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Doctoral Thesis

Robust control of an industrial high-purity distillation column

Author(s): Musch, Hans-Eugen

Publication Date: 1994

Permanent Link: https://doi.org/10.3929/ethz-a-000959203

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Diss. ETHNo. 10666 20. JUll KWH

Ma,

Robust Control of an

Industrial High-Purity

Distillation Column

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

for the degree of

Doctor of Technical Sciences

presented by

HANS-EUGEN MUSCH

Dipl. Chem.-Ing. ETH

born June 19,1965

citizen of Germany

accepted on the recommendation of

Prof. M. Steiner, examiner

Prof. Dr. D. W. T. Rippin, co-examiner

1994

Leer - Vide - Empty

3

To my grandparents

4

5

Acknowledgments

This Ph. D. thesis was written during my years as a research and educa¬

tional assistant of the Measurement and Control Laboratory at the

Swiss Federal Institute of Technology (ETH) at Zurich. I would like to

take this opportunity to thank the numerous persons who have

supported this project.

First of all I express my gratitude to Prof. M. Steiner. He arranged this

project and helped to overcome many difficulties with the industrial

environment. Many thanks are also due to him and to Prof. D. W. T.

Rippin for the critical examination of this thesis, which essentially

improved its clarity.

The numerous discussions with my colleagues and their uncountable

suggestions gave rise to important contributions to this work. In this

context, E. Baumann, U. Christen, and S. Menzi must be speciallymentioned.

Last but not least I should emphasize the support of B. Rohrbach. She

never lost her patience with my never ending questions concerning the

English language. Without her willingness to correct the manuscript,the choice of the English language for this thesis would have been

impossible.

6

7

Content

Symbols 13

Abstract 15

Kurzfassung 17

1 Introduction 19

1.1 "Modern Control: Why Don't We Use It?" 19

1.2 Scope and significance of this thesis 21

1.2.1 Distillation as a unit operation example 21

1.2.2 Earlier research 21

1.2.3 Robust control and nonlinear plants 22

1.2.4 Contributions of this thesis 22

1.3 Structure of the dissertation 23

1.4 References 26

2 The Distillation Process —

An Industrial Example 29

2.1 Introduction 29

2.2 Column design and operation 29

2.3 Steady-state behavior 32

2.4 Composition dynamics 35

2.5 Control objectives and configurations 37

2.5.1 The 5x5 control problem 39

2.5.2 Control design steps 40

8

2.6 Tray temperatures as controlled outputs 41

2.6.1 Pressure-compensated temperatures 42

2.6.2 Temperature measurement placement 44

2.7 References 45

3 A Rigorous Dynamic Model of

Distillation Columns 47

3.1 Introduction 47

3.2 Conventions 48

3.3 The objective of modelling 48

3.4 Simplifying assumptions 48

3.5 Balance equations 51

3.5.1 Material balances 51

3.5.2 Energy balance equations 52

3.6 Fluid dynamics 55

3.6.1 Liquid flow rates 55

3.6.2 Pressure drop 57

3.7 Phase equilibrium 59

3.7.1 Vapor phase composition 59

3.7.2 Boiling points 60

3.8 Volumetric properties 60

3.8.1 PVT relations 61

3.8.2 Density 61

3.9 Enthalpies 62

3.10 Numerical solution 63

3.10.1 The dependent variables and the equation system... 63

3.10.2 Formal representation of the DAE 66

3.10.3 The index 66

3.10.4 Solution methods and software 67

3.11 Notation 71

3.12 References 74

4 Linear Models 77

4.1 Introduction 77

4.2 How to linearize the rigorous model? 78

4.2.1 The state, input, and output vectors 78

4.2.2 Handling of the algebraic equation system 80

4.3 Linearization of a simplified nonlinear model 80

4.3.1 The simplified model 80

4.3.2 Analytical linearization 84

4.4 Linearization of the rigorous model 86

4.4.1 Model modifications 86

4.4.2 Numerical linearization 88

4.5 Comparison of the linear models 89

4.5.1 Open loop simulations 89

4.5.2 Singular values 92

4.6 Order reduction 94

4.7 Summary 96

4.8 Appendix: Model coefficients 97

4.9 Notation 101

4.9.1 Matrices and Vectors 101

4.9.2 Scalar values 102

4.10 References 103

5 A Structured Uncertainty Model 105

5.1 Introduction 105

5.2 Limits of uncertainty models 106

5.3 Input uncertainty 107

5.4 Model uncertainty 110

5.4.1 Column nonlinearity 110

5.4.2 Unmodelled dynamics 117

5.5 Measurement uncertainty 118

5.6 Specification of the controller performance 119

5.7 Summary 120

5.8 References 122

6 |0,-Optimal Controller Design 123


6.2 The structured singular value 124

6.2.1 Representation of structured uncertainties 124

6.2.2 Definition of the structured singular value 126

6.2.3 Robustness of stability and performance 128

6.3 The design model 130

6.4 Controller design with u-synthesis 133

6.4.1 Synthesis algorithms 134

6.4.2 Applying the DK-Iteration 137

6.4.3 Applying the uK-Iteration 137

6.5 Design of controllers with fixed structure 148

6.5.1 Diagonal PI(D) control structures 149

6.5.2 PI(D) control structures with two-way decoupling ...156

6.5.3 PID control structures with one-way decoupling 161

6.6 Summary 164

6.7 References 166

7 Controller Design for

Unstructured Uncertainty —

A Comparison 169


7.2 Diagonal Pl-control 170

7.2.1 The BLT method 170

11

7.2.2 Sequential loop closing 172

7.2.3 Optimized robust diagonal Pi-control 174

7.3 Pi-control with decoupling 177

7.4 H„ optimal design 182

7.5 Summary 187

7.6 References 187

8 Feedforward Controller Design 189


8.2 The design problem 190

8.2.1 The design objective 190

8.2.2 One-step or two-step design? 190

8.3 Hro-minimization 192

8.4 Optimization approach 196

8.5 Summary 199

8.6 References 200

9 Practical Experiences 203


9.2 Controller implementation 204

9.3 Composition estimators 207

9.4 Controller performance 208

9.5 Economic aspects 214

9.6 Summary 214

10 Conclusions and

Recommendations 217


12

10.2 Controller synthesis 218

10.3 State-space or PID control? 219

10.4 How many temperature measurements? 220

10.5 Column models 221

10.6 Recommendations 221

10.6.1 Academic research 221

10.6.2 Decentralized control systems 222

10.6.3 Cooperation industry—university 223

Curriculum vitae 225

Symbols

8 Uncertainty scalar value

A Uncertainty matrix or deviation from nominal operating point

8 Parameter vector

k Condition number, k = ov /o_.ind.x nun

X Eigenvalue

(j, Structured singular value

p Spectral radius

a Singular value

B Bottom product stream (mol/s)

D Distillate stream (mol/s) or diagonal scaling matrix

d Disturbance signals

e Control error

F Feed flow rate (mol/s)

7t Lower fractional transformation

G(s) Transfer function

Gu Transfer function from control signals u to output signals y

I Identity matrix

K(s) Controller

L0 Reflux (mol/s)

M Joint weighted plant and controller, M (P, K) = ^(P, K)

P Weighted plant

p Pressure (N/m2)

r Reference signals

Se(s) Sensitivity function at e, Se (s) = [I + G (s) K (s) ] -1

Su(s) Sensitivity function at u, Su(s) = [I + K(s)G(s)]_1

T Temperature (°C)

Tr Transfer function from reference signals to output signals

u Control signals

V51 Boilup (mol/s)

W(s) Diagonal matrix ofweighting transfer functions

w(s) Weighting transfer function

xrj Top product composition (mol/mol)

xg Bottom product composition (mol/mol)

xF Feed composition (mol/mol)

y Output signals

15

Abstract

It is well known that high-purity distillation columns are difficult to

control due to their ill-conditioned and strongly nonlinear behavior.

Usually distillation columns are operated within a wide range of feed

compositions and flow rates, which makes a control design even more

difficult. Nevertheless, a tight control of both product compositions is

necessary to guarantee the smallest possible energy consumption, as

well as high and uniform product qualities.

This thesis discusses a new approach for the dual composition control

design, which takes the entire operating range of a distillation column

into account. With the example of an industrial binary distillation

column, a structured uncertainty model is developed which describes

quite well the nonlinear column dynamics with several simultaneous

model uncertainties. This uncertainty model forms the basis for feed¬

back controller designs by |x-synthesis or u-optimization. The resultingcontrollers are distinguished by a high controller performance and highrobustness guaranteed for the entire operating range. This method

enables the synthesis of state-space controllers as well as the u-optimal

tuning of advanced PID control structures.

The already satisfactory compensation of feed flow disturbances can be

improved even further by use offeedforward control. Even for the designof the feedforward controllers the basic ideas of the feedback controller

design can be employed. A simultaneous feedforward controller designfor two column models representing the extreme column loads yields

outstanding results. Similar to the feedback controller design, a designof state-space controllers by Hm-minimization or an optimal tuning of

simple feedforward control structures by parameter optimization is

possible.

Control engineers working in an industrial environment are conscious

of the high effort needed for the implementation of state-space control-

16

lers in a distributed control system. Therefore a controller design based

on PID or advanced PID control structures is of high relevance for the

industrial practice. Usually, the performance ofthese PID control struc¬

tures is expected to lag significantly behind the performance of high-

order state-space controllers. However, comparing the performances of

the state-space controllers with those of the advanced PID controllers,

merely slight advantages of the state-space controllers are detected.

This surprising result, achieved with an unconventional tuning of the

PID control structures, allows the simple implementation of advanced

PID control structures in a decentralized control system without a

significant loss of controller performance.

The good robustness properties and the high performance of the control

schemes are confirmed by the implementation of an advanced PID

control scheme on a real industrial distillation column. An estimation of

the economic benefits made by this project much more than justifies the

effort expended.

17

Kurzfassung

Bekanntermafien sind Rektifikationskolonnen mit hohen Produktrein-

heiten wegen ihres schlecht konditionierten und stark nichtlinearen

Verhaltens schwierig zu regeln. Haufig werden sie in einem weiten

Bereich unterschiedlicher Zulaufkonzentrationen und -mengen

betrieben, was den Entwurf von Regelungen zusatzlich erschwert.

Dennoch ist eine gute Regelung beider Produktkonzentrationen

notwendig, um einerseits einen moglichst kleinen Energieverbrauchund andererseits hohe und einheitliche Produktqualitaten sicher-

zustellen.

Diese Arbeit beschreibt einen neuen Ansatz fur den Entwurf von

Konzentrationsregelungen, der den gesamten Arbeitsbereich einer

Rektifikationskolonne berucksichtigt. Am Beispiel einer industriellen

binaren Rektifikationskolonne wird ein strukturiertes Unsicherheits-

modell entwickelt, welches das nichtlineare dynamische Verhalten der

Rektifikationskolonne durch mehrere Modell-Unsicherheiten gut

beschreibt. Dieses Unsicherheitsmodell bildet die Basis fur den

Entwurfvon Reglern mittels u-Synthese oder u-Optimierung. Die resul-

tierenden Regler zeichnen sich durch eine - iiber den gesamtenBetriebsbereich garantierte - hohe Regelqualitat bei sehr grosser

Robustheit aus. Dieses Vorgehen erlaubt sowohl den Entwurf von

Zustandsregelungen als auch die Berechnung u-optimaler Einstel-

lungen fur erweiterte PID-Regelstrukturen.

Die bereits zufriedenstellende Unterdriickung von Storungen der

Zulaufmenge wird durch den Einsatz einer Storgrofienaufschaltungnoch verbessert. Auch fur ihren Entwurf kdnnen ahnliche Konzepteverwendet werden. Ein Entwurf von Storgrossenaufschaltungen, bei

dem gleichzeitig zwei Modelle der Rektifikationskolonne berucksichtigtwerden, welche die extremen Kolonnenbelastungen wiedergeben, fuhrt

zu hervorragenden Ergebnissen. Vergleichbar mit dem Regelungs-entwurf konnen sowohl Storgrossenaufschaltungen mit der Struktur

18

von Zustandsregelungen (durch Minimierung der H^-Norm) als auch

Storgroflenaufschaltungen mit einfacher Struktur (durch Parameter-

optimierung im Zeitbereich) berechnet werden.

In der industriellen Praxis tatige Regelungstechniker sind sich der

Schwierigkeiten, die mit der Realisierung von Zustandsregelungen auf

dezentralen ProzelJleitsystemen verbunden sind, sicherlich bewufit.

Daher ist der Regelungsentwurf aufder Grundlage von PID- oder erwei-

terten PID-Regelstrukturen von hoher praktischer Relevanz. Meist

bleibt die mit solchen Strukturen erzielbare Regelgiite hinter der von

Zustandsregelungen deutlich zuriick. In dieser Arbeit werden die

entworfenen Zustandsregelungen und die optimal eingestellten fortge-

schrittenen PID-Regelstrukturen verglichen. Dabei zeigt sich, dafi auch

mit einfachen Regelstrukturen, die entsprechenden unkonventionellen

Regler-Einstellungen vorausgesetzt, eine Regelqualitat erzielt wird, die

der von Zustandsregelungen nahekommt. Dieses iiberraschende

Resultat erlaubt die einfache Implementierung von erweiterten PID-

Regelstrukturen in dezentralen ProzelJleitsystemen ohne wesentlichen

Verlust an Regelgiite.

Die Erprobung eines Regelungsentwurfs auf der Grundlage fort-

geschrittener PID-Strukturen an der industriellen Rektifikations¬

kolonne bestatigt die grofie Robustheit und die hohe Regelgiite in der

Praxis. Dabei zeigt eine Abschatzung der Wirtschaftlichkeit, dafi der bei

einem solchen Projekt notwendige Aufwand mehr als gerechtfertigt ist.

1.1 "Modern Control: Why Don't We Use It?" 19

Chapter 1

Introduction

1.1 "Modern Control: Why Don't We Use It?"

"Modern Control: Why Don't We Use It?" is the title of a paper written

by R. K. Pearson in 1984 [1.4]. In the first section of that paper Pearson

states: "Advanced control systems utilizing multivariable strategies

based on process models can outperform traditional designs in broad

classes of application. Yet, in spite of market forces demanding better

process performance and ample evidence showing that the improve¬

ments can be achieved, the gap between theory and practice in the

industrial sector is not narrowing appreciably."

Ten years later the situation has not changed. The modern control theo¬

ries provide the process control engineer with increasingly sophisticated

tools for a robust, model-based controller design. The advantages of

these controllers over the PID control structures which are usually

tuned on-line, have been shown in numerous publications. Neverthe¬

less, more than 90% of all control loops in the process industry use PID

control, while only a few applications of the modern control theories can

be reported [1.10]. Therefore the mismatch between theory and practice

is still evident. Some of the reasons for this situation are discussed

below.

20 1 Introduction

Distributed Control Systems

For a control engineer in the process industry, process control in the first

place is a hardware problem. His perspective is the installation and

configuration of a Distributed Control System (DCS) [1.1]. Even the

modern DCS are often limited to PID and advanced PID control. For the

DCS, an implementation of modern state space controllers requireseither the coupling with an external computer or the programming of

software modules. Both ways are troublesome and expensive. The

university research pays little attention to this situation. The design of

robust controllers with fixed structures (e.g., PID control structures) is

a largely unexplored field.

Dynamic Models

Linear dynamic models are the foundation of a modern, robust

controller design. However, no general dynamic models are available for

unit operations. For each plant linear dynamic models must be devel¬

oped, based on either linearization of nonlinear models or on system

identification methods. Both ways are often expensive and very time-

consuming ([1.5], [1.6]). Furthermore, most plants in the process

industry show a strongly nonlinear dynamic behavior, which is unsatis¬

factorily described by a single linear model.

Economic benefits

The economic benefits of improved control tend to be significantly

underestimated. A benchmark study by ICI "indicated that the effective

use of improved process control technology could add more than one

third to the worldwide ICI Group's profits" [1.1]. Another study shows

smaller, but still massive benefits [1.2].

Of course it is not necessary to replace all PID-controllers by modern

advanced control structures. Most control problems in the process

industry are handled well with simple PID control. However, strongly

nonlinear or/and ill-conditioned plants require advanced control tech¬

niques for a high controller performance.

1.2 Scope and significance of this thesis 21

1.2 Scope and significance of this thesis

1.2.1 Distillation as a unit operation example

Distillation is one of the most widely used unit operations in the process

industry. In the simplest case, a distillation column separates a feed of

two components into a top product stream (with a high fraction of the

low-boiling component) and a bottom product stream (with a high frac¬

tion of the high-boiling component). In an industrial setting, the feed

flow rate and the feed composition may vary within a wide range ofoper¬

ating conditions.

This separation consumes a huge amount of energy. A minimization of

the energy consumption and an economic optimal operation usually

require (1) a tight control of both product compositions (dual composi¬

tion control) and (2) often small fractions of impurities in the product

streams (high purity distillation). However, the strongly nonlinear and

ill-conditioned behavior makes high-purity distillation columns difficult

to control. Therefore high-purity distillation columns have become an

interesting test case for robust control design methods.

1.2.2 Earlier research

Without any doubt the distillation process is the most studied unit oper¬

ation in terms of control. Skogestad estimates that new papers in this

field appear at a rate of at least 50 each year [1.7]. It is practically

impossible to give a review of all these publications. The interested

reader is advised to consult the reviews of Tolliver and Waggoner [1.8],

Waller [1.9], MacAvoy and Wang [1.3], and the recent review of

Skogestad [1.7].

If we focus our interest on the design of linear, time-invariant control¬

lers, we must state that all the well-known model-based and robust

control design methods (LQG/LTR, H^, Normalized Coprime Factoriza¬

tion, u-synthesis, etc.) have been applied to distillation columns.

However, all these publications discuss the controller design forjust one

operating point. The problem designing a robust controller which maxi-

22 1 Introduction

mizes the controller performance for the entire operating range has not

been addressed as yet.

1.2.3 Robust control and nonlinear plants

The well-known robust control design methods like HM -minimization or

LQG/LTR are based on the assumption of an unstructured, frequency

dependent uncertainty at one location in the plant. Such an unstruc¬

tured uncertainty may be a multiplicative uncertainty at plant input or

output, or an additive uncertainty.

A controller design for the entire operating range of a distillation

column using one of these well-known methods has two inherent prob¬lems:

• Due to the high nonlinearities an estimation of unstructured

uncertainty bounds will lead to very large bounds, prohibiting

any acceptable controller design.

• A controller design using any arbitrary, smaller uncertaintybound guarantees robust performance (RP) and robust stability

(RS) for the actual operating point, but not for the entire oper¬

ating range.

1.2.4 Contributions of this thesis

This thesis presents a new approach for the composition control designof a binary distillation column (Figure 1.1). The design concept is based

on a structured uncertainty model which describes the column dynamicsfor the entire operating range quite well. The resulting controller

designs using u-synthesis (for state-space controller) or u-optimization(for controllers with fixed structure), respectively, lead to results which

guarantee robust performance and robust stability for the entire oper¬

ating range of the distillation column. Special emphasis is placed on the

optimal tuning of easy-to-realize PID-control structures. It will be

shown that extraordinary controller performance can be achieved even

with these relatively simple controller structures.

1.3 Structure ofthe dissertation 23

Standard approaches

Linear model for a

single operating point

Robust control design

IL LQG/LTR,

Weak point:

Improved approach

Uncertainty model

describing column dynamics

for entire operating range

(i-synthesis

(X-optimization

Advantage:

RS & RP guaranteed

for whole operating range

Figure 1.1: Robust control design approaches

1.3 Structure of the dissertation

A robust, model-based controller design for a distillation column

consists of several steps. A typical course is illustrated in Figure 1.2.

The results and methods of each step influence all the following steps.

The consideration ofjust one of these design steps, disengaged from all

others, neglects the conceptional coherence. Therefore all of the design

steps are discussed within this thesis. The sequence orients itself to the

natural course of the controller design.

24 1 Introduction

Nonlinear Model

Uncertainty structure

Controller synthesis

Nonlinear simulations

Tests on plant

Implementation in DCS

Figure 1.2: Steps of a model based controller design

The following chapter consists of three parts: The first part describes

the design and operating data of the distillation column, followed by an

overview of the steady-state and dynamic column behavior. The second

part discusses the control objectives and control configuration for this

column, while the third part describes the use of pressure-compensated

temperatures as controlled outputs.

Rigorous nonlinear dynamic models are the basis for simulation studies

and for linearization. They are discussed in Chapter 3.

1.3 Structure of the dissertation 25

The main subject of Chapter 4 is the derivation of linear models. Two

different methods are presented which lead to linear models which

neglect and include flow dynamics, respectively.

A structured uncertainty model which describes the nonhnear behavior

of the distillation column for the entire operating range is developed in

Chapter 5.

Based on that structured uncertainty model, controllers can be designed

within the framework ofthe structured singular values. In the first part

ofChapter 6 the theoretical background ofthe structured singular value

\i is summarized. While the second part of that chapter presents the u-

optimal design of state-space controllers, the third part is dedicated to

the u-optimal design of PID control structures. Simulation studies

confirm the theoretical results.

In Chapter 7 the results ofthe (i-optimal controller design are compared

with results obtained by more common design methods, based on an

unstructured uncertainty description.

Usually the feed flow rate is a measured disturbance input to a distilla¬

tion column. Therefore, feedforward control can significantly improve

the compensation of feed flow disturbances, which is discussed in

Chapter 8.

A controller design should yield a satisfactory control quality not only in

dynamic simulations but also in the real plant. The results of the prac¬

tical implementation are presented in Chapter 9.

The conclusions and the recommendation for further research in

Chapter 10 complete this thesis.

The literature references and, if necessary, the special notations are

given at the end of each chapter.

26 1 Introduction

1.4 References

[1.1] Brisk, MX.: "Process Control: Theories and Profits," Preprints of

the 12th World Congress of the International Federation ofAuto¬

matic Control, Sydney, July 18-23, 7, 241-250 (1993)

[1.2] Marlin, T. E., J. D. Perkins, G. W. Barton, and M. L. Brisk: "Ben¬

efits from process control: results of a joint industry-university

study," J. Proc. Cont, 1, 68-83 (1991)

[1.3] McAvoy, T. J. and Y. H. Wang, "Survey of Recent Distillation

Control Results," ISA Transactions, 25,1, 5-21 (1986)

[1.4] Pearson, R. K: "Modern Control: Why Don't We Use It?," InTech,

34, 47-49 (1984)

[1.5] Schuler, H., F. Algower, and E. D. Gilles: "Chemical Process

Control: Present Status and Future Needs— The View from Eu¬

ropean Industry," Proceedings of the Fourth International Con¬

ference on Chemical Process Control, South Padre Island, Texas,

February 17-22, 29-52 (1991)

[1.6] Schuler, H.: "Was behindert den praktischen Einsatz moderner

regelungstechnischer Methoden in der Prozess-Industrie," atp,

34, 3, 116-123 (1992)

[1.7] Skogestad, S.: "Dynamics and Control of Distillation Columns -

a Critical Survey," Preprints of the 3rd IFAC Symposium on Dy¬

namics and Control of Chemical Reactors, Distillation Columns

and Batch Processes, April 26-29, College Park, Maryland, 1-25

(1992)

[1.8] Tolliver, T. L. and R. C. Waggoner: "Distillation Column Control;

a Review and Perspective from the CPI," Advances in Instrumen¬

tation, 35, 1, 83-106 (1980)

[1.9] Waller, K. V.: "University Research on Dual Composition Con¬

trol of Distillation: A Review", Chemical Process Control 2, Sea

Island, Georgia, January 18-23, 395-412 (1981)

1.4 References 27

[1.10] Yamamoto, S. and I. Hashimoto: "Present Status And Future

Needs: The View from Japanese Industry," Proceedings of the

Fourth International Conference on Chemical Process Control,

South Padre Island, Texas, February 17-22, 1-28 (1991)

28 1 Introduction

2.1 Introduction 29

Chapter 2

The Distillation Process —

An Industrial Example

2.1 Introduction

A distillation column is not just any mass-produced article such as a

toaster or a washing-machine. Each distillation column is a unique

process unit, specially designed for the separation of a particularsubstance mixture. Nevertheless, the thermodynamic principles and

basic dynamics are always the same. Therefore it is possible to demon¬

strate ideas for the controller design by the example of one column

without extensive loss of generality.

First in this chapter, the design and operating data of the industrial

distillation column are outlined, followed by a brief description of the

composition dynamics. The further two sections outline the control

objectives, the control structures, and the use of tray temperatures as

controlled outputs. The literature references terminate the chapter.

2.2 Column design and operation

The distillation column described in this thesis is an industrial binarydistillation column. A synopsis ofthe most important data for this distil-

30 2 The Distillation Process — An Industrial Example

lation column is given in Table 2.1. The distillation column (Fig. 2.1) is

equipped with 50 sieve trays, a total condenser, and a steam-heated

reboiler. The subcooled feed F enters the column on tray 20 (counted

from the top) and for the greater part consists of a mixture of two

substances. Because of the small fraction of impurities, these are

neglected and the distillation column is considered to be a binary distil¬

lation column. The desired product compositions are 0.99 mol/mol (low

boiling component) for the top product D and 0.015 mol/mol for the

bottom product B. As these product purities are relatively high, this

distillation column can be classified as a "high purity distillation

column."

Table 2.1: Steady-state data

Column data

No. of trays 50

Column diameter (m) 0.8

Feed tray 20

Murphree tray efficiency =0.4

Relative volatility a 1.61

Operating data

Top composition x-q (mol/mol) 0.99

Bottom composition xg (mol/mol) 0.015

Feed composition xp (mol/mol) 0.7-0.9

Feed flow rate F (mol/min) 20-46

Top pressure (mbar) 60

Nominal operating point

Feed composition (mol/mol) 0.8

Feed flow rate (mol/min) 33

Reflux L0 (mol/min) 65

Boilup V51 (mol/min) 104

2.2 Column design and operation 31

Feed

F,xp

20

47

48

49

50

Reflux

Boilup

Vacuum

Condenser

Top product (Distillate)

D,xD

Reflux accumulator

Bottom product

Figure 2.1: The industrial distillation column


Feed disturbances

The distillation column is connected in series following two other distil¬

lation columns, which operate in parallel. The bottom product streams

of these two columns are buffered by a tank and fed into the column

considered here. The level of the buffer tank is measured periodically

(typical period: 2 hours) and the feed of the column is set to keep the

tank level within specified bounds. Therefore, the feed flow is varied not

continuously but stepwise. In contrast to that, the variations of the feed

composition are always smooth. Even a shutdown ofone of the other two

columns cannot cause a sudden increase of the buffer tank's composi¬

tion.

Top pressure control

The boiling points ofthe entering substances are high at standard atmo¬

spheric pressure. Because of a thermal decomposition of the light

component at higher temperatures, the column is operated under

vacuum. Correspondingly, the cooling water flow rate for the condenser

is kept constant and the top pressure is controlled by a vacuum pump.

Top level control

The reflux accumulator level is controlled by overflow. Hence the top

product flow rate D is not available as a manipulated variable for a

composition control system.

2.3 Steady-state behavior

Let us assume a composition control scheme with integrating behavior,

e.g., one PI controller which controls the top composition by manipu¬

lating the reflux and one which controls the bottom composition by

manipulating the boilup. Then, in steady-state, the product composi¬

tions are kept perfectly at their set-points, and an S-shaped composition

profile is developed within the distillation column. Figure 2.2 shows the

simulated composition profiles for different feed flow rates and compo¬

sitions. While these steady-state profiles are nearly independent of the

2.3 Steady-state behavior 33

i 1 1 1 1 1—i 1—i—r

xp = 0.7 mol/mol

xp = 0.8 mol/mol

xp = 0.9 mol/mol

F = 20 mol/min

F = 33 mol/min

F = 46 mol/min

i i i i i i i i i i i i i i

0.0 0.2 0.4 0.6

Composition (mol/mol)

0.8 1.0

Figure 2.2: Simulated composition profiles for the industrial distillation column


feed flow rate, they depend essentially on the feed composition. This has

a high significance for a controller design: Ifwe want to keep the product

compositions close to their setpoints, we must allow profile variations in

the middle of the column. Consequently, we cannot control any composi¬

tion in the middle ofthe column.

The internal flow rates can be illustrated in a similar manner. Figure

2.3 shows the simulated liquid and vapor flow rates for the nominal

operating point. As previously mentioned, the reflux as well as the feed

are subcooled, i.e. they enter the column at a temperature below the

boiling point. A fraction of the vapor flow is condensed at the trays

where these two streams are fed into the distillation column. The two

discontinuities of the vapor flow profile at trays 1/2 and 20/21 result

Liquid flow

Vapor flow

Figure 2.3: Simulated vapor and liquid

flow rates at nominal operating point

60 80 100 120

Flow rate (mol/min)

2.4 Composition dynamics 35

therefrom. The reason for the slopes of the two profiles within the strip¬

ping and rectifying section of the column is the different heat of evapo¬

ration of the two substances.

