (628) as follows
4u4 -([jampol 4M 2 2
+ M 3 3 + - ) bull feu bull tS322pound] 4t]) ESJ33 J bull||- 1 gt bull )
X n=l n=l (b31)
where the two lines in (631) correspond with the two terms in (628) As N becomes large since 0 lt lttbj lt 1 i = 23raquo all the terms in the top lin anish except the first which remains unchanged with N For large N the first term 1n the second line grows continuously at a rate [SJn P e r l 1 m e s teP while according to the asymptotic relationshyship (520) all the other terms approach steady-state constants over N The meanings of Conclusions I and II are clear in (631) in that at time t K + the only term of Tr[P[+N(zbdquo)] which is still a function of z K is [P^Zj)]- none of the other terms effect the optimization over values of z K
Heuristically the response of the surface of Tr[Pv+M(i|()] o v e l a H values of zK as t K + N grows can be thought of as follows
167
EUK)] = T f | ] + [ e ^ ) + Nig] (632)
which may be studied schematically as in Figure 69 For successive values of N the contour of the surface of T r rPjJ + N (i K ) I I over z R is com-
i posed of the contour of [ P pound ( Z bdquo ) ] plus a constant value of Tr[ pound2] plus ~K ~K i i s s
a value which grows with t ime NEgJ^ The shape of the contour
Tr[ppound + f J ( K )3 should be exaatlythe same as the shape of the [P j^ (z | lt ) ] 1 1
surface and the value of a point anywhere on those two contours should
d i f f e r only by a constant
Figure 69 Asymptotic growth of TrlE^J
As a simple verification compare the values on the two surfaces for the global minimum itself the point plotted with a From the calculations for time t K = 009
[Pfc)]u deg- 0 1 2 6 4 5- (6-33) For fifteen steps after the sample at t K + 1 5 = 024 from Figure 67E
168
Tr -K+15 ( z ) j = 0044224 (634)
To estimate the stsady-state constant in (632) and Figure 69 hand ca l shy
culate the series in (631) by using only the f i r s t few terms and use
values for Q (called WKP1) from Figure 62 to obtain
11 = 1 N - 1 5 fl = O00125O N nn - 001875
bull 2 = 09060 0 22 + 22 + bullbullbull) ~ 55485 fl22 = 0001568 n 22 E 22 = bull 00080
33 bull= 0673B ( 1 + 4 raquo + 3 3 + - ) l-am ( 1 3 3 = 0000330 n 33 E 33 - 000060
44 = 04114 ( l + 44 + 44 - ) - 12037 n 4 4 = 000215 4444 bull 000255
hs bull= nraquo06Z ( + 55 + 55 + bullbullbull) Umdeg
Npoundgt11 + T j s 8 s] =
poundlg5 = 0000992 n 55 r 55
+
000104 bull= nraquo06Z ( + 55 + 55 + bullbullbull) Umdeg
Npoundgt11 + T j s 8 s] =
poundlg5 = 0000992 n 55 r 55
+ 003163
( + 55 + 55 + bullbullbull) Umdeg
Npoundgt11 + T j s 8 s] = n=l (635)
~gt W ~ 001288
(636)
Thus from (633) and (635) approximate (632) at z as
[ P K ( Z K 3 I + N a + T r L | ] + N f i 1 1 + T r | Ci = 004428 (637)
I t is thus seen from a simple hand calculation that (634) and (637) are V
in close agreement thus values on the two surfaces nP K(z K)] and Tr[Ppound+ls(Z|)] do in fact differ only by a constant the constant in (635) For increasing values of N t K + M tbdquo N etc as in Figure 69 for N T+N K+N large any point on the Tr[Pbdquo+f(g1)J contours would then simply consist of Tr[ 8] from (636) added to Nfn] plus the value at the same point
The Tr[Pbdquo + N(zbdquo)j surface is just a trans-on the surface of [Mzj)] 11 lation in time of the [Ppound(z)] surface for N large ~K ~K bdquo
Another way of interpreting the asymptotic growth of the trace sur-face to that of the (11)-element of K as N becomes large is as follow
169
At the time of the f i r s t sample for t bdquo = 009 decompose the surface
for Tr[Ppound(z K)J into surfaces for each element of the trace that i s
[ E K ( Z K ) ] [E|^(z K) l poundPpound(zK)J as shown in contour plots of
Figue 610 The f u l l t race as in Figure 66 is shown in Figure 610A
with the individual elements shown on succeeding p lots As time t K + N
becomes large the formula for the trace in (631) may be rearranged as
fol lows
T r [ amp laquo ] [EK(K)]bdquo + B9nN
n=l
n=l
Each line in (638) represents what happens to each diagonal element of ppound + N comprising the trace as time goes on Since 0 lt lt 1 i = 23 45 as N becomes large all the terms except the first loose their funcshytional relationship with the positions of the measurement device given in zbdquo In terms of the plots for [pound + NJ through [ P pound + N ] in Figures 610B through 61 OF as time goes on these surfaces become flat with constant values equal to the steady-state values of the right-hand terms in (638) Thus for large time the surface Tr[P K + N(z K)] is made up of a number of steady-state slices a flat surface growing at the rate [pound]bdquo per time step and the surface [PD(z)]
CONTOUR PLOT OF TRACErPCKK+NMZfK) )3 AS FUNCTION OF tZtK)31 HORIZ IZ(K)32 VERT EXAMPLE TO SHOW GROWTH OF T R A C E C P ( K K N ) ] SURFACE WITH T I K E T C K N ) I TS SHAPE APPROACHES THAT OF [ P ( K K ) 3 1 1 SURFACE ASYMPTOTICALLY FOR LARGE N
+553 555 555
[ZCKJ12 0 9
44 33 222 44 33 44 3 3
444 3 3 444 3 3
444 33 444 33
4444 33 44-14 33 4444 33 44-14 33
bull 4444 33 4444 33 4444 33 4444 33
444444 33 bull44444 31
4444 33 44 333
033 i 3333333 2 333 22=
22222 2222322222
222 222 222 pound22 22 222 222 22 222 222 222
2222 2222 2222 2222 2222 2222 2222 2222 2222 222 222 222 222 222
33 4 5 e 77 33 44 9 6 77 33 44 5 G 77 33 44 5 66 777 33 44 59 56 77 33 44 5S 66 77 33 44 55 6 7
1888 99999999 B308 9SU99999 nS86 9^999999 9889 93399399 80083 99399999
66 33 4 33 4 33 44 95 tit 3 44 95 66 33 44 5 666 33 4 55 66 666 23 33 44 55 66-222 3 44 55 66 22 3S 44 50 laquo 22 33 4 55 222 33 44 55 22 3 44 955 22 33 44 5555 222 1111111111111 22 33 444 222 33 4444 22 33 44444 222 333 2222 3333333 22222 222r J222222222 22222222222 22222222222222222 22222222222222222222222 1111
22222222 2222222 111111 22222 22222 1111111 2222 333333 2222 111111 222 3333333333 222 222 333333333333 222 223 33333333333 222 222 333333333 222 2222 2222 222222 22222
9999999999 77 eeeeaeae 777 688066668 77777 6803068885
77777 088308888088860 7777777 8886Ce068e386
777777777 680688588 bulli 7777777777 56 7777777777777 gtSli 77777777777777 gt6iS6 77777777777777
51JS666666 777777 16666666666666666
666666666G666666666+ J5ii5555
55555555555555555555555955959 144
4444444444444444444444444444 JM333333333333333333 2322222222222222222222222222222
22222222222222 222332
bull333333 2223 3333 2222
44 333 222 44444 333 222
444 33 222 444 33 222
11111111111111 m i n i m u m 1 m i n i
1111
n i m 111111111111111 m m i i i i m i 111111
i n
2222222 2222 33333
222 333333 2222 3333 222 333 4444 222 3 44444
21222222222222222222222222222222+ 131
11333333333333333333333333333333
SYMB LEVEL RANSE
(6)375341E 62 (9) (9) 34616E-02 33891E-02 (8) ltegt
3316CE-02 32440E-02 (7) (7) 31715E-P2 30990E-02 (6) (6)
3Q265E-02 29340E-02 (9) (5) 26814E-02 2608SE-02 C4gt (4) 27364E-02 26639E-02 (3) (3) 25914E-02 25103E-Q2 (2) lt2gt 24463E-02 23730E-02 (1) (1 ) 23013E-02 22268E-02 fQgt 2-15C3E-02
ESTIMATION ERROR CRITERION CONSTRAINT = 75000E-02
Figure 610A Contour plot of Tr [K) at first measurement time tbdquo = 009
CONTOUR PLOT OF T R A C E [ P ( K K + N gt ( Z ( K ) gt 3 AS FUNCTION OF t Z ( K ) J l HORIZ t Z lt K 1 3 2 VERT EXA11PLE TC SHOW GROWTH OF T R A C E I P ( K K + N ) 3 SURFACE WITH T IME T ( K + N gt ITS SHAPE APPROACHES THAT OF [ p ( K K ) 3 1 T SURFACE iSYMPTOTICALLY FOR LARGE N
TJKE= 9 0 0 0 0 E - 0 2 F I R S T MEASUREMENT ELEMEhTt 1 11
+ 4 4 4 3 3 2 2 2 2 2 2 2 2 2 2 2 2 - ^ 2 2 4 4 4 3 3 2 2 2 2 2 2 2222i i 2
4 4 4 4 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
4 4 4 4 4 3 3 222222f 22L 2pound 2222
4J444 33 2pound2pound22 - 2222222raquo + 4 4 4 4 3 3 2 2 2 2 2 2 2 2 2 2 22pound2222 4 d 4 3 3 2 2 2 2 2 i 2 2 2 2 3 2 P 2 2 2 2 2
3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 - i 2 i gt 2 2 2 2 2 3 3 3 2 2 2 2 ~ 333 2222 333 222
333 44 S 66 333 44 S 66 333 4 0 6B 333 44 59 66 33 44 9 66 ___ 55 661-33 44 55 m 333 44 55 6
Ik 939999 D 999999 999909 999999 99999939
9999990999 oeeoae 9999999
222 222 222
CZ(K)32 09
33333
33333
33333
30393 +33333 22 33333 22 33333 222 3-33 22 1 3333 22 1 bull33 22 11
222 111 22222 111 2222 111
bull iiitm in
22222lt222i pound22 2 2 2 2 2 2 2 2 2 2 2 3 3 3 44
2 2 2 2 2 2 2 2 2 3 3 4 4 2 2 2 2 2 2 3 3 3 4 4
2 2 2 2 4 4
03236 7777 7777
77777 5 7 7 7 7 7 8808(1088 S6 7 7 7 7 7 7 8 3 8 0 6 8 6 0 3 3 3 6666 7777777
66666 7777777 535 S6656 777777777
2 2 2 3 3 3 4 4 4 CSS 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 1 U U U 1 U 1 2 2 2 3 3 4 4 5H3 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 4 4 4 5 5 5 3 6 6 6 6 6 6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 4 4 5 5 5 5 5 66G666666 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22 33 4 4 5 0 5 0 5 5 5 5 5 666666SG6G6 11 1 1 1 1 1 1 1 2 2 3 3 44 5 5 5 5 5 5 5 5 S G S f 1666
1111 2 2 3 3 4 4 4 1 4 5 5 5 5 5 5 5 5 5 5 1111 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 +
111 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 1111 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
_ 1111 2 2 2 2 2 0 3 3 3 3 3 3 3 3 3 3 3 3 3 A 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 111111111111111
111111111111111111111 222 333333333333 22 11111 11111111111111111111111111
222 333 333 222 111 U 1 111 1111ll 111111111 1 1111111111
11111 1111111111111111
11111 1111 + 11 111 2222222222222
111111 2222 222 n n i i - mdash 111111 Mil 222 333 4444444 33 22
222 33 444444444 333 22 222 333 444444444 333 225 222 33 4444444 33 22 222 333 333 22
222 33333 333333 222 11111 +111 Hill 222 333 2222 1111 1111111111 2222222 222222 11111 1111111111111 2222 11111
11111111111111 1 111111111111111111111
+222222 11 111 1111111111111 2222 111111111111111
33333 222 1111111111111 333 222 111111111111 33 222 11111111111
i +44 333 222 1111111111
111111111111111111 1111111111111111 11111111111111111 lllllllllllllllllltll 111111111 1111111 1111 11111111111111111 till 1111
11111 1111111111111111111111
111 1 11111111111111111111111111111111111111111111 1 1 1 i i 1 1 1 1 111111 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1111 2 2 2 2 2 2 2 2 111 2 2 2 2 3 3 C 3 3 3 3 3 3 3 3 3 3 3 111 2 2 2 3 3 3 C J 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 +
SYMB
CO) LEVEL RAN3E 2 2 2 0 0 E - O 2
( 9 ) ( 9 ) ( 8 ) ( 6 )
2 2 2 2
1 6 9 7 E - 0 2 1 ^ 3 4 E - 0 2
0 6 9 1 E - 0 2 0 1 8 S E - 0 2
C7J ( 7 )
1 1
9 6 B 6 E - 0 2 9 1 8 J E - 0 2
(G) ( 6 )
1 1
6 6 8 0 E - 0 2 6 1 7 7 E - 0 2
lt 5 ) ( S )
1 1
7 6 7 4 E - 0 2 7 1 7 1 E - 0 2
C4gt t 4 1 1
6 6 6 S E - 0 2 6 1 6 5 E - 0 2
( 3 ) ( 3 )
1 1
5 6 6 3 E - 0 2
5 1 6 0 E - 0 2 t Z ) (2)
1 1
4 6 5 7 S - 0 2
4 1 5 4 E - 0 2
( 1 ) ( 1 )
1 1
3 S 5 1 E - 0 2
3 1 4 0 E - 0 2
tcopy) 12645E-02
ESTTMATTOM ERROR CRITERION CONSTRAINT =
75000E-02
I25OOE-01]
Figure 61GB Contour plot of first term of Tr Ppound (z K) raquo K(JK)
CONTOUR PLOT OF T R A C t [ P ( K K N ) ( Z t K ) )3 AS P J N C T M N OF [ Z ( K ) 1 1 H O R I Z C Z ( K ) J 2 VERT EXAMPLE TO SHOW GROWTH 3F T R A C E P ( K K raquo N ) 1 SURFAi^ WITH TIME TCf + H) I TS SHAPE APPROACHES TH-T OF t P lt K k ) 1 1 1 SURFACE AMP10T1CALLY FOR LARGE N
TIME= 9 0000E-02 FIRST MEASUREMENT ELEMENTC Z 2)
2 2 2
660 gas
i w 22-1
33 4 S 6 77 80 _ 03 H 55 G 77 OB
Qpound2 3 4 5S 6 77 31 22 3 4 o 6 77 H5 bullPAV 33 4 s P6 7 (iO
33 44 5 56 7 03 33 4 5 6 bull BO
33 4 55 iS 77 faD 33 4 5 G 7 83 3 3 41 5 (iS 7 SO
3a 5 6 7 amp 33 4 amp 6 7 88
333 44 St (J 7 OS 323 44 6 77 00
3333 4 5 5C 7 mdash m 77 777 777 +7777 777 77777
-1 3 l l l | f JJ | II II
444 ri-14 44I 4441 444
bullM44 4144 55 GG 14144 444-14 5gt fi bullbull44-444-14 lili (it
5 5 aa
444444 333333333 444pound 4444gt
44444144444 ^^TI^-^^^ 444 ^ ^44
99999099 9 9999S9999S999 )y99999999C99999G999999939999S9 - 199999990999999593993 + amp939929309 000003008306000 10^83090803006060 laquo 777777777777777 i 77777777777 igtwC6C6+
eeeeccccecc Ii oiiSSBSS 4 -14444444 4 4 4 4 4
4^ 4444444 3 3 3 3 3 3 3 3 3 44 3 3 3 3 3 3 3 3 3 bull
3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 222222 1111111111111 2-S22 11111111111 c- 1111111111 + 111111111 11111111 copybull
111111111 US 1 1 1 1 1 1 1 1 1 2J222 1 1 1 1 1 1 1 1 1 1 1 1 +
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 nl 2 2 2 2 S 2 2 2 2 amp 2
3 3 3 3 3 3 0 3 2 2 2 9 2 2 2 2 2 2 Ain 3 3 3 3 3 3 3 3 3 3
4444444-14444 333333+ EiftSti 4444-1444444
S^bSOjEbSriSbS 4144 pound 55SS0 rt55iS I16G b55riij555
SYMamp LEVEL RANGE
CO) 8 9 S 2 7 E - 0 3
( 9 1 8 7 6 2 6 E - 0 3 8 5 6 2 S E - 0 3
( 8 ) ( 8 )
B 3 6 ^ 5 E - 0 3 6 1 6 2 5 E - 0 3
( 7 ) ( 7 )
7 lt1 i24E-03 7 VigtK3E-03
( 6 ) ( 6 )
7 5 6 2 3 E - 0 3 7 3 6 2 2 E - 0 3
( 5 1 (5
7 1 6 2 E - 0 3 6 0 S 2 1 E - 0 3
( 4 ) ( 4 )
5 7 f = 2 0 E - 0 3 C 5 S 2 D E - 0 3
( 3 ) ( 3 )
6 3 6 1 0 E - 0 3 G 1 6 1 9 E - 0 3
( 2 1 ( 2 )
5 9 amp 1 6 E - 0 3 5 7 0 1 7 C - 0 3
(1 ) t l )
S 5 G 1 7 E - 0 3 5 3 t i 1 6 E - 0 3
(0) 5 1 6 I 6 E - 0 3
E S I M A I ION ERCHR Ct l TERION CONSTRAINT =
7 H 0 0 Q E - O 2
1-2500E-01J
Figure 6IOC Contour plot of second term of Tr P ( K ) K(K) -
0 6
t Z l K ) J 2
C3NT0UR PLOT O F TRACECPCK^K-Ni t Z ( K U l AS FUNCTlC- t OF I Z t M H H C R I Z t Z ( K 1 1 2 VERT EXAMPLE TO SHOW GROWTH OF T R A C E [ P ( K K N ) ] SURFACE U I T H TIME T C K N ) ]Tlt SHAPE APPROACHES THAT OF [ P lt K K i 3 1 1 SURFACE XSVMPTOTCALLY FOR LARGE r
bull raquo + 4 4 + bull9-J19 8 0 7 5 4 3 272 3 4 5 6 7 0 3 9 0 bull 0 0 9 e a fi 5 4 3 2 2 2 2 2 3 3 4 5 6 7 8 ltlaquoltraquo laquo laquolf q 6 6 r b 5 lt1 3 2 2 2 2 2 3 3 A 3 6 7 O
6 0 7 6 5 1 3 3 2 2 2 2 2 2 2 2 3 3 4 5 7 7 8 s 7 7 5 U raquo3 2 pound 2 gt P 2 2 3 4 4 5 6 7 Q - - - laquo bull laquo bull - - - - -1 L o i B i a 3 6 0 7 6 S 4 3 6 0 7 6 5 A 3 a a y 6 5 lt 3 3 M 0 5 4 33 60 7 6 5 4 33 SB 7 6 5 4 33 80 7 6 5 J 33 03 7 E 5 4 33 B8 i amp 5 1 33 CB 7 6 3 A 33 e i 7 6 3 J 13 80 7 G 5 4 33
i 8 1 6 5 4 3 3
I 22
U3 83 7
Lgt A
iSP5
3 3 4 5 G 7 B 9 3 9 3 3 A 5 C 7 8 0 9 9 3 3 4 5 6 7 8 0 9 9 3 4 5 6 7 8 9 P 9 3 4 5 6 7 8 9 P 9 3 4 5 6 7 8 9 3 9 3 A S 6 7 O 5 9 9 3 4 5 6 7 3 G pound 9 3 3 4 5 6 7 0 9 9 0 3 3 4 S 6 7 8 SD9 3 4 5 6 7 8 9 9 3 4 5 6 7 8 0 9
