• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 List of Symbols
 Abstract
 Problem definition, literature...
 Hardware-software system, distillation...
 The Optimal control problem
 Experimental technique
 Results, conclusions, recommen...
 Appendix
 Basic data for linear model...
 User software
 Matrix multipliers for the optimal...
 Data
 Literature cited
 Biographical sketch
 Copyright














Title: Experimental studies in optimal computer control of a continuous distillation column
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00086011/00001
 Material Information
Title: Experimental studies in optimal computer control of a continuous distillation column
Alternate Title: Distillation column, Esperimental studies in optimal computer control of a continous
Physical Description: xiii, 130 leaves. : illus. ; 28 cm.
Language: English
Creator: Eschenbacher, Robert Carl, 1941-
Publication Date: 1970
 Subjects
Subject: Distillation apparatus   ( lcsh )
Computer programs   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida, 1970.
Bibliography: Bibliography: leaf 129.
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00086011
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000955790
oclc - 17010459
notis - AER8419

Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
    List of Symbols
        Page ix
        Page x
        Page xi
    Abstract
        Page xii
        Page xiii
    Problem definition, literature discussion
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    Hardware-software system, distillation column and computer linkup
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
    The Optimal control problem
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Experimental technique
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
    Results, conclusions, recommendations
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
    Appendix
        Page 74
    Basic data for linear model derivation
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
    User software
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
    Matrix multipliers for the optimal contral law
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
    Data
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
    Literature cited
        Page 129
    Biographical sketch
        Page 130
        Page 131
    Copyright
        Copyright
Full Text









EXPERIMENTAL STUDIES IN OPTIMAL COMPUTER

CONTROL OF A CONTINUOUS DISTILLATION COLUMN














By
ROBERT CARL ESCHENBACHER















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
1970











A CKN OWL EDGIMENTS


The author wishes to thank Dr. F. P. May, chairman of his

supervisory committee, for his helpful advice and personal interest;

Drs. M. Tyner, M. E. Thomas and R. G. Selfridge for serving on his

supervisory committee; Drs. A. W. Westerberg and H. E. Schweyer for

their interest and help for the duration of this work; and Mr. D. H.

Pike for his discussions and assistance in accomplishing the research.

The author wishes to acknowledge the financial support of the

Rinker Materials Corporation of West FPai Beach, Florida, the

University of Florida, and the National Science Foundation during the

author's undergraduate and graduate tenure at the University of

Florida.

'1Thanks go to the University of Flurida Computing Center and

its personnel who aided this project with time and advice whenever

needed.

Special thanks go to Messrs. Jack Kalway, Myron Jcnes, and

Tracy Lambert for their :ssjstance throughout ethe author's tenure,

and especially on this project, and to u!rs. Karen Wjalker for her
/
first-rcae tVypi-g oF tbe r.'_iija, -ipt.

Deep, p.risonal r:a,.::S go' o to the l.:hor's fr.ily, especially

his faithful wife, Virginia, whio continually s-crificed during the

graduate program.











TABLE OF COhl .i ;


ACKNOWLEDGMENTS ...............................................

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

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

LIST OF SYMBOLS ...............................................

ABSTRACT......................................................

CHAPTERS:

I. PROBLEM DEFINITION, LITERATURE DISCUSSION...........

II. HARDWARE-SOFTWARE SYSTEM, DISTILLATION COLUMN AND
COMPUTER LINKUP .....................................

A. General Interface for Process Systems
Instrumentation ............................. ..

B. GIPSI Software.................................

C. Distillation Column: Design and Control........

III. THE OPTIMAL CONTROL PROBLEM........................

A. Derivation of the Performance Index.............

B. Derivation of a Suitable Constraint Set........

C. Obtaining the Optimal Control Law..............

IV. EXPERIMENTAL TECHNIQUE..............................

A. Building and Debugging the System:
Distillation Column, Analog-Digital Interface
Software Package...............................

