NONLINEAR MODEL PREDICTIVE CONTROL OF A DISTILLATION COLUMN USING

HAMMERSTEIN MODEL AND NONLINEAR AUTOREGRESSIVE MODEL WITH

EXOGENOUS INPUT

RAMESH KANTHASAMY

UNIVERSITI SAINS MALAYSIA

2009

NONLINEAR MODEL PREDICTIVE CONTROL OF A DISTILLATION COLUMN USING HAMMERSTEIN MODEL AND NONLINEAR

AUTOREGRESSIVE MODEL WITH EXOGENOUS INPUT

by

RAMESH KANTHASAMY

Thesis submitted in fulfillment of the requirements for the degree of

Doctor of Philosophy

March 2009

ACKNOWLEDGMENTS

First and foremost, I would like to express my deepest gratitude to my parents

for providing me with all the best in life though it would mean lots of sacrifice on

their behalf. Thank you for your unconditional love, encouragement, support and

giving me the absolute freedom to choose my path of life.

I would also like to convey my sincere gratitude to my main supervisor, Dr.

Norashid Aziz, for the expert guidance, constant attention, invaluable comments and

encouragement throughout the course of my study. My deepest thanks also go to my

co-supervisor Dr. Syamsul Rizal Abd. Shukor, for providing incessant support and

guidance throughout my studies. My heartfelt appreciation is forwarded to Dr. M.

Ramasamy, for providing me an opportunity to carry out my research and also for his

care, support, guidance and motivation. His contribution will be remembered

eternally.

I would also like to thank Prof. Dr. Abdul Latif Ahmad, Dean of the School

of Chemical Engineering USM for his continuous motivation, cultivated briefing on

the postgraduate project requirements and most importantly for his concern on my

future. I would also like to extend my sincere appreciation to all lecturers in this

school for giving me support and guidance, especially to Prof. Subhash Bhatia,

Assoc. Prof. Dr. James Noel Fernando, Assoc. Prof. Dr. Mohamad Zailani Bin Abu

Bakar, Assoc. Prof. Dr. Mashitah Mat Don, Dr. Zainal Ahmad, Dr. Suhairi Abdul

Sata and Dr. Tan Soon Huat, who have shared their precious knowledge and

experience with me. I extend my gratitude to all the laboratory technicians and

ii

administrative staff of the School of Chemical Engineering USM, especially to Mrs.

Aniza, Mrs. Azni, Mr. Aziz Ismail, Mrs. Yusnadia, Mr. Shamsul Hidayat, Mr.

Shaharin Mohamed, Mr. Osmarizal Osman and Mr. Mohd Roqib for the assistance

rendered.

I am indeed very thankful to the few kind souls for their prayers, patience and

emotional support. They are always beside me, through my ups and downs, thin and

thick as well as giving me encouragement to pursue my dreams. My deepest thanks

to my grandfather Karumannan, grandmother, uncle Kumar and family, aunty Guna,

Thulasi akka and family for their unconditional love and care for making me feel

warm and welcomed in Malaysia. I wish to immortalize my sincere thanks to my

dearest friends Aravind, Aziah, Baskar, Chai, Choe Peng, Choi Yee, Chua, Derek, Ee

Mee, Fakhroni, Gokul, Irvan, Ivy, Jayaprakash, Jia Huey, Kelly, Mohan, Murugesh,

Nurul Ameen, Ooi Boon Seng, Pei Ching, Sakthivel, Siew Chun, Sim Yee, Satis,

Siva Kumar, Sumathi, Tan, Thiam Leng, Yin Fong and Zalizawati for their endless

assistance, support and encouragement.

Last but definitely not least, I am very much indebted to the Ministry of

Science, Technology and Innovation (MOSTI), Malaysia for providing me the

financial support through the IRPA – 8 (Project No: 03-02-4279EA019) and to

Universiti Sains Malaysia for offering me the USM fellowship.

Once again, I would like to thank everyone from the bottom of my heart.