3 3 4 5 6 7 8 9-J j 3 3 A 5 6 7 8 8 9ltJ 3 3 3 4 5 6 7 C 8 S9raquo0 9 9 9 9 3J3C-S33 bull 5 I) 7 (J T J 9 L 9 0 9 t i 9 9 9 9 9 3 9 9 9 9 9 9 3 9 9 9 S 9 9 9 9 9 9 9 9 9 9 9 9
-I 3 - ^ 3 -14 ti 6 7 flJ i - 3 9 9 9 9 y 3 y 3 3 3 deg 9 9 9 3 9 9 9 9 9 9 9 9 9 9 9 9 9 9 41 3 3 4 3 6 7 7 0 8 0 B B B 8 8 8
-14 4 4 5 5 6 7 7 8 8 ( 1 0 8 8 8 6 8 6 3 ^ 3 3 8 3 3 8 8 8 8 8 8 8 8 8 6 8 0 8 8 4 4 4 4 4 5 3 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 -laquo4 4-14 5 5 6 7 7 7 7 7 7 6666ltgt6C6 6 5 G G G G G 6 6 C 6 6 e G 6 G 6
4 4 4 4 pound S tC6GE(JC6-J6 ampK35 5 3 5 S 5 5 5 gt t W 3 5 5 3 4 4 4 4 5 5 3 55455 ampAAamp - - - - - - -
3 3 3 3 3 3 3 4 4 4 4 3 4 4 3 3 3 3 3 3 3 3 4 4 4 4 4 3 3 3 3
i^Sa^^S1i bull 2 2 2 22 2222J2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22222 j S2laquolaquo2laquo S333 3 3 3 3 av^ raquo J laquo J U ) raquo raquo raquo raquo raquo J S
^rf11^4 233a33333 dd^-J^ 3 3 3 33333 2 - 2 2 2 - 2 Z 2 2 2 2 2 2 2 2 2 2 2 2 2
bdquo 3 3 3 3 3 4 4 1 4 4
5 3 5 6 6 6 C 6 b
7 7 7 7 7 7 7 7 7 7 7 7 7 r0 i 0 0 3 ( i O B f gt pound n O O - 8 6 8 8 G P 0 8 6 6 6 6 0 e 8 8 3 Q O Q J 6 7 7 HO 8 8 0 0 6 77 0 0 S1099lt E U 3 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 mdash 9 0 J 9 0 J - lt i j J 9 9 1 - 9 9 9 9 i S 9 9 9 3 9 3 9 9 9 9 9 9 9 9 9
9B0igtD0 9 gt ) 3 9 e G 3 9 9 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 S S 9 9 9 amp 9
Tl| f lE= 9 O 0 0 0 E - O 2 F l f S T MEASUREMENT ELEMENT 3 3 )
JYflB LEVEt RANGE (0gt 6 042ZE 04
S1 3 5 9133E 7CB4E 04 04
5 6S15E 534GE 04 04
tfi 5 5 4077E 2G00C 04 04
s 3 5 1339E 027OE 04 04
II A A
9001E 732E
04 04
(jJ) 4 4
64F3F S 1 04 E
04 04
iSJ f 393E 2G5GE
04 04
S A 1387E 04
il 3 3
6849 75301T 04 04
ltbull 3 6311E 04 EStMATION ERlIOR CRITERION CONSTRAINT = 7e000E-02 SampiJRCE IMI-JT CQVARIANCE [WJi r 1 2300E on MEASURfiMCNT ERlJOR COVAR IV3 = [ 050 -0] 0231
Figure 610D Contour plot of th i rd term of Tr )] [4
CONTOUR PLOT OF TRACETP(KK4N)CZ(K))1 AS FUNCTION OF tZ(K)J1 HORIZ tZ(K)J2 VERT EXAMPLE TO SHOW GROWTH OF TRACEtP(KK+N)] SURFACE WITH TIME T(KlaquoN) ITS SHAPE APPROACHES THAT OF [P(KKgt111 SURFACE SMPT0YI5Ai-LY FOR LARGE N
TIME 9O0O0E-O2 FIRST MEASUREMENT ELEMENT 4 4)
IUIAL 33 A 5 67 38 93 3 4 5 7 08 99 3 4 56 7 88 99
33 44 6 7 8 99 3 3 4 5 6 7 8 99
333 4 5 6 7 r mdash 39 8 76 S 1 333333 4 5 6 7 8 99 99 8 7 9 4 333333 4 3 6 7 8 99 99 6 7 6 5 44 223333 44 5 67 88 99 99 6 7 SS 44 C53333 44 5 7 88 99 99 8 7 6 4-1 3333H3 lti4 5 7 OS 39 99 B 7 6G 44 333333 44 5 7 tiS 99 99 8 7 5 4 333333 4 5 67 86 39 99 8 7 5 4 333333 4 5 6 7 6 99 99 8 76 5 4 33 33 4 5 6 7 8 99 9 8 G 5 4 33 33 4 5 6 7 0 99 9 0 7 6 4 33 3 4 6 7 8 - 5
99 8 7 OS 4 33 22 33 4 5 7 8 gg a 7 es 4 3 222 - - - - -99 8 7 65 4 O 222
9 8 7 65 4 3 22 9 87 6 54 3 S9 8 76 S - __ 99 6 7 6 44 333333 4 5 6 7 8 __ 8 76 S 44 44 S E 7 S 999
0 7 6 5 444 444 5 6 7 88 ~~ 69 7 6 55 4444 5 6 7 laquolaquo
fiSSeoe 7 66 5 55 06 7 66 7 7 77 S 5a 55 6 77 7777 5 5 5 copy6 6 8 5 5 65 666 666
1-4444 55 106 55 4444 5U 6665 555 3333 4 S5P5503 4-14444444 555503S5 444
2222 33 44 44- i 4444 444
3 4 5 7 6 99 3 4 5 7 8 99
33 4 56 7 0 9 3 43 6 7 8 99
33 4 5 5 88 99
8 6 8
1199999 9939999999i9pound999S9999g999999999g9g99g
esoossBBe aaeeeew
i n
t i
77777 UfcSB 55 33 13-333 44444 3333333333333 _- 333333 444 33 222 22222222 2222222222222 11 22 323 3333333333 333 22 111 11111 22222222
t 2 333 333333333333333 33333 22 11111111111 H i t 11 2 333 33333333333333 3333 22 H I T 111 1)11 11
11 2 33 3333333333 33 22 1 1 1 22222222222222 22 33 444444 4444444 33 ZZZZZZ222222 2222222 3 44 444 444 444 3333 3333333 333333333333
mdash mdash 4444 44444444 4444444444444 53363 U555S5355 35555V-3rraquo550
GCOC 6fo665G6 665GCSG6 777 77 77777 7777777 03C yi300C6P8 (-88831130008
6fi6
444 555 4444-14 355 555 oeeebf-Gb 55 15 seceeSSGe 777777 6 55 55 C 77777777
77777 BflaS 7 6C 5 E i 66 77 80C98 8S00amp 88 7 6 55 44444 3 0 7 88 __ bull ampSgt39399amp 3 7 6 E 44 44 3 6 7 06 939999999^ raquo9jiC0l-3 J999Ci999999S93asaampS9
99 Oft 0 it 4 3333 4 5 6 7 8 339 99993 99 6 76 0 4 33 33 4 5 7 99 9 87 65 4 3 222 33 4 6 7 8 99 9 0 7 5 33 222222 3 4 5 7 8 99
93 8 76 54 3 222222 3 4 5 7 8 3
SYKB LEVEL RANGE (0 25437E-03 (9) (9) 25Q05E-03 2455pound-03 (81 (81 24101E-03 23649E-03 17) (7) 23197E-03 22745E-03 (61 (6) 222H3E-03 2 1841E-03 (5) (51
213S9E-03 20937E-03 (4) 14) 20-135E-03 20033E-03 (31 (3)
19561E-03 10129E-O3 (2gt (2)
10677E-03 1S225E-03 lt1 ) (1 1
17773E-03 17321E-03
lcopyl_I 66 i3E-03 ESTIMATION ERROR Cftt tERION CONSTRAINT =
75000E-02
12300E-Oil
Figure 610E Contour plot of fourth term of Tr (4 [0 44
CONTOUR PLOT OF TRACEtP(KKNl li(K)) J AS FUNCTION OF tJIIOlt HPRIZ t2(KJ3Z VERT EXAMPLE TO SHOW OROUTH OF TRACECP(KKN)J SURFACE WITH TIME T(KN) ITS SHAPE APPROACHES THfl flF [P(KK)111 SURFACE 3VlaquoPT0T|CALLV FOR LAROE N
02
S3 0 76 5 44 99 6 76 S 4 99 8 7 5 4 99 0 7 C 3 44 99 B 7 6 5 44
4 5 6 7 6 09 4 5 6 7 8 99 4 3 O 7 9 99 44 3 6 7 00 99 44 3 6 7 r OB bull 9 8 7 6 3 444 444 5 6 7 8 9 1 8 7 6 3 444444 5 6 7 8 Q9 I 87 6 55 444444 53 6 7 8 99 I 6 76 55 44444 S 6 7 C 99 J 8 76 3 4444 5 6 7 8 39 08 + 99 8 76 5 4444 5 CS 7 6 99 9 8 7G 55 4 1444 3 6 7 t S3 9 87 6 3 444444 35 6 7 O 99 9 8 7 6 5 444444 3 i5 7 8 99 a 8 7 6 0 44 44 5 6 7 8 9 99 8 7 5 4 4 5- 7 GB 99 99 8 6 3 4 33 44 5 6 7 9 99 9 87 6 3 4 33333 4 5 G 7 9 99 9 6 7 65 4 333333 44 56 7 8 9 9 6 7 5 4 333 33 4 5 8 99 06 9 8 7 3 4 23 33 4 9 7 0 9 9 fl 7 6S 4 33 313 44 6 7 fl 91 9 ) 8 6 3 4 33333 4 3 6 7 0 99 bullJ 8 7 3 44 3 4 56 7 0 09 99 87 6 S 44 44 5 6 7 0 09 03 999S9 OB 8 7 6 5 4444 5 6 7 4 99 1 999amp9US 8 7 65 S3 S3 6 77 O S999999999 88 8D 7 6 305553 F6 7 8 9 888 77 8B8O0B 7 65 3355 7 83080388 77 66 7 77 G6 55 GG 77 777 66 04 444 0 6 77 66 553S G6 777 G6 530 333 44 5 eCGGGC 5C553t55 6666606 53 444 pound22 33 4 53 C555 5v53 553 4 It 2 33 4 335 44 5533 44 33 _bdquo 112 3 44 441444444 444 33 2222 03 -ltgt 11 2 S3 444 444^1444444444 4444 31 222 11 2 33 444 444444444444 4444 33 222 112 3 44 44444444 44 33 222 11 22 3 4 SSSiVS 3535555 44 333
222 3 44 gtZgt 3555 5555 555 44
199999
555 114444 1333
999 806888 888388 7777777777777 66666006066666 3535550555553353 4444444 44444444444 33333 3333333
ZPgt2 33333333330333333 22222 3333333333 222 333333333333 222 33333333 333333333333
bull33 44 05 tgt5 66G S33555 G66 656 35 6 77777 SS 555 65 77777777 6G6 6666 77 EOC 77 66 S5fgt 66 77 O03C9 777 777 68 EB 7 6 SS3rS5 66 7 8 803081 830 taiUQ 8 7 6 5 53 6 7 O 2999301)99 99939 93 C 70 5 444441 3 6 7 0 09 9 0 7 5 4 33raquo 44 U6 7 O 99 9 0 7 5 4 3 33 4 56 7 0 39 99 0 65 4 3 22222 3 4 6 7 6 99 9 8 7 34 3 22 S 34 3 7 6 99 99 3 76 4 3 2 22 3 3 67 0 99
33333 44444 55355 663066 7777777
iGFtlOUampUOOB
444444 4444444 55555055+ b0666666 7777777 88080608 93 990999999999999999999999999
TIKE 90000E-02 FIRST MEASUREMENT ELEMENT 3 5)
(0) LEVEL RANGE 1 0362E-03~
it 10I98E-03 1 -0035E-03
3GTI2E-04 97076E-04
95441E-04 93806E-04
sect 92170E-04 9053SE 04
ii 6e899E-04 872D4E-04
S3 B5G^9pound-04 83993E-04
sect G2358E-04 00722E-04
79037E-04 77452E-04
7S816E-04 74181E-04 (0) 72545E-04
ESTIMATION ERROR CRITERION CONSTRAINT =gt
75000E-02
to00E-O1J
Figure 610F Contour plot of fifth term of Tr [bull (4 [^L
176
622 Optimality of Measurement Locations - In Figure 64 was i
shown the trajectory TrlP K + N(z K)J where the optimal choice cf measureshyment positions was used at each measurement time In contrast suppose the designer felt that an intuitively good choice for the measurement positions would be to place the two statistically independent sensors right at the position of the source that is z = zbdquo = z = 03 Figshyure 611 compares the optimal trajectory Tr[ppound+f(zp)] of Figure 64 using
i
min [Pbdquo(z)] as the criterion at each measurement with the case with z K ~ K ~ K 11 z K = [0303] that is with measurements positions at the source The optimal case is plotted with the symbol 1 that with measurements at the source with the symbol 2 Clearly Case (1) is optimal since over a larger time interval it would result in fewer measurements necesshysary to maintain the estimation error below its bound
623 Comparison of Performance Criteria - Moore L 9 5 ] suggests that the minimization of the trace T rEPpound(z K)] at a sample time t K mey not be the best thing to do to lead to the fewest number of samples necshyessary over some time interval To demonstrate that this is in fact a true conjecture consider a slight modification to the problem of Section 61 Let
I 04 W
002 (639) -^ 000001
J^ 000001_ oioio
to -
lim and
(bull K)= 0 001
(640)
(641)
6 7 S 0 0 E - 0 2
5 5 0 0 0 E - 0 2
42300E-02
30000E-02
1 7 0 0 D E - 0 2
C mdash r ~ - rmdashU raquo mdash - bull bull r J V- mdash bull mdash a a t 2 1
2 i pound i I 2 1 2 1
2 2 1 2 1 1 2 1 2 1
2 2 1 2 1 pound J 2 1
2 1 2 1 2 1 2 1
2 1 2 1 2 1 2 1
2 1 2 1 2 1 - 2 1
2 1 2 1 -2 1 2 1
2 1 2 1 2 1 2 1
2 1 2 1 2 1 2 1 2
2 1 2 1 2 2 1 2 1 2
2 1 2 1 2 2 1 2 1 2
2 1 Z 1 2 1 1 bull pound
2 1 2 1 2 2 1 2 1 _2
2 1 2 2 1 2 1 2 1
2 1 2 1 2 1 2 1 2 1 2 1
2 1 2 1 2 1 bull 2 1 1
2 1 2 1 2 1 1 2 1 2 1
2 1 2 1 2 1
1 2 1 1 1 2 1 2 1
2 1 2 1
Figure 611 Time response of T r [P^ + H ( z )J for (1) z the result of the minimization min [ p ^ z K j j M bdquo + bdquo H i t h s y m b o 1 a n d ( 2 ) Ln = r| = z ^ f b o t h m e a s u r e m e n t s a t tKe source
plotted with symbol 2 L J2 plotted wit locat ion
178
The other problem parameters are as before To measurement strategies are contrasted The first is at each
measurement time t K finding z K such that
as before The second is finding zbdquo such that 2 N
x T 4 Tr = min Trj Ppound(z) | (643)
In ti1s problem measurements are necessary at t 0 the initial time and it is found that immediately after the first measurements strategy number 2 using zj appears superior to that using ir The two trajectories
5 U l u
are plotted with symbols 1 and 2 in Figure 612 However it is seen that at t - 0021 the two curves cross afterwhich Criterion 1 remains superior leading to a second measurement at t = 0078 vs t = 0071 for Criterion 2 At the end of the interval 0 lt t lt 01 Criterion 1 clearly possesses the lower estimation error Thus it is not optimal to minishymize the trace of the estimation error covariance matrix at the time of
the sample but 1t is optimal to minimize its value for large time N which by Collusion II is equivalent to minimizing the (ll)-element of the covariance matrix at the time of the measurement
624 Effect of Instrument Accuracy - To study the effect of the quality of the measurement instruments upon the evolution of the Tr[PK+N(zj)] contours in the above problem consider the measurement error covariance matrix
005 O
001 (644)
93000E-02
76000E-02
59000E-02
42000E-02
23000E-02 I OE+00
222 111 222 111 22 111 222 111 22 111 222 111 222 111 22 111 22 HI 222 1 I 22111 221 11 2211
122 11222 1 1222 1122 1122
22111 2111 1111 321
22 22 1 2 1 2 1 pound 2 2
22 2 22 11 22 11 22 11 2 t 22 11 2 11 laquo2 1 2 2 2 2
1 2 1
1 1 1 1 1
B000E-02 1000E-01
Figure 612 Time response of 7r| P^ + ( j (z j j for (1) z the result of the minimization min P K ( K ) plotted
with symbol 1 and (2) zpound the result of the minimization min Tr |ppound(z K )J plotted with symbol
2 note how after the f i r s t measurement at t K =00 (2) possesses lower estimation error but with t ime the curves cross such that (1) is superior at the end of the time interval shown and thereafter
180
This accounts for a 51 difference in variances in the two sampling deshyvices in contrast to the 21 difference in the problem above The evo-
i
lution of T r L P ^ + N ] is shown in Figure 613 The contour plot of Tr[Ppound i (z K)] at t K = 009 is shown in Figure 614 Contour plots of Tr[ppound+f
(Z|)] are shown for t bdquo + 1 t K + 5 t K + ( | and t K + 1 5 in Figure 615 and finally that for [P(zbdquo)J in Figure 616 In this case since the two -K -K ii measurements are of much different quality than those in the previous case the error contour is much less symmetric showing where the more accurate sensor [z]o is preferred over the more inaccurate poundz] Notice the large motion that the global minimum can make over time in a particular problem the positions of zt the global minima can change greatly as a function of t+ for the surfaces TrpoundP K + N(z K)]
63 Problems with Bound on Output Estimation Error
In the monitoring problem with bound on the maximum allowable error in the estimate of the pollutant throughout the medium it is necessary to make a measurement whenever for a time t K +bdquo
T 4JhZ) Aim ( 6 4 5 gt
a 2K + N(z Kz) S c(z) TP + N (z K) c(z) (646)
where
as in Section 541 Suppose the first time (645) is satisfied is at sample time t K gt
It is required to select the best set of measurement locations zt such that
0 K + N ( 4 Z ) = m l nK mx deg K + N ( 2 K Z ) (6-47)
EXAMPLE TO SHOW QROWTH OF T R A C E I P t K - K + N H SlRi-ACE WITH T IME T ( K N ) I T S SHAPE APPROACHES THAT OF t P l K K J 5 1 1 SURFACE ASYMPTOTICALLY FOR LARGE N
I XX I X I X bull X 1 X
X X
X
x x
X X
X X
X
IX
X X
X X
gtbull X
X
X X
X X
X X XX
X
s X X
XX X
X X
X X
X
X X
X X
X X
X X
X
I X 1 X I X I X I X I X
X X
X X
X X
X
x x
X
I X I X
I i
X
X X
X
X X
X X
X
Figure 613 Time response of Tr 096
ppound + N(z^j] showing three sample times at t R = 009 052 and
CONTOUR PLOT OF TRACECP(KK+N) (ZIK)) 1 AS FUNCTIC-J OF CZCKUI HORIZ [2CK)1Z VERT EX^tfPLE TO SHOW GROWTH OF TRACEEPCKKN)1 SURFACE WITH TIME T(KN) ITS SHAPE APPROACHES THAT OF tP(KK)J11 SURFACE ASYMPTOTICALLY FOR LARGE N
95 44 33 55 44 33 55 44 33
S55 44 33 6 5
5 5 5 5 5 5 5 5 5 5 5 5 5 S 5 5 5 5 5 5 5 bull 5 5 4
4 4 444 444 444 444
444 444 3
4444 3 44laquo4lt44 3 44444 33
333 33333
333333 2222
22122 222222
2222222 3 222^2222
22J2222 222J2C22
222igt22lt222 2222222S2Z
222222222222 222222222222
22222 2222 222 222 222
2 2 2
33 44 55 66 77 OSS 999339999 33 44 55 66 77 868 S939933999 333 44 55 66 77 88 9 9999993999 333 44 5 66 77 68 38 999^9999099 333 4 5 6 77 8 380 99999-JS999999 333 4 5 66 777 -36488 999D9999999S999999+ 33 44 55 66 777 809863 939999999999 33 44 55 66 777 686368888 33 44 5 66 7777 6880860680300 333 4 55 66 777 8385600068866880888888