B. Collecting Data for the Linear Model............

C. Obtaining the Optimal Control Law...............

D. On-Line Computer Control Runs ..................

V. RESULTS, CONCLUSIONS, RECOMMENDATIONS ...............


Page

ii

V

vi

ix

xii




1


9


9

13

17

27

27

30

40

44



44

46

47

48

50











TABLE OF CONTENTS (Continued)

Page

A. Results......................................... 50

B. Conclusions ..................................... 68

C. Recommendations ................................. 71

APPENDICES:.................................................... 74

A. Basic Data for Linear Model Derivation .............. 75

B. User Software........................................ 90

C. Matrix Multipliers for the Optimal Control Law....... 105

D. Data............................................... .. 114

LITERATURE CITED............................................... 129

BIOGRAPHICAL SKETCH ......................................... .. 130











LIST OF TABLES


Table Page

2.1 Column Variables Read by Computer...................... 17

2.2 Maximum Column Variable Operating Capability........ 19

3.1 Performance Index Cost Coefficients, A and B.......... 29
i
3.2 Time Constants T. for Equation (3.8) for all State
Variables.. ............................ 34
i i
3.3 Constants K. and K. for.Equation (3.8) for all
1j 2j
State Variables....................................... 34

3.4 Computed Values of i./. +i At, 'K. and K............ 37
i i lj 2j............

4.1 Control and Load Variable.............................. 46

4.2 Load Variable Upset Pattern........................... 48

5.1 Nois e Levels of Column Variables.................... 52

5.2 Maximum Values of Performance Index Per Stage......... 59

5.3 Corltrol and Load Variable Deviations for I1l Percent
Controller Set Point Deviations....................... 67

C.1 High Control Costs ................................... 106

C.2 Equal Costs ........................................... 108

C.3 Low Control Ccsts...................................... 110

C.4 Extra Low Control Costs............................. .... 112










LIST OF FIGURES


Figure Page

2.1 GIPSI: General Interface for Process Systems
Instrumentation, Front Layout ........................ 14

2.2 Execution and Delay Phases of Software Package........ 18

2.3 Typic'.l Computer Control Loop Showing Alternate
Configuration......................................... 21

2.4 Contro.l Scheme, Feed System.......................... 23

2.5 Total Column Control Scheme........................... 25

2.6 Computer-Column Connections........................... 26

3.1 Time-Dynamic Programming Stage Relationships ......... 39

5.1 Perforo;ance Index Values for Various Control Cases
for Cast Case One........- ..... ..................... 53

5.2 Performuance Index Values for Various Control Cases
for Cost Case Two.................................... 54

5.3 Performance Index Values for Various Control Cases
for Cast Case Three.................................. 55

5.4 Perform:ance Index Values for Various Control Cases
for Cost Case Four ................................... 56

5.5 Control Variable, Steam, for Composition Control and
All Optimal Control Cost Cases....................... 61

5.6 Control Variable, Reflux, for Cost Cases One, Two,
Three................................................ 62

5.7 Control Variable, Reflux, for Cost Case Four and
Composition Control.................................. 63

A.1 Top CoGmposition: Column-Model Response to Step
Change in Feed Rate.................................. 78

A.2 Bottom Composition: Column-Model Response to Step
Change in Feed Rate.................................. 79

A.3 Top Flow Rate: Column-Model Response to Step
Change in Feed Rate .................................. 80










LIST OF FIGURES (Continued)


Figure Page

A.4 Top Flow Rate: Column-Model Response to Step
Change in Feed Composition ........................... 80

A.5 Top Composition: Column-Model Response to Step
Change in Feed Composition.......................... 81

A.6 Bottom Composition: Column-Model Response to
Step Change in Feed Composition...................... 81

A.7 Top Composition: Column-Model Response to Step
Change in Steam Rate ................................. 82

A.8 Bottom Composition: Column-Model Response to Step
Change in Steam Rate ................................. 82