RAMESH KANTHASAMY

March 2009

iii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii

TABLE OF CONTENTS iv

LIST OF TABLES ix

LIST OF FIGURES x

LIST OF PLATES xvi

LIST OF ABBREVIATIONS xvii

LIST OF SYMBOLS xx

ABSTRAK xxiv

ABSTRACT xxvii

1CHAPTER 1 INTRODUCTION 1

1.1 Distillation 1

1.2 Distillation equipment 1

1.3 Need for distillation control 3

1.4 Distillation control techniques 4

1.4.1 Multivariable controllers 6

1.4.2 Model Predictive Control 7

1.4.3 Nonlinear Model Predictive Control 9

1.5 Problem statement 15

1.6 Research objectives 18

1.7 Scope of work 18

1.8 Organization of thesis 21

2

CHAPTER 2 LITERATURE REVIEW 23

2.1 Importance of distillation control 23

2.2 Major disturbance in distillation control 24

2.2.1 Feed composition upsets 24

2.2.2 Feed flow rate upsets 25

2.2.3 Loss of reboiler steam pressure 25

2.2.4 Feed enthalpy upsets 26

2.2.5 Column pressure upsets 26

iv

2.3 Nonlinear characteristics 27

2.4 Nonlinear process models 28

2.4.1 Fundamental models 28

2.4.1.1 Equilibrium model 29

2.4.1.2 Rate-Based Model 32

2.4.2 Empirical models 34

2.4.2.1 Hammerstein model 40

2.4.2.2 Wiener model 43

2.4.2.3 Nonlinear auto-regressive model with exogenous input 45

2.4.2.4 NARMAX models 47

2.4.2.5 ANN models 48

2.4.2.6 Volterra models 49

2.4.3 Hybrid models 50

2.5 Principle of Nonlinear Model Predictive Control 53

2.6 Mathematical formulation of NMPC 54

2.7 Computational aspects of NMPC 56

2.7.1 Sequential quadratic programming 57

2.7.1.1 Sequential solutions 58

2.7.1.2 Simultaneous solutions 59

2.7.2 Successive linear programming 61

2.7.3 Generalized reduced gradient method 62

2.7.4 Other optimization algorithms 63

2.8 NMPC applications to distillation column 64

3

CHAPTER 3 MATERIALS AND METHODS 74

3.1 Materials 74

3.1.1 Properties of Methanol 74

3.1.2 Properties of water 75

3.2 Distillation column design 75

3.3 Experimental setup 78

3.3.1 Stacked plate construction 79

3.3.2 Reverse flow plate 80

3.3.3 Sieve plate 80

3.3.4 Downcomer and weir 81

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3.3.5 Condenser 82

3.3.6 Reflux drum 84

3.3.7 Reflux inlet arrangement 85

3.3.8 Feed inlet arrangement 86

3.3.9 Reboiler return inlet arrangement 87

3.3.10 Reboiler specifications 88

3.3.11 Control panel 89

3.3.12 Temperature indicators 89

3.3.13 Temperature controllers 90

3.3.14 Digital Refractometer 91

3.4 First principle modeling and simulation strategy 91

3.4.1 Development of first principle model 91

3.4.2 Simulation strategy 94

3.5 Experimental procedure 97

3.5.1 Composition calibration using refractive index 98

3.6 First principle model validation 98

3.7 Development of NMPC and closed loop control studies 100

3.7.1 Nonlinear identification of wavenet based Hammerstein model 100

3.7.2 Nonlinear identification of sigmoidnet based NARX model 104

3.7.3 NMPC problem formulation 105

3.7.4 Unscented Kalman Filter 106

3.7.5 NMPC Optimization algorithm 106

3.7.6 Closed loop simulation studies 107

4

CHAPTER 4 RESULTS AND DISCUSSIONS 108

4.1 Experimental validation of first principle model 108

4.1.1 Steady-state experiments 108

4.1.2 Dynamic experiments 110

4.1.2.1 Positive step change in reflux flow rate 111

4.1.2.2 Negative step change in reflux flow rate 113

4.1.2.3 Positive step change in feed flow rate 116

4.1.2.4 Negative step change in feed flow rate 118

4.1.2.5 Positive step change in feed composition 121

4.1.2.6 Negative step change in feed composition 123

vi

4.2 Nonlinearity studies 125

4.2.1 Steady-state analysis 126

4.2.2 Effect of feed flow rate 128

4.2.3 Effect of feed composition 130

4.2.4 Effect of reflux flow rate 131

4.2.5 Effect of reboiler heat load 133

4.3 Nonlinear model identification of wavenet based Hammerstein model 136

4.3.1 Wavenet based Hammerstein model structure 136

4.3.2 Parameter estimation of Hammerstein models 137

4.3.3 Hammerstein models validation using process model 142

4.3.4 Experimental validation of Hammerstein models 143

4.3.5 Model analysis 145

4.3.5.1 Effect of inputs on non-linearity 145

4.3.5.2 Step response curve 147

4.3.5.3 Impulse response curve 149

4.3.5.4 Bode plot 150

4.4 Nonlinear model identification of sigmoidnet based NARX model 153

4.4.1 Sigmoidnet based MISO NARX model structure 153

4.4.2 Parameter estimation of sigmoidnet based MISO NARX models 155

4.4.3 NARX models validation using process model 159

4.4.4 Experimental validation of NARX models 160

4.4.5 Regressor analysis 162

4.5 Development of NMPC 168

4.5.1 The objective function 168

4.5.2 State estimation 170

4.5.3 Process constraints 170

4.5.4 Optimization algorithm 173

4.6 Closed loop control studies using Hammerstein model NMPC 174

4.6.1 Regulatory control of Hammerstein NMPC 174

4.6.2 Servo control of Hammerstein NMPC 178

4.6.3 Control performance of Hammerstein NMPC for simultaneous

changes in load and set point 179

4.7 Closed loop control studies using NARX model NMPC 181

4.7.1 Regulatory control of NARX NMPC 181

vii

4.7.2 Servo control of NARX NMPC 184

4.7.3 Control performance of NARX NMPC for simultaneous changes

in load and set point 186

4.8 Comparison of Hammerstein NMPC and NARX NMPC 187

5

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 191

5.1 Conclusions 191

5.2 Recommendations 194

REFERENCES 196

APPENDICES

Appendix A MATLAB Program for McCabe and Thiele Graphical Method 209

Appendix B Mathematical Modeling of Distillation Column 212

Appendix C MATLAB Functions for Solving First Principle Model 226

Appendix D Euler Method 233

Appendix E Composition Calibration using Refractive Index 235

Appendix F Experimental Results for First Principle Model Validation 240

LIST OF PUBLICATIONS 246

viii

LIST OF TABLES

Page

Table 2.1 Table 2.2 Table 2.3 Table 3.1 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6

Summary of block-oriented models and NARX models applied for tray distillation column Summary of nonlinear model predictive controller applications to distillation column Recent applications of NMPC to other chemical processes Distillation column specification sheet Calibrated tray efficiency values Comparison between steady-state experimental and simulation results Parameters of MISO Hammerstein models Parameters of MISO Hammerstein models Effect of MISO NARX model regressors on nonlinearity IAE performance index of closed loop simulation studies

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LIST OF FIGURES

Page

Figure 1.1 Figure 1.2 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 4.1 Figure 4.2

Schematic of continuous binary tray distillation column Basic NMPC control loop Flow chart for nonlinear model development (Doyle III et al., 2002) Hammerstein model structure Wiener model structure Principle of nonlinear model predictive control (Findeisen et al., 2003) Stacked plate construction Reverse flow plate Equilateral Triangular hole pitch Segmental downcomer Horizontal in-shell condenser Reflux Drum Reflux inlet arrangement Feed inlet arrangement Reboiler return inlet arrangement Flowchart for systematic modeling of distillation column Flow chart for distillation column simulation strategy Comparison between experimental and simulated responses in top product composition for positive step change in reflux flow rate Comparison between experimental and simulated responses in bottom product composition for positive step change in reflux flow rate

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10

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Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12 Figure 4.13 Figure 4.14

Comparison between experimental and first principle model results with varying tray temperatures for positive step change in reflux flow rate Comparison between experimental and simulated responses in top product composition for negative step change in reflux flow rate Comparison between experimental and simulated responses in bottom product composition for negative step change in reflux flow rate Comparison between experimental and first principle model results with varying tray temperatures for negative step change in reflux flow rate Comparison between experimental and simulated responses in top product composition for +10% step change in feed flow rate Comparison between experimental and simulated responses in bottom product composition for +10% step change in feed flow rate Comparison between experimental and first principle model results of varying tray temperatures for +10% step change in feed flow rate Comparison between experimental and simulated responses in top product composition for -10% step change in feed flow rate Comparison between experimental and simulated responses in bottom product composition for -10% step change in feed flow rate Comparison between experimental and first principle model results of varying tray temperatures for -10% step change in feed flow rate Comparison between experimental and simulated responses in top product composition for +10% step change in feed composition Comparison between experimental and simulated responses in bottom product composition for +10% step change in feed composition

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Figure 4.15 Figure 4.16 Figure 4.17 Figure 4.18 Figure 4.19 Figure 4.20 Figure 4.21 Figure 4.22 Figure 4.23 Figure 4.24 Figure 4.25 Figure 4.26 Figure 4.27 Figure 4.28 Figure 4.29 Figure 4.30

Comparison between experimental and first principle model results of varying tray temperatures for +10% step change in feed composition Comparison between experimental and simulated responses in top product composition for -10% step change in feed composition Comparison between experimental and simulated responses in bottom product composition for -10% step change in feed composition Comparison between experimental and first principle model results of varying tray temperatures for -10% step change in feed composition Simultaneous changes in manipulated variables L and QB

Responses in top and bottom product compositions for simultaneous changes in L and QB Responses in top product composition for ±10% changes in feed flow rate Responses in bottom product composition for +10% changes in feed flow rate Responses in top product composition for ±10% changes in feed composition Responses in bottom product composition for ±10% changes in feed composition Responses in top product composition for ±10% changes in reflux flow rate Responses in bottom product composition for ±10% changes in reflux flow rate Responses in top product composition for ±10% changes in reboiler heat load Responses in top product composition for ±10% changes in reboiler heat load MISO Hammerstein model structure Input data u1(k) for MISO Hammerstein models parameter estimation

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Figure 4.31 Figure 4.32 Figure 4.33 Figure 4.34 Figure 4.35 Figure 4.36 Figure 4.37 Figure 4.38 Figure 4.39 Figure 4.40 Figure 4.41 Figure 4.42 Figure 4.43 Figure 4.44 Figure 4.45 Figure 4.46 Figure 4.47 Figure 4.48 Figure 4.49

Input data u2(k) for MISO Hammerstein models parameter estimation Output data y(k) for first MISO Hammerstein model parameter estimation Output data y(k) for second MISO Hammerstein model parameter estimation Comparison between the responses of first MISO Hammerstein model and process model Comparison between the responses of second MISO Hammerstein model and process model Comparison between the responses of first MISO Hammerstein model and experimental results Comparison between the responses of second MISO Hammerstein model and experimental results Effect of input u1 on output nonlinearity of the nonlinear static element Effect of input u2 on output nonlinearity of the nonlinear static element Step response for input u1