33 44 55 66 7777 8888886808088883-33 44 55 66 7777777
22222 33 4 55 666 777777777777 2222 33 44 5 666 777777777777777777777777777
pound22 33 44 55 0666 777777 777777777777777 222 33 4 55 6066 56 77^77777-
22 33 44 55 66E JEiS66 555 6Le0j66660CCCG666C66666S
555 S66G5eeUf=i6e6G-eSB6666S666666 5555
4 5555555535555555055555555555555555555-2K araquo 444 222 33 4444444434444444444444444444444444444444
22 333 222 333333333^ 53333333333333033333333333333333
222
111 11111 11111 111111 m m 111111 1111111 i m m - m i n i
2222222222222 222222222gt222
222-fc222 2222222222222 222-22222222222222222222222222
222222 22222 22222 333333333333333333 2222
22222 333333 33333 222 2222222 3333 3333 222
222232 333 444444144444 333 2222 +222222 333 4444444444444 333 2222
222222 333 444444-^44444444 33 2222 222222 333 44444-44444 333 222 222M22 333 333 222 222^2222 333333 333333 2222
22222 333333333333333333 2222 I i i 222222222 22222 1111 22pound22ii222 222222222 222222222 111 2222222 22222222222222222222222222
2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 bull 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 gt2
3 3 3 3 2 2 2 2 2 2 2 pound 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 J 3 4 4 4 4 4 4 4 3 3 3 2 2 1 - 2 2 2 2 2 2 2 3 3 3 3
2222 2222222222222222 2gt22222222Z222222
22^222222222 n m m m i ii
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111 111 1 1 1 1 1 1 bull 1 1 1 1 1 1 11II - 1 1 1 1 1 1 11111 i i i n n n n n m i i m i i i i m m 11 m i n i m u m m i i n i i m i n i m u m i m i n m i m n m m i m m i m m i i i m
2222222222
11 m m i i i n u m n n n m i i i u m i t i 1 U 1 1 1 U H m m i i i i i i
444 333 555553 444 333 5S5iiti55 44 333
-i222222 22C222 22222
333 333 333 4444444444144 4 444444 144344444444 4444 4444444444444
1111111111111111 1111111111111111111 52222222222 22222222222222 33333333333333333033330
TtKN)= 90000E-02 T(K) = 90000E-02 N - 0 STEPS AFTER FIRST MEASUREMENT CONTOUR LEVELS AND SYMBOLS SYMB LEVEL~RANGE (0) 2 9993E 02 (9) (9) 2 wm 02 02 lb) (0) 2 2 5poundI 02 02 (7) (7) 2 2 sectisectSe 02 02 (51 CO) 2 2 m 02 02 (5) (5) 2 2 poundpoundi 02 02 C4) (4) 2 2 iiaE
02 02 (3) pound3 2 1 g|pound 02 02 (2) (2) 1 SJ3i 02 02 (1) (1 ) 1 1 Z2TJ 02 02 (0) 1 flf 02
ESTIMATION ERROR CRITERION CONSTRAINT = 7 Slt gtgtbullbull)pound-02
Figure 614 Contour plot of T r l g ^ A ] a t f 1 r s t measurement time for case with d i f ferent measurer-gtnt error covariance matrix V
t bdquo - 009 compare with Figure 66 K
CONTOUR PLOT OF TRACEtPCK K+Nl t2(Kgt 11 AS FUNCTIOt- Cl= CZltK)11 HORIZ [2CK)J2 VERT EXAMPLE TO SHSW GROWTH OF TXACEtr(KKNgt3 SUff AGE WITH TIME T(KNgt ITS SHAPE APPROACHES THAT OF CP(KK)311 SURFACE rSVPTOTICALLY FOR LARGE N
EZ(K)J2 09
555 44 44 44 444 3555 5355S 5555 5555 535 444 44 444 444 4444 44-44 44-14-14 444 bull144 bull444-144 3 444-J4 3 444-14 3 44444 4444
333 333 44 333 333 44 333 3333 44 333 3333 pound4 333 3333 44 333 333 At 33 3333 4 333 333 4-333 22 333 333 222222 333 333 222222222 333 33 222222J2222 333 (33 222222222222 33 13 2232 22222222252 333
6 77 bull CS 77
dec oec oota
eteo cae
999Q99S99 5359929999 SC339^-99
S999i)J99399 D999399SP9999
333 4 33 222 333 222 3333333 222 353 222 22222 22222222
JPPZZ 2222 2222 222 222 11
m i 1M11 H i l l
n n i i 11111111 11111111
777 euseoe 77 BSEBSC3
777 acaoseesee 6 777 7 see8fJ8633888888 6S 77777 6R6 7777777
rgt0G 777777777777 56G6 777777777777777777777
_J 6G6E6 777777777777777777 22222 33 4-1 555 66E6t5poundS
22222 333 44 550 EGtmejGGGSS 222 33 44 555 C5e6tweampe6u66eGfl0^6eS666666666 2222 33 444 55tgt3 666666o6666S6GG6666l3S
222 33 44 5ti055amp 222 33 44 555S5iij555S555555SS555555555 222 33 444 55355555555555
222 33 444444^44 444444444 V2Z 3333 2222 33333333233333333333333333333333^3333333
2222 2222222222222 pound22222222222222222222222222
1 1 1 1 - -
1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 111 111 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
11111 1 1 1 1 1 1 1 1 1 1 1 1 n u n
f i i t u r n i i 2222222222222222222222222222 11111111 222222222 222222 111111 22222222 33333333333333333333 22222 111 22222 33333 3333 2222 3333 444444444444444 3333 222322 3333 44444 4444 Clt33 22222^2 33333 4444 4444 333 2222222 3333 4444 4444 333 22222pound2 3333 --4444 44444 3333 222222 33333 444444444444444 333 2222 22 3333 3333 2222 bull222222 3J333333333333333333 2222 2222222222 22222 2222^222^2222222 2222222222 2222i2ii22222222222222222222e22222pound222 22222222pound22-i2222222 2222222222222 +33333 222H2222222222222222222222222222222pound 111111111111 333333 222222222222222 222222222222222222222222222 444444 3333 2222222 33333333333333Ct3333 44444 3333 33333 333333333333333333333333 35 444 3333 3333 444444444^4-la 5555 444 333 33333 4444444444444-1444
22222222
111111111111111111111111111 1111 111111 111111111 1 i i m i u m i i n t i n i i a
m i m u n i n i i i i i n i i i m i i i i
T(K+N)= 1OOOOE01 T(KJ = 90000E-O2 N s 1 STEPS AFTER FIRST MEASUREMENT
^ =^ i f (91 (9) l^llgl lt8) IIg3f|gl (7) (7gt lSiil tS) pound6) i83I--8 (5) t5gt i3^igi (4) (4) l8sSgi f3I (3) lf^gl C21 (2 li5SIgl ( 1 ) (1) P | (0) _l18537E 02_
ESTIMATION ERROR CRITERION CONSTRAINT = 75000t-02
12500E-O13
Figure 615A Contour plot of Tr measurement
p K ~K+1 M at time t K+l 010 one time step after first
CONTOUR PLOT OF T R A C E C P f K K + N ) lt Z ( K ) ) 3 AS FUNCTION OF t Z t K U l HORIZ pound Z ( K ) ] 2 VERT EXAMPLE TO SHOW GROWTH OF T R A C E [ P ( K K + N ) 3 SURFACE WITH TIME T I K + N ) I T S SHAPE APPROACHES THAT OF C P ( K K gt ] 1 1 SURFACE ASYMPTOTICALLY FOR LARGE N
5S3 44 333333 555 444 333333
5555 44 33333 S5SSS 44 3333
_ S555S 444 3333 +555 44 333
44 3333 444 3333
4444 333 444444 333
CZ(K)12
09
3333333 333333 3333333
333333 33333
33333 44 55 65 777 3333 44 55 66 777 0888G888BS
3333 44 55 66 777 660688886888 3333 444 55 6S 77777
3333 44 55 G66 777777777
4 4 4 55 6 77 889 pound39999999 0 5 6 77 8C8 993399999
4 4 5 66 77 860EI 9999999999 4 4 55 66 77 eSEIS 9999999999 4 4 55 66 77 009688 999999999999S999
44444 U33 222222222 333 44 55 4444 333 22222222222222 333 444 55 444 333 2222222222222222 333 44 51 44 33 222222 22222222 333 44
333 2222 22222 33 444 333 2222 2222 33 44
333 222 2222 333 222 1111111 222
3333 222 11111111111 222 333
$656 777777777777 66666 7777777777777777777
lta 6563566 777777777 555 66666GS66666
555 666666666656666666666 G66666666666666-555E5
14 55SS5o335 444 5553S5555amp5S55SS5555
_ _ _ _ _ 444 amp55555lgt535555555555555 33333 222 1111111111111 222 333 4444444
333333 222 111111111111111 222 333 444444444444444444444444444444444+ 33 2222 1 1 1 1 11 1111111 222 33333
2222 111111 11111 2222 3333333333333333333333333333333333 222222 1111 11111 pound22222222 22222222222
11111 1111111 1111111111 1111 H i l l 111 1111111111111111111 1111111111111111-11111111 111111111111111 1111111 11111111111111111111111 1111111111t 111111111111111111111111 I -bull 111 11111111 2222222222 111111111
222222 22222 11111111111111111111111111111 2222222 3333333333333333333 22222 11111111111111111111111111111111
22222 3333 4444444144 333 22222 3333 4444 4444 333 2222223222222222222222222222
33333 444 555555555 444 333 222i2222222222222222222222222222222 +3333333 444 555555b555555 44 333 22Ppound2222222poundpound222222222222222222222-3333333 444 5555Si5o555355 44 333 22^ZV32222222222222222222222222222 33333333 444 55S55L555 444 333 2222 213222222222222222222222222
3333 4444 4444 333 2222ZT22Z 22222222 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 ^ 4 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1
+ 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 P 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 Q 2 2 2 2 2 2 2 2 2 2 2 2 2 pound 2 2 2 2 2 2 t 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2pound222222
3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 3 3 2 J 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 33C-333333333
4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 AAA 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 -
T ( K + N ) laquo 1 4 0 0 0 E - 0 1 T ( K ) = 9 0 0 0 0 E - 0 2 N = 5 STEPS AFTER F I R S T MEASUREMENT
SYMB
( 0 )
LEVEL RANGE
3 6 1 1 7 E - 0 2
( 9 ) ( 9 )
3 5 5 5 5 E - 0 2 3 4 9 9 2 E - 0 2
( 8 1 ( 8 )
3 4 4 2 S E - 0 2 3 3 0 5 6 E - 0 2
( 7 ) (7)
3 3 3 0 4 E - 0 2 3 2 7 4 1 E - 0 2
( 6 ) ( 6 )
3 J 17BE-02 3 I 6 1 6 E - 0 2
( 5 ) (5gt
3 1 0 5 3 E - 0 2 3 0 4 9 0 E - 0 2
( 4 ) lt4)
2 9 9 2 7 E - 0 2 2 9 3 3 5 E - 0 2
( 3 ) ( 3 )
2 8 6 0 2 E - 0 2 2 8 2 3 9 E - 0 2
( 2 ) ( 2 )
2 7 6 7 C E - 0 2 2 7 1 1 4 E - 0 2
( 1 ) ( 1 )
2 6 - 5 1 E - 0 2 2 5 9 0 8 E - 0 2
(copygt 2 5 4 2 5 E - 0 2
ESTIMATION ERROR CRITERION CONSTRAINT =
7 3 0 0 0 E - 0 2
Figure 615B Contour plot of Tr measurement amp 5 (0] a t t in tbdquo = 014 five time steps after first LKt5
CCM-OUR PLOT OF T R A C E t P ( K K N K 2 ( K ) I AS FUNCTION OP t Z ( K ) 7 1 HORIZ EZ fKJJS VERT EXAMPLE TO SHOW GROWTH OF TRACECP(KKN)3 SURFACE WITH TIME T ( K + H ) I TS SHAPE APPROACHES THAT OF [ P ( K K ) 3 U SURFACE ASYMPTOTICALLY FOR LARGE N
4 4 4 46 AC A
r5 66 - 7 7 7
GG 7 7 7 PSb 77
6G6 5 66 55 666
0 bull 555 144 333333333333 55f 44 333333333333
555 44 03333333333333 _ 55555 444 33333353333333 55555 44 333^333033333333 bull555 444 333333333333333333
4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 XH M 4 4 4 3 3 3 3 3 3 3 3 3 3 3 4 4 5
4 4 4 4 3 3 3 3 3 3 3 3 3 3 4 4 4 4 1 4 4 4 3 3 3 3 3 3 3 3 4 4 4
1 + 4 4 4 4 3 3 3 3 3 3 3 4 4 4 4 ^ 4 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 44 5 5 5
3 3 3 3 222222222222P gt 33 4 4 5 5 5 333 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 4 5 5 5 5
3 3 3 3 2 2 2 2 2 2 2 2 2 3 3 4 4 5 5 3 3 3 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 g
3 3 3 3 2 2 2 2 2 2 333 4 4 4 4 3 3 3 3 2 2 2 1 I t 11111 2 2 2 33 4 4 4 4
3 3 3 3 3 3 3 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 444
3 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 4 bull 3 3 3 2 2 2 U 1 1 M 1 1 1 U 1 U 1 1 2 2 2 3 3 3 3
2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 2 2 2 2 2 2 2 1111 11111 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 raquo I 1 1 1 ) 1 1 1 1 1 1 bull 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 111111 1 gt
2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2
2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 2 2 2 2 2 2 3333 4444 53535 444 333 2222
3333333 444 5555555 5555555 444 333 33333 444 555 555 444 333 33333 444 5555 5555 444 333 333333 44 55555 55555 444 333 33333333 444 555555555 444 333 222
3333 444444 44444 330 221222 222 33333 T^33 22222 222222222 3333333333333 22222
2222222222222222 2222222 22222222222222222222
2222222222222 222222222222
333333 222222222222 222222222222222222 33333 2222222222222222222
4444444 333 22222222222 33333333333 4444 3333 333333
4444 3333 3333
JSiJ 3Sfl e raquo 3 8
9 9 9 9 9 9 9 9 9 9 9 9 9 S 9 S 9 9
9 9 9 9 9 9 9 9 9 9 9 9 3 9 9 9 9 9 9 9
iSBraquolaquo 9 9 9 9 9 9 9 S 9 9 9 9 S 9 9 858cea3e 999999999-
7777 7777777
7777777777 iGi i 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 ei5666 7777777777777
S6666666666 6666G66666666666
S35 SGS6S066666666B )5i S55555
HJ5555555S5U555555 5555555555^555555555
14 55555 1444444444444444444444
4 4 4 4 4 4 4 4 4 4 4 J 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 I222222222222222222222222222222
r i u i u i u i u i i i u n u n i i i i i i
1 i n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 bull m i n i
2 2 2 2 2 2 2 2 2 2 2 gt 2 2 2 2 2 2 2 2 2 2 2 2 2 lt 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 pound
2 2 2 2 2 2
u u i n 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1
m i n i m i m i n i m i 1 1 1 1 1 1 1 m 11 1111 111 1111111111
1 1 1 1 1 1 1 1 1 1 m i m m 1 1 m 2J22222
222222222222222222222 i33333333 3303
33333333333333333333332 3 333333333333333333
T(KraquoN)= ISOOOE01 TIK) = 90000E-02 N = 10 STEPS ftFTE F IRST MEASUREMENT
CONTOUR LEVELS ANO SYMBOLS
SYMS LEVEL RANGE
t O ) 4 2 3 1 9 1 1 - 0 2
( 9 ) ( 9 )
4 1 7 9 7 E - 0 2 4 1 2 7 4 E - 0 2
3 ) t e gt
4 0 7 5 1 E - 0 2 4 0 2 2 0 E - 0 2
(7gt ( 7 )
3 9 7 0 5 E - 0 2 3 9 l a 2 E - 0 2
( 6 ) (Ggt
3 6 amp 3 9 C - 0 2 3 amp 1 3 C E - 0 2
( 5 ) ( 5 )
3 7 t e l 3 E - 0 2 3 7 0 9 1 E - 0 2
( 4 ) ( 4 )
3 6 5 G R E - 0 2 3 6 0 4 5 E - 0 2
C3gt ( 3 )
3 5 5 2 2 E - G 2 3 4 S amp 9 pound - 0 2
( 2 ) 3 4 4 7 6 C - 0 2 3 3 S b 3 E - C 2
(1 ) ( 1 )
3 3 4 C O H - 0 2 3 2 9 U 6 E - 0 2
(0) 3 2 3 0 5 E - Q 2
EST) MAT 1 Oi l EKROR CRITERION CONSTRAINT =
7 5 O 0 C F - 0 2
1 - 2 5 0 Q E - 0 1 1
Figure 615C Contour plot of Tr measurement
bullK+10AK (h) at time t K+10 019 ten time steps af ter f i r s t
cz(Kgtia 03
CONTOUR PLOT OF T R A C E t P t K K N ) t Z ( K gt ) 3 AS FUNCTION OF t Z ( K ) ] T HOR1Z t Z ( K H 2 VERT EXAMPLE TO SHOW GROWTH OF TRACEEPCKKraquoNgt1 SURFACE WITH TIME T ( K N ) ITS SHAPE APPROACHES THAT OF [ P lt K K ) ] 1 1 SURFACE SVYPTOTICALLY FOR LARGE N
555 44 33323333 555 4 333023333 555 444 333333(333
5b55 44 3333tngt33333 5S55S 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 55L5 444 333333333333333
444 33333333333333333 444 33333333333333333333
444 55 6 444 55 444 55 444 S 5
77 BE 6 77 OEGfl
7 7 pound9118 777 ease
4404 33333 444444 3333 44444 3333 444 3333 222
33333333 444 5 333333 444 3333 444 333 444
55 66 777 44 55 66 777 444 55 666 7777 666 77777
999999999 999S90999 9S9SS39999 99999999999 99999999999999 99999999
333 2222P222222222 333 22222222222222222 3333 222222 22222222 3333 22222 2222 3333 222 222
680e88666038B68 6S6 7777777 BC3QBQSBBB gtamp 66GC 7777777777 555 6i6fiS 77777777777777 777 bull 555 6056666 77777777777
3333 222 333333 222 11111111111 33333 222 11111111111111 33 2222 111111111111111111 2222 11111 111111 222222 1111 11111
444 5555 666666366666 I3 444 555S 66GS66666S6666666 33 444 5amp05S5 6666666666666 333 444 t5Sy555555S5 333 444 555555555555555555 55555555S55555555 222 333 4444 222 333 444444444 222 3333 44 14444444444444444444441 mdash2 333333 44444444 222 333313333333333333333333333333 222222 111111 221-22222222222222222222222222222 111111111111111111111111111111 1111111111111
