An Analysis of Model Parameter Uncertainty on Online Model ...

An Analysis of Model Parameter Uncertainty on Online

Model-based Applications

by

Yingying Chen, M.S.

A Dissertation

In

Chemical Engineering

Submitted to the Graduate Facultyof Texas Tech University in

in Partial Fulfillment ofthe Requirements for

the Degree of

Doctor of Philosophy

Approved

Karlene A. Hoo

Uzi Mann

Mark W. Vaughn

Xiaochang Wang

Zdravko Stefanov

Peggy Gordon MillerDean of the Graduate School

May, 2012

Texas Tech University, Y. Chen, May 2012

Acknowledgements

I am greatly indebted to my research advisor, Dr. Karlene A. Hoo, for her sup-

port, advice and guidance during my doctoral study at Texas Tech University. I

appreciate Dr. Hoo for introducing me to the areas of multivariate statistical anal-

ysis, process control and optimization, process modeling, and the study of model

parameter uncertainty. I am very grateful to Dr. Hoo for the time she spent on re-

viewing and proofreading my various manuscripts and reports. It has truly been a

privilege for me to be her student.

I would like to thank Dr. Uzi Mann, Department of Chemical Engineering

at TTU, for his kindness in guiding my early study of reaction systems; and Dr.

Zdravko Stefanov for his assistance during my internship period. I also would like

to thank Dr. Mark Vaughn and Dr. Xiaochang Wang for agreeing to serve on my

doctoral committee. A special thank you goes to Dr. Shameem Siddiqui (Petroleum

Engineering) who gave me guidance on reservoir engineering. I also want to recog-

nize my fellow graduate students in the chemical engineering department for their

enthusiastic discussions and friendship during my studies at TTU.

Financial support for my graduate studies have been provided by TTU Process

Control & Optimization Consortium, National Science Foundation (#0927796) and

the Petroleum Research Fund (#49545-ND9).

ii


Finally, this research could never have been completed without the continuous

support, love and encouragement of my parents, Xinmin Chen and Qin Zhang. This

dissertation is entirely dedicated to them. My very special thanks to my husband,

Le Gao, for his love, patience and support in the past four years.

iii


Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . x

I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 13

II Preliminaries on Uncertainty Propagation . . . . . . . . . . . . . . 16

2.1 Sampling Techniques . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Monte Carlo Sampling . . . . . . . . . . . . . . . . . . 17

2.1.2 Latin Hypercube Sampling . . . . . . . . . . . . . . . . 18

2.1.3 Latin Hypercube Hammersley Sampling . . . . . . . . . 22

2.1.3.1 Hammersley sequence points . . . . . . . . . . . 22

2.1.3.2 Combination of Latin hypercube sampling and

Hammersley sequencing . . . . . . . . . . . . . . 24

2.2 Example: HDA Process . . . . . . . . . . . . . . . . . . . . 28

iv


2.2.1 HDA Process . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.2 Propagation with Different Sampling Methods . . . . . 30

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

III Real-time State Prediction . . . . . . . . . . . . . . . . . . . . . . 41

3.1 PLS Regression . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 KL Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 State Prediction with ROM . . . . . . . . . . . . . . . . . . 46

3.3.1 State Prediction of HDA Process . . . . . . . . . . . . . 48

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

IV Preliminaries on Uncertain Parameter Updating . . . . . . . . . . 55

4.1 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . 56

4.1.1 Adaptive Metropolis Algorithm . . . . . . . . . . . . . 57

4.2 Ensemble Kalman Filter . . . . . . . . . . . . . . . . . . . 60

4.2.1 Forward Step . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.2 Assimilation Step . . . . . . . . . . . . . . . . . . . . . 62

4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

V Real-time Model-based Optimization with Parameter Updating . . 68

5.1 Uncertain Parameter Updating in a Model-Based Optimiza-

tion Framework . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Updating in a Reservoir Management Framework . . . . . . 68

v


5.2.1 Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.2 Basic Optimization Problem . . . . . . . . . . . . . . . 74

5.2.3 Markov Chain Monte Carlo Updating . . . . . . . . . . 77

5.2.4 Ensemble Kalman Filter Updating . . . . . . . . . . . . 80

5.2.5 Optimal Oil Production Results . . . . . . . . . . . . . 82

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

VI Robust Estimates of the Uncertain Parameters . . . . . . . . . . . 102

6.1 Robust Statistics Estimates . . . . . . . . . . . . . . . . . . 103

6.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3 Mathematical Foundation . . . . . . . . . . . . . . . . . . . 112

6.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

VII State Estimation & Model Predictive Control with a Maximum

Likelihood Model . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.1 Tubular Reactor . . . . . . . . . . . . . . . . . . . . . . . . 121

7.2 Determination of Robust Estimation of Uncertain Parame-

ters for State Estimation . . . . . . . . . . . . . . . . . . . . 123

7.2.1 Maximum Likelihood Model for Model Predictive Con-

trol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

vi


VIII Summary, Contribution and Future Work . . . . . . . . . . . . . . 142

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 145

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

I Preliminaries on Model-based Control . . . . . . . . . . . . . . . 160

II Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . 165

B.1 Hammersley Points Generation . . . . . . . . . . . . . . . . 165

B.2 Karhunen-Loeve (KL) Expansion . . . . . . . . . . . . . . . 167

B.3 Partial Least Squares Regression [1] . . . . . . . . . . . . . 167

B.4 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . 169

B.5 Generation of Monte Carlo Samples . . . . . . . . . . . . . 170

B.6 Ensemble Kalman Filter . . . . . . . . . . . . . . . . . . . 172

B.7 Robust Statistics . . . . . . . . . . . . . . . . . . . . . . . . 173

B.8 Model Predictive Control . . . . . . . . . . . . . . . . . . . 174

B.9 HDA Process . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.10 Tubular Reactor . . . . . . . . . . . . . . . . . . . . . . . . 182

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Abstract

It is important to predict the behavior of an engineering process accurately and

timely. The predictions are usually achieved using a first-principles-based model

that describes the complex phenomena embodied in the process. However, no

model is an exact representation of the complex process for multiple reasons. The

primary goal of this research is to investigate one of the possible reasons, the un-

certainty of the model parameters from the viewpoint of their effect on the accuracy

of the model’s predictions. Other secondary goals of this research are updating the

uncertain parameter values and determination of robust estimates of the uncertain

parameters to improve the accuracy of a model.

The methodologies applied to understand propagation of the uncertain param-

eters through a model are Latin hypercube sampling coupled with Hammersley

sequencing (LHHS). These methods are selected because of their efficiency and

effectiveness when there are multiple uncertain parameters in a model.

Real processes experience unmeasured and unplanned disturbances. Even though

a model may come arbitrarily close to estimating the output of the process, because

of these types of disturbances there always will be process/model mismatch. This

study addresses this issue by investigating updating of the model uncertain param-

eters to minimize this mismatch. The updating methods designed in this research

viii


come from the class of particle filters (also referred to as sequential Monte Carlo

filters); they include a Markov chain Monte Carlo filter and an ensemble Kalman

filter.

As the number of uncertain parameters increase so does the computational bur-

den. While updating is one solution to improve model accuracy another potential

solution is to determine a robust estimate of the uncertain parameter using the the-

ory of robust statistics. This research will provide the theoretical proof that the

maximum likelihood estimate is the best statistic to provide a robust estimate.

The operational side of this research focuses on online model-based applications

such as model-based control and monitoring with processing of uncertain model

parameters. To demonstrate these research concepts, we employ simulations of a

continuous reactor system and an oil producing reservoir system. The results are

analyzed and discussed.

ix


List of Tables

2.1 Dimensionless parameters for the HDA process [2]. . . . . . . . . . 30

2.2 Computation time and number of samples at three spatial locations

that achieves 0.5% error of the true mean and variance of the ben-

zene concentration and reactor temperature. . . . . . . . . . . . . . 33

3.1 Maximum relative errors in the outputs between the physical model

and the ROM with and without uncertainty. . . . . . . . . . . . . . 49

5.1 Definition of Model Variables and Parameters. . . . . . . . . . . . . 71

5.2 Measurement uncertainties. [3] . . . . . . . . . . . . . . . . . . . . 74

5.3 Porosity, permeability and water injection ratio of four parts of

reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1 Estimates and errors . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Robust estimates. y =2∑

j=1P j. . . . . . . . . . . . . . . . . . . . . . 118

6.3 Robust estimates. y = exp(P1) + P2 . . . . . . . . . . . . . . . . . . 118

7.1 Dimensionless parameters for the tubular reactor [4]. . . . . . . . . 123

7.2 Nominal values and robust estimates of the uncertain parameters . . 127

7.3 Closed-loop performance comparison betweenMr andMn . . . . . 132

7.4 Closed-loop performance comparison betweenMr andMn . . . . . 133

x


List of Figures

1.1 Road map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Selection of Pri. Top: MCS random selection. Bottom: LHS selec-

tion from equal probable intervals. . . . . . . . . . . . . . . . . . . 20

2.2 One-dimensional uniformity analysis with 20 sample points. Top:

MCS. Bottom: LHS. . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 One hundred sample points on a unit square with X,Y ∈ (0, 1). Top:

MCS. Bottom: LHS. . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 One hundred sample points (sampled with LHHS) on a unit square

with x, y ∈ (0, 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 One hundred sample points on a unit square with correlation 0.9.

Top: MCS. Middle: LHS. Bottom: LHHS. . . . . . . . . . . . . . . 36

2.6 Plug Flow Reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 Standard deviation of benzene concentration (left) and reactor tem-

perature (right) as a function of sample size. +: MCS. : LHS. ?:

LHHS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.8 Distribution of dimensionless benzene concentration and temperature. 39

3.1 Output of the ROM and physical model in the presence of a +5%

bias in the mean value of the uncertain parameters. Left: Benzene

concentration. Right: Reactor temperature. 4: Physical model. :

ROM generated with uncertain data. ∗: ROM without uncertainty. . 50

3.2 Output of the ROM and physical model in the presence of a -5%

bias in the mean value of the uncertain parameters. Left: Benzene

concentration. Right: Reactor temperature. 4: Physical model. :

ROM generated with uncertain data. ∗: ROM without uncertainty. . 51

xi


3.3 Output of the ROM and physical model in the presence of a +20%

error in the standard deviation of the uncertain parameters. Left:

Benzene concentration. Right: Reactor temperature. 4: Physical

model. : ROM generated with uncertain data. ∗: ROM without

uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Outputs of the logistic map. . . . . . . . . . . . . . . . . . . . . . . 65

4.2 State estimation with MCMC. . . . . . . . . . . . . . . . . . . . . 65

4.3 State estimation with EnKF. . . . . . . . . . . . . . . . . . . . . . 66

5.1 Model-based optimization framework with parameter uncertainty

updating. y∗: set-points; u: outputs from the optimizer; d: dis-

turbances; ym: measurements; y(Θ): model outputs; Θ: model

uncertain parameters; e: errors between measurements and model

outputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Schematic of a two-dimensional reservoir and wells. ↓: water in-

jection well; ↑: oil production well. . . . . . . . . . . . . . . . . . . 69

5.3 Schematic of the closed-loop management framework [5]. n: con-

trol step; u: outputs from optimal controller;D: disturbances; dobs:

observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 Schematic of a two-dimensional reservoir and wells. ↓: water in-

jection well; ↑: oil production well. . . . . . . . . . . . . . . . . . . 78

5.5 Prior distributions of porosity and permeability. . . . . . . . . . . . 78

5.6 Posterior distributions of porosity and permeability. . . . . . . . . . 88

5.7 Initial, true and updated permeability of reservoir. . . . . . . . . . . 89

5.8 Initial, true and updated porosity of reservoir. . . . . . . . . . . . . 90

5.9 Uncertainty in the porosity parameter after 30, 120, 180, 240, 300,

and 390 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.10 Uncertainty in the permeability parameter after 30, 120, 180, 240,

300, and 390 days. . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.11 Forecasts of the measurements after 30, 120, 180, 240, 300, and

390 days. ’’: true; ’∗’: initial; ’’: 30 days; ’4’: 180 days; ’+’:

390 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xii


5.12 Forecasts of the measurements after 30, 120, 180, 240, 300, and

390 days. ’’: true; ’∗’: initial; ’’: 30 days; ’4’: 180 days; ’+’:

390 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.13 Final water saturation when the water cut is 0.9. Top: non-optimized.

Middle: optimized with EnKF. Bottom: optimized with MCMC. . . 95

5.14 Top: Comparison of cumulative production of oil and water. Op-

timized case: EnKF, oil () and water (5) production; MCMC, oil

(+) and water (4) production. Non-optimized case: oil () and (∗):

water production. Bottom: Cumulative NPV. Optimized: EnKF,

blue; MCMC, green. Non-optimized: red. . . . . . . . . . . . . . . 96

5.15 Optimal dimensionless injected water flow rate in the first well as

a function of the updating steps, and the average permeability (top)

and porosity (bottom) values. . . . . . . . . . . . . . . . . . . . . . 97

5.16 Optimal dimensionless injection water flow rate in the second well

as a function of the updating steps and average permeability (top)


5.17 Optimal dimensionless injection water flow rate in the third well as

a function of the updating steps and average permeability (top) and

porosity (bottom) values. . . . . . . . . . . . . . . . . . . . . . . . 99

5.18 Optimal dimensionless injection water flow rate in the fourth well

as a function of the updating steps and average permeability (top)


6.1 Skewed Gaussian distribution. . . . . . . . . . . . . . . . . . . . . 104

7.1 State estimation framework with a maximum likelihood model. . . . 124

7.2 State estimation. Top: dimensionless temperature. Bottom: dimen-

sionless concentration. +: measured values; : PLS estimates. . . . 126

7.3 State estimation. Dimensionless concentration. +: plant measure-

ments; : PLS estimates. q: PLS estimates that violate the ±3σ

limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xiii


7.4 Outputs of the nominal and maximum likelihood models. Top: di-

mensionless temperature. Bottom: dimensionless concentration. :

states estimates from M ; ∗: states estimates fromMN . . . . . . . . 128

7.5 MPC framework with a MLM. . . . . . . . . . . . . . . . . . . . . 129

7.6 Model predictive control performance in the presence of distur-

bances: Top: Mr. Bottom: Mn. ∗: set-point. . . . . . . . . . . . . . 135


bances and with 3% decrease in the value of the model’s param-

eters. Top: Mr. Bottom: Mn. ∗: set-point. . . . . . . . . . . . . . . 136


bances and with 3% increase in the value of the model’s parameters.

Top: Mr. Bottom: Mn. ∗: set-point. . . . . . . . . . . . . . . . . . 137

7.9 Closed-loop tracking. Top: Mr. Bottom: Mn. ∗: set-point. . . . . . 138

7.10 Closed-loop tracking with a 3% decrease in the values of the model’s

parameters. Top: Mr; Bottom: Mn. ∗: set-point. . . . . . . . . . . 139

7.11 Closed-loop tracking with a 3% increase in the values of the model’s

parameters. Top: Mr. Bottom: Mn. ∗: set-point. . . . . . . . . . . 140

A.1 Internal model control framework [6]. y∗: desired set-points trajec-

tory, u: controller outputs, ym: process measurements, y: model

outputs, d: measured or unmeasured disturbances. . . . . . . . . . . 161

A.2 A MPC scheme [7]. y: controlled variable; u: controller output. . . 162

xiv


Chapter 1

Introduction

1.1 Background

Uncertainties exist inherently in real situations. Traditionally, cognitive heuris-

tics is applied to ascertain the appropriate management decisions. However, as real

systems have become increasingly complex the decisions to be made may have

higher associated risks, thus, heuristics while useful may not engender the required

level of certitude. An alternative is to use a mathematical model that describes the

dominant phenomena of these systems. Even though no model is an exact repre-

sentation of the system this approach may be more reliable and the confidence in

the model’s predictions can be quantified.

There are multiple reasons why a model is not exact. Primary among this is

the uncertainty associated with critical or sensitive model parameters that influ-

ence the final decisions. The study of the effect of model parameter uncertainty

on the model’s outputs is the primary objective of this research. Model parameter

uncertainty is common to many areas including, chemical, petrochemical, pharma-

ceutical industries, energy planing, power generation system planning, and so forth

[8, 9, 10].

Because of the widespread existence of uncertainties, their propagation has been

1


studied in many fields. The study of uncertainty propagation involves the develop-

ment of a mathematical model of the physical process and its numerical solution.

Xiu presented an algorithm of polynomial chaos to model the input uncertainty

and its propagation in incompressible flow simulations [11]. Polynomial chaos is a

non-sampling based method to determine evolution of uncertainty in the dynamical

system. Xiu stated that the polynomial chaos generalized for uncertainty promised

a substantial speed-up compared with Monte Carlo methods. However, in the case

study presented, there were only two uncertain parameters in a micro-channel flow

model. The precision depends on the order of the polynomials. For large numbers

of uncertain variables, polynomial chaos becomes very computationally expensive

making Monte Carlo methods more attractive. Tatang proposed a probabilistic col-

location method to analyze the uncertainty and its propagation in a geophysical

model [12]. With the orthogonal polynomials as functions of the uncertain pa-

rameters, the order of the model is reduced, which mitigates the computational

burden. Actually, the probabilistic collocation is a type of polynomial chaos that

approximates the response of the model with polynomial functions of the uncer-

tain parameters. For systems where a polynomial expansion cannot provide a good

approximation, sampling methods are needed.

Sampling of the uncertain parameter distributions followed by propagation of

the samples is a common and effective way to study the uncertainty effects on a

model’s outputs. The prior distribution of the uncertain parameters should be known

2


or assumed for sampling. Probability theory provides the foundation for sampling

uncertainty [13]. With a sample space that relates to the set of all probable outcomes

denoted by Ω, probability theory assumes that for each element θ ∈ Ω, an intrinsic

”probability” value from a function f (θ) satisfies the following properties:

• f (θ) ∈ [0, 1],∀θ ∈ Ω; and

•∑θ∈Ω

f (θ) = 1.

A probability distribution is a function f (θ) that describes the probability of a

random variable taking certain values. Based on the probability distribution func-

tion, various sampling techniques can be applied to sample the probable values of

uncertain variables for propagation.

Monte Carlo sampling (MCS) is one of the methods commonly used and is

considered to have high accuracy, but high computational burden. Because MCS

samples uncertain variables randomly, it needs a large number of sampling points

to cover the uncertain ranges. Thus, the model should be executed many times to

propagate these points. The corresponding computational efficiency is low. Related

to MCS is the Quasi Monte Carlo (QMC) sequences and the stratified technique

of Latin hypercube sampling (LHS). Both QMC and LHS have high accuracy but

they reduce the computational burden and improve the propagation efficiency [14].

However, as the number of uncertain variables increase, their efficiency decreases

noticeably.

3


Multiple dimensional uniformity of the samples can be applied to reduce the

number of samples and thus efficiency. Hammersley sequencing of points have

been shown to distribute the multiple variables samples regularly [15], and when

combined with LHS, together they provided an efficient sampling and sequenc-

ing technique called the Latin hypercube Hammersley sequence sampling (LHHS)

technique [16]. LHHS has been employed to study uncertainty propagation in var-

ious fields, such as health risk assessment process [14], estimation of greenhouse

gases emissions [17], off-line quality control of a reaction process [16], state esti-

mate of chemical processes [18, 19].

For unmeasurable parameters, updating is a necessary step to obtain an accurate

estimate of the state of the system by way of a model. Bayes’ theorem has been used

widely to estimate a variable from known conditions by determining the inverse

probability of this variable [20].

In probability theory and applications, Bayes’ theorem relates a conditional

probability to its inverse [21]. Consider A and B two events and denote the proba-

bility for each event happening by ℘(A) and ℘(B). Joint probability ℘(A, B) designs

the probability of both A and B occurring,

℘(A, B) = ℘(A|B)℘(B) = ℘(B|A)℘(A)

where ℘(A|B) and ℘(B|A) are conditional probabilities. ℘(A|B) is the probability of

4


A occurring given that B has already occurring. Bayes’ theorem in its simplest form

is given by,

℘(B|A) =℘(A|B)℘(B)℘(A)

To further elaboration on the use of this theorem, let Θ = (θ1, θ2, · · · , θd) be a

vector of model parameters with length d. Assume there are m observations y =

(y1, y2, · · · , ym). It follows that Bayes’ theorem in terms of probability distributions

is given by

℘(Θ|y) =℘(y|Θ)℘(Θ)℘(y)

℘(Θ|y) is the distribution of model parameters posterior to the observations,

y, and represents the probability that the model is correct given observations y.

℘(y|Θ) is called a likelihood function. Before y is observed, ℘(y|Θ) represents

the probability density function associated with the probable data realizations for

a fixed parameter vector Θ. Following an observation, ℘(y|Θ) is the likelihood of

obtaining the realization that was actually observed as a function of the parameter

vector Θ.

Based on Bayes’ theorem, a number of related approaches have emerged includ-

ing Markov chain Monte Carlo (MCMC) [22, 23, 24] and the ensemble Kalman

filter (EnKF) [25, 26, 27, 28, 29], especially to address updating in the presence

of multiple dimensional uncertain parameters in complex nonlinear models with

non-Gaussian distributions.

5


Uncertain parameter estimation and updating in a rainfall runoff model was

studied by [30, 31]. They introduced an adaptive Metropolis (AM) algorithm within

the MCMC framework. The AM algorithm was found to have several advantages

compared with the traditional Metropolis-Hastings algorithm. In the AM scheme,

the use of the parameter covariance matrix in the proposed distributions allows

the sampling of several uncertain parameters together, which provides a more effi-

cient exploration of the posterior distributions. Hassan used the AM algorithm in

the MCMC framework to update two uncertain parameters, recharge and hydraulic

conductivity, to quantify the parameters’ effect on the predictions of a groundwater

flow model [32]. The AM-based MCMC scheme was used successfully to obtain

the posterior distributions of the two uncertain parameters. However, a large num-

ber of model executions was required.