Step response for input u2

Impulse response for input u1 Impulse response for input u2 Bode plot for input u1 Bode plot for input u2 MISO NARX model structure Input data u1(k) for MISO NARX models parameter estimation Input data u2(k) for MISO NARX models parameter estimation Output data y(k) for first MISO NARX model parameter estimation

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Figure 4.50 Figure 4.51 Figure 4.52 Figure 4.53 Figure 4.54 Figure 4.55 Figure 4.56 Figure 4.57 Figure 4.58 Figure 4.59 Figure 4.60 Figure 4.61 Figure 4.62 Figure 4.63 Figure 4.64 Figure 4.65 Figure 4.66 Figure 4.67

Output data y(k) for second MISO NARX model parameter estimation Comparison between the responses of first MISO NARX model and process model Comparison between the responses of second MISO NARX model and process model Comparison between the responses of first MISO NARX model and experimental results Comparison between the responses of second MISO NARX model and experimental results Effect of regressors y(k-1) and y(k-2) on nonlinearity Effect of regressors y(k-1) and u1(k-1) on nonlinearity Effect of regressors y(k-1) and u2(k-1) on nonlinearity Effect of regressors y(k-2) and u1(k-1) on nonlinearity Effect of regressors y(k-2) and u2(k-1) on nonlinearity Effect of regressors u1(k-1) and u2(k-1) on nonlinearity Hammerstein NMPC responses of product compositions to a feed flow rate disturbance Hammerstein NMPC responses of manipulated variables to a feed flow rate disturbance Hammerstein NMPC responses of product compositions to a feed composition disturbance Hammerstein NMPC responses of manipulated variables to a feed composition disturbance Hammerstein NMPC responses of product compositions to -3% change in xD at t =10 min and -3% change in xB at t = 50 min Hammerstein NMPC responses of manipulated variables to -3% change in xD at t =10 min and -3% change in xB at t = 50 min Hammerstein NMPC responses of product compositions for simultaneous changes in set points and disturbances

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Figure 4.68 Figure 4.69 Figure 4.70 Figure 4.71 Figure 4.72 Figure 4.73 Figure 4.74 Figure 4.75 Figure 4.76 Figure 4.77 Figure 4.78

Hammerstein NMPC responses of manipulated variables for simultaneous changes in set points and disturbances NARX NMPC responses of product compositions to a feed flow rate disturbance NARX NMPC responses of manipulated variables to a feed flow rate disturbance NARX NMPC responses of product compositions to a feed composition disturbance NARX NMPC responses of manipulated variables to a feed composition disturbance NARX NMPC responses of product compositions to -3% change in xD at t =10 min and -3% change in xB at t = 50 min NARX NMPC responses of manipulated variables to -3% change in xD at t =10 min and -3% change in xB at t = 50 min NARX NMPC responses of product compositions for simultaneous changes in load variables and set points NARX NMPC responses of manipulated variables for simultaneous changes in load variables and set points Comparison between Hammerstein NMPC and NARX NMPC responses of product compositions for changes in both set points and disturbances Comparison between Hammerstein NMPC and NARX NMPC responses of manipulated variables for changes in both set points and disturbances

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xv

LIST OF PLATES

Page

Plate 3.1

Pilot plant distillation column set up

78

xvi

LIST OF ABBREVIATIONS

ADMC Adaptive dynamic matrix control

AIC Akaike information criterion

ANN Artificial neural network

ARMA Autoregressive moving average

ARMAX Auto-regressive moving average model with exogenous inputs

ASME American society of mechanical engineers

AT Assay transmitter

BFGS Broyden-Fletcher-Goldfarb-Shanno

BIBO Bounded input-bounded output

BIC Bayesian information criterion

CPI Chemical process industries

CPU Central processing unit

CSTR Continuous stirred-tank reactor

DMC Dynamic matrix control

EKF Extended Kalman filter

FCC Fluid catalytic cracking

FIR Finite impulse response

FPE Final prediction error

GRV Gaussian random variables

GWN Gaussian white noise

HJB Hamilton-Jacobi-Bellman

IAE Integral of absolute value of error

IDCOM Identification and command

xvii

IMC Internal model control

IWMPC Inverse Wiener model predictive control LC Level controller

LED Light emitting diode

LI Level indicator

LILC Law of iterated logarithms criterion

LMPC Linear model predictive control

LP Linear programming

MAP Method of approximate programming

MATLAB Matrix laboratory

MCB Miniature circuit breakers

MFNN Multilayer feed forward neural network

MIMO Multiple-input multiple output

MISO Multiple-input-single-output

MPC Model predictive control

MSE Mean square error

MVC Multivariable control

NAMPC Nonlinear analytical model predictive control

NARMAX Nonlinear auto-regressive moving average model with exogenous inputs NARX Nonlinear auto-regressive with exogenous inputs

NC Number of components

NEQ Non-equilibrium stage

NGA Narendra Gallman algorithm

NLP Nonlinear program

NMPC Nonlinear model predictive control

xviii

NT Number of trays

ODE Ordinary differential equation

OE Output Error

PEM Prediction error method

PID Proportional-integral-derivative

PRBS Pseudo random binary signal

PT Pressure transmitter

PV Process value

PWMPC Polytopic Wiener model predictive control

QDMC Quadratic dynamic matrix control

QP Quadratic program

RDNN Recurrent dynamic neuron network

RI Refractive index

RMSE Relative mean square error

RPEM Recursive prediction error method

SISO Single-input single-output

SLP Successive linear programming

SQP Sequential quadratic programming

SVD Singular value decomposition

TEMA Tubular exchanger manufacturers association

UKF Unscented Kalman filter

UNIFAC UNIQUAC functional group activity coefficients

VLE Vapor liquid equilibrium

xix

LIST OF SYMBOLS

amn Group interaction parameter (K)

an Output coefficients vector in sigmoidnet function

Apc Amplitude ratio

ask Scaling coefficients

awk Wavelet coefficients

B Bottom product flow rate (l/min)

bn Dilation matrix in sigmoidnet function

bsk Scaling dilation coefficients

bwk Wavelet dilation coefficients

C Third virial coefficient

cn Translation vector in sigmoidnet function

csk Scaling translation coefficients

cwk Wavelet translation coefficients

d Output offset scalar in sigmoidnet function

Output offset scalar in wavenet function

D Distillate flow rate (l/min)

e(k) Model prediction error

F Feed flow rate (l/min)

f(u) Scaling function

F(x) Nonlinear regression function

FL Liquid flow rate over weir (m3/s)

g Gauge pressure (bar)

g(u) Wavelet function

h Liquid height (m)

H(z) Linear dynamic element in Hammerstein model

hD Enthalpy of distillate (J/kmol)

hF Enthalpy liquid in feed (J/kmol)

hN Enthalpy of liquid in Nth tray (J/kmol)

HN Nth tray vapor enthalpy (J/kmol)

hNF Feed stage liquid enthalpy (J/kmol)

HNF Feed tray vapor enthalpy (J/kmol)

How Height over outlet weir (m)

xx

i Time instant

k Current sampling instant

l Dummy indices

L Linear coefficients vector in sigmoidnet function

Linear term coefficients in wavenet function

Reflux flow rate (l/min)

LN Nth tray liquid flow rate (l/min)

LNF Feed tray liquid flow rate (l/min)

lw Weir length (m)

M Control horizon.