llll1111111111 111111111 1111 111111)1111 22222222222 11111111 22222 22222 11111111 222222222 3333 3333 22222 2 3333 444444 444444 333S 222222221 3333 144 555555535 444 3333 ZZpoundZ 333233 444 5555 5555 444 333 3333 444 555 555 444 3333 333 444 5556 555 444 3333 3333 44 5555 5555 444 3333 333333 444 5555555555555 444 3333 Zt 33333 4444 4444 333 2222222 33333 444444 3333 22222 22222222 3333333333333333 22222 111 22rgt2pound222222222 222222 11111111 2^2 2e2Sgtpound22222222222222222 1111111111 2gt2212222Ve^^-^2^222 1111111 222222poundZi2222
3333333 22222222222222222222222222222222222222 33333 222222222222222222 4441444 0333 22222222 3333333333333 444 3333 33333
111111111111111111111111111111
444 3333 33333
111111111111111111111111111111 11111111111II 111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 = 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
T t K N ) = 2 4 0 0 0 E - 0 1 TCKl = 9 0 0 0 0 E - 0 2 N = 15 STEPS AFTER F IRST MEASUREMENT
CONTOUR LEVELS AND SYMBOL5 SYM0 LEVEL RAN3E tOgt 46551E-02 (9gt (9
4 4 9039E-7D27E--02 02
4 4 701-1E-eao2pound-02 -02
lt7 (7raquo
4 4 59fSE-5477E--02 -02 lt6J (6gt
4 4 49GEE 44S2E--02 -02
(5J 4 4 39C0E-34pound7E--02 -02
(4j (4J 4 4 291
rJE-2-103 E-
bull02 -02 I3J (3)
4 4 1 830E-I37SE- 02 -02 (2gt 12)
4 4 06C5E- 03L3E--02 -02 J (1)
3 3 93-IIE-3323E--02 -02 lt0 36310E-02
EST 1 HAT I ON ERRPR CRITERION CONSTRAINT = 7taOOOE-02
Figure 615D Contour plot of T r EK+^^K) a t time t K +_ = 024 fifteen time steps after first measurement L J
CONTOUR PLOT OF TRACpound[PCKKNgtCZ(KgtgtJ AS FUNCTION pff C Z lt K ) ] 1 HORIZ t Z ( K gt 1 2 VERT EXAMPLE TO SHOW GROWTH OF T R A C E t P ( K K + N H SURFACE WITH TIME T lt K N ) I T S SHAPE APPROACHES THAJ OF C P f K K l l U SURFACE AgtV1PT0TICALLY FOR LAROE N
TJME= 9 0 0 0 0 E - 0 2 FIRCT MEASUREMENT ELEMENT 1 1)
555 444 444 55 6G 55 444 33 444 53 66
555 44 0333 4444 55 66 555 444 3333333 444 55 66
553555 AAA 3333333333 4444 55 6pound 5555 444 3 3 33 33 i 133333 444 553 S 444 333333333333333 444 ~
6D3 8 0 3e
3 3 F 9 7 7 7 3poundJt
939909039 9999S9999
990030099 39J999999
7 7 7 1 3 8 8 0 8 6 9 9 9 9 D 9 9 S 9 9 9 9 9 9 66 777 eaiaaena 99999999-
_ 666 7 7 7 7 8 6 6 e 8 8 - 8 8 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 55 6 6 77777 8e38688C8O880OO(38
4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 5 3 6 6 6 7 7 7 7 7 7 7 7 8 8 0 6 8 8 8 8 8 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 4 4 4 5 5 656G 7 7 7 7 7 7 7 7 7 7 7 4 4 4 4 4 3 3 3 3 3 3 3 3 4 4 4 5 5 5 6G3E-6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 - -4 4 4 3 3 3 3 3 3 3 144 sect55 5pound-SG6666 7 7 7 7 7 7 7 7 7
333 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 = 5 5 666665G5GGG6 3 3 3 2 R a R a raquo K 2 2 2 S 3 3 3 4 4 4 505 CGtJ6ampo6-6GGGCrGCGfiC6
3333 r y 2 2 2 2 r i 2 2 L 2 2 2 2 2 33 4 4 4 SS55 -gtb 66Gl5CCftgtG0tgt5 3 3 3 3 2gtZ2 2 2 2 2 Z 33 4-14 5E- 3 j ^ S S r i S W S 3 3 3 3 2r-22 2 2 2 2 333 4 4 4 4 55555503555511555555
3 3 3 3 3 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 0 5 5 5 5 5 amp 5 5 5 5 5 5 5 5 3 3 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 333 4 4 4 4 4lt 4-14444
3 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 I I I 11 2 2 2 2 333 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 + 3 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4
2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 V3Z 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 11111 1 1 1 1 1 1 2 2 2 2 2 2 2 ^
1111T1 111111 2 2 2 2 2 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111
11 1 11 111 1 1 1 -
11111111111111111111111111 1111111111 2222222222222 11111111
2222^ 22222 111117 1 22222222 33i^3 3333 2222
333 4-S44 44444 333 pound2222222 333333 444 55555555555 444 333 1
33333 444 5555 S555 444 3333 33 44 55 3 6G666 D55 44 393333
444 505 6665066 555 44 33333 333 444 555 555 444 3333 333333 444 55555555555555S 444 333 3333333 4444 4444 333 2222221
33333 4444444444 3333 222222 2222722 3333333333^3333333 22222 2222222222222 222222 111 1111 22P222i2-22l22P22222222222222 U11 U 1111 2ir2ai22-222i22irr2222 1111 11
22222r2-2Ki2 22 3333333 22pound2J22222Z22222222222222222Jai
3333 22222222222222222 4344444 3333 2222222 333333333 33C-
4444 3333 33333 444 33333 33333
11111111111111111111111111111 22222222222222222222222222222
33333333 3333333333333333333 333333333333033333
33353 22222
bull22222222222222222222222222222 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22222222
2 2 2 2 2 2 2 2 2 1 3 3 3 3 3 3 3 3 3 ^ 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 J
( 0 )
LEVEL RAKCE
1 6 0 S 3 E - 0 2
13) ( 9 )
1 6 3 4 S E - 0 2 1 5 0 4 0 E - O 2
1 5 3 3 4 E - C 2 1 4 C 2 E - 0 2
it ( 7 )
1 4 r 2 U - 0 2 1 3 t 1- t 02
( S ) ( 6 )
i sacaoos 1 2 8 0 2 E - 0 2
(5gt ( 5 )
1 2 2 9 5 f - 0 2 1 1 7 6 9 c - 0 2
( 4 ) ( 4 )
1 l pound P E - 0 2 1 0 7 7 C E - 0 2
( 3 J 133
1 O27OE-02 9 7 o 3 ^ pound - 0 3
(2) ( 2 )
9 2 5 t t l E - 0 3 8 7 j O ^ E - 0 3
(1 ) (1 )
BZnopound-03 7 7 3 7 5 E - 0 3
tOgt 7 2 3 1 2 E - 0 3
ESTMATUN ERtiR CRITERION C L l t T R U I H =
7 t r n o e - 0 2
SOURCE NPUr COVAKlANCE I W 1 - 1 2 5 0 f E - 0 1 1
Figure 616 Contour plot degl [amph at f i r s t measurement t ime t bdquo = 009 compare with asymptotic
response of Tr [ppound + N (z K )1 surface at t K + l g = 024 in Figure 615D
188
at the next sample at time t K + N when (645) is next satisfied From Conclusion X the minimax problem in (647) separates into finding zt
such that
[ E^4i = IK L - ^ that z which
^n-lr-1 $ 5$ pound
and independently findino that z which leads to
4 T max c(z) c(z)
(648)
(649)
for N large Various properties of the solution of th is problem are
demonstrated by example in what fol lows
631 Asymptotic Responses of Output Estimation Error - to demonshy
strate the asymptotic separation of the minimax problem in (647) into
the independent problems of vector minimization in (648) and scalar
maximization 1n (649) the problem of Section 61 was solved but as a
monitoring problem of the second kind with
~005 p 002
000001 (650) 000001
^ 000001 _
and with thi bound on maximum variance in the output estimate
Pdeg = ~0
lim 01 (651)
For this case a plot of the evolution of o^+(j(S((z) t n e gtin1max probshylem statement In (647) as a function of time t K + N 1s shown in Figure 617
The asymptotic separation of the minimax problem is demonstrated in Figures 618 and 619 The former 1s a plot of a^[z0z) as a function of the position 1n the medium z for values of time t R = 0 T 2T 9T
1OOOOE-01
6BO0OE-O2
S2000E-02
OeOOOE-02
4C000E-DZ
X X
X X
X
X
X X
XX
gt XX
X
X X
X XX
X X
X X
X X
X X
X X
X X
X X
X XX
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
Figure 617 Time response of aLwU((laquoz)gt t h e P e l f deg r m a n c e criterion for the optimal monitoring probshylem with bound on error in the output estimate for a = 010 samples occur at t = 011 047 and 085
EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTFUT ESTIMATE WITH TIME POSITION OF MAXIMUM VARIANCE APPROACHES STEADY-$ I At IT VALUE FOR LARGE TIME
80000E-02
74000E-02
96 7777 6 7 709 9 76666 e 876 6 7 9 976 555 6 78 6 55 56 9
1 0 0
865 4444 56 S 87 44 4 9 7654 4 5676
8654 33 9 754 33 33 4 567
3 SZZ100 965 3333g2H00 754
6BOOOE-02 4444pound2110 8343 5 55-JJgt3322 1002533222 777ii -514332293222 S^SS tiS314i65 0111
g- 03779S7 0 S99 (
62O00E-O2 1 2 3458
1 6 1 2 34579
36 1 2 4 79
1 35 8 2 6
i 1 34576 O I 2 6 J
C 1 23457) 6 9
12345 B 1 234 i7f)
1 gt579 0 123 13539
00 12 J4M5S9 41 OC 1 3-Ti67 9 9567
00 i345 6 6300 OOOOOO 0 00
SYMB TIME TK+N (0) 0E00 CI) 5000CE-C3 (2) 0001^-03 CD GOOCOE-03 (4) 80000C-03 t5 ) IOuOCE-02 (6) 12000E-02 ( 7 lJidOOC-02 0000 (6) 1600CC-02 000 I1 (9) 1 OOOOE-02 00 111
00111222222 0122 30333 P0112233344444 011223S4445SS-3 01 23341553 C306 012g34J50tt b67777 01254553077770360
12334Lpound67736999 12345My88393 12345677S99 12345^769 123b67699 1245S7S9 12456099 1245789 1246709 1246G99 134689 135799 13579 14R89 i99 2589 04799 2599
4000E-01 PtSl ION Z
Figure 618 Plot of performance criterion oilaquo[z) as a function of position z in the medium for K + N- -- - 2 _ _ _ J times t K+N 00 002 004 018 note how position z
changes with time of o + N(z ) = max a K + N U)
130O0E-O1
1 32O0E-O1
1 1 4 0 0 E - 0 1 ODDOnOOOCOO
raquoe00Dpound-02
oooooouooo
60D00E-02
Figure 619 Plot of asymptotic shape of performance c r i te r ion deg K + N ( z ) as a function of position z in the medium as N-raquo compare posit ion z =
totic position of maximum in Figure 618
the medium as N+degdeg compare posit ion z = 03 for Urn r j x apound + M (z) in th is curve with asymp-im n x N-~gt z
192
where T = (t K + - tbdquo) = 0002 zbdquo was taken as the initial guess at the best measurement locations z Q = [015015] The latter plot is a plot of
lti(z)T a c(z) (652) SS
2 the steady-state term in the asymptotic response of crJ + N fo r N large
Thus comparison of the asymptotic approach in time of the curves in
Figure 618 to the steady-state curve in Figure 619 shows that
N
c ( z ) T V n 1 M n 1 d(z) - c ( z ) T a c(z) (653) imdash S~S~ n=l
As a special case it shows that
max o+fzz)mdashgt max c(z) q c(z) SS
(654)
at the position of maximum variance z Note here that as expected the position of maximum variance is directly over the source position
(655)
632 The Effect of a priori Statistics mdash To demonstrate the efshyfect of the uncertainty in the initial state estimate x = m upon the optimal monitoring design problem consider variations in the a priori
statistics given in the initial state estimate error covariance matrix Pg = M- For this example fix the time interval of interest at 0 lt t lt 20 and set o | i m 5 02
(656A)
Compare the f i r s t case for which
000001 o E o s 8 o
0 0 00001
193
with the case where
E g - H o
oi 000001
o
o
000001
(656B)
The first choice results in the evolution of obdquo+bdquo(ztz) shown in Figure 620 resulting in one measurement at t = 126 The corresponding con-tour plot of [ E K ( K ) ] ] I as a function of [ z j and [jd for that meashysurement is shown in Figure 621
The plot of o^+f(zJz) for the second choice of M as in (656B) 2 is shown in Figure 622 where owing to the higher initial value of aQ
two sample times result at t = 046 and t = 160 The corresponding conshytour plots for those measurements are shown in Figure 623
Study of Figures 621 and 623 show that the locations of optimal measurement positions are not effected by the a priori statistics given in MQ provided that the time to the firsc sample is sufficiently long for the infrequent sampling approximations to apply
For the first case the time to the first sample is t = 126 for the second case the first sample occurred at t K = 046 Thus the only
effect that the choice of Mbdquo has upon the optimal monitoring design probshylem is the detirnrination of the time of the first sample
Thus the results of Conclusion V are substantiated here within the context of a monitoring problem with bound jn output estimation error
To illustrate the transient effects at play in the general monitorshying problem effects that exist before the infrequent sampling requireshyments of (518) and (520) are met consider the same problem as in the
20000E-01
16000E-01
taoooE-oi
raquo XX XX X XX X XX XX X
X Xt XX X XX X XX XX X
X X XX XX X XX X XX
I XX
X sx
XX X XX
X XX XX X XX X XX X 1 XX 1 X I X I XX I X I X I X I X I X
XX X X X X X X
X 1 X IX
X
X
1 600E+CO
2 2 0 Figure 620 Time response of ai+ufivtZ J f o r degi- = 0- 2 with initial covariance matrix P Q H H Q given in (656A) one sample occurs at t = 126
CONTOUR PLOT OF CP(KK) tZ(K)) J11 AS A FUNCTION CF CZCOU HORIZ AND EZtKgt32 VERT
bull4444 33 22222222222222222 4444 333 222222222222222222 4444 33 222222222222222 444 33 22222222222222222J 333 22222 2222222222222 333 2222
fZCKHZ
03
3333 __ 3333 22
33333 222 3333 2222 333 222 333 222 33 222 3 222
222222222E222 222222222222 2222222222222
2222222222222 222222222222
222 222
222 1 2222 11
22222 t11 1111
11111 bull1111111
22222222222 2222 31
1111 2222 31 11 111111 222 11(1111111111111 222
111111111111111111111 222 1111111111111111111111 22
1111111111 22 1 1111 I 22
11111 1111
1111
33 AA RK 7 aesss 999939 0 33 AA UK 7 7 eaaeo 99999 333 AA KH 7 a ieaa 333 A fifi 77 888908 999999
33 Ai HH 333 A 55 6t 7777 CAB 188 99999999
33 44 bullgt B 77777 888883 9953 333 AA Vgt 6(i 77777 0888883
3 3 44 Hfgt lies 777777 8880088885 3 3 3 AA 55 8SS6 777777 889Pd3S8
66666 7777777 44 555 6G6666 77777777 444 E5gt3 6666666 7777T777777
I 44 5SS5 66D6666 7777777 I 44 i5555 666G665 13 444 5555S5 666G66G66 J3 444 55055555 6665666666
1111 1111111111 22 33 AApoundA 5555555555 66666 22 333 J4I44 555555555 222 333 44AA4A4AAA 55555555553
222 3333 4444444444444 222 3323gt33 444444444444
111 222 33333333333333333 11U1 222E222 3333333333
11111 222222222222222222222 11111 1111111111 2222222-
H I 11 i i i i m i i i i i i i n i i i i n i u i u n i i i m i n 11111 m m 111111111111111111111
11111 222222222222 1 1 m m m i m - m m 1111111 222 33333333 222 11111 11111111111111111111 11111111 22 33 444 33 22 111111 11111111111111111111111111 1111 2E2 33 44 444 33 222 1 11 11 11 1 1 11 1 11 1 1111 1111 222 33 44 555 555 4 33 222 1111)11 2222 3 4 55 66666666 55 44 3 222 22222222222222 222222 33 4 5 G6 666 55 ltJ 33 222 222bull22222222222222222222 bull22222 33 44 55 66 777 66 35 44 33 22222 2222222222222222pound22222222-22222 33 44 53 66 777 6 5S 4 33 2222 2222222222222222222222222 22222 33 4 5 66 666 55 44 3 222 2222222222222222 222222 33 A 55 6666666 35 44 33 222 1 2222 33 44 655 555 44 33 22 11111111111111111 1111111111 222 33 444 44 33 22 1111111 1111111111111111111111111 222 333 333 222 11111 1111 ^
bull11 O 111111111111111111111 1111111 111111 111111111111111111111 1 22222222 22222 222222 22222222222222222222222222222 2222 222222222222 222 2222222
11111 bull2222 1 11 11
2222 1111 333 2222 11
3333 224 333 222 333 222
222222 222 111 m m
i m i m i i 111111 1111111111 111111 m i m 111 m m i i 11111 n
m m i i m n
CONTOUR LEVELS AND SYMBOLS
SYMB LEVEL RAIiGE
~76) iTs^ ie -o i 19) (9)
2 2
4972E 4402E-