An EnKF was used to assimilate thickness (the difference in height between two

pressure levels in a weather forecast) data for operational numerical weather predic-

tion [33]. The quasi-geostrophic model was grided on a 64×32 two-dimension grid.

A series of 30-day data assimilation cycles were performed using ensembles of data

of different sizes. The result indicated that ensembles having an order of 100 mem-

bers or more were sufficient to describe the ensemble covariance accurately. In the

study by Heemink et al, an EnKF was designed to solve atmospheric chemistry data

assimilations problems [34]. The studied domain was divided into 30×30 grids and

the EnKF was able to provide a solution to the data assimilation problem. In recent

6


years, because of the successful use in the the above described complex geophys-

ical areas, the EnKF was introduced in the area of reservoir uncertain parameter

updating. In this application, the EnKF also showed good performance of updat-

ing critical geographic parameters of permeability and porosity of very large scale

reservoirs [3, 35, 36, 37, 38].

Updating of the uncertain model parameters is essential for the case where the

parameters are unmeasurable. However, the updating process is very time consum-

ing since it requires a large number of observations to make the uncertain param-

eters converge to their stationary distributions or stationary values. Even if there

is a parameter whose sampled values are known, it remains a nontrivial task to de-

termine an estimate of this uncertain parameter from its samples. The key point is

that an estimate of the uncertain parameter should be robust to prevent the effects

of outliers, insufficient number of samples, and so forth.

Robust statistics [39, 40, 41] is widely applied to obtain a robust estimate of an

uncertain variable. Olive did a study on the calculation of robust estimates based

on robust statistics for different kinds of usual distributions and nonparametric data

sets [42]. Daszykowski introduced robust statistics in chemometrics for data explo-

ration and modeling [43]. It was pointed out that when data contained outliers, the

data mean as well as its standard deviation were no longer reliable estimates, and

therefore, data preprocessing was required. The median and the absolute deviation

around the median were selected as robust estimates of a variable and its deviation;

7


these statistics then were used to scale the data sets and to screen for outliers.

The analysis of uncertain parameters’ propagation and updating are relevant

for model-based applications. In this work, the study of parameter uncertainties

is applied to the real-time application of model-based control, in particular model

predictive control (MPC).

Model-based control is a framework that explicitly uses a model of the process

to determine the optimal control input to regulate the real process. It is a well known

fact that the ideal feedback controller is the inverse of the process. This indirect

construction of the controller is known as internal model control (IMC) [6]. The

underlying assumption is that the inverse of the model is realizable. For models that

are not realizable, other model-based control designs have been proposed. Notable

ones are, model predictive control (MPC) [44], dynamic matrix control (DMC)

[45, 46], quadratic dynamic matrix control (QDMC), and generalized model-based

control (GMC) [47, 48]. The basic concepts remains the same – that of using a

model of the process in a constrained optimization formulation to determine the

optimal controller movers.

Real-time applications impose the requirement of reliable, stable, and timely

computational solutions. A complex, nonlinear, high-dimensional process model

usually in the form of a system of partial differential-algebraic equations, may not

be suitable for these types of applications. One means of addressing this limi-

tation is to employ an appropriate reduced-order model (ROM). Zheng and Hoo

8


combined the characteristics of singular-value decomposition and the Karhunen-

Loeve (KL) expansion to arrive at a ROM that captured the dominant characteris-

tics of a distributed parameter system [2, 4, 49, 50]. A proportional-integral (PI)

controller, a dynamic matrix controller (DMC) and a quadratic dynamic matrix

controller (QDMC) were designed based on this ROM and all showed good per-

formance for disturbance rejection. Astrid studied reduction of process simulation

models using a proper orthogonal decomposition (POD) approach for missing point

estimation and model-based control [51]. In Astrid’s work, heat conduction models

and a computational fluid dynamics (CFD) model of an industrial glass melt feeder

were used to demonstrate the model reduction technique. Based on a principal com-

ponent analysis (PCA), Lang reduced CFD models of a gas turbine combustor and

an entrained-flow coal gasifier to ROMs for process simulation and optimization

[52]. As a data-based latent variable method, partial least squares (PLS) regression

was analyzed and applied for process analysis, monitoring and control [53]. The

implementation of control policies for autoregressive moving average exogenous

(ARMAX) models was studied [54]. The control scheme implemented was a feed-

back strategy, with white noise control. The ARMAX models were shown to be

valuable for examining the closed-loop stability.

9


1.2 Motivation

In order to manage a physical process, it is very important to understand and

accurately predict the phenomena of this process fully. It would involve developing

a process model consisting of mass, moment and energy balances that can be ex-

pressed in a mathematical framework. The accuracy of the model parameters will

affect the solution results. Studying the effect of parameter uncertainties and their

validation in a simulation (or in silico) model play a critical role in managing the

process.

There are multiple reasons why model parameter uncertainties exist. For in-

stance, measurement error (uncalibrated sensors), the parameter cannot be esti-

mated reliably (bulk effective value versus local values), the parameter simply can-

not be measured given the current state-of-the-art sensors, or the experimental con-

ditions to carry out the measurements are dangerous. Since model parameter uncer-

tainties affect the numerical results, state estimates and other process management

applications (e.g., model based control, online monitoring, process optimization), it

would be very prudent to quantify the effect of parameter uncertainty on the model

outputs by way of propagation of the uncertainty, developing criteria and meth-

ods that provide robust parameter values, and designing an online framework for

efficient model parameter updating to maintain model accuracy.

10


1.3 Objectives

As stated previously, there are multiple reasons that contribute to model uncer-

tainty. These include, the model assumptions, the model parameter uncertainties,

the functional forms assumed to represent the phenomena, and so forth. The focus

of this research is to investigate the model parameter uncertainties and the road map

in the Figure 1.1 is meant to illustrate the focus of this work.

Model Uncertainty

De elo e t A u tio /Uncertain Parameters Development Assumptions/ Functional Forms/…

Propagation Updating Robust EstimationPropagation Updating Robust Estimation

Quantify Effect of Q yUncertain Parameters on Model Outputs

Improve Model Accuracy

Real‐time State Prediction from Parameters

Real‐time Model‐based Optimization

Real‐time State Estimation/

Model based ControlParameters Optimization Model‐based Control

Figure 1.1: Road map.

The objectives of research focus on the following items.

1. Investigate the effect of the model parameter uncertainties on a mathematical

model’s predictions.

11


By propagating uncertain parameters through a model, the effect of the model

parameter uncertainties on a mathematical model’s predictions can be quan-

tified. By capturing the relationships between the uncertain parameters and

the real-time state prediction, more accurate state estimation can be real-

ized. In this work, Monte Carlo sampling (MCS), Latin hypercube sampling

(LHS) and Latin hypercube sampling with Hammersley sequencing (LHHS)

are used to establish propagation efficiency. Partial least squares regression

is applied to capture the relationships between the uncertain parameters and

the model outputs. Once the relationships are determined, the PLS model

can be used to predict the process states from known values of the parame-

ters. Usually, the process states are of a higher dimension than the number of

uncertain parameters. To make the PLS feasible and efficient, an application

of the Karhunen-Loeve expansion is employed to reduce the dimension but

retain the dominant temporal and spatial characteristics of the process.

2. Update the uncertain parameters efficiently.

When the uncertain parameters are not readily measurable, an initial guess

of their values is unavoidable. However, for accuracy of the application, up-

dating the initial guess of the parameter uncertainties is justified. One of the

objective of this work is to update the uncertain parameters and thus improve

the estimate of the system states. For very complex and nonlinear models, as

12


described above, MCMC and EnKF have been used successfully. They will

be embedded in a model-based optimization framework in this work and their

performance will be compared.

3. Estimate robust values of uncertain parameters.

Robust statistics seeks to provide estimating methods that emulate popular

statistical methods, but which are not unduly affected by outliers or other

small departures from model assumptions. Three types of robust estimates

will be studied for robust estimation. They are, the maximum likelihood type

estimates (MLE), the linear combinations of order statistics (L-Estimate), and

the rank estimates derived from suitable rank tests (R-Estimate). This work

will demonstrate that models parameterized by robust estimates of the uncer-

tain parameters can provide more accurate state estimates.

1.4 Organization

The rest of the dissertation is organized as follows. Chapter 2 presents Monte

Carlo Sampling (MCS), Latin hypercube sampling (LHS) and Latin hypercube

sampling with Hammersley sequence (LHHS) techniques for uncertainty propa-

gation. The production of benzene from hydro-dealkylation (HDA) of toluene is

employed in this chapter to compare the propagation efficiency of these sampling

techniques. To study real-time state prediction from uncertain parameters, partial

13


least squares (PLS) regression and Karhunen-Loeve (KL) expansion are introduced

in Chapter 3. The performance of the real-time state prediction from uncertain pa-

rameters also is demonstrated using the HDA process. Chapter 4 introduces and

compares two updating techniques, Markov chain Monte Carlo (MCMC) and en-

semble Kalman filter (EnKF). In Chapter 5, the updating techniques are embed-

ded in a real-time model-based optimization framework. A five-spot pattern oil

reservoir system is introduced to demonstrate and compare the MCMC and EnKF

updating methods. Chapter 6 presents an overview of the theory of robust statis-

tics. Based on robust statistics, a new theorem about a maximum likelihood model

is introduced and proven. In Chapter 7, the robustness feature of the maximum

likelihood model is demonstrated in a model predictive control (MPC) framework.

Lastly, Chapter 8 summarizes the contributions of the research and suggests future

work.

14


Nomenclature

AM adaptive Metropolis

ARMAX autoregressive moving average exogenous

CFD computational fluid dynamics

DMC dynamic matrix control

EnKF ensemble Kalman filter

GMC generalized model-based control

HDA hydro-dealkylation

IMC internal model control

L-estimate linear combinations of order statistics

KL Karhunen-Loeve

LHS Latin hypercube sampling

LHHS Latin hypercube Hammersley sequence sampling

MCMC Markov chain Monte Carlo

MCS Monte Carlo sampling

MLE maximum likelihood type estimates

MPC model predictive control

PCA principal component analysis

PLS partial least squares

POD proper orthogonal decomposition

QDMC quadratic dynamic matrix control

QMC quasi Monte Carlo

R-estimate estimates derived from rank tests

ROM reduced-order model

15


Chapter 2

Preliminaries on Uncertainty Propagation

This chapter presents an overview of sampling methods of Monte Carlo, Latin

hypercube sampling, and Latin hypercube sampling with Hammersley sequencing

for uncertainty propagation. The production of benzene from hydro-dealkylation

(HDA) of toluene is then introduced to compare the propagation efficiency of these

sampling methods.

With known distributions, for example from historical data, of the multiple un-

certain variables, efficient and effective sampling are reasonable expectations to

propagate the large number of sequences through the complex model to determine

the state estimates and their distributions.

The conventional sampling method of Monte Carlo sampling (MCS) is known

to have low efficiency for uncertainty propagation. As a stratified sampling tech-

nique, Latin hypercube sampling (LHS) technique, is more accurate and efficient

than MCS when there is a single uncertain variable. But in the case of multiple

uncertain variables it has been demonstrated that the efficiency of LHS decreases

noticeably. Another sequencing method, Hammersley sequencing, when combined

with Latin hypercube sampling (LHHS) has been shown to be very efficient when

there are multiple uncertain variables [14].

16


2.1 Sampling Techniques

2.1.1 Monte Carlo Sampling

There is no consensus on how Monte Carlo should be defined. For example,

Ripley [55] reserved Monte Carlo for most probabilistic modeling as stochastic

simulation. Sawilowsky [56] distinguished between Monte Carlo method and a

Monte Carlo simulation. A Monte Carlo method can be used to solve a mathe-

matical or statistical problem but a Monte Carlo simulation repeated sampling to

investigate the properties of a phenomenon. Anderson [57] defined Monte Carlo

as the art of approximating an expectation by the sample mean of a function of

simulated random variables.

Monte Carlo sampling (MCS) method is one of the best known methods for

sampling a probability distribution that is based on the use of a pseudo-random

number generator [16]. This simple random sampling involves repeatedly forming

random values of a variable from a prescribed probability distribution. Because the

characteristic of Monte Carlo samples is random, to sample the distribution of an

uncertain variable means a very large number of sample points to cover the distri-

bution range and approximate the real expectation. The large number of sample

points is the primary reason that causes the computational burden for uncertainty

propagation.

To sample an uncertain variable X with MCS, the cumulative distribution func-

17


tion (CDF) F of X should be known. The values of the CDF are Pri = F(xi), where

F is the cumulative density function. The procedure is as follows.

• Randomly sample N values of Pri with uniform distribution ranging from 0

to 1, U (0, 1) (i = 1, · · · ,N), where N is the number of sample points and Pri

is a value of the cumulative probability.

• Transform the probability values Pri into the value xi using the inverse of the

distribution function F−1 : xi = F−1(Pri).

2.1.2 Latin Hypercube Sampling

Latin hypercube sampling (LHS) is a stratified-random 1 procedure which pro-

vides an efficient way of sampling a variable from its distribution. The LHS in-

volves sampling N values from the prescribed distribution of the variable X. The

cumulative distribution for this variable is divided into N equally probable intervals.

From each interval a value is selected randomly. Unlike simple random sampling

of MCS, this method ensures a full coverage of a variable by maximally stratifying

its marginal distribution 2 The procedure is as follows.

• Divide the cumulative distribution of the variable into N equal probable in-

1Stratified means that the range of a variable’s distribution has been separated in several intervals.Stratified-random means from each of the divided interval a sample is collected randomly.

2A distribution function may be for more than one variables for example f(x,y) which is the jointdistribution of x and y, the marginal distribution refers to the distribution of one of them f (x) =∫

y( f (x, y)dy. Stratifying the marginal distribution means separating the marginal distribution rangeaveragely according to the number of samples that we want.

18


tervals.

• Randomly select a value from each interval. The sampled cumulative proba-

bility can be written as [58]

Pri = (1/N)ru + (i − 1)/N

where ru is a uniformly distributed random number, ru ∼ U (0, 1).

• Transform the probability value Pri into the value Xi using the inverse of the

distribution function F−1 : Xi = F−1(Pri).

Assume 5 sample points of a normal distribution variable X are sampled with

MCS and LHS. X ∼ N (0, 1). Figure 2.1 shows how MCS and LHS select values

for Pri.

It is easy to demonstrate that LHS can use less sample points than MCS to

cover the full range of a variable distribution. Better one-dimensional uniformity

is indicated by the closeness of the sample points to the 45 line with a uniform

interval between the adjacent sample points [14]. As shown in Figure 2.2, for 20

sample points selected by MCS and LHS, the latter shows better uniformity than

the former.

The attractive uniformity property of the LHS method can provide efficient

propagation of LHS sample points for a one dimension uncertain parameter. How-

19


-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

X

Cum

ulat

ive

Pro

babi

lity

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

X

Cum

ulat

ive

Pro

babi

lity

Figure 2.1: Selection of Pri. Top: MCS random selection. Bottom: LHS selectionfrom equal probable intervals.

20


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X

Cum

ulat

ive

Pro

babi

lity

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X

Cum

ulat

ive

Pro

babi

lity

Figure 2.2: One-dimensional uniformity analysis with 20 sample points. Top:MCS. Bottom: LHS.

21


ever, it has been shown that for multiple uncertain distributions, LHS cannot retain

this advantage. Assume there are two variables X and Y and both have uniform dis-

tributions between 0 and 1, that is, X ∼ U (0, 1), Y ∼ U (0, 1). Figure 2.3 compares

the uniformity property of MCS and LHS in two dimensions. One hundred sample

points are generated on a unit square by each sampling method. The figure shows

that the sample points of both MCS and LHS are not ordered.

2.1.3 Latin Hypercube Hammersley Sampling

To combine sample points of multiple dimension variables, the conventional

approach is to pair them randomly. It is in this step that Hammersley sequencing

has better multi-dimensional uniformity [15, 59].

2.1.3.1 Hammersley sequence points

The definition of Hammersley points and an explanation of the procedure to

generate the Hammersley points are as follows [15, 16].

Let any nonnegative integer K be expanded using a prime base p,

K = k0 + k1 p + k2 p2 + · · · + km pm (2.1)

where each k j, j = 1, · · · ,m, is an integer in [0, p-1], m = [logKp ] (square brackets

denote the integral part).

22


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

Figure 2.3: One hundred sample points on a unit square with X,Y ∈ (0, 1). Top:MCS. Bottom: LHS.

23


Inverse of prime numbers can be used to define a function φp(K):

φp(K) =k0

p+

k1

p2 + · · · +km

pm+1

The sequence of Hammersley points of dimension d is given by,

(KN, φp1(k), φp2(k), · · · , φpd−1(k)

), K = 1, · · · ,N

where N is the number of samples and p1, · · · , pd−1 are the first d-1 prime numbers.

2.1.3.2 Combination of Latin hypercube sampling and

Hammersley sequencing

In order to retain the uniformity of LHS for multiple dimensions variables,

Hammersley sequence is applied to arrange the LHS sample points of each vari-

able [14, 17]. This sampling method is Latin hypercube Hammersley sampling

(LHHS). Based on the use of rank correlations [60], the LHS sample points for

multi-dimensional variables can be arranged according to the order of the multiple

dimensional Hammersley sequence points.

Suppose X is an N × d matrix that consists of N sets of d uncorrelated pa-

rameters (sampled with LHS). Then the correlation matrix is an identity matrix I .

24


Let

C = Γ × Γ′

be the desired correlation matrix ofX where Γ is a lower triangular matrix obtained

by Cholesky factorization. The transformed matrixXΓ′ then has the desired corre-

lation matrix C. This is the theoretical basis for transforming a desired correlation

to an uncorrelated matrix.

Because X is to be rearranged according to the Hammersley sequence points,

these points should be a matrix with the same dimension as X , N × d. Denote

the Hammersley sequence points by the matrixH . With the desired correlation C,

transformH to the rank matrixH∗ whose correlation matrix is C.

To avoid the problem that the correlation matrix of H is not I but R, a matrix

S should be found out that

SRS′ = C

With Cholesky factorization,

R = QQ′

Therefore, the solution of S can be obtained,

S = ΓQ−1

25


and correspondingly the transformation factor for the rank matrix is S and the rank

matrix becomes

H∗ =HS′

As a result the correlation matrix of H∗ is exactly equal to the desired matrix

C. The sample matrixX therefore can be paired according to the rank matrixH∗.

In the case of LHS, H is not used and the sample matrix X is rearranged ac-

cording to the correlation matrixC. In the case of the LHHS method ,the matrix of

Hammersley sequence points H is used to rearrange X . According to the defini-

tion of the Hammersley sequence points, these points are distributed uniformly in a

multiple-dimensional space. However, for LHS, uniform distribution is for each di-

mension. When these dimensions are combined the property of uniformity is weak.

In the case of MCS, since it is based on a random score, the uniformity property is

not applicable even when considering one dimension.

Revisiting the two variables case, X ∼ U (0, 1) and Y ∼ U (0, 1). Figure 2.4

shows 100 sample points on a unit square sampled with the LHHS method. When

compared with Figure 2.3, it is observed that these 100 sample points cover the unit

square evenly. In other words, there are no sparse spaces or clot points visible when

compared to Figure 2.3.

If we compare the uniformity property of the MCS, LHS, and LHHS methods in

two-dimensions, Figure 2.3 and 2.4 show that the LHHS generated points have bet-

26


ter uniformity properties. The main reason is that the Hammersley sequence places

the sampled points on a multi-dimensional hypercube in an ordered fashion. In con-

trast, the LHS method while designed for uniformity along a single dimension in the

case of multiple dimensions, it randomly pair the sampled points for placement on a

multi-dimensional hypercube. In the case of MCS, even for a single dimension, the

uniformity property is not applicable. Therefore, the likelihood that the MCS and

LHS methods can provide good uniformity property on multi-dimensional cubes is

extremely small.

In the case that there are correlations between the multiple parameters, the

LHHS method still shows the best uniformity property when compared to the MCS

and LHS methods. Once again revisiting the two variables X and Y case, X ∼

U (0, 1), Y ∼ U (0, 1). Assume the correlation matrix between X and Y is

C =

1 0.9

0.9 1

A rank correlation H∗ = HC can be used to design each dimension. Figure

2.5 shows the uniformity property when there is a desired correlation matrix C for

a two-dimensional uniform distribution of 100 sample points with the imposed 0.9

correlation. As before, it is not difficult to conclude that the LHHS generated points

have better uniformity property when compared to the other methods. The reason

27


for this is that the Hammersley sequence preserves the uniformity property.

2.2 Example: HDA Process

The chemical process of benzene production from hydro-dealkylation (HDA)

of toluene [61] is introduced to demonstrate the sampling methods.

2.2.1 HDA Process

The hydro-dealkylation of toluene occurs in a nonlinear plug-flow reactor (PFR)

system (see Figure 2.6) which has been studied extensively by Zheng and Hoo

[7, 2].

Two reactions are known to occur, namely

C7H8 + H2 → C6H6 + CH4

2C6H6 C12H10 + H2

The first reaction is irreversible and the second is an equilibrium reaction. A

28


first-principles model can be developed to describe the major reaction phenomena.