MB Liquid holdup in reboiler (l)

MC Liquid holdup in condenser (l)

MNF Feed tray liquid holdup (l)

N Tray number

N(·) Nonlinear static element in Hammerstein model

na Number of past output terms

nb Number of past input terms

ne Delay from input

P Nonlinear subspace matrix in sigmoidnet function

Nonlinear subspace parameters in wavenet function

Prediction horizon

p Integer order parameter

Pc Critical pressure (bar)

Pr Reduced pressure

Q Linear subspace matrix in sigmoidnet function

Linear subspace parameters in wavenet function

Positive-definite weighing matrix

q Fraction of liquid in feed

QB Reboiler heat load(kW)

QC Heat removed in condenser (kW)

qi Relative molecular surface area

R Positive-definite weighing matrix

r Regressor mean vector in sigmoidnet function

Regressor means in wavenet function

xxi

ri Relative molecular volume

S Positive-definite weighing matrix

T Temperature (K)

t time (min)

Tc Critical temperature (K)

tf Final time (min)

Tr Reduced temperature

u Manipulated input

u(k) Scalar input (manipulated variable)

U(k) Vector of manipulated input variables

u1 First manipulated input (Reflux flow rate)

u1(k) Input to the first nonlinear static block of the Hammerstein

model

u1(k-1), u2(k-1) Past input regressors

u1max Maximum value of the reflux flow rate

u1min Minimum value of the reflux flow rate

u2 Second manipulated input (Reboiler heat load)

u2(k) Input to the second nonlinear static block of the Hammerstein

model

u2max Maximum value of the reboiler heat load

u2min Minimum value of the reboiler heat load

us Steady-state input

∆u1max Maximum value of rate of change of reflux flow rate

∆u1min Minimum value of rate of change of reflux flow rate

∆u2max Maximum value of rate of change of reboiler heat load

∆u2min Minimum value of rate of change of reboiler heat load

Vc Critical molar volume (cm3/mol)

VN Nth tray vapor flow rate (l/min)

VNF Feed tray vapor flow rate (l/min)

x(k) Output of the nonlinear static block

X(k) Vector of state variables

x1(k) Output of the first nonlinear static block of the Hammerstein

model

xxii

x2(k) Output of the second nonlinear static block of the

Hammerstein model

xB Bottom product composition

XBj Liquid mole fraction of jth component in bottom product

xD Top product composition

xDj Liquid mole fraction of jth component in distillate

xij Liquid mole fraction of jth component in ith stage

Xm Mole fraction of group m in the mixture

ŷ Predicted output

y(k) Output of the linear dynamic block

y(k) Scalar output (response variable)

Y(k) Vector of controlled output variables

y1(k-1), y2(k-1) Past output regressors

y1max Maximum value of the top product composition

y1min Minimum value of the top product composition

y2min Minimum value of the bottom product composition

y2max Maximum value of the bottom product composition

yij Vapor mole fraction of jth component in ith stage

ys Steady-state response

Zc Critical compressibility factor

ZF Feed composition

zF,j Mole fraction of jth component in feed

Greek letters

θm Area fraction of group m

Ψmn Group interaction parameter

φ Fugacity coefficient

γ Activity coefficient

ω Acentric factor

ΩT Ellipsoidal or polytopic set

ωgc Gain crossover frequency

ωpc Phase crossover frequency

xxiii

KAWALAN RAMALAN MODEL TAK LELURUS BAGI TURUS

PENYULINGAN MENGGUNAKAN MODEL HAMMERSTEIN DAN

MODEL AUTO MUNDUR TAK LELURUS DENGAN MASUKAN LUAR

KAWALAN

ABSTRAK

Turus penyulingan adalah unit proses penting dalam industri penapisan

petroleum dan kimia. Ia perlu dikawal hampir dengan keadaan-keadaan pengendalian

yang optima demi insentif-insentif ekonomi. Kebanyakan turus penyulingan industri

pada masa kini dikawal oleh pengawal berbilang gelung yang berasaskan model-

model lelurus yang mengakibatkan beberapa kekurangan. Skim kawalan berasaskan

model tak lelurus merupakan salah satu pilihan terbaik untuk diselidiki bagi

mencapai pengawalan turus penyulingan yang baik. Dalam kerja ini, dua skim model

ramalan kawalan tak lelurus (NMPC) yang menggunakan model Hammerstein dan

model NARX telah dibina untuk mengawal turus penyulingan. Turus penyulingan

perduaan untuk pemisahan metanol-air telah digunakan untuk mengesahkan prestasi

skim-skim kawalan yang dibangunkan.

Turus penyulingan loji pandu yang bergaris pusat 10.2cm dan 15 dulang-

dulang ayak telah direkabentuk, difabrikasi dan digunakan dalam kajian ini. Model

matematik berasaskan jumlah imbangan jisim, imbangan komponen dan imbangan

entalpi telah dibangunkan berasaskan prinsip-prinsip pertama. Pengiraan kegiatan

dan fugasiti telah dimasukkan dalam model tersebut untuk mengambil kira

ketakunggulan sistem tersebut. Satu algoritma yang sesuai telah dibina untuk

menyelesaikan persamaan-persamaan model dalam persekitaran MATLAB.

Eksperimen-eksperimen telah dijalankan dengan menggunakan turus penyulingan

xxiv

loji pandu pada keadaan-keadaan mantap dan dinamik untuk mengesahkan model

prinsip pertama yang dibina. Nilai-nilai kecekapan dulang telah ditala dengan

menggunakan hasil-hasil eksperimen pada keadaan mantap. Hasil-hasil model

menunjukkan tahap konsistensi yang tinggi dengan hasil-hasil eksperimen. Model

prinsip pertama yang telah disahkan digunakan sebagai proses model dalam

pengenalpastian sistem tak lelurus dan kajian-kajian kawalan.

Pengenalpastian tak lelurus bagi turus penyulingan telah dibuat dengan

menggunakan dua model tak lelurus iaitu model Hammerstein berasaskan wavenet

dan model auto mundur tak lelurus dengan model input-input luar kawalan (NARX)

berasaskan sigmoidnet. Parameter untuk kedua-dua model tersebut dianggarkan

dengan menggunakan kaedah peminimuman ramalan-ralat berlelar. Penganggaran

parameter, pengesahan model dan analisis model telah dijalankan dengan

menggunakan kotak perkakas bagi sistem pengenalpastian dalam MATLAB dan

keupayaan model-model untuk mewakili dinamik model tak lelurus untuk turus

penyulingan telah disahkan.

Dua jenis teknik NMPC iaitu NMPC model Hammerstein dan NMPC model

NARX telah dibangunkan. Masalah NMPC telah dirumuskan dengan menimbangkan

fungsi objektif, kekangan dikenakan oleh model tak lelurus dan juga pembolehubah-

pembolehubah masukan dan keluaran. Turas Unscented Kalman (UKF) telah

digunakan untuk menganggar pembolehubah keadaan dan permasalahan NLP telah

diselesaikan dengan menggunakan kaedah program kuadratik berjujukan (SQP)

dalam kedua-dua teknik NMPC. Kajian-kajian kawalan gelung tertutup telah

dikendalikan dalam persekitaran MATLAB untuk mengesahkan prestasi teknik-

xxv

teknik NMPC dalam penolakan gangguan-gangguan dan penjejakan titik set. Kajian-

kajian kawalan gelung tertutup ini menunjukkan bahawa prestasi Hammerstein

NMPC adalah lebih baik daripada NARX NMPC dalam pengawalan turus

penyulingan.

xxvi

NONLINEAR MODEL PREDICTIVE CONTROL OF A DISTILLATION COLUMN USING HAMMERSTEIN MODEL AND NONLINEAR

AUTOREGRESSIVE MODEL WITH EXOGENOUS INPUT

ABSTRACT

Distillation column is an important processing unit in petroleum refining and

chemical industries, and needs to be controlled close to the optimum operating

conditions because of economic incentives. Most of the industrial distillation

columns are currently controlled by multiloop controllers based on linear models

which are penalized by several shortcomings. Nonlinear model based control scheme

is one of the best options to be explored for proper control of distillation columns. In

the present work, two nonlinear model predictive control (NMPC) schemes using

Hammerstein model and nonlinear autoregressive model with exogenous input

(NARX) were developed to control distillation column. The binary distillation

column separating methanol-water was used to verify the developed control schemes

performance.