02 02
( 8 ) 2 2
303i 3263E
02 02
C7) pound7)
2 2
2S94I-2124b
02 02
(61 (6)
2 2
155ipound 0985g
02 02
(5) (5)
2 1
011 5pound 98-562
02 02
t4 ) (4)
1 1
927ampE 87071J
02 02
(3) (3)
1 1
6137E 75S8E
02 02
(2) (2)
1 1
6996E S428E
02 02
(1 ) n 1
1 1
5059E 52QUE
02 02
(copy) 1 J720E 0 2
EST1 HATION ERROR CRITERION CONSTRAINT =
SOOCC^-Ol
12500E-O13
F i g u r e 6 2 1 C o n t o u r p l o t o ^ F K ^ K ^ l n w 1 t h i n i t i a l cdegvariance matrix E Q = - 0 9 i v e n i n ^ 6 5 6 A f o r
the sample at t j 126
20000E-01
95000E-02
6 OOOOE-OS
SS000E-02
Figure 622 Time response of C J | + N ( Z Z ) for ltm = 02 with i n i t i a l covariance matrix P 0 i MQ given in (656B) two samples occur at t K = 046 and 160
CONTOUR PLOT OF t P ( K K ) ( Z ( K ) ) 311 AS A FUNCTION OF CZCfOJ I HORIZ AND r Z ( K gt 1 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT ESTIMATE WITH T IME P O S I T I O N OF MAXIMUM VARIANCE APPROACHES STEADY-SiTATE VALUE FOR LARGE T I M E
CZ(Kgt]2 05
4444 33 222222222222222222 4444 333 222222222222222222 444 33 222222222222222222 444 33 222222222222222222 333 22222 2222222222222 333 2 22 333 2222 3333 2222 33333 222 3333 222 333 222 222
222 t 222 11
2222222222222 2222222222222 2222222222222
22 22 22 22 111
2222222222222 pound22222222222 2222222222 22222
23 44 55 6G 77 33 44 5 66 777 333 AC 5 66 777 333 4 55 66 777 33 44 55 C3 777 333 4 55 56 7777 33 44 5 e3 77777 333 4 55 i36 77777
999999 99999 93999 999999 99999999 99989999 9999 8888866
0
2222 222 222
111 222 222 222 2222 111 22222 111 111 1 11111 1111111
11111 11111111111 11111111111111 1111111111111111
1111111111 111 1 I I
11111 1111
111
55 666 77777 4 53 6666 777777 68688688 4 tgt55 66666 7777777 44 3E5 666666 77777777 444 5J55 6666666 77777777777
44 S55S 66665C6 777777 44 5555 6666666 Aamp1 555555 66666666
2 a 3 J14 555555555 6666666666 2 33 4144 555555555 66666 22 333 44444 555555555 222 333 4444444444 55555555555
222 3331 4444444444444 222 3133333 444444444444
1111 222 333333333333333333 11111 22 2^22 3333333333
111111 322222222222222222222 222222 11111
111111111111111 -11111 111111 111111111111111111111
11U1 222222222222 11111 1111111)11111111 1111111 222 33333333 222 11111 11111111111111111111
bull11111111 22 33 4444 33 22 111111 11111111111111111111111111 111 222 33 44 44 33 222 11111111111111111111 11111
222 33 44 555 555 4 33 222 11111111 22222 33 4 55 66666666 55 4 33 222 22222222222222
222222 33 4 5 66 66 5 4 33 2222 2L 1222222222222222222222 22222 33 44 55 66 7777 66 55 44 33 22222 2222222222222222222222222 2222 33 44 5 66 7777 66 SS 44 33 2222 2222222222222222222222222 22222 33 4 5 66 666 55 4 3 222 2222222222222222 222222 33 4 55 66666666 55 44 3 222
2222 33 44 555 555 44 33 22 111111111111111111 111 11 111 111 222 333 44 444 33 22 1111111 1111111111111111111111111
2222 333 333 222 1111 11111 2222 3333333 222 1111 11 22222222222222 22222 22222 3333 2222 333 222 333 222 333 222
111 11 0 11111111111111111111 1111111 111111 111111111111111111111 1 2222222222 22222 222222 22222222222222222222222222222 2222 22222222222 2222 222222
SYMB
t b i
LEVEL RANGE
z 5 5 1 9 pound - b 2 _
( 9 ) ( 9 )
2 2
4952E 4384E
0 2 C2
I B ) ( 8 )
2 2
3816E 3248E
0 2 0 2
( 7 ) ( 7 )
2 2
2G60E 2112E
0 2 0 2
( 6 ) ( 6 )
2 2
1544E 0977E
0 2 0 2
( 5 ) lt5gt
2 1
0409E 984 I E
0 2 0 2
( 4 ) ( 4 )
1 1
9273E 8705E
0 2 0 2
( 3 ) ( 3 1
1 1
8137E 7570E
0 2 0 2
( 2 ) t 2 )
1 1
7002E E 4 3 4 E
0 2 0 2
( 1 ) ( 1 )
1 1
5 8 6 6 E 5298E
- 0 2 - 0 2
( 0 ) 1 4 7 3 0 E - Q 2
ESTIMATION ERROR CRITERION CONSTRAINT =
2 0 0 0 Q E - O 1
1Z300E-011
Figure 623A Contour plot of Ppound( K )1 with in i t ia l covariance matrix f 0 = MQ given in (656B) and ulim = 02 for the first sample at tbdquo = 046
CONTOUR PLOT OF t P I K K lt ^ C K J gt 1 1 AS A FUNCTION O t 2 ( K ) J 1 HOBI2 AND t Z ( K gt ) 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPU ESTIMATE WITH T I M E POSIT ION OF KAXtrUlK VARIANCE APPROACHES STEADY-31A7E VALUE FOR LARGE T I M E
09
333 44 14 33 44J 33 333 333 2222 3333 2222 3333 222 33333 222 3333 2222 06 +333 222 333 222 222 I 222 11
07
CZCK132 O S
222 222 222 2222 22222 1i till lllli bull1111111
22222222222232222 22222222222222222 222222222222222222 222222222pound2222Z222 22222 2222^2^222222 2L2 222 222222 222222222222 2222222222222 2322222222222 222222222222 22222222222 22222 1 2222 1111 1 111111 Mil 111111111111111 11111111111)111 11111111 111
3 3 4 4 7 7 7 3 3 3 4 4 S 3 3 4 59 66 7 7
3 53 6 7 aar 4 4 ss i6 3 4-1 tgt
333 44 S3 tgttgt6
777
333
222 222
44 SS 66 77 USB66 993939 0- 3 G 6 99999 8388 59939 eeeoas gposgfl 00866 99999599 ltft aeoeoo 9999399s-77777 888068 3999 77777 8638880 665 777777 6608800888 4 OS 6SE6 777777 86000680 4 55= 66666 7777777 44 ESi 66SCC6 77777777 444 5i3 60EG666 77777777777 44 iSC5 6GGGGG6 7777777 44 35355 G61606G 1 44t 555553 GoGG66G66 22 33 114 355553C3 G6GG66G6G6 I 22 33 44 4 5355330553 CS666 II 22 333 4444 535555553 III 222 333 1444444444 33353515533 111 222 3333 4444444444444 till 222 3313J33 444444444444 1111 222 33333333333333333 11111 222122 3333333333 11)11 222222222222222222222 11111 Hill It II 222222 111 M1111111111111 11 II111111 I 1111 III11111I1 11111 111111 111111111111 11M111 11111 222222222222 11111 1 It 1M111111111 111111 222 33333333 222 11111 bull 1 111 11 1111 11111M 11111111 22 33 44 33 22 111111 11 111 11 I 111 11111 1111 I till 22 33 44 444 33 22 11111 11 bull 111111111 11 II 222 33 44 553 335 4 33 222 1 1111 III 2222 3 4 55 C666666G 53 44 3 222 222222raquo22222 222222 33 4 5 G6 666 53 4 33 222 22222222222^222272222222 22222 33 44 55 C 777 5 05 44 33 2222 2S222Wr2S2222222222 22222 33 44 5E 66 777 6 53 4 33 2222 22L-22rT22E22222 222222 22222 33 4 5 66 6SG 55 44 3 222 2227222222222222 2222222 33 4 55 G66GC66 53 44 33 222 11 2222 33 44 555 533 44 33 22 111 111 1 11 I I 111 111 11 I11 1 11 111
1111 111 HI
222 33 444 222 333 -111111 222 333333 222 11P1H 22222222222222
33 22 22
11111
2222 111 t i n 11 t i n 1 3traquo3 222 1111111111111
3333 222 11111111111 333 222 1111111111
2 111 m m
m n i u l i i i i i n m i n i i m i 11111
m m 11111 mi 11 1 11111 m 1 m 111111 1 22222gt222
222222 2222 222
m m 111 m 1111111111111
111111 m i n i m i m m i i m t 22222 222222222Z3222222222232222222 2222222222222 2222222
SYI-3 LEVEL RANGE (0) 25540E-02
l 2 2
4970E-02 440IE-02
2 3B31E-02 32G1E-02
i l l 2 2
2GXE-02 21225-02
1 2 1352E-02 0963E-02
11 2 1
O4I3E-02 9843E-Q2
i I I
9274E-02 8704E-02
II 1 8I3-JE-02 -756-S-02
si 1 1 6S93E-02
GJ25 -i-02
1 I
3333pound-02 GZilLC-02
lt0gt
g trade -12uorE-oil
Figure 623B Contour plot of [ P pound ( Z K j L with i n i t i a l covariance matrix PQ = HQ given in (656B) and
degl lim = 02 for the second sample at t R = 160
199
2 last case above with HQ defined in (656B) but with a = 016 instead
This results in the curve for o K + N(zJJz) shown in Figure 624 for the
shorter time interval 0 lt t lt 10 Two sample times result at t bdquo = 011
and t K + r ) = 086 Corresponding plots for [pound(lt)] and [ P pound + [ | ( Z K + H ) ]
are given in Figure 625 Notice how in this case that the optimal meashy
surement positions it and z bdquo + N at the two samples are different The o
reason for this is that here the estimation error l i m i t o is so low
that the infrequent sampling approximations do not apply at the f i r s t
sample t ime This is inferred by the response of degV+N^K Z^ i 9 U r e
624 where i t is seen that zhe steady-state slope [ f tJ i i = 000125 for
this problem has not been reached yet at the f i r s t sample whereas i t has
at the second thus the steady-state simpl i f icat ions 1o not apply at the
f i r s t sample For th is reason in practical applications step (3) of the
algorithmic procedure given in (572) is important where at each sample
i t is necessary to check whether or not steady-state conditions have
been adequately approached for the infrequent sampling approximations to
apply
833 Problems with a Fixed Number of Samplers aid Constant Error
Bound - Consider a problem withm = 2 samplers to be used in every 2
measurement with a time-invariant error bound o = 0075
The i n i t i a l covariance matrix
000001 O 1
eS = y 0 (657)
O 000001 Conclusion V and XI are substantiated in the context of this problem with bound on output error
laquo vV
X X K
- w XX XX XX XX XX XX XX XX
X
XX XX XX XX XX XX XX XX XX
xx m
X XX XX
gt X X X X
X X XX X X X X X
S5QQ0E-Opound
X X
X
X
X X X x
X
Figure 624 Time response of a K + fzpoundz) for a = 015 with initial covariance matrix P Q = M Q
given in (656B) Two samples occur at t = 011 and 086 compare with Figure 622 for case with a = 02
CONTOUR PLOT CF CP(KKMZltKgt1311 AS A FUNCT0^ t r [ZC EXAMPLE TO SHOW EVOLUTION OF VARIANCE ID C - J _ P C rSrl POSITION CF MAXIMUM VARIANCE APPROACHES S T C ^ V bullpound ATE i
Ji HOTIZ AND tZ(Kgt]2 VERT E WITH TI ME LUE FOR LAHGE TIME
tZ(K)12 05
aa 33 44 4 -_ -
4444 33 222 4 4 4 33 2222 4 4 333 222
33 222i 33 222
3333 222 33333 222 33333 222 33333 222
2222222222222222222 2222222222222r222 22222 2 t2222^^22
22222 2 2222 2f P 22 2222^22P2 22j2^^2r^22
22 ^lt7ih
3333 3333 3333 3333 3333 33 33 3333 333
22 22
2 2
i n n i m n n i n 11111111111111
2222 222 222
77 7 A C e R B 9C99 0 77 c-rrc-rs 90909 77 SCT638 S3^99 7 77 0^036 099999 777 CC3C36 92999999 7777 G363G3 99999999-
7777 eee 9999 i j 7777 e^cr pound33 (--bull 77777 iJZWrampec V G 7777 7 6^000833
j GMJ-5 7777777 o U -CG 777777 gtbullgt Ev -ro 777777777777 bulljT -5 CCSG^GS 7 7 7 7 7 7 7 7
11111111111 11111 j
1111 m i
22 33 AA
2 2 2 2 2 2 2 2 111
1111 bull i m m 1111
1511 2 2 33
i l l
111111111111 11111 11111
1111 222222222222 1111 111111 222 3333333 222 1111
11111111 22 33 4444 33 222 11111 i m 222 33 44 44 33 22 11111
222 33 44 55 555 4 33 22 11 22222 33 4 5 666666666 55 A 3 222
22222 33 44 55 G6 66 5 4 3 2Z2 2222 33 4 5 6 7777777 CO 55 4-1 33 2222 2222 33 4 5 66 7777777 56 55 44 33 2222 22222 33 4 4 5 5 6 6 7 66 5 4 3 222 222222 33 44 55 G66S C666 55 4 3 222
22222 33 4 555 55J 4 33 22 11 2222 33 444 44 33 pound2 11111
222 33 44444 T3 222 111 11
4l4 4fCltits44-44 53355-ltt44-144444
J333333333 r333533 33333333333
2222^^^^22^22222222 11111 1 I I 2222
m m m m i 11111 m 11 m m m i i m u m
m m m t m u u u u-u m i m 1111 m m i m m m m 11 n m M TVZ
222ytgt gtr 222222 2 2 2 - f v SW2V2vbullgt222222
22 - ^ ^ 2 ^ 2 2 2 2 2 2 2 2 2 ^ V 2 2 2 2 2
11111 bull m i m i m u m m m M U U 1 1 1 1 1
i raquo i 11 I I 111 m i 2 2 2
2 2 2 2 2 2 2 2 2 2
333333 222 333 222
4444 33 222 44444 33 222
2 2 2 2 2 2 111 m m
11 M l 111 1
111 M i l l 1 1 1 11 t m i l i u m m u i 11 U U 1 1 U 1
1111 u
22222 2222
2222 33 222 333
1111111111111111 1111111111111111-1111111
2222222222222 22222222222222 222222222
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3133333333 3333333333
CONTOUR LEVELS AND SYMBOLS
SYMB LEVEL RANGE
( 0 ) 2 C ^2E-02
( 9 1 113^151 ca t I-I13II--S1 pound71 pound71 iiS51ESf ( 6 ) (6) flIIlsecti ( 5 ) ( 5 1 UI|g| ( 4 ) pound41 i laquoSIS ( 3 ) ( 3 ) ^IIsectI ( 2 ) ( 2 ) sectvSgSI pound1 ) pound1 1 ssiis (0) 1 4302-02
ESTIMATION ERROR Jraquo TERION C0NampTR i - r =
C W 1 =
pound - 0 1 )
Figure 625A Contour plot of te)]u wi th initial covariance matrix P = H given in (656B) and cC HO15 for the first sample at t K = 011 case with a s 02
Lim
Compare with Figure 623A for
CONTOLR PLOT OF tPCKK) CZIK)) 311 AS A FUNCTION 3F [ Z ( m W3R1Z AND tZ tK) )2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTPUT ESTIMATE UTH TIME POSITION OF MAXIMUM VARIANCE APPROACHES STEADY-STATE VALUE fOR LARGE TIME
tZ(K)12 os
44444 333 22222222 44444 333 2222222 44444 333 22222-222 4444 33 2222222 4-14 333 2222222 A 33 22PZZZ
333 22ZS-K^^2222 333 22222^ 22222 333 2222222-222 22222
333333 222222222222222222222 33333 22222 33333 2222 3333 2222 3333 222 333 222 bull333 222 333 22 333 222
222
9333 At 3333 A 3333 A 333 gt 3333 333 bull 3333 333 333 33
22 222
2222 11 pound2222 1111
22222222222222 2222222222222 3
22222222222 22222 2222
111 11 222 11111111111 222
111111111 222 1111111111111111 1 11111111111
1111111 1111111
11111
99299 0 909999
S3 GG TIT B06BB 939999 55 66 77 85BG03 993299
A 5 65 77 03BBB 99999999 4 55 66 7777 66G86 99D999999 AA 5 6 3 7777 BBB30G 999999 AA 55 6 56 77777 QBOB600
44 55 56U 77777 eeBSBBBO AA 555 6GS6 777777 8008806008
44 S5S 0666 777777 66800 44 55 i 666C6 7777777
i3 44 5-iS 666666 777777777 33 44 550 6GG6666 7777777777 33 444 raquo5Si5 6G666SS6 77777 333 44 S)iS35 GGGGGGG6 33 444 3555535 6GG6660GG0 333 444 5555555555 66G666666
222 33 44 14 5555555S5S5 22 33 14144 5555555555 22 333 4444444444444 5535555 222 333 1 4444lt1444444 2222 I3lt13333333333 4144444
33333333333333
111111111111 1111111111111111111
1111 111111 1111 2222222222222 11111
11111 222 33333 2222 11 11111 222 333 333 222
1111111 222 33 44444444 333 111111 22 33 444 444 33 ez 1111 222 33 44 5555 44 33 Zi 11 22 33 44 55555555 44 33 2 11 22 33 44 55SS5 444 33 f 1111 222 33 444 444 33 22i 111111 222 33 3444 4444 233 222 11111111 22 333 44 333 222
111111 222 3333333333 222 11 11111 2222 22L1
11111 22222 1111111 111111
11111 11111111111 111111111
11111 222222 11111
2222 1111 222 11111
33 222 11111
11111 2222 11111 222222222222222 1111111 222222222222222222
i i11111 i 11111 n i i i I 11 i m i n i i i i i i
n i n m i i i i i i i i i n i n i i i i i i 111U1111111U1111111111111
m i l i i n u n i i i n i i 1111 i i m i m i i-1111111 I 111 i i n i n i 11111 11 1 111
1 1 2222222222
1111
1111 11111 11111 111111
222222 222W222222222 1 2222P222 HiP2222222222 2 2222222i^22222222222
1 111 222 1111 1 I I 1 1 1 1 1
1 1 1 1 1 1 1 lt i m i l m 1111 in 1111 m i ii 1111 ii i i lt i i i i i i i i i i i i i i i i i
m i i i i i i i
n m i n i m i i 222
222 - 2222222222222222222222222222Z22 222222 222222222222222222
22222 2222222222222
SYK3 LEVEL KAKEJE
(01 25171E-02
l 2 2
d570E-02 397CE-02
2 2
33G3E-02 27tiOE-02
2 2
21amp8E-02 15G7E-02
i 2 2
OQti7E-02 OatiiSE-QZ
i 1 0765E-02 9163E-0Z
1 05G4E-C2 79G4E-02
1 1
73G H-02 O7r3E-02
sect 1 1
eir2pound-02 55G1E-02
1 1
49G1E02 43G0E-0Z
tQl 137G0E-02
ESTIMATION EMWJ3 CPlTpoundRtCN CCNS^MNT laquo
I SOJSt-Ot
HIAfCL IWJ