∂ε1

∂τ= −υ

(∂ε1

∂τ1+ε1

θ

∂θ

∂τ1

)− ε1ε

0.52 θ

1.5eγ(θ−1/θ)1

∂ε2

∂τ= −υ

(∂ε2

∂τ1+ε2

θ

∂θ

∂τ1

)− ε1ε

0.52 θ

1.5eγ(θ−1/θ)1 + κ2(ε3θ)2eγ

(θ−1/θ)2 − κ3ε2ε5θ

2eγ(θ−1/θ)3

∂ε3

∂τ= −υ

(∂ε3

∂τ1+ε3

θ

∂θ

∂τ1

)+ ε1ε

0.52 θ

1.5eγ(θ−1/θ)1 − 2κ2(ε3θ)2eγ

(θ−1/θ)2 + 2κ3ε2ε5θ

2eγ(θ−1/θ)3 + FBm

∂ε4

∂τ= −υ

(∂ε4

∂τ1+ε4

θ

∂θ

∂τ1

)+ ε1ε

0.52 θ

1.5eγ(θ−1/θ)1

∂ε5

∂τ= −υ

(∂ε5

∂τ1+ε5

θ

∂θ

∂τ1

)+ κ2(ε3θ)2eγ

(θ−1/θ)2 − κ3ε2ε5θ

2eγ(θ−1/θ)3

∂θ

∂τ=

1ζ

[Hr1∂ε1

∂τ− Hr2

∂ε5

∂τ+ Q(θF − θ) − v

(ζ∂θ

∂τ1− Hr1

∂ε1

∂τ1+ Hr2

∂ε5

∂τ1

)]− FBmζB

(2.2)

where ζ = (Cp/CP0)(P0/Pr), υ(z) = (Fin + Fin j)/(P/RT ), Fin is the feed to the reac-

tor, εi is the concentration of the ith component and θ is the dimensionless reactor

temperature. The initial feed concentrations (mole fraction) of toluene, hydrogen,

and methane are ε1,0 = 0.0807, ε2,0 = 0.4035, ε4,0 = 0.5158. Table 2.1 provides

the definition and values of the parameters and variables, and subscript 0 is the

reference condition.

The boundary conditions are:

z = 0, εi = εi(t = 0), i = 1, · · · , 5, θ = θ(t = 0)

z = 1,∂εi

∂z= 0, i = 1, · · · , 5,

∂θ

∂z= 0

The pure benzene stream is injected at the start of the reactor. The finite difference

solution of this system serves as the true solution.

29


Table 2.1: Dimensionless parameters for the HDA process [2].

Parameter Nominal values Definition

P = Pr/P0 1.0 Reactor pressure

FBm = Fin j/Fin 0.008 Dimensionless benzene injection

θF = TF/T0 1.0 Jacket temperature

κ1 = k1(T0)/k1(T0) 1.0 Dimensionless reaction 1 rate constant

κ2 = k2(T0)P20/k1(T0)P1.5

0 0.995 Ratio of reaction 1 to reverse reaction 2

κ3 = k3(T0)P20/k1(T0)P1.5

0 5.34 Ratio of reaction 1 to reverse reaction 3

Hr1 = HR1/Cp0T0 -1.51 Heat of reaction 1

Hr2 = HR2/Cp0T0 -0.473 Heat of reaction 2

Q 0.0 Heat transfer coefficient

γ1 = Ea/RT0 29.26 Reaction 1 activation energy



τ ∈ <+ > 0 Dimensionless time

z ∈ Ω [0,1] Dimensionless space

2.2.2 Propagation with Different Sampling Methods

Each parameter in the HDA process is not exact, but not all of the parameters

have a pronounced effect on the system solution. Using a parametric sensitivity

analysis, the outputs are found to be most sensitive to the rate constant of reaction 1

(κ1), the fresh benzene injection rate (FBm), and the heat of reaction 1 (Hr1). These

parameters are uncorrelated.

Assume that a Gaussian distribution can represent each of these three param-

30


eters with mean values 1.0, 37.34, -1.51 and error levels of 20% ((σi/ui)100%),

where ui and σi represent the mean and standard deviation of the ith uncertain pa-

rameter, respectively. Here, the three sampling methods, Monte Carlo sampling

(MCS), Latin hypercube sampling (LHS), and Latin hypercube Hammersley sam-

pling (LHHS) are implemented and their efficiency and accuracy are compared.

The true mean and variance are needed to make a worthwhile comparison.

These statistics are estimated by propagating a very large number of samples, for

example MCS 50,000 samples, LHS 10,000 samples and LHHS 10,000 samples.

When these three methods provide the same (or similar) estimates for the mean

and variance, these values are accepted as the true values. Otherwise, the number

of samples is increased until this criterion is satisfied. Once the true values are

obtained, the efficiency of the different sampling techniques is compared by esti-

mating the number of samples required to obtain the true mean and variance to be

within a 0.5% error.

The finite difference solution of the above system of partial differential equa-

tions gives the spatial and temporal values of the dimensionless reactor temperature

and component concentrations. Consider three spatial locations, z = 1/3, 2/3, 1,

the number of samples and the computation time needed for the three methods to

be within a 0.5% error of the true mean and variance of the benzene concentra-

tion and reactor temperature are tabulated in Table 2.2. The results show that the

LHHS method requires a smaller number of samples (approximately 7 times less)

31


when compared to the LHS and is superior to the MCS (approximately 60 to 70

times less samples). For the same accuracy, LHHS is most efficient of these three

methods.

Figure 2.7 shows the standard deviations of the dimensionless benzene concen-

tration and reactor temperature that result with the implementation of the different

sampling methods at the exit of the reactor. The upper and lower solid lines give

the error limits of ± 0.5% of the true standard deviation. True standard deviation

of the benzene concentration is 1.033 × 10−3 and that of the reactor temperature is

1.782 × 10−3. It can be found that the LHHS method uses least number of sample

points to be within the error range and to have the fastest convergence rate.

The effect of the propagation of the parameter uncertainties through the model

is shown in Figure 2.8, which shows the distributions of the dimensionless ben-

zene concentration and reactor temperature at the reactor exit. The probability of

the model output uncertainties caused by uncertain parameters can be obtained by

propagation.

2.3 Summary

This chapter introduced three sampling techniques, Monte Carlo sampling (MCS),

Latin hypercube sampling (LHS), and Latin hypercube sampling with Hammersley

sequencing (LHHS). Their uniform property are explained to compare their sam-

pling efficiency. These three sampling techniques are applied in hydro-dealkylation

32


Tabl

e2.

2:C

ompu

tatio

ntim

ean

dnu

mbe

rofs

ampl

esat

thre

esp

atia

lloc

atio

nsth

atac

hiev

es0.

5%er

roro

fthe

true

mea

nan

dva

rian

ceof

the

benz

ene

conc

entr

atio

nan

dre

acto

rtem

pera

ture

.

z=1/

3z=

2/3

z=1

Met

hod

MC

SL

HS

LH

HS

MC

SL

HS

LH

HS

MC

SL

HS

LH

HS

Tim

eB

enco

nc15

2.1

15.6

2.1

114.

113

1.9

86.9

131.

9(m

in)

Tem

p18

7.3

15.6

2.1

182

15.6

1.9

14.1

15.6

1.9

Sam

ple

Ben

conc

5600

600

9042

0050

080

3200

500

80Te

mp

6900

600

9067

0060

080

4200

600

80

33


(HDA) of toluene process which illustrates the advantage of LHHS over MCS and

LHS. The product of LHHS technique is the generation of the outputs and their

uncertainty distributions that are functions of the uncertainty in the parameters.

34


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

Figure 2.4: One hundred sample points (sampled with LHHS) on a unit square withx, y ∈ (0, 1).

35


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

y

Figure 2.5: One hundred sample points on a unit square with correlation 0.9. Top:MCS. Middle: LHS. Bottom: LHHS. 36


Feed

FB BenzeneQuench

z = 0 z = 1

51 L=iiε

Outputcomponents

C7H8

H2 CH4

C7H8 H2

C6H6 CH4

C12H10

Figure 2.6: Plug Flow Reactor.

37


0 1000 2000 3000 4000 50001.015

1.025

1.035

1.045

1.055

1.06 x 10-3

Number of sample points

Sta

ndar

d de

viat

ion

of b

enze

ne c

once

ntra

tion

0 1000 2000 3000 4000 50001.72

1.74

1.76

1.78

1.8

1.82

1.84 x 10-3

Number of sample points

Sta

ndar

d de

viat

ion

of d

imen

sion

less

tem

pera

ture

Figure 2.7: Standard deviation of benzene concentration (left) and reactor temper-ature (right) as a function of sample size. +: MCS. : LHS. ?: LHHS.

38


0.04 0.05 0.06 0.07 0.08 0.090

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Dimensionless benzene concentration

Pro

babi

lity

1.02 1.04 1.06 1.08 1.1 1.120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Dimensionless temperature

Pro

babi

lity

Figure 2.8: Distribution of dimensionless benzene concentration and temperature.

39


Nomenclature

CDF cumulative distribution function


LHHS Latin hypercube Hammersley sampling

LHS Latin hypercube sampling

MCS Monte Carlo sampling

PDE partial differential equation

C correlation matrix H matrix of Hammersley pointsH∗ rank matrx K integer

N number of sample points Pr probability value

X random variables Y random variablesd dimension of uncertain variables ru uniformly distributed random valueφp function of Hammersley points N normal distributionU uniform distribution

40


Chapter 3

Real-time State Prediction

Chapter 3 investigates real-time state prediction as a function of the set of un-

certain parameters using partial least squares regression (PLS) and the Karhunen-

Loeve (KL) expansion. These concepts are demonstrated using the HDA process

that was introduced in Chapter 2.

As stated previously, a first-principles model that describes the behavior of a

process is a system of nonlinear partial differential equations (PDEs) in which the

variables are function of space and time. Solving such systems is time consuming

and the solutions are an infinite series that cannot be used readily for real-time

applications such as real-time state prediction. However, if there are data that carry

the relationships between the process state variables and the uncertain parameters,

then a technique such as partial least squares (PLS) regression may be suitable to

capture these relationships in a less computationally-burdensome regression-type

model to enable real-time prediction.

3.1 PLS Regression

In order to consider the effects of uncertain parameters on the state variables, it

is necessary to determine the relationships between the uncertain parameters and the

41


data set of solutions generated from the first-principles model. Partial least squares

(PLS) regression is an often used method to determine the relationships between

the prediction variables and response variables.

The following is excerpted from [62]. Let XN×I be a matrix of I predictors

collected on N observations that describe J response variables, Y N×J. Decompose

both X and Y as a product of a set of orthogonal factors (T ) and a set of loadings

(P ),

X = TP ′ +E

Y = TBC′ + F

(3.1)

The columns of T are the latent vectors, P is the coefficient matrix of X , the

diagonal elements ofB are the regression weights,C represents the weights of the

response variables, and E and F are the matrices of residual errors.

To specify the latent vectors in T , two sets of weights w and c are needed to

create a linear combination of the columns of X and Y such that their covariance

is maximized. The goal is to obtain a pair of vectors t = Xw and u = Y c with

constraints such that w′w = 1, t′t = 1. It then follows that p =X ′t.

Procedurally, let Q = X and R = Y . Then column center and normalize R

andQ.

Step 1: Initialize the vector u with random values

Step 2: Estimate weights forX , w ∝ Q′u

42


Step 3: EstimateX factor scores, t1 = Qw1

Step 4: Estimate weights for Y , c1 ∝ R′t1

Step 5: Estimate Y scores, u1 = Rc1

Step 6: Return to step 2 if t1 has not converged. Otherwise continue.

Step 7: Calculate b = t′1u1

Step 8: Compute the loadings forX: p1 = Q′T .

Step 9: Subtract the effect of t1 from both Q and R: Q = Q − t1p′1 and R =

F − bt1c′1. b is a diagonal element ofB.

Step 10: Repeat from step 1 until the matrixQ becomes null.

The symbol ∝ represents a normalization of the result. The above relations

show thatw1 is the first right singular vector ofX ′Y and c1 is the first left singular

vector of X ′Y . Similarly, t1 and u1 are the first eigenvectors of XX ′Y Y ′ and

Y Y ′XX ′, respectively [62].

The prediction of the dependent variables is based on a multivariate regression

given by,

Y = TBC′ =XBPLS (3.2)

whereBPLS = (P ′)−1BC′.

43


3.2 KL Expansion

In some chemical processes such as a reactor, the states, for example, temper-

ature, concentration and so forth are functions of time and space whose data sets

have a larger dimension than the dimension of the uncertain parameters. To make

the PLS model feasible initial pre-processing of the data with a method such as

Karhunen-Loeve (KL) expansion is necessary, since the KL method can reduce the

dimension of the state data sets.

The basic concept behind the KL expansion is to find those modes that repre-

sent the dominant character of the system such that the basis set corresponding to

these modes comprise what is called the empirical eigenfunctions (EEFs) of the

system. The number of EEFs is usually small, which brings about a dimensionality

reduction without a loss in the complexity that is inherent to distributed parameter

systems [63].

Consider Equations (3.3) and (3.4) that represent the state and measured out-

puts, respectively of a simple one-dimensional (1D) PDE that can be used to repre-

sent a variety of nonlinear phenomena. Here, z is the spatial variable defined on a

closed and compact domain Ω, t is time, y(z, t) are the outputs, x(z, t) are the state

44


variables, and u(z, t) are distributed forcing functions.

∂x(z, t)∂t

= Ax(z, t) + u(z, t) (3.3)

y(z, t) = C(z)x(z, t) z ∈ Ω, t ≥ 0 (3.4)

A = α∂2

∂z2 − υ∂

∂z+ b α, ν > 0

Equation (3.3) usually is solved numerically. The solution, y(z, t), is the data from

which to develop the KL expansion. When t is fixed, the data points at this time are

said to represent a snapshot of the state of the process.

Let y(z, t) denote the mean of the y and v(z, t) = y(z, t)− y(z, t). The covariance

function of these data can be obtained from

R(z, ζ) = limM→∞

1M

∫ M

0,Ωv(z, t)v(ζ, t) dt z, ζ ∈ Ω (3.5)

For D snapshots, the empirical spatial correlation function is given by [64],

R(z, ζ) =1D

D∑k=1

vk(z)vk(ζ) (3.6)

where k is the time sequence number. The eigenvalue, λk ∈ Λ, and corresponding

eigenfunction, ψk ∈ Ψ, of the covariance matrix can be found. The KL expansion

45


of the system defined in Equation (3.3) is given by,

y(z, t) = y(z, t) +∞∑

k=1

√λk ψk(z)ςk(t) (3.7)

where ςk are the projections of y onto ψk. It has been shown that a small number

of EEFs can capture more than 95% of the system’s energy [65] in many instances

indicting a high degree of correlation among the data or a small number of degrees

of freedom in the data [60].

3.3 State Prediction with ROM

In this work, PLS regression is applied to determine the relationships between

the uncertain parameters and the state data sets obtained from the first-principles

model. The dimension of the state data sets is reduced using a KL expansion to

build a reduced-order model (ROM). Since Latin hypercube Hammersley sequence

(LHHS) has good multiple-dimensional uniformity property, it is used to sample

the set of uncertain parameter distributions. The sample sequences are propagated

through the first-principles model. By averaging the resulting state data sets, the

dominant empirical eigenfunctions (EEFs) of the averaged data matrix can be de-

termined. The EEFs serve as the basis for propagation of the state data. Usually a

small number of EEFs can capture the dominant characteristics of the data. With

a small number of dominant EEFs, the corresponding coefficients also is small.

46


These coefficients are the response variables for PLS regression. With the sampled

sets of uncertain parameters as predictors, the relationships between the uncertain

parameters and the coefficients of the corresponding propagated data matrices can

be determined with PLS regression to identify a predictive model. Then, given any

set of uncertain parameters, the ROM coefficients can be calculated from the iden-

tified PLS model. It then follows that the ROM’s outputs which are the states can

be predicted directly by projection of the coefficients onto the EEFs. Thus, with

the combination of PLS regression and KL expansion execution of the complex,

nonlinear first-principles model is avoided. To summarize, the procedural steps are

as follows.

• Step 1: Sample multiple uncertain parameters with LHHS technique.

• Step 2: Propagate samples of uncertain parameters (sampled in Step 1) through

first-principles model. Solve models with samples of uncertain parameters to

obtain the corresponding model outputs.

• Step 3: Average the model outputs and use KL expansion to reduce the di-

mension of averaged matrix to determine its dominant EEFs.

• Step 4: Calculate the corresponding coefficients of all the model outputs ob-

tained in step 2 based on the EEFs calculated in step 3.

• Step 5: Determine the relationships between the samples of uncertain param-

47


eters (sampled in Step 1) and the corresponding coefficients of model outputs

(calculated in Step 4) to build a PLS model.

• Step 6: Predict the process states from known values of the parameters with

the PLS model obtained in step 5.

3.3.1 State Prediction of HDA Process

The HDA process that was introduced in Chapter 2 will be used to demonstrate

state prediction using a ROM generated from the combination of PLS regression

and KL expansion. The uncertain parameters are the rate constant of reaction 1

(κ1), the fresh benzene injection rate (FBm) and the heat of reaction 1 (Hr1). The

parameter distributions assumed in Chapter 2 remain unchanged. The data sets are

the numerical solutions to the system of PDEs that describe the HDA process. This

set includes five component concentrations and the reactor temperature.

A means of validating the generated ROM is as follows. For the same set of in-

put parameters compare the outputs of the ROM to the outputs of the first-principles

model. Furthermore, in order to test if the outputs of the ROM generated from the

uncertain data can track the outputs of the first-principles model better than the out-

puts of a ROM generated from data without uncertainty, a ±5% bias in the mean

values and +20% error in the standard deviations of the uncertain parameters are

introduced.

48


Figures 3.1 to 3.3 compare the benzene concentration and the reactor tempera-

ture, determined using the ROMs with and without uncertainty propagation against

the results of the physical model in the presence of the bias and error in the stan-

dard deviation of the uncertain parameters, respectively. From these figures, it is

observed that the outputs of the ROM generated from data with uncertainty can

track the outputs of the physical model satisfactorily.

The maximum relative errors between the first-principles model and the ROM

are listed in Table 3.1. It is shown that with the same bias of the parameters mean

values and errors in their standard deviations, the ROM generated with uncertain

data has less error.

Table 3.1: Maximum relative errors in the outputs between the physical model andthe ROM with and without uncertainty.

µi: +5% bias µi: -5% bias σi: +20% std

ROM Benz Temp Benz Temp Benz Temp

With uncertainty +3.47% +0.22% -9.95% -0.49% +2% +0.51%

Without uncertainty -12.21% -0.74% -23.56% +0.88% +9.22% +0.81%

3.4 Summary

Partial least squares (PLS) regression is applied to determine the relationships

between the uncertain parameters and the model outputs. To reduce the high di-

mension of the data sets, the Karhunen-Loeve (KL) expansion is employed. Two

49


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

z (m) - Dimensionless length of reactor

Ben

zene

con

cent

ratio

n

0 0.2 0.4 0.6 0.8 10.98

1

1.02

1.04

1.06

1.08

1.1


Dim

ensi

onle

ss te

mpe

ratu

re

Figure 3.1: Output of the ROM and physical model in the presence of a +5% biasin the mean value of the uncertain parameters. Left: Benzene concentration. Right:Reactor temperature. 4: Physical model. : ROM generated with uncertain data. ∗:ROM without uncertainty.

50


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07


Ben

zene

con

cent

ratio

n

0 0.2 0.4 0.6 0.8 10.97

1

1.03

1.06

1.09


Dim

ensi

onle

ss te

mpe

ratu

re

Figure 3.2: Output of the ROM and physical model in the presence of a -5% biasin the mean value of the uncertain parameters. Left: Benzene concentration. Right:Reactor temperature. 4: Physical model. : ROM generated with uncertain data. ∗:ROM without uncertainty.

51


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


Ben

zene

con

cent

ratio

n

0 0.2 0.4 0.6 0.8 10.98

1

1.02

1.04

1.06

1.08

1.1


Dim

ensi

onle

ss te

mpe

ratu

re

Figure 3.3: Output of the ROM and physical model in the presence of a +20% errorin the standard deviation of the uncertain parameters. Left: Benzene concentration.Right: Reactor temperature. 4: Physical model. : ROM generated with uncertaindata. ∗: ROM without uncertainty.

52


reduced-order models (ROMs) are generated, one with and the other without knowl-

edge of the uncertainties. The results of these two ROMs are then compared to the

outputs from the first-principles model. In the presence of parameter uncertainties,

the ROM that is generated by knowledge of the uncertainties has a smaller error

when compared against the outputs from the first-principles model. The hydro-

dealkylation (HDA) of toluene process was used to demonstrate these concepts.

53


Nomenclature

EEF empirical eigenfunction



KL Karhunen-Loeve

PDE partial differential equation



X predictors Y response variablesx state variables y outputsz space variable λ eigenvalueψ eigenfunction ς coefficientN normal distribution U uniform distribution

54


Chapter 4

Preliminaries on Uncertain Parameter Updating

As stated before, some parameters are not measurable in some complex condi-

tions, such as geography, meteorology and oceanography. In many instances there

is almost no information about the value of these parameters. To obtain accurate

estimates of the process behavior from a descriptive model, the parameters of the

model are updated at some frequency using information from the measured vari-

ables.

Based on Bayes’ theorem, a number of approaches called particle filters or se-

quential Monte Carlo techniques are used to estimate values of the uncertain pa-

rameters. This chapter introduces and compares two recursive filters, Markov chain

Monte Carlo (MCMC) filter and the ensemble Kalman filter (EnKF) for uncertain

parameters updating. The contents of this chapter are excerpted from [38, 66].

One of the reasons that Bayesian methods are so attractive to use in uncertain

parameter updating is that they can incorporate prior knowledge through the as-

sumed known information, which is then combined with actual observed data to

update and eventually converge to a final and accurate estimation of the parameter

values.

MCMC methods are a class of algorithms for sampling from a proposed param-

55


eter probability distribution based on constructing a Markov chain for each parame-

ter that has the desired distribution as its stationary distribution. The final variables’

stationary distributions give the estimates of the uncertain parameters. The EnKF

is a Monte Carlo implementation of the Bayesian update problem, basically using

an ensemble of model parameter values to evaluate the necessary statistics.

4.1 Markov Chain Monte Carlo

The Markov chain Monte Carlo (MCMC) methods generate parameter values

from a constructed Markov chain, which converges to a stationary distribution. Al-

though there are many different MCMC algorithms, the general steps are as follows.

1. At iteration i = 0; arbitrarily choose an initial parameter value Θ = Θ0

2. (a) Generate a candidate value Θ∗ for Θ from a proposed distribution de-

pending on the current value Θi.