The pilot plant distillation column of 10.2 cm diameter with 15 sieve trays

was designed, fabricated and used in this work. A mathematical model based on total

mass balance, component balance and enthalpy balance was developed based on first

principles. The activity and fugacity calculations were included in the model in order

to account for the non-ideality of the system. A suitable algorithm was developed to

solve the model equations in MATLAB environment. The experiments were carried

out in pilot plant distillation column under steady-state and dynamic conditions to

validate the first principle model. The tray efficiency values used in the first principle

model were tuned using the steady state experimental results. The model results

xxvii

showed a high level of consistency with the experimental results. The validated first

principle model was used as a process model in nonlinear system identification and

control studies.

The nonlinear identification of distillation column was done using two

nonlinear models namely wavenet based Hammerstein model and sigmoidnet based

NARX model. The parameters of both the models were estimated using an iterative

prediction-error minimization method. The parameters estimation, model validation

and model analysis were carried out using system identification toolbox in MATLAB

and the capability of the models to capture the nonlinear dynamics of the distillation

column was verified.

Two types of NMPC techniques namely Hammerstein model NMPC and

NARX model NMPC were developed. The NMPC problem was formulated by

considering the objective function, constraints imposed by nonlinear model as well

as input and output variables. The Unscented Kalman Filter (UKF) was used to

estimate the state variables and the nonlinear programming problem was solved

using sequential quadratic programming (SQP) method in both the NMPC

techniques. The closed loop control studies were conducted in MATLAB

environment to verify the performance of the NMPC techniques in disturbances

rejection and set-point tracking. The closed loop control studies indicated that the

performance of Hammerstein NMPC was superior than NARX NMPC in controlling

the distillation column.

xxviii

1

1 CHAPTER 1

INTRODUCTION

1.1 Distillation

Distillation is one of the most important unit operations in chemical

engineering. The aim of a distillation column is to separate a mixture of components

into two or more products of different compositions. The physical principle of

separation in distillation is the difference in the volatility of the components. The

separation takes place in a vertical column where heat is added to a reboiler at the

bottom and removed from condenser at the top. A stream of vapor produced in the

reboiler rises through the column and is forced into contact with a liquid stream from

the condenser flowing downwards in the column. The volatile (light) components

are enriched in the vapor phase and the less volatile (heavy) components are enriched

in the liquid phase. A product stream taken from the top of the column therefore

mainly contains light components, while a stream taken from the bottom contains

heavy components.

1.2 Distillation equipment

A simple continuous binary tray distillation column for separating a feed

stream into two fractions, an overhead distillate product and a bottoms product is

shown in Figure 1.1. The column is normally provided inside with horizontal plates

or trays.The liquid mixture to be separated is introduced more or less centrally into a

vertical cascade of trays. A reboiler is provided at the bottom of the column to supply

2

Figure 1.1: Schematic of continuous binary tray distillation column

the heat required for the vaporization involved in distillation and also to compensate

for heat loss. A water-cooled or air-cooled condenser is provided at the top of the

column to condense and cool the overhead stream. The purity of the top product can

be improved by recycling some of the externally condensed top product liquid as

reflux from the upper part of the column. The more reflux that is provided, the better

is the column separation of the lower boiling from the higher boiling components of

the feed. The feed tray divides column into two parts namely rectifying section and

stripping section. In rectifying section, the vapor rising is rectified with liquid

flowing down from top to remove less volatile component and in stripping section

the liquid is stripped of volatile components by vapor produced at bottom by partial

LC

AT

AT

Vapor

LCPT

QB

QC

Rectifying section

Stripping section

Reflux drum

Condenser

ReboilerBottom product

Distillate product

Feed

Reflux

Levelcontroller

Assay transmitter

Pressuretransmitter

3

vaporization of bottom liquid in reboiler. The condensed liquid that is removed from

reflux drum is known as distillate or top product and the liquid removed from

reboiler is known as bottom product.

1.3 Need for distillation control

Distillation is used in many chemical processes for separating feed streams

and for purification of final and intermediate product streams. There are many

reasons for the interest in distillation control. From an academic point of view

distillation control is an interesting multivariable problem, and from an industrial

point of view improved distillation control has a potential to substantially increase

the profit. Distillation accounts for approximately 95% of the separation systems

used for refining and chemical industries (Humphrey et al., 1991). It has a major

impact upon the product quality, energy usage, and plant throughput of these

industries. It consumes enormous amounts of energy, both in terms of cooling and

heating requirements. It can contribute to more than 50% of plant operating costs.

The energy requirement may be reduced significantly through improved operations.

This is achieved not only through optimal column design, but requires, in addition, a

control system which is able to maintain the optimal conditions.

Distillation control is a challenging endeavor due to (1) the inherent

nonlinearity of distillation, (2) multivariable interaction, (3) the non-stationary

behavior and, (4) the severity of disturbances (Shinskey, 1984). Tighter control of

distillation columns is consequently important for energy savings, and will also yield

increased profit through improved product recovery. The major benefits of improved

4

distillation control are reduced energy consumption, increased yield and higher

throughput.

1.4 Distillation control techniques

Distillation columns provide a very challenging example within the field of

process dynamics and process control. Traditionally PID controllers were used in the

process industries for control of the distillation column. The main drawback of the

conventional feedback PID control is that corrective action for disturbances does not

begin until the controlled variable deviates from the set point (Skogestad, 1997b). In

industry, most of the columns are operated by single-input single-output (SISO)

controllers and usually only one composition is automatically controlled (one point

control). This leads to waste of valuable products and excessive energy. However,

automatic control of both compositions may be very difficult to obtain due to strong

interaction between top and bottom product compositions (Shinskey, 1984).

Skogestad et al.(1988) have reported that high purity columns, i.e. columns where

both top and bottom compositions are very pure; suffer from strong interaction which

makes the system very sensitive to small changes in the manipulated variables (input

uncertainty). Without a rigorous method for dealing with uncertainty it may be

practically impossible to tune a two point controller for a system with strong

interaction. This may in fact be one of the reasons to why one point control is so

commonly used. Another disadvantage with such a decentralized (multiloop) control

is that the control performance may seriously deteriorate if the system hit some

constraints. For example, if a stabilizing loop saturates, the system goes unstable. To

avoid this, the plant has to be operated sufficiently far away from the constraints, or

5

facilities for reconfiguration have to be installed ‘on-top’ of the SISO controllers

(Lundstrom and Skogestad, 1995).

Configuration selection is an important aspect in the case of multiloop

controller design. Control configuration for a distillation column can be selected

from the knowledge of the thermodynamic parameters, reflux ratio, vapor boil-up

rate and distillate to bottoms ratio for binary and multicomponent distillation

(Stilchmar, 1995). Improper choice of manipulated/controlled variable pairings can

result in poor control performance. Decouplers are introduced into the multiloop

configuration to compensate for the process interactions and reduce the control loop

interactions. Hurowitz et al.(2003) have used decouplers to control the top product

composition using reflux flow rate, and bottom product composition using vapor

boil-up rate for the xylene/toluene column and the depropanizer. In both cases, the

decouplers resulted in improved control performance compared to the feedback

controllers without a decoupler.

The insufficient performance of SISO controllers leads to the development of

specialized single loop control strategies such as feedforward control (Broll et al.,

1995), inferential control (Zhang and Agustriyanto, 2001), cascade control (Kano et

al., 2000), adaptive control (Natarajan et al., 2006) etc. The abilities of the

specialized single-loop control strategies and multiloop controllers were not

satisfactory for increasingly stringent performance requirements of the chemical

processes which led to the development of multivariable control techniques.