2.4 Composition dynamics

The composition dynamics within a distillation column is effectively

described by movements and shape alterations of the composition

profile. In order to illustrate this, let us control the reboiler level of the

distillation column by the bottom product flow rate B, and let us keep

the reboiler heat duty constant. The simulated step responses of the

composition profile to a 5% increase and a 5% decrease ofthe reflux flow

rate are shown by Figure 2.4. An increase ofthe reflux (Fig. 2.4 a) raises

the fraction of the light component in the column bottom. Consequently,

the composition profile of the light component moves towards the

column bottom, degrading the bottom product composition from 1.5% to

more than 30% impurity. The opposite effect is observed for a decrease

of the reflux flow rate (Fig. 2.4 b): The composition profile moves

towards the column top, which improves the bottom product composi¬

tion and debases the top product composition.

These plots illustrate two important properties of the composition

dynamics:

• Column nonlinearity: The product compositions are a nonlinear

function of the reflux, boilup, and the feed condition: A 5%-

increase of the reflux flow rate improves the top product compo¬

sition by 0.007 mol/mol, but a 5% decrease degrades it by more

than 0.2 mol/mol.

• Strong interactions: A change of reflux or boilup alters both

product compositions.

The interaction between both product compositions and reflux and

boilup has a severe consequence for the composition dynamics, usually

called


0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Composition (mol/mol) Composition (mol/mol)

a) b)

Figure 2.4: Simulated composition profiles (light component) for a step change oi

the reflux. Reboiler heat duty, feed flow rate and composition are kept at their

nominal values (see Table 2.1)

a) L0=1.05*L0>nom b) L0=0.95*L0inom


• Ill-conditioned behavior.

This is best explained by another two examples. If we like to increase

both product purities simultaneously, we have to increase reflux and

boilup by an exact quantity, for example the reflux by +26.5% and the

boilup by +19% (Figure 2.5 a). This keeps the composition profile's posi¬

tion constant, but it slowly intensifies the S-shape of profile. However a

slightly smaller step size for the reflux completely alters the dynamic

behavior (Fig. 2.5 b): The purity ofthe top product decreases, the purity

of the bottom product increases, and the dynamic response is much

faster. Therefore an exact direction of the input vector [L, V]T is

required in order to achieve a simultaneous increase of both product

purities. Consequently, even a small uncertainty of the input vector

[L, V]T may lead to undesired results. High condition numbers

K.°-.t°q«»>

(2.„<Jmi„{G(jo)))

of the plant model G indicate such a behavior.

2.5 Control objectives and configurations

The control of distillation columns has three objectives [2.2]:

• Control of the material balance (inventory control)

• Product quality control

• Satisfaction of constraints

The first objective includes the control of the vapor holdup (top pres¬

sure), the reflux accumulator level, and the reboiler level. Generally,these control objectives are easily achieved by simple PI controllers.

The second objective is the most important objective. It is strongly

related to the economic and ecological optimal operation of a distillation

column. Tight control of both product qualities minimizes the energy

consumption and the amount of products being off the specifications. It

is not a simple task to keep both product compositions close to their


0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Composition (mol/mol) Composition (mol/mol)

a) b)

Figure 2.5: Simulated composition profiles (light component) for a step change of

the reflux and the reboiler heat duty. The feed is kept at nominal condition (see

Table 2.1).

a) Lo=1.265*L0>nora b) L0=1.260*L0,nom

V51=1.19*V51inom V51=1.19*V51>noln


setpoints, especially in the presence of disturbances such as variations

of feed flow rate and feed composition. Tight composition control

requires sophisticated control schemes. Their design is the main topic of

this thesis.

Reflux, boilup, and pressure drop are allowed to vary within a

predefined range. Any operation of a distillation column outside of this

range may cause insufficient separation or even damage of the column.

Each control system must handle such constraints to enable safe opera¬

tion. This topic is well discussed by Buckley et al. [2.2] and Shinskey

[2.4].

2.5.1 The 5x5 control problem

A simple distillation column, such as the industrial example discussed

here, presents a control problem with the five control objectives

• Top composition• Bottom composition

• Reflux accumulator level

• Reboiler level

• Top pressure

and the five manipulated variables

• Reflux

• Boilup (indirectly controlled by reboiler duty)

• Top product flow rate

• Bottom product flow rate

• Cooling water flow rate (or vapor flow rate to vacuum)

This problem is often called the 5x5 control problem. As mentioned

above, the top pressure is controlled by a vacuum pump and the reflux

accumulator level by overflow. Thus the 5x5 control problem is reduced

to a 3x3 control problem. These relations are illustrated in Figure 2.6.


Controlled outputs Manipulated inputs

3x3 control problem

Top product xp Reflux L

Bottom product xB Boilup V (Reboiler duty Q)

Reboiler level Mb Bottom product flow rate B

Condenser level MD + »» Top product flow rate D

Top pressure p •* » Overhead vapor Vp

(Cooling water flow rate,

vacuum pump)

5x5 control problem

Figure 2.6: The distillation control problems

2.5.2 Control design steps

In principle, the design of a MIMO controller for the 5x5 or in this case

the 3x3 control problem does not cause any particular difficulties.

However, the failure of just one actuator or sensor disables all control

loops. Due to the high sensitivity ofMIMO controllers to sensor or actu¬

ator failure, the inventory control and the composition control usually

are independently designed, thus improving the robustness of the

control system and simplifying the controller design. The corresponding

design approach consists of three steps [2.5]:

1. Choosing the control configuration

In a first step the two manipulated variables for the composition control

are to be chosen. This choice names the control configuration. For

example, if the top composition xrj is controlled by reflux L and the


bottom composition xjg is controlled by boilup V, the control configura¬

tion is called L,V control configuration. After the choice of the manipu¬

lated variables for composition control, the remaining three

manipulated variables are available for level and pressure control.

The choice of the control configuration is often based on configurationselection methods such as Relative Gain Array (RGA), Niederiinski

Index, or Singular Value Decomposition (SVD). The application of these

indices may lead to very different results (see [2.1], [2.6]), and the reli¬

ability seems to be low. One reason for the limited reliability may be the

neglect of inventory control: Yang et. al. [2.9] point to the substantial

influence of inventory control on the composition control dynamics.

Most indices for control configuration selection are based on steady-

state gains. Consequently, perfect inventory control is assumed and

dynamic effects due to the interaction of inventory and compositioncontrol are neglected.

The most common control configuration in the chemical industry is the

L,V configuration [2.7]. This control structure is rather independent of

inventory control dynamics [2.9] and has shown good results within an

experimental comparison of different control structures [2.8].

2. Inventory control design

In general, tight inventory control can be achieved with three simple PI

controllers. Some distillation columns show an inverse response of the

reboiler level to an increase of boilup. In this case, tight level control

with boilup as manipulated variable may be difficult.

3. Composition control design

A 2x2 controller for composition control is to be designed as a third step

of the design. This step is discussed in chapters 5-8.

2.6 Tray temperatures as controlled outputs

On-line composition analyzers are frequently used to determine product

compositions. However, their investment and maintenance costs are


prohibitive for distillation columns below a certain size. Provided that

substances with a boiling point difference ofat least 10 °C are separated

and that the product purity specifications are not extremely stringent,

pressure-compensated temperatures may substitute composition

measurements ([2.2], [2.4]).

2.6.1 Pressure-compensated temperatures

For binary mixtures a definite correlation exists between boiling

temperature, pressure, and composition

T = f(p,x) (2.1)

This correlation is illustrated in Figure 2.7 for the two components

entering the industrial distillation column. A substitution of the compo¬

sition measurements by temperature measurements requires a

compensation for the effect of pressure variations.

If the pressure variations are small, the temperature measurement can

be compensated by a linear function. The nominal pressure and compo¬

sition are denoted by the index N.

(P-PN) (2.2)

N

In case of larger pressure variations, a second-order term has to be

supplied:

(p-pN)2 (2.3)

N

Estimation oftray composition

It is possible to infer the tray composition directly. By regression of

{x, T, p} data, the coefficients of a simple polynomial expression can be

calculated. An example is given by

T = T +—Compensated gp

T = T + —-

Compensated Qp N<p-PN>+5aprT

x = e] + Q2(T: + TCon)+e3p + Q4p2 (2.4)


Figure 2.7: Boiling points of the two-component-mixture


Such an equation in terms of the absolute temperature and pressure is

simpler to implement in a distributed control system than an equationin terms of deviations from reference values

x = e1 + e2(T-TN)+e3(p-pN)+e4(p-pN)2 (2.5)

One problem of the tray composition estimate is a potential bias of the

temperature measurements. Practical experience has shown that a bias

of up to 2 °C is to be expected due to heat transport phenomena. In (2.4)

the bias is corrected by the parameter TCoTT •In practice, however, this

correction is difficult to estimate. In principle, it would be possible to

include cross terms such as 0Tp in the regression model. However,

errors in the absolute temperature may lead to incorrect numerical

values of these cross terms. Therefore, in the regression model, cross

terms should be avoided.

Pressure compensation as well as the estimation of tray composition are

easily implemented in a process control system. Without a pressure

compensation, it is impossible to use tray temperatures in a vacuum

column as controlled variables and expensive composition analyzers are

necessary. For temperature measurements close to the column top, a

linear eompensation is usually sufficient. For trays close to the column

bottom, we have to expect higher pressure variations, and a compensa¬

tion with a second-order polynomial is recommended.

2.6.2 Temperature measurement placement

The sensitivity of the tray temperatures near the ends of the column to

changes ofthe product compositions is very small. To make the temper¬

ature measurement sensitive enough, it has to be located at some

distance from the column ends. Figure 2.2 shows simulated steady-state

composition profiles for the industrial distillation column. These profiles

illustrate the fact that the effect of a change of operating conditions

increases with growing distance from the column ends. On the other

hand, a deterioration of the correlation between product composition

and tray temperature results from an increasing distance from the

2.7 References 45

column ends. A compromise between correlation with product composi¬

tion and sensitivity must thus be found. Eister discusses the most

important rules and tools in [2.3].

In the case of the industrial distillation column, the temperatures on

trays 10 and 44 are chosen as controlled outputs. Additionally, the

temperature on tray 24 is measured. Since tray 24 is close to the feed

tray, it is expected to be sensitive to any change offeed composition and,

dynamically, to the feed flow rate.

2.7 References

[2.1] Ariburnu, D., C. Ozge, and T. Gurkan: "Selection of the Best

Control Configuration for an Industrial Distillation Column,"

Preprints of 3rd IFAC Symposium on Dynamics and Control of

Chemical Reactors, Distillation Columns and Batch Processes,

April 26-29,1992, College Park, MD, 387-392 (1992)

[2.2] Buckley, P. S., W. L. Luyben, and J. P. Shunta: Design ofDistil¬

lation Column Control Systems, Instrument Society of America,

Research Triangle Park, NC (1985)

[2.3] Kister, H. Z., Distillation Operation, McGraw-Hill, New York

(1990)

[2.4] Shinskey, F. G., Distillation control for productivity and energy

conservation, 2. ed., McGraw Hill, New York (1984)

[2.5] Skogestad, S., and M. Morari: "Control Configuration Selection

for Distillation Control," AIChE J., 33,10,1620-1635 (1987)

[2.6] Skogestad, S., P. Lundstrbm, and E. W. Jacobsen: "Selecting the

Best Distillation Control Configuration," AIChE J., 36, 5, 753-

764 (1990)

[2.7] Skogestad, S.: "Dynamics and Control of Distillation Columns —

A Critical Survey," 3rd IFAC Symposium on Dynamics and Con-


trol of Chemical Reactors, Distillation Columns and Batch Pro¬

cesses, April 26-29, 1992, College Park, MD, 1-25 (1992)

[2.8] Waller, K. V., D. H. Finnerman, P. M. Sandelin, K. E. Haggblom,

and S. E. Gustafsson, "An Experimental Comparison of Four

Control Structures for Two-Point Control of Distillation," Ind.

Eng. Chem. Res., 27, 624-630 (1988)

[2.9] Yang, D. R., D. E. Seborg, and D. A. MeUichamp: "The Influence

of Inventory Control Dynamics on Distillation Composition Con¬

trol," Preprints of the 12th World Congress of the International

Federation ofAutomatic Control, Sydney, 18-23 July 1993,1, 71-

76(1993)

3.1 Introduction 47

Chapter 3

A Rigorous Dynamic Model of

Distillation Columns

3.1 Introduction

The rigorous dynamic process simulation has become an accepted and

widespread tool in process and even more so in controller design [3.11].

Increasing competition and environmental protection provisions

require an optimization of process and control structures, which can be

obtained only by a substantial knowledge of process dynamics. At the

same time, dynamic experiments on a running plant are less and less

desired. Rigorous dynamic modelling and simulation can replace such

expensive and time-consuming measurements. This has special signifi¬cance for high-purity distillation columns. Due to their long time

constants and varying feed flow rates and feed compositions, reproduc¬ible operating conditions are difficult to guarantee. Therefore, new

controllers are usually tested thoroughly by dynamic simulation for the

full operating range of the distillation column. The rigorous models of

distillation columns used for that purpose match the reality to a largeextent [3.17].

In this chapter, a rigorous dynamic model for distillation columns is

discussed. This model is used in all nonlinear dynamic simulations

48 3 A Rigorous Dynamic Model of Distillation Columns

within this thesis. In a special section, the numerical treatment of the

resulting system of algebraic-differential equations is outlined. The

modelling and control fields use very different notations. Therefore the

notation used within this chapter is explained in section 3.11.

3.2 Conventions

Figure 3.1 shows a schematic representation of a distillation column

equipped with nt trays. The column top (condenser and reflux accumu¬

lator) is denoted by the index 0, the trays with the indices 1, 2,... nt, and

the column bottom (including the reboiler) with the index nt+1. To

simplify the formal mathematical description the reflux stream R is

designated as liquid flow (L0).

The feed of the industrial distillation column, as described in Chapter 2,

is in liquid phase and subcooled. The top pressure is controlled by a

vacuum pump and the condenser is operated with a constant cooling

water flow rate. Flash calculations for the feed stream as well as

dynamic models for the top pressure of the column are therefore not

considered here. For other applications, the model presented is easily

extended with appropriate model equations.

3.3 The objective of modelling

The control or process engineer is interested in the dynamic behavior of

various important process variables (e.g., tray temperatures, product

compositions) as a function of the time-varying column inputs. The

objective of a dynamic model is an approximation of the real process

input/output behavior by a system of differential and algebraic equa¬

tions. These model equations are based on material and energy balances

as well as on thermodynamic and fluid dynamic correlations.

3.4 Simplifying assumptions

Within a distillation column many different physical phenomena occur.

Although it would be possible to include models for the fluid streams on

3.4 Simplifying assumptions 49

•nt-2

.1.

.2.

.3.

4

V;

nt-2

nt.:!

nt

R

(=L0)

Si,

5v,nt-l

Vnt+1

QoCondenser

1 Reflux accumulator

D

Qnt+1Reboiler

nt+1

&B

Figure 3.1: Distillation column


the trays, for the dead time caused by the transport time of vapor flow

from one tray to the next one above, or for the heat exchange with the

environment, the resulting model would be of very high order. As

mentioned earlier, the aim of modelling the distillation column

dynamics is a sufficient description of the real macroscopic behavior.

This means that we are interested primarily in the dynamics of tray

compositions, temperatures, and pressures etc. rather than in the fluid

streams on the trays. Experience shows that no substantial improve¬ment can be achieved with models including effects with more micro¬

scopic characteristics. Hence the following assumptions are usuallyintroduced in order to achieve a compromise between model accuracy

and order ([3.3], [3.13], [3.17]):

• The holdup of the vapor phase is negligible compared to the

holdup of the liquid phase.

• Liquid phase and vapor phase are each well mixed on all trays,

i.e., the composition of the liquid and of the vapor phase are inde¬

pendent of the position on the tray.

• The residence time of the liquid in the downcomer is neglected.

• The variation of the liquid enthalpy on a tray can be neglected on

all trays. (This assumption is not applicable to the evaporator.)

In the literature so far, uniform liquid flows and constant holdups for all

trays have often been assumed (equimolar overflow). This assumptionis problematic because it implies a neglect of flow dynamics. Essential

dynamic effects may remain unmodelled, e.g., a non-minimum phasebehavior (inverse response) of the reboiler level and the tray composi¬tions in the lower section of the column to an increase in reboiler heat

supply.


3.5 Balance equations

3.5.1 Material balances

The differential equations describing the dynamics of the holdup for

each component on a tray are derived from a material balance for each

component. The balance border is the single tray with its ingoing and

outgoing streams (Figure 3.2).

Figure 3.2: Balance border for the material balances

Material balance for component k on trayj (k=l, ..., nc;j=l, ..., nt)

dnVi d(n-xt-)

"dT= —dT1^ = pixF,kj +Vi*kj-i- (VSy)^

(3.

+ (Vj + 1-SVij + 1)yk)j + 1-Vjyk>j


In the same way, the balance equations for the column top and the

column bottom are formulated:

Material balance for component k in condenser (k=l, ...,nc)

dnk0 d(n0xk0)dt dt (Vi-Sv,.)yk,i-(Lo + D)xk,o (3.2)

Material balance for component k in the evaporator (k=l, ...,nc)

Usually the liquid phases in the column bottom and the reboiler are

mixed either by natural convection or by a pump. Assuming perfect

mixing we obtain

dnk,nt+1=

d(nnt+lxk,nt+l>dt dt (3.3)

= *-'ntXk, nt ~ "Xk, nt + 1~~

%t + 1 ^k, nt + 1

The total holdup on tray j equals the sum of the holdups of the indi¬

vidual substances:

nc

nj= X nk, (3.4)

k= 1

3.5.2 Energy balance equations

The vapor flows within a distillation column are calculated by an energy

balance. The balance border is the same to the border in Figure 3.2,

which was used for the material balance equations.

Energy balance for tray j:

SW=F^ +V.hH+(VJ + «-Sv,] + ,)h"] + ,(35)

-(S^ + L^-V^

For the left-hand side of this equation the following holds


d dni dh'irt(nih'i)=h'jdF+nniF (3-6)

If in (3.6) we substitute the expression for the differential term dn-/dt

according to

^ = VLj-i+vj+i-svj+i-si,rLrvi w

the following energy balance equation holds

A "h'

W= tFi<hVj-h'P+LJ-i^j_1-h-j) (3g)

+ (Vj + 1-SVfj + 1)(h"j + 1-h'j)-Vj(h»j-h'j)]

Usually, the assumption n- (dh'./dt) = 0 is permissible, except for cases

with large temperature variations on the trays, a large heat of mixing,

or a large tray holdup. With this assumption we can rewrite equation(3.8) as an algebraic expression for the vapor flow rate V-

^ = h-i^[Fi(hF,rhi)+Li-1(hj-1-hj) (39)

A similar balance equation is formed for the evaporator. Because of the

large inventory, the derivative n- (dh'./dt) cannot be neglected. Since

an increase in vapor flow causes an increase in bottom pressure and

consequently an increase of boiling temperature in the evaporator, the

vapor flow follows any change in reboiler heat supply with a time lag.

Hajdu et al. [3.9] present a model for this vapor flow lag. We can imagine

that an energy stream Q supplied to the evaporator is subdivided into

two fractions: One part causes an evaporation of liquid, the other

increases the bottom temperature. Written as a differential equation we

obtain the energy balance equation


AQ= AH

,,AV

,.^

v, nt + 1 nt+.

dAT+ n

nt + lVnt+lPnt+lCp,nt+l Jt

nt+1(3.10)

To achieve a first-order differential equation in AVnt+1, the differential

term dATnt+1/dt has to be substituted by a differential term in AVnt+1.The increase of the pressure drop due to a changing vapor flow rate

(assuming a constant total holdup on the tray) can be estimated with

A(APj) =

K+ JAV.

j + l(3.11)

Hence the pressure change in the evaporator can be approximated for a

distillation column with nt trays by

A<Pnt+l) =nt( 8APj )

UVj + JAV,

nt+1(3.12)

The increase in boiling point temperature caused by the increase in

bottom pressure can be calculated according to

ATnt+1 3Pnt+l

A(Pnt+l) (3.13)

Mi, nt + 1

Substituting (3.13) in equation (3.10), the following differential equa¬

tion is obtained:

AQ-AHVjnt+1AVnt+1

nnt+lVnt+lPnt+lCp, nt+1

^nt+l3p

nt

V Hnt+17

r9APj^dAVnt+ 1

(3.14)

UVi + J dt

Therefore, the vapor flow lag at an increase in reboiler heat supply can

be described by the first-order lag


lag

with the time constant

nnt+lVnt+lPnt+lcp,nt+l

^-g = T3(Q-Qlag) (3.15)

9Pnt+lnt

UVj + Jlag AH^

(3'16)

If we substitute the total bottom holdup balance equation in the energy

balance equation

dn„+, ,

"nt+l-ir1 = Lnth'nt + Qlag-Bh'nt+I-Vnt+1h"nt+1 (3.17)

the following equation holds:

Energy balance for the evaporator

V _

Lnt(nnt-h'nt+l> + ^lag ,„ 1SxVnt+1 V5 Iv l ;

"nt+1 nnt + 1

The parameters (e>Tnt+))/(9pnt+1) and (3Ap)/(3V-+ 1) canbeeval-

uated numerically or analytically from the appropriate equations (see

sections 3.6.2 and 3.7.2)

3.6 Fluid dynamics

In the previous sections, the equations describing composition and total

holdup dynamics, as well as the vapor flow rates have been derived.

Here the calculation of the liquid flow rates and of the pressure drop is

discussed.

3.6.1 Liquid flow rates

The volumetric liquid flow rate over the weir on tray j can be calculated

according to the Francis weir formula ([3.16], [3.10]):


LV;j = u^2i|bwh^;j(3.19)

For sharp-edged weirs jo. = 0.64 holds. Perfect mixing on the trays,

including the liquid in the downcomers, is assumed. Nevertheless, if we

calculate the effective liquid head hLW ,above the weir edge, we have

to take the liquid phase fraction ej and the liquid volume in the down-

comer into account (Figure 3.3). The liquid level in the downpipe is the

sum ofthe liquid head on the tray and ofthe hydrostatic level due to the

pressure drop according to

p- -p-

Hydrostatic liquid level in downcomer = —

Pjg(3.20)

The liquid head hL • of the pure liquid on a tray (without a vapor phase

fraction) is equal to the total liquid volume on the tray n-v'- minus the

°

°o °o °

o °o o°

Pj

"LWJ

Pj-Pj -l

Pj*1

thLJ

Figure 3.3: Liquid levels on a tray


liquid volume in the downcomer due to pressure drop

AB (P: - Pj _ j) / (Pjg) >both divided by the total area AA + AB:

Vj

Ki =

Pj-Pj-1,

AA + AB(3.21)

For the application of the Francis weir formula, we have to evaluate the

liquid level of the pure liquid (liquid without vapor phase fraction). For

that purpose, first the height of the two-phase layer is to be evaluated

and second the liquid phase fraction £j must be taken into account. The

effective liquid level becomes

Ti.W.j-hw

Vj-Pi-Pi_L-i

£j =Pjg

AA + AB -£jhw (3.22)

Substituting (3.22) into the Francis weir formula (3.19), we obtain the

volumetric liquid flow rate of the two-phase mixture. The flow rate from

tray j in molal units is calculated by:

u-v^tv

Lj =

VrPj-p'izi.

pjg

AA + AB £jhw

3/2

(3.23)

In many industrial distillation columns, calming zones exist in front of

the weir. For this case, e- = 1 holds at the weir edge. Otherwise, we

have to estimate the liquid phase fraction on the trays. The Stichlmair

correlation is well suited for that purpose [3.18].

3.6.2 Pressure drop

A vapor flow through a tray in a distillation column suffers a pressure

drop. Its amount depends on the vapor flow rate, the tray holdup, and


the geometry of the tray. Usually, the pressure drop is assumed to

consist of three different parts ([3.7], [3.12]):

• Dry pressure drop occurring at the flow through the tray without

liquid (Aptr j)

• Hydrostatic pressure drop due to liquid head and liquid density

(ApLJ)

• Pressure drop by bubble-forming due to surface tension of liquid

(APa;i>

The pressure drop by bubble-forming usually is insignificant and can be

neglected.

Dry pressure drop

With sufficient accuracy, the dry pressure drop can be approximated by

the following well-known expression:

AptrJ = ^(Re)^V Ao J

(3.24)

The orifice coefficient £(Re) either can be evaluated by measurement

on comparable trays, or it can be estimated with experimentally verified

correlations. During the simulations, the following correlation for sieve

trays is used [3.19]:

Aptr,j1-

aaJ+ 0.211f

v Ao ;

(3.25)

Hydrostatic pressure drop

The hydrostatic pressure drop results from the liquid head and the

liquid density. We have to take the liquid volume in the downcomer into

account (see 3.6.1).

3.7 Phase equilibrium 59

ApL,i' A.+L p> (3'26)

The total pressure drop consists ofthe sum ofthe two parts dry pressure

drop and hydrostatic pressure drop:

APj = Pj + i-Pj= Aptr>j + ApLj (3.27)

3.7 Phase equilibrium

All equations we have discussed in the previous sections are explicitly

or implicitly interrelated with the vapor phase composition. In this

section, the most important correlations concerning the vapor phase

compositions and boiling points are presented.

3.7.1 Vapor phase composition

The liquid on each tray and in the evaporator is at boiling-point. Phase

equilibrium thus can be assumed. At moderate pressures up to some few

bar, the concentration of a substance in the vapor flow leaving tray j can

be obtained according to

yEquilibrium = M*Jx = Kk .xk j(3.28)

If the substance mixture exhibits ideal behavior, the activity coefficient

y becomes one, and the vapor phase compositions are equal to the ratios

of the partial pressures of the substances and the absolute pressure on

the tray.

The vapor pressures of the pure substances pk can be calculated with a

high level of accuracy by the Antoine equation (3.29). The parameters A,

B, and C are listed in many tables of substance properties (e.g., [3.5]).

^M^tTC (3"29)


The calculation of the liquid phase activity coefficients yk . can be

effected by one of the well known correlations (Wilson, NRTL,

UNIQUAC etc.).

Murphree tray efficiency

In a distillation column only little contact time exists on each tray for

the mass transfer between liquid and vapor phase. Therefore no perfect

phase equilibrium can be achieved, and the tray efficiency will deviate

from the unit value. This effect can be modelled by the Murphree tray

efficiency for the vapor phase.

-Êquilibrium ,.

yk,j ~yk,j + l

3.7.2 Boiling points

The vapor phase composition according to (3.28) is a function of the tray

temperature Tj. At boiling point, the sum of the vapor phase mole frac¬

tions calculated becomes one. Hence for a tray j, the following boiling

point equation holds:

X yEquilibrium = £ ^ . ^ p.,^^ . = , (3.31)

k=l k=l

The Murphree tray efficiency is not considered for the boiling point

calculation, because it relates to the mass transfer between vapor and

liquid phase rather than to the equilibrium composition.

3.8 Volumetric properties

The fluid dynamic models discussed are interrelated with the molar

volumes of the vapor phase and of the liquid phase, and with the corre¬

sponding densities. Their calculation is the subject of this section.

3.8 Volumetric properties 61

3.8.1 PVT relations

The molar volumes ofthe liquid phase v'- or the vapor phase v". dependson the pressure pj, the temperature Tj, and the actual compositions x^jor ykj. A great number ofdifferent equations ofstate has been developedto describe this behavior. They are extremely well documented ([3.5],

[3.6]), and a discussion of their properties is not repeated here.

The PVT behavior is described here by the Soave-Redlich-Kwong equa¬

tion (SRK equation, [3.15], [3.6]) with the Peneloux correction. This

correction improves the estimate of the molar volumes of the liquid

phase, which is overrated by 10-15% using the SRK equation.

If measurement data of the PVT behavior of the pure substances exist

and their mixing behavior is nearly ideal, a different possibility has

shown good results for the liquid phase:

We can correlate the molar volumes measured with the temperature by

a polynomial regression. The molar volume v'- of the substance mixture

can be approximated as a weighted sum of the individual molar

volumes:

nc

v'j = I xk,/k,j (3-32)

k=l

3.8.2 Density

The densities ofliquid and vapor phase can be computed from the molar

volume, the molar mass, and the mole fractions.

nc

I xk,jMkLiquid phase density: o'- =

k= ],

(3.33)

nc

I yk>JMkVapor phase density: p" •

=„

(3.34)Vj


3.9 Enthalpies

The quantity not discussed so far is the enthalpy of a substance mixture

in liquid or vapor phase. The enthalpy of a real fluid is estimated by the

sum of an ideal part and the value of a departure function Ahâpdescribing the deviation of the enthalpy from the enthalpy of the ideal

gas state:

T

h = h° + j cjfdT + Ahp (3.35)

T

The ideal part can be calculated by summing the ideal parts for each

component:

( T

KddT= I xkHdkdTT k=l

0 "_iV *0Tn

(3.36)

The ideal heat capacities c are often approximated by a third-order

polynomial for each component:

cj,dk = Ak + BkT + CkT2 + DkT3 (3.37)

The parameters for equation (3.37) are listed in many tables of

substance properties, or they can be estimated with very high accuracy

by Joback's method ([3.15], p. 154-156).

The real part Ah^ pdescribes the departure of a mixture from the ideal

behavior. It can be evaluated using one of the well-known equations of

state, e.g., the SRK equation ([3.15], [3.6]).