Figure 625B Contour plot of | EK(^K) w i t h i n i t i a l covariance matrix p[j = HQ given in (656B) and a =015 for the second sample at t K = 086 Compare with Figure 623B for
case with a 7 - ~ 02
203
Supoose the problem starts at time tbdquo As discussed in Section 63 and according to Conclusion XI the position z of maximum variance in the estimate of the pollutant concentration at all measurement times is independent of time and is thus calculated at the beginning of the problem With this value z relationships among the various optimal measurement position vectors z at Ihe measurement times are to be conshysidered
Assume that the time the first measurement is required is at timj t iy is found to maximize Ktt) the time the next measurement is reshyquired Then at t K + N gt it+bdquo is found to maximize the next time interval to a measurement etc A typical plot of a (zz) over values of tbdquo is shown in Figure 626 For each measurement time t bdquo + N gt zJ +bdquo is to be
found to minimize [ P S ( Z K + N ) ] so that to corroborate the optimizations K+N over K + N contour plots are made at every measurement time for [ P K + N
(z K + N)] as a function of [ji+N] horizontally and [Zj+NJ vertically Plots for the four resulting measurement times in this problem at t = 027 048 069 and 090 are shown in Figure 6-7 Notice that the contours at all samples are the same leading to the eame optimal design for z] + N at all measurement times t K +^ thus Conclusion VI is demonshystrated
Comparing the first two measurement time intervals in Figure 626 that is (t K - t Q) = 027 compared with ( t K + N - t K) = 021 shows that for N large the only effect that the choice of U Q has upon the optimal measurement design at the first sample at time t is in determining the time of the required measurement t K it has no effect upon the optimal locations zt which demonstrates again Conclusion V
RUN N3 1 EAMfgtLE 7 0 - T I C W IPOLUTION OF VARIANCE I N O U T f U I ESTIMATE WITH T I M r S I G ( t ) POSIT ION OF r A X I M W I VARIANCE Prf iOACHES STFAIV -STATE VALUE FOR LArtCE T I M
60000E-02
4B0DEE-02
1-6000E-02
x x raquo X
X X
X X
X
X X
X X
X X
I X
I X I X I X I X I X I X
X
X X
X X
X
x
x x x X X X x x x
x x
I X I X
I X
X X
X
X
X X
X
X
X X
X
X
X X
X
X
1 X
i x
IX
X X X laquo(
X
l - f y r s ^ - ^
Figure 626 Time response of o K + t Yz z ) fcr obdquo - 0075 fojr samples occur at t f deg-7 048 069 and 090
205
deg gK Slt1
1 ss rjti on OO OO s
Vr gK Slt1
1 is 5 1 T 3
ore 2-5--
co iZ ^ pound3 Sm mdash SS raquo N
T 3
ore 2-5--
o tfgt W laquo WWttWW r-r- bullft w laquo NWWWW r-- ID n v ^ n WWVWftl r-f^ o m raquoltT f t WWWWCd S lO V o WWWWW
o rt V WWfV-W N T iT o ftiwwcvw N r u w N M V N N
bulla L i V laquo ltj laquobull IV o V o n wywcvcv
t o o i n lt o n WflWftWfti bull bull M O O m T WltoeJW
O t f rt V WWftftftiftJ O O w T o r a OlttKiV-jAiAW p laquo T WWMMftAlMW - N L I V WftiXFMAiiVOi
- N 1 bulli l V OCT L i ft
pound o irw 7 o ft ltt -v
t ID o ttvfitirv i m laquo w bullcjftCnWW
^ tvft fNPJVWPi o Ift W o W f - gt bull laquo ( raquo gt laquo OHO ifl bull o laquo c (M^Cft(M lOul n ^r Vi Nfftl O O - iv iww
(D^-gt bull c- laquo wwv luWNUi 10OO - 1 n n wwcv vwni
ww o o bull
mdash mdash mdash CJW
mdash mdash - mdash mdash Wftl
- ^ N N N N r v
www bull inmdash
bull (Oioininraquo-))0in
H 5 S 5 2
ftjft www Mftt
WiMCU
mdash ^ - w
c^v fJSJCl mdash - mdash -
iiiisis mdashmdash WW bull O mdashmdash (M J bull bull bull bull o
CONTOUR PLtff OF EP(KK)(Z(K))311 AS A FUNCTION F rZ(K)J1 H0R1Z AND CZ(Kgt32 VERT EXAMPLE TO SH3W EVOLUTION OF VARIANCE IN OUTPU1 ESTIMATE WITH TIME POiilTION Of MAXIMUM VARIANCE APPHOACHES STEADY- J T M T E VALUE FOR LARQE TIME
1
CZ0012 05
444 444
4444 4344 4VV
4144 444
4444 bull4444
44444 44-14-14
444-144 444444 44 14 4 4414
444444 5 444444 5 444444 5 4lt14lti44 SI 444M 44444 4444 44-144 4444 4 114
777 777 777
66
114 333 3333 3333 3333 3333 3333333 3333 22JU22 2222 2222222 22221-2
3333 333C333 333S$33333 I33333J373333 $33313raquo33S33333 4-144 555 666 3$3333amp3i33333333 444 55 666 33$^33J33J33333333333 444 55fgt 66J 3333 33333 333333 44 55 6E-6 333$3 3J33C333 444 55 GS 05S33 444 505 33333 44 b-S 222222 3333 444 555
0080 ueeo H388 SC30
OC038P occecoo
9990339 99093999 9S9S999 99303399 1S999 _ 9999S99raquo 999959999b 88663098 S99999 77 388833083 7777 8063000068 777777 808EJ8C88380860 7777777 0S03C3SQyC8B 777777777 6838008 777777777 Jifi 77777777777 ltJ 0C6 7777777777777777 fgtiit36SC 77777777777 66b5Eil3S^GC6 222222222222 333 44ltJ 555 22222-gt^I22amp2pound22 3333 bull 222pound222222222222 33 222222222 333 2222222 3333 -4^4ltM1414444
222222 33333 4441444444444444444444 222222 333333333
50305555355555 555555505555555555
222222 pound22222-2 2222
2222 2222222 ^ 2 2 2 2
1111111 1111 111 1111 111 II i n i n m i n i m m m i i n m m i m m i m i m i m i 1111
i n i m n
m m n n m i i
11111 m i l
2222
22
111
n
m i n i I n 11iin11 I I ii i m i n i m i IinI1111 n i n m 111111111 in-1
111 Ull 22222222-111111 22222222poundK22222 t 22222222222
11111 2222 2222 2222222222222 11111 2222222
11111 22222 33333333333 J3333 oo ^22222 ii i n ^ ^ i m i i H2222 333333c
SYMamp LEVEL RANGE
tO 21520E-6pound
(6t C6gt lISISi (5[ (5f l3ililgl (4) 14) 15SfI8i
(2J 1026oE-02
ita
I250UE-01J
F i g u r e 6 2 7 B C o n t o u r p l o t o f fe)jn for the second sample at tv = 048 K
CONTCLrt PLOT OF I P f K K ) ( Z ( K ) ) J 1 1 A3 A FUNCTION O f [ Z t K l l l HOR12 AND t Z ( K gt 3 2 VERT EXAMPLE Tr- SiTOW EVO^UTIDN OF VARIANCE I N OUTPUT r S l M A T E WITH T IME POSIT ION Oi MAXIMUM VARIANCE APPROACHES STEADY-G ATI VALUE FOR LARG T IME
444444 55 66 777
41 V pound4 tgt5 SS6 77 SS 66 777
44 444 oL-5 06 77
4 4 4 4 4 4
4 4 4 4
4 4 4 44-11 33333
444 1 3 3 3 3 3 3 3 A4-aA 3 3 3 3 3 3 J 3 3 3 [4ltii 3333Cgt J3073033 44 4 3 3 3 5 J i 3 J 3 3 3 3 3 3 3 IJ44 33 3V333o3-raquo3333333 M4 3 3 1 3 3 i 3 3 ^ J j J 3 a 3 3 3 3 a 3 144 3 3 J 3 3 3 3 3 3 3 3 3 3 3 3 44 5 S 3 6(gt 14 3 2 3 3 3 3 3 3 3 3 3 444 5S 5C I 3 3 3 3 3 3 2 3 3 3 AAA C 5
3 3 S 3 3 3 3 3 3 4 4 035 3 3 3 3 E222222 3333 -144 j-5
3 3 3 3 ZZZsrlte22 C3C3 4 4 4 5555 bull 3 3 3 3 322 2 22SV2222 3 3 3 3 4 4 4 5 3 3 3 3 3 3 3 23222gt2222-2raquoPgtpound22 3i3 4 4 4 4 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 1 2 3133 4-1444-i
2 2 2 R 2 r t 2 2 2 3 3 3 3 4- 1
2222-222 2 2 2 2 ^ 2 3 3 3 3 3 bull 2 2 2 2 2 2 222 3 3 3
1 1 1 1 1 1 1 1 1 1 2J222
-1-114 J 44-1
4 4 4
m i n i i i i i i i n i m i i i i i i i i t i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 bull i i i n m m 11 I n i i
m i i n n i m i m l i m n
u i m 11 i i t 1 1 I 11 II 1111 1111111
bull111111 111111111
11111 m i l
2222 1111 +pound2222 1111
111
22222 22222222 2222222
m u m 111111111111111 11 111 1111111111111111111111
11111111111111111111111111111 m m i u n i i i i i
m m i i m n u m n m
l i m n m m
11 11 1 22-gt22 11111 2222222 11111 22222 11111 22222
1 C8 9 3 9 9 9 9 9 0+ B t3 9 9 5 9 3 9 9 9 U CBS 93S--99
EU3S 3SJSJ3U39 E-r-so 9 r j099S99
CC30a 33S-SSE9 CfiSBOO 9999 pound999999
383S3S8 9 9 9 9 2 9 9 3 9 9 77 8S33C308 9 9 9 9 9 9 bull717 GC^raaraquoSB 7T777 amp088 iS9QeS
777 777 e8oSSr 30808388 777777 6 3 a 0 8 3 8 3 3 8 0 a
7 7 7 7 7 7 7 7 7 8 8 8 8 6 0 8 7 7 7 7 7 7 7 7 7
Ht 7777777777
CCfiSS 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 i l Egt6 -amp3S 7 7 7 7 7 7 7 7 7 7
S-j^tiGfcG666SG
0 j55 6C6eSCi66e666 _x^CJ50tgtSS555553
S5Cgt5055C55DS5oS5S5 -4M44444444A
4444444444--1444444444 3333333
33333333033333333333133333 3333oJ33333 2r^222 2- i^^22222222
22 pound 3ft laquoraquoamp 2 22222P2S2 222 22i^lamp r PP-2-2222^22222e2
2222 vr^- amp2222222 2 r ^ g 2 L - - ^ 2 2 pound 2 2 2 2 2 2 2 2 2
2^2 r 22 gt22222HS222222S22 P22^252i-pound-HSpoundHS-222i 12K 22c
2222^222gt2222222P22 22222222 2 222 222^22
pound22222222222 m i 1 bull m i n 11111111111111
i i i 1 1 1 1 1 1 i i i 1111111 11H11111 i l l 111
22222222-2^222222222222222^22222222222 bullbulliiiiL22ZZgt2Z-ZZZt
SYMB (01 mm (91 i OC03E-02 0152E-Q2 (8) (B)
9450E-02 B748E-02
(7) (7)
C04SZ-02 7344E-02
(6) (6)
GC43E-02 5341C-02
15) (5)
5235E-02 4 33E-ca
14) (4)
3S36E-02 3134E 02
(3) 13)
2432E-02 1730E-02
(2) C21
103pound-C2 Oj27t--02
(1) (1 J i 6252E-03 9234pound-03 (copyJ 8 - 2 2 1 7 E - 0 3
ESTIMATION ERROR C-RIrEKl f lN CONSTRAINT =
7 0 0 0 P S - 0 3
KlANCE [WJe
1 2 S 0 0 E - 0 1 1
Figure 627C Contour plot of [bullft M i l for the th i rd sample at t K = 069
CONTOUR degLOT OF tPCKK)(2(K))I1 AS A FUNCTIOM CF [ Z C K U I HORI2 AND (Z(K)13 VERT EXANPLF TO SiampU EVOLUTION OF VARIANCE N OUTPUT ESllMATE WITH TIME POSITION OF MAXIMUM VARIANCE APPROACHES STEADY-STATE VALUE FOR LARGE TIKE
3b55 5Sgt3 S5S6 555
444 4444 444 AAH 444
aaaa aaa
4444 44 3-4
lt4444 44- 114 444-44 44441
444444 444444 444 4 11 444414 4444-1 44444 1444-1 -14414 4444
53 G6 777 55 66 777 55 66 777 55 (JPS 77 GSS SS 77 55 GG 7 55 S6 7
I o
4 t44 Sco SG$
535
IZ(K)J2
05
33333 3333333
333333J333 333333330
33-raquor-ltgt3^ii333 V J 33ogt-i333ampJ^33333 444
3 3 3 3 S 3 3 S S 3 3 3 3 3 3 3 3 3 3 J 3 4 4 4 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 4 4 4
I 3 3 3 3 3 3 3 3 3 3 3 4 4 4 3 3 3 3 3 3 3 3 3 4 4 4
3 3 3 3 3 3 3 3 3 44 3 3 3 3 2 2 2 2 2 2 2 2 3 3 3 3 4lt
3 3 3 3 222222gt22 3 2 y a 4 4 4 3333 27-1- 2222Z 3333
3333333 S2Sk4gtgtZSfgtamp2lrfS32 033
3 5 0
4 4 4
3 3 3 3 22222 2 2 2 2
2222222 + 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 U U 1 1 1 1 1 + 1 1 1 1 1 1 1 1 1
l l t t l l l 11 1 1 1 1 1 1 1 ) 1 1 1 1 1 1 1 1 1 1 1 - 1 1 1 U I U M 1 i i i m n 1111111 n u n 11111 bull i i i n 11111 m m
m m i m m
Z-2222P2 3333 414 2222222 3333
222222 33333 222222 3
pound222322 22222r
222
222222 2222J222 2222222
1 1 1 1 1 1 1 1 M M M M M M I
1 1 1 1 1 1 1 1 1 1 111111 111 111111 1 1 M l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M I M t n i 11 M l
m m m 11111 m 0 1 1 m
m m 1 1 1 1 1 1 1 1 1
u r n 11111
2 2 2 2 11111 +22222 Mill
1111M1M 1111111 1 M 1 M Mill ZZM 11111 222raquo222 11111 22222 Hill 22222
9S39399 0393339 UvV9 9S0S999 8bamp3 30S0S3999 B08CSS S99SS3S999 Oer668 9999999999999 6800836 939S3S9939 77 8SC8PC03 999999
777 08SS bull-iOPOS 7777 uoaac^osae 777777 5031^GOBpound3338
7 7 7 7 777 8S08S3 l 38J 08 7 7 7 7 7 7 7 6080668
bullrraquo 7 7 7 7 7 7 7 7 7 7 G6 7 7 7 7 7 7 7 7 7 7 t-se66 77777777777777
coorgts6eeu 7777777777
iJ amplaquo053 660CC666C666 i J5S5055oj5C55
14 5535555S^0li055555 --444444444444
4 4 4 4 4 4 4 4 4 4 ^ 4 4 4 4 4 4 4 4 4 4 r )33333339
33333333333333333332233333 33pound-3333333 gt22222222 22P22 gt2222222
222222gtpound2222222222 2P^222 igt222222222222 22-222poundgt ^22^22^2^22222 2gt=-r^^c-^i7iVgt^y2^2
2poundf 2222 pound laquo 2t222-poundT2222 222222pound^2 222222
222L22222222P2^22222 1 22222-222222 1 1 1 M 1 1 1 1 1 1 1 1 1 1 1 111 M l 11 1111111
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111 M l 111 111 111 1 I M M
22222222-gt22r222gt2222222 12222222222222
^2^22222^2222222
CONTOUR LEVELS AND SYMBOLS
SYC1 LEVEL BADGE
( 0 ) 2 1 5 6 2 E - 0 2
( 9 ) ( S I Isectlil81 ( 8 ) ( 8 ) i l^Ig| 17gt ( 7 ) SMIgI ltS) (6gt lWSUi ( 5 ) ( 5 ) iI5SIsectI ( 4 ) 1 4 ) V^f-Si ( 3 ) (3gt f^gl C2gt 12 ) JSISi ( M ( 1 ) lIii8i lt0gt 8 2 3 3 E - 0 3
E-STIMAIICI-I E R O J CUTEFUQN CONSTPAlMT =
7 i i C 3 C E - 0 2
12oOCE-013
Figure 627D Contour plot of [laquo)] for the fourth sample at tbdquo = 090
20
634 The effect of Level of Estimation Error Bound upon the Opti-niaJ_jhpoundrtoring Problem - In the examples of the previous two sections a comparison is now made of the effect of the level of the estimation error limit upon Jie outcomes of the optimal monitoring problems of design and management In both cases start with H given in (656A) or (657) In the first example in Section 632 o r 02 whereas in that of Section 633 j v 007b
In the first case o+(zjtz] is shown in Figure 620 in the secshyond in Figure 626 Notice immediately that there is a diieat effect upon the bullbullbullbull bullt- problem a lower estimation error limit leads to higher sampling frequency as would be expected
However a more interesting point comes in the effect of the value of o v upon the optincl design problem the optimal placement of moni-
tors Comparison of the contour plots of [P^(zbdquo)l for sample times 2 2
tbdquo in Figure 621 for a r 02 with those in Figure 627 for a = 0075 shows that the optimal design problems are vastly different leadshying to entirely different positions zt for the global minima in the two problems
Notice also that the shape of the contour in Figure 621 is differshyent from those in 627 the predominant difference being the cmaller height of the rise around the source location z = 03 This can be exshyplained as fallows la the case of the flrst samples far the problem with a = 0075tbdquo = 027 whereas for o = 020 tbdquo = 126 Thus
urn J K ivn K
the stochastic source has longer to act upon the system with te higher error bound The effect of this can be seen by considering ihe form of the predicted covariance matrix P^ in (624) and (628) For the asympshytotic case of infrequent sampling from Section 532
210
Pdeg Mbdquo Ktg]
(628)
o o n s~s
(Jo] + K C ^n)
L ss
(658)
Thus as K grows the first element of fdeg get larger relative to the other steady-state terms in Pdeg as seen on the right-hand side of (658) This results in different values for the inverse [ pound ( 2 K ) P S C ( J K ) T + V] in the equation for the corrected covanance matrix in (626) Thus with T = (t K + 1 - t R) = 001 oZ
tim = 02 leads to K = 126 for the probshylem in Section 633 whereas that in 632 with cr = 0075 leads to K = 27 this results in the different contours in Figures 621 and 627 Thus the optimal design of the measurement locations is seen to be a function of the level of the error bound which substantiates Conclusion IV
635 Examples of Various Levels of Bound upon Output Error -The same problem as in the last examples was solved but with a range of error bound levels as follows o ^ H 005 0075 01 0125 015 02 and 05 Resultant contours of [Ppound(Z)]bdquo at the first sample time tbdquo for each case are shown in Figure 628