6

1.4.1 Multivariable controllers

Processes which are multivariable in nature, i.e. processes where the variables

to control and the variables available to manipulate cannot be separated into

independent loops where one input only would affect one output, constitute a major

source of difficulty in process control. These processes show a certain degree of

interaction, i.e. one control loop affects other loops in some way. The complexity of

the control problem raises as this interaction increases (Luyben, 1992).

Multivariable processes in industrial and other applications are often of higher order,

where there are many, possibly tens or hundreds, of control loops interacting

(Skogestad and Postlethwaite, 1996).

The term multivariable control refers to the class of control strategies in

which each manipulated variable is adjusted on the basis of errors in all of the

controlled variables, rather than the error in the single controlled variable, as in the

case of multiloop control. Multivariable control is particularly well-suited for

controlling processes with several interacting controls which need to be

simultaneously decoupled (Liptak, 2007).

An adequate model is generally considered as a prerequisite for multivariable

controller design. The model is used to predict the behavior of the controlled

variables with respect to changes in the input variables (Sagfors and Waller, 1998).

Established multivariable control techniques rely on the availability of the linear

system models. This is to ensure that the resulting control scheme is closely matched

to the dynamics of the process. The multivariable system must therefore first be

7

modeled either analytically using set of differential equations to describe the system

behavior or empirically by fitting experimental obtained data to an assumed structure

of the process i.e. black-box modeling. Obviously, how well the resulting control

strategy performs depends on the accuracy of the model. In applications where the

physical and/or chemical characteristics of the system are well known, usually the

former approach is adopted. In the process industries, where the higher degree of

uncertainty about the process behavior emprical modeling approach is often

employed. However for control system design purposes, the input-output (transfer

function) model obtained using later approach is generally adequate (Boling et al.,

2004). Multivariable controls strategies can also be developed that include integral,

derivative and feedforward control action. Among the multivariable controllers,

Model Predictive Control (MPC) is an important advanced control technique which

can be used for difficult multivariable control problems (Goodwin et al., 2001).

1.4.2 Model Predictive Control

The term MPC describes a class of computer control algorithms that control

the future behavior of the plant through the use of an explicit process model. At each

control interval the MPC algorithm computes an open-loop sequence of manipulated

variable adjustments in order to optimize future plant behavior. The first input in the

optimal sequence is injected into the plant, and the entire optimization is repeated at

subsequent control intervals (Henson, 1998). MPC technology was originally

developed for power plants and petroleum refinery applications. However, at present

MPC is used in wide variety of manufacturing environments including chemicals,

8

food processing, automotive, aerospace, metallurgy, and pulp and paper (Qin and

Badgewell, 1998).

The success of MPC technology as a process control paradigm can be

attributed to three important factors. First and foremost is the incorporation of an

explicit process model into the control calculation. This allows the controller, in

principle, to deal directly with all significant features of the process dynamics.

Secondly the MPC algorithm considers plant behavior over a future horizon in time.

This means that the effects of feedforward and feedback disturbances can be

anticipated and removed, allowing the controller to drive the plant more closely

along a desired future trajectory. Finally the MPC controller considers process input,

state and output constraints directly in the control calculation. This means that

constraint violations are far less likely, resulting in tighter control at the optimal

constrained steady-state for the process. It is the inclusion of constraints that most

clearly distinguishes MPC from other process control techniques (Qin and Badgwell,

2003).

It is interesting to note that in the early usage of MPC technology, the

nonlinear process behavior was addressed using a linear dynamic model in the

control algorithm. Richalet et al. (1978) have described how nonlinear behavior due

to load changes in a steam power plant application was handled by executing their

Identification and Command (IDCOM) algorithm at a variable frequency. Prett and

Gillette (1980) have applied a Dynamic Matrix Control (DMC) algorithm to control

a fluid catalytic cracking unit and model gains were obtained at each control iteration

by perturbing a detailed nonlinear steady-state model.

9

The original IDCOM and DMC algorithms provided excellent control of

unconstrained multivariable processes. However, constrained handling was not

planned in those two algorithms. Engineers at Shell Oil addressed this weakness by

posing the DMC algorithm as Quadratic Program (QP) in which the input and output

constraints appear explicitly and is known as Quadratic Dynamic Matrix control

(QDMC). Its key features include linear step response model for the plant, quadratic

performance over a finite prediction horizon and future plant output behavior

specified by trying to follow the set point as closely as possible subject to a move

suppression term (Cutler et al., 1983). Even though MPC is having many advantages,

many processes are sufficiently nonlinear to preclude the successful application of

MPC technology. This has led to the development of nonlinear model based

controllers such as nonlinear model predictive control (NMPC) in which more

accurate nonlinear model is used for process prediction and optimization.

1.4.3 Nonlinear Model Predictive Control

NMPC can be applicable to the areas where process nonlinearities are strong

and market demands require frequent changes in operating conditions. There are

cases where nonlinear effects are significant enough to justify the use of NMPC

technology. These include regulator control problems where the process is highly

nonlinear and subject to large frequent disturbances (example: pH control); and servo

control problems where the operating points change frequently (example: Polymer

manufacturing and distillation column) and span a sufficiently wide range of

nonlinear process dynamics (Qin and Badgewell, 1998). The overall basic structure

10

of NMPC control loop is depicted in Fig. 1. The NMPC algorithm (Findeisen et al.,

2000) can be summarized as follows.

The nonlinear dynamic model of the process is used to predict the future

values of the output from the current measurements. Then the appropriate changes in

the input values can be calculated based on both predictions and current

measurements. Based on the predicted values and constraints the optimal control

problem is solved online in the dynamic optimizer. Set of control moves will be

calculated and the first part of the optimal input signal will be implemented until new

measurements or estimates of the state are available. The above procedure will be

repeated for next time instant.

Figure 1.2: Basic NMPC control loop

u y

NonlinearSystemModel

Cost function+

Constraints

DynamicOptimizer

NMPC Controller

Plant

StateEstimator

11

The key characteristics of NMPC are as follows.

i. It allows the use of nonlinear model for prediction

ii. It allows an explicit consideration of state and input constraints

iii. A specified performance criteria is minimized online

iv. In general, the predicted behavior is different from the closed loop

behavior

v. The online solution of an open-loop optimal control problem is necessary

vi. The system states must be measured or estimated to perform the

prediction

The major steps in implementation of NMPC includes development of a

suitable nonlinear process model to be used with NMPC, formulating NMPC

problems with inherently better computational characteristics and finding out an

efficient and reliable solution methods for the nonlinear programming problem with

better computational efficiency.

In their survey of industrial applications of NMPC, Qin and Badgwell (1998)

listed nonlinear model development as one of the three most significant obstacles to

NMPC application, by noting that there is no systematic approach for building

nonlinear dynamic models for NMPC. Lee (1998) drawn the same conclusion by

arguing that the inability to construct, a nonlinear model on a reliable and consistent

basis is the most important reason that nonlinear MPC has less influence on

industrial control practice than linear MPC. He also has a view that nonlinear

dynamics are significant in industrial manufacturing processes.

12

The nonlinearity of the process should be measured in order to select suitable

nonlinear model structure for the process to be used with NMPC. The information

about the nonlinearity of the process will be helpful to make a decision as to whether

it would be worthwhile attempting to identify a nonlinear model of the process and

also to select the model structure. In this context, Pearson (1995) classified processes

using a degree of nonlinearity, i.e. mild, intermediate, or strongly nonlinear based on

the qualitative nature of process nonlinear behavior such as asymmetric response to

symmetric changes in input, input multiplicities i.e., the same output could be

generated by different input magnitudes, output multiplicative behavior, chaotic

dynamics etc. There are three types of nonlinear models available namely

fundamental models, empirical models and hybrid models.