If measurement data for the heat capacities and for the heat of vapor¬

ization are available, a simple solution is possible in a manner similar

to that mentioned in section 3.8.1:


f T,

Liquid phase enthalpy: ti = V

k=ll Tft

k,JcP,J,kdT + h° (3.38)

Vapor phase enthalpy: h". = V Yk j

k = l VTn

+ h° (3.39)

3.10 Numerical solution

The complete rigorous dynamic model for distillation columns, as intro¬

duced above, consists ofa system of differential and algebraic equations

(DAE). The complexity of the model is illustrated by Figure 3.4. It illus¬

trates the interconnection of the model equations for three adjoining

trays. The solution of the differential equations obviously depends on

the solution of the algebraic equation system. Therefore an efficient

numerical integration using standard integration methods is not

possible. This requires special adapted integration algorithms, as

outlined in section 3.10.4.

3.10.1 The dependent variables and the equation system

As a first step for the numerical treatment, we have to decide which

variables should form the vector of the dependent variables. This vector

of dependent variables must at any time completely describe the state

of a distillation column and should be of minimum size to avoid exces¬

sive computation times.

The vapor phase composition is an illustrative example for the complete

description of the distillation's state: If we know the tray composition,

the tray temperature, and the tray pressure, then the vapor phase

composition in equilibrium is easily calculated by an explicit algebraic

equation. Consequently, it is not necessary to insert the vapor phase

composition into the vector of the dependent variables.

crc?3oCO


As one vector which satisfies the requirements of a complete descriptionand of minimum order, the following vector is proposed (as a modifica¬

tion of the vector proposed by Holland & Liapis [3.10]):

y - [QlCond> D> nl,0» •••> nnc,0> T0> P0> L0>

(Vj, nxj,..., nncJ, (Sy), (SvJ), Tj, pj( Lj}j=1> 2>..., nt

Qlag» Q> Vnt+1, nlnt+1, ..., nncnt+1,

B, Tnt+1, pnt+i,

States of the control system]

(): Value is inserted only if it physically exists

The Jacobian matrix of the equation system (as described below) corre¬

sponding to these dependent variables has a numerically advantageousblock diagonal dominant structure.

For the calculation of these dependent variables y, the following equa¬

tions are to be solved

Differential equations

nc material balance equations (3.1)

Algebraic equations1 equation for vapor flow rate (3.9)

1 equation for liquid flow rate (3.23)

1 equation for boiling point (3.31)

1 equation for pressure drop (3.27)

Total: nc +4 equations per tray

and in addition the equations for the evaporator, the condenser, and the

control system. Considering industrial distillation columns which are

often equipped with more than 50 trays, the resulting algebraic differ¬

ential equations amount to several hundred equations. The model for

the industrial binary distillation equipped with 50 trays gives an

impression of these numbers: It consists of a system of 107 differential

and 210 algebraic equations.


3.10.2 Formal representation of the DAE

We can formally represent the entire dynamic model by the semi-

explicit equation system

^ = f (t, n (t), z (t)) n (t0) = n0 (3.40a)

0 = g(t,n(t),z(t)) z(t0)=z0 (3.40b)

The vector n consists of all tray holdups (for all components), while the

vector z contains all other dependent variables. A different but equiva¬

lent formal representation is the implicit form:

F(t,y(t),y'(t)) =0 y(t0) = y0 (3.41)

Here the vector y contains all the dependent variables. A simulation of

the dynamic behavior requires a simultaneous solution of the whole

equation system.

3.10.3 The index

The index of a set of differential-algebraic equations (DAE) character¬

izes the integration problem. The higher the index, the more difficult is

a solution of the DAE. The differential index is the most common defini¬

tion:

The differential index m ofthe system F (t, y (t), y' (t)) = 0 is the min¬

imal number m such that the system ofF (t, y (t), y' (t)) =0 and ofthe

analytical differentiations

d(F(t,y(t),y'(t)))_

A dm(F(t,y (t), y'(t)))_

dt-U'""

dt

can be transformed by algebraic manipulations into an explicit ordinary

differential system [3.8].

Consequently, a system of ordinary differential equation has an index of

m=0.


3.10.4 Solution methods and software

The first general method for the numerical solution of semi-explicit

DAE with index 1 was proposed by C. W Gear in 1971 [3.4] and was soon

extended to the solution of implicit index 1 problems. The method is

based on a special class of the linear multistep methods entitled the

backward differentiation formulas, which are standard algorithms for

the integration of stiff systems. The most important convergence results

may be found in [3.1]. In theory, it is possible to solve problems ofhigher

indices with the backward differentiation formulas. However, the neces¬

sary software is not available as yet. The apparently very frequently

used integrators DASSL and LSODI are based on Gear's method. These

methods are distinguished for their effectiveness in solving continuous

problems. However, the computational effort grows significantly for

systems with discontinuities arising, for example, during the simulation

ofthe response to several feed flow or feed composition step changes. For

such cases, the one-step methods find more and more interest [3.11].

The one-step methods are extensions of the well-known Runge-Kutta,

Rosenbrock, or extrapolation methods. An extensive discussion of the

properties ofthese methods is found in [3.8]. However, the development

of the integrators (RADATJ5, LIMEX) is in an early stage, and no imple¬

mentations are found in any of the widespread Fortran libraries.

For the simulation studies the DASSL integrator, as implemented in the

NAG Fortran Library is used with good success. The differential-alge¬

braic equations (DAE) are solved in an implicit manner according to

(3.41).

The calculation sequence

During the integration, the right-hand sides of the differential and alge¬

braic equations repeatedly have to be evaluated for a given vector y of

the dependent variables and for a given time t. The algebraic equations,

and often the differential equations as well are solved in an implicit

manner. The equation errors, which have to be supplied to the integra¬

tion, are the difference between the right-hand sides of the equations


(that means the calculated vapor flow rates, liquid flow rates, etc.) and

the corresponding values within the vector y.

A correct calculation sequence evaluating these terms is stringent: If,

for example, we calculate the right-hand side of the boiling point equa¬

tion (3.31), we must know the vapor phase composition. Therefore we

first have to calculate the vapor phase composition and subsequentlythe error of the boiling point equation. If this basic idea is applied to the

whole model, the calculation sequence illustrated in Figure 3.5 results.

The vector y, which is supplied by the integration routine (Step a),

contains the component holdups in liquid phase, the tray temperatures,

tray pressures, as well as liquid and vapor flow rates. With the data

supplied, all vapor phase compositions in equilibrium can be calculated

(Step b). In a next step (Step c), using the distribution coefficients Kkobtained in the previous step, the errors of the boiling point equations

are calculated. Since all the feed data are known, its enthalpy, molar

volumes and densities are computed in step d. The vapor phase compo¬

sitions deviate from the equilibrium compositions. Applying the

Murphree tray efficiency, the effective vapor phase compositions are

computed (Step e). Since for the computation of the effective vapor

phase composition for a tray the effective vapor phase composition ofthe

next lower tray must be known, the computation starts at the column

bottom, assuming n = 1 for the reboiler. Now all necessary data are

known to calculate the enthapies, molar volumes, and densities for all

trays, the condenser and the evaporator (Step f). In step g, the energy

balance equations for the trays are applied, and the differences between

the resulting vapor flow rates and the flow rates supplied by the integra¬

tion routine are calculated. In step h, the same is done for the evapo¬

rator. Similarly to the error of the vapor flow rates, the errors of the

liquid flow rates and the tray pressures are computed in steps i and j.

Using the flow rates and compositions calculated in the previous steps

rather than the data supplied by the integration routine, the differential

terms (left-hand sides) of the equations describing the vapor flow lag

(Step k) and the holdup of the substances in liquid phase (Step 1) are

calculated. In a last step (Step m), all differential terms, the errors


a)

b)

c)

d)

e)

( Vector of dependent variables y J

Vapor phase composition for evaporator and trays

(Equation (3.28))

Error for boiling point at evaporator and trays

k

Calculation of the thermodynamic states

h', v', v", p', p"

for the feed

Murphree tray efficiency for trays nt, nt-1,..., 1

yk,J = <yk^"lbnum-yk,J+iJ+yk,]+,

Calculation of the thermodynamic states

hVhVv'rv>>"jfor condenser, all trays, and evaporator

Error for vapor flows

hVh*j

Figure 3.5: Calculation sequence

Explanation: see text


u

h)

Error for vapor flow leaving evaporator

Lnt(hnt-hnt+l>+Qlag „

h" -h' nt+1"nt+1 "nt+1

V

i)

Error for liquid streams

13 p,gB

,

3/2

3A + A

J W

AA+ABJ

I )

If

j)

Error for pressure drop

P]+1-P3-AP](Equation (3.27))

' '

k)Differential equation for vapor flow lag

(Equation (3.15))

< '

1)Differential equations for holdup of substances

(Equations (3.1), (3.2) and (3.3))

' '

m) C Vect or of differeiitials and errors J

Figure 3.5 continued

3.11 Notation 71

between supplied and calculated flow rates and pressures, and the

errors of the boiling point equations are combined in one vector and

supplied back to the integration routine.

3.11 Notation

A0 [m2] Hole area in tray

AA [m2] Tray area without downcomer area

Ab [m2] Downcomer area

bw [m] Length of weir

pidLP

[J/mol-K] Ideal gas heat capacity

CP,1 [J/kg-K] Liquid heat capacity

do [m] Diameter of holes of sieve tray

Fj [mol/s] Feed flow rate to tray j

h [J/mol] Molar enthalpy

h'j [J/mol] Molar enthalpy of liquid phase

h"j [J/mol] Molar enthalpy of vapor phase

hL [m] Liquid level above upper edge of weir

hw [m] Weir height

AHv,k,j [J/mol] Heat of evaporation of component k on tray j

AHvj [J/mol] Heat of evaporation of liquid on tray j

Kkj [mol/mol] Distribution coefficient for comp. k on tray j

LJ [mol/s] Liquid flow leaving tray j

Wj [m3/s] Volumetric flow from tray j

Mk [g/mol] Molar mass of component k

nt H Number of trays in column


nj [mol]

nkj [mol]

nc [-]

Pj [N/m2]

APj [N/m2]

K [N/m2]

P [N/m2]

Q [J/s]

Qlag [J/s]

s [m]

SU [mol/s]

Jvj[mol/s]

t [s]

T [K]

TJ [K]

Vj [mol/s]

VVj [m3/s]

xkj [mol/mol]

XF,ko [mol/mol]

ykj [mol/mol]

Yk [-]

Total holdup on tray j

Holdup of substance k on tray j

Number of components

Pressure on tray j

Pressure drop over tray j

Steam pressure of pure component k

Pressure

Heat supply to evaporator

"active" heat supply

Thickness of sieve tray

Side product flow rate from tray j,

liquid phase

Side product flow rate from tray j,

vapor phase

Time

Temperature

Temperature on tray j

Vapor stream from tray j

Volumetric vapor stream from tray j

Liquid phase mole fraction of

component k on tray j

Mole fraction of component k

in feed to tray j

Vapor phase mole fraction of

component k above tray j

Liquid phase activity coefficient

of component k

3.11 Notation

£j [m3/m3] Liquid phase fraction on tray j

Tl [mol/mol] Murphree tray efficiency for vapor phase

V [m3/mol] Molar volume

V'j [m3/mol] Molar volume of liquid phase on tray j

V"j [m3/moI] Molar volume ofvapor phase on tray j

% H Orifice coefficient

P'j [kg/m3] Liquid density on tray j

P"j [kg/m3] Vapor density on tray j


3.12 References

[3.1] Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical so¬

lution of initial-value problems in differential-algebraic equa¬

tions, North-Holland, New York (1989)

[3.2] Byrne, G. D., P. R. Ponzi, Differential-Algebraic Systems, Their

Application and Solution, Comp. Chem. Eng., 12, 5, 377-382

(1988)

[3.3] Gani, R., C. A. Ruiz, and I. T. Cameron: "A Generalized Model for

Distillation Columns," Comp. Chem. Eng., 10, 3, 181-198 (1986)

[3.4] Gear, C. W.: "Simultaneous Numerical Solution of Differential-

Algebraic Equations," IEEE Trans, on Circuit Theory, CT-18, 1,

89-95 (1971)

[3.5] Gmehling, J. and U. Onken: "Vapor-Liquid Equilibrium Data

Collection;' 1, Part 1, XI-XXII, DECHEMA, Frankfurt (1977)

[3.6] Gmehling, J. and B. Kolbe: Thermodynamik, Georg Thieme Ver-

lag, Stuttgart (1988)

[3.7] Grassmann, P. and F. Widmer, Einfiihrung in die thermische

Verfahrenstechnik, 2nd ed., de Gruyter, Berlin (1974)

[3.8] Hairer, E. and G. Wanner: Solving Ordinary Differential Equa¬

tions II — Stiff and Differential-Algebraic Problems, Springer

Verlag, Berlin (1991)

[3.9] Hajdu, H., A. Borus, and P. Foldes: "Vapor Flow Lag in Distilla¬

tion Columns," Chem. Eng. Sc, 33, 1-8 (1978)

[3.10] Holland, C. D. and A. I. Liapis, Computer Methods for Solving

Dynamic Separation Problems, Chapter 8, McGraw-Hill, New

York(1983)

3.12 References 75

[3.11] Marquardt, W.: "Dynamic Process Simulation — Recent

Progress and Future Challenges," Fourth International Confer¬

ence on Chemical Process Control, South Padre Island, Texas

(1991)

[3.12] McCabe, W. L., J. C. Smith, and P. Harriott: Unit Operations of

Chemical Engineering, 4th ed., McGraw-Hill, New York (1985)

[3.13] Najim, K. (Editor): Process Modeling and Control in Chemical

Engineering, Marcel Dekker, New York (1989), Chapter III, 145-

211, S. Domenech, L. Pibouleau, "Distillation"

[3.14] Petzold, L.: "Differential/Algebraic Equations are not ODE,"

SIAMJ. Sci. Stat. Comput, 3, 3, 367-384 (1982)

[3.15] Reid, R. C, J. M. Prausnitz, and B. E. Poling: The Properties of

Gases and Liquids, 4th ed., McGraw-Hill, New York (1988)

[3.16] Retzbach, B.: "Mathematische Modelle von Destillationskolon-

nen zur Synthese von Regelungskonzepten," Fortschritt-Berichte

VDI, Reihe 8: Mess-, Steuerungs- und Regelungstechnik, Nr. 126,

VDI Verlag (1986)

[3.17] Rovaglio, M., E. Ranzi, G. Biardi, and T. Faravelli: "Rigorous Dy¬

namics and Control of Continuous Distillation Systems — Simu¬

lation and Experimental Results," Comp. Chem. Eng., 14, 8, 871-

887 (1990)

[3.18] Stichlmair, J.: Grundlagen der Dimensionierung des GaslFliis-

sigkeit-Kontaktapparates Bodenkolonne, Verlag Chemie, Wein-

heim (1978)

[3.19] Weiss, S. et. al.: Verfahrenstechnische Berechnungsmethoden,

Teil 2: "Thermisches Trennen", VCH Verlagsgesellschaft, Wein-

heim (1986)


4.1 Introduction 77

Chapter 4

Linear Models

4.1 Introduction

Robust controllers are designed on the basis of linear process models.

Therefore the elaboration of linear dynamic models for the distillation

column is a central part of control system synthesis. These models

should describe the dynamic behavior of the process within a wide

frequency range. They can be obtained in two ways:

• System identification

• Linearization of a nonlinear model

It is a big advantage of the system identification that it avoids a compli¬

cated and expensive nonlinear model. Nevertheless, this approach has

some severe drawbacks, for example:

• The time-constants of the composition dynamics are large. A

recording of input/output data for the real plant is very time-

consuming.

• Due to the high sensitivity of distillation columns to changes of

the internal flow rates, even for small magnitudes of the input

variation (e.g., 5% of the steady-state value) the response may

far exceed the linear region.

78 4 Linear Models

• Each experiment causes undesired disturbances of the product

qualities.

• It is practically impossible to obtain models for the entire oper¬

ating range of the distillation column

These disadvantages and some other fundamental problems ofthe iden¬

tification itself (see Jacobsen et al. [4.5]) lead to a strong recommenda¬

tion of the second method (Skogestad, [4.12]) that means the

linearization of nonlinear column models.

Two linear models are evaluated within this chapter, which are based on

the linearization of different nonlinear column models. The first linear

model is obtained by an analytical linearization of a simplified

nonlinear model neglecting flow dynamics. The other model is obtained

by a numerical linearization of the rigorous model presented in Chapter

3. In further sections the accuracy of these linear models and the role of

the flow dynamics are discussed. Different mathematically order reduc¬

tion methods are compared at the end of this chapter. The notation is

listed in section 4.9 on page 101, the literature references are collected

in section 4.10.

4.2 How to linearize the rigorous model?

4.2.1 The state, input, and output vectors

The complete rigorous dynamic model as discussed in Chapter 3

consists of a high-order system of coupled differential and algebraic

equations (DAE). A linearization of this large system would be possible

in principle. However, the resulting linear state-space model would be

of the same order as the DAE. Such a high order causes high computa¬

tion times for a controller design or even for a model reduction. Conse¬

quently, a compromise between model order and accuracy must be

sought. This means we have to decide, which dependent variables are

very important for the composition dynamics and should be included in

the state vector x of the linear model. Most important, of course, are the

tray compositions themselves. Because flow dynamics have a high influ-

4.2 How to linearize the rigorous model? 79

ence on the composition dynamics in the high-frequency range, the tray

holdup is a candidate as well. Assuming a perfect level control of the

reboiler and the reflux accumulator, it is not necessary to include their

holdup. The corresponding candidates for the state vector of the linear

model are

x =

dxc

dx,

dx51

or

dx0

dx,

dx51

dn,

dn50

(4.1)

The dynamics of the distillation column are stimulated by the manipu¬

lated variables (reflux L0 and boilup V51) and the several disturbance

sources. Most important disturbances are variations of the feed compo¬

sition xp and the feed flow rate F. Other disturbances such as variations

of the reflux temperature or the feed temperature have significantly less

influence and can be neglected for the composition control design. Hence

we define the input vector according to

dxF

d dF

u dL0

dv51

(4.2)

The output vector y follows directly from the temperature measure¬

ments. It represents the deviations of the pressure compensated temper¬

atures on tray 10, 44, and 24:

80 4 Linear Models

dTPio

dTP44

dTP24

(4.3)

4.2.2 Handling of the algebraic equation system

The algebraic equation system of the rigorous column model defines

dependent variables such as tray pressures, vapor flows, and liquid

flows, which are not included in the state vector x of the linear model.

Nevertheless the algebraic equation system represents algebraicconstraints for the composition and holdup dynamics. These equationscan be handled in two ways:

• elimination by idealizing assumptions, or

• numerical solution during linearization

The first method allows an analytical linearization of the resultingmodel. This has the advantage that merely one steady-state data set

must be supplied, which can be calculated by one of the common flow-

sheeting programs (e.g., PROCESS, ASPEN PLUS). In contrast to

that, the second method requires a numerical linearization of the

rigorous model, which is discussed in section 4.4.

4.3 Linearization of a simplified nonlinear model

4.3.1 The simplified model

In this section we will derive a simplified nonlinear column model

neglecting the holdup and thus the flow dynamics. For that purpose

idealizing assumptions are formulated which allow to dispense with all

flow dynamics and with most ofthe energy balance equations. However,the subcooling of reflux and feed have a significant influence on the

internal flow rates (see section 2.3) and are explicitly taken into account.


Idealizing assumptions

The algebraic constraints of the rigorous model and the holdup

dynamics can be eliminated by the following idealizing assumptions:

• constant pressure drop

• constant and equal enthalpies on all trays

• constant total holdup on all trays (equimolar overflow)

Of course, all these assumptions do not agree with the real conditions.

The first assumption means a neglect of the correlation between tray

pressures, holdups, and boilup rates. The second assumption impliesuniform vapor flows within the stripping section and within the recti¬

fying section of the column. The assumption of a constant tray holdupcontains a neglect of flow dynamics. The error in the high frequency

range introduced by that is discussed in section 4.5.

It has to be emphasized here that these assumptions concern only the

simplified nonlinear model as a basis for an analytical linearization.

The steady-state operating points must be calculated using a model,which includes the energy balance equations, as well as the flow

dynamic models.

The composition dynamics

For a column separating a binary mixture, it is sufficient to formulate

the material balance equation for the light component of the substance

mixture. If we assume constant total holdup, the following material

balance equation holds for the tray j:

^ = Lj-i+Vj + i+Fj-VVj = 0 (4.4)

Similar balances are obtained for the reboiler and the condenser. Substi¬

tuting these balance equations in the material balances (3.1)-(3.3), we

obtain the following differential equations describing the composition

dynamics:

82 4 Linear Models

Condenser

7>ays (Feed is liquid phase, j = 1.. .50)

^j[V.(vi-v+Vi^-v(4.6)

Evaporator

"aT= 4[L50(x50-x5i)-V51(y51-x5])] (4.7)

Effect ofsubcooled reflux and feed

Feed and reflux ofthe distillation column are subcooled. A portion of the

vapor flow is condensed at the trays where these two streams enter the

column. The effect of an additional condensation of the vapor stream

caused by increasing the flow rates of these streams must be considered

to avoid large model errors. The two energy balance equations for the

reflux tray (tray 1) and for the feed tray (tray 20) become part of the

nonlinear model:

0 = Loh'o-Ljh'j+Vjh'^-V^", (4.8)

0 = L19h'19~L20h'20 + V21h"2]-V20h"20 + Fh'F (4.9)

Tray temperatures as model outputs

The model outputs are the deviations of the pressure compensated tray

temperatures. These temperatures are correlated with the tray compo¬

sition by the boiling point equation


2

X (y£) -1=0 (4.10)

k=l

Consequently the boiling point equation for tray 10, 44, and 24 are part

of the simplified column model.

The vapor phase composition

The differential equations for the composition dynamics are in terms of

the vapor phase composition yj, too. Usually, the tray efficiency is

smaller than one and the vapor phase compositions deviate from the

equilibrium compositions. As described in Chapter 3, this can be

modelled by the Murphree tray efficiency n

Yj= O-TDyj^+Tiy'j (4.11)

Primarily the vapor phase composition in equilibrium y*j is a function

ofthe liquid phase composition x-. For example, ifwe assume a constant

relative volatility a. on tray j, the vapor phase composition in equilib¬

rium is calculated by

ct X-

y*- = i-J (4.12)

To simplify the analytical linearization, it is convenient to substitute the

vapor phase composition y• by a correlation exclusive in terms of the

equilibrium compositions y*. Then the calculation of the derivatives

3 (...) /dx for each these terms cause no particular problem.

Such an equation in terms of the equilibrium compositions y* is derived

by subsequent substitutions of the vapor phase compositions in (4.11)

from the evaporator up to actual tray. For example (n=l assumed for the

reboiler)

y50 = (l-Ti)y*gi+r|y*50 (4.13)

84 4 Linear Models

y49 = (l-Ti)y50 + 1iy*49(4.14)

= (i--n)V5i+Ti(1-1i)y*5o + 7iy*49

y48 = (i-Ti)y49 + 1iy*48,A,E,

(4.15)

= (l-*i)3y*5i+Ti(i-'ri)2y*5o + 'rl(i-n)y*49 + riy*48

For a binary distillation column with nt trays the following generalizedformula (with vapor/liquid equilibrium (n=l) in the evaporator) is

obtained:

y. = (l-Tl)at+1-ynt+1

nt+l-j (4>16)+ X lKl-M)nt+1-j-Vnt+,-n

n= 1

This equation demonstrates the strong influence on the composition of

the trays below the actual one, presuming the tray efficiency n is

substantially smaller than one.

4.3.2 Analytical linearization

Let us formally represent the simplified nonlinear model as the vector

functions f and g:

d^l = f[x(t),u(t),d(t)] (4.17a)

y(t) = g[x(t)] (4.17b)

Then the matrices A, B, and C of the linear state space model

d^ = Ax + Bd (4.18a)

y = Cx (4.18b)


are evaluated as partial derivatives of the vector functions f and g at a

steady-state operating point (OP). The OP is calculated either solving

the equation system of the complete rigorous model for steady state, or

using a steady-state flowsheeting program. The following relations hold

for the partial derivatives:

A =

3f

3xOP

*0 <*0

dx0 dx,

3fj 3fj

3x0 dx.

dx,0

dx51

3f.51

3x51 IOP

(4.19)

B =

3f

d

u

OP

"»03xF

3f03F

df0

3L0

3f0

3V51

3f, 3f, 3f, 3f,

3xF 3F 3L0 av51

*5i

3xF

3f513F

^51

3L0

3f51

IJ [OP

(4.20)

L~diOP

9g] 3g, *i

3xQ 3xj '3x51

3g2 3g2 3g?

3x0 3xj '^Sl

3g3 3g3 3g3

3xQ 3x, '^51 IOP

(4.21)

86 4 Linear Models

Combining the idealizing assumptions, we can conclude that a deviation

of reflux Lo or boilup V51 causes the same deviation of the liquid flow

and vapor flow rates within the whole column:

dL. = dLnJ

(4.22)

dVj = dv51

The resulting coefficients of the matrices A, B, and C are listed in the

appendix of this chapter (page 97). Although important interactions in

the column model are suppressed by the idealizing assumptions it turns

out that this model coincides within acceptable bounds with the

rigorous nonlinear model. This aspect is discussed in detail in section

4.5 below.

4.4 Linearization of the rigorous model

In this section the linearization of the rigorous dynamic column model

is discussed. This model includes the dynamics of the tray holdups and

thus flow dynamics. Hence it describes the high-frequency dynamics

much better than the simplified nonlinear model (section 4.3.1).

4.4.1 Model modifications

The desired outputs of the linear model are the deviations of pressure

compensated tray temperatures, which are functions of the tray compo¬

sitions, but not of a component's holdups. Therefore the material bal¬

ance equations (3.1)-(3.3) are replaced by the following, equivalent

differential equations:

Condenser

^-iv.to,-^) (4.23)

equations:vectorfollowingthebycomprisedbecanequations

theseAllequations.differentialtheforconstraintaissolutionwhose

systemequationanformequationalgebraicThese(3.31).pointsboiling

theand(3.23),flowsliquid(3.9),flowsvaporthe(3.27),droppressure

theforequationsalgebraictheand(4.24),dynamicsholduptheand

(4.5)-(4.7)componentlightthefordynamicscompositionthedescribing

equationsdifferentialtheofconsistssystemequationwholeThe

(4.28)L50-V51=B

and

(4.27)V,-L0=D

toaccordingcalculatedarestreamsproductbottomandtop

theevaporator,theandaccumulatorrefluxtheforholdupconstantfore

there¬andcontrollevelperfectWithloops.controlcompositionthethan

fastermuchtunedbecanevaporatortheandcondensertheforloops

controllevelthethatfactthebyjustifiedisassumptionThisorator.

evap¬theinandaccumulatorrefluxtheincontrollevelperfectaoftion

assump¬thefromfollowsmodelrigoroustheofmodificationsecondThe

(4-26)-V51(y51-x51)](*»-**>5^L»=

T

Evaporator

-vj(yrxP+Fj(xF,rxP]

1-xj)

+

1(yj

+^[Lj_1(xj_1-xj)+Vj=^

(42g)1-xj;+1iyj+vj+

^-ii*j-i-'yidx.

.50)1..=jphase,liquidis(FeedcompositionTray

(4-24)Li-'-VV--VFi=S

holdups:Tray

87modelrigoroustheofLinearization4.4

88 4 Linear Models

di|P = f[x(t),n(t),u(t),z(t),v(x,n,u,z)]

^P = l[x(t),n(t),u(t),z(t),v(x,n,u,z)]

y(t) = g[x(t)]

0 = k[v(x, n, u, z)]

(4.29a)

(4.29b)

(4.29c)

(4.29d)

The vector v (x, n, u, z) represents the solution of the algebraic equa¬

tion system k and consists of the tray pressures, the vapor flow rates,

the liquid flow rates, and the boiling points.

4.4.2 Numerical linearization

The matrices of the linear state space model including flow dynamics

d_dt

X

n

= AX

n

+ Bd

u

y = Cx

(4.30a)

(4.30b)

can be numerically evaluated column by column using a finite difference

approach. After solution of the whole equation system for a steady-state

operating point (OP), each composition or tray holdup can be varied by

a small increment, the algebraic equation system k can be solved, and

each column of the state dynamic matrix A can be calculated according

to

a

i=l 102,j + lfor j = 0 .51 (4.31a)


a1=1 102J + 52

(dxdt

V

dx

„ adt opj

fdndt

V

Anjdn

« adt

VAnjopJ

An,

for j = 1...50 (4.31b)

In a similar manner the input matrix B can be evaluated.

4.5 Comparison of the linear models

4.5.1 Open loop simulations

Most important for the control design is a good representation of the

dynamic behavior in the mid-frequency range. The steady-state

behavior as well as the high-frequency behavior are less important.

Some idea of a linear model's quality is obtained by a simple qualitative

comparison of the various models. Nevertheless, a definitive judgement

requires a comparison of control designs based on the different models.