As the time interval grows before a sample is made the uncertainty in the estimate of the state in the area near the source z w s 03 beshycomes large relative to that elsewhere in the medium These plots further
CONTOUR PLOT OF t P ( K K ) ( Z ( K gt ) 3 I 1 AS A FUNCTION C CZ(K )31 HORIZAND t Z ( K ) J 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT E-STlMATE WITH T l W E POSIT ION OF KAXir iUM VARIANCE APPUOACHES STfeADY-STATE VALUE FOR LARGE T I M E
CZ(K)32 05
555 555 553 555 555 S55 555
444444 444444 44444 44444 44444 4444 41444 4444 4444 4444 4444 4444 444
4444444 4444444 4444144 4444144 AAA 144 44 1-144 444144
55 G6 77 083
4444 444 444 444 444 444 44
44444 444444 44444 raquo5 et 4444 555 I 44444 55 I 4444 55 I 4444 55 lt 33 444 55 3333333 4444 55 33333333333 444 555 33333333333333 444 33J333333333333 44 33333 3333333 444
999999999 992919339 53 66 i i JBB 53993399 55 66 77 CSS 099S9S939 55 66 77 608 999329999 55 66 77 copySi 9029099993 555 66 77 CU-iS 09 Oji 309999 555 f-6 77 BCe 9S23DS99S9999 55 66 77 Or60 999990999999999 55 56 777 FEd9 99993999999999993999 535 GB 77 C0U98 9S9P9999992999993-777 8U930O 99999999999999 77 03311388 939999939 6 777 S0ii008338
6 7 7 7 7 s a o a a a a e e a s 6 7 7 7 7 Q880aBCelt23688e tiG 7 7 7 888dC0e0LC388Ca8C338888
_ 6 6 6 7 7 7 7 7 8 8 8 8 3 8 0 3 8 0 8 8 3 8 8 8 9 5 5 66S 7 7 V 7 7 7 7 7
665 777777777777 4444 3333 33333 144 555 6666 777777777777777777777777
4444 3333 3333 444 553 6C6C866 7777777 444444 333 2222 3333 44 5555 6666666565606066666
3333 222222222222 3333 444 S55t3S 566S66666 33333 22222222222222222 3333 4444 55D55555555S555555j55555555
3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2
pound 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 i m i m t m u
bull 1 1 1 1 1 1 1 I M 1 1 1 1 K 1 1 1 1 1 1 1 1 1 U 1 1 11111
i i i
n u n 1 1 U 1 1 I 1 1 1
m i l l 2 2 2 2 2 11111 2 2 2 2 2 2 2 2 1 1 1 1 1 1
2 2 2 2 2 1 1 1 1 1 1 1 1
22222222 333 4444 222222 33333 4444444444444444444444444444
22222 333353333 333333333333333333333333333333
222 333333 22222222222
2S 25 722222222222222222222 2^2222222^22222222222222222222
22222lt222222222227222222 Z22222222222232222222
22222222222222222-11 1111 111111 1111111 11111111 111111
1111111 22222222222222222222 111111 22222222222222222222222222222222222 11111 2 2 222-2 pound2 111111 22222 333333333333333333333333d 1111111 2222 33333333333333333333333333 11111111 222 3333333
1111111 2 2 2 2 2 2 u i m u n 1 2 2 2 2 2 2 1 1 1 1 U 1 1 1 1 1111 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111111 n n u m i i i 1111111111
1111111111111111 i m i n t t i i i i i i
l i m n l i m i t 11111111
SYM3 LEVEL RANGE (6) 13141E-02 ( 9 ) ( S
1 2 6 8 7 E - 0 2 1 2 2 3 4 E - Q Z
( 0 ) ( 6 )
1 1 7 6 1 E - 0 2 1 1 3 2 8 E - 0 2
(7gt (7gt
1 0 8 7 4 E - 0 2 1 0 4 2 1 E - 0 2
( 6 ) ( 6 )
9 9 3 7 0 E - 0 3 9 5 1 4 5 E - 0 3
( 5 ) ( 5 )
9 O 6 1 2 E - 0 3 8 6 0 7 9 E - 0 3
( 4 ) ( 4 )
8 1 5 4 6 E - 0 3 7 7 0 1 3 E - D 3
(33 lt3gt
7 2 4 3 0 E - 0 3 6 7 3 4 7 E - 0 3
( 2 1 ( 2 )
6 3 4 1 5 E - 0 3 5 6 0 9 2 E - 0 3
( 1 ) ( 1 1
5 4 3 4 9 E - 0 3 4 9 8 1 6 E - 0 3
(Q) 4 5 2 3 3 E - 0 3
ESTIMATION ERROR CRITERION CONSTRAINT =
5 0 0 0 0 E - 0 2
12500E-01J
Figure 628A Contour plot of B ^ ( z K ) l 1 1 at f i r s t sample tirr t K = on for o ^ = 005
CONTOUR PLO T OF [P(KKIZ(K))JM AS A FUNCTION O r Z(K)11 H3RIZ AND tZ(K)J2 VERT EXAILE TQ SIOW EVOLUTION or VARIANCE I N OUTPUT E I M A T E WITH T I M E POSITION OF MAX MUH VARIANCE APTtOACKES SrCADY-SrAE VALUE FOR LAHQE TIME
C Z lt K gt J 2
0 5
4 4 4 1 4 4 4 5 5 5 6G 4 4 4 4 1 - 1 4 gtSgt 6 6 4 4 4 4 1 4 1 SOS G5 7 7 7 4 4 I - 4 - 4 0 5 eC 7 7
4 1 4 4 4 4 5 5 GC 7 7 7 444 - = i14 5 5 5 5G 7 7 7
4 4 4 4 - 1 4 5 5 5 GS 7 7 7 4 4 4 4 4 5 5 6 S 7 7
3 3 3 3 4 3 144 5 5 5 0 5 6 77 3 3 3 3 3 3 3 4 - 1 4 4 4 5 5 6 C 7
3 3 3 2 3 3 3 3 3 3 1444 5 5 5 tgteuro6 3 3 raquo 3 3 - j ^ - 3 i 3 3 4 4 4 4 S 5 6 3 6
333333-gt gtraquo3 -gt3333 4 4 4 4 5 5 5 C S G I - ^ v 3 3 3 o 3 3 5 - j 3 3 3 3 3 3 3 3 4 4 4 5 S 5 6GGS 4 4 3 1 r--ijgt333 3 5 3 3 3 0 3 3 1 3 4 1 4 5 J 3 6 -4 4 4 3 3 3 S 3 3 3 i 3 3 r ^ 3 3 4 1 4 -i CC 4 4 4 4 33T-2 3 3 i J 3 3 454 j ^ 5 f 4 4 3 ^ 3 3 3 ^ J 3 4 4 4 5 3 5
3 3 3 3 3 3 3 4 4 4 5 5 1 5 3 3 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 555E-
3 3 3 3 2 2 2 r - i ^ 2 2 2 3 3 3 3 4 4 4 5 S 3 3 3 3 3 3 22 laquo - - yraquo jraquo2 3 3 3 4 4 4 4
_ _ - - - r ^ amp ^ 2 ^ i 2 2 3 3 1 3 4 4 4 4 2 2 2 2 2 C 2 r 3 3 3 3 4 4
2222ltgt2 3 3 3 3 3 2 2 laquo 2 2 2 2 3 3 3
P 2 2 2 gt
5555 444 5555 444 555 4-i4 5-5 444 i 55 444
44-14 4444 4444 4444 44414 44444
4444- 14 444444
33333 222222
22222 222 -2
1111111 1111111 1UI1 11
2 - 2 pound bull
11111 11111111111 11 n m i n i i i n n i n m m n i
1111 n m 111111 m m m 1111 111111m 111111
m i
1111 11111111
111111 11111 ftfraquofgti- bull
1 1 1 1 1 WWZZZ
JErJSe pound 1 9 3 9 9 9 9 0lt
S L B 3 9 9 0 i T 9 - 9 f - a 3 D O - bull s - s s
bull i 3 3 3 O 3-3999 eccose ss-v9S3999
8 t S S C 8 9 9 0 9 9 0 9 9 9 9 9 9 9 9 8tt81B8 99S999999S9
V e t J f i380 t i 9 9 9 9 9 9 9 7 7 c s s o e r G O y77 e o s u c c - i i s n o
7 7 7 7 7 fcampceooaaeoeoe 7 7 7 7 7 7 7 a p 3 3 C 8 e e e e e 3 9
7 7 7 7 7 7 7 7 o c e o B e o s i 777f77777 Jo 77771(1777 3 3 77tn7777 pound 0 6 5 3 6 7 7 7 7 7 7 7 7 7 7
iGeampampG6CgtGS6 3poundGC66SC(GpoundGQ
i 5 i 3 6 G amp a amp 6 6 G 6 6 6 6 C G 5 5 J 3 5 5 5 5 5 W S 5 5
3 5 5 5 5 5 1 J S C - 5 5 5 5 5 G 5 5 5 5 5 1 1 - 1 1 4 4 4 4 4
44444 44 44444444-T444444444 J333 4444
3-3Cn3S333J3L--J33333 3 3 3 J J 3 3 3 3 3 J 3 3 3 3 0 3
pound 2 - 2 2 2 r i - 1 H i i 2 2 2 2
2-raquo i- raquogtr---2igt2 j j - r gt V ^ - l 2322
222 - bullgtbullbull2 raquo2222222raquoa 2 2 gt V 2 ^ gt i gt - S P 2 2 2 2 2 2 2 2
- 2 r ^ - gt 2 K 2 2 2 2 2 2 2 2 2 2 ^ 2 - ^ - - V 2 ^ 2 2 2 2 2 2
2fc i 2^22^ -2lt i 222
m m bull m m 1111 m 1 1 m 11111 i i i i m - i i
222222222 bull bull bull 2 i r - ^ 2 2 r ^ 2 R 2 2 2 2 2 2 2 2 2 2 2 - 2 ^ r ^ - ^ ^ 2 2 2 2 2 2 2
SYM3
( 0 ) mm ( 9 ) ( 9 - iJiiI8i ( 8 ) ( 6 ) Wiiiii lt 7 t ( 7 ) J5JiSi ( C ) ( 6 ) I8Sf8 ( 5 ) ( 5 ) 3i5i|g| ( 4 ) 4 gt lHIgI ( 3 ) ( 3 ) lfJ|8i ( 2 ) ( 2 ) HSSiSi ( I ) ( 1 gt I2iJIsect lt0gt 7 0 S W E - O J
ESTt l - ATITN tlrila C1C TCR10N C C K - r r A f T =
7 S 0 J C E - 0 S
SOURCE 1VPUT CUVAFUANCE [ W l raquo t 1 2 6 0 0 E - 0 1 J
Figure 628B Contour plot of fell at f i r s t sample time t 027 for o- i 0075
CONTOUR PLOT OF [ P f K K X Z C K ) ) 111 AS A FUNCTION CF I Z O O J 1 H 0 R I 2 AND t Z ( K 1 1 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT ESTIMATE WITH T I M E P O S I T I O N OF MAKIKUH VARIANCE APPROACHES S T E A D Y - E T A T E VALUE FOR LARGE T I M E
t Z lt K gt 3 2
0 0
44 444 AAA
4114 44444 A 4 4 L I 41 44 44 4 4 4 4 444
33333333333 33333333333 35 S 3 3 3 3 laquo33
SiJ^JyS gtlt33 32 i i - - 3^ - gt33 33-gt3- bull -
05 66 77
33 444 444
444 333 313 r i 33 laquo - i n333 3 2 ^ 3 3 J i3i bullbullraquo33333
3 3 3 3 3 3 3 3 gt t j r 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 444 3 3 3 3 3 3 3 3 3 3 3 3 3 3 AAA
3 3 3 3 3 3 3 3 3 3 3 3 Ad 3 1 3 3 3 3 3 3 4 4 4
3 3 3 3 2 2 2 2 2 3 3 3 3 44 2 V 2 2 P 2 3 3 3 3
63 66
i 66G
8000 0 3336
7 60G0B 7 Q6CS0 77 seaoe 777 flSJSi
777
9999939 9999999
9 9 9 9 9 3 9 93939999
S9Q09999 osaa 9993099999S9 80COB0B 999999999ltgt
533 3333
3i33 333333 33333 ZtZZ 333 gtZZ
2^22 222P2222 22222 1
1 M I 11111 11111
111111111111 11H1M 111
111 - 1111
11111 111111
ifpFte gt222 -gt22222 a 2pound-2P2
22222222 2S2ii2^
2 22
7777 O00C36 66 7777 0050008
gt 06 77777 6G03Ceea iS 656 777777 0030830888088-i5 6tGti 7777777 060088308 53 CM a 77777777 BB 555 6006 777777777
^1 533 ( J6GS6 77777777777 144 Su fJ3 60695096 777777777777
44 5355 6660CC-66S6 777777 3 44 G3C3 6CS56GG=S06 3 ^ 4 4 4 ^ s s - s s e e i i c c s e e s G c s 3laquo3 444 o35S355SS 66Gpounde66666
333 441 Sb5335rgtS55j5 333 444441 igtS5Sgt5SS55S55535
1 1
2 2 2 2 3 3 3 lti 1 4 4 4 4 4 4 4 4 4 4 4 4 4 111 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 111111 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 M 1 I 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 2 2 2 2 2 - 2 2 2 2 2 2 2 11 2 pound 2 2 2 2 2 2 2 2 2 2 2 gt2222
2222r-V j2222222222
22222222222 222i-rt 2 222r-222222 2222222222222222 2222--i2 22poundPamp22
1 1 111111
111 euro 3333
222222 222222 22222 22222 22222
22222 222222
1 1 1 1 1 1 1 1 11111
1111 2 2 2 2 2 bull 2 2 2 2 2 1111
222222- V222JV222J-P22222 22^22 -- ^^22222laquo22
22--V-J W J2gt2gtJ 22
222f Pr - gt 225r^laquo2J 2222 2222raquo fi 2r-2^igt22222
11111111111 22222 1111 222222222222 11111111111111111111 Kill 11 II 11111 111 11111 1 i 111111111 111
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 m i l 111 m i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
g) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111 2 2 2 2 2 2 2
11111 2Pgt 2222 2222 =V 22222222222 222 11111 22l- bull 22Vv22222
11111 222222 11111 222222 333
SYtu
( 0 )
LEVEL RANGE
2 4 0 S J E - b 2
9 ) 9 )
2 2
33Z 1E-02 J 6 2 C E - 0 2
( 6 ) ( 3 )
2 1 9 2 7 E - 0 2 1 2 2 r i E - 0 2
( 7 ) ( 7 1
2 05271E-02 0 C 2 3 pound - r ) 2
t o ( 6 i
1 1
9 i r 2 r - o e - S ^ l E - 0 2
( 5 ) ( 5 )
1 1
7 7 2 G E - 0 2 7 0 1 3 E - 0 2
( 4 ) ( 4 ) 6 3 1 7 E - 0 2
S61 (3pound -02
f 3 ) ( 3 ) 1
4 9 1 5 E - 0 2 4 2 1 4 E - 0 2
(2gt lt2J 1 331 3- -02
2 8 1 I E - 0 2
( 1 ) ( 1 )
1 1
21 I O E - 0 2 1 4 0 9 E - 0 2
(0) 1 0 7 0 8 E - 0 2
s^fc 1 2 Q 0 0 E - 0 1 ]
Figure 628C Contour plot of fe)]n at f i r s t sample time t bdquo = 046 for a = 010
CONTOUR PLOT OF tPltXK)(Z(K))J1 1 AS A FUNCTION 01 tZ(K)I HORIZ AND [Z(K)J2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTPUT KSUMATE WITH TIME POSITION OF MAXIMUH VARIANCE APPROACHES S1EA0Y-SAVE VALUE FOR LARGE TIME
333 444 4444 4444 333 44444 333 44444 333 +4444 333 444 333 333 222 333 2222222 3333 222222222 3333
33C3
CZ(K)]2 OS
333TJj3 333333 33333 33333 33333 3333 3333 333
33333333 44 6 68 77 33333333 44 S3 66 77 3333333 44 55 65 7 3333333 44 55 66 7 33333333 444 S ^6 3333333 44 55 J6 3333333 44 55 666 33333 44 55 666 33333 444 55 G6i 3333 4-1 55 6-222K2222222 333 44 55 i 222222222222222 333 44 2^2222222222222222222 22222 2222222222222 333 44 222 22222222222 2222 222 222 111 222 111 1111111
222 m n i i i i i i 222 1 HI 11 11 111 1 22 11ll 1111ll 111
33 33 333
44 444 55
11111 1111
11
2222222 2222 2222 33 444 222 333 4444 222 333 444 222 33C 4 222 333 2222 3333 2222
mil limiii ii i i 1111 2222222 1111 22222 22222 Mill 222 3 2222 22
22
222222 2 1111 11111111 11111111 11111111 11111111 11111111 1111111
68BG8 999999 eSCfiS 093999 86838 999999 bull7 8SC83 9399999 77 eoooee 99999999 777 7777 77777 i 77777 S 777777 S58ECSBQBC30 bull SM5 7777777 60830860+ 6ilaquoC6 7777777 66666 77777777 66666GE 77 77777777777 i 6SG6C666 777777777 iSf 6pound 6666566 7-i5amp05 666666666 50555595 6666666666666
555Q5555C35 6666666 I 5 5 U 5 ^ 5 5 5 5 5 14lt144 5555553555555-
444444444444444 13 4laquoi444444444444 333333333333333333 3333333333333 22222222222222222
22222222222222222 1111 11111111111 11 imiimt
222 33333333333 222 333 333 2222 iiii 33 4 333 2222 333 44444 333 222 33 4444 333 2222 333 3333 222 11111 J 33333 333333 222 11111111 222 3333 2222 1111111111
+11111 1111m mi 22222 111 222222 111 2222 111
11111
copy
22222 222222 1111 m m m m m i m
urn
m m m i 2222 m i 222222 1111 2222222
I 2222222222222222222 222222222222222222222 222222222222222222222 22222222222222222222 II 1 11111111 111111111111111111111111111)1 1 m u m m i n i m u m
i i n m m m i m m m m m i l i m u m i m m m 2222222
2i22222222222222222222222222222 22222222222222222e22
22222222222222
TIME laquo 66D00E-O1 FIRST MEASUREMENT
CONTOUR LEVELS AND SYMBOLS
SYM3 LEVEL RANGE (01 2 4793E 02 (9gt 2 pound9gt 2 4158E 3523E 02 02 (0gt 2 (8) 2 2363E 2252E 02 02 (7) 2 (7) 2 1617E 0982E 02 02 (6) 2 (6) 1 0347E 9712E 02 02 (51 1 (5) 1 9077E 9441E 02 02 (4) (4) 1 7806E 7171E 02 02 (3) 1 (3) I
6536E 5901E 02 02 (2) 1 (2) 1 52S5E 463DE 02 02 (1) 1 (1) 1 39S5E 3350E -02 02 (0) 1 2725E 02
ESTIMATION EPROR CRITERION CONSTRAINT = 1-2500E-01 SOURCE COVARi INPUT AHCE Wl-
12500E-01J
Figure 628D Contour plot of feMi at f i r s t sample time t K = 066 for o ^_ = 0125 2
CONTOUR PLOT OF E P ( K K H Z lt K gt ) 3 1 1 AS A FUNCTION 0 Z ( K ) 1 1 KORIZ AND C Z ( K ) 3 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT i S T I M A T E WITH T I M E P O S I T I O N OF MAXIMUM VARIANCE APPROACHES STEADY-S A f t VALUE FOR LARGE T I M E
bull 4 4 4 4 4 3 3 3 2 2 2 2 2 2 2 2 46640 3 3 3 2 2 2 2 2 2 2 44444 3 3 3 V2Z2Z9ZZ 4444 33 22222c J 22 4 4 4 3 3 3 2 2 2 2 2 2 2 2 bullA 3 3 2 gt 2 2 2 V 2 2 2
3 3 3 2 2 2 2 ^ ^ 2 2 2 2 3 3 3 2 2 ^ ^ f - 2 2 2
3 3 3 22^V22^ 2 2 2 2 2 3 3 3 3 ^ 3 2 2 2 2 2 i 2 ^ f r 2 2 2 2 2 2 2
[ 2 1 K gt 3 2
05
3333 4 3333 4 3333 4
333 3333
333 3333
3^3
11 65308
2 2 2 33333 2222i 33333 3353 3333 333 333 E33 33 222
222
3 3 3 3 3 3 3 3
44
2 2 2
2 2 I t 2 2 2 111
2 2 2 2 1 11 2 2 2 2 2 1111
1111 l i n t
2 2 2 2 2 2 2 2 2 2 2 2 2 pound 2 2 2 2 2 2 2 2 2 2 2 2 2 3 a
2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 3
2222 I 11111 2 2 2 2 11111111111 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2
1 1 1 1 1 1 1 1 1 1 1 2 2 11111111
3 9 9 9 9 9 9 3 9 9 9
9 S 9 9 9 9 9 9 9 9 9 9
t i s s u e 9 9 9 9 9 9 9 9 7 7 7 7 09888 993S99399-
6 7777 esesao 999999 5 77777 8300886
flfl -Jigt 66 77777 88030688 44 5 5 5 ltSlt~C 777777 8008808888
44 SS5 liCSS 777777 86665+ 44 S55 66S66 7777777 44 5SE 6G66G6 77777777
44 556 666S666 7777777777 13 444 5t 5raquo 66666666 77777 3 44 SJ55 66666666 33 444 pound5555555 6666066666 333 444 55505S5555 666666666
33 444fl 53555555555 33 44-AV 555555S555
333 4144444444444 5555335
1111111 m i l l m i 111 n I mi mm in
It T1111 222 3333 44444444444 111111 2222 3(333333333333 4444444
111111 2222 333333333333333 111111 222J 2222222222222
1111111 2222222222222222222-111111111111 1111111111-11111
1111111111111111111 11111i111111111111111111 1111 111111 1111111111111111111111111111
1111 2222222222222 111111 111111111111111111111111111 11111 222 33333 222 11111111 1T11111111111111111111111111111
11111 222 333 333 222 11111111111111111111 222 33 22 33 bull 222 3 44 22 39 44 22 33 44
33 44444444 333