Fundamental model are derived by applying transient mass, energy and

momentum balances to the process. In the absence of spatial variations, the resulting

models have the general form

),( uxfx (1.1)

),( uxg0 (1.2)

),( uxhy (1.3)

where x is a n-dimensional vector of state variables, u is a m-dimensional vector of

manipulated input variables and y is a p-dimensional vector of controlled output

variables. The ordinary differential equations (1.1) and algebraic equations (1.2) are

derived from conservation laws and various constitutive relations, while the output

equations (1.3) are chosen by the control system designer (Pearson, 2003). Since

NMPC is most naturally formulated in discrete time, it is necessary to discretize the

13

continuous time differential equations. This is usually achieved by orthogonal

collocation or finite elements.

Fundamental models for processes of realistic complexity tend to involve on

the order of 102 - 103 nonlinear differential equations and a comparable number of

algebraic relations (Michelsen and Foss, 1996). Further, in many cases it is not

realistic to simplify these models by excluding subtle details. Gross et al.(1998)

made this point strongly, by noting that even construction details of particular

equipment sometimes can have a profound influence on process dynamics. Another

drawback of fundamental model is that the lack of process knowledge often leads to

disappointing results, since it is hard to capture all relevant phenomena in the model.

In many cases, the fundamental models are too complex to be used for

control purposes. Empirical models, also called black-box models are useful in this

scenario (Sjoberg et al., 1995). In empirical modeling, a model structure is first

selected and the model identification problem involves determining the model

parameters that best fit the input-output data. The development of empirical

nonlinear models from plant data is known as nonlinear system identification. In

contrast to linear models, the identification problem is considerably more complex

for nonlinear systems. To begin with, nonlinear models exhibit a diverse range of

nonlinear behavior, and unlike linear models, nonlinear model structures are

generally not equivalent. Because of this diversity, selection of an appropriate model

structure becomes critical. Also, selection of an appropriate input is also considerably

more challenging for nonlinear models than their linear counterparts. As an example,

the PRBS sequence that is widely used for linear model identification is inadequate

14

for identification of a broad class of the block-oriented models (Doyle III et al.,

2002). The input should also possess enough energy to exercise the full range of

process nonlinear behavior, and this persistence of excitation condition, well

established for linear systems, and does not have a well-defined nonlinear equivalent.

The types of discrete time nonlinear models utilized for NMPC in the recent

literature includes Hammerstein models (Jurado, 2006; Harnischmacher and

Marquardt, 2007; Huo et al., 2008), wiener models (Lazar et al., 2007; Shafiee et al.,

2008), NARX model (Lee and Lee, 2005), Nonlinear auto-regressive moving

average model with exogenous inputs (NARMAX) (Zeybek et al., 2006), Volterra

models (Wang and Zhu, 2008), neural network models (Nagy, 2007; Al Seyab and

Cao, 2008a; Al Seyab and Cao, 2008b) and fuzzy models (Cetinkaya et al., 2006;

Prakash and Senthil, 2008).

Hybrid models are developed by combining the fundamental and empirical

modeling approaches. This is the case when some physical insight is available, but

several parameters remain to be determined from observed data. It is useful to

consider two sub cases in this hybrid models namely physical modeling and semi-

physical modeling. In physical modeling, a model structure can be built on physical

grounds, which has a certain number of parameters to be estimated from data. In

semi-physical modeling, physical insight is used to suggest certain nonlinear

combinations of measured data signal. These new signals are then subjected to model

structures of emprical character (Sjoberg et al., 1995).

15

After suitable nonlinear model is developed for the process, the NMPC

problem is formulated by considering the constraints on input and output variables,

as well as constraints imposed by the nonlinear model equations. One of the key

obstacles for a successful application of NMPC in practice is that most existing

NMPC schemes require the explicit state information for the prediction. Since in

practice not all states are available by measurements, a suitable observer for the

estimation of the system states must be used (Findeisen et al., 2003).

NMPC requires the repeated on-line solution of a nonlinear optimal control

problem. In the case of linear MPC the solution of the optimal control problem can

be cast as the solution of a (convex) quadratic program and can be solved efficiently

even on-line. This can be seen as one of the reasons why linear MPC is widely used

in industry. For the NMPC problem the solution involves the solution of a nonlinear

program. In general the solution of a nonlinear (non-convex) optimization problem

can be computational expensive. One could address the solution of nonlinear

programming problem for the purpose of model predictive through successive

linearization of model equations, sequential model solution and simultaneous model

solution.

1.5 Problem statement

NMPC has been around for many years and has been scientifically discussed

extensively, but several issues that affect the industrial practice of NMPC are yet to

be resolved. The NMPC approach assumes availability a suitable nonlinear dynamic

model of the controlled process. In most application studies of NMPC, the nonlinear

16

model is readily obtained due to the simplicity of the process considered. The

nonlinear modeling problem is significantly more challenging for large scale

complex processes. Consequently, the development of nonlinear model is of highest

importance to the continued advancement of NMPC. Foss et al. (1998), in their case

study on process modeling in Germany and Norway concluded that despite the

commercially available modeling tools, the effort spent for all kinds of modeling

activities is the most time consuming step in an industrial project where model based

process engineering techniques are applied.

Many researchers (Eskiant et al., 1991; Sriniwas et al., 1995; Fruzzetti et al.,

1997) already proved that the performance of linear models is insufficient in

capturing the dynamics of the distillation column due to its nonlinear nature. Henson

(1998) also has drawn the same conclusion by arguing that many nonlinear processes

including distillation column are sufficiently nonlinear to preclude the successful

application of linear models. Hence the development of nonlinear process models is

tremendously essential due to the unavoidable nonlinearity of the process and

complexity of nonlinear system.

The practical difficulty of nonlinear dynamic model development arises from

several sources, of which the following two are fundamental. First is the fact that

model utility can be measured in several, generally conflicting ways. Second, the

class of nonlinear models does not exhibit the unity that the class of linear models

does. The four extremely important measures of model utility are approximation

accuracy, physical interpretation, suitability for control and ease of development.

17

Fundamental model is generally far superior to empirical moel and hybrid

model with respect to the first two of these criteria, but they also suffer badly with

respect to the last two. On the other hand, the empirical model does not require the

detailed process understanding for model development and also, complexity of the

model can be avoided. The main drawback of empirical model is that the nonlinear

model identification problem is very tedious. In the case of hybrid models, it is very

difficult to distinguish the particular part of the process to be modeled using

fundamental model or empirical model.

The NMPC problem formulation involves online computation of a sequence

of manipulated inputs which optimize an objective function and satisfy process

constraints. The development of NMPC techniques for large scale systems may

require problem formulations which exploit the specific structure of the nonlinear

model. Finally, NMPC requires online solution of a nonlinear program (NLP) at each

iteration. The solution of such NLP problems can be very time consuming, especially

for large scale systems. An additional complication is that the optimization problem

generally is nonconvex because the nonlinear model equations are posed as

constraints (Cannon, 2004). Consequently, NLP solvers designed for convex

problems may converge to local minima or even diverge. So it is necessary to find

out an improved solution algorithm for nonconvex NLP problems. The vital parts of

the present research are to develop suitable nonlinear models for distillation column,

formulate NMPC problem and to identify an efficient optimization algorithm to be

used with NMPC.

18

1.6 Research objectives

The objectives of this study are

1. To design a pilot plant distillation column, condenser and reboiler, and

to fabricate the experimental set up based on design specifications.

2. To identify a suitable first principle model to be used as process model

in nonlinear model identification and in NMPC, and to develop an

algorithm to solve the model equations in MATLAB environment.