A simple method to compare the two linear models with the complete

rigorous model is the simulation of step responses to the model inputs

(reflux L0, boilup V5i, feed composition xp, and feed flow rate F). These

are shown by the Figures 4.1- 4.4. During the nonlinear simulations, the

bottom level was controlled by the bottom product flow rate B. Exceptfor the denoted input, all other column inputs are kept constant at their

steady-state values. The changes of flow rates and feed composition are

very small to maintain the column close to the steady-state and to avoid

large nonlinearities.

The coincidence of the step responses with the rigorous nonlinear model

is acceptable for both linear models. However, the linear model obtained

by a linearization of the rigorous model is distinguished by a somewhat

better representation of the low-frequency gains.

90 4 Linear Models

Tray 10 Tray 44

Nonhnear model

Anal linearized model

Num linearized model

Figure 4.1: Step response to a 0.3 mol/min (0.46%) increase in reflux

Tray 10 Tray 44

Nonlinear model

Anal linearized model

Num linearized model

2 4 6

Time (h)

Figure 4.2: Step response to a 0.3 mol/min (0.29%) increase in boilup


Tray 10 Tray 44

Nonlinear model

Anal, linearized model

Num. linearized model

Figure 4.3: Step response to a 0.005 mol/mol increase in feed composition

Tray 10 Tray 44

2 4 6

Time (h)

Tray 24

2 4 6

Time (h)

Nonlinear model

Anal, linearized model

Num. linearized model

Figure 4.4: Step response to a 0.3 mol/min (0.91%) increase in feed flow rate

92 4 Linear Models

Surprising is the high coincidence for the analytically linearized model

as well. Apparently the influence of the algebraic constraints on the

composition dynamics is substantially smaller than the interactions

within the composition dynamics themselves.

4.5.2 Singular values

An important difference between the two linear models is the high

frequency behavior due to unmodelled and modelled flow dynamics,

respectively. Best suited for a comparison are the singular values of the

transfer functions G, >y(jco) andGu. , (jco) ,shown in Figure 4.5.

in3Disturbance inputs

10

1

MagnitudeS

3

O

i "ii .

10" ioJ 10 10

Frequency (rad/min)

Control inputs

10 10

10 10' 10" 10

Frequency (rad/min)

10 10'

Figure 4.5: Singular values of the linear models

Upper plots: GA^y(i<Si) ,lower plots: Gu->y

Solid lines: Analytically linearized model

Dashed lines: Numerically linearized model

(jo)


Both models show the typical course of the singular values for a high

purity distillation column. In the low frequency range, the maximum

and minimum singular values of the transfer functions Gu (jo) are

very different and the condition numbers

<WGu-»yfj«>)}K(jco) = (4.32)

WGu_>y(J©)}

are high. With increasing frequency, the maximum and minimum

singular values approach and the corresponding condition number

decreases, but never falls below 20. This is illustrated in more detail by

Figure 4.6 for the transfer functions from the control inputs to the pres¬

sure compensated temperatures on tray 10 and 44. The large condition

numbers indicate a high sensitivity of the column outputs to the direc¬

tion of the control inputs u. Consequently, the performance of a control

system can be very sensitive to uncertainty at the control inputs.

Significant for the numerically linearized model is the double as bigcondition number in the low-frequency range and the completely

different course in the high-frequency range. These differences of the

high-frequency range between the models can be explained from the

structure of the nonlinear models they are calculated from:

As mentioned above the analytical model neglects the flow dynamics.Thus the high frequency behavior is determined only by the first-order

equations of the composition dynamics. Therefore, the singular values

in the higher frequency domain (above 0.1 rad/min) are dominated by a

negative slope of one magnitude per decade.

The numerically linearized model takes the flow dynamics into account.

Thus, considering the reflux as column input, additional lags for the

composition dynamics are introduced, and for the minimum singular

value a negative slope of several magnitudes per decade for the

frequency range above 1 rad/min results therefrom. The effect of the

flow dynamics considering the boilup as the column input is different:

An increase of the boilup increases the vapor fraction on the tray, which

94 4 Linear Models

10J

v ,„oT3 10

3

'i§10

Singular values

10

'

-

_ -

——.—___^^

: *^^^^^V

^

-

\

\ -

\

10' 10 10" 10

Frequency (rad/mm)10

10 10 10 10

Frequency (rad/min)10"

10'

io3Condition number

1—~ -

-~^

^

^^^^--

^ x

'i

J.'.

1'

1

r

/ :

a) 2"3 io3

2 io

0

r

1

N

\

\

\ /

/

/ -

/

10'

Figure 4.6: Upper plot: Singular values of the transfer function from the control

inputs u to the tray temperature T10 and T44Lower plot: Condition number K of the same transfer function

Solid lines: Analytically linearized model

Dashes lines: Numerically linearized model

causes higher liquid flow rates leaving the trays. Because the composi¬

tion of the light component is higher in the upper part of the column,

more of the light component is transported down, and the expected

decrease of the light component's composition is retarded in the

frequency range between 0.2 and 1 rad/min.

4.6 Order reduction

The orders of the linear models developed above are 52 for the analytical

model and 102 for the numerically evaluated model. During modern

4.6 Order reduction 95

robust control synthesis procedures such as H^ or (x-synthesis, the order

of the model is enlarged by frequency-dependent weights for the model

inputs and outputs. Since the computation time for the controller design

strongly depends on the model order, order reduction is ofutmost impor¬tance. Many methods exist to approximate the state-space representa¬

tion of a linear system with a lower-order state-space approximation

[4.13]. Most of the mathematical methods which are available in

MATLAB or MATRLXX toolboxes are based on computing the

Hankel Norm singular values and subsequent removing of states corre¬

sponding to relatively small Hankel Norm singular values. Jacobsen et

al. [4.5] compared the following four methods, with a reduction of a

column model of 82 states to 2 states. These methods are implementedin one of the MATLAB toolboxes:

I Balanced Truncated Approximation [4.7] (Robust Control Tool¬

box [4.2] and u-Tools [4.1])

II Balanced Truncated Approximation without balanced minimal

realization [4.9] (Schur method, in Robust Control Toolbox)

III Hankel Norm Approximation [4.3] (n-Tools [4.1])

IV Optimal Hankel Norm Approximation without balancing [4.8]

(Robust Control Toolbox)

Jacobsen et al. conclude that the methods II and IV gave significantlybetter models than the other two methods. These results have to be

considered carefully: It is not necessary to reduce the column models to

such an extremely low order. Models of an order 10-15 are absolutelysuitable for control synthesis and show a very good coincidence with the

full-order linear model.

As an example, a numerically evaluated linear model of order 102 was

reduced to an order 10 using each of the four methods mentioned above.

All step responses to the different inputs and calculated with the

different reduced-order models have shown a perfect coincidence with

the full-order linear model. This fact is supported by the singular-value

plots of the models. Figure 4.7 shows that all reduced order models

96 4 Linear Models

10J

10

3

I

10

10

10

Full order model

Methods I + II

Method III

Method TV

10 10* 10

Frequency (rad/min)

10 10'

Figure 4.7: Singular value plots of the transfer functions G (jco) of the

full order model and the different reduced order models

approximate the low and medium frequency behaviors up to 1 rad/min

very well. However, in the high frequency range the singular values are

best approximated by the models derived with a Balanced Truncated

Approximation (Method I or II).

4.7 Summary

This chapter presented two methods to obtain linear models for the

industrial distillation column. The first model is derived by an analyt¬

ical linearization of a simplified nonlinear model neglecting flow

dynamics and most of the energy balance equations. The second linear

model is obtained by a numerical linearization of the complete rigorous


model. Both linear models exhibit an acceptable approximation of the

process dynamics. The singular value plots indicate a high coincidence

of the linear models in the mid-frequency range, but significant differ¬

ences in the low and high-frequency range. Comparing step responses

with those of the rigorous nonlinear model, a slightly better representa¬tion of the column dynamics by the numerically evaluated model is

demonstrated. The relatively high order of the linear model (52 for the

analytically, and 102 for the numerically linearized models) can be

reduced essentially by one of the well known order reduction methods.

All tested methods yielded a nearly perfect approximation of the Unear

model oforder 102 up to a frequency of 1 rad/min by a model oforder 10.

4.8 Appendix: Model coefficients

For all coefficients the following holds:

Analytically differentiating the equation (4.12), the actual numerical

values of k. may be calculated.

A-Matrix

Condenser (k=l. ..50)

a, ,i=

dx0

-v,

no

,k + :

9f0

'"arknd-rDk-iVj^

aino

ai ,52:_ô_. (1- n)50Vlk51

no

(4.34)

(4.35)

(4.36)

98 4 Linear Models

Trays (j=l...50, k=j.. 50)

^ _v.j+,-j_3xJ-r ni

(4.37)

T + i.J + i dx_

*j_^VZiîLVZ&^V (4.38)

(4.39)

at- (V^.-a-TDvp

aJ+,,52 = axi=(1-Ti)50-Jkkox51

(4.40)

Evaporator

_

3f51_^50

a52,51 -

ax_

nox50 n51

(4.41)

df„ -(B + V„ksl)x51 51""51J

l52,52~

ax-

nox51 n51

(4.42)

B-Matrix

We have to consider the portion of the vapor flow which is condensed by

an increase of the feed flow rate because of the subcooled feed. For the

decrease of the vapor flow in the rectifying section of the distillation

column

dV =

( h' -h' '*n20 nF

Vn 20_n20ydF (4.43)

The liquid flow rate in the stripping section of the column is increased

by the same amount.


Condenser

b,,=

at,

L '

"

3x,= 0 (4.44)

h3fo

J.2"

3F~

/ V,' V,' Ah

20~

n

F

Vn 20 n207

yi-xo

nn(4.45)

Rectifying section (j = 1... 19)

af.(4.46)

af.h = —IDJ + '.2 dF

n20

X1F

Vn 20 n2oy

yj-n-yj(4.47)

Feed tray (20)

b"*20 *20

21> ' d-x noxF n20

(4.48)

_

5f20521,2 "

"9F

,

n20~^F/ ,

xF x20 + ,„ ,, (y20 x20j11

20 n20

x20

(4.49)

Stripping section (j = 21... 51)

(4.50)

af.b. = -i =

J + 1.2 9F

h'„n-h

1+rrr20 "F

"20 n2o;

Vl-*j

nj(4.51)

If the reflux is subcooled, the vapor stream in the column is condensed

partially at the first tray. The liquid stream leaving the first tray and all

100 4 Linear Models

trays below is thereby increased by the same amount. From (4.8) we

obtain

dV,'h'i-V

vh"i-h'iydL„ (4.52)

dLt =

f,.h'i-ho1+CV7

h"i-h'iy

dLn (4.53)

With (4.52) and (4.53) there follows for the elements of the B matrix:

Condenser

h'j-h'g

_

af0 h"1-h'1(xo_yi)bL3 =

3L= n^(4.54)

_

9fo_yi-xoDl>4"av" nA

(4.55)

Tray 1

h',-h'0

b ^2,3

"

9L-

x0-x1+(y1-x1)K^—j-l

"

l

n,

(4.56)

af,i y2-yi

J2,4 av' n,

(4.57)

Trays (j = 2... 50)

3f,x-

,- x.

j + i.3 3L•j-"3-i "j

nj

h'.-h' ^1 + CT

1 "0

h",-h'iy

(4.58)

b; _

9fj_

yj+i-yj'j + 1,4 gy

(4.59)

4.9 Notation 101

Evaporator

bf)f X -x

' u' u' N

0I51 x50 x51

52,3- 9L- „5f

1 +hj-h'0

1l_n

V h"l-h'l7(4.60)

af«i -(y«i-x,,)

b»-w-^r (4-61)

C-Matrix

The coefficients of the measurement output matrix are numericallyevaluated by solving the boiling point equation for small increments in

tray composition.

4.9 Notation

4.9.1 Matrices and Vectors

A State dynamic matrix

B Control input matrix

C Measurement output matrix

G Transfer function matrix

Gu Transfer function matrix from

control signals u to output signals y

n Vector of holdup deviations from operating point (OP)

n = [dnpdn2, ...,dn50]T

n Vector of holdupsiv = [n,,n2, ...,n50]T

u Vector of the manipulated variables (L0, V51) deviations

u= [dL0,dV5,]T

u Vector of column inputs*= [L0>V5l]T

102 4 Linear Models

Vector of composition deviations from OP

x= [dxQ, dxp ...,dx50,dx5,]T

Vector of tray compositions

X = [Xq, Xj, ..., XjjJ

Vector of the deviations of the pressure-comp. temperatures

y=[dTP10>dTP44'dTP24]TVector of pressure compensated tray temperatures

y [TPio' TP44' TP24.]d Vector of the disturbance input deviations from OP

z = [dxp,dF]T

d Vector of disturbance inputs

i= [xF,F]T

4.9.2 Scalar values

B [mol/s] Bottom product flow rate

D [mol/s] Top product flow rate

F [mol/s] Feed flow rate

hJ [J/mol] Molar enthalpy of liquid phase on tray j

h"i [J/mol] Molar enthalpy of vapor phase on tray j

Lo [mol/s] Reflux

LJ [mol/s] Rate of liquid flow leaving tray j

nJ [mol] Holdup on tray j

nt [-] Number of trays in column

T [K] Temperature

Tpj [K] Pressure compensated temperature

V [mol/s] Boilup

Vi [mol/s] Rate of vapor flow leaving tray j

4.10 References 103

Xj [mol/mol] Liquid phase composition on tray j

xB [mol/mol] Composition in column bottom

xF [mol/mol] Feed composition

yj [mol/mol] Vapor phase composition on tray j

y£ [mol/mol] Equilibrium vapor phase composition on tray j

a [-] Relative volatility

n [-] Murphree tray efficiency

c [-] Singular value

k [-] Condition number

4.10 References

[4.1] Balas, G. J., J. C. Doyle, K. Glover, A. Packard, and R. Smith: u-

Analysis and Synthesis Toolbox (\i-Tools), The MathWorks, Inc.,

Natick, MA (1991)

[4.2] Chiang, R. Y., and M. G. Safonov: Robust Control Toolbox, The

Mathworks, Inc., Natick, MA (1988)

[4.3] Glover, K.: "All optimal Hankel-norm approximations of linear

multivariable systems and their L„ error bounds," Int. J. Con¬

trol, 36,1115-1193(1984)

[4.4] Haggblom, K. E.: "Modeling of Flow Dynamics for Control of Dis¬

tillation Columns," Proc. 1991 American Control Conference,

Boston, USA (1991)

[4.5] Jacobsen, E. W., P. Lundstrom, and S. Skogestad: "Modelling

and Identification for Robust Control of Ill-Conditioned Plants —

a Distillation Case Study," Proc. 1991 American Control Confer¬

ence, Boston, USA (1991)

104 4 Linear Models

[4.6] Kapoor, N., and T. J. McAvoy: "An Analytical Approach to Ap¬

proximate Dynamic Modeling of Distillation Towers," IFAC Con¬

trol of Distillation Columns and Chemical Reactors,

Bournemouth, UK (1986)

[4.7] Moore, B.C.: "Principal Component Analysis in Linear Systems:

Controllability, Observability and Model Reduction," IEEE

Trans. Automatic Control, 32, 115-122 (1981)

[4.8] Safonov, M.G., R. Y. Chiang, and D. J. N. Limebeer: "Hankel

Model Reduction without Balancing — A Descriptor Approach,"

Proc. IEEE Conf. on Decision and Control, Los Angeles, CA, Dec.

9-11(1987)

[4.9] Safonov, M. G. and R.Y. Chiang: "Schur Balanced Model Reduc¬

tion," Proc. American Control Conference, Atlanta, GA, June 15-

17 (1988)

[4.10] Skogestad, S. and M. Morari: "The Dominant Time Constant for

Distillation Columns," Comp. Chem. Eng., 11, 6, 607-617 (1987)

[4.11] Skogestad, S. and M. Morari: "Understanding the Dynamic Be¬

havior of Distillation Columns," Ind. Eng. Chem. Res., 27, 1848-

1862 (1988)

[4.12] Skogestad, S.: "Dynamics and Control of Distillation Columns —

A Critical Survey," 3rd IFAC Symp. on Dynamics and Control of

Chemical Reactors, Distillation Columns, and Batch Processes,

April 26-29, College Park, MD, USA (1992)

[4.13] Troch, I., P. C. Muller, and K.-H. Fasol: "Modellreduktion fur Si¬

mulation und Reglerentwurf," at, 40, 2, 45-53 (1992)


Chapter 5

A Structured Uncertainty Model

5.1 Introduction

Each linear or nonlinear dynamic model can only approximatelydescribe the behavior of a real distillation column. While a nonlinear

model may be valid for a wide range of operating conditions, the error of

a linear model rapidly increases with the distance from its steady-state

design point due to process nonlinearity. Since stochastic effects influ¬

ence the process behavior as well, the error of a linear model comparedto the real process can never be exactly determined. Lacking an exact

error description, the error between the process model and the process

itself is modelled as a single frequency-dependent uncertainty bound

(unstructured uncertainty) or as several frequency-dependent uncer¬

tainty bounds (structured uncertainties).

Typical sources of uncertainty for a distillation column are measure¬

ment errors, limited actuator speed, unmodelled high-frequency

dynamics, and process nonlinearity. All these sources of uncertainty

occur simultaneously and can be classified into three different groups:

• Uncertainty of the manipulated variables (input uncertainties)

• Model uncertainty due to process nonlinearity and unmodelled

high-frequency dynamics

106 5 A Structured Uncertainty Model

• Uncertainty of the temperature measurements (output uncer¬

tainties)

This grouping corresponds to the principle that uncertainty should be

modelled where it physically occurs.

In this chapter an uncertainty model for the industrial distillation

column is developed. The complete uncertainty model covers not only a

single operating point but the entire operating range of the column. It is

the basis for the analysis and synthesis of controllers using the struc¬

tured singular value u\

5.2 Limits of uncertainty models

Before we start to model the uncertainty in the frequency domain, we

must be conscious of its limits: An uncertain model in the frequency

domain is a model, which is time-invariant, but uncertain in its coeffi¬

cients.

This statement is best explained by an example: Let us model a ±10%

uncertainty at the input of any plant and design a controller which

guarantees closed-loop stability and a certain performance for all plants

within the specified bounds. Then the stability and performance proper¬

ties of the controller are not guaranteed for a time-varying plant, that

means e.g. for variations of the input error between -10% and +10%.

Consequently, using uncertainty models in the frequency domain, the

excitation ofthe controller by the time-variation ofthe plant is not taken

into account. If time-varying uncertainties are assumed, nonlinear

simulations must be used for a validation of the robustness properties.

However, the experience shows that for most cases uncertainty descrip¬

tions with frequency dependent and hence time-invariant uncertainty

bounds are sufficient.

This holds especially for our distillation column: The main disturbances

are step changes of the feed flow rate. Each step change alters the

steady state operating point and defines a new linear model describing

the dynamic behavior up to the next step change. Each of these linear


models is one of the models within the set of all models. This set is

defined by the specified uncertainty bounds.

5.3 Input uncertainty

The actual values of the manipulated variables reflux and boilup will

never match exactly the values requested by the control system. The

error between the setpoints for the boilup or the reflux and the true

streams will be frequency dependent. The main causes are

• static and dynamic measurement errors of reflux and

reboiler duty

• changing heat of evaporation due to pressure and

temperature variations

• reboiler lags

• actuator lags

• effects of sampling

The bounds for the relative errors of the column inputs u can be

modelled by a multiplicative uncertainty description with the

frequency-dependent error bound wu for the reflux L and the error

bound wu for the boilup V These bounds are combined in the diagonalmatrix Wu. As illustrated in Figure 5.1 the following uncertainty model

holds:

u(jco) = {I + Au(jco)Wu(j<o)}u(j(») (5.1)

1— W Au

u1 1

Figure 5.1: Multiplicative uncertainty description for column input


with

|Au(jco)| <1u loo

(5.2)

Wu(jco)wUl(joo) 0

0 wUy(jco)(5.3)

The frequency-dependent complex matrix Au(jco) is limited in magni¬

tude. It shapes only the spatial direction of the error and is chosen to be

the worst case during u-analysis (see Chapter 6). Therefore the phase

behavior of the individual uncertainty bounds wu. is not significant.

They should be chosen to be stable and minimum phase.

If we assume that the reflux and the boilup errors are independent, the

matrix Au (jca) becomes a diagonal matrix with two single perturbations

8 yielding the following uncertainty model:

u (jco) 1 +8Ul(jo» 0

0 5U (jco)uv

Wu(jco) u(jco) (5.4)

with

|5u.(jco)| <1 (5.5)

Both models have been used for u-synthesis with very similar results.

For two reasons, the model (5.1) is preferred in this study:

• The number of uncertainty blocks (Aj or 8j) is reduced by one

compared to (5.4). This simplifies the fi-synthesis.

• Any change in reflux may cause a change of the vapor flow rate

within the column and vice versa. The interactions due to flow

dynamics and to the energy balance are to be considered here.


Shaping the input error bounds

It has been shown by Skogestad et al. [5.4] that the controlled system's

performance for a high purity distillation column is very sensitive to

errors in the manipulated variables. For controller design or analysis

the error bounds Wu should be estimated as exactly as possible. This

holds especially for the low-frequency range, where the condition

number of the column models is high. Otherwise potential controller

performance is given away in case of an overestimation.

In the lower-frequency range the errors of the manipulated variables at

the plant input are strongly dominated by flow measurement errors and

parameter variations. As an example for a parameter variation we

consider the heat of evaporation in the reboiler. The boilup is controlled

indirectly by steam flow rate. Therefore a change in heat of evaporation

will cause an error in vapor flow rate leaving the reboiler.

Skogestad and Morari [5.4] assume a conservative 20% error in steady

state, which is fairly high. If all flow measurements are carefully cali¬

brated the error bounds should be less. An error bound of 10% for the

lower frequency range is assumed to be large enough.

The effects of reboiler lags, actuator lags, dynamic measurement errors,

and sampling time concern the higher frequency range. The errors

caused by these uncertainty sources increase with the frequency and

easily exceed more than 100% of the nominal value for frequencies

above 0.5-1 rad/min.

The steady-state error, together with the high-frequency error, is well

approximated by the first order lead/lag transfer function

1 + s/coM

G(s)=KTT^7uf withcoN<coD (5.6)

The gain K represents the steady-state error. The cut-off frequencies are

typically chosen according to coD > 10coN.


5.4 Model uncertainty

5.4.1 Column nonlinearity

The highly nonlinear behavior of distillation columns is observed at

varying operating points (varying feed flow rate and feed composition)

and at transients during disturbance rejection. Ifwe consider the simpli¬

fied composition dynamics (without feed or side-product stream) of a

tray (see (4.6))

nj(S) = LJ-i(xJ-i-xJ)+vj+i^+!-V-vJ(yJ-xj) (5-7)

we recognize that the composition dynamics and thus the nonlinear

behavior depend on

• the varying internal flow rates (L and V), and

• the composition profile within the distillation column (repre¬

sented by the liquid and vapor phase compositions)

Effect of varying operating points

Any control system for a distillation column must exhibit large gains in

the low-frequency range to achieve small control errors at steady-state.

Therefore, at steady-state both product compositions (or the tempera¬

tures on trays 10 and 44) can be kept at their setpoints. Thus transients

have no significant influence in the low-frequency range and the internal

vapor and liquid flow rates as well as the composition profile within a

column become a function of feed flow rate and composition only.

However, the dynamic behavior of a distillation column depends

substantially on the actual composition profile and on the actual

internal vapor and liquid flow rates. Normally the operating range of a

distillation column can be bounded with a maximum and a minimum

feed flow rate and composition. Ifwe consider the whole operating range

defined in this way, we can observe the largest internal flow rates for the

smallest feed composition and largest feed flow rate and, vice versa,


smallest internal flow rates for the largest feed composition and

smallest feed flow rates. The composition profiles for these two steady

states bound the domain of all steady state composition profiles (see

Figure 2.2).

Hence we can conclude that the low-frequency behavior of a binary

high-purity distillation column is bounded by the models for maximum

and minimum column load. As a basis for further discussion the

following three linear models are introduced:

Model N column at nominal load

Model I column at maximum feed flow rate and

minimum feed composition (increased load)

Model R column at minimum feed flow rate and

maximum feed composition (reduced load)

The feed data of the different models are listed in Table 5.1.

Table 5.1: Operating conditions for design purposes

Operating point Feed flow rate

(mol/min)

Feed composition(mol/mol)

OP-N 33 0.8

OP-I 46 0.7

OP-R 20 0.9

The simplest way to represent the column nonlinearity due to varying

operating points would be by a multiplicative output uncertainty. If we

assume that the uncertainty for each model output is independent ofthe

actual value of the other two model outputs, the following form for the

output uncertainty holds (Figure 5.2):


Figure 5.2: Multiplicative uncertainty at output

y(jco) ="

8 (jco) 0 0•M

1 + 0 8y2 (jco) 0 Wy(jco)

.

0 OS (jco)3 .

y(jco) (5.8)

with

and

Nl--1

y(jco) = GN(jco)d(jco)

u(jco)_

The transfer matrix Wy is a diagonal matrix with the uncertainty

bounds for each output (w,r ,w„

,wv ) on the main diagonal. An upper

^1 ^2 -^3

bound for these uncertainties can be obtained by a calculation of the

standardized errors AGj (jco) and AGR (jco) for each channel ui -^ yj or

d- -» y- of the models Gj (jco) and GR (jco) , respectively:

AGj(jco) = [GI(jco)-GN(jco)]G^1(Jco) (5.9)


AGR(jco) = [GR(j(o)-GNa<«))]GN1afl» (5.10)

The upper bound for the uncertainty weights w is the maximum of allj

standardized errors for the output y •.

In earlier papers it has already been recognized that column nonlin¬

earity is not well represented by simple multiplicative uncertainty

bounds at model output (McDonald [5.2]). This fact is confirmed by the

uncertainty bounds for the two numerically evaluated linear models Gjand Gr which include the flow dynamics (Figure 5.3). The multiplicative

output uncertainty exceeds 80% (for Gu ) in the low-frequency range.

It is significantly smaller in the medium frequency range, but increases

sharply for frequencies above 0.1 rad/min, where the flow dynamics

influence the dynamic behavior. An uncertainty description with such a

high multiplicative uncertainty in the low-frequency range is prohibi¬

tive for any control design.

Fortunately the errors are highly correlated: The variation of the

steady-state operating points causes a simultaneous increase or

decrease of the singular values ofthe transfer functions from the control

signals u (L and V) to the model outputs y (T10, T44, T24). This is illus¬

trated by the Nyquist plots for the individual channels Uj -» y. (Figure

5.4). It clearly shows that the variation of the column load causes a

simultaneous increase or decrease of the open-loop gains in the low-

frequency domain. Thus we can assume that the dynamic behavior of

the distillation column must lie "somewhere between OP-I and OP-R."

This can be represented by a linear combination of the two column

models Gz (jco) and GR (jco) (Figure 5.5)

GT (jco) + GR (jco) GT (jco) - GR (jco)G(jco) =

lU

2

R+ 5G(jco)

lU

2

R(5.11)

with

|6g|L-i 8Ge C or 8Ge R


Standardized error of T10, Gj(s) Standardized error of T44, Gj(s)

10 10 10 10 10 10

Frequency (rad/mm)

Standardized error of T24, Gj(s)

10 10 10 10 10 10

Frequency (rad/min)

Standardized error of T10, Gg(s)

10 10 10 10 10 10

Frequency (rad/min)

Standardized error of T24, Gr(s)

10 10 10 10 10' 10"

Frequency (rad/min)

Standardized error of T44, Gr(s)1

1

"""%, /

0.5\ /

\ /

\ s^.—-

. 1

.1

. !

0 -4li1

111 \ ' i-N

10 10 10 10 10

Frequency (rad/min)

Legend

xF

F

L

V

Frequency (rad/mm)

Figure 5.3: Standardized model errors at operating points OP-I and OP-R


GL-»T10(Jffl) GV^T (J«>)

GL_T44(jco)

-250

GL^ T CJ0»

61

1

V\\

f

1

/

/

/

\

\

/

/

/

y

<z>

/

500

Figure 5.4: Nyquist plots for different column loads -

solid lines: Model N;

dash-dotted lines: Model I;

dashed lines: Model R;

x: 0) = lxlO"3 rad/min;

o: co = lxlO"4 rad/min


r-H.Gl J~^-^'

u

T+ r-

1/2

5G*3x y

Figure 5.5: Uncertainty model due to nonlinearity in the low-frequency range

The uncertainty parameter §G may be either complex or real. If we

define it to be complex we allow a phase shift for all models between GRand GT, that means the points of all models in the defined set in the

Nyquist plots and for a fixed frequency are not required to be on a

straight line.

In this way we generate a plant which covers the properties of the distil¬

lation column at low and at high feed compositions, and at different feed

flow rates without introducing additional conservatism. It is impossible

to model such a behavior with an unstructured uncertainty description!

Effect of transients

While an appropriate uncertainty model for different operating points

requires a highly structured uncertainty description, the effect of tran¬

sients is rather unstructured: During disturbance rejection, the compo¬

sitions on tray 10, tray 44, and tray 24 as well as the product

compositions will deviate from their steady-state values, caused by a

movement of the composition profile toward one column end. Due to the

nonlinear vapor/liquid equilibrium, the singular values of the transfer

functions Gu may change in different directions, e.g., towards higher

singular values of G„ . Tand lower singular values of G„ . T .