(b) Compute an acceptance probability α that depends on Θ∗, Θi, the pro-

posed distribution, the model, and the observed data.

(c) Accept Θi+1 = Θ∗ with probability α; otherwise, Θi+1 = Θi.

3. Increment i and repeat step 2 until a stationary distribution of the parameter

is achieved.

Marshall et al discussed the study of MCMC methods for conceptual rainfall-

runoff modeling [30]. Two MCMC algorithms were compared: a conventional

56


Metropolis-Hastings algorithm and an adaptive Metropolis (AM) algorithm. The

study showed that the AM algorithm was superior in many respects and can offer a

relatively simple basis for assessing parameter uncertainty.

4.1.1 Adaptive Metropolis Algorithm

The AM algorithm can generate all the uncertain model parameter values in a

single iteration [30].

1. At iteration i = 0 choose an arbitrary variable vector Θ = Θ0. Candidate

values of this vector, Θ∗, are generated from a proposed probability density

function according to the current values Θi of Θ.

2. Compute the acceptance probability, α that is a function of Θi, Θ∗, the model

and observed data. If the candidate value Θ∗ is accepted with acceptance

probability

α = min

1,P(y|Θ∗)P(Θ∗)P(y|Θi)P(Θi)

(4.1)

then Θi+1 = Θ∗; otherwise Θi+1 = Θi. Here P(Θ) is the prior distribution of

Θ and P(y|Θ) is the likelihood function of the observed data y with model

M,

P(y|Θ) = (2πσ2ε)−N/2exp

−∑N

j=1[y j − M(Θ)]2

2σ2ε

(4.2)

N is the number of data points, M(Θ) is the in silico model outputs, and Θ

57


is the vector of uncertain model parameters. The error term is ε = y − M(Θ)

and σ2ε is its variance.

The proposed distribution of the parameters is a multi-normal distribution that has

mean values at the current point, Θi, and a covariance matrix [30]. The covariance

matrix, Ci, has a fixed value C0 for the first i0 iterations and is updated according to

the following rule,

Ci =

C0 i ≤ i0

sdCov(Θ0, · · · ,Θi−1) i > i0

(4.3)

where sd is a scaling parameter that depends on the dimensionality, d, of Θ. A large

value of i0 will result in a slower adaptation [23]. Thus, the size of i0 reflects our

trust in the initial covariance C0. The initial covariance C0 is an arbitrary strictly

positive-definite matrix chosen according to the best prior knowledge.

The empirical covariance of Θ1, · · · ,Θk ∈ Rd, is given by,

Cov(Θ1, · · · ,Θk) =1

k − 1

k∑i=1

Θi(Θi)′ − kΘk(Θk)′ (4.4)

where

Θk =1k

k∑i=1

Θi

58


and the elements Θi ∈ Rd are column vectors. For i ≥ i0 + 1, the covariance Ci

satisfies the recursion formula:

Covi+1 =i − 1

iCovi +

sd

i

[iΘi−1(Θi−1)′ − (i − 1)Θi(Θi)′ +Θi(Θi)′

](4.5)

When the dimension d (the number of uncertain parameters) is large, the prior

distribution will converge at a slower rate to the stationary distribution. The work

of [67] has shown that a value of sd = (2.38)2/d will yield asymptotic optimality of

accepting rate for d ≤ 6.

In the AM algorithm, the covariance of the first i0 iterations is unchanged, but

thereafter the covariance is updated according to Equation (4.3). The proposed val-

ues of the uncertain parameters are generated from Equation (4.6) using the ith pa-

rameter values as the mean. The initial number of iterations in the MCMC method

can be selected to be large, however, the tradeoff is a high computing burden.

As an example, suppose that Θ is a vector with two elements Θ = [Θ1 Θ2]′.

The following function f is the probability density function of uncertain parameters.

f (Θ1,Θ2) =1

2πσ1σ2

√1 − ρ2

×

exp

−(Θ1 − µ1)2

σ21

−2ρ(Θ1 − µ1)(Θ2 − µ2)

σ1σ2+

(Θ2 − µ2)2

σ22

2(1 − ρ2)

(4.6)

59


where µi, σi, i = 1, 2 are the mean and standard deviation of Θi and ρ is the corre-

lation coefficient of Θi j. The covariance matrix for this bivariate normal is given

by

Cov(Θ1,Θ2) =

σ11 σ12

σ21 σ22

=σ2

1 ρσ1σ2

ρσ2σ1 σ22

(4.7)

4.2 Ensemble Kalman Filter

The EnKF is a recursive filter. When compared to the standard Kalman filter, the

notable difference is that the covariance matrix is replaced by the sample covariance

computed from the ensemble data. An advantage of the EnKF method is that it can

do updating in very nonlinear systems. The EnKF is a Monte Carlo implementation

of the Bayesian update problem, basically using an ensemble of model parameters

to evaluate the necessary statistics. This method consists of a forecast step and a

assimilation step in which the state variables and uncertain parameters are corrected

based on the current observations. The assimilation step has been described as the

step that attempts to update the uncertainty based on the measurements.

The EnKF is developed by sequentially running a forecast step followed by an

assimilation step. The inputs to the forecast (y fk+1) step are the results obtained

from the assimilation (yak ) step that are updates of the uncertain parameters after

the inclusion of the current set of observed data. The forecast step is the simulator

advanced in time.

60


4.2.1 Forward Step

The EnKF is initialized by generating an initial ensemble described by,

ya0,i = y

a0 + e0,i i = 1, . . . ,Ne (4.8)

where Ne is the total number of members in the ensemble; ya0 is the initial mean of

the ensemble members; and e0,i is the error.

The state vector yk,i consists of two parts: all the variables Θ that are uncertain

and the in silico data denoted by d,

yk,i =

s

−

d

k,i

= [Θ1,Θ2, · · · ,ΘD, d1, d2, · · · ]Tk,i (4.9)

where D is the number of uncertain parameters. The index i means the ith ensemble

member; and d1, d2, · · · are the individual datum in the observation vector d.

The forecast step includes execution of the process model and providing an

expression for the uncertainty in the model output. In the forecast step, the model

is executed from time k − 1 to k when the next observation data are available,

y fk,i = f

(ya

k−1,i

)i = 1, . . . ,Ne, k = 1, . . . ,N (4.10)

61


Each member of the ensemble is used in one execution of the process model.

The relationship between the observed and predicted state vectors is given by,

dobs,k,i =Hkytrue + εk,i i = 1, . . . ,Ne, k = 1, . . . ,N (4.11)

where εi is assumed to be Gaussian distributed with zero mean and covariance

Qd,k; Hk ∈ RNd,k×Ny,k is an operator that relates the actual state to the theoretical

observation [36]; Nd,k is the number of observations; and Ny,k is the number of

variables in the state vector at time k. Since the in silico data d are a part of y (see

Equation (4.9)), the elements in Hk can be assigned values of 0 or 1, thus Hk can

be arranged as

Hk =

[0 | I

](4.12)

where 0 is an Nd,k × (Ny,k − Nd,k) matrix of zeros, and I is a Nd,k × Nd,k identity

matrix.

4.2.2 Assimilation Step

The assimilation step incorporates y fk . The state vector y(k) can be updated

based on a difference between the observed data and the in silico data. The weight-

ing matrix at time step k is the Kalman gain,Kk,

yak,i = y

fk,i +Kk

(dobs,k,i −Hky

fk,i

)k = 1, . . . ,N i = 1, . . . ,Ne (4.13)

62


Let Y fk ∈ RNy,k×Ne denote the ensemble of forecast state vectors at time k, and

Y fk ∈ RNy,k×1 as the mean of the state variables calculated from the current ensemble

members,

yk,i =(yk,1,yk,2, · · · ,yk,Ne

)(4.14)

The covariance matrix for the state variables at any step k is estimated from the

forecast ensemble,

P fk =

1Ne − 1

(Y f

k − Yf

k

) (Y f

k − Yf

k

)Tk = 1, . . . ,N (4.15)

where the covariance between mth and the nth variables (m, n = 1, . . . ,Ny and i =

1, . . . ,Ne), pm,n ∈ Pf

k , is obtained from

pm,n =1

Ne − 1

Ne∑i=1

(y f

m,i − yfm

) (y f

n,i − yfn

)(4.16)

Here, ym and yn are the means of the mth and nth variables, respectively calculated

from the ensemble members of the state vector. The Kalman gain matrix is a func-

tion of the covariances,

Kk = Pf

k HTk

(HkP

fk H

Tk +Qd,k

)−1(4.17)

63


4.3 Example

An example, a nonlinear logistic map equation, is presented to demonstrate

these concepts. MCMC and EnKF are used to update the state according to the

measurements. The logistic equation is Equation (4.18) [68]

xk+1 = axk(1 − xk)

yk+1 = xk+1

(4.18)

where a is a parameter, 1 ≤ a ≤ 4; 0 < x < 1 is the state variable and y is the output

that can be measured. When a = 3.5, the outputs of the logistic map equation are

four distinct values 0.875, 0.8269, 0.5009, 0.3828 as shown in Figure 4.1. Figures

4.2 and 4.3 show the estimated state based on the MCMC and EnKF methods. The

initial guess values of x for both MCMC and EnKF are the same. For MCMC, 1000

points are used to form a Markov chain and for EnKF 100 elements are used as an

ensemble. In 100 steps, both MCMC and EnKF show satisfactory state updating

and estimation for the nonlinear logistic map.

4.4 Summary

In this chapter, two updating methods for estimating the uncertain parameters in

a nonlinear model are presented. A simple example is employed to demonstrate the

updating performance of MCMC and EnKF. In the next chapter, the Markov chain

64


0 10 20 30 40 50

0.4

0.5

0.6

0.7

0.8

0.9

1

Step k

Out

puts

from

logi

stic

equ

atio

n

Figure 4.1: Outputs of the logistic map.

0 20 40 60 80 100

0.4

0.5

0.6

0.7

0.8

0.9

1

Step k

MC

MC

sta

te e

stim

atio

n

Figure 4.2: State estimation with MCMC.

65


0 20 40 60 80 100

0.4

0.5

0.6

0.7

0.8

0.9

1

Step k

EnK

F st

ate

estim

atio

n

Figure 4.3: State estimation with EnKF.

Monte Carlo (MCMC) and the ensemble Kalman filter (EnKF) methods will be

demonstrated and compared when there are a large number of uncertain parameters

to be updated.

66


Nomenclature

AM adaptive Metropolis



C covariance of Markov chain H matrix operatorI identity matrix K Kalman gain matrixN number of observations in MCMC Nd number of observationsNe number of ensemble members Ng number of grid blocksNy number of state vector elements P state variables covarianceP likelihood function Qd measurement covarianceT transpose Y ensemble of state vectorsd model output data dobs measured data in EnKFe error term k time steps model parameters sd scaling parametery measured data in MCMC ya assimilated estimates of the statesy f forecast of the states ε measurement errorΘ uncertain parameters vector σ standard deviation

67


Chapter 5

Real-time Model-based Optimization with Parameter Up-

dating

In this chapter, parameter updating is embedded in a real-time model-based

optimization framework. A five-spot pattern oil reservoir system is introduced to

demonstrate and compare the Markov chain Monte Carlo (MCMC) and ensemble

Kalman filter (EnKF) updating methods. The contents of this chapter are excerpted

from [38, 66].

5.1 Uncertain Parameter Updating in a Model-Based Optimization

Framework

Figure 5.1 shows a framework that combines model-based optimization with

model parameter updating.

At each step, the uncertain parameters in the first-principles model are updated

using the measurements from the nonlinear system and a new reduced-order model

(ROM) is identified from the updated first-principles model.

5.2 Updating in a Reservoir Management Framework

Figure 5.2 is a simple schematic of a reservoir.

68


Nonlinear System

d

u

ym

First Principles Model(Θ)

Optimizer

Low-orderModel

MCMCy(Θ)

e

++

+-

y*

Nonlinear System

d

u

ym

Optimizer

Reduced OrderModel

Update

y(Θ)

e

++

+-

y*

First Principles Model(Θ)

Figure 5.1: Model-based optimization framework with parameter uncertainty up-dating. y∗: set-points; u: outputs from the optimizer; d: disturbances; ym: mea-surements; y(Θ): model outputs; Θ: model uncertain parameters; e: errors be-tween measurements and model outputs.

Injection well 2

Injection well 4

Injection well 1

Injection well 3

Production well

I II

III IV

Figure 5.2: Schematic of a two-dimensional reservoir and wells. ↓: water injectionwell; ↑: oil production well.

5.2.1 Reservoir

Many different patterns can be selected to represent the reservoir. Here, one of

the most well-known patterns, a five-spot pattern is chosen [69]. The water injection

wells are located at the four corners of the reservoir and the oil well is located in

the middle of the reservoir. The reservoir covers an area of 630×630 ft2 and has a

69


thickness of 30 ft. This region is modeled by a 9×9×1 horizontal two-dimensional

grid blocks. The fluid system consists of two phases, oil and water, with 0.1 connate

water saturation and 0.3 residual oil saturation.

A first-principles model for this two-phase reservoir with immiscible porous

media flow and isotropic permeability can be described by,

∇

(krokµoBo

∇po

)=∂

∂t

(φS o

Bo

)+

qo

V

∇

(krwkµwBw

∇pw

)=∂

∂t

(φS w

Bw

)+

qw

V

S o + S w = 1

po − pw = Pc(S w)

(5.1)

where ∇ is the gradient operator; the subscripts o and w represent oil and water,

respectively; and S o and S w are oil and water saturation; Pc is the capillary pressure

which is the force to squeeze oil droplets through porous media. It works against

the interface tension between oil and water phases. The definition of the variables

and parameters can be found in Table 5.1.

The first-principles model is a system of nonlinear partial differential equations

that is not trivial to solve numerically. However, there are commercial reservoir

simulation software that are available and may provide a solution that represents

the temporal and spatial behavior of the system. The solution provided by the com-

mercial software also can be used as in silico data to supplement historical data so

70


Table 5.1: Definition of Model Variables and Parameters.

Variable Definition Unit

k absolute permeability miliDarcykr relative permeability %µ viscosity centipoiseB formation volume factor RB/STB (real barrels/standard barrels)p pressure psiPc capillary pressure psiφ porosity %qo oil production rate STBD (standard barrels/day)qw water injection rate STBDS fluid phase saturation %V reservoir volume ft3

t time day

that an input-output model of the reservoir can be identified. Such models are usu-

ally preferred in real-time model-based optimization since they are computationally

less burdensome. In this work, the reservoir software ECLIPSE (Schlumberger Co,

Houston, TX), version 2009.1, is used to provide in silico data to identify a ROM

for the purposes of optimal management of an oil producing reservoir.

In order to solve a reservoir model, the values of parameters in the model should

be known. As described before, the values of the parameters may contain uncer-

tainty. The especially serious problem is when there is no consensus on values for

some of the reservoir parameters because of the changing geographic conditions.

Porosity and permeability are two such parameters that suffer from this problem.

Porosity is a fraction of the void volume over the total volume of a material. How

much oil or water that is contained in a reservoir totally depends on the porosity.

71


Permeability is a measure of the ability of a porous material to allow fluids to pass

through it. Thus, how much oil can be pushed out of the reservoir mainly depends

on the material’s permeability. We can conclude that porosity and permeability

are two of the key parameters whose values are essential to providing an accurate

simulation of oil production from a reservoir.

Because of the changing geographic structure, a single effective value of either

porosity or permeability ought not to be assumed. In this work, the reservoir is

divided into several grids and each grid is assumed to have a single effective value of

porosity and permeability. As the number of grids increase so too does the accuracy

of the simulation but at the expense of a huge computational burden.

The values of porosity and permeability are measured by experimenting on the

rock cores which can be obtained by drilling a reservoir. The common way to mea-

sure the porosity is to use imbibition method. The weight of vacuumed rock sample

is measured at beginning. The vacuumed rock sample is immersed in a liquid envi-

ronment and then weighted again. The weight difference between the rock sample

after soaking up the liquid and the vacuumed rock sample is then calculated. With

the density of the liquid, the volume of the void spaces in the rock sample can be

computed. The porosity is the ratio between the volume of the void spaces and

the volume of the rock sample. Permeability is typically determined by applying

Darcy’s law to experiment on a rock sample. Darcy’s law describes the flow of a

fluid through a porous medium. Darcy’s law (see Equation (5.2)) is a proportional

72


relationship between the discharge rate of a fluid through a porous medium, the fluid

viscosity and the pressure difference over a given distance of the porous medium.

Q =−kAµ

∆PL

(5.2)

To measure the permeability k of a rock sample, a liquid is forced to flow through

the sample. In Equation (5.2) Q is the discharge rate of the liquid; µ is the liquid

viscosity; A is the cross-sectional area of the rock sample; ∆P is the pressure drop,

which is the difference between the exit liquid pressure from the rock sample and

its input liquid pressure; and L is the length of the rock sample. The permeability k

of the rock sample can be calculated with the measurements of Q, A, ∆P, L and the

liquid viscosity according to Equation (5.2) .

The rock samples are obtained by drilling wells in a reservoir. It is impossible

and expensive to drill everywhere in a reservoir to obtain all the samples that repre-

sents the entire geographic structure. The porosity and permeability at the position

where there is a well can be measured. But their values everywhere else are not

known. We can conclude that in a reservoir, the values of porosity and permeability

have a large amount of uncertainty. Then, in an in silico model of the reservoir that

is constructed as a grid, in almost every grid the values of porosity and permeabil-

ity are not known with any certainty. It is reasonable that these model parameters

should be updated whenever measurements from the reservoir are known to main-

73


tain model accuracy.

Assume that the true porosity and permeability distributions are in the intervals

[0.04, 0.33] and [6, 600] miliDarcy, respectively [70] (see Figures 5.7 and 5.8). The

outputs of the reservoir model are generated by execution of the ECLIPSE model

with the true parameters and the addition of noise to the oil and water production

values, water cut (the ration of the production of water over the total production of

oil and water), and bottom hole pressures (BHPs, BHP is the pressure at the bottom

of the hole of a well). The measurement uncertainties are presented in Table 5.2.

Table 5.2: Measurement uncertainties. [3]

Quantity ±σ

Oil production 5%

Water production 5%

Bottom hole pressures 1 bar

5.2.2 Basic Optimization Problem

Maximizing the potential production capability of an oil or gas reservoir is the

aim of the real-time model-based reservoir management framework. This frame-

work, referred to as a closed-loop management approach, is modified from Figure

5.1 to include parameter updating and illustrated in Figure 5.3.

The ECLIPSE commercial software provides a fundamental model of the reser-

voir that when executed with feasible initial conditions and known reservoir size

74


Nonlinear System(reservoir, wells)

OptimalController

D(n)u(n)

dobs(n)

EnKFARMALow-order Model

NonlinearEclipse Model yf(n)

ya(n)

Nonlinear System(reservoir, wells)

OptimalController

D(n)u(n)

dobs(n)

UpdateARMALow-order Model

NonlinearEclipse Model Model

Outputs

UpdatedParameters

Figure 5.3: Schematic of the closed-loop management framework [5]. n: controlstep; u: outputs from optimal controller;D: disturbances; dobs: observations.

will output oil production data, the BHPs and the oil and saturation. For reservoirs

that rely on water addition to force the oil out of the ground, the flow rates of the

water are regulated not only to maximize oil production but also to minimize the

water production rate, because the production of water costs money as well. By

varying the water addition rates, the oil and water production rates can be estimated

by the ECLISPE model until the net present value (NPV) is equal to 0. The in sil-

ico model outputs can be used to identify an input-output ROM of the production.

Here, we employ a ROM in the form of an autoregressive moving average (ARMA)

type model,

y(n + 1) = A(n)y(n) +B(n)u(n − kd) (5.3)

where y is a vector that includes the oil and water produced; u is a vector of water

injection rates; A and B are properly sized coefficient matrices for y and u, re-

spectively; n is the nth optimization step; and kd is the time delay. Disturbances are

75


not considered in this study.

To generate the ARMA model, at each optimization step, different sets of water

well injection rates (u = u(0), . . . ,u(n)) are used by the ECLIPSE model to

generate estimates of the oil and water production (y = y(0), . . . ,y(n)).

Maximizing the NPV is the objective of the optimization,

L = maxN∑

n=1

Ln (5.4)

where N is the total number of optimization steps. The NPV for each step is defined

as [71]

Ln =∆tn

(1 + a)tn

Nprod∑j=1

(roqn

o, j − rwqnw, j

) (5.5)

where ∆t is the length of the optimization step; Nprod is the number of producers;

and ro and rw are respectively, the benefit and cost coefficients of oil and water

production. The denominator of the first term represents the effect of discounting

where a is the discounting factor and tn is the time expressed at the nth optimization

step. Because the value of the discounting factor is constant over a fixed interval

of time, it does not affect either the water or oil production rates. In this work, it

is assumed that the effect of discounting is neglected; i.e., that a = 0 according to

[71, 72]. Maximizing NPV is equivalent to maximizing cumulative oil production

(qo) and minimizing cumulative water production (qw) by adjusting the injected

76


water flow rates to each well. The controller outputs are the optimum injected

water flow rates that maximize NPV.

5.2.3 Markov Chain Monte Carlo Updating

As shown in Figure 5.4, the reservoir is divided into three sections (labeled as

I, II, or III), with each section containing three 3×3 connected grids. As mentioned

before in Chapter 4, scaling parameter sd is dependent on the dimensionality and

valid at 6 dimensions or lower. Thus, within each section, the selected uncertain

parameters, porosity and permeability, are assumed to have the same distributions

for a total of 6 uncertain parameters.