444 444 ~ S555 44
553SS3 444 555055 444
444 11 22
33 4444 33
222 4444 333
333 2 3333333333 222
222 u m m uui 222 11111 222 222 222 222 1111
33 2222222222 2222222Zamp22amp222222222 -2222222222222222222222 222222222222222222Z22
222 11111111111111 111111
1111 2222 2222 11111 11111 22222 11111 copy
11111111 1111111 11111 11111111111 11H11111 niituut nnniniv mu mmiimi i m mimiim urn m 222222 11111 111111 222
2222 1111 11ll 1 2222222222222222222222222222222222222 222 1111 11111 222222 222222222222222222
33 222 IHtl 11111 22222 2222222222222
TIME 6 6 0 0 O E - O f 1RST MEASUREMENT
CONTOU LEVELS AND SYMBOLS
SYHB LEVEL RANGE
lt0) 2 5 1 6 G E 0 2
( 9 ) ( 9 1
2 4 5 6 5 E 2 3 9 6 4 E
0 2 0 2
( 8 ) ( 6 )
2 3 3 6 2 E 2 2 7 6 1 E
0 2 0 2
( 7 1 lt71
2 2 1 6 0 E 2 1 5 5 S E
0 2 0 2
( 6 ) (6gt
2 0 0 5 7 E 2 0 3 5 6 E
0 2 0 2
lt5) ( 5 )
I 9 7 5 5 E 1 9 I 5 4 E
0 2 0 2
( 4 ) 14 )
1 0 5 5 3 E 1 7 9 5 1 E
0 2 0 2
( 3 ) ( 3 )
1 7 3 5 0 E 1 6 7 4 9 E
0 2 0 2
12) ( 2 )
t 6 1 4 G E 1 5 0 4 7 E
0 2 0 2
1 ) ( 1 )
1 4 9 4 5 E 1 4 3 4 4 E
0 2 0 2
l O ) 1 3 7 4 3 E - 0 2 ESTIMATION ERROR CRITERION CONSTRAINT =
1 5 0 D 0 E - 0 1
SUUSCE INPUT COVTMANCe pound 1 2300E
MEASUREMENT ERR03 COVAR
I 0 5 0 I - 0
W]=
on tv)laquo - 0 1 D233
Figure 628E Contour plot of [ P ^ K J I a t f i r s t Spoundp1e time t K = 086 for a l i m = 015
CONTOUR PLOT OF I P ( K K ) ( Z ( K ) ) 1 1 1 AS A FUNCTION ( F t Z I K I I I HCRIZ AND t Z ( K ) 1 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT ESTIMATE WITH T I M E P O S I T I O N OF MAXIMUM VARIANCE APPROACHES STEADY- ITAVE VALUE FOR LARGE T I M E
CZltK)J2 0 3
^IPllI 33 44 55 6G 7 J
3330 2222 3333 222 33353 222 3333 2222 333 pound22 222 pound22 pound22 333
pound2 22 22 pound2
22-gt222 2222Z_222Z 22222 T-K222 2222- 0272ZZ 33
2d2i7gt2922 33 22lt2gt-222 3 22222 1 2222 1111 222 11111111111 222 111111111111111 222 1111111111111111 22 1111111111 22 111111
333 44 5 lt 333 4 55 I 33 44 55 333 44 55
0CCSO 0S8GO 83808
333 44 55
7 i 777 bull 777 til bulllt 7777 pound C 77777 t Gi 77777 rgt66 777777
999999 03099 9D399 999399 99939999 99999999 686830 9999 8608069 8088366368
111 222 222 222 II 2222 111 22222 111 1111 11111 1111111 11111
111 111111111111111 11111 11 11111 222222222222 1111111 222 33333333 222 11111111 22 33 444 33 1111 222 33 44 444 3 222 33 44 555 555 4 2222 3 4 5 66666C66
6665 777777 44 55 66G66 7777777 3 44 55 GSG666 77777777 3 444 5-5 66GCCCC 77777777777 33 -14 5555 6605666 777777 33 44 gt5535 666G66G 33 444 555555 606660666 33 AAe 5tgt5lgt5555 666G666G66 z 33 44I4 5553355S55 6GG66 22 333 4-144 555555555
bull 33 506 55 4 33 222 777 66 55 A- 33 22222 777 6 55 4 33 2222 i 66 665 55 44 3 222 55 6666G6G 55 44 33 222 2222 33 44 555 555 44 33 22 1111 222 33 444 44 33 22 1111111 222 333 333 222 11111 11111 222 333333 222 1111
11111 222 333 -1-544444444 55555555555 Ill 222 3333 4444444444444 1111 222 33C3323 444444444444 1111 222 33333333333333333 11111 2222 pound2 3333333333 11111 222222222pound22222222222 11U11IMI 2222222 1111-111111111111111111111 11 1111111 111 1111 111111 11111 11111
111111 111 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 2 2 2 1 1 1 1 1 1 1 1 3 2 2 2
111 H I 1 111 1111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1111
33 44 53 66 33 44 55 66 33 4 ~
223J222222222 22222 ^2poundf22^2 2222222
22XgtM2V-gtpoundlt2V2Z_WW2PZZZ 22222 e222222gt22222
22222L-2222222222
1 1 1 1 1 1 11111
2 2 2 2 11 2 2 2 pound
333 2 2 2 2 3 3 3 3 222
3 3 3 2 2 2 3 3 3 2 2 2
22H22222222222 11 1 uiiinninniniii mini iniii iiiinmniiinimi 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
SYK3 LEVEL RANGE
oi 2 5 5 4 1 E 0 2
sect I 4 3 7 2 E 4402E
0 2 0 2
sect I 383 3S 32C5F
0 2 0 2
n I 2 0 9 E 21 EZ
0 2 0 2
ni I 1554E 0ampC6E
0 2 0 2
fl 0 7 1 5 E 9S45E
0 2 0 2
n 927SE 67D7E
0 2 - 0 2
sect 8 I 3 7 E 7560E
0 2 - 0 2
I 6 9 M E 6 4 2 J E
- 0 2 - 0 2
i 55C5E 5a5C
- 0 2 - 0 2
a -L 4 7 2 0 E 0 2
ESTIMATION ERHCrt CR f E i d O N CONSTRAINT =
2 C 0 0 0 E - 0 1
1 2500E-01]
Figure 628F Contour plot of P ^ i O m a t f i r S t s a m p l e t i m e tK = 1 2 G f o r deglw = deg 2 0
217
C O i O O bull O O i O O ss OO i
i mdash tfgt i W mdash 1 mm gt turn CUM I bull n n 55 flH
^ w J I
H U J U O
Si mdashbull- ltgtjltvwlaquotvw
O l o r -
E D gt o o O C O O f -
KM (-^-gt -gt - 3 V J mdash w n n laquo j - mdash mdash o o bull O D H W o o o n W - - o bull Z 10 - ltl O O O O WftJ wv 3 K - - lti o o ft l L - ^ 0 - W O laquo ^ 1 1 laquo W M fu
HI - W gt T 1 gt O N bull t U T n -v i i i o bull=bull w w
o o - w T I m i l i i c raquo ltgt l i v - w n igt t i v W C J bullVft -lt lt o - o v i n I O O O O ifgt n w i i y bull
laquo mdash W m t o I D T O laquo w - e n mdash W O f ( N - M v i 3 laquo J t ^ - laquo o - w n v m o huraquo n laquo ^ (
bull-gtlt - N 0 ( 0 0 (OTTO ft-lt bull - laquo (0 h - U J i f l W gt
w _ O O N raquo t u r n o r n ftikM w bull o o ftlt - 2 laquo o ^ E a N lt 0 sect W lt n sect rt N T ^ lt WCgtVtfgt 0) O N V O - o - ftt-gtv P - M i laquo i i laquo r ^ mdash o N laquo I O O O ^ V L I T C K I gt I - ( w o v O X - N O ^ V c
o gt p P - n ogt O N I - gt T c x -i
- - - - - - O R - n v o laquo o o r - T o n
- D E - - - - - - O w f t v a o s o c o t a T I laquo - D E - - - - - - O laquo W O N ) lt O O O - laquo r o N O i 1 o
o U 1 X
- laquo r o N O i 1 o
o OO
l u - w B i o N N ifgt o o o o -- - W O ^ r i O o m i T O
O u W O 10 Q U O igt T O O J O O [ j bdquo _ _ mdash _ _ _ _ _ - - M V i f t O 3 ( i o o D-t- - w w w w w _ _ _ _ _ _ bdquo _ - - - W 0 gt T - W u l l O L I T O
z ( C W O n i z ( C W O +_laquolaquoOKV f t JgJlaquo l ~ _ W w o Slt n T5 SS lt- n i 3 _ 1 ~ ftftjftjlt) _ ft O - 3 1 T V [J laquo 0 C H mdash _ j o W T S J - C o o o 1 laquoSp ^ojci^S^^Jv^^^NN^ bullbull w ^ v i - j ^ 2 5 ^ laquo laquo - gt laquo laquo W W ft I j - W N W l ^ C f l J W O T o o o o L1U1 bull o x o 0 - ~ 0 W M M ( laquo gt N A i M mdash - M W O O O O O t O f i -O a J t t laquo f ^ O U N T W W W - - - w w o o o o o in1) bull
0 0 ( 0 W W W W W bdquo _ _ (u Pgt n n o n laquo laquo raquo bull
218
substantiate the existence of a functional relationship between the optishymal measurements zt and the level of the output error bound o
636 The Effect of Time-Varying Error Bound upon the Optimal Meashysurement Design - Consider here an example where the output estimation error limit cC is allowed to vary in time For this problem let
lim 01 (659)
at the first sample time and then
Aim - degL + deg- 0 2 5 (660) for each sample thereafter
The resultant plot of o^ + N(jtz) over time for the interval 0 lt t S 2 is shown in Figure 629 where the initial covariance P^ E M n is as before in (657)
Notice how the curve asymptotically approaches the slope [Q]- =
00025 just before each sample in accordance with the infrequent samshypling approximations
v
At each samplecontour plots of lEDU^)] a r e 9 e n e r iraquoted and preshysented in Figure 630 for sample ti mes t| - 046 104 180 As can be seen from these plots the contours change with the error level as shown in the previous sections in fact they directly compare with those of the previous section Thus the converse of Conclusion VI may be stated as
Conclusion VIB The optimal measurements found at one measurement time may not in general be optimal for other measurement times if the bound on estimation error varies with time (CVIB)
Further verifications of the effects of the a priori statistics and level of estimation error bound upon the optimal design problem can be
1 2 0 0 0 E - 0 1
6 0 0 Q O E - 0 2
1 X
X
x x X
XX x
X X X
X X
x x X
XX x
X X
X X
X X
X
x X
X X X
X X
X X
X
x X
X
X
XX 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
c
X X X
Figure 629 Time response of ^+n(K z) f o r t lt n e v a r y i n S estimation error l imit o z ^( t) = 010 0125 and 0150 at sample times t K = 046 104 and 180 respectively
CONTOUR PLOT OF t F ( K K ) ( 2 ( K gt ) i 11 AS A FUNCTION = I Z I K H I HOfIZ AND I Z i K ) ] 2 VERT EXAMPLE TO S1ICW EVOLUTION OF VARIANCE I N OUTPUT r l 11 MATE WITH T IME POSITION CF MAXIMUM VARIANCE APPROACHES S I EADY- -T TE VALUE FOR LAKOE T I M E
C6
tZltKgt12
444 444 4444 444
44 33333333 444 444 3333333lt33 444 3333333J333 444 33C-^rS3J3333 444 33333S3333333 4444 3333333i333333 444444 3333333333333333 444444 333333333333333333 44444 33333333333333333333 4444 33333 3333333333333 444 33333 333333333333 3333 3333333333
55 6G 77 bulljV 66 77 eoaee 9900J 0 3
93 li9
3333 3333 3333 333333 33333 333 2222 2222 22222222 22222 1 1111 1111111111 m m i m i l 1U1111 m
i n m i
i n n m m
111111 m m 111111
i i i m i n n
i n n i m
i n
2222 333 lt 4444444444-1414
33333333 i 33333 I 22222 3333 2^^-^^2-222 3333 2222222222222222 333 2222J2222222222222 333 22222222 2222222222222 333 2222 2222
22222222222 22222222^22222222 2222 222222 222 222322 222 33 22222 222 3333 22222 2222 22222 2222 22222 222222 222222 1111111111 222222222222 111111111111111111
^222iV-2v_iV bullbull VJlaquo
222 2 L 22 2 2 r-^ gt L2 22I-22 22222
11111 11111111111U11111111
1111111111111
11111 11111111
u r n 22 11 11 22222 1111 22222 1111
11111111111 111111111111111
11111
n u n i m i n i i i i i i i i i i n m i i n bull m 11111 n i i i m i m i - i i i i i n m i i i m 1111151111111111111 ill 1 1 2222222 222222222222222222222222222 22222J-=2 2222222222222222r 222222 222222 333
t o t z i-o-
( 9 ) ( 9 )
2 KiSi ( 0 )
2 bulltJi-ll ( 7 1 (7)
2 1 degri-pound
ltegt 1 -vmii lt5gt ( 5 )
1 1 STSIgl
( 4 gt ( 4 )
1 -mii-n ( 3 ) ( 3 ) bullm-E ( 2 ) ( 2 )
i i if8f
C 1 ) ( 1 )
i i bullVW-ll
) O70 pound e ii ON
- 0
lwAa v i i E U T [W] =
C 5 C 0 C 3 E bull 3 1 1
ESamp sr EV3-
I -5g =pound
Figure 630A Contour plot of Figure 628C [4i a t f i r s t s a m p l e t i m e t bull deg 4 6 f 0 r deglin 0 1 0 compare w i t h
CONTOUR PLOT OF I P ( K K M Z ( K ) ) ] U AS A FUKCUOH poundlPLE TO SHDW evOLUT ON OF VUiJAhCE IN OHIrJ COS TIOM OF MAXIMUM VARIANCE APliCACULi STL-HY
pound2(KH2 03
d4At 33 4444 333 44444 333
444 44 333 44-1dfl 333 4J44 333 3^3
3333lt33 4 3333333 4-0333333 4 3333333 J 3333333 333333 333333 333J3
3333
bull ^ 3 9lti9nlaquo
33333 33333 33333 33333 3333 3333
32 2p||p-gtill p 044 55
2222 222 222
2222222222 333 222222 33 444 22222 333 44 33 444 1 J-2 333 44 2^2 333 laquo 222 333 2232 333 2222
11111 222 222 111111111111 22 33 22 1111111111111111 2 222 1111111 111 11 1 1 ] 1 111 222 i n u n u u u i u n n
222222 111 11 111111111 222 11111 1111111 111111 1 1 1111 11111111 1 I U U 1 U 1111 111111111111 11111111111111111111111111111111 inn i m n 11 n
1111 2222222 111 111 Tll 22222 22222 1 1 1 1111 222 3 1 2222 111 11 222 333333333333 222 111111 22 333 333 2J-22 m m m u 22 33 44pound 333 2222 bull11111111 22 333 444-144 333 222 11111111 22 333 44444 333 2EKpound 1 lllllll 222 333 333 222 lilt 1 1111 22 33333 33333 222 UUUi 1111 222 33333 222 111111111 111 22222 222222 11111 1111 22 11111 111111111111111111 11111111 1 1 1
bull4444444I4444444 C _ r 4^44444444444
m 1 r i i m 111 m
illllll
111 111 22222 111
I 1 M 111 ill 11 1 1 1 111 111 1111 1111 1 111 111111111111 m m m
2222222 bullit bull-222222^SfTl - 2222222222222 bullZ 222222222222 2 ^222 22222222222222
Figure 630B Contour plot of [ l $ (z K ) with Figure 628D
at second sample time t ^ = 104 for ^lln
CONTOUR PLOT OF tP(KK)(Z(Kgt)311 AS A FUNCTION V IZ(K)JI IflRIZ AND tZltKgt12 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTPUT EI-M HATE WITH TIHE POSITION OF MAXIMUM VARIANCE APPROACHES STEAD-li TATE VALUE FOR LARGE TIHE
i 444d4 333 22222222 44444 333 22222222 44444 333 22222222 4444 33 22222222 444 333 2222222222 I a 33 22222222222 333 222222222222 333 22222222222222 333 2222222222222222 333333 222222222222222222222 bull33333 22222 33333 2222 3333 2222 3333 222 333 222 bull333 222 333 22 333 222 1 222 1
39399 999939 999939 999399
CZ(K))2 06
3333 44 5 66 77T 6BI 3333 44 0 66 777 861 3333 44 55 66 777 81 333 4 55 66 777 I 3333 44 9 66 7 77 333 44 55 66 7777 3333 44 5 60 7777 333 44 55 665 77777 333 44 53 CiSe 77777 33 44 35^ St 66 777777 333 44 555 6666 777777 2222222222222 33 44 555 66666 7777777 22222222222 33 44 555 666666 777777777 222222 333 44 535- 6666666 7777777777 2222 33 444 55S-5 66666666 77777 111111 2222 33 44 515555 66666666 111111111111 222 33 444 5555555 111111111111111 222 333 444 5555555555 1111111111111111 222 33 4444 555555555SS 1 11111111111 22 33 444lt44 5555555555 11111111 22 333 444444444444 5555555 1111111 222 3333 44444444444 11111 2222 33G33333333333 4444444 111111 2222 333333333333333 11111 22221222222222222 1 11111 2222222222222222222+ 111111111111 1111111111111111 1111111111111111111 111111111111111111111111 1111 bdquobdquobdquobdquobdquo A 111111 1111111111111111111111111111 1111 2222222222222 111111 111111111111111111111111111 11111 222 33333 222 11111111 111111111111111111111111111111+ 11111 222 333 333 222 11111111111111111111 1111 222 33 44444444 333 222 1111111111111 111 22 33 444 444 33 222 11111 2222222222 1 222 3 44 5555 44 33 222 222322222222222222222 22 33 44 55555555 444 33 222 2222222222222222222222+ 22 33 44 055555 444 33 222 222222222222222222222 222 33 44 444 33 222 11111 222
M 33 4444 4444 333 222 U l l l l i m u U 33 44 333 222 1111111111111111111111111111111111111111 222 3333333333 222 11111 111111111111111111111 2222 2222 1111
222 111 2222 111 22222 1111 1111 11111 +111111
111111 11111111 2 bull 111111 1111 11111 22222 11111 1111111 1111111 11111 11111111111 +111111111 1111111111 11111111111+ 11111 111111111111111111111111111111111111 11111 111111 222 2 2 1 1 1 1 11111 2222222 2222222222222222r 222222222222 n n n 1111 11111 222222 222222222222222222
CONTOUR LEVELS AND SYMBOLS SYMBLEVEL RANGE (0) 25168E-02 (9) (9) 24567E-02 239G6E-02 (6) (6) 23365E-02 22764E-02 17) (7) 22164E-02 21563E-02 (6) (6) 20962E-02 20361E-02 (5) (5)
19760E-02 19159E-02 C4gt (4) 18558E-02 1795SE-02 (3) (3) 17357E-02 16756E-02 (2) (2) 16155E-02 15554E-02 (1) 14953E-02 14353E-02 (reg) 1375EE-02
ESTIMATION ERROR CRITERION CONSTRAINT =
15000E-01
5Q000E-0J1
^ 2 2 11111 111111 22222 2222222222222
Figure 630C Contour plot of [ p ^ z ^ at third sample time t K - 180 for o ^ = 0150 compare with Figure 628E