Due to the high controller performance in the low-frequency range, tran¬

sients do not affect the low-frequency range. However, they cause


nonlinearity in the middle and higher frequency range, which can be

described with a multiplicative uncertainty description as in equation

(5.8). The uncertainty weights w are chosen to have large singular

values in the higher-frequency range and low singular values in the low-

frequency range. It is not possible to calculate these uncertainty bounds

exactly. Each disturbance input will cause a variation of the operating

point, but the magnitude of the deviation from a steady-state operating

point cannot be predicted. The selection of appropriate transfer func¬

tions is discussed in Chapter 6.

5.4.2 Unmodelled dynamics

It has been shown in Chapter 2 that flow dynamics affect the high-

frequency behavior of distillation columns. If linear models which

neglect the flow dynamics are used for control design, an appropriate

uncertainty model is necessary.

Most authors treat the effect of flow dynamics in the same way as the

effect of an input time delay t with 0 < x < 1 minute ([5.1], [5.3], [5.4],

[5.5]). The corresponding input uncertainty is often modelled with a

multiplicative uncertainty, using a first order Pade approximation for

the uncertainty bound ([5.4], [5.5]):

1 _I5

e-B (5.12)

This uncertainty can be combined with the other input uncertainties

(Chapter 5.3). Lundstrom et al. [5.1] point to the fact that some combi¬

nations of gain uncertainty and time delay uncertainty are not repre¬

sented using simple uncertainty weights. They developed new and more

complicated uncertainty bounds, which cover the whole domain of

combined gain uncertainty and time delay uncertainty. However, the

control design studies in this research (Chapter 6) show very goodresults using simple first-order weights for the input uncertainty

description.


Model uncertainty due to flow dynamics could be represented by a multi¬

plicative output uncertainty, as well. This approach has the disadvan¬

tage that the uncertainty bounds can no longer be approximated by time

delays.

5.5 Measurement uncertainty

An additional source of uncertainty are the temperature measurements.

The dynamic behavior of a temperature sensor is well approximated by

a first-order lag:

GT(8) =~L- (5.13)

The time constant TT of this transfer function depends on the tempera¬

ture measurement position. While the time constant will usually be

clearly below 1 minute if the sensor is placed in the liquid phase, we have

to expect time constants up to 10 minutes if it is placed in the vapor

phase. In the case of the industrial distillation column under investiga¬

tion, a position in the liquid phase cannot be guaranteed because of the

small head on the plates. Therefore we have to consider time constants

up to 10 minutes for the control design.

The gain KT of the sensor model GT depends on the sensor calibration

and on the heat loss to the environment. The sensor can easily be cali¬

brated with high accuracy. However, the dynamic effects of the heat loss

due to variations of the environment temperature are difficult to esti¬

mate. They concern mainly the low-frequency range and cause a slow

bias variation of the temperature measurements. This effect is compa¬

rable to variations of the setpoints for the control system. The stability

of the control system is not affected if large bias variations are avoided.

They would lead to product compositions which are very distinct from

those at the design operating points. A good thermal isolation of the

temperature sensors is thus recommendable.

Because the u-analysis and u-synthesis guarantees the performance for

the worst case, it is proposed to include the model for the temperature

5.6 Specification of the controller performance 119

sensors with a gain KT = 1 and a time constant TT = 10 min into the

column model. Thereby further uncertainty blocks can be avoided. It

will be easily recognized later, that shorter time constants TT will not

endanger the closed-loop system's stability due to the large output

uncertainties w specified for the controller design in the upper

frequency range.

5.6 Specification of the controller performance

The uncertainty model discussed above is structured. A controller

design or a robust performance analysis requires the framework of the

structured singular value u, which expects the disturbance inputs d

(feed composition, flow rate), the reference inputs r, and the control

error to be in a H^-norm bounded set. This is illustrated by the

frequency shaped plant in Figure 5.6.

i!We(8)

/ I

Wd(s) GA(s)+o—K(s)

*\J

-L 1 [—*T24

T10>T44

Figure 5.6: Performance specification for the uncertain plant

The uncertain plant GA(jco) describes the nonlinear behavior of the

distillation column. The performance objective is defined as making the

weighted control error e to be in the set


sup [jWe (j co) e (j co) J 2 < 1 Vco e R+

LrJ 2

The following H^ bound is equivalent to this specification:

Td

r

(jco)^P

We (s) is a (usually diagonal) matrix of transfer functions which shapesthe maximum allowed amplitude of the transfer function from [d, r]

T to

e. If We is large in a certain frequency range, only a small control error

is allowed there.

The matrix Wd (s) shapes the frequency content of the disturbances and

setpoint changes. In the case of our distillation column, variations of the

feed composition and feed flow rate will affect the medium and lower

frequency range. First order lags shape the frequency content of these

two disturbances quite well. Because measurement noise enters the

control loop at the same position as the reference inputs, the corre¬

sponding weights are chosen to be constant. The weighting functions

chosen are discussed in the following chapter.

5.7 Summary

The complete uncertainty model is shown in Figure 5.7. It consists of the

input uncertainty (5.1), the model uncertainties (5.8) and (5.11), and the

performance specifications. Simple dynamic models of the temperature

sensors are included in the column models. This relatively complex

uncertainty model has the advantage that the entire operating range of

the distillation column is covered. The large conservatism of an unstruc¬

tured uncertainty description is avoided. Therefore, with design proce¬

dures based on the u-synthesis or u-optimization, we can expect high

controller performance for the entire operating range.

5.7 Summary121

6

Eh

rV)^«**+

Us

iL iL

s

I

o

0)

O

•at

u

as

a

ao

O

IO

O

Q.


The input uncertainty bounds wu are easily shaped. Only a few reflec¬

tions are necessary about the steady-state error and the frequencywhere a 100% error is to be expected. However, the output uncertainties

wy. are more difficult to shape. During the controller design procedureit is often necessary to adjust them iteratively until nonlinear simula¬

tions show a satisfactory closed-loop dynamics. This problem is

discussed further in Chapter 6.

5.8 References

[5.1] Lundstrom, P., S. Skogestad, and Z.-Q. Wang: "Uncertainty

Weight Selection for H-Infinity and Mu-Control Methods," Proc.

30th Conference on Decision and Control, Brighton, U. K. (1991)

[5.2] McDonald, K. A.: "Characterization of Distillation Nonlinearity

for Control System Design and Analysis," The Shell Process Con¬

trol Workshop, ed. D. M. Prett and M. Morari, Butterworth, Bos¬

ton, 279-290 (1987)

[5.3] Postlethwaite, I., J.-L. Lin and D.-W. Gu: "Robust Control of a

High Purity Distillation Column Using u-K Iteration," Proc. 30th

Conference on Decision and Control, Brighton, U. K (1991)

[5.4] Skogestad, S., M. Morari, and J. C. Doyle: "Robust Control of 111-

Conditioned Plants: High-Purity Distillation," IEEE Trans. Auto¬

matic Control, 33,12, 1092-1105 (1988)

[5.5] Skogestad, S., and P. Lundstrom: "Mu-Optimal LV-Control of

Distillation Columns," Comp. Chem. Eng., 14, 4/5, 401-413 (1990)


Chapter 6

ja-Optimal Controller Design

6.1 Introduction

While the synthesis and analysis of controllers using the structured

singular value a (SSV) has attracted considerable attention among

aerospace and electrical engineers (e.g., [6.8], [6.9]), it has been less

commonly considered by process control engineers. One reason for that

might be the lack of adequate structured uncertainty models for chem¬

ical processes. The uncertainty model discussed in the previous chapterforms a suitable basis for a u-optimal controller design. Since this uncer¬

tainty model covers the dynamic behavior of the industrial distillation

column for the entire operating range, the resulting controllers guar¬

antee stability and performance for all operating points.

This chapter presents the results of a u-optimal controller design for the

LV control structure of the distillation column. After a summary of the

most useful aspects of the SSV, the design of state-space controllers by

u-synthesis is demonstrated. Because the implementation of state-

space controllers in a distributed control system is a troublesome

project, the design of controllers with fixed and easy-to-implementstructures (PID control structures) is considered in a special section. A

comparison of the controller's performances in the time-domain termi¬

nates this chapter.

124 6 |i-Optimal Controller Design

6.2 The structured singular value

The uncertainty model approximating the nonlinear dynamic behavior

of the industrial distillation column (see Chapter 5) includes several

simultaneous uncertainty blocks (8i; A-), thus representing a structured

uncertainty model. Most of the well-known robust control design

methods (e.g., H^, LQG/LTR) are based on unstructured uncertainty

descriptions. The application of these methods on such uncertainty

models often introduces unnecessary conservatism in controller design,

because these methods combine all the uncertainties in one large, fully

occupied uncertainty block. Thus the special structure of the uncertain¬

ties is neglected. This conservatism can be avoided by the use of the

structured singular value |i, which was introduced in 1982 by J. C.

Doyle ([6.5], [6.6]).

The structured singular value (t so far has seldom been discussed in

textbooks. Therefore the most important facts about |i are summarized

within the following three sections. The discussion is restricted to

complex uncertainty blocks. Results for mixed real/complex uncertain¬

ties can be found in [6.15]. The references [6.4], [6.12], and [6.14]

contain additional informations.

6.2.1 Representation of structured uncertainties

The definition of the structured singular value presumes that the uncer¬

tainty model for a plant is rearranged into a special form, as shown in

Figure 6.1. The plant P consists ofthe process models and the weighting

functions. It has three sets of inputs and outputs:

The first set of inputs and outputs is highly important. Within the

uncertainty model, this set represent the output and input signals ofthe

uncertainty blocks. In our case the inputs to the uncertainty blocks are

the signals po,£,0 and no. The corresponding outputs are the signals

p.î.andTii.

The second set of inputs consists of all external signals (disturbances d,

reference inputs r), while the third set of inputs consists of all manipu¬

lated inputs u. The corresponding set of outputs contains the outputs p,


= z

A

Uncertainties

PPlant

K

Controller

v =

Figure 6.1: Standard representation of an uncertain plant.

The definition of the vectors z and v is related to Figure 5.7

subject to any performance measure (e.g., the weighted control error),

and the measured plant outputs y, respectively.

If the uncertainty model is structured (i.e., it contains more than one

uncertainty block) the matrix A is a block diagonal matrix with all

uncertainty blocks on the main diagonal. In case of the uncertainty

model for the distillation column considered in this thesis, the following

block structure holds:

A = diag (Au, 6GI3,5y], 5^, 6^|Au e C2 * 2, 8G e C, 8y. C )

or, alternatively,

A = diag (A u, 8GI3,8 8 ,8 IAu e C2 x 2,8G e R, 8 e C ) (6.1)y,' y2' y3 î

with M-^ftl-*1

For an unstructured uncertainty model, the matrix A is a fully occupied

matrix without predefined structure.

126 6 (i-Optimal Controller Design

The rearrangement of an uncertainty model into the standard form is

always possible. The MATLAB p-Analysis and Synthesis Toolbox

[6.1] as well as the Robust Control Toolbox [6.3] provide efficient tools

for that purpose.

6.2.2 Definition of the structured singular value

Let X be the set of all A matrices with a given, fixed block-diagonal

structure:

X- {diag 8]Ir,...,8sIr,A1,...,Af 1^ e C, A. e CmJxmJ} (6.2)

The structured singular value [6.7] of the Matrix Me Cmxm with

m = Yfj + Ym. (Fig. 6.2) is defined by

HA(M) ='

1

min {omax(A)|(det(I + MA)=0)}Ae X

0 ifno(AeX) solves det (I + MA) =0

(6.3)

Hence 1/u (M) is the size of the smallest matrix A which moves a pole

of the system shown in Figure 6.2 onto the imaginary axis. In the case

of a nominally stable system M, 1/li(M) is the size of the smallest

destabilizing matrix A. In case of a nominally unstable system M,

Figure 6.2: M-A feedback connection


however, |i(M) is not defined, and the numerical results are

misleading.

Some important properties of\l [6.7]

Let D be the set of diagonal scaling matrices:

(6.4)

D= {diagfD! ,...,Ds,ds + 1Imi,...,ds + FImF]

|DiCrixrsDi=D*>0,ds+jR,ds+j>0}

and let U be the set of block-diagonal unitary matrices

U = {diag(Ulf U2, ...,Un) |Ui6 CÛÛ^} (6.5)

With these definitions the following properties of \i hold:

p(M)<u(M)<omax(M) (6.6)

H(DMD->) = u(M) (6.7)

max p(UM) <p(M) < inf ofDMD-1) (6.8)UeU DSD

mM

Property (6.6) reflects the advantage of the structured singular value u:

In the presence of structured uncertainty, usually the inequality holds.

Therefore, u is smaller than the maximum singular value.

The invariance of \i to diagonal scaling is indicated by property (6.7),

which is essential for the approximate calculation of the structured

singular values as well as for the DK-iteration for u-synthesis.

No direct way has been found yet to calculate u exactly. All algorithms

for the numerical computation calculate upper and lower bounds

according to property (6.8).

Both bounds represent an optimization problem. The optimization

problem for the upper bound is convex. For simple block structures with

2S + F < 3 the upper bound is guaranteed to be equal to nA (M).

128 6 fi-Optimal Controller Design

The optimization problem for the lower bound is not convex and its

calculation may converge to local maxima. Nevertheless numerical

experience indicates that usually the difference between upper and

lower bounds is within 5%, and almost always within 15% ([6.12],

[6.14]).

6.2.3 Robustness of stability and performance

Before we start to discuss robust stability and performance within the

framework of the structured singular value, we have to join the plant P

and the known controller K of the standard configuration in order to

close the control loop (Fig. 6.3). This is easily done by a linear fractional

transformation [6.13]:

M(P,K) = J,(P,K) = Pn+P]2K(I-P22K)-iP12(6.9)

The resulting plant M has two sets of inputs and outputs:

=

M„ M12

M21 M22|

z

[d]|_rj

(6.10)

d

r

Z (

A ^ 1

) V

1

M(P,K)

Figure 6.3: Representation ofuncertain control system with controller K and plantP combined into the system M


The input sets are (1) the outputs from the uncertainty block A, and (2)

the disturbance and reference inputs. The outputs, in turn, are the

inputs to the uncertainty block A and the set of performance measures

P-

Theorem 6.1: Robust stability

Let BX be the set of all block diagonal matrices with a particular struc¬

ture and with infinity-norm-bounded submatrices:

BX = {diagr81Iri,...,8sIvAI,...,AflV

(6.11)

|6i6c,Ajec,».m8i|î.hL:si}

The system shown in Figure 6.3 remains stable for all A e BX if and

only if

sup uA(Mn)<l (6.12)co

Proof: see [6.6]

Theorem 6.1 allows the stability analysis of control systems with struc¬

tured uncertainties. If we plot the upper and lower bounds of |A (Mu)for enough frequency points in the frequency range of interest and find

that the maximum value of u is smaller than one, the control system is

stable for the uncertainties specified with the assumption |oJ <. 1,

||Aj| < 1. If p, (M) exceeds one for any frequency, the control system is

not guaranteed to be stable. However, for smaller uncertainties with

II5J <l/(sup uA) and |A.|| <l/(sup \iA) stability is guaranteed.ii iii~

ma ii Jiioo

wa

Theorem 6.2: Robustperformance

The performance of the control system is robustly achieved ifand only if

supp-(M)<l with A = diag[A,Ap] (6.13)

Proof: see [6.6]

130 6 u-Optimal Controller Design

For the application of theorem 6.2 we have to add one uncertainty block

Ap to the uncertainty structure A (Fig. 6.4). Imagine that the perform¬

ance specification of the control system is met for all allowed distur¬

bance matrices A in the set BX. In this case the output p is bounded by

HpIL < 1 for all inputs [dT, rT]T with || [dT, rT] T|L < 1. If we close the

loop from p to [dT, rT]T by introduction of the block Ap with |Ap|[ < 1,

the system will be stable. But ifany block Ap with ||Ap| < 1 destabilizes

the loop p -» [dT, rT]T, the specified performance cannot be achieved

for all possible plants within the specified set. Therefore, in the frame¬

work of the SSV, the robust performance problem is handled like a sta¬

bility problem. A test for robust performance will be similar to a test for

robust stability. Because the test for robust performance includes robust

stability, it usually will be sufficient.

A 0

0 AP

M(P,K)

Figure 6.4: The robust performance setup for the SSV-framework

6.3 The design model

The design of u-optimal controllers for the industrial distillation column

is based on the uncertainty model developed in Chapter 5 (see Figure

5.7 on page 121). For this model, the weighting functions for the

<)

6.3 The design model 131

• input uncertainties,

• output uncertainties,

• reference inputs, disturbance inputs, and

• controller performance

are to be selected.

All weighting functions are chosen as diagonal matrices:

Wd(s) = diag[wXF(s),wp(s),wrio(s),wr44(s)] (6.14)

Wu(s) =diag[wUL(s),wUv(s)] (6.15)

Wy (s) = diag[wyifl (s), wy^(s), wy^(a)] (6.16)

We(s) = diag|"we (s),we (s)l (6.17)

The selection of the weighting functions is primarily done on the basis

of physical considerations:

Feed disturbances: The variations of the feed composition and the feed

flow rate will affect the lower frequency range. The frequency contents

of these disturbances are modelled by first-order lags. Typical weights

chosen here for the control design are

wv (s) = K -—i—withK = 0.1 mol/mol, T = 180 min (6.18)Xp XF1 + 1 S XF XF

wF (s) = KF*

with Kp = 6 mol/min, Tp = 120 min (6.19)1 + lps

Reference inputs: The reference inputs r can be used to model setpoint

changes as well as measurement noise. They are chosen as constant

weights, representing setpoint changes and measurement noise of

±0.2 °C:


wr (s) = wr (s) = 0.2 (6.20)

Input uncertainties: An uncertainty of 10% is assumed for both manip¬

ulated variables within a wide frequency range. For higher frequencies

the uncertainty is expected to be much higher. An uncertainty of more

than 100% is assumed for co > 0.5 rad/minute:

wUl(s),wUv(s)=0.1i^ (6.21)

Output uncertainties: The resolution of the temperature measurements

is limited. The maximum deviation of the pressure-compensated

temperatures from their setpoints is usually significantly smaller than

1 °C. A measurement error for AT of 10% seems to be reasonable. In the

higher frequency range the output uncertainty is affected by model

mismatch. An assumption of a 100% error for co ~ 1 /16 rad/minute has

shown good results in controller design. Adjusting of this 100% cross¬

over is one of the possibilities to influence the high-frequency behavior

of the resulting controller. Typical uncertainty weights are

wy (s) = w (s) = w (s) = 0.li±if£ (6.22)yiO y44 ylA 1 + 1.0/S

Performance weights: The performance weights "punish" the control

error in the frequency domain. These weights have been selected as

first-order lags with a large steady-state gain, which forces nearly inte¬

grating behavior ofthe controller. The cut-off frequency of these weights

is a matter of optimization: If the frequency is too high, robust perfor¬

mance cannot be achieved. On the other hand a cut-off frequency speci¬

fication significantly lower than the maximum attainable frequency

may lead to an unsatisfactory controller design. This holds especially for

the uK-iteration which is discussed later. A typical performance specifi¬

cation, which allows a 0.01 °C steady-state offset, is given by

^-^-^T^Sooi (6-23)


All weights above are illustrated by Figure 6.5

Uncertainty weights Input and performance weights

10 10 10 10

Frequency (rad/min)10" 10 10 10"

Frequency (rad/min)

Figure 6.5: Weights for u-synthesis

6.4 Controller design with {i-synthesis

The objective of the (i-synthesis is the calculation of a stabilizingcontroller K without restriction on the controller order and its structure,

which minimizes the SSV for all frequencies:

K= arg inf |Ui-(M (P, K)) IK stabilizing " >

(6.24)

As it is not possible to calculate the SSV exactly, the design task (6.24)

is usually approximated by the upper bound for u (6.8)

K=arg inf | inf amax(DM(P,K)D-1)Kstabilizing" De D

(6.25)

The aim of the u-synthesis is perfectly reached, if the maximum value

of |i- for the closed-loop system (Figure 6.4) is below one.


6.4.1 Synthesis algorithms

The u-synthesis is not a trivial task. Yet no algorithms have been devel¬

oped which allow a one-step solution of the u-synthesis problem (6.24).

The known algorithms require the repeated calculation of an H^problem, alternating with a scaling of the plant. These algorithmscannot guarantee convergence of the iteration.

DK-Iteration

The synthesis problem (6.25) is a simultaneous optimization problem of

the frequency-dependent scaling matrices D and the controller K.

Because no direct solutions exist, Doyle [6.7] proposes an iterative

approach: Ifwe keep the diagonal scaling matrices D constant, the mini¬

mization of 1 inf o(DMD-1) || forms the convex H problem||

DgDmax^ Ml-

K = arg infllDTUP.KJD-1! (6.26)Kll 1 II-

If we fix the controller K, equation (6.26) represents a convex optimiza¬tion problem for the diagonal scaling matrices D. These scaling matrices

are optimized by a u-analysis of the closed-loop system:

u- [?i (P, K) ] = inf o (DMD-1) (6.27)A

DeD

The frequency-dependent scaling matrices D are approximated with

stable, rational transfer functions D (s) . Alternating the HM controller

synthesis and the optimal scaling, convergence is achieved for most u-

synthesis problems after several iterations. The iteration procedure is

illustrated in Figure 6.6. The DK-iteration is finished either if the solu¬

tion does not show any further improvement or if u < 1. However,

convergence cannot be guaranteed: Both ofthe single optimization prob¬lems are convex, but not the overall optimization problem (6.25). The

optimized scaling matrices D are an optimal solution for the local opti¬

mization problem (6.27), but they are not optimal for the global optimi¬zation problem (6.25). Therefore, the DK-iteration may converge to local

minima.


K0 = arg infl^P.K)^

ZZZ3ZZI^[^(P.Ko)] =D>nfDamax(DMD-i)

Fit D(s) with stable, mm. phase transfer functions D (s)

TK, =arg irfJDfsj^fP.KjD-'fB)!.

|

^[JjCP.K,)] =D.nfD0-max(DMD-i)

Figure 6.6: DK-Iteration

ui<L-/terafJon

A new algorithm for u-synthesis has been proposed by Lin et al. [6.11].

Instead of fitting the scaling matrices D, this algorithm is based on a fit

of the frequency-dependent SSV with a stable rational transfer func¬

tion. At each iteration step, the plant is premultiplied with a diagonalmatrix of the u-approximating transfer function. Thus the peaks of u

within the frequency range of interest are punished, and the algorithmtries to flatten the u-curve. The convergence of this algorithm is not

proved. The authors present "a reasoned argument for believing that

the sequence will converge" [6.11]. A scheme of the uK-algorithm is


Kârg inff?, (P,K)|m

u(jco) = ^[^(P.Kq)]

Fit |10 (jco) =., .

'.. , ji0(s) stable, miminum phase

z.iz'vi" :"

K, = arg inf|jio(s)7'1(P>K)||oo

;::_ :.l."._::..:u(jco) = u-fJiCP.K,)]

:::::: ::\:' '

»•*.«=[?&

iK- = arg infNji. (s)(l0(s)?1(P, K)||

'

J..." '

Figure 6.7: uK-Iteration

usually converges more slowly than the DK-iteration, and the conver¬

gence properties are strongly dependent on the fit of the u-curve. Even

if convergence is achieved, u is not minimized for all frequencies.

6.4.2 Applying the DK-iteration

The apphcation of the DK-iteration to our design problem was not

successful because convergence of the algorithm is not attainable. Most


likely, the main problem is the fit of the fully occupied 2x2 or 3x3

(including or excluding the measurement of T24, respectively) block of

the D-scaling matrices. This block results from the repeated scalar

uncertainty block 8GI. Ifwe fit each position ofthis block with a scalar,

stable, and minimum-phase transfer function, a minimum-phasebehavior for the resulting MIMO system is guaranteed. However,

unnecessary conservatism is introduced thereby, since a minimum

phase behavior is only required for the MIMO system, but not for the

single scalar transfer functions. This problem remains to be solved.

6.4.3 Applying the uK-Iteration

The apphcation of the uK-Iteration does yield convergence. However, it

is necessary to slightly modify the algorithm. The premultiplication of

the u-curve-fitting transfer function jlj (s) increases the order of the

design model at each iteration step. This easily leads to models with

more than 200 states. Unacceptable calculation times and numerical

problems result therefrom. This problem is avoided by an order reduc¬

tion step after the augmentation ofthe plant and before the HM design.The order reduction method utilized is a balanced truncated realization

[6.1].

Experiences

A typical course of the iteration is shown by Figure 6.8. In the frequency

range where the performance specification u- (ja>) < 1 is not achieved,the upper bound of the SSV is forced down at each iteration step. After

six steps, the solution is reached for which no further improvement is

possible. If we look at the frequency range between IO-2 and 10_1 rad/

min, we discover that the first controller exhibits much better robust

performance than the final design. This results from the "flattening"behavior of the uK-Iteration, which leads to any solution of the design

task, but not necessarily to the optimal one.

This behavior of the iteration scheme may lead to strange results. Even

if the robust performance criterion is achieved, the simulation of the

closed-loop behavior may exhibit an insufficiently damped oscillation. It


10 10 10 10 10

Frequency (rad/min)

10 10

Figure 6.8: Convergence of the uK-Iteration

has to be emphasized here that such an oscillation is consistent with the

performance specification. This performance ofthe closed-loop system is

specified in the frequency-domain rather than in the time-domain!

Slightly increasing the performance requirement or the uncertainty

specifications usually removes this problem.

Another problem of the uK-iteration is the small convergence area: The

performance specification We(s) (see (6.17)) has to be close to the

maximum achievable performance, otherwise the iteration does not

converge. As a last drawback the long computation times have to be

mentioned. The CPU time for a design usually exceeds 2h on a SUN

SPARC 2 workstation!


Analytically or numerically linearized models?

In chapter 4 two different types of linear models have been developed.The main differences between these model types consist of the low

frequency gains and the representation of flow dynamics. In the case of

the analytically linearized models, the relative uncertainty in the low

frequency range due to variation of the steady-state operating points is

essentially smaller (see Chapter 5).

For both types ofmodels, state-space controllers have been designed. In

order to achieve an acceptable controller design (oscillation free) with

analytically linearized models, the low-frequency gains of the distur¬

bance weights (6.18) and (6.19) must be approximately doubled. With

respect to the higher frequency range the fact of unmodelled flow

dynamics does not dominate the shape of the output uncertainty

weights w for the tray temperature T10 and T44. Both weights may be

kept equal for both types of linear models. However, the uncertainty

weight for the temperature measurement in the middle of the column

T24 has to be increased for the analytically linearized models due to the

unmodelled flow dynamics. In accordance with these adaptations of the

weighting functions, the resulting state-space controllers yield nearlyidentical closed-loop behavior with a small advantage from using the

numerically linearized models.

Complex or real uncertainty block 8GWithin the structured uncertainty models, the uncertainty 8G may be

chosen as a complex or real uncertainty. The choice as a real uncertaintyreduces the uncertainty for the entire frequency range. However, the

resulting closed-loop behavior exhibits insufficiently damped oscilla¬

tions. To avoid this problem, the performance weights and the distur¬

bance weights must be increased. The state-space controller designedwith the modified weights are not superior to the design obtained with

a complex uncertainty block 8r.


\\K-Iteration results for three temperature measurements

The input vector of the controller may consist either of the pressure-

compensated tray temperatures T10 and T44, or of all three tempera¬tures. The temperature T24 is close to the feed tray and its response to

feed flow disturbances is faster than that of the other two temperatures.

Therefore, an improved controller design should result from this addi¬

tional temperature measurement.

Several synthesis attempts have shown that it is possible to increase the

performance specification by circa 30% up to

w„ 1001

1+ 20580s(6.28)

After the convergence of the uK-Iteration, the final controllers were

reduced to an order 20 using a balanced truncated realization of the

control system [6.1]. The u-plot for the reduced-order controller (Figure

6.9) using the uncertainty, input, and performance weights (6.14)-(6.23)

demonstrate the excellent robustness properties of this controller.

10 10

Frequency (rad/min)

10

Figure 6.9: Robust performance

and stability for the \i-optimal

state-space controller (controller

inputs: T10, T44, T24)

An analysis of the nominal closed-loop system (with plant model G^) is

shown in Figure 6.10 and Figure 6.11. The singular values of the

transfer function from the reference signals r to the controlled outputs

(Fig. 6.10 a) y indicate a good set-point tracking for the frequency range

of interest. The singular values of the individual transfer functions from


10J 10J 10' 10

Frequency (rad/min)

a)

Frequency (rad/min)

b)

Figure 6.10: Singular values for the nominal closed-loop system with the u-optimal

state-space controller (controller inputs: Tig, T44, T24)

a) Transfer function from reference signals to controlled output signals

b) Transfer functions from disturbance signals to controlled output signals

Dash-dotted line: TF ,solid line: Tx

10 f 1 1 I HUM T IIJJIL c—r-T-rrr 1 J 1 nil 1—I Mini.