With known porosity and permeability data obtained from the analysis of rock

cores, initial distributions of these property values are estimated and assigned to

the reservoir. By comparing the results with known historical data the most ap-

propriate combinations of three grids (see Figure 5.4, which is divided in a 3×3

grid) are selected. Porosity is usually found to have a normal distribution while

permeability is characterized as having a log-normal distribution [73]. As shown

in Figure 5.5, the assigned prior distributions of porosity are: φI −N (0.22, 0.05),

φII −N (0.17, 0.04), φIII −N (0.13, 0.03); and the assigned prior distributions of

permeability are: kI −Nlog(6.17, 0.25), kII −Nlog(5.02, 0.24), kIII −Nlog(3.58, 0.23).

Updates of the parameter distributions require a comparison between the ob-

served data (oil and water production, water cut and BHPs) and the ECLIPSE model

77


III

II

I

Injection well 1 Injection well 3

Injection well 2 Injection well 4

Production well

Figure 5.4: Schematic of a two-dimensional reservoir and wells. ↓: water injectionwell; ↑: oil production well.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

Porosity I

Pro

babi

lity

dens

ity

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

Porosity II0 0.1 0.2 0.3

0

0.1

0.2

0.3

0.4

Porosity III

100 550 1000 14500

0.1

0.2

0.3

0.4

Permeability I

Pro

babi

lity

dens

ity

0 100 200 300 4000

0.1

0.2

0.3

0.4

Permeability II10 26 42 58 74 900

0.1

0.2

0.3

0.4

Permeability III

Figure 5.5: Prior distributions of porosity and permeability.

78


predictions. This means that a very large number of different combinations of the

samples of the uncertain parameters are input parameters in the ECLIPSE model. In

other words, the ECLIPSE model needs to be executed thousands of times, which

translates to a huge computational burden [66]. To overcome this obstacle, PLS

regression described in Chapter 3 can be applied to find the relationships between

the uncertain parameters and the ECLIPSE model outputs. Thus, execution of the

ECLIPSE reservoir model is supplanted because the outputs can be calculated di-

rectly from the values of uncertain parameters by the PLS regression model. The

execution time to generate the PLS results is at least 10 times less than one execu-

tion of the ECLIPSE model. Thus, the MCMC combined with PLS can improve

the updating efficiency.

Figure 5.6 shows the updated distributions of ki, φi, i = 1, 2, 3. The updated

distributions of φi are: φI−N (0.25, 0.027), φII−N (0.2, 0.018), φIII−N (0.15, 0.015);

and the updated distributions of ki are: kI−Nlog(6.38, 0.088), kII−Nlog(5.23, 0.082),

kIII −Nlog(3.78, 0.086). It is noted that the updated distributions have smaller stan-

dard deviations than those of the prior distributions. This is not unexpected since

with the MCMC technique, the distributions are guaranteed to attain a stationary

distribution.

79


5.2.4 Ensemble Kalman Filter Updating

The reservoir state vector consists of all the reservoir variables that are uncer-

tain. There are two parts to the reservoir state vector yk,i: the model parameters s

(including porosity, permeability, pressure and phase saturation) and the in silico

data d (generated by the simulator including oil and water production, water cut

and BHPs),

yk,i =

s

−

d

k,i

=[φ1, · · · , φNg , ln(k1), · · · , ln(kNg), p1, · · · , pNg , S w1, · · · , S wNg , d1, d2, · · ·

]T

k,i

(5.6)

where i is ith ensemble member; Ng is the number of grids in a reservoir; d1, d2, · · ·

are individual datum in the observation vector d; and T denotes transpose of a vec-

tor.

In this work, the ensemble set is chosen to be of size one hundred [26]. Thus,

the assignment of the initial estimation of the porosity and permeability is much

finer with EnKF than that with MCMC. The generation of the error term for the

permeability parameter is based on a spatially correlated Gaussian model that as-

sumes that the permeability in grid block (i1, j1) is correlated with the permeability

80


in grid block (i2, j2) [3],

exp[−

( i1 − i2

`

)2

−

( j1 − j2

`

)2](5.7)

The correlation length ` is a normally distributed stochastic variable. A correla-

tion matrix C is computed from Equation (5.7). The covariance matrices used for

generating noise for the permeability ensemble is computed as σ2C where σ is the

standard deviation in the permeability of each grid block.

Based on known values of porosity and permeability obtained from the well’s

rock cores, the initial ensemble values are generated by linearizing the critical pa-

rameter values along the length and width of the reservoir [35]. The measurements

are assimilated everyday for the first 10 days, then every 10 days for 50 days, and

once every month for 11 months. This schedule is chosen according to the work of

[71, 74] but this should not imply that the schedule itself could not be an optimiza-

tion parameter. Results with control steps of 25, 50, and 70 days also were carried

out but not shown here. It was found that the oil and water production differences

are within 0.4% when compared to a 30-day control step.

For comparison, the mean values of porosity and permeability of the initial en-

semble is used as reference to compare the updated results of parameters. Figures

5.7 and 5.8 show k, φ for the true and initial cases and also the updated cases after

30, 180, 240, and 390 days.

81


Figures 5.9 and 5.10 show the uncertainties associated with k, φ after 30, 120,

180, 240, 300, and 390 days. The uncertainty associated with the grid blocks that

are closer to the wells is smaller when compared to those that are farther away. Ad-

ditionally, the overall uncertainty decreases as the number of days increases since

more measurements are assimilated.

In Figures 5.11 and 5.12, the forecasts of the measurements for daily oil and

water production, cumulative oil and water production, water cut and BHPs are

shown based on 30, 120, 180, 240, 300, and 390 days. As expected, as more and

more data are assimilated the quality of the forecasts is improved asymptotically.

The mean values of the ensemble can be used as the updated values of k, φ

at each time step. These updated values are inputs to the in silico model to obtain

more data to identify the input-output model (see Equation (5.3)).

5.2.5 Optimal Oil Production Results

In the reservoir, water is injected in the wells, which affects the reservoir’s pres-

sure to displace oil from the reservoir and out the production well. In this work,

800 STBD (standard barrels/day) of water are specified for the well with a min-

imum BHPs of 4000 psia. Additionally, there are minimum and maximum con-

straints for each injection well, in this case 160 and 240 STBD, to ensure that the

maximum BHPs are within 4350 psia. These pressures are chosen to maintain a

pressure balance between the underground and ground surfaces.

82


The optimized solution is compared to a non-optimized case. The oil price, ro,

is set at 110$/bbl according to the crude oil price at the end of April, 2011 and water

production costs, rw, are set at 12$/bbl according to [72]. To guarantee that NPV

> 0, the water cut target, qw/(qo + qw) = 0.9 so that the equality roqo − rwqw = 0 is

satisfied.

In this work, the optimizer has a 30 days basis to make a decision. At each

optimization step, the uncertain parameters are updated and an ARMA model is

identified as the ROM in the model-based optimization framework (cf Equation

(5.3)) until the target water cut value of 0.9 is reached. To maximize the NPV in

Equation (5.4), water production (qw) should be minimized while simultaneously

maximizing oil production (qo).

Figure 5.13 shows the topology of the injected water when the water cut equals

0.9 for the non-optimized and optimized cases (with the EnKF and MCMC meth-

ods updating the uncertain parameters respectively. With the updated values of the

uncertain parameters, the ROM is re-identified and used by the optimizer at the next

step. Clearly, the water has saturated the reservoir.

Comparing the graphs in Figure 5.13, the water sweep as determined by the op-

timal closed-loop controller is an improvement over the non-optimized case. More

oil is produced by adjusting the amount of water added in the optimized case; and

at the same time the water saturation values are higher than in the non-optimized

case.

83


The top panel of Figure 5.14 shows the cumulative oil and water production

rates of the non-optimized and optimized cases at each step until the water cut is

0.9. For the optimized case with EnKF, the final cumulative oil production increases

7.1% and the final cumulative water production decreases by 10.5% when compared

to the non-optimized case. For the optimized case with MCMC, the final cumulative

oil production increases 5.3% and the final cumulative water production decreases

by 3.4%.

Since the oil production has increased and water production has decreased, the

cumulative NPVs (with EnKF and MCMC updating) also increases significantly

(bottom panel of Figure 5.14), 9.0% (EnKF) and 8.2% (MCMC) more than the

final NPV of the non-optimized case.

Although the geological properties are different for different reservoirs, it is still

meaningful to analyze the effects of permeability and porosity on the adjustment

of the injection water rate with this five-spot pattern reservoir. The reservoir is

divided into four equal size parts with the five wells as shown in Figure 5.2. The

following analysis is based on the EnKF results since the EnKF provides better

updating performance.

The average permeability and porosity in the four parts of reservoir in each

optimization steps are calculated. Figure 5.15 shows a graph of the changes in the

optimal injected water flow rate in the first well as a function of the updating steps

and the average permeability and porosity values at each step. Figures 5.16 to 5.18

84


provide a similar graph for the second, third and fourth wells. The z-axis in the

graph is the proportion of the water flow rate in the corresponding well to all the

water flow rates in four injection wells.

Table 5.3 shows the ranges of porosity, permeability and water injection ratio of

the four parts of the reservoir.

Table 5.3: Porosity, permeability and water injection ratio of four parts of reservoir.

Porosity (%) Permeability (miliDarcy) Water injection ratio (%)

I 0.21–0.28 280–350 0.27–0.3

II 0.12–0.16 190-245 0.2–0.25

III 0.21–0.28 209-248 0.27-0.3

IV 0.06–0.10 22-32 0.2

From the graphs and Table 5.3, it can be concluded that the optimal injected wa-

ter flow rates have a strong relationship to the values of permeability and porosity.

When the permeability and porosity are larger, more water can be added to force

the oil out of the reservoir. This is reasonable behavior since a larger porosity value

indicates there is more oil present in the reservoir and a larger permeability value

indicates easier flowability.

The ranges of the porosity and permeability in the fourth part of the reservoir

are smallest when compared to the others, thus the injected water flow rate hovers

around the injected water flow rate lower constraint. In this case, it also can be con-

cluded that porosity has a larger effect on the calculated values of the optimal water

85


flow rates than permeability. The first and third parts of the reservoir have the same

range of porosity but different ranges of permeability. However, the optimal flow

rates of the first and third wells are similar to each other. Although the permeability

range in the second part is similar to that of the third part, the optimal water flow

rates in the second well are different from the third well because of the difference

in their porosity ranges.

5.3 Summary

An optimal management approach combined with model parameter updating

was presented. Two updating filtering methods, Markov chain Monte Carlo (MCMC)

and ensemble Kalman filter (EnKF) are designed and compared. Although the

MCMC combined with partial least squares (PLS) can improve the updating ef-

ficiency, the main disadvantage of MCMC is the limitation on the number of un-

certain parameters to be updated. In contrast, the EnKF can update a large number

of uncertain parameters. To make the closed-loop optimization framework com-

putationally manageable while remaining accurate, a reduced-order computational

model (ARMA, autoregressive moving average) is identified that relates the in-

jected water flow rate to oil production. By optimizing the injection water flow

rate, increased oil production and decreased water production resulted in a increase

(9.0% EnKF, 8.2% MCMC) in the final cumulative net present value. An analysis

of the results revealed that the values of permeability and porosity have important

86


effects on the optimal adjustment of the injected water flow rates. Thus, updating

the values of these two uncertain parameters played an important role in optimal

management of the reservoir.

87


0.1 0.18 0.26 0.340

0.1

0.2

0.3

0.4

Porosity I

Pro

babi

lity

dens

ity

0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

Porosity II0.07 0.11 0.15 0.190

0.1

0.2

0.3

0.4

Porosity III

400 550 700 8500

0.1

0.2

0.3

0.4

Permeability I

Pro

babi

lity

dens

ity

120 170 220 2700

0.1

0.2

0.3

0.4

Permeability II30 42 54 660

0.1

0.2

0.3

0.4

Permeability III

Figure 5.6: Posterior distributions of porosity and permeability.

88


True

50100

150

200

250

300

350

400

450

500

550

Initi

al

30 d

ays

120

days

180

days

240

days

300

days

390

days

Figu

re5.

7:In

itial

,tru

ean

dup

date

dpe

rmea

bilit

yof

rese

rvoi

r.

89


True

00.05

0.1

0.15

0.2

0.25

0.3

Initi

al

30 d

ays

120

days

180

days

240

days

300

days

390

days

Figu

re5.

8:In

itial

,tru

ean

dup

date

dpo

rosi

tyof

rese

rvoi

r.

90


30 days

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14120 days

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

180 days

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14240 days

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

300 days

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14390 days

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Figure 5.9: Uncertainty in the porosity parameter after 30, 120, 180, 240, 300, and390 days. 91


30 days

0

1

2

3

4

5

6120 days

0

1

2

3

4

5

6

180 days

0

1

2

3

4

5

6240 days

0

1

2

3

4

5

6

300 days

0

1

2

3

4

5

6390 days

0

1

2

3

4

5

6

Figure 5.10: Uncertainty in the permeability parameter after 30, 120, 180, 240, 300,and 390 days. 92


0 50 100 150 200 250 300 350 4000

100

200

300

400

500

600

700

800

900

Days

Dai

ly o

il pr

oduc

tion

STB

D

0 50 100 150 200 250 300 350 4000

100

200

300

400

500

600

700

800

Days

Dai

ly w

ater

pro

duct

ion

STB

D

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Days

Cum

ulat

ive

oil p

rodu

ctio

n M

STB

0 50 100 150 200 250 300 350 4000

50

100

150

200

250

Days

Cum

ulat

ive

wat

er p

rodu

ctio

n M

STB

0 50 100 150 200 250 300 350 4004100

4250

4400

4550

4700

4850

5000

Days

BH

P o

f wel

l 1 P

sia

0 50 100 150 200 250 300 350 4004100

4250

4400

4550

4700

4850

5000

Days

BH

P o

f wel

l 2 P

sia

Figure 5.11: Forecasts of the measurements after 30, 120, 180, 240, 300, and 390days. ’’: true; ’∗’: initial; ’’: 30 days; ’4’: 180 days; ’+’: 390 days.

93


0 50 100 150 200 250 300 350 4004600

4800

5000

5200

5400

5600

5800

6000

6200

Days

BH

P o

f wel

l 3 P

sia

0 50 100 150 200 250 300 350 4004100

4250

4400

4550

4700

4850

5000

Day3

BH

P o

f wel

l 4 P

sia

0 50 100 150 200 250 300 350 4004050

4150

4250

4350

4450

4550

Days

BH

P o

f wel

l 5 P

sia

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

Days

Wat

er c

ut

Figure 5.12: Forecasts of the measurements after 30, 120, 180, 240, 300, and 390days. ’’: true; ’∗’: initial; ’’: 30 days; ’4’: 180 days; ’+’: 390 days.

94


0.45

0.5

0.55

0.6

0.65

0.45

0.5

0.55

0.6

0.65

0.45

0.5

0.55

0.6

0.65

Figure 5.13: Final water saturation when the water cut is 0.9. Top: non-optimized.Middle: optimized with EnKF. Bottom: optimized with MCMC.

95


0 30 60 90 120 150 180 210 240 270 300 330 360 3900

40

80

120

160

200

Time (days)

Cum

ulat

ive

prod

uctio

n (M

STB

)

30 60 90 120 150 180 210 240 270 300 330 360 3900

2000

4000

6000

8000

10000

12000

14000

Time (days)

Cum

ulat

ive

NP

V (t

hous

and

dolla

rs)

Figure 5.14: Top: Comparison of cumulative production of oil and water. Opti-mized case: EnKF, oil () and water (5) production; MCMC, oil (+) and water(4) production. Non-optimized case: oil () and (∗): water production. Bottom:Cumulative NPV. Optimized: EnKF, blue; MCMC, green. Non-optimized: red.

96


05

1015

280300

320340

3600.1

0.15

0.2

0.25

0.3

StepsPermeability

Dim

ensi

onle

ss w

ater

flow

rate

05

1015

0.2

0.25

0.30.1

0.15

0.2

0.25

0.3

StepsPorosity

Dim

ensi

onle

ss w

ater

flow

rate

Figure 5.15: Optimal dimensionless injected water flow rate in the first well asa function of the updating steps, and the average permeability (top) and porosity(bottom) values.

97


05

1015

180200

220240

2600.1

0.15

0.2

0.25

0.3

StepsPermeability

Dim

ensi

onle

ss w

ater

flow

rate

05

1015

0.12

0.14

0.16

0.180.1

0.15

0.2

0.25

0.3

StepsPorosity

Dim

ensi

onle

ss w

ater

flow

rate

Figure 5.16: Optimal dimensionless injection water flow rate in the second well as afunction of the updating steps and average permeability (top) and porosity (bottom)values.

98


05

1015

200

220

240

2600.1

0.15

0.2

0.25

0.3

StepsPermeability

Dim

ensi

onle

ss w

ater

flow

rate

05

1015

0.2

0.25

0.30.1

0.15

0.2

0.25

0.3

StepsPorosity

Dim

ensi

onle

ss w

ater

flow

rate

Figure 5.17: Optimal dimensionless injection water flow rate in the third well as afunction of the updating steps and average permeability (top) and porosity (bottom)values.

99


05

1015

20

25

30

350.1

0.15

0.2

0.25

0.3

Dim

ensi

onle

ss w

ater

flow

rate

StepsPermeability

05

1015

0.06

0.08

0.1

0.120.1

0.15

0.2

0.25

0.3

Dim

ensi

onle

ss w

ater

flow

rate

StepsPorosity

Figure 5.18: Optimal dimensionless injection water flow rate in the fourth well as afunction of the updating steps and average permeability (top) and porosity (bottom)values.

100


Nomenclature

ARMA autoregressive moving average

BHP bottom hole pressure

EEF empirical eigenfunction



NPV net present value


A coeff matrix of water injection rates B coeff matrix of productionD(n) disturbance N total number of optimization stepsNprod number of produce well

a discounting factor d model outpute error term k absolute permeabilitykd time delay kr relative permeabilityl permeability correlation length n optimization stepsd scaling parameter t timeu water injection rates u(n) input at time step ny oil and water production φ porosity

Nlog log-normal distribution N normal distribution

101


Chapter 6

Robust Estimates of the Uncertain Parameters

As described in Chapter 4, when the parameters in a model are totally unknown,

to obtain the accurate estimates of process behavior from the model, it is necessary

to update the uncertain model parameters from measurements. However, updating

is a time consuming process. Once we have some information about the uncertain

parameter values, it is reasonable to use these values to generate a statistical esti-

mate based on some formal arguments. Traditionally, the mean value of the data

samples is used as a statistical estimate. However, considering the existence of out-

liers, insufficient sample size, and so forth, the statistical mean value may not the

best statistical estimate. In light of this argument, this chapter presents an overview

of the theory of robust statistics to provide a formal basis for the generation of a

statistical estimate. Based on robust statistics, a new theorem about a maximum

likelihood model parameterized by a maximum likelihood estimate of the uncer-

tain parameters is introduced and a formal proof of its existence is provided. The

contents of this chapter are excerpted from [5].

102


6.1 Robust Statistics Estimates

Given an uncertain parameter in a model, without loss of generality, it is fair

to assume that its values are constrained to be within a well-defined feasible range.

It is important to discuss how to choose a value from within this range because a

good estimate of this parameter may have a strong relationship to the real items of

interest, the estimates of the process states (i.e, the model outputs).

There are essentially three types of robust statistic methods to obtain a satisfac-

tory choice of the parameter value from its uncertain data set [39]. Assume X is an

uncertain variable with N possible values xi ∈ X : x1, x2, · · · , xN

1. Maximum likelihood estimate — MLE.

Let xM be an estimate of X. If xM is such that for any arbitrary function F,∑F(xi; xM) is minimum, then xM is the maximum likelihood estimate of X.

To obtain a value of xM, take the derivative of F with respect to xM and let∑ ∂F(xi; xM)∂xM

= 0.

2. Linear combination of order statistics — L-estimate.

L-estimates include the mean, trimmed mean, minimum and maximum val-

ues, and so forth.

3. Rank test estimate — R-estimate

For a variable X with samples xi, i = 1, 2, · · · ,N, if an estimate xR can make

103


the original samples x1, x2, · · · ,N and (2xR−x1), (2xR−x2), · · · , (2xR−xN)

have the same rank, then xR is the rank estimate.

6.1.1 Example

Because samples are a subset of the population, the distribution of a variable is

often not perfect. Assume there is a variable X whose true value is 1.31 and that

its measured values are distributed as a skewed Gaussian distribution as shown in

Figure 6.1.

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

Figure 6.1: Skewed Gaussian distribution.

If the mean value is taken as the estimate, it is 1.64. To determine its MLE

104


value, let F(xi; xM) = |xi − xM |. Thus, xM is the median of the measurements. To

obtain its L-estimate, assume a 20% trimmed mean. Discard 10% of the ordered

samples at the bottom and the top of the samples. By so doing the L-estimate is

1.57. To find the R-estimate, apply the Wilcoxon rank test.

Table 6.1 lists the MLE, L-estimate, R-estimate and the errors between these

estimates and the true value. From this simple example and the data in the table,

we can determine that the MLE value is closest to the true value for this skewed

Gaussian distribution. We also note that the mean value has the largest error. We

may conclude with some confidence that robust statistics can be used to provide a

better estimate.

Table 6.1: Estimates and errors

xM xL xR mean

value 1.40 1.57 1.47 1.64

errors 0.09 0.26 0.16 0.33

6.2 Preamble

Although robust statistics can provide satisfactory choices of the values of the

uncertain parameters, these estimates are dependent on the feasible parameter ranges

and the choice of their distribution. Usually a process model contains more than one

parameter that the model outputs are sensitive to. The propagated combinations of

105


the samples of these parameters through the model must satisfy the constraints of

the state variables.

Definition 6.1. Parameter uncertainty

If a model parameter P j, j = 1, · · · ,m has more than one value: p1j , p

2j , · · · , p

nj .

Then, this set reflects the uncertainty of P j.

Remark 6.1. A parameter is uncertain if the parameter can have more than one

value.

Definition 6.2. Model parametrization,M

Let FM be a space of operators and FM ⊂ FM , Ø. Here FM can be linear or

nonlinear time-varying operators. We define a model based on a parameter set by

M (p) = fM(p)

where fM ∈FM.