3. To validate a first principle model through experimentation using pilot

plant distillation column, and to conduct open loop simulation studies

under steady- and unsteady-state conditions.

4. To develop suitable nonlinear emprical models and to identify an

efficient optimization algorithm to be used with NMPC.

5. To design and to evaluate the performance of NMPC for different

changes in disturbances and set-points using closed loop simulation

studies.

1.7 Scope of work

The main focus of the present work is to develop suitable nonlinear models

and to identify an efficient optimization algorithm to be used with NMPC to control

a distillation column.

19

The detailed design of binary sieve tray distillation column is carried out

based on the VLE data of methanol-water system. Based on the tray spacing and the

plate hydraulics, the design details of the tray such as tray thickness, weir length,

weir height, hole size, hole pitch, number of holes etc. are calculated. A reboiler and

a total condenser are designed based on the condenser duty and heat duty for the

system. A detailed process design and mechanical design of plates are also carried

out. Stainless steel is the material of construction for the column, condenser, reboiler

and column internals, and rock wool with aluminum foil is used as insulation

material. Provisions are provided to take the sample and to measure the temperature

in each tray. Horizontal in-shell TEMA E-type condenser with vertical baffle cuts is

used and water is used as a cooling medium in the condenser. A kettle type reboiler

with resistance type electrical heaters is used. The column is commissioned and the

required instruments like flow meters, temperature measuring devices, pressure

measuring device, level measuring devices, and transmitters are installed. A few test

runs are made to calibrate the instruments and also to check the proper functioning of

the various signal processing units.

A mathematical model based on total mass balance, component balance and

enthalpy balance is developed based on first principles. The activity and fugacity

calculations are included in the model in order to account for the non-ideality of the

system. A suitable algorithm is developed to solve the model equations using

MATLAB. Simulation studies are carried out in the MATLAB environment for

steady-state and unsteady-state conditions.

20

The experiments are carried out to study the steady-state and dynamic

characteristics of the column. The steady-state experimental data is used to fix

optimum operating conditions. Then dynamic studies involving changes in feed flow

rate, feed composition and reflux flow rate are carried out and from those results the

dynamic behavior of the system is obtained. The result of simulation studies are

compared with experimental data. The tray efficiencies used in the simulation studies

are calibrated to match the first principle model results with experimental results and

the developed first principle model is validated.

Suitable nonlinear empirical models are developed and used with NMPC. The

main factor considered for nonlinear model development is that the model developed

for binary distillation column should have the flexibility to extend to the

multicomponent distillation column. Two nonlinear models namely wavenet based

Hammerstein model and sigmoidnet based NARX mode are found to be good in

capturing the nonlinear dynamics of the distillation column and also they can be

easily modified for multicomponent distillation column. The data required for

nonlinear model parameter estimation and validation are generated from

experimentally validated first principle model. Two types of NMPC techniques

namely Hammerstein model NMPC and NARX model NMPC are developed. The

NMPC problem is formulated by considering the constraints imposed by nonlinear

model, input and output variables. The sequential quadratic programming (SQP)

method is used in both the NMPC techniques to solve the NMPC problem.

The control problem is solved in MATLAB environment. The top and bottom

product compositions are the controlled variables in distillation column. The reflux

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flow rate and reboiler vapor boil up rate are used as manipulated variables, whereas

feed flow rate and feed composition are considered as disturbances. The closed loop

simulation studies are carried out in MATLAB environment to verify the

performance of both NMPC techniques for different disturbances, changes in set

points and simultaneous changes in disturbances and set points.

1.8 Organization of thesis

This thesis consists of five chapters. Chapter 1 provides a brief description of

distillation process, distillation equipment, need for distillation control, and

distillation control techniques including the advanced control strategies like MPC

and NMPC. This chapter also includes the problem statement that provides

foundation to identify research directions and objectives. The objectives and scope of

study are then elucidated followed by the organization of the thesis.

Chapter 2 summarizes the past research works in the field of nonlinear

modeling of distillation column including nonlinear characteristics, fundamental

models, empirical models and hybrid models. The NMPC problem formulation and

various optimization algorithms used with NMPC were discussed. Finally, current

applications of NMPC technology were discussed along with the advantages and

drawbacks. This chapter serves as the background information about the specific

problems that have to be addressed in this research work.

Chapter 3 presents the details of the materials, chemicals and research

methodology used in the present study. The design specifications of pilot plant

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distillation column are explained and the experimental set up is elaborated along with

the specifications of other instruments. The development of first principle model for

distillation which is used as a platform in this research work is described along with

solution algorithm. Finally, the methodology for the development of NMPC and

closed loop control studies are presented.

Chapter 4 is the main part of the thesis in which all important findings and

results of this research work are discussed. This chapter includes experimental

validation of fist principle model, steady-state and unsteady-state simulation results

using first principle model, nonlinear identification of Hammerstein model and

NARX model, NMPC control problem formulation, sequential quadratic

programming (SQP) optimization algorithm as well as results of closed loop

simulation studies to validate NMPC controller performance.

Chapter 5 summarizes the results reported in the previous chapters and also

some concluding remarks are made based on those results. The conclusions are

obtained from each individual study carried out in the present research work. This

chapter also suggests the ways to improve the present work and recommend the

possible future studies in this field. These recommendations and suggestions are

given after taking into consideration of significant findings, limitations, the

conclusions obtained as well as difficulties encountered in this study.

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2 CHAPTER 2

LITERATURE REVIEW

2.1 Importance of distillation control

Malaysia is a significant Southeast Asian producer of oil and natural gas.

According to Oil & Gas Journal (OGJ), Malaysia held proven oil reserves of 3.0

billion barrels and 75 trillion cubic feet (Tcf) of proven natural gas as of January

2007 (Country analysis brief: Malaysia, 2007). Most of the separation processes

employed in petroleum refineries and other chemical processing industries (CPI) are

distillation columns for separating feed streams, and for purification of final and

intermediate product streams. The separation needs relatively large amount of energy.

Close control of distillation column improves the product quality, minimizes energy

usage and maximizes the plant throughput and its economy (Hurowitz et al., 2003).

Also producing products with low variability is many times crucial for the success of

CPI (Downs and Doss, 1991). For most high-value added products, low variability is

a primary customer concern and can determine the market demand of a product. The

reduction in the variation in the products can also be used to increase production

rates or decrease utility usage (Riggs, 2001). Clearly, reduced variability is

economically important to the CPI and can be achieved by proper control of unit

operation in the plant, especially the distillation column. Hence, the development of

nonlinear model based control system for distillation column would be beneficial for

CPI.

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2.2 Major disturbance in distillation control

The type and magnitude of disturbances affecting a distillation column have

a direct effect on the resulting product variability. An analysis of the major types of

disturbances encountered in distillation column are mentioned in the following

sections.

2.2.1 Feed composition upsets

Changes in the feed composition represent the most significant upsets with

which a distillation control system must deal on a continuous basis. Most industrial

columns do not have a feed composition analyzer; therefore, feed composition upsets

usually appear as unmeasured disturbances. When a feed composition analyzer is

available, a feed forward controller can be applied using the on-line measurements of

the feed composition (Stichlmair, 1995). Feed composition changes represent a

major disturbance for distillation control, thus the sensitivity of potential control

configurations to feed composition upsets is a major issue for configuration selection.

Luyben (2005) has studied the effect of feed composition on the selection of control

structures for high purity binary distillation column using methanol-water system. He

concluded that feed composition changes largely affect the product purities compare

to other disturbances. Zhang et al. (2006) noted that a feed composition change shifts

the composition profile through the column resulting in a large upset in the product

compositions.