10 10' 10'

Frequency (rad/min)

Figure 6.11: Singular values of the sensitivity function at u for the nominal closed-

loop system with the \i-optimal state-space controller (controller inputs: Tig,

T44, T24)


the two disturbance inputs to the controlled outputs (Fig. 6.10 b) show

a maximum of the sensitivity to these disturbances in the mid-

frequency range. While in the high-frequency range the sensitivity is

smaller due to the low-pass characteristics of the plant, the large

controller gains cause an effective compensation in the low-frequency

range. The plot of the sensitivity at u (Fig. 6.11)

Su(s) = [I + K(s)G(s)]-1 (6.29)

confirms the good robustness properties in the common unstructured

uncertainty representation. The maximum value (=1.6) guarantees a

stability phase margin of at least 35° [6.4].

The simulation of step responses using the rigorous dynamic model

described in Chapter 3 demonstrates the closed-loop behavior in the

time-domain. Two disturbances are simulated: An increase of the feed

composition from 0.8 to 0.9 mol/mol, and an increase in the feed flow by

3.6 mol/min. Figure 6.12 shows the top and product impurities as well

as the control errors for these disturbances and for maximum and

minimum feed flow rates. To estimate the sensitivity to errors in the

manipulated variables, a 10% error of the controller outputs AL and AV

for the same test bench has been simulated. The results are represented

by the thin lines in Figure 6.12. The steady-state offsets of the product

compositions are caused by controlling pressure-compensated tempera¬

tures on trays 10 and 44 instead of the product compositions.

The simulation results confirm the good robustness properties, espe¬

cially the low sensitivity to errors in the manipulated variables. At both

operating points, the overshoot of the control error is small.

For the compensation of the first disturbance — an increase in feed

composition — reflux and boilup must be reduced. The second distur¬

bance is an increase in feed flow rate, which has to be compensated by

an increase in reflux and boilup (see Figure 6.13). The large difference

between reflux and boilup even at steady-state has various reasons.

First, the reflux and feed are subcooled and a partial condensation of the

vapor flow thus increases the liquid flow rates below the corresponding


0.020Ft=0=20 mol/min

0.005

Top composition

Bottom composition

o 10 20 30

Time (h)

40

©

Igao

O

a

So

O

0.020

0.015

0.010

Ft=0=46 mol/min

0.005

Top composition

— — Bottom composition

o 10 20 30

Time (h)

40

-8

a.

B

Ft=0=20 mol/min

0.4 }'

0.2

0.0

1:

!!1^

/KW'""}i

0.2 !!

0.4ii*

Control error T-10

06

— Control error T-44

Ft=0=46 mol/min

10 20 30 40

Time (h)

s

I

0 10 20 30 40

Time (h)

Figure 6.12: Simulation results with \i-optimal state space controller (controller

inputs: Tjn, T44, T24) for an increase in feed composition (0.8 -> 0.9 mol/mol)

at t=0 h and an increase of feed flow rate (+ 3.6 mol/min) at t=20 h

Upper plots: Product composition

Lower plots: Control error

—^—— L, V equal to controller output

AL with +10% error, AV with -10% error


70

60

•^ 50

s

u 40

o

Ft=0=20 mol/min Ft=0=46 mol/min

30

20

1

\ /' :

V.

/'

— — Boilup

\

Reflux

V-J ]V

10 20 30

Time (h)40

140

10 20 30

Time (h)

Figure 6.13: Simulation results with \i-optimal state-space controller (controller in¬

puts: Tiq, T44, T24) for an increase in feed composition (0.8 —» 0.9 mol/mol) at

t=0 h and an increase of feed flow rate (+ 3.6 mol/min) at t=20 h

^—^—-— L, V equal to controller output


trays. Second, the major part of the feed leaves the column as top

product.

If we compare the plots for the minimum and maximum feed flow rate,

we recognize an essentially slower rejection of the feed composition

disturbance at the maximum feed flow rate. A distinct improvement of

the performance at maximum feed flow rate is not possible using a

linear time-invariant feedback controller. Higher controller gains would

improve the disturbance compensation at this operating point, but

simultaneously destabilize the control loop at the minimum feed flow

rate. A closer look at these figures demonstrates that especially at high

feed flow rates the controller response is more sluggish for changes in

feed composition but not for disturbances in the feed flow rate. This fact


is explained by the course of the manipulated variables in Figure 6.13.

An increase in feed composition at minimum feed flow rate forces the

controller to reduce the reflux and the boilup by =11 mol/min, while at

maximum feed flow both flow rates have to be reduced by =30 mol/min!

Since in practice a step change of feed composition is improbable, the

rejection of feed flow variations has much higher significance.

[iK-Iteration results for two temperature measurements

When only the pressure-compensated temperature on trays 10 and 44

are used as the controller inputs, it becomes extraordinarily difficult to

achieve convergence of the uK-Iteration and an oscillation-free closed-

loop dynamics. For design purpose, the same weights as in the previous

design for all three temperature measurements have been used.

The final controller was reduced to order 20 by a balanced truncated

realization of the control system [6.1]. The u-plots of the reduced order

controller (Figure 6.14) shows worse robustness properties of this

controller in the higher frequency range (compared to the controller

with three measured temperatures as input), only just matching the

robustness and stability criteria.

10 10

Frequency (rad/min)

10

Figure 6.14: Robust performanceand stability for the \i-optimal

state-space controller (controller

inputs: T10, T44)

An analysis of the sensitivity functions (Figure 6.15) exhibits a small

maximum sensitivity at the control error e, but an evidently reduced


Sensitivity at e

ICO

I

10 10" 10'

Frequency (rad/min)

Sensitivity at u

10 10' 10 10 10 10'

Frequency (rad/min)

Figure 6.15: Singular values of the sensitivity functions at e {upper plot) and at u

{lowerplot) for the nominal closed-loop system with the \i-optimal state-space

controller (controller inputs: Tig, T^


stability margin at u. This illustrates the direct relationship between

the SSV and the common unstructured robustness measures.

Nevertheless the simulation results in Figure 6.16 demonstrate a high

controller performance, paired with a larger sensitivity to input errors.

If we compare the controllers with 3 temperatures and 2 temperatures

as input, we must state that the "control qualities" in the time-domain

are very similar. As mentioned before, the intuition ofa control engineer

is to expect a better performance for more measurements due to a faster

state-estimation. This is obviously not the case! It will be possible to

give an explanation for this result in the further course of this chapter.

0.020

Ft=0=20 mol/min

o

Io

B

ao

03

o

a

So

O

0.015

0.010

0.005

- Top composition

— Bottom composition

o 10 20 30

Time (h)

40

0.020

o

elo

ID

oa.

So

Ft=0=46 mol/min

0.015

0.010

0.005

Top composition

Bottom composition

o 10 20 30

Time (h)

40

Figure 6.16: Simulation results with \i-optimal state-space controller (controller

inputs: Ti0, T^ for an increase in feed composition (0.8 -> 0.9 mol/mol) at

t=0 h and an increase of feed flow rate (+ 3.6 mol/min) at t=20 h

-"——— L, V equal to controller output



6.5 Design of controllers with fixed structure

In the process industry PID or advanced PID control structures are very

common. Therefore, the implementation of a controller design and its

acceptance are substantially improved if the design is based on PID

control structures. The corresponding design objective is the u-optimal

tuning of simple control elements (such as PID controllers, first-order

lags) within a fixed control structure:

K = arg inf |MM)||(6 30)

K stabilizing II A II~ <,o.ou;

K with fixed structure

The solution of this design objective is extremely difficult. Because no

synthesis methods exist, (6.30) must be solved by a parameter optimiza¬tion approach. During this optimization the SSV has to be calculated

repeatedly for a number of frequency points. However, the maximum of

the SSV may be very sensitive to the number of frequency points calcu¬

lated. In order to simplify the numerical treatment, the design objectivecan be approximated by a summation of the cube of the SSV for all k

frequency points:

k

8 = arg inf £ u| {Jr[P, K(0) ] } (6.31)e

i = l

Summing the cube, large values of the SSV have much more weight and

the design objective becomes closer to (6.30).

The calculation of the SSV presumes nominally stable control loops.

Within u-synthesis, the controllers are calculated by solving an H^

problem, which always guarantees nominal stability. However, during a

parameter optimization, nominally unstable control loops may be gener¬

ated. Therefore, the design objective (6.31) must be supplemented with

the boundary condition for nominal stability:

Re Kn *,«*!. K} <0 (6.32)


A second boundary condition is the robust stability criterion, which

should be fulfilled for the final parametrized controller:

uA{^[P,K(8)]} <1 (6.33)

This constrained parameter optimization problem is solved by sequen¬

tial quadratic programming [6.10].

In contrast to the u-synthesis methods, this approach has shown a high

reliability, at the price of even higher computation times. However, the

excellent results justify the effort.

6.5.1 Diagonal PI(D) control structures

The diagonal PI(D) control structure (Figure 6.17) is the simplest and

most frequently used composition control structure for distillation

columns. Due to the high interaction between the two control loops, this

structure is difficult to tune, and the response to setpoint changes or

disturbances is known to be very sluggish.

TioL„

Distillation

Column

with

inventorycontrol

r10 SPID!:^j

V„

r44 f>! PID2T44

+ sJi_

Figure 6.17: Diagonal PID control structure

The design model

The design model for this optimization is the same uncertainty model as

that used for the u-synthesis, excluding the temperature measurement


on tray 24. The weighting functions are the same transfer functions as

discussed in section 6.3.

Results for PI control

The matrix transfer function of the diagonal PI control structure is

given by

L(s)

V(s).

Table 6.1 summarizes the results of the parameter optimization for the

analytically and numerically linearized column models, as well as for a

complex and mixed real/complex u-analysis.

A comparison of the different optimization results shows quite similar

parameters for a complex u-analysis and a mixed real/complex u-anal¬

ysis. However, a significant difference exists between the numericallyand the analytically linearized models: Using the analytically linearized

models, the time constants TIj are much smaller and the corresponding

low-frequency gains are much higher. The reason for that are the

smaller low-frequency gains of these linear models. An underestimation

of low-frequency uncertainty results therefrom. Simulations with these

controller designs show a faster, but insufficiently damped closed-loopbehavior. Of course, with an increase in the output uncertainty of the

Table 6.1: Results for the diagonal PI control structure

Model linear¬

ization

U-analysis KRt(mol/min/°C)

TIi

(min)

KR2

(mol/minTC)

TI2(min)

NumericalComplex -14.09 137 2.49 34

Mixed R/C -11.27 141 3.15 52

AnalyticalComplex -11.52 49 6.92 56

Mixed R/C -14.58 60 6.36 41

KRl+TI,s

1 TI,s

KRl+TI2s

2 TI2s

e10(s)

e44(s)(6.34)


design model, the design can be improved. This leads to results compa¬

rable to those obtained with the numerically linearized models. These

experiences corresponds to those of the u-synthesis.

Consequently, the design with the analytical linearized models is not

further discussed. Due to the extremely large computation times using

mixed real/complex u-analysis and the very similar optimized tuning,

the further discussion will focus on the optimization with complex u-

analysis and numerically linearized models.

The upper bounds for robust stability and performance (numerically

linearized models, complex u-analysis) using the diagonal PI control

structure are shown in Figure 6.18. While stability is guaranteed for the

specified uncertainties and for the entire frequency range, the perfor¬

mance specification is not met in the lower frequency range. However,

robust performance is achieved within the upper frequency range.

a

CO

2 /T\1.5 j RP

1 \/\0.5 RS-

^ / \^

Figure 6.18: Robust performanceand stability for diagonal PI

control

10 10" 10" 10'

Frequency (rad/min)

The transfer functions from the reference and disturbance inputs to the

temperature measurements for the nominal closed-loop systems (Figure

6.19) shows a high condition number for the tracking behavior within

the most important frequency range. This means a high sensitivity of

the tracking behavior to the direction of the reference inputs.


io1r->y

•8 io°3

1S io

_____^"\

^\,n-2

\10 10 10

Frequency (rad/min)

a)

10 10 10 10

Frequency (rad/min)

b)

Figure 6.19: Singular values for the nominal closed-loop with

diagonal Pi-controller

a) Transfer function from reference to output signals

b) Transfer functions from disturbance to output signalsDash-dotted line: TPj„, solid line: T

_. „

These conclusions are confirmed by the results of the nonlinear simula¬

tions (Figure 6.20). They demonstrate the sluggish disturbance rejec¬

tion of the optimally tuned diagonal PI control. However, as expected

from the robust performance plot, the maximum control error is suffi¬

ciently small. Another positive result is the small sensitivity to input

uncertainty.

Results for PID control

The use of real PID control instead ofPI control gives additional degrees

of freedom for the controller design. Since true differential behavior is

not realizable, the parameters for PID controllers with a first order lag

in series are optimized (real PID controllers). The following transfer

function for the controllers holds:

L(s)

V(8)_=

GK1 o"

0 GK2

e,0(s)

644(8)(6.35)


0.020

0.015

0.010

Ft=0=20 mol/min

0.005

Top composition


0 10 20 30 40

Time (h)

Ft==o=20 mol/min

0.4

k

0.2i \

-

0.0I

^T^=

0.2 !! -

0.4- Control»I error T-10

0,6

— Control, • error

.ItT-44

0 10 20 30 40

Time (h)

o

Io

6

ao

CO

o

o.

6o

O

Ft=o=46 mol/min0.020

h

0.015 ^ HI i v.--

V--'

0.010 ^ :

\r0,005

Bottom composition

0.4

0.2

0 10 20 30 40

Time (h)

Ft=0=46 mol/min

W

IUCD

P.

a

0.0 ''

-0.2

-0.4

-0.6

f •--

! r> :

Control error T-10

— Control error T-44

0 10 20 30 40

Time (h)

Figure 6.20: Simulation results with diagonal PI control for an increase in feed

composition (0.8 -» 0.9 mol/mol) at t=0 h and an increase of feed flow rate

(+ 3.6 mol/min) at t=20 h

Upper plots: Product compositionLower plots: Control error

——— L, V equal to controller output



with

1+TIs + TI.TD.s2

GKi^=KRi TI^l + sTL,)(6.36)

The optimal tuning results show unacceptably large controller gains in

the high-frequency range. A high amplification of the measurement

noise can be avoided by various methods:

• "Punishment" of high frequency controller output by

additional weighting functions

• Limitation of high frequency gains by additional

boundary conditions

In order to keep the uncertainty model invariant, the differential

behavior ofthe controller was limited by a minimum bound of2 min for

the time constants TL of the first-order lags. The resulting tuning

constants are given in Table 6.2.

Table 6.2: Results for the diagonal real PID control structure

Controller KR

mol/min/°C

TI

(min)

TD

(min)

TL

(min)

PID1 -15.97 101 7.41 2.00

PID 2 4.40 39.0 15.2 7.16

Results achieved with numerically linearized model and complex u-analysis

The u-plots (Figure 6.21) for the diagonal PID control structure show an

improvement of the robust performance. However, the design objective

of robust performance

u-{Jr[P,K(0)]} <1 (6.37)

is by far not reached. The simulation results in Figure 6.22 illustrate the

same sluggish behavior as was obtained with the PI controllers. The im¬

provement is a slight reduction of the settling time.


10"J 10" 10

Frequency (rad/min)

Figure 6.21: Robust performanceand stability for diagonal real

PID control

10

0.020

Ft=0=20 mol/min

0.005

- Top composition

- Bottom composition

o 10 20 30

Time (h)

40

o

Io

S

ao

CO

oft

6o

o

0.020

0.015\

Ft_o=46 mol/min

0.010

0.005

0

A

lt

! ^ '_

**".'s*-

jr

Top composition

i,

Bottom composition

10 20 30

Time (h)

40

Figure 6.22: Simulation results with diagonal PID control for an increase in feed



—-— L, V equal to controller output



An analysis of the controller's singular values shows large high-

frequency gains despite a limitation ofthe minimum filter time constant

TL (Figure 6.23). This makes a first-order filter for the reference inputs

,„2

Figure 6.23: Singular values of the

diagonal PID controller

'"lO"5 103 10' 101Frequency (rad/min)

necessary. A reduction of these high-frequency controller gains would be

possible. However, decreasing high-frequency limits annihilate the

improvements achieved over the diagonal PI control structure.

Summarizing the results for the diagonal PI(D) control structure, we

can conclude that this control structure is absolutely not suited for a

high performance.

6.5.2 PI(D) control structures with two-way decoupling

The major disadvantage of the diagonal PI(D) control structures is the

neglect of the interactions between the two control loops. These interac¬

tions can be partially cancelled by use of decoupling techniques.

A simple controller structure with decoupling is shown in Figure 6.24.

The decoupling elements can be static (static decoupling) or dynamic

(dynamic decoupling).

The tuning of the decoupling control structure for a distillation column

is difficult. Often decouplers are based on an inversion of the plant's

transfer function G(s) .The resulting closed-loop behavior is very

sensitive to input uncertainty and decoupler errors. Summarizing the

research results, Skogestad in [6.17] concludes that (two-way) decou-


r10 *"

+

PIDi

PID, o

Distillation

Column

with

inventorycontrol

•10

l44

Figure 6.24: PID control structure with static decoupling

piers should never be used for high-purity distillation columns with the

LV-configuration. On the other hand one-way decoupling seems to be

less sensitive to input uncertainty and should be preferred [6.18].

Results for static two-way decoupling

The simplest decoupling structure is static decoupling. Here the two

decoupling elements Cj and C2 are constant factors. The results for

this structure are obtained with the same weighting functions and with

the same uncertainty model as used for the diagonal PI(D) control struc¬

ture.

Table 6.3 summarizes the u-optimal parameters for PI and real PID

control with static decoupling. The high-frequency gains of the PID

controller are small enough that no boundary conditions concerning this

criterion were necessary.

The results for the decouplers are somewhat surprising. They indicate

that the optimal decoupling is very close to a one-way decoupling! Let

us examine this control structure in detail:


Table 6.3: u-optimal parameters for PI(D) control with

static decoupling

Controller or

decoupler No.

KR

(mol/min/°C)

TI

(min)

TD

(min)

TL

(min)

C

(-)

1 -5.21 22.8 - - -0.0240

2 3.71 46.8 - - 1.11

1 -13.1 51.6 7.83 8.43 -0.217

2 4.56 62.1 5.11 3.07 1.03


The decoupler parameter C2 is close to one. Therefore any variation of

the output of the top composition controller causes a simultaneous

increase or decrease ofreflux and boilup by almost the same magnitude.Thus this controller shapes the composition profile within the column by

an adaptation of the separation.

The other decoupler parameter C} is small. Consequently the output of

the bottom composition controller has a small effect on the reflux. This

controller moves the composition profile within the column.

In light of this interpretation, the limited advantage of an additional

temperature measurement in the middle of the distillation column is

easily explained. Since no setpoint is available for such a temperature

measurement, an improved feedback may be calculated neither for the

composition profile nor for the composition profile's position.

This special behavior of the control system has its significant advan¬

tages for the closed-loop behavior. The u-plots in Figure 6.25 demon¬

strate the superior robust performance of the decoupling control

structures. While the optimal tuning for PI control shows a peak of the

robust performance plot within the medium frequency range, the PID

control structure shows nearly flat and significant smaller structured

singular values. Using a decoupling control structure, the additional

degrees of freedom in the controller design allow a significantly better

controller performance, especially in the important mid-frequency


a

-a>

a

0.5

03 0 tn 0

,RP-

///\

- -RS-/ \

10 10 10

Frequency (rad/min)

10 10 10 10

Frequency (rad/min)

10'

Figure 6.25: Robust performance and stability for PI control {left) and

real PID control {right) with static decoupling

range. Because of the better performance using PID controllers, the

further discussion focuses on that control structure.

The singular value plots of the loop transfers from the reference and

disturbance signals to the output signals (Figure 6.26) illustrate the

r-»yio1

oitude 10° _^\

"^\\C8

IO"'

,n-2

\10 10 10

Frequency (rad/min)

a)

10 10 10 10

Frequency (rad/min)

b)

Figure 6.26: Singular values for the nominal closed-loop system for PID control

with static decouplinga) Transfer function from reference to output signals

b) Transfer functions from disturbance to output signalsDash-dotted line: Tp^,,, solid line: T

^„r —* y ^f ~* y

160 6 U-Optimal Controller Design

better controller performance as well. The condition numbers of Trare much smaller than those ofthe diagonal PI(D) control structure, and

the tracking behavior is significantly improved.

The simulation results (Figure 6.27) confirm the fundamentally

improved controller performance. The sluggish behavior has vanished,

and the maximum control errors are comparable to those obtained with

the diagonal PI(D) control structures. While the sensitivity to input

uncertainty has increased, it is still small.

0.020

o

Io

gao

XD

o

a

6o

O

Ft=0=20 mol/min

0.015

0.010

0.005

- Top composition


o 10 20 30

Time (h)

40

0.020

Ft_0=46 mol/min

0.005

Top composition

Bottom composition

o 10 20 30

Time (h)

40

Figure 6.27: Simulation results for PID control with static decoupling for an in¬

crease in feed composition (0.8 —> 0.9 mol/mol) at t=0 h and an increase of feed

flow rate (+ 3.6 mol/min) at t=20 h

^^^^^— L, V equal controller output



PID Control with dynamic decoupling

Using lead-lag transfer functions for the decoupler elements Cx and C2

Ci(B) = Kci±|g (6.38)

a dynamic decoupling structure is realized. The additional degrees of

freedom allow a further improvement of the control design. The

resulting optimal tuning constants are listed in Table 6.4.

Table 6.4: u-optimal parameters for PID control with

dynamic decoupling

Controller KR

(mol/min/°C)

KC

(-)

TI

(min)

TD

(min)

TL

(min)

PID1 -22.2 - 80.2 19.6 44.8

PID 2 5.68 - 59.4 12.6 24.7

CI - -0.138 - 117 7.42

C2 - 1.07 - 53.0 71.43


The simulation results exhibits a performance which is insignificantly

worse than that of the u-optimal state-space controllers (Figure 6.28).

However, the more difficult initialization of a control structure with

dynamic decoupling in a distributed control system is a disadvantage.

6.5.3 PID control structures with one-way decoupling

The results for two-way decoupling have shown optimal results for

decoupling structures which are close to one-way decoupling. In this

section the optimal tuning results for one-way decoupling are discussed.

This control structure is particularly easy to implement in a distributed

control system and simple to initialize. In order to keep the decoupler as

simple as possible, the discussion is limited to static one-way decou¬

pling.

162 6 jl-Optimal Controller Design

0.020

Ft=0=20 mol/min

0.005

- Top composition


o 10 20 30

Time (h)

40

O

CO

o

a

So

O

Ft=0=46 mol/min

0.020

fi

a !i0.015 '\ i\

{\/~. -' '-'

TV—0.010 . f :

0.005


10 20 30

Time (h)

40

Figure 6.28: Simulation results for PID control with dynamic decoupling for an in¬

crease in feed composition (0.8 -> 0.9 mol/mol) at t=0 h and an increase of feed

flow rate (+ 3.6 mol/min) at t=20 h

——^—~~ L, V equal controller output


Two different decoupler structures are possible ifwe set either Cj or C2of the control structure shown in Figure 6.24 to zero. While the results

for the two-way decoupling lead us to expect a good performance for the

firstcase(C, = 0), no inference is possible for the second case (C2 = 0).

In fact, the optimization results show insufficient performance for the

second case (C2 = 0). Therefore a reversal of the decoupling control

structure with shaping of the composition profile by the bottom compo¬

sition controller and moving the composition profiles position by the top

composition controller does not lead to results comparable to those

obtained with the other decoupling structure.

The tuning parameters for the controller with Cj= 0 can be found in

Table 6.5. The corresponding u-curves (Figure 6.29) let us expect a


Table 6.5: u-optimal parameters for PID control with

static one-way decoupling

Controller or

decoupler No.

KR

(mol/min/°C)

TI

(min)

TD

(min)

TL

(min)

C

(-)

1 -10.5 45.7 2.18 5.01 0

2 5.35 67.4 13.4 13.9 1.05


CD

73>

a.3 0.5

T3CD

w

Figure 6.29: Robust performance and

stability for real PID-control with

one-way decoupling

Frequency (rad/min)

performance somewhere between that of the PI control with static two-

way decoupling and that of the real PID control with static decoupling.

The simulation results in Figure 6.30 support this interpretation.

Therefore this controller represents a structure which is simple and

easily implemented in a distributed control system, distinguished by a

sufficiently high controller performance.


Ft=0=20 mol/min

0.020 r"1 ' '

0.015

0.010

0.005

— Top composition


0.020

Ft=0=46 mol/min

0.015

0.010

0.005

• Top composition


o 10 20 30

Time (h)

40 10 20 30

Time (h)

40

Figure 6.30: Simulation results for PID control with static one-way decoupling for

an increase in feed composition (0.8 -> 0.9 mol/mol) at t=0 h and an increase

of feed flow rate (+ 3.6 mol/min) at t=20 h

—^—^— L, V equal controller output


6.6 Summary

The comparison of the state-space controllers obtained by u-synthesis

with PID control structures obtained by u-optimization leads to

surprising results. The frequently heard opinion that state-space

controllers are much superior to PID control structures apparently is

not true for this distillation column. The PID control structures with

decoupling exhibit nearly the same performance as that achieved with

state-space controller of a higher order, provided that the PID control

structures are optimally tuned. The visual results of the u-curves and

6.6 Summary 165

simulation plots shall be supported by numerical measures. For

purposes of comparison, the integral square of the control errors

t=40h

ISE = J [e20(t)+e|4(t)]dt, (6.39)

o

and the integral of the time-multiplied absolute control errors

t=40h

ITAE = | C|eio<t>| + |e44(t)|] * dt (6-40>

0

have been calculated and summed up for both operating points and all

controllers. While ISE punishes especially large control errors, the

ITAE performance measure has a higher importance for the process

industry because it punishes any undesirably sluggish disturbance

rejection. Both criteria, relative to the result for the state-space

controller using 3 temperature measurements, can be found in Table

6.6. The last two columns in this table state the maximum absolute

value of the SSV (RP) and the value of the optimization criterion

k

f(0) = £u|{^[P,K(0)]} (6.41)

i= 1

relative to the value for the state-space controller using 3 temperature

measurements. The high correlation of the ITAE and the optimizationcriterion are obvious. The single exception is the state-space controller

using 2 temperature measurements, which may be caused by the

convergence problems mentioned before.

This table effectively illustrates the high performance achieved with

simple and easily realized PID-control structures. The u-optimization

approach has proved to be an efficient tool for the optimal design of

controllers with fixed structure.


Table 6.6: Comparison of controllers in time-domain

Control structure Relative

ISE

Relative

ITAE

Max. u- Relative

2>f (j<»)

State-space controller,

3 temp, measurements1.00 1.00 0.85 1.00


2 temp, measurements0.82 1.05 1.04 1.54

Diagonal PI control 3.13 2.89 2.14 9.03

Diagonal PID control 2.08 1.87 1.53 3.98

PI control with static two-

way decoupling2.42 1.74 1.13 1.55

PID control with static

two-way decoupling1.44 1.23 0.91 1.19

PID control with dynamic

two-way decoupling1.18 1.12 0.88 1.12

PID control with static

one-way decoup. (Ci=0)1.99 1.51 0.97 1.34

6.7 References


Analysis and Synthesis Toolbox, MUSYN Inc., Minneapolis MN,

and The MathWorks, Inc., Natick, MA (1991)

[6.2] Balas, G. J., A. K. Packard, and J. T. Harduvel: "Application of u-

Synthesis Techniques to Momentum Management and Attitude

Control of the Space Station," Proc. 1991 AIAA Guidance, Navi¬

gation and Control Conference, New Orleans, LA (1991)

[6.3] Chiang, R. Y., M. G. Safonov: Robust Control Toolbox User's

Guide, The Mathworks Inc., Natick, MA (1992)

6.7 References 167

[6.4] Dailey, R. L.: "Lecture Notes for the Workshop on H„ and u

Methods for Robust Control," IEEE Conference on Decision and

Control, Brighton (1991)

[6.5] Doyle, J. C: "Analysis of Feedback Systems with Structured

Uncertainties,"IEEProc., 129, Pt. D., No. 6, 242-250 (1982)

[6.6] Doyle, J. C: "Performance and Robustness Analysis for Struc¬

tured Uncertainty," Proc. of the 21st Conference on Decision and

Control, (1982)

[6.7] Doyle, J. C: "Structured Uncertainty in Control System Design,"

Proc. ofthe 24th Conference on Decision and Control, Ft. Lauder¬

dale, FL (1985)

[6.8] Doyle, J., K. Lenz, and A. Packard: "Design Examples Using u-

Synthesis: Space Shuttle Lateral Axis FCS During Reentry,"

NATO ASI Series F: Computer and Systems Science, 34,128-154

(1987)

[6.9] Enns, D. F.: "Rocket Stabilization as a Structured Singular

Value Synthesis Design Example," Control Systems, 11, 4, 67-73

(1991)

[6.10] Grace, A.: Optimization Toolbox — User's Guide, The Math-

Works, Inc., Natick, MA (1990)

[6.11] Lin, J.-L., I. Postlethwaite, and D.-W. Gu: "u-K Iteration: A New

Algorithm for u-synthesis," Automatica, 29, 219-224 (1993)

[6.12] Maciejowski, J. M.: Multivariable Feedback Design, Addison-

Wesley Publishing Company, Wokingham, England (1989)

[6.13] McFarlane, D. C, and K. Glover: "Robust Controller Design

Using Normalized Coprime Factor Plant Descriptions," Lecture


Notes in Control and Informations Science, 138, Springer-Verlag,

Berlin (1990)

[6.14] Packard, A., J. Doyle, and G. Balas: "Linear Multivariable

Robust Control With a u Perspective," Trans. oftheASME, 115,

426-438 (1993)

[6.15] Packard, A., and J. Doyle: "The Complex Structured Singular

Value," Automatica, 29 1, 71-109 (1993)

[6.16] Shinskey, F. G., ''Distillation control for Productivity and Energy

Conservation," 2nd ed., McGraw-Hill, New York, 194-203 (1984)

[6.17] Skogestad, S.: "Dynamics and Control of Distillation Columns -

A Critical Survey," Preprints of the 3rd IFAC Symposium on

Dynamics and Control of Chemical Reactors, Distillation Col¬

umn and Batch Processes, April 26-29, 1992, College Park, MD,

1-25 (1992)

[6.18] Skogestad, S., and M. Morari: "Implications of Large RGA Ele¬

ments on Control Performance," Ind. Eng. Chem. Res., 26, 2323-

2330 (1987)

[6.19] Skogestad, S., and P. Lundstrom: "MU-Optimal LV-Control of

Distillation Column," Comp. Chem. Eng., 14, 4/5, 401-413 (1990)


Chapter 7

Controller Design for

Unstructured Uncertainty —

A Comparison

7.1 Introduction

A controller design for the entire operating range of the distillation

column (see Chapter 6) requires a structured uncertainty model incor¬

porating two linear models, and a huge computational effort. Naturally,the question arises what controller performance and robustness proper¬

ties can be achieved ifwe use simpler design methods, based onjust one

plant model for the nominal operating point (Model Gn) and classical

design methods or simple unstructured uncertainty bounds.