Remark 6.2. A model is complete if the model parameters P j, j = 1, · · · ,m are all

assigned values.

Definition 6.3. Evaluation models,M (P )

Let P be a set of evaluation model parameters andM be a model parametriza-

tion. The set of evaluation models is

M (P ) = M (p)

where p are values assigned to P

106


Definition 6.4. State estimates, y

Let P be a set of evaluation model parameters andM be a model parametriza-

tion M (P ). The set of state estimates, y ∈ Rk, is a solution of the evaluation

models, y = M (p), where p are values assigned to P .

Remark 6.3. The set of evaluation model parameters P and the set of sensitive

model parameters Ps is distinguished; Ps ⊆ P . Let Pn be the set of non-sensitive

parameters, Pn ⊂ P . Thus, Ps ∪ Pn = P and

∣∣∣∣∣ ∂y∂Ps

∣∣∣∣∣ ∣∣∣∣∣ ∂y∂Pn

∣∣∣∣∣Remark 6.4. Since Ps ⊆ P , thenM (Ps) ⊆M (P ).

Definition 6.5. Plant measurements, yM

Let yM : yM,`, ` = 1, · · · , nm be a set of plant measurements,

Remark 6.5. The target values y∗M are assumed to be stable equilibrium states of

the plant but constrained

y∗M − δyl ≤ yM ≤ y

∗M + δy

u, δyl ≥ 0, δyu ≥ 0.

In the case of the ideal plant, yP ≡ y∗M.

Remark 6.6. The set of disturbances (measured or unmeasured) is denoted by d.

The true outputs of the plant (unaffected by disturbances) yP is related to the mea-

sured values by, yM = yP + d.

107


Remark 6.7. y = M (p). The outputs of the model should be close to that of the

outputs of the undisturbed plant yP when perturbed by the same inputs,

‖ yP − y ‖2=‖ ε ‖2≤ ε0

Definition 6.6. Set of models with sensitive parameters,M (Ps)

Let Ps be a set of sensitive model parameters andM be a model parameterized

by Ps,

M (Ps) = M (ps)

where ps are the values assigned to Ps

Remark 6.8. The setM (Ps) will be used to determine feasible combinations of Ps

constrained by the range of the plant measurements yM. Because the set Pn has

negligible effects on y, the values assigned to the members of the set Pn are fixed

at their nominal values and are not considered when determining feasible combi-

nations of Ps. For simplicity, in the development that follows, the term ’uncertain

parameters’ means ’sensitive uncertain parameters’.

Definition 6.7. Feasible ranges of ps, [pls,p

us]

Let ps be a set of the values of Ps in the feasible ranges, [pls,p

us], and M be a

model parametrization such that y = M (ps). The feasible ranges satisfy

y∗M − δyl ≤ y ≤ y∗M + δy

u

108


Definition 6.8. Robust estimates of model uncertain parameters, ps

Let [pls,p

us] be the feasible range of the uncertain parameter andM be a model

parametrization. State estimates, y, determined by a model parameterized by the

robust estimates are closer to the states from the actual plant than state estimates

found from a model parameterized by anything other the robust parameter estimates

‖ yP − y ‖2 = ‖ yP − M (Ps) ‖2 = ε

‖ yP − y ‖2 = ‖ yP − M (Ps) ‖2 = ε ≤ ε

Remark 6.9. Robust estimates of the uncertain parameters must be within their

feasible ranges, pls ≤ ps ≤ p

us .

Definition 6.9. Errors, e

The set of errors is the difference between the measured values and the model’s

estimates,

e = yM − y

Remark 6.10. Since disturbance may not be measurable, the error between the

model and the measurement is one means of estimating the disturbance,

‖ d − e ‖2 = εd

Definition 6.10. Model-based controller operator, C(M , e)

Let FC be a non-empty space of operators and FC ⊂ FC. A model-based con-

troller operator is defined as

109


C := FM × e→ FC : (M , e) 7→ fC = C(M , e)

where fC ∈ FC. The absence of a controller is given by, e : fC = 0

Definition 6.11. Model-based controller outputs, U

According to the target values, y∗M, the controller outputs are the solution of

C(M , e) that forces the measured values ym to follow the trajectory of y∗M in finite

time, T .

U = C(M , e) × y∗M : limt→t+T

yM → y∗M

Remark 6.11. The model-based controller outputs, U , can provide compensation

in the face of disturbances,

limt→t+TU × P = yP − e + d = y

∗M + d − e = y

∗M + εd

where P represents the actual plant.

Definition 6.12. Nominal model,MN

The nominal model MN is a model whose parameter values are the designed

nominal values.

Definition 6.13. Maximum likelihood model, M

A maximum likelihood model can provide robust estimates of the disturbances

such that compensation from the maximum likelihood model-based controller makes

110


the errors between the measurements and the targets smaller than any other model-

based design controller.

‖ yP − M ‖2 ≤ ‖ yP −MN ‖2

dn = yM −MN d = yM − M

‖ d − dn ‖2= εd ‖ d − d ‖2= εd ≤ εd

From definition 6.9 and remark 6.10,

en = yM − y = yM −MN = dn

e = yM − y = yM − M = d

Let U and andUN be controller outputs based on a maximum likelihood model-

based controller design and a nominal model-based controller design, respectively

U = C(M , e) × y∗M

UN = C(MN , en) × y∗M

The compensation determined by the nominal model-based controller design

results in

limt→t+T

y∗M − yM = y∗M − UN × P = y

∗M − (yP − en + d) = εd

In contrast, the compensation determined by the maximum likelihood model-

111


based controller design results in

limt→t+T

y∗M − yM = y∗M − U × P = y

∗M − (yP − e + d) = εd ≤ εd

6.3 Mathematical Foundation

Lemma 6.1. The feasible ranges of the uncertain parameters are distribution inde-

pendent.

Proof: Assume ps ∈ [pLs ,p

Us ] and the feasible ranges [pl

s,pus] ⊆ [pL

s ,pUs ]. From

the preamble, [y∗ − δyl,y∗ − δyu]′ = M (pls,p

us). Since the values of pl

s and pus

are distribution independent, it follows that the feasible ranges also are distribution

independent. QED

Lemma 6.2. The distributions of the set of uncertain parameters affect their robust

estimates according to the constraints imposed by the real plant.

Proof: There are two cases.

1. Uniqueness. There is only one combination of parameter values, pos , such that

y =M (pos) ∈ [y∗ − δyl,y∗ + δyu]. In this case, the value of each uncertain

parameter is the robust estimate of the model parametrization that gives the

smallest error.

2. More than one combination of the uncertain parameter values in the feasible

range, pls ≤ ps ≤ p

us , are admissible and satisfy y =M (ps) ∈ [y∗ − δyl,y∗ +

112


δyu].

Different distributions of the uncertain parameter values will give different

values of P

P(a ≤ ps ≤ b) =∫ b

apd f (ps) dps

where pd f is the probability density function of an uncertain parameter.

Different distributions of the uncertain parameters will give different values

of P(ps).

The uncertain values ps ∈ [pls, p

us] of the parameters must satisfy their fea-

sible range (see lemma 6.1). Clearly, each assigned distribution may yield a

different probability value as a potential estimate of the value of the uncertain

parameter. Therefore, the assigned distributions of the uncertain parameters

will affect their robust estimates. QED

Theorem 6.1. A model parameterized with robust estimates of the uncertain pa-

rameters is the maximum likelihood model,M (ps) = M .

Proof: Assume y =M (ps) and y =M (ps).

From definition 6.8,

‖ yP − y ‖2 = ‖ yP −M (Ps) ‖2 = ε

‖ yP − y ‖2 = ‖ yP −M (Ps) ‖2 = ε ≤ ε

113


Estimates of the disturbances obtained from MN = M (ps) are given by (see

remark 6.6),

‖ d − dN ‖2 =‖ d − (yM − y) ‖2

=‖ d − yM + yP − ε ‖2

‖ d − dN ‖2 =‖ d − d − ε ‖2= ε

For the nominal model parametrization,M (ps),

limt→t+T

y∗M − yM = ε

Estimates of disturbances obtained fromM (ps),

‖ d − d ‖2 =‖ d − (yM − y) ‖2

=‖ d − yM + yP − ε ‖2

=‖ d − d − ε ‖2

‖ d − d ‖2 = ε ≤ ε

For the robust estimate parametrization,M (ps),

limt→t+T

y∗M − yM = ε ≤ ε

This means that M (ps) can provide estimates of the disturbances such that

the compensation from this model-based controller design makes the plant outputs

114


track their target values with less offset than one based on M (ps). It then follows

thatM (ps) = M . QED

Lemma 6.3. The estimates y obtained from the maximum likelihood Model, M ,

are closer to the real plant estimates yP than those obtained from any other model

parametrization.

Proof: According to definition 6.13,

‖ d − (yM − y) ‖2= ε

‖ d − (yM − y) ‖2= εd ≤ ε

However,

‖ y − yP ‖2 = ‖ y + d − (yP + d) ‖2

= ‖ d − (yM − y) ‖2

‖ y − yP ‖2 = ε

and

‖ y − yP ‖2 = ‖ y + d − (yP + d) ‖2

= ‖ d − (yM − y) ‖2

‖ y − yP ‖2 = εd ≤ ε

QED

Lemma 6.4. The error between the target values, y∗M, and the plant measurements

forced by U , the outputs of a maximum likelihood model-based controller design is

115


smaller than the error between y∗M and the plant measurements forced by Un from

any other model-based controller design.

Proof: For a nominal model-based controller design (see definition 6.11 and remark

6.11),

limt→t+T

(y∗M −Un × P)′(y∗M −UN × P) = ε′dεd

For a maximum likelihood model-based controller design,

limt→t+T (y∗M − U × P)′(y∗M − U × P) = ε′dεd

εd ≤ εd

ε′dεd ≤ ε′dεd

QED

6.3.1 Example

There are two cases, linear and nonlinear models. Assume both linear and non-

linear models have two uncertain parameters P j, j = 1, 2. The uncertain original

ranges are 1 ≤ P1 ≤ 12, 11 ≤ P2 ≤ 20. The constraint on the model output is

14 ≤ y ≤ 22. The largest feasible ranges for all uncertain parameters should be

considered together.

Case 1: the model is a linear combination of parameters, M(P ) : y =2∑

j=1P j.

• Assume P j2j=1 are uniformly distributed, that is, P1 ∼ U (1, 10), P2 ∼

116


U (11, 20). There are more than one feasible combination of uncertain pa-

rameters that satisfy the constraint on y:1

13

≤

P1

P2

≤

2

20

,

3

11

≤

P1

P2

≤

12

14

,2

12

≤

P1

P2

≤

6

16

, · · ·• P j

2j=1 are normally distributed. P1 ∼ N (5.5, 1.44), P2 ∼ N (15.5, 1.44).

Sample N points for each parameter in their original ranges. the number of

sample points for each parameter can be chosen according to the efficiency of the

sampling techniques [18, 19]. In this example, the feasible ranges of P j2j=1 are

2 ≤ P1 ≤ 6, 12 ≤ P2 ≤ 16. After the feasible ranges of the uncertain parameters are

determined, the samples are used to calculate the robust estimates.

M –, L –, and R – robust estimates of P j2j=1 are listed in Table 6.2. In these

estimates, the function F(xi; xM) = |xi − xM | is used to find the M-estimate); the

mean value is used for an L-estimate; and the Wilcoxon test is use to obtain an

R-estimate.

Case 2: the model is nonlinear, M(P ) : y = exp(P1) + P2.

The feasible ranges for P j2j=1 are 1 ≤ P1 ≤ 2, 11.3 ≤ P2 ≤ 14.7.

• Uniform distributions, P1 ∼ U (1, 10), P2 ∼ U (11, 20)

• Normal distributions, P1 ∼ N (5.5, 1.44), P2 ∼ N (15.5, 1.44).

117


Table 6.2: Robust estimates. y =2∑

j=1P j.

Uniform Normal

Estimate M±1.155 L±1.155 R±1.155 M±0.757 L±0.748 R±0.750

P1 4 4 4 5.04 4.92 4.97

P2 14 14 14 15.04 14.92 14.97

Robust estimates of P j2j=1 are listed in Table 6.3. Robust estimates of the

uncertain parameters are both distribution and model-type dependent (see lemma

6.2).

Table 6.3: Robust estimates. y = exp(P1) + P2

Uniform

Estimate M±σ L±σ R±σ

P1 1.5±0.289 1.5±0.289 1.5±0.289

P2 13±0.9815 13±0.9815 13±0.9815Normal

Estimate M±σ L±σ R±σ

P1 1.79±0.2226 1.74±0.2231 1.76±0.2177

P2 14.19±0.543 14.06±0.528 14.12±0.531

From this example, Lemma 6.2 is further proven. The robust estimates of the

uncertain parameters are both distributions and model-type dependent.

6.4 Summary

Robust statistics is introduced in this chapter to obtain robust estimates of the

uncertain parameters from their samples. A theorem is developed and proven to

118


show that the model parameterized with robust estimates of the uncertain param-

eters is the maximum likelihood model. This model then can be employed to ob-

tain robust estimates of disturbances and states of the plant. Thus, the maximum

likelihood model can improve the performance of model-based applications when

compared to the nominal model-based applications.

119


Nomenclature

MLE maximum likelihood estimation

pd f probability density function

L-estimation linear estimation

R-estimation rank estimation

C model-based controller M model parametrizationP parameters Pn non-sensitive parametersPs sensitive parameters U controller outputsX uncertain variabled disturbances e errorxL linear estimate xM maximum likelihood estimatexR rank estimate y state estimates from modelyM plant measurements y∗M target valuesyp plant state without disturbances N normal distributionU uniform distribution

120


Chapter 7

State Estimation & Model Predictive Control with a Max-

imum Likelihood Model

Based on the knowledge of robust statistics and consideration of state variable

constraints, a maximum likelihood model which is parameterized with the robust

estimates of model uncertain parameters was introduced in Chapter 6. This chap-

ter demonstrates the robustness features of the maximum likelihood model (MLM)

namely, accurate state estimation and improve the model-based application per-

formance, in a model predictive control (MPC) framework1. The content of this

chapter is excerpted from [5].

7.1 Tubular Reactor

To demonstrate the performance of the MLM in a real-time application that in-

volves state estimation and optimal control, a chemical reaction that occurs in a

tubular reactor is introduced [4]. A dimensionless first-principles model that de-

1The concepts of model-based control and MPC are reviewed in Appendix A.

121


scribes this reaction is given by,

∂T∂t=

1Peh

∂2T∂z2 −

1Le∂T∂z+ ηCeγ(1−1/T ) + µ[Tw(z, t) − T (z, t)]

∂C∂t=

1Pem

∂2C∂z2 −

∂C∂z− DaCeγ(1−1/T )

(7.1)

The boundary conditions are:

z = 0

∂T∂z= Peh(T (z, t) − Ti(t))

∂C∂z= PeM(C(z, t) −Ci(t))

z = 1

∂T∂z= 0

∂C∂z= 0

The state variables T and C are dimensionless temperature and concentration,

respectively. The dimensionless wall temperature Tw is regulated to control the

reaction temperature by changing the flow rate of the cooling fluid. The reference

temperature is 500C. The parameters values for the fluid are listed in Table 7.1.

In this reactor model, the outputs are found to be most sensitive to Peh, η, and

µ. A 10% change in Peh and η will increase the reactor temperature by 2.9% and

1.8%, respectively; and a 10% increase in µ will decrease reactor temperature by

1.5%. Assume they are the uncertain parameters and that their distributions are Peh

– U (4, 6), η – N (0.84, 0.007) and µ – Nlog(log(12.6), 1.3).

122


Table 7.1: Dimensionless parameters for the tubular reactor [4].Parameter Nominal value Definition

Peh 5.0 Heat Peclet number

Pem 5.0 Mass Peclet number

Le 1.0 Lewis number

Da 0.875 Damkohler number

γ 15.0 Activation energy

η 0.8375 Heat of reaction

µ 13.0 Heat transfer coefficient

Tw(z, 0) 1.0 Dimensionless wall temperature

Ti(0) 1.0 Dimensionless inlet temperature

Ci(0) 1.0 Dimensionless inlet concentration

7.2 Determination of Robust Estimation of Uncertain Parameters

for State Estimation

According to lemma 6.3 in Chapter 6, the MLM can provide more accurate

estimates than the nominal model. Moreover, in the absence of disturbances, the

errors between the MLM estimates and the actual plant measurements, yM, will be

the smallest.

In the present case, one of the state variables, concentration, is not measured as

frequently as the other. To circumvent this sparseness, the first-principles model can

be used to provide in silico data to develop the partial least squares (PLS) regression

model. An important issue is how to determine the feasible ranges of the uncertain

123


parameters to enable calculation of the robust estimates. In this application of open-

loop state estimation of the reactor, a threshold value of three times the standard

deviation (this value can be determined from the historical data) of the measured

plant values is selected to develop the PLS model, the outputs yPLS ∈ [yM−3σ,yM+

3σ].

Measurements from the plant will be compared to the state estimates from the

PLS model. If the absolute errors between them are within ±3σ, ‖ yM − yPLS ‖2≤

3σ, the values assigned to the uncertain parameters are assumed to be the correct

robust estimates. Otherwise, the feasible ranges are revised, as shown in Figure

7.1. This revision can be accomplished by sampling the uncertain parameter dis-

tributions and propagating the sampled values through the first-principles model.

The sets of parameter values giving y ∈ [yM − 3σ,yM + 3σ] determine the feasible

ranges. Based on the work in Chapter 2, the Latin hypercube Hammersley sampling

(LHHS) technique is used since it is known to be very efficient when sampling mul-

tiple uncertain parameter distributions.

PlantInput

)ˆ(ˆ Θ= MM PLS modely PLSy

My

+-

σ3|| >eLHHS

updateΘ

Figure 7.1: State estimation framework with a maximum likelihood model.

124


In the reactor example, the prediction variables to build PLS model are the reac-

tor feed and wall temperatures and the feed concentration. The response variables

are the reactor exit temperature and exit concentration.

When the feed reactor temperature decreases below the optimum reaction condi-

tions, the reactor exit temperature decreases, but the exiting reactant concentration

increases. This is because at a lower reaction temperature, less reactant is con-

verted. Figure 7.2 shows the change in the exiting temperature and concentration.

The top and bottom lines show the range of the state estimates from the PLS model

(yM ± 3σ). The plus-symbols (+) are the measured values from the real plant. The

circle-symbols () are the estimates from the PLS model developed using robust

parameter values.

From Figure 7.2 it is observed that the PLS state estimates are within the ±3σ

range after two updates of the uncertain parameter values. The exit temperature is

within the ±3σ range, but the exit concentration violates its threshold twice. This

violation triggers updating of the uncertain parameter values. Note, the concen-

tration graph does not show this violation clearly due to the size of the change,

which is over a wide range. Figure 7.3 more clearly shows the updates based on the

plant measurements. The diamond-symbol (q) represents the PLS state estimates

that violate the ±3σ range. Once the uncertain parameter values are updated, the

subsequent PLS state estimates are found to be within the ±3σ range.

Table 7.2 lists the nominal and updated uncertain parameter values. The robust

125


0 5 10 15 201.02

1.025

1.03

1.035

Dimensionless time

Dim

ensi

onle

ss T

0 5 10 15 200.18

0.19

0.2

0.21

0.22

0.23

Dimensionless time

Dim

ensi

onle

ss C

Figure 7.2: State estimation. Top: dimensionless temperature. Bottom: dimension-less concentration. +: measured values; : PLS estimates.

estimates are the maximum likelihood estimates.

The first-principles model parameterized with the updated estimates is the MLM

(see theorem 6.1 in Chapter 6). Figure 7.4 shows the mismatch between the MLM

(M ) and the nominal model (MN) estimates of temperature and concentration

along the length of the reactor.

126


0 1 2 3 40.182

0.184

0.186

0.188

0.19

0.192

0.194

0.196

Dimensionless time

Dim

ensi

onle

ss C

Figure 7.3: State estimation. Dimensionless concentration. +: plant measurements;: PLS estimates. q: PLS estimates that violate the ±3σ limits.

Table 7.2: Nominal values and robust estimates of the uncertain parameters

Peh η µ

Nominal Θ 5.0 0.8375 13.0

Robust Θ1 4.83 0.859 12.54

Robust Θ2 4.78 0.856 12.49

7.2.1 Maximum Likelihood Model for Model Predictive Control

Assume the robust estimates of the model uncertain parameters are as listed in

Table 7.2. Assume that a model, called the MLM, based on these robust estimates,

will be used in a MPC framework. Figure 7.5 shows the MPC framework with the

MLM.

Discrete linear time-invariant (DLTI) models have been used successfully in

many MPC applications. Bageshwar and Borrelli [75] proposed an offset-free MPC

127


0 0.2 0.4 0.6 0.8 11.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

1.11

Dimensionless length

Dim

ensi

onle

ss T

0 0.2 0.4 0.6 0.8 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Dimensionless length

Dim

ensi

onle

ss C

Figure 7.4: Outputs of the nominal and maximum likelihood models. Top: dimen-sionless temperature. Bottom: dimensionless concentration. : states estimatesfrom M ; ∗: states estimates fromMN .

128


Model PredictiveController

Optimizer

M

Plant

MaximumLikelihood Model

M

py

d

my++

+-y

d

*y u

Model PredictiveController

Optimizer

M

Plant

MaximumLikelihood Model

M

py

d

My++

+-y

d

*y u

Figure 7.5: MPC framework with a MLM.

framework which includes a Kalman filter and an output disturbance model. The

feasibility, stability and optimality of a MPC framework for constrained discrete-

time linear periodic systems was studied based on a DLTI model [76]. In this work,

two DLTI models are identified from step tests of the open-loop first-principles

model parameterized with either the robust estimates or the nominal estimates of

uncertain parameters. Denote Mr and Mn as two DLTI model identified from M

and MN . Step testing can be applied to understand the effect of a process variable

on other process variables of a dynamic system. With a step test, the time behavior

of the outputs of a system can be obtained [77].