A few of these simpler methods are discussed in this chapter. They are

applied in a straightforward manner, and the design results are not

guaranteed to represent the optimum achievable controller perfor¬

mance. However, the results give an impression of the limits and

inherent problems of the application of design methods based on

simpler uncertainty concepts, and they allow a comparison with the u-

optimal results presented in the previous chapter. The weighting func-

170 7 Controller Design for Unstructured Uncertainty — A Comparison

tions of the structured uncertainty model used for the u-analysis are the

same as those used in the previous chapter.

7.2 Diagonal Pi-control

A diagonal Pi-control scheme seems to be most frequently used in

conventionally controlled distillation columns. Usually these PI control¬

lers are tuned on-line. Due to the large time constants of the composi¬

tion dynamics, we cannot expect this on-line tuning approach to lead to

a controller performance close to the optimum. The attempt to use

tuning rules such as Ziegler-Nichols for the individual SISO loops often

results in an unstable MIMO closed-loop system, because these tuning

rules do not take the interaction between the two control loops into

account.

While the following two simple and model based tuning methods make

use of the classical design methods, they try to pay attention to the loop

interactions. Both methods lead to a nominally stable controller design.

However, sufficient stability margins for the closed-loop system at all

possible operating points cannot be guaranteed.

7.2.1 The BLT method

The Biggest Log Modulus Tuning was proposed by Luyben in 1986

([7.5], [7.6]). This method is a multivariable extension of the classical

Nyquist stability criterion. The closed-loop system (Tr_^y) with a square

nominal model G (s) = Gu (s) and a diagonal PI control law K(s) is

given by

y(s) = [I + G(s)K(s)]-1G(s)K(s)r(s) (7.1)

The characteristic equation of the multivariable system is the scalar

equation

det(I + G(s)K(s)) = 0 (7.2)

7.2 Diagonal Pl-control 171

Ifwe plot (7.2) as a function offrequency, the number of right half-plane

zeros of the closed-loop characteristic equation are determined. In order

to make this multivariable plot like the SISO scalar Nyquist plot,

Luyben introduces a new function W(s):

W(s) = -l+det(I + G(s)K(s)) (7.3)

The closer this function approaches the (-1,0) point in the Nyquist plot,the closer the MIMO system is to closed-loop instability. The design

objective is defined as

L. = 20 log W(J(»)1+W(ja»

< 2p Voe R+ (7.4)

where p is the number of inputs/outputs of G(s). The proposed tuning

procedure starts with independent Ziegler-Nichols settings for PI-

controllers of the individual control loops. In a second step these

settings are detuned by a factor F

K,

Ki =

ZN;

Tli = F TIm (7.5)

in order to achieve the design objective (7.4).

Results ofthe BUT tuning

The tuning results for the nominal model GN (s) of the distillation

process are listed in Table 7.1. A detuning factor F of3.82 was necessary

to achieve the design objective (7.4). The proportional gain KR ofthe top

composition controller is too large for satisfactory setpoint tracking and

Table 7.1: Tuning constants with BLT-method

Controller KR

(mol/min/°C)

TI

(min)

PI1 -47.1 95.2

PI 2 6.74 171.8


measurement noise attenuation. A plot of the structured singular

values (with the same uncertainty and performance weights as used in

the previous chapter) illustrates the insufficient robust stability and

robust performance of this composition control design (Fig. 7.1).

However, any further detuning would reduce the low and high-

frequency gains of the bottom composition controller to an absolutely

insufficient level.

Figure 7.1: u-plots for a diagonal

Pi-control law tuned with BLT-

method

10 10

Frequency (rad/min)10

7.2.2 Sequential loop closing

The idea of the sequential loop closing was introduced by Mayne ([7.8],

[7.9]). First, a SISO controller is designed for one pair of input and

output variables. When this design has been completed, the corre¬

sponding control loop is closed and the next pair of input and output

variables is chosen. Thus the interaction between the control loops is

taken into account. This design procedure is illustrated in Figure 7.2.

It is an advantage of this method that each single loop can be designed

using classical methods. However, this method has some severe draw¬

backs: First, the selection of the first one or two input/output pairs may

have a deleterious effect on the behavior of the remaining loops [7.7].

There exists little help for this sequence problem. Second, this method

cannot guarantee robustness for the entire operating range. Especially

if the plant G(s) is not diagonal dominant, that means the condition

7.2 Diagonal Pi-control173

G3(s) G2(s) G^s)

Figure 7.2: Sequential loop closing

iGâco)] >|Gij(jco)| VcoeR+ (7.6)

is not satisfied, we have to expect robustness problems.

Design Results

The sequential loop closing idea has been applied to the composition

control problem represented by the nominal model GN(s) .For each

SISO loop, a phase margin of at least 60 degrees and for both controllers

a maximum high frequency gain of 18 mol/min/°C has been required.

The results ofboth possible design sequences and with a minimal inte¬

gral absolute error (LAE) for the rejection of feed composition and feed

flow disturbances (with respect to the linear model) are summarized in

Table 7.2.

An analysis of the robustness to unstructured peturbations shows

maximum values for the sensitivities of Se=2.6, and Su=2.1 for the

Top —»Bottom design sequence, and of Se=2.4 and Su=1.9 for the


Table 7.2: Results of the sequential loop closing

Design sequence KR1

(mol/min/°C)

Til

(min)

KR2

(mol/min/°C)

TI2

(min)

Top -> Bottom -18.0 101.9 10.09 55.8

Bottom -» Top -18.0 52.6 8.78 214.5

sequence Bottom -> Top. These stability margins are insufficient. The

results of the analysis using the structured uncertainty model are illus¬

trated by the u-plots in Figure 7.3. Both controller designs can neither

guarantee robust performance nor robust stability.

CO10 10 10 10

Frequency (rad/min)

a)

a

a

to 10"' 10 10"' 10'

Frequency (rad/min)

b)

Figure 7.3: |l-plots for the sequential loop closing designs

a) Top -> Bottom design sequence

b) Bottom —> Top design sequence

7.2.3 Optimized robust diagonal Pi-control

The objective of this controller design is a maximization of the distur¬

bance rejection capabilities with the boundary conditions of sufficient

stability margins. As a measure of the disturbance rejection capabilities

the IAE as defined by

LEnd

IAE = J []e10(t)| + |e44(t)|]dt (7.7)

7.2 Diagonal Pi-control 175

is a suitable measure. It is calculated for step responses to feed compo¬

sition and feed flow rate of the closed-loop system. If we tune both PI-

controllers in order to minimize the IAE-criterion, the robustness prop¬

erties of the closed-loop system form boundary conditions for the

minimum achievable IAE. Stabihty bounds in terms ofthe sensitivity at

the plant input and output are well established. If we require a phase

margin of at least 35 degrees (which is relatively small), the following

sensitivity bounds hold

Se(jco) = [I + G(jco)K(jco)]-1 <1.7 VcogR+ (7.8)

Su(j<o) = [I + KCJoojGGffl)]-1 <1.7 VcoeR+ (7.9)

The optimal parameters which minimize the IAE criterion are found

either by trial and error or by a constrained parameter optimization.

Results

The results for this design approach are given in Table 7.3. The corre¬

sponding u-plots (Fig. 7.4) illustrate the improved robust stabihty prop¬erties compared to the previous two methods. While design guarantees

robust stability, the robust performance is substantially worse than the

u-optimal design of a diagonal Pi-controller design (see Figure 6.18,

page 151). An analysis of the controller behavior in the time domain

(Figure 7.5) shows extremely sluggish disturbance rejections.

Table 7.3: Tuning constants with optimizing method

Controller KR

(mol/min/°C)

TI

(min)

PI1 -5.10 600.0

PI 2 4.92 86.2


Figure 7.4: u-plots for diagonalPi-controller designed by op¬

timizing method

10" 10

Frequency (rad/min)

10

0.020

o

gao

TO

o

o

O

0.015

0.010

0.005

0

Ft=0=20 mol/min

h

V

i \! 5

i \\.


^^^

10 20 30

Time (h)

40

0.020

Ft=0=46 mol/min

o

Io

gao

o

a

so

O

0.015

0.010

0.005

"1 ,MUMIIIM"MMI "

M1 ;

\

11

I1,

—fr

- Top composition

-


l",

yj

0 10 20 30

Time (h)

40

Figure 7.5: Simulation results with diagonal PI controller (designed by optimiz¬

ing method) for an increase in feed composition (0.8 —» 0.9 mol/mol) at t=0 h

and an increase of feed flow rate (+ 3.6 mol/min) at t=20 h



-^^^^^ L, V equal to controller output



7.3 Pi-control with decoupling

The basic idea of decoupling is a reduction of the loop interactions. If we

increase the diagonal dominance of the system, the design task takes on

more the characteristics of a multiloop SISO design problem. However,

as emphasized already in the previous chapter, a reduction of the loopinteractions does not automatically imply better control. Due to an

increased sensitivity to model and input errors, the maximum perfor¬

mance of a controller exhibiting sufficient stability margins may be

strongly reduced even compared to that of a diagonal PI controller.

In the simplest case, as discussed in this section, the plant behavior is

altered by a pre- or postmultiplied constant "compensating" or "decou¬

pling" matrix. Different approaches for the selection of these interaction

reducing matrices are proposed:

Davison [7.3] recommends a steady-state decoupling of the process. For

the "decoupled" process G* (s) holds

G*u_y(s) =Gu^y(s)G-'u^y(0)or (7.10)

G*u_y(s) =G-'u_y(0)Gu^y(s)

With a state space representation of the process, the decoupling matrix

is calculated according to

G"1u_>y(0) = (CA^B)-1 (7.11)

The choice of a premultiplication or postmultiplication of this interac¬

tion reducing matrix is another degree of freedom for the controller

design.

Mayne [7.9] proposes a reduction of the high-frequency interactions of

the plant. The corresponding decoupling matrix is calculated by

G-»u_y(j~) = (CB)-1 (7.12)


As before, the choice of a pre- or postcompensation has to be decided

during the controller design.

Ryskamp [7.11] suggests a decoupling scheme which is based on the

idea of a composition profile control: The difference in the temperature

deviations should be used to set the reflux ratio, and the sum of the two

temperature deviations should be used to set the reboiler heat duty.

This scheme is called "implicit decoupling."

Another interesting approach, based on a singular value decomposition(SVD) of the process at steady state, is presented by Brambilla et al.

[7.1]. Let the SVD of the steady-state transfer matrix of the process

Gu_y(0) be

Gu^y(0) = UIVT, (7.13)

where U and V are unitary matrices and X is a diagonal matrix

containing the singular values £ = diag(Oj,a2). A plant-inverting

compensator D (at plant input) according to this SVD is the matrix

D = VZ~'UT (7.14)

In order to avoid a high sensitivity to input errors due to the perfect

decoupling at steady state, Brambilla et al. [7.1] introduce a matrix F

F - al+ (l-a)E (7.15)

and define a new compensation matrix D as

D = VFIÛT (7.16)

The single parameter a with a = 0...1 allows a continuous shift

between a plant-inverting compensator (a = 1) and a compensator

which does not remove the effect of the directionality of the process

(a = 0). The tuning parameter a has to be chosen on the basis of (1) the

magnitude of the assumed errors in the model, (2) the sensitivity of the

process to the model errors (Relative Gain Analysis of D), and (3) the


required performance in terms of reduction of interactions and direc¬

tionality (Relative Gain Analysis of G"1^ (0) D).

Design results

The four proposed compensation matrices are summarized in Table 7.4.

In order to calculate "optimal" controllers, the optimization approachdescribed in section 7.2.3 has been applied to the different compensated

plants. However, it was not possible to achieve any acceptable control¬

lers using the proposed compensation matrices, except for the SVD-

based compensator. This SVD-based compensator is distinguished byalmost the same one-way decoupling structure as we obtained as a

result of the u-optimal decoupling (see Chapter 6).

Table 7.4: Compensator matrices

Type of compensator Position of com¬

pensator

Compensatormatrix

Decoupling at

co = 0

Plant input or

plant output

-0.636 0.168

-0.728 0.195

Decoupling at

0) = oo

Plant input or

plant output

0.380 -0.295

0.875 -0.193_

Implicit decoupling Plant output-1 -1

-1 1

SVD-based compen¬

sation (a = 0.8)Plant input

0.901 0.082

0.955 0.391

The parameters of the IAE-optimal PI controllers (with respect to an

additional boundary condition for the proportional gains IKRJ < 18 mol/

min/°C) are given in Table 7.5. The n-plots for this controller design

(Figure 7.6) demonstrate good robust performance and robust stability.However, the simulation results (Figure 7.7) show an insufficiently

damped oscillation at higher frequencies for the minimum feed flow

rate. The damping ofthese oscillations is significantly better for a +10%

error in the change of the reflux L and a -10% error in the change of


Table 7.5: Optimal PI tuning constants for plant with

SVD-based compensation

Controller KE

(mol/min/°C)

TI

(min)

PI1 -18.0 47.4

PI 2 18.0 116.0

3

1S-c

03

1

1

0.5

sCO

RP-

//

-y

<7\

- - RS-** Figure 7.6: u-plots for SVD-based

compensation with optimallytuned diagonal PI control.

10 10 10

Frequency (rad/min)

10'

boilup V. These unwanted oscillations are allowed by the performance

specification in the frequency domain! They require a detuning of the

controllers' proportional gains which on the other hand, reduces the

controller performance.


0.020

| 0.015

o

g

"ao

en

op.

so

O

Ft=0=20 mol/min

0.010

0.005

\

'"1

.J v..\ /

«"

- Top composition

— - Bottom composition

0 10_

20 30

Time (h)40

0.020

o

Io

a

aa

x

o

eo

O

Ft=0=46 mol/min

0.015

0.010

0.005

l /x

i J -

X*

•/

i

:y\^

'


0 10_

20 30

Time (h)

40

Figure 7.7: Simulation results with SVD based compensator and diagonal PI con¬

trol for an increase in feed composition (0.8 —» 0.9 mol/mol) at t=0 h and an

increase of feed flow rate (+ 3.6 mol/min) at t=20 h


———— L, V equal to controller outputAL with +10% error, AV with -10% error


7.4 H^ optimal design

The H^-norm minimizing design ([7.2], [7.4], [7.7], [7.10]) of multivari¬

able controllers have proved to be a powerful method for robust, model-

based controllers.

HM Design specification

The closed-loop system with the plant G(s) and the controller K(s),

augmented with the weighting functions Wd (s) , We (s) , Wu (s) ,and

W (s) is outlined in Figure 7.8. This scheme is often called S/KS/T-

weighting scheme. The matrix Wd (s) is a diagonal matrix of transfer

functions and represents the frequency content of the feed composition,

feed flow rate, and reference input signals. The selection of these input

weights is discussed in section 6.3. The same weighting functions are

applied here.

d(s)-

rOO-

zfi(s)

zu(s)

• Zy(s)

Figure 7.8: Augmented closed-loop system with weighting functions

for the H^ design

H^ optimal design 183

All other weighting functions are chosen as diagonal frequency-depen¬

dent weights because the performance and robustness properties are

equal for all channels:

We (s) = diag [we (s), we (s) ] (7.17)

Wu (s) = diag [wu (s), wu (s) ] (7.18)

Wy(s) = diag[wy(s),wy(s)] (7.19)

The performance of the closed-loop system is specified in terms of the

sensitivity function by the weighting function We (s). A first-order lagwith a static gain of 100 has been specified to achieve a nearly inte¬

grating behavior.

The bandwidth of the closed-loop system is limited by the weighting

function W (s) ,which punishes the transfer function T[dT)rT]T_>y

from the disturbance and reference signals to the plant outputs. A first-

order lead-lag transfer function is suitable for this task.

A weighting of the plant inputs allows a frequency-dependent limitation

ofthe control energy and helps to achieve sufficient stability margins for

the sensitivity function at u. As done with W (s), a first-order lead-lagtransfer function has been selected.

The poles and zeros of the weighting functions were adjusted until the

sensitivity functions at e and at u ofthe closed-loop system had attained

approximately the same peak values as the u-optimal controller design

(with 2 temperature measurements), a high performance, and

«1 (7.20)

were achieved. The best weighting functions are given by

(jco)

We(s> = 100i+^20i (7.21)


,

, „cl+ 520s

Wu(s)=°-5T7T3T(7.22)

,, „,1 +1500s

wy(s)=0JT+T5T(7.23)

Design results

Despite the fact that the singular values of sensitivity functions for the

H^- design (Figure 7.10) and for the u-synthesis (Figure 6.15) are nearly

identical, the u-analysis shows significant differences. The p>plots ofthe

H^ design (Figure 7.9) show much higher peak values in the low and

mid-frequency ranges. The simulation results (Figure 7.11) allow a

conclusion with respect to the larger structured singular values: The

sensitivity of the closed-loop performance to errors in the manipulated

variables is large. A reduction of this sensitivity to plant input errors

was not possible using the common S/KS/T weighting scheme.

Figure 7.9: u-plots for H^ op¬

timal controller

10" 10" 10

Frequency (rad/min)

10

H„ optimal design185

Sensitivity at e

3

cs

2

10" 10 10 10

Frequency (rad/min)

10"

10'Sensitivity at u

r. 1—iiii mi 1—j—i' i ' j 'il 1—i—i i 11 nj j—i i i 11 in 1—i—r-rrrm 1—r-i i 11 ra

m° /_>=-

-

itude//~ :

a io

2 __X / :

102 / "

10 10 10

Frequency (rad/min)

Figure 7.10: Singular values ofthe sensitivity functions at e {upper plot) and at u

{lower plot) for the nominal closed-loop system with the H^ controller


0.020

Ft=0 =20 mol/min

(mol/mol) 0.015v..

If

h... J \ _

osition Af-Comp 0.010

\\ 7

- Top composition


0.005 1,,

10 20 30

Time (h)

40

0.020 r

Ft=0=46 mol/min

0.005

Top composition

Bottom composition

10 20 30

Time (h)

40

Figure 7.11: Simulation results with the HM-controller for an increase in feed





^^^^^— L, V equal to controller output


7.5 Summary 187

7.5 Summary

The application of design methods for unstructured uncertainty to the

composition (or temperature) control problem shows that it is extraordi¬

narily difficult to obtain performances which are comparable to those of

the u-optimal controllers. Despite the high effort for a robust tuning of

the Pi-control structures, it was not possible to achieve any satisfactory

result.

Better results were obtained using the H^-minimization approach. The

resulting state-space controller guarantees stability for the entire oper¬

ating range and the singular values of the sensitivity functions (Se, Su)

are nearly identical to those of the u-optimal state-space controller.

Nevertheless, the high sensitivity to input uncertainty demonstrates

the limits of simple unstructured uncertainty bounds. Even a robust

controller design based on an unstructured uncertainty model tends to

be very sensitive to input uncertainty at operating points different from

the design point. The advantages of a |x-optimal controller design as

presented in Chapter 6 are obvious.

7.6 References

[7.1] Brambilla, A., and L. D'Elia: "Multivariable Controller for Distil¬

lation Column in the Presence of Strong Directionality and Mod¬

el Errors," Ind. Eng. Chem. Res., 31, 536-543 (1992)

[7.2] Dailey, R. L.: "Lecture Notes for the Workshop on H„ and ]i Meth¬

ods for Robust Control," 1991 IEEE Conference on Decision and

Control, Brighton, December 9-10 (1991)

[7.3] Davison, E. J.: "Multivariable tuning regulators: The feedfor¬

ward and robust control of general servomechanism problems,"

IEEE Trans. Aut. Control, AC-21, 35-47 (1976)

[7.4] Glover, K., and J. C. Doyle: "A State Space Approach to HM Opti¬

mal Control," Lecture Notes in Control and Information Sciences,

135, 179-218, Springer-Verlag, Berlin (1989)


[7.5] Luyben, W. L.: "Simple Method for Tuning SISO Controllers in

Multivariable Systems," Ind. Eng. Chem. Process Des. Dev., 25,

654-660 (1986)

[7.6] Luyben, W. L.: Process Modeling, Simulation, and Control for

Chemical Engineers, 2nd ed., McGraw-Hill, New York (1990)

[7.7] Maciejowski, J. M.: Multivariable Feedback Design, Addison-

Wesley Publishing Company, Wokingham (1989)

[7.8] Mayne, D. Q.: "The design of linear multivariable systems," Au-

tomatica, 9, 201-207 (1973)

[7.9] Mayne, D. Q.: "Sequential design of linear multivariable sys¬

tems," Proc. IEE., 126, 6, 568-572 (1979)

[7.10] Raisch, J., L. Lang, und E.-D. Gilles: "H^-Reglerentwurf fur

Zwei- und Dreistoffdestillationsprozesse", at, 41, 6, 215-224

(1993)

[7.11] Ryskamp, C. J.: "Explicit vs. implicit decoupling in distillation

control," Chemical Process Control II, American Institute of

Chemical Engineers, New York, 361-375 (1982)


Chapter 8

Feedforward Controller Design

8.1 Introduction

It is a drawback offeedback control that a corrective action necessitates

a deviation of the controlled variables from their setpoints. This disad¬

vantage can be overcome by the use offeedforward control. A major and

probably the most frequent disturbance of a distillation column is a

change in the feed flow rate. Because the feed flow rate is always

measured, it can be used as a controller input. An appropriately

designed feedforward controller takes most of the necessary corrective

action before the product compositions and the controlled tray temper¬

atures change. However, because ofmodel errors and other unmeasured

disturbances a feedforward controller alone will never be able to yield

perfect control so that feedback control will still be needed.

Within this chapter, the design of linear time-invariant feedforward

controllers for our distillation column is discussed. The proposed design

methods take into account the wide operating range of the distillation

column and the unmeasured feed composition.

190 8 Feedforward Controller Design

8.2 The design problem

8.2.1 The design objective

The objective of feedforward control is a reduction of the control error in

presence of feed flow rate disturbances. The main problem is the

nonlinear behavior of distillation columns. The perfect control action for

a rejection of a feed flow disturbance depends on the actual and

measured feed flow rate and the unmeasured feed composition. A

controller design for one operating point may be unsatisfactory at any

others. Consequently, it is impossible to design a perfect linear time-

invariant feedforward controller for the entire operating range of a

distillation column. Hence the design objective is a feedforward

controller which improves the compensation of feed flow disturbances

for the largest possible part of the operating range, but never makes it

worse.

A perfect solution of this design objective would be an enormous task.

Fortunately, the ideas discussed in Chapter 5 lead to very good results:

If we design the feedforward controller simultaneously for the models

GR(s) (representing minimum feed flow rate and maximum feed

composition) and GT(s) (representing maximum feed flow rate and

minimum feed composition), we obtain a design which improves the

compensation of feed flow disturbances for the entire operating range.

8.2.2 One-step or two-step design?

The design of feedforward controllers is feasible either in a one-step

design, simultaneously with the feedback controller, or as a second step

for the closed-loop system (Fig. 8.1) [8.3]. The design of a feedforward

controller for the open-loop system is not recommended because the

feedback controller shifts the poles and, consequently, affects the

dynamics of the system.

A u-optimal one-step design using the uncertainty structure presented

in chapter 5 is tempting. For that purpose, the uncertainty structure is

slightly modified by the additional input to the controller, i.e., the

8.2 The design problem 191

a) One-step design

xF

F-f-

wF(s) KF(s)

K(s)

b) Two-step design

Step 1: Feedback design

A «*—

A

1 *-

P

1

K(s)

Step 2: Feedforward design

F-fr

wF(s) Kp(8)

K(s)

-»*P

Figure 8.1: Design of feedforward controllers

a) Simultaneous design with feedback controller

b) Design as a second step for the closed loop system. The weighted plant P*

may be a simpler uncertainty structure than the plant P.


weighted feed flow signal. However, this approach has certain draw¬

backs:

• Convergence is unattainable using the uK-Iteration in our case

• Using the u-optimization approach, the one-step design needs

significantly more computation time than the two-step design.(The computing cost is proportional to {number of parameters)11with k>2.)

• For acceptable results, the weighting function for the feed flow

signals must be modified: Small improvements in the compensa¬

tion of feed flow disturbances cause a dominance of the reference

and the feed composition inputs with regard to the performance

specification. Very small gains in the feedforward part result

therefrom.

Consequently, the discussion is focused on the design of feedforward

controllers for the closed-loop system, i.e., as a second design step. Since

feedforward control does not affect any stability properties of the closed-

loop system, the design is relatively simple. It is discussed by means of

two examples.

8.3 H^-minimization

The H^-minimization [8.4] is well suited for a feedforward controller

design. Before we use the numerical tools available (e.g., [8.1], [8.2]), we

have to build up a closed-loop plant with a previously designed feedback

controller K(s). As an example, the u-optimal state-space controller

using all 3 temperature measurements is selected (see section 6.4.3). If

we wish to improve the compensation of feed flow disturbances for the

plant models GR (s) as well as for GT (s) ,we have to close the feedback

loops for both models separately, define the desired performance, and

limit the high-frequency output of the feedforward controller KF (s) .

The design plant is outlined in Figure 8.2.

The performance weight We(s) is a diagonal matrix of the transfer

functions we(s)

H^-minimization 193

KF(S)

F

u

Gr(b)

Wu(8)

K3uF Jl +

K(s) 6

u

GT(s)

K3 K(s) tO

We(s)

Figure 8.2: The augmented plant for a design of the feedforward

controller KF(s) by H^-minimization

Wp(s) = diag[w„(s),wp(s),wp(s),w„(s)] (8.1)

It demands the same performance for both column models and both

controlled temperatures. The transfer function we(s) is chosen as a

first-order lag with a high static gain. The pole of we (s) is adjusted

until |TF _»J ~ 1 is achieved. The final transfer function becomes

We(S> = 100TT2380i (8.2)

If we do not specify any high-frequency limits of the feedforward

controller output, we obtain a controller with large high-frequency

gains. This is undesirable because measurement noise and short-time

feed flow fluctuations cause unnecessarily, large control actions. Using

a diagonal transfer function matrix Wu(s) for the feedforward

controller output uF according to


Wu(s) = diag[wUF(s),wUF(s)]

with the lead-lag transfer function

(8.3)

w„ (s) = 0.51 + 104s

l+2.5s(8.4)

a controller behavior similar to a first order lag is obtained. The singularvalues of the controller and the transfer functions from the disturbance

inputs to the control error (for the nominal model) are shown by Figure8.3. If we compare Figure 8.3 b with Figure 6.10 b, we recognize the

significant improvement of the feed flow disturbance compensation

(dash-dotted lines).

10J 10J 10" 10

Frequency (rad/min)

a)

icr io io io

Frequency (rad/min)

b)

Figure 8.3: a) Singular values of the feedforward controller

b) Singular values ofthe transfer function from the disturbance inputs d to the

controlled output signals y for the nominal model G^ with feedforward and

feedback control. Solid line: T_,.

dash-dotted line: Tw_. „Xr, —> y r —> y

Nonlinear simulations confirm these expectations (Figure 8.4). In the

interest of consistency, the same disturbances are simulated as in all

previous chapters. Of course, the response to the step changes in the

feed composition remains identical to the one shown in Figure 6.12.

However, the maximum control error during the compensation of the

H^-minimization 195

0.020

Ft=0=20 mol/min

o

Io

gao

oa

So

O

0.015

0.010

0.005

- Top composition


0 10 20 30 40

Time (h)

Ft=0=20 mol/min

0.4 }'

(K)0.2

1 .

. J.

i! -

Temperature0.0

-0.2

i

i-0.4

Control error T-10

-0.6— — Control error T-44

0 10 20 30 40

Time (h)

0.020

Ft=0=46 mol/min

o

Io

gao

0.015 r

o

P.

Bo

O

tfl

0.010

0.005

i

0.4

0.2

0.0

-0.2

-0.4

-0.6

- Top composition


0 10 20 30 40

Time (h)

Ft=0=46 mol/min

Control error T-10

Control error T-44

0 10 20 30 40

Time (h)

Figure 8.4: Simulation results with \i-optimal state space controller (controller in¬

puts: Tig, T44, T24) and feedforward controller for an increase in feed composi¬tion (0.8 -» 0.9 mol/mol) at t=0 h and an increase of feed flow rate (+ 3.6 mol/

min) at t=20 h


———— L, V equal to controller output



feed flow disturbance is approximately halved, and for the maximum

feed flow rate it is reduced even more.

8.4 Optimization approach

The implementation of state-space controllers in a distributed control

system is difficult. Of course, this holds for feedforward controllers as

well. Most desirable are feedforward controllers with a simple and

easily implementable structure.

The singular values of the H„, norm-minimizing state-space controller

suggest a feedforward controller structure with a first-order lag and

different gains for the outputs to the reflux L and the boilup V:

KF(s) =

The parameters of this simple control structure are computed by a

constraint parameter optimization [8.5]. The objective may be of

different kind: One possibility is the minimization of the H norm of the

transfer function TF _^for the plant shown in Figure 8.2. This design

objective has certain disadvantages, however:

• Due to the few degrees of freedom resulting from using this

simple controller structure, it is not possible to obtain a

controller which is close to design specifications for a wide

frequency range.

• The H^-norm minimizing parameters strongly depend on the

allowed maximum for y, with y = |TF_^J .If we allow y>5

controller designs with large enough gains (KRL, KRy) are

obtained. But for a performance specification allowing y ~ 1, we

attain small controller gains and the improvement of the distur¬

bance compensation is insufficient

Most of the feed flow disturbances entering this distillation column are

step changes. Consequently, we are able to define an appropriate design

KRj

KR,

1

1+Ts(8.5)

8.4 Optimization approach 197

objective in the time domain. It is the minimum absolute control error

for a step change in the feed flow rate. The design objective becomes

[T, KRL, KRV] = arg inf E

[T, KRL, KRV](8.6)

with

E = f {|e10 (tOl + le^ (t)| + |e10 (t)\ + \eu (t)|}dt. (8.7)J | 1UR I I Ê I I 1UI I | ^*I I

The performance measure E is calculated for a step response to the

plant input F, employing the plant illustrated by Figure 8.5.

'io.

'44t

'10,

"44,

Figure 8.5: Plant structure for the optimization of

feedforward controller parameters

Ifwe select the u-optimal PID-controller with one-way decoupling as the

feedback controller K (see section 6.5.3), and limit the time constant T

by a lower bound of5 minutes, the following simple optimal feedforward

controller results:


KF(s) =

1.5

2.6

1

l + 5.0s(8.8)

The singular values of the feedforward controller are shown in Figure

8.6 a. In Figure 8.6 b we find the singular values of the transfer func¬

tions Td for nominal closed loop system with this feedforward

controller. It demonstrates the low sensitivity of the feedback and feed¬

forward controlled distillation column to variations of the feed flow rate.

10 10 10 10

Frequency (rad/min)

a)

T,

10'd -*y

<D 0'V 10

i \

=1 ' \\*>

6 / / \ \/ /

s 1U / /

/ /

/ i

\v\ \

\ \

,n2 / i \ \

10 10 10

Frequency (rad/min)

b)

10

Figure 8.6: a) Singular values of the feedforward controller with fixed structure

b) Singular values of the transfer functions for the nominal closed loop system

from the disturbances inputs d to the controlled output signals y (Feedback

and feedforward control). Solid line: T,dash-dotted line: T

F-*y

The simulation results (Figure 8.7) demonstrate that the maximum

deviation of the product compositions for a step change of the feed flow

rate is very small. A comparison with the simulation results for the

same feedback controller but without feedforward control in Figure 6.30

confirms the substantial improvements by this simple feedforward

controller.

8.5 Summary 199

Ft=0=20 mol/min

0.0201 ' '

0.005

Top composition


o 10 20 30

Time (h)

40

0.020

o

Io

s

ao

oa.

Bo

O

Ft=0=46 mol/min

0.015

0.010

0.005

0

— Top composition


10 20 30

Time (h)

40

Figure 8.7: Simulation results with ^-optimal PID controller with one-way decou¬

pling and a simple feedforward controller for an increase in feed composition

(0.8 -> 0.9 mol/mol) at t=0 h and an increase of feed flow rate (+ 3.6 mol/min)

at t=20 h

~——-"-— L, V equal to controller output


8.5 Summary

The compensation of feed flow disturbances can be improved by using

feedforward controllers. H«,-norm minimization and the minimization

ofthe control errors in the time domain (for feedforward controllers with

fixed structure) are efficient design methods. Frequency domain as well

as time-domain results demonstrate the pleasing improvements which

are obtained even by a feedforward controller oforder one. A comparison

ofthe ISE and ITAE criteria (see section 6.6) in Table 8.1 demonstrates


improvements up to 50%! As mentioned previously, the maximum struc¬

tured singular value |i is not a good performance measure if we include

the feedforward control in the structured uncertainty model.

Table 8.1: Comparison of controllers in time-domain

Control structure Relative

ISE

Relative

ITAE

Max n-


3 temp, measurements1.0 1.0 0.85


3 temp, measurements

and feed forward control

0.63 0.51 0.86

PID control with static one-way

decoupling (C 1=0)1.99 1.51 0.97

PID control with static one-way

decoupling (C 1=0) and simple

feedforward control

1.13 0.87 1.05

8.6 References


Analysis and Synthesis Toolbox, MUSYN Inc., Minneapolis MN,

and The MathWorks, Inc., Natick, MA (1991)

[8.2] Chiang, R. Y., and M. G. Safonov: Robust Control Toolbox — Us¬

er's Guide, The MathWorks, Inc., Cochituate Place, Natick, MA

(1992)

[8.3] Christen, U., M. F. Weilenmann, and H. P. Geering: "Design of

H2 and H„ Controllers with Two Degrees of Freedom," Proc. of

the 1994 American Control Conference, Baltimore, MA (1994)

8.6 References201

[8.4] Glover, K, and J. C. Doyle: "A State Space Approach to H„, Opti¬

mal Control," Lecture Notes in Control and Information Science,

135,179-218, Springer-Verlag, Berlin (1989)

[8.5] Grace, A.: Optimization Toolbox — User's Guide, The Math-

works, Inc., Natick, MA (1990)



Chapter 9

Practical Experiences

9.1 Introduction

In simulations the performance of controllers is tested in a sterile envi¬

ronment. Lacking measurement noise, operator actions, and varying

environmental conditions, the results of these simulations represent a

well established basis for a comparison of different controller designs.

However, only the implementation of a controller in the real plant

proves its performance. While in the literature a great number ofdesignmethods has been proposed and the resulting controllers have been

tested by simulations, only very few results of an implementation at a

real industrial distillation column have been reported.

This chapter complements the simulation results presented in previous

chapters with the results of a controller implementation in the distrib¬

uted control system (DCS) which is coupled with this distillation

column. The first section describes the implementation including the

handling of constraints. Further sections discuss the use of pressure

compensated temperatures, the controller performance observed, and

economic aspects. A short summary concludes the chapter.

204 9 Practical Experiences

9.2 Controller implementation

In the research field the objective of any control design is a highcontroller performance. A control design implemented in an industrial

environment must consider many additional aspects. A few of them are

listed below.

Simple implementation: As mentioned previously, state-space control¬

lers are difficult to implement in a DCS. Therefore the control scheme

should be based on fixed low order structures, e.g. on PID control or on

advanced PID control structures.

Robustness: The control design must guarantee stability for the entire

operating range of the column, including time variations due to corro¬

sion of trays, transmitter drifts, etc.

Easy to initialize: The switch from manual to automatic control must be

simple and easy to understand. Operators often are semiskilled

workmen who cannot and should not be expected to have an engineering

background. A complex initialization procedure of a control scheme

unnecessarily increases the risk of errors and requires an intensive

operator training.

Handling of constraints: Constraints are necessary to prevent the

column from flooding, weeping, overpressure, overtemperature, etc.

Often it is sufficient to limit reflux and reboiler heat duty.

Performance: Despite the requirements listed above, the performance of

the control scheme should still be high.

Comparing the different control schemes proposed within this thesis,

the PID control structure with one-way decoupling including the simple

feedforward controller evolves as the best compromise among all these

requirements. The control scheme is simple to initialize1, robust to

1. Initialization of the control scheme (see Figure 9.1): First, the output of the top

composition controller in manual mode is adjusted to achieve r^ = Lactuai. Then the

top composition controller is switched to automatic mode. Second, the output ofthe

bottom composition controller in manual mode is adjusted to achieve rq = Qactuai-

After that the controller is switched to automatic mode.

9.2 Controller implementation 205

plant uncertainty, it allows a simple handling of constraints, and it

exhibits a high performance in simulations.

This control scheme has been implemented in the DCS installed at the

plant considered here (i.e., an Eckardt PLS 80E). The controller inputs

are estimated tray compositions Xj, which for the operators have proved

to be easier to understand than pressure compensated temperatures.

The proportional gains of the controllers are easily converted for these

controller inputs.

A scheme of the implementation is shown in Fig. 9.1. The handling of

constraints is realized by using the anti-windup facility of the standard

PID controller blocks within the DCS. The following ideas have been

realized:

• If the setpoint for the reflux controller rR becomes smaller than

its minimum limit Rmjn, the top composition is allowed to rise

above the setpoint (=> top composition purer than required), and

the top composition controller must be prevented from windup.

• If the setpoint for the reflux controller rR exceeds its maximum

limit Rmax, the top composition is allowed to decrease (=> top

composition less pure than required), and the top composition

controller again must be prevented from windup. However, if at

all possible, this case should be avoided.

• Equivalent constraints hold for the bottom composition

controller.

This policy establishes individual constraints for the top composition

control loop as well as for the bottom composition control loop. Since we

have to include the feedforward control and the one-way decoupling, the

outputs of the composition controllers are limited by the following four

signals entering the anti-windup facility of the PID controllers:

RFBl,max = Rmax_RFF (9<1)

RFBl,min = Rmin ~ RFF (9-2)

implementation

Controller

9.1:

Figure

_Q"Valv

PIDQ

i9_

Valve

Decoupling

FBI

Q

Kq/R

"

R

-

^

%

PIDR

rR»

+

FF

Q

Kqf/rf

controller

Feedforward

VFF

Qmin-^FF-^FBl

LAG

mm

FB2,

Q

Qmax^FF^FBl

max

FB2,

Q

FB2

Q

n,

iiPID2

^*

04p2

+0,p

+

TCml

)+

0]+02(T

constraints

with

controllers

Feedback

Rmin_RFF

vFBl,min

RFF

Rmax

max

FBI,

R

"FBI

estimation

^44

Composition

estimation

Composition

niiPIDl

11Q.

5o(P51-P

0)+

Po

PlO

^(P51-

Po)

+Po

P51

Po

e3p

+

TCor

r)+

92<T

+ei

L10

9.3 Composition estimators 207

°-FB2,max " Qmax- °«FF- °-FBl (9-3)

Q-FB2,min = Qmin" ^FF- °-FBl (9-4)

These individual constraints make unnecessary the configuration of

variable structure control in the DCS. However, the maximum

constraint of the reflux may lead to a top product quality significantlybelow the product specification, which is much more undesirable than a

deterioration of the bottom product quality. Fortunately, simulation as

well as practical experiences have shown that the reboiler heat dutyexceeds its maximum limit Qm,v first. In this case, the behavior of the

control scheme is identical to that ofa single composition control scheme

with reboiler heat duty set at maximum Qmax and top composition

controlled by reflux flow rate.

If the reflux as well as the reboiler heat duty reach their minimum

constraints, both products become purer than desired.

9.3 Composition estimators

While the implementation of the controllers did not cause any partic¬

ular problems, the correct parametrization of the composition estima¬

tors was very troublesome. In a first step the parameters of the

estimators were calculated by regression of {Tpx} data (see Chapter 2).

However, the correlation of the estimated compositions on tray 10 and

44 with the product compositions analyzed once a day proved to be

unsatisfactory.

Hence operating data were recorded for two weeks. Since the feed

composition was almost constant, it was possible to compare these

measurement data with tray compositions calculated by steady-statesimulations. Minimizing the errors between the estimated and the

calculated tray compositions, a correction of the estimated tray pres¬

sures by 20% was necessary to correct the estimates. Since pressure

sensors on tray 10 and 44 are not installed, the pressures on these trays

are calculated by a linear interpolation between top pressure and


bottom pressure (see Fig. 9.1). The error in the pressure compensation

might have been caused by this interpolation. Other error sources could

have been incorrect {Tpx} data or pressure measurements. Once the

parameters of the estimators had been adjusted, these simple estima¬

tors worked fairly satisfactorily. Nevertheless, the compensation of the

pressure variations' influence on the tray temperatures is the limiting

factor for the overall performance of the control scheme. This will be

shown in more detail in the following section.

Of course, the effort for the parametrization of the estimators is fairly

high and the performance of the control scheme is limited by them. In

view of these two points, the installation of on-line gas chromatographscould be preferable. However, in our case the light component polymer¬

izes at temperatures exceeding a certain level. Since polymerization

plugs a gas chromatograph in a short time, the use of pressure compen¬

sated temperatures or estimated tray compositions as controller inputs

is indispensable.

9.4 Controller performance

The controller performance observed matched the simulation results

quite well. Figures 9.2 and 9.3 depict the recorded deviations of the esti¬

mated tray compositions from their setpoints in the presence of several

feed flow disturbances and at two different feed flow rates.

The large measurement noise of the estimated tray compositions is

caused by the noisy pressure readings in the column bottom. Using a

first-order low-pass element in series with the bottom pressure

measurement, the noise could be significantly reduced. Unfortunately,

at the time of the installation of the control scheme, the capacity of the

DCS was exhausted. Even for the configuration of this simple element,

there was no space left. As soon as new capacity is available, the bottom

pressure measurement will be filtered.

During the recording of these operating data, the setpoints were kept

constant. In Figure 9.2 the feed flow rate was increased by 401/h in four

steps. The feed flow rate at t=0 h was 260 1/h (49 mol/min), while the


Feed

Tray 10

40 60Time (h)

100 Q

Tray 44

{

L5 •>

Ml

i ;,V

.

ik'3.' M§Li >S Jss ieS ills

53S ,*£ *

t??^ 7 ?' "

165 SflftSf WBt3B%jff"WB5f t HB'Sff

1 1jj' j;

20 40 60

Time (h)

80

H-osS 8

9 3

0.5

100

ao

•i-t

+a

CO

"gQ

a.

a0)

Figure 9.2: Recorded operating data with installed PID control scheme including

one-way decoupling and feedforward control.

Top: Deviation of feed flow rate from 260 1/h (49 mol/min)

Middle: Deviations of estimated tray composition and of pressure

compensated temperature from setpoint on tray 10

Bottom: Deviations of estimated tray composition and of pressure

compensated temperature on tray 44


Feed

P «3

30 40

Time (h)

Tray 10 -a

30 40

Time (h)

-0.5 ga<a

a,

0.5 V

70 P

Tray 44 13

l^^diAilw-0.5

- 05

10 20 30 40

Time (h)

50 60 70

ao

>a)

P

Figure 9.3: Recorded operating data with installed PID control scheme including

one-way decoupling and feedforward control.

Top: Deviation of feed flow rate from 170 1/h (32 mol/min)

Middle: Deviations of estimated tray composition and of pressure

compensated temperature from setpoint on tray 10

Bottom: Deviations of estimated tray composition and of pressure

compensated temperature on tray 44


feed composition was approximately 0.85 mol/mol. Although the feed

flow rate was out of the design range, the reflux and boilup remained

within the range covered by the controller design. In Figure 9.3, the feed

flow rate at t=0 h was 170 1/h (32 mol/min) and it was increased only

once by 101/h.

The control errors in presence of these feed flow disturbances remain

extraordinary small. In fact, it is almost impossible to separate the

control error from the measurement noise and the effect of all other

unknown disturbances. This proves the high performance ofthis simple

advanced PID control scheme.

The advantages ofthe controller implementation are demonstratedbest

by a comparison of the product compositions analyzed once a day before

and after the installation. At the beginning of this project, top and

bottom composition were controlled manually. The results are shown on

the left-hand sides ofFigure 9.4 and 9.5. Obviously, the average product

compositions are found far from their setpoints, and the variations of

the product compositions are very large.

The right-hand sides of Figure 9.4 and 9.5 show the analysis results

beginning after the adjustment of the composition estimators. Clearly,

the variations of the product compositions are much smaller and the

average product compositions are close to the desired results. However,

despite the high performance of the control scheme as illustrated by

Figure 9.2 and 9.3, significant variations of the product compositions

can still be detected. Please remember that pressure measurements of

tray 10 and 44 are lacking. Therefore the influence of the large pressure

variations (bottom pressure: 120-190 mbar) to the tray temperatures

cannot be perfectly compensated and an adjustment of the controller

setpoints depending on the feed flow rate is necessary. Since the results

presented are achieved with almost constant setpoints, the results will

improve even further as the operators gain more extensive experience

with the setpoints.


Manual operation Controlled

0.3

~ 0.25

0.2

•£ 0.15

0.1

0.05

xxxXx

x

Xx

X x

xx x

X

X

XX

X

xx

x*-x- -

v-

x—"

X

XX XX

X X*

x £ >** ^35

Days70

0.25

0.15

Manual operation

0.25 r

0.05

Controlled

Figure 9.4: Analysis data of top and bottom product

Top: Top composition 1-xjjBottom: Bottom composition xg

Dashed line: Average composition



0 0.1 0.2 0.3

Top composition (mol/mol)

0 0.1 0.2 0.3

Top composition (mol/mol)


0 0.1 0.2 0.3

Bottom composition (mol/mol)

0 0.1 0.2 0.3

Bottom composition (mol/mol)

Figure 9.5: Histograms of analyzed product qualities

Top: Top composition 1-xjjBottom: Bottom composition xg


9.5 Economic aspects

The management decision for or against the installation of a control

system depends primarily on the economic feasibility and to some

degree on ecological improvements. In our case the installation of the

control scheme yields the following most important improvements:

• More uniform product qualities

=> Less overpurification necessary

=> Energy savings (which is an ecological advantage, too)

• Reduced mean of light component in bottom

=> More top product with a market value of > 250000 $/a

• Increased maximum column load

=> The installation of an additional column can be avoided

These pay-offs are complemented by side effects, for example a deeper

understanding of column dynamics by the operating staff, which as a

consequence achieved a better operation of other columns in the same

plant.

These financial benefits must be weighed against the investment costs.

Hardware and software expenses exclusively for this project total

approximately 50000 $. It is not unreasonable to estimate the necessary

engineering effort for a similar project to be less than half a man year.

Therefore the economic benefits are on a very positive side.

9.6 Summary

The results ofthe implementation of the PID control structure with one¬

way decoupling and feedforward control on the real plant confirm the

high performance of this simple control scheme indicated by simula¬

tions. The main problem of the implementation was, except for over¬

coming high psychological resistances, the correct parametrization of

the composition estimators. A solution of this problem never would have

been possible without an extensive comparison of simulation and oper¬

ating data. Nevertheless, the use of pressure compensated tempera-

9.6 Summary 215

tures or estimated tray compositions remains the limiting factor of the

overall performance of the implemented control scheme. The economic

advantages achieved by this simple control scheme exceed the financial

effort by far.



Chapter 10

Conclusions and

Recommendations

10.1 Introduction

This thesis treats all the necessary steps for a composition control

design for an industrial binary distillation column. Each of these steps

produced new insights into various aspects of the control design. Since

a chronological discussion of these steps would lead to a thematic confu¬

sion, they are summarized in the four sections

• Controller synthesis

• State-space or PID control?

• How many temperature measurements?

• Column models

This thesis does not presume to present a final solution to all distillation

control problems. The ideas presented come up against many gaps in

research, limits of distributed control systems, and problems of cooper¬

ation between industry and university. In the last section the most

important aspects of these topics are discussed.

218 10 Conclusions and Recommendations

10.2 Controller synthesis

This thesis discusses the design ofrobust controllers for the dual compo¬

sition control problem of an industrial binary distillation column. Distil¬

lation columns are usually operated over a wide range of feed

compositions and feed flow rates. Consequently, a controller must guar¬

antee stability and a high performance not only at a single operating

point, but for the entire operating range of the distillation column.

The common robust controller design methods are based on unstruc¬

tured uncertainty models, for example a multiplicative uncertainty at

plant output. An estimate of the corresponding uncertainty bounds has

shown that these bounds are too large to allow any controller design.

Nevertheless a solution of the design problem is possible. It is based on

a structured uncertainty model which to a large extent avoids the unnec¬

essary conservatism of an unstructured uncertainty description. This

model treats the nonlinear column behavior as several simultaneous

uncertainties and quite well describes the column dynamics for all oper¬

ating points within the predefined operating range.

Utilizing this uncertainty model, a feedback controller synthesis

requires the framework of the structured singular value \i. The appro¬

priate design methods are the uK-Iteration for the synthesis of state-

space controllers and a constraint parameter optimization for the

synthesis of controllers with fixed structure. These methods lead to

feedback controllers which are distinguished by a high controller perfor¬

mance and guaranteed stability within the entire operating range,

paired with a low sensitivity to errors in the manipulated variables.

A drawback of this design approach is the high effort for uncertainty

modelling and computation of the controllers. In principle, comparable

results could be obtained and the computational effort could be signifi¬

cantly reduced by using design methods based on arbitrary small

unstructured uncertainty bounds. However, it has been shown that

these common design methods are not well suited for ill-conditioned

plants such as high-purity distillation columns.

10.3 State-space or PID control? 219

The ideas of the feedback controller synthesis can be extended to the

feedforward control design. A simultaneous controller design for the

closed-loop models at maximum and minimum column load using HM-minimization (for state-space feedforward controllers) or optimizationin the time-domain (for feedforward controllers with fixed structure)

yield controllers, which greatly improve the compensation of feed flow

disturbances.

The theoretical and simulation results are confirmed by the results of

the practical implementation of a simple PID control structure with

one-way decoupling and a simple feedforward control scheme. The very

satisfactory controller performance achieved without any expensive on¬

line composition analyzers leads to high economic and ecologic benefits

which justify the effort of the control design and implementation.

10.3 State-space or PID control?

A comparison of the different state-space controllers with optimallytuned advanced PID control structures has demonstrated an unex¬

pected result:

• The performance of u-optimally tuned advanced PID control

structures is only insignificantly worse than the performance of

high-order state-space controllers

This statement is of great significance for industrial practice. It holds

for the feedback as well as for the feedforward control design. The imple¬mentation of advanced PID control structures in a distributed control

system requires much less effort than that of state-space controllers and

increases the acceptance of the control design by the operators. It must

be emphasized, however, that the high performance of the PID control

structure is achieved with unconventional controller settings.

The optimal tuning of PID control structures with decoupling for this

distillation column caused an additional insight. The optimal controller

performance is achieved with an implicit decoupling scheme where in

essence


• the bottom composition is controlled by moving the composition

profile, and

• the top composition is controlled by intensifying or weakening

the S-form of the composition profile.

Since the position and shape of the composition profile at steady-state

depends essentially on the actual and unmeasured feed composition, it

is difficult to make any inference from a composition or temperature

measurement in the column middle to the manipulated variables.

Very similar considerations hold for the relative performance of the

state-space controllers. For the same reason, the estimation of the

composition profile by the inherent observer has no advantages. The

better performance results only from the higher degree offreedom in the

controller design, which allows a higher performance in the low- and

mid-frequency range without destabilizing the closed loop system in the

high-frequency range.

10.4 How many temperature measurements?

A comparison of a control design including a temperature measurement

in the middle of the column with a design excluding this measurement

leads to the following statement

• Additional temperature or composition measurements in the

middle of the distillation column have no significant influence

on the maximum controller performance.

The reason for the very limited advantage of additional temperature

measurements for the control design is their unknown setpoint, which

depends on the actual, unmeasured feed composition. The high perfor¬

mance of the control design can be achieved with just two pressure-

compensated temperatures or two estimated tray compositions.

Dispensing with additional measurements reduces the installation

costs of the control system and increases its economic viability.

However, if regression models are used to estimate the product compo-

10.5 Column models 221

sitions based on temperature and flow measurements, additional

temperature measurements are of great advantage.

10.5 Column models

All results of this thesis are based directly or indirectly on models ofthe

distillation column. Especially the model-based adjustment of the

composition estimators clearly proved that such process models are

absolutely necessary. However, the control design may be based on

linear models that include or exclude flow dynamics.

Within the structured uncertainty model, a multiplicative uncertaintyis included for each measured tray temperature, whose uncertainty

bounds exceed 100% for frequencies above 1/16 rad/min. Since the flow

dynamics affect the high-frequency range, the following statement is

justified:

• Including or excluding flow dynamics in the linear models

is insignificant for the controller design.

This has an impact on the design effort. If a controller design can be

based on an analytical linearization of a simple model for the composi¬tion dynamics at particular steady states, a rigorous dynamic model is

not absolutely necessary. The steady states of a column may be calcu¬

lated with common flowsheeting programs such as ASPEN PLUS or

PROCESS and the controllers designed can be tested using a simpli¬fied nonlinear model without flow dynamics.

10.6 Recommendations

10.6.1 Academic research

Multicomponent distillation: The results of this thesis are based on the

example of a single binary distillation column. While the adaptation of

these results to other binary columns is expected to be straightforward,


the uncertainty modelling of multicomponent distillation columns

needs additional research.

[i-synthesis: The robustness analysis of controllers using the structured

singular value î has shown to be a reliable and outstanding tool.

However, the convergence properties of the corresponding algorithms

for u-synthesis (DK-iteration, uK-iteration) are insufficient. More

robust algorithms are absolutely necessary.

Decentralized control: Generally, the design of robust controllers with

simple structures is at an early stage of development. In the case of this

distillation column, it was relatively easy to propose potential control

structures and to solve the design objective with a constrained param¬

eter optimization (u-optimization). However, dealing with many more

control loops simultaneously, the problem of the loop pairing is still not

solved. For example, the high performance of the controllers in this

thesis has been obtained using the LV control configuration. Since

common methods for control structure selection (single loop pairing of

controlled variables and manipulated inputs) try to minimize the inter¬

actions between the individual control loops, certainly these methods

favor other control configurations. Therefore methods for the selection

of control structures are necessary, which include simple multivariable

control schemes. Similar arguments hold for the controller tuning. The

current methods for the tuning of multiloop SISO control schemes are

known to be either very conservative or else to lack robustness. Better

methods would be very desirable.

10.6.2 Decentralized control systems

Today a control engineer in the research field is familiar with modern

and flexible software tools such as MATLAB or MATRLXX. His first

contact with a decentralized control system (DCS), even with a modern

one, arouses feelings of working in the analog computing era. The

replacement of the old consoles with a computer seems to be the only

idea for the development of the DCS. The inherent possibilities for a

faster, more flexible, and simplified controller implementation are not

exhausted yet.

10.6 Recommendations 223

10.6.3 Cooperation industry—university

Often the industry complains of the inadequate cooperation between

university and industry. Some typical problems of such a cooperation

were encountered during the course ofthis project. The main problem is

the divergence between the interest of the partners in the project.

University researchers are interested in deeper insights into basic prob¬

lems and their solution, while the process industry wants a rapid solu¬

tion of the actual problem. Additionally the contact persons in industry

are chronically overworked with everyday problems, thus unable to

spend enough time to concern themselves with such a project. This leads

to an insufficient flow of communication. Consequently, both partners

speak different languages: the university researcher does not under¬

stand the industrial needs, while the industrial counterpart does not

understand the mathematical methods. Therefore it is of high impor¬

tance that

• the aims and responsibilities of both partners in the project are

spelled as clearly as possible

• at least one control engineer of the industrial partner actively

follows the progress of the project

If these two points could be kept in mind, many problems between

industry and university could be avoided.


Curriculum vitae

Name Hans-Eugen Musch

Date of birth June 19,1965

Place of birth Freiburg im Breisgau, Germany

Nationality German

1971-1975 Primary school

1975-1984 Humanistic gymnasium Kolleg St. Sebastian

at Stegen near Freiburg

1984 Abitur

1984-1985 Military service

1985-1989 Chemical engineering studies at the ETH Zurich

1989 Masters degree in Chemical Engineering CDiplom")

Since 1990 Research assistant at the Measurement

and Control Laboratory, ETH Zurich

Documents

In Copyright - Non-Commercial Use Permitted Rights ......It is well knownthat high-purity distillation columns are difficult to control due to their ill-conditioned and strongly nonlinear