The sampling time is chosen as 10% of the dominant time constant, which is

0.5 dimensionless reference time units. The limits on the change of the manipulated

variable are −0.02 ≤ ∆u ≤ 0.02; and the set-point for the maximum temperature

is: T ∗max = 1.099. The control and prediction horizons are m = 3 and p = 6,

respectively. Neither the control nor the prediction horizons are optimized; instead

129


general rules are applied to arrive at these values.

To compare the closed-loop performance withMr– andMn– based controllers,

the following closed-loop performance criteria of the rise time, settling time, and

deviation from set-point are used[77].

Definition 7.1. Rise time: the first time the response reaches the set-point value

after a perturbation. A short rise time is often desired.

Definition 7.2. Settling time: the minimum time that the closed-loop response en-

ters and remains within a ±5% error band. A short settling time is usually favored.

Definition 7.3. Deviation from the set-point: the first peak value reached by the

closed-loop response. It also can be expressed as the percentage of the first peak

deviation of the closed-loop response from its set-point to its set-point value.

Random disturbances occur every 6 control steps; the disturbance is assumed

to be uniformly distributed, du ∈ [-0.01,0.01]. Figure 7.6 shows the closed-loop

performance with the MLE model, Mr, and the nominal model, Mn. For com-

parison, the disturbances are the same for both Mr– and Mn– based controllers.

From the figure, it can be observed that with the MLE-based controller, T reaches

its set-point with fewer control steps than withMn-based controller. Moreover, the

Mn-based controller output has more frequent changes.

Figures 7.7 and 7.8 show the closed-loop performance for the same disturbance

and also when the uncertain parameters’ values are decreased or increased by 3%,

130


respectively. The performance with either model-based controllers when the pa-

rameters’ values are decreased is similar with theMn-based controller response re-

quiring 3 more control steps to settle to the set-point value. The response has both

shorter rise and settling times; and the deviations from the set-point are smaller. We

can conclude that the Mr-based controller gives a better closed-loop performance

when the parameters’ values are increased.

To compare the errors between the set-point and the two model-based controller

responses, the integral of the time-weighted absolute error (ITAE) is calculated,

ITAE =∫ t

0t | e(t) | dt (7.2)

where t is time and e(t) is the error between the set-point and the closed-loop re-

sponse at time t.

Table 7.3 shows the ITAE and other controller performance criteria. In the

presence of the random unmeasured disturbances, theMr-based controller response

has smaller rise and settling times and set-point deviations. When the parameters’

values are decreased by 3%, theMr-based controller has a slightly longer rise time

but a much shorter settling time. When the parameters’ values are increased by

3%, the Mr-based controller response exhibits both shorter rise and settling times.

The set-point deviations also are smaller. In every case, the ITAE values associated

with the Mr-based controller are smaller than the ITAE values associated with the

131


Mn-based controller.

Table 7.3: Closed-loop performance comparison betweenMr andMn

Case Model ITAE Rise time Settling time Set-point dev(%)

random duMr 0.7407 1.8 1 1.7Mn 1.9530 2.8 3 2.4

∆Θ=-3%Mr 1.5204 3 1.6 2.2Mn 2.2610 2 6.8 2.2

∆Θ=+3%Mr 0.9815 2 0.4 1.5Mn 2.4677 4.5 2.5 2.4

Figure 7.9 shows the closed-loop performance when the set-point trajectory is

a stable first-order response. From the figure, it can be observed that both the Mr–

andMn– based controllers track the reactor temperature trajectory satisfactorily. In

the case of theMn-based controller, there is a small overshoot before settling to the

final value.

Figures 7.10 and 7.11 show the closed-loop controller performance when the

parameters’ values are decreased and increased by 3%, respectively. In the former

case, the Mn-based controller response shows more deviations from the set-point.

In the latter case, the reactor temperature response with the Mr-based controller

has a slightly larger overshoot from the set-point than the response with the Mn-

based controller. However, the rise and settling times are larger with theMn-based

controller and the changes in the manipulated variable are latched at the upper con-

straint.

132


Table 7.4 shows the ITAE and controller performance criteria when tracking a

first-order response in the presence of ±3% changes in the uncertain parameters’

values. In the case of set-point tracking, the Mr-based controller has smaller rise

and settling times and set-point deviations. When the values of the uncertain pa-

rameters are decreased by 3%, the Mr-based controller has a longer rise time but

a shorter settling time. The set-point deviations also are less. When the values of

the uncertain parameters are increased by 3%, the Mr-based controller has shorter

rise and settling times. In every case, the ITAE values associated with the Mr-

based controller are smaller than the ITAE values associated with the Mn-based

controller.

Table 7.4: Closed-loop performance comparison betweenMr andMn

Case Model IASE Rise time Settling time Set-point dev(%)

random duMr 0.1676 5.9 4 0.19Mn 0.2966 6.2 6 0.40

∆Θ=-3%Mr 0.2601 7 6 0.08Mn 0.5727 4.7 9 0.74

∆Θ=+3%Mr 0.2882 5 5 0.2Mn 0.4359 7 6 0.11

7.3 Summary

The maximum likelihood model is parameterized with robust estimates of the

uncertain parameters. To generate this model, the constraints imposed by the practi-

133


cal measurements must be considered to determine the feasible ranges of uncertain

parameters. Robust estimates of the uncertain parameters are calculated from the

feasible ranges using robust statistics theory. A maximum likelihood model was

developed for an example chemical reaction process and used in a model predictive

control framework to demonstrate its robust properties. It was found that accu-

rate state estimation resulted in satisfactory closed-loop control performance with

this particular model when compared to the traditional approach of using a model

parameterized with nominal values of the uncertain parameters.

134


0 5 10 15 20 25 30 35 401.06

1.08

1.1

1.12

1.14T

0 5 10 15 20 25 30 35 40-0.015-0.01

-0.0050

0.0050.01

0.015

Steps

Δ u

0 5 10 15 20 25 30 35 401.06

1.08

1.1

1.12

1.14

T

0 5 10 15 20 25 30 35 40-0.015-0.01

-0.0050

0.0050.01

0.015

Steps

Δ u

Figure 7.6: Model predictive control performance in the presence of disturbances:Top: Mr. Bottom: Mn. ∗: set-point.

135


0 5 10 15 20 25 30 35 401.07

1.09

1.11

1.13

T

0 5 10 15 20 25 30 35 40-0.01

-0.005

0

0.005

0.01

Steps

Δ u

0 5 10 15 20 25 30 35 401.07

1.09

1.11

1.13

T

0 5 10 15 20 25 30 35 40-0.01

-0.005

0

0.005

0.01

Steps

Δ u

Figure 7.7: Model predictive control performance in the presence of disturbancesand with 3% decrease in the value of the model’s parameters. Top: Mr. Bottom:Mn. ∗: set-point.

136


0 5 10 15 20 25 30 35 401.06

1.08

1.1

1.12

1.14

T

0 5 10 15 20 25 30 35 40-0.01

-0.005

0

0.005

0.01

Steps

Δ u

0 5 10 15 20 25 30 35 401.06

1.08

1.1

1.12

1.14

T

0 5 10 15 20 25 30 35 40-0.02

-0.01

0

0.01

0.02

Steps

Δ u

Figure 7.8: Model predictive control performance in the presence of disturbancesand with 3% increase in the value of the model’s parameters. Top: Mr. Bottom:Mn. ∗: set-point.

137


0 5 10 15 20 25 30 35 401.09

1.1

1.11

1.12

1.13

1.14T

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

Steps

Δ u

0 5 10 15 20 25 30 35 401.08

1.1

1.12

1.14

T

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

Steps

Δ u

Figure 7.9: Closed-loop tracking. Top: Mr. Bottom: Mn. ∗: set-point.

138


0 5 10 15 20 25 30 35 401.09

1.1

1.11

1.12

1.13

1.14T

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

Steps

Δ u

0 5 10 15 20 25 30 35 401.09

1.1

1.11

1.12

1.13

1.14

T

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

Steps

Δ u

Figure 7.10: Closed-loop tracking with a 3% decrease in the values of the model’sparameters. Top: Mr; Bottom: Mn. ∗: set-point.

139


0 5 10 15 20 25 30 35 401.09

1.1

1.11

1.12

1.13T

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

Steps

Δ u

0 5 10 15 20 25 30 35 401.09

1.1

1.11

1.12

1.13

T

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

Steps

Δ u

Figure 7.11: Closed-loop tracking with a 3% increase in the values of the model’sparameters. Top: Mr. Bottom: Mn. ∗: set-point.

140


Nomenclature

DLTI discrete linear time-invariant

ITAE integral of time and absolute error


MLM maximum likelihood model



C concentration M maximum likelihood modelMN nominal first-principles model Mn DLTI model identified fromMN

T temperatured disturbances d estimates of disturbancese errors u manipulated variablesy model outputs y∗ set-point

yM plant measurement yp plant outputs without disturbancesyPLS outputs from PLS model σ standard deviationΘ uncertain parameters U uniform distribution

Nlog log-normal distribution N normal distribution

141


Chapter 8

Summary, Contribution and Future Work

8.1 Summary

A process model in the form of a system of equations can be used to predict

the states of a process in real-time applications such as monitoring, control, fault

detection, scheduling, supply chain management, and so forth. The uncertainty in a

model’s parameters will affect the accuracy of the predictions. Thus, it is justifiable

to study model parameter uncertainty, the subject of this work.

It is desirable to analyze the effects of multiple sensitive parameter uncertainties

on the solutions of a computational model. The distributions of the model’s outputs

can be calculated based on the distributions of the input parameter uncertainties.

There are certain processes, especially large complex systems, where information

about the critical parameters are not known even in a gross manner. In such cases,

initial guesses of these parameters must be updated if the model’s outputs are to be

used for making decisions about the operation of the process. Related to this issue

is the case where some information is known about the parameter values so that

a potential robust estimate can be derived. A model with robust estimates of the

uncertain parameters can provide accurate predictions.

The objectives of this dissertation were to address these issues. Specifically, (i)

142


analyze the effect of multiple model uncertain parameters on the model’s outputs,

(ii) update the uncertain parameter values efficiently to improve the prediction ac-

curacy for real-time applications, and (iii) develop a model parameterized by max-

imum likelihood estimates of the parameters to improve the model’s performance.

Following an approach of sampling of the distributions and propagation of the

samples within the model the Latin hypercube Hammersley sampling (LHHS) tech-

nique was applied. By permuting the Latin hypercube samples according to the or-

der of Hammersley points’ sequence, LHHS provided efficient and effective propa-

gation results for multiple dimension uncertain parameters.

To represent the relationships between parameter uncertainties and the model

outputs, partial least squares (PLS) regression was applied. The solution space of a

distributed model of a process is usually of a large dimension. To focus the infor-

mation on the independent directions of variability, a reduced-order model (ROM)

was identified using a Karhunen-Loeve (KL) expansion. This technique has been

shown to concentrate the independent directions of variability in a smaller dimen-

sion whose basis set is made up of empirical eigenfunctions. Thus, the response

variables in the PLS regression were not the large dimension of the solution space

associated with the distributed model but rather the smaller dimension of coeffi-

cients of the empirical eigenfunctions identified by the KL expansion. The use of

the KL expansion to identify the coefficients of the empirical eigenfunctions so that

they can be used as the response variables in the creation of the PLS model is one

143


of the contributions of this work.

Updating the uncertain parameter values is necessary to improve the model’s ac-

curacy. In this work, two updating methods, Markov chain Monte Carlo (MCMC)

and an ensemble Kalman filter (EnKF), were designed as a function of the problem

space and compared. The updating methods then were embedded in an optimal con-

trol framework in which the system to be optimally managed was an oil producing

reservoir. It was found that the EnKF method can provide better parameter updat-

ing performance when compared to the MCMC when there are multiple uncertain

parameters to be updated.

Updating is a time consuming process, which is a critical limitation for some

real-time processes. Once there is some information about the uncertain parameter,

such as its sampled values, it is attractive to estimate a likely value with some con-

fidence. This work developed the concept of a maximum likelihood model as that

model which is parameterized with robust estimates of the uncertain parameters.

This work developed the necessary theoretical underpinnings to prove the existence

of this model. Since real process have finite and feasible ranges of the variables,

these ranges are imposed when determining the robust estimates of the uncertain

parameters using both LHHS and the theory of robust statistics. The use of the

maximum likelihood model was demonstrated in an application of state estimation

and model predictive control.

144


8.2 Contributions

The contributions of this work are as follows.

1. Use of the Karhunen-Loeve expansion to identify the coefficients of the em-

pirical eigenfunctions so that they can be used as the response variables in the

creation of a partial least squares regression model.

2. It is demonstrated that the maximum likelihood estimate can provide more

accurate estimates of the model uncertain parameters than any other kind of

estimates (e.g., mean, linear estimate, and/or rank estimate).

3. Theoretical underpinnings are developed to establish the existence of a maxi-

mum likelihood model. A maximum likelihood model is parameterized with

robust estimates of the uncertain parameters, found using robust statistics,

that are within the uncertain parameters’ feasible ranges.

8.3 Future Work

This work analyzed model parameter uncertainty for model-based applications.

Many issues remain; below are some suggestions following aspects.

1. The uncertainty analysis in this work was based on the probability theory.

Since possibility theory is another statistical theory for dealing with certain

types of uncertainty it would be worthwhile to investigate the study of uncer-

tainty with a combination of probability and possibility theories.

145


2. Real processes are nonlinear, complex, and of high dimension. In this work

the nonlinearity was approximated by assuming that the operation of the pro-

cess is around the designed operating conditions, thus a linear approximation

is justified from which to study small perturbations. As a consequence of

this approximation, in the model-based control framework, a linear controller

was designed. Future work should be focused on developing some form of a

nonlinear controller to regulate the nonlinear process.

3. Although a nonlinear controller may be more appropriate to regulate a non-

linear process, the issue of closed-loop stability cannot be overlooked. Estab-

lishing general closed-loop stability criteria is another study that should be

undertaken.

4. Another important issue is to understand fully the area of particle filters (this

includes MCMC and EnKF) for robust state estimation. These methods suf-

fer from degeneracy and re-sampling. A study that investigates these issues

with the goal of developing the maximum likelihood model for robust state

estimation is intriguing.

5. As stated at the outset of this study, the inaccuracy of a mathematical model

of a process is not only due to model parameter uncertainty. A future study

should focus on other sources of model inaccuracy and then combinations of

reasons for the inaccuracies.

146


Nomenclature


KL Karhunen-Loeve




147


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Appendix A

Preliminaries on Model-based Control

The regulation of the dynamic behavior of the process to some desired target is

the goal of the control system. This goal must be achieved within the constraints

of the process and any other limits imposed by the environment, health and safety

policies. Model-based control is concerned with the explicit use of a model of

the process to determine the control inputs to regulate the process. Model-based

control has roots in optimal control theory and linear and nonlinear system theory.

The simplest optimal controller is the linear quadratic regulator (LQR) [78] or a

state-feedback controller. In contrast, the solution to the control of uncertain linear

systems disturbed by additive white Gaussian noise is the linear quadratic Gaussian

(LQG) controller [79].

There are multiple the model-based control frameworks but they can be classi-

fied in two distinct categories. First is the direct synthesis approach. A trajectory is

specified for the desired plant output behavior. The process model is used directly

to synthesize the controller required to cause the process output to follow the tra-

jectory exactly. The most common approach is what is known as internal model

control [77].

Figure A.1 illustrates an internal model control framework. In this figure, a

process model which describes the process behavior is used. The controller outputs

u are input to both the process and the model. The difference between the plant

measurements ym and model outputs y is an estimation of disturbances (d) and any

process/model mismatch. Given a desired set-point trajectory, y∗, the controller

160


outputs are calculated based on the difference between the process measurements

and the model outputs.

du ym+

+

+-

y* Controller Plant

Modely

-

+

Figure A.1: Internal model control framework [6]. y∗: desired set-points trajectory,u: controller outputs, ym: process measurements, y: model outputs, d: measuredor unmeasured disturbances.

Second is the optimization approach. An objective function such as Equation

(A.1) is formulated for the desired output behavior. The model is used to derive

the controller required to minimize (or maximize) the objective function. In the

optimization objective, operating constraints are almost always included.

J = (y∗ − y)TQ(y∗ − y) + ∆uTR∆u

sub ject to :

y = M (x,u,Θ)

y ∈ [ymin,ymax]

x ∈ [xmin,xmax]

u ∈ [umin,umax]

∆u ∈ [∆umin,∆umax]

(A.1)

where x ∈ RM are the state variables which are in the constraint [xmin,xmax]; y ∈

RN are controlled variables which are in the constrain [ymin,ymax]; y∗ are the set-

points of y; u ∈ RL are controller outputs which are in the constrain [umin,umax];

∆u are the changes of the input variables between the previous and current control

161


step and are in the constrain [∆umin,∆umax]; M is the process model. Q is the

weighting coefficient matrix reflecting the relative importance of y and R is the

weighting coefficient matrix penalizing relative big changes in u.

Model Predictive Control

Since this work is concerned with a form of model-based control, model pre-

dictive control (MPC), the overview that follows is focused on MPC.

Figure A.2 illustrates the basic principle of model predictive control.

k k+1 k+2 k+pk+mk-1k-2k-3

umin

umax

Control Horizon, m

Input Variable

Past MovesPlanned Moves

ymax

Set-point

ymin

Prediction Horizon, p

Controlled Variable MeasurementsEstimations

Figure A.2: A MPC scheme [7]. y: controlled variable; u: controller output.

With the current process measurements at step k, the control inputs of the system

162


from step k to a control horizon, M, that is, k, . . . , k+M are determined and based

on these inputs the dynamic behavior of the system from step k to a prediction

horizon P (k, . . . , k + P) is predicted by minimizing an objective function (A.2).

Only the optimal control inputs determined for step k are implemented. At the next

step, k+1, new measurements are obtained and the objective function is resolved to

generate the optimal control inputs that would cause the outputs to meet the desired

behavior k + 1, . . . , k + 1 + P steps into the future.

J =∑P

k=0(y∗ − y)TQ(y∗ − y) +∑M

k=0 ∆uTR∆u

sub ject to :

yk+1 = M (xk,uk,Θ)

y ∈ [ymin,ymax]

x ∈ [xmin,xmax]

u ∈ [umin,umax]

∆u ∈ [∆umin,∆umax]

(A.2)

where the symbols in this equation has the same meaning as Equation A.1. ∆uk =

uk − uk−1. M and P are the control and prediction horizons, respectively. The

difference between this MPC objective function and previous model-based control

objective function (A.1) is Equation (A.2) optimizes the controller outputs over

the control horizon; while Equation (A.1) only is concerned with estimation of the

controller outputs at the current step without regard to future behavior.

163


Nomenclature

IMC internal model control

LQG linear quadratic Gaussian

LQR linear quadratic regulator


M process model T transposed disturbance k control stepm control horizon p prediction horizonu controller outputs y process model outputsym plant measurements y∗ set-pointsΘ model parameters

164


Appendix B

Computer Programs

In this appendix, the programs for the following algorithms and models are

listed.

1. Hammersley points generation

2. Karhunen-Loeve (KL) expansion

3. Partial least squares regression

4. Markov chain Monte Carlo

5. Generation of Monte Carlo samples

6. Ensemble Kalman filter

7. Robust statistics

8. Model predictive control

9. HDA Process

10. Tubular reactor

B.1 Hammersley Points Generation

function POINT = HHSmatrix(N, k)

% Inputs:

% N: the number of samples for each dimension;

165


% k: the number of dimensions

% Outputs:

% POINT: N*k Hammersley Points matrix

p = primes(410); BOUNDS = repmat([0 1.0],k,1); z = zeros(N,k);

for n=1:N

z(n,1)=n/N;

for i=2:k

R=p((i-1)); %Convert n into base R notation

m=fix(log(n)/log(R)) ; %Ensure m is an integer

base=zeros(1,m+1);

phi=zeros(1,m+1);

coefs=zeros(m+1,1);

for j=0:m

base(j+1)=Rˆj;

phi(j+1)=Rˆ(-j-1);

end

remain=n;

for j=m+1:-1:1

coefs(j)=fix((remain)/base(j));

remain=remain-coefs(j)*base(j);

end

z(n,i)=phi*coefs;

end

end

POINT = zeros (N, k);

for j = 1:k

LOWER = BOUNDS(j, 1);

UPPER = BOUNDS(j, 2);

LEN = UPPER - LOWER;

for i = 1:N

r = LOWER + (z(i,j) * LEN);

POINT (i, j) = r;

end

end

166


B.2 Karhunen-Loeve (KL) Expansion

load dataset; \% load matrix for expansion;

[m,n] = size(dataset);

for i = 1:n

datasett(:,i) = dataset(:,i)- mean(dataset(:,i));

end C = datasett’*datasett;

% covariance [eigfun lambda] = eig(C);

%eigfun is the eigenvectors of data covariance

%lambda is the

eigenvalues of data covariance

cef = datasett*(eigfun(:,n-3:end));

% assume first 4 largest eigenvalues can capture almost all

% the characteristics of

% the dataset cef is the coefficients of the

% eigenfunctions correspond to the first

% four largest eigenvalues

B.3 Partial Least Squares Regression [1]

function [t p u q w b] = PLSI (x,y)

% ( standard PLS by using NIPALS algorithm.

% Inputs:

% x: n*m matrix; prediction variables

% y: n*l matrix; response variables

% Outputs:

% t: n*max(m,l) matrix; score for x

% p: m*max(m,l) matrix; loading for x

% u: n*max(m,l) matrix; score for y

% q l*max(m,l) matrix; loading for y

% b: max(m,l)*max(m,l) matrix; regression coefficient

% important properties:

% x = t*p’;

% y = u*q’;

167


% ti’ * tj = 0;

% wi’ * wj = 0; )

[nX,mX] = size(x);

[nY,mY] = size(y);

nMaxIteration = max([mX,mY]);

nMaxOuter = 10000;

for iIteration = 1:nMaxIteration

% choose the column of x has the largest square of sum as t.

% choose the column of y has the largest square of sum as u.

[dummy,tNum] = max(diag(x’*x));

[dummy,uNum] = max(diag(y’*y));

tTemp = x(:,tNum);

uTemp = y(:,uNum);

% iteration for outer modeling

for iOuter = 1 : nMaxOuter

wTemp = x’ * uTemp/ norm(x’ * uTemp);

tNew = x * wTemp;

qTemp = y’ * tNew/ norm (y’ * tNew);

uTemp = y * qTemp;

if norm(tTemp - tNew) < 10e-15

break

end

tTemp = tNew;

end

% residual deflation:

bTemp = uTemp’*tTemp/(tTemp’*tTemp);

pTemp = x’ * tTemp/(tTemp’ * tTemp);

x = x - tTemp * pTemp’;

y = y - bTemp * tTemp * qTemp’;

% save iteration results to outputs:

t(:, iIteration) = tTemp;

p(:, iIteration) = pTemp;

u(:, iIteration) = uTemp;

q(:, iIteration) = qTemp;

w(:, iIteration) = wTemp;

b(iIteration,iIteration) = bTemp;

168


% check for residual to see if we want to continue:

if (norm(x) ==0 || norm(y) ==0)

break

end

end

B.4 Markov Chain Monte Carlo

function x = mh(N,M,xmean,xsd)

% Inputs:

% N: the number of samples generated to be tested to form

% Markov chain;

% M: the number of variables

% xmean: initial mean of x

% xsd: initial standard deviation of x

% Outputs:

% x: Markov chain

x = zeros(N,M);

x(1,:) = MC(xmean,xsd,1,M);

for i = 1:N

u = rand();

xstar = MC(x(i,:),xsd,1,M);

lpha = min(1, P(d_Theta_star)/P(d_Theta(i)));

% P(d_Theta_star): probability of observation data d

% given Theta_star

if u < alpha

x(i+1,:)= xstar;

else

x(i+1,:)=x(i);

end

169


xsd = 2.38/sqrt(M)*std(x(1:end-1,:);

end

B.5 Generation of Monte Carlo Samples

function s = MC(xmean,xsd,N,k)

% Inputs:

% xmean: initial mean of x

% xsd: initial standard deviation of x


% k: the number of dimensions

% Outputs:

% s: N*k Monte Carlo samples

ran = rand(N,K);

s=zeros(N,K);

for j=1: K

s(:,j) = xmean(j) + ltqnorm(ran(:,j)).* xsd(j);

% assume variables in x are distributed normally

end

% z = ltqnorm(p) returns the lower tail quantile for the

standard normal

% distribution function. I.e., it returns the Z

% satisfying Pr(x <z) = P,

% where x has a standard normal distributions

The function ltqnorm is cited from [80].

function z = ltqnorm(p)

% Inputs:

% p: probability

% Outputs:

% z: values that satisfying Pr(x<z) = p

% Coefficients in rational approximations.

a = [ -3.969683028665376e+01 2.209460984245205e+02 ...

170


-2.759285104469687e+02 1.383577518672690e+02 ...

-3.066479806614716e+01 2.506628277459239e+00 ];

b = [ -5.447609879822406e+01 1.615858368580409e+02 ...

-1.556989798598866e+02 6.680131188771972e+01 ...

-1.328068155288572e+01 ];

c = [ -7.784894002430293e-03 -3.223964580411365e-01 ...

-2.400758277161838e+00 -2.549732539343734e+00 ...

4.374664141464968e+00 2.938163982698783e+00 ];

d = [ 7.784695709041462e-03 3.224671290700398e-01 ...

2.445134137142996e+00 3.754408661907416e+00 ];

% Define break-points.

plow = 0.02425;

phigh = 1 - plow;

% Initialize output array.

z = zeros(size(p));

% Rational approximation for central region:

k = plow <= p & p <= phigh;

if any(k(:))

q = p(k) - 0.5;

r = q.*q;

z(k)=(((((a(1)*r+a(2)).*r+a(3)).*r+a(4)).*r+a(5)).*r+...

a(6)).*q./(((((b(1)*r+b(2)).*r+b(3)).*r+b(4)).*...

r+b(5)).*r+1);

end

% Rational approximation for lower region:

k = 0 < p & p < plow;

if any(k(:))

q = sqrt(-2*log(p(k)));

z(k) = (((((c(1)*q+c(2)).*q+c(3)).*q+c(4)).*q+c(5)).*q+...

c(6))./ ((((d(1)*q+d(2)).*q+d(3)).*q+d(4)).*q+1);

end

% Rational approximation for upper region:

k = phigh < p & p < 1;

if any(k(:))

q = sqrt(-2*log(1-p(k)));

171


z(k) = -(((((c(1)*q+c(2)).*q+c(3)).*q+c(4)).*q+c(5)).*q+...

c(6))./((((d(1)*q+d(2)).*q+d(3)).*q+d(4)).*q+1);

end

% Case when P = 0:

z(p == 0) = -Inf;

% Case when P = 1:

z(p == 1) = Inf;

% Cases when output will be NaN:

k = p < 0 | p > 1 | isnan(p);

if any(k(:))

z(k) = NaN;

end

k = 0 < p & p < 1;

if any(k(:))

e = 0.5*erfc(-z(k)/sqrt(2)) - p(k); % error

u = e * sqrt(2*pi).* exp(z(k).ˆ2/2); % f(z)/df(z)

z(k) = z(k) - u./( 1 + z(k).*u/2 ); % Halley’s method

end

B.6 Ensemble Kalman Filter

load y1a; % intial guess or the data that generated from the

%previous EnKF step

load ob; % practical observation data

load dEn1;% observation data calculated from fundamental model

load c;

% uncertain parameters’ noise which is cov(previous uncertain

% parameters - previous previous uncertain parameters) for

% the first step c is guessed for the second step

% c = Cov(previous uncertain parameters -

% initial guess of uncertain parameters)

mc = MC(xmean,xsd,N,k); % Monte Carlo samples

% xmean: mean of variables

172


% xsd: standard deviation of variables


% k: the number of observations

obi = zeros(k,N);

for i = 1:N

obi(:,i) = ob + mc(i,:)’;

end

R = cov(mc);

mm = mean(y1a’);

y1aa = zeros(N,m); % m: the number of uncertain parameters

for i = 1:N

y1aa(i,:) = c(i,:)+mm(1:m);

end

y1aa = y1aa’;

y1f = [y1aa; dEn1];

C = cov(y1f’);

H1 = zeros(k,m);

H2 = eye(k);

Hk = [H1 H2];

Kk = C*Hk’*inv(Hk*C*Hk’+R);

y2a = y1f + Kk*(obi - Hk*y1f);

% the values of uncertain parameters in y2a will be applied to

% fundamental model to generate the observation data from model

% as ’dEn2’ y2a will be saved and used in the next step

B.7 Robust Statistics

load X; %samples of a variable X

N = length(X); % number of samples

%MLE, assume f = |MLE-Xi|;

173


MLE = median(X)

%L-estimate, assume 20% trimmed mean

XS = sort(X);

l = round(N)*0.2;

u = round(N)*0.8;

LE = mean(X(l:u));

%R-estimate, Wilcoxon rank test

RE = []; n = 1;

for i = 1:N

for j = 1:N

RE(n) = X(i)+X(j);

n = n + 1;

end

end

RE = median(0.5*RE);

B.8 Model Predictive Control

load diss; %load disturbance

%the plants are the same for both robust model and nominal model

steps_m = 3;

steps_p = 6;

Wu = 0.5e0;

Wy = 0.56e0;

%%%%robust plant

A=[0.71968 -0.69594

0.23729 0.24044];

B=[58.053

-114.75];

C=[0.0033509 -0.0074329];

174


Am=[0.71968 -0.69594

0.23729 0.24044];

Bm=[58.053

-114.75];

Cm=[0.0033509 -0.0074329];

x0 = [0;0];

AA = zeros(steps_p,2);

BB = zeros(steps_p,steps_m);

for i = 1:steps_p

AA(i,:) = C*Aˆi;

end

for j = 1:steps_m-1

for i = j:steps_p

BB(i,j) = C*Aˆ(i-j)*B;

end

end

BB(steps_m,steps_m) = C*B;

for i = steps_m+1:steps_p

BB(i,end) = sum(BB(i-1,end-1:end));

end

dis = -0.01;

u(1) = 0; x(1:2,1) = [0;0]; xm(1:2,1) = [0;0]; KK = 36;

ym = zeros(KK,1); u = zeros(KK,1);

for i = 1:KK-1;

dis = diss(i);

x(:,i+1) = A*x(:,i)+B*u(i);

ym(i+1) = C*x(:,i+1)+dis;

xm(:,i+1) = Am*xm(:,i)+Bm*u(i);

yM = C*xm(:,i+1);

175


dist = ym(i+1) - yM;

uu = ss_model(dist,steps_m,steps_p,Wu,Wy);

u(i+1) = uu(1);

end

function u = ss_model(dis,steps_mm,steps_pp,Wu,Wy)

steps_m = steps_mm; steps_p = steps_pp;

ystar = zeros(steps_p,1);

A=[0.71968 -0.69594

0.23729 0.24044];

B=[58.053

-114.75];

C=[0.0033509 -0.0074329];

x0 = [0;0];

AA = zeros(steps_p,2); BB = zeros(steps_p,steps_m);

for i = 1:steps_p

AA(i,:) = C*Aˆi;

end

for j = 1:steps_m-1

for i = j:steps_p

BB(i,j) = C*Aˆ(i-j)*B;

end

end

BB(steps_m,steps_m) = C*B;

for i = steps_m+1:steps_p

BB(i,end) = sum(BB(i-1,end-1:end));

end

dist = ones(steps_p,1)*dis;

176


H = BB’*BB+Wu;

f = -2*Wy*BB’*(ystar-AA*x0-dist);

Auneq = BB;

buneq = ones(steps_p,1)*0.021-AA*x0-dist;

lb = -0.02;

ub = 0.02;

u = quadprog(H,f,Auneq,buneq,[],[],lb,ub);

B.9 HDA Process

clear;clc;

% parameter

k1 = 0.95; k2 = 0.995*k1; k3 = 5.34*k1; HR1 = -1.4345;

HR2 = -0.473; r1 = 29.26; r2 = 29.68; r3 = 33.49;

T0 = 894.26; P = 1.0; D = 0.1;

K = 0.00; h = 0.0000;

Tinj = 318.15; %K

Tinjvap = 533.15; %K vaporization Temperature at 34.47 bar

Cpinjliq = 8.314*(-0.747 + 0.06796*(Tinjvap+Tinj)/2...

+ 37.78E-6*((Tinjvap+Tinj)/2)ˆ2);

%J/(mol K) liquid benzene mean heat capacity

%between 318.15 K and 533.15K

Cpinjvap = 8.314*(-0.206 + 0.039064*(Tinjvap+T0)/2...

+ 3.301E-6*((Tinjvap+T0)/2)ˆ2);

%J/(mol K) vapor benzene mean heat capacity between

%318.15 K and 533.15K

Hlatentinj = 13567.956; %J/mol latent heat of benzene

%at 34.47 bar

cp = 67.1866; %J/(mol K)

%mixture cp at T0 = 894.26K and 34.47 bar without

% injection of benzene

177


FC7H8 = 376.67; Ffre = 484; FfreH2 = Ffre*0.95;

FfreCH4 = Ffre*0.05;

Fcyc = 3806.77; FcycH2 = Fcyc*0.374; FcycCH4 = Fcyc*0.626;

F0 = FC7H8 + Ffre + Fcyc;

Finj = 39.206475;%0.008*F0;

Ftot = F0 + Finj;

% Heat capacity parameter

A1 = 47.375; B1 = -0.22014; C1 = 0.0024826; D1 = -4.9176E-06;

E1 = 4.604E-09; F1 = -2.12E-12; G1 = 3.8505E-16; A2 = 19.671;

B2 = 0.069682; C2 = -0.0002001; D2 = 2.8949E-07;

E2 = -2.2247E-10;

F2 = 8.8147E-14; G2 = -1.4204E-17; A3 = 40.445; B3 = -0.2629;

C3 = 0.0024569; D3 = -4.9004E-06; E3 = 4.6368E-09;

F3 = -2.1527E-12;

G3 = 3.9333E-16; A4 = 44.357; B4 = -0.14623; C4 = 0.00060025;

D4 = -8.7411E-07; E4 = 6.7812E-10; F4 = -2.7538E-13;

G4 = 4.5807E-17; A5 = 148.132; B5 = -1.1072; C5 = 0.0068544;

D5 = -0.000013214; E5 = 1.2449E-08; F5 = -5.7904E-12;

G5 = 1.0614E-15;

m = 100; %interval on z direction

n = 3000; % interval on t time

km = m + 1; % number of knots on z direction

kn = n + 1; % number of knots on t time

dz = 0.01; % step on z direction

dtau = 0.0005; % step on t time

% initialize

c1 = zeros(km,kn); % C7H8 toluene concentration

c2 = zeros(km,kn); % H2 hydrogen concentration

c3 = zeros(km,kn); % C6H6 benzene concentration

c4 = zeros(km,kn); % CH4 methane concentration

c5 = zeros(km,kn); % C12H10 diphenyl concentration

178


theta = zeros(km,kn); % Temperature

thetaF = 0.95; % Temperature of jacket

Cp1 = zeros(km,1); Cp2 = zeros(km,1); Cp3 = zeros(km,1);

Cp4 = zeros(km,1); Cp5 = zeros(km,1); Cp = zeros(km,1);

kexi = zeros(km,kn); kexi3 = zeros(km,kn);

Fbm = zeros(km,kn); Fbm(1,:) = 0.008;

% initialize variables, when time = 0 at the inlet of

%the reactor

c1(1,1) = FC7H8/Ftot; c2(1,1) = (FfreH2 + FcycH2)/Ftot;

c3(1,1) = Finj/F0; c4(1,1) = (FfreCH4 + FcycCH4)/Ftot;

c5(1,1) = 0;

%theta(1,1) = 1; % initial temperature = 894.26 k

theta(1,1) = (Tinjvap*Finj*Cpinjvap+T0*F0*cp-Finj*...

(Cpinjliq*(Tinjvap-Tinj)+Hlatentinj))/...

(Finj*Cpinjvap+F0*cp)/T0;

c1(km-1:km,1) = 0.019713; c2(km-1:km,1) = 0.34377;

c3(km-1:km,1) = 0.058473; c4(km-1:km,1) = 0.576787;

c5(km-1:km,1) = 0.001257; theta(km-1:km,1) = 1.075;

deta1 = (c1(km,1)-c1(1,1))/(m-1);

deta2 = (c2(km,1)-c2(1,1))/(m-1);

deta3 = (c3(km,1)-c3(1,1))/(m-1);

deta4 = (c4(km,1)-c4(1,1))/(m-1);

deta5 = (c5(km,1)-c5(1,1))/(m-1);

deta6 = (theta(km,1)-theta(1,1))/(m-1);

for k = 1:km-3

c1(k+1,1) = c1(k,1) + deta1;

c2(k+1,1) = c2(k,1) + deta2;

c3(k+1,1) = c3(k,1) + deta3;

179


c4(k+1,1) = c4(k,1) + deta4;

c5(k+1,1) = c5(k,1) + deta5;

theta(k+1,1) = theta(k,1) + deta6;

end

for l = 1:km

Cp1(l,1) = A1+B1*theta(l,1)*T0+C1*(theta(l,1)*T0)ˆ2+...

D1*(theta(l,1)*T0)ˆ3+E1*(theta(l,1)*T0)ˆ4+...

F1*(theta(l,1)*T0)ˆ5+G1*(theta(l,1)*T0)ˆ6;











D5*(theta(l,1)*T0)ˆ3 +E5*(theta(l,1)*T0)ˆ4+...


Cp(l,1) = c1(l,1)*Cp1(l,1)+ c2(l,1)*Cp2(l,1)+...

c3(l,1)*Cp3(l,1)+c4(l,1)*Cp4(l,1) + c5(l,1)*Cp5(l,1);

kexi(l,1) = Cp(l,1)/Cp(1,1);

kexi3(l,1) = Cp3(l,1)/Cp(1,1);

end

for j = 1:kn-1

for i = 2:km-1

c1(i,j+1)= c1(i,j)+D*dtau/dzˆ2*(c1(i+1,j)-2*c1(i,j)+...

c1(i-1,j))-dtau/dz*(c1(i+1,j)-c1(i,j))/P -...

dtau*k1*c1(i,j)*c2(i,j)ˆ0.5*theta(i,j)ˆ1.5*...

exp(r1*(1-1/theta(i,j))) ;


180


c2(i-1,j))-dtau/dz*(c2(i+1,j)-c2(i,j))/P - ...


exp(r1*(1-1/theta(i,j)))+ dtau*k2*(c3(i,j)*...

theta(i,j))ˆ2*exp(r2*(1-1/theta(i,j))) - dtau*k3*...

c2(i,j)*c5(i,j)*theta(i,j)ˆ2*exp(r3*(1-1/theta(i,j)));


c3(i-1,j))-dtau/dz*(c3(i+1,j)-c3(i,j))/P + ...


exp(r1*(1-1/theta(i,j))) - dtau*2*k2*(c3(i,j)*...

theta(i,j))ˆ2*exp(r2*(1-1/theta(i,j)))...

+ dtau*2*k3*c2(i,j)*c5(i,j)*theta(i,j)ˆ2*...

exp(r3*(1-1/theta(i,j)))+Fbm(i,j)*dtau;


c4(i-1,j))-dtau/dz*(c4(i+1,j)-c4(i,j))/P +...


exp(r1*(1-1/theta(i,j))) ;


c5(i-1,j))-dtau/dz*(c5(i+1,j)-c5(i,j))/P + ...

dtau*k2*(c3(i,j)*theta(i,j))ˆ2*...

exp(r2*(1-1/theta(i,j))) - dtau*k3*c2(i,j)*c5(i,j)*...

theta(i,j)ˆ2*exp(r3*(1-1/theta(i,j)));

theta(i,j+1) = theta(i,j)+K*dtau/dzˆ2*(theta(i+1,j)-...

2*theta(i,j)+theta(i-1,j)) + dtau/dz*(kexi(i,j)*...

(theta(i+1,j)-theta(i,j))-theta(i+1,j)*...

(kexi(i+1,j)-kexi(i,j))) + HR1*(c1(i,j+1)-c1(i,j))-...

HR2*(c5(i,j+1)-c5(i,j))-kexi3(i+1,j)*Fbm(i,j)*dtau+...

h*(thetaF - theta(i,j));

end

c1(1,j+1) = c1(1,j);

c2(1,j+1) = c2(1,j);

c3(1,j+1) = c3(1,j);

c4(1,j+1) = c4(1,j);

c5(1,j+1) = c5(1,j);

theta(1,j+1) = theta(1,j);

181


c1(km,j+1) = c1(km-1,j+1);

c2(km,j+1) = c2(km-1,j+1);

c3(km,j+1) = c3(km-1,j+1);

c4(km,j+1) = c4(km-1,j+1);

c5(km,j+1) = c5(km-1,j+1);

theta(km,j+1) = theta(km-1,j+1);

end

z = dz.*(1:km); t = dtau.*(1:kn);

% Plot the results

figure;

subplot(3,2,1); plot(t,theta(km,1:kn)); xlabel(’t (n)’);

ylabel(’theta’);

subplot(3,2,2);plot(z,c3(1:km,kn)); xlabel(’z (m)’)

ylabel(’benzene’);

subplot(3,2,3); plot(z,c1(1:km,kn)); xlabel(’z (m)’);

ylabel(’toluene’);

subplot(3,2,4); plot(z,c2(1:km,kn)); xlabel(’z (m)’);

ylabel(’hydrogen’);

subplot(3,2,5); plot(z,theta(1:km,kn)); xlabel(’z (m)’);

ylabel(’theta’)

B.10 Tubular Reactor

clear all;clc;

%load initial conditions

load T0.mat;

load C0.mat;

m = 50; dz = 0.02;

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n = 6000; dt = 0.0005;

Tw = ones(m,n);

% % %%Plant

Peh = 5.0; Pem = 5.0; Le = 1.0; Da = 0.875; gamma = 15.0;

eta = 0.8375; miu = 13.0;

T = zeros(m+1,n+1); C = zeros(m+1,n+1);

T(:,1) = T0; C(:,1) = C0;

T(1,:) = T0(1); C(1,:) = C0(1);

for j = 1:n

for i = 2:m

T(i,j+1) = T(i,j) + dt * ( 1/Peh/dz/dz*(T(i+1,j)...

-2*T(i,j)+T(i-1,j))- 1/Le/dz*(T(i+1,j)-T(i,j)) ...

+ miu*(Tw(i,j) - T(i,j)) + eta*C(i,j)*...

exp(gamma*(1-1/T(i,j))));

C(i,j+1) = C(i,j) + dt * ( 1/Pem/dz/dz*(C(i+1,j)...

-2*C(i,j)+C(i-1,j)) - 1/dz*(C(i+1,j)-C(i,j)) ...

- Da*C(i,j)*exp(gamma*(1-1/T(i,j))) );

end

T(m+1,j+1) = T(m,j+1);

C(m+1,j+1) = C(m,j+1);

end

z = dz*(0:m); t = dt*(0:n);

figure;

subplot(2,2,1); plot(t,T(m+1,:)); xlabel(’t (n)’); ylabel(’T’) ;

subplot(2,2,2); plot(t,C(m+1,:)); xlabel(’t (n)’); ylabel(’C’) ;

subplot(2,2,3); plot(z,T(:,n+1)); xlabel(’z (m)’); ylabel(’T’) ;

subplot(2,2,4); plot(z,C(:,n+1)); xlabel(’z (m)’); ylabel(’C’) ;

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An Analysis of Model Parameter Uncertainty on Online Model ...

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