A.9 Top Flow Rate: Column-Model Response to Step
Change in Steam Rate................................. 83

A.10 Top Composition: Column-Model Response to Step
Change in Reflux Rate ................................ 84

A.11 Bottom Composition: Column-Model Response to Step
Change in Reflux Rate ................................ 85

A.12 Top Flow Rate: Column-Model Response to Step
Change in Reflux Rate................................ 86

A.13 Top Composition: Column-Model Response to
Simultaneous Step Changes............................ 87

A.14 Bottom Composition: Column-Model Response to
Simultaneous Step Changes............................ 88

A.15 Top Flow Rate: Column-Model Response to
Simultaneous Step Changes............................ 89

D.1 Load Variables for Two Cost Cases to Illustrate
Effect of Filters and Noise.......................... 115

D.2 State Variable Response for Cost Case One for
Optimal Control...................................... 116

D.3 State Variable Response for Cost Case Two for
Optimal Control...................................... 117

D.4 State Variable Response for Cost Case Three for
Optimal Control...................................... 118










LIST OF FIGURES (Continued)


Figure Page

D.5 State Variable Response for Cost Case Four for
Optimal Control...................................... 119

D.6 State Variable Response for Composition Control...... 120

D.7 State Variable Response for Uncontrolled Case......... 121

D.8 Control Variable Response for Cost Case Four for
Optimal Control. Try at Reproduction................ 122

D.9 Control Variable Response for Cost Case Four for
Optimal Control. Try at Reproduction................ 123

D.10 Load and Control Variable Response for No Upset,
Optimal Control Noise Level Test for Cost Case One... 124

D.11 Stage Variable Response for No Upset, Optimal Control
Noise Level Test for Cost Case One.................... 125

D.12 Comparison of Performance Index Values of Runs for
Reproduction for Cost Case Two....................... 126

D.13 Comparison of Load and Control Variables of Runs
for Reproduction for Cost Case Two ................... 127

D.14 Comparison of State Variables of Runs for
Reproduction for Cost Case Two ....................... 128


viii











LIST OF SYMBOLS


Capital Letters

A Matrix of state variable quadratic cost coefficients

B Matrix of control variable quadratic cost coefficients

B Bottom product flow rate, !bm/hr

C Vector of state variable linear cost coefficients

D Vector of control variable linear cost coefficients

D Top product flow rate, Ibm/hr

E Matrix of coefficients for previous state variables in
linear model

F Vector of load variables

F Load variable, feed rate, lbm/hr

G Matrix of coefficients for present load variables in linear
model

H Matrix of coefficients for previous load variables in linear
model

J_ Matrix of coefficients for present control variables in
linear model

I 2Matrix of coefficients for previous control variables in
linear model

N Number of stages in the dynamic programming solution

R Concrol variable, reflux rate, ibIi/ihr

S Control variable, steam rate, ibm/hr

U Vector of control variables

X Vector of state variables

XB Bottom composition

XD Top composition

XF Feed cor)po-:ition











Small Letters

d Derivative operator

m Present time stage in linear model

n Previous stage in dynamic programming solution

s Laplacian operator

t Time


Subscripts

D

F

i

j

R

S

XB

XD

XF

1

2


Variable with this subscript represents top rate

Variable with this subscript represents feed rate

Denotes variables in an array

Denotes variables in an array

VariabLe with this subscript represents reflux ra

Variable with this subscript represents steam rat

Variable with this subscript represents bottom cor

Variable with this subscript represents top compo

Variable with this subscript represents feed comp

Signifies particular constant

Signifies particular constant


te

e

position

sition

position


Superscripts

D Variable with this ui~rc.-L.pi ieresents top product rate

i Denotes variable in an array

T Denotes transpose of a vector

XB Variable with this ;superscript represents bottom composition

XD Variable with this superscript reprne~ets top composition

ss Steady state











Subscripted and Superscripted Letters

I. Represents control and load variables in li
derivation

i
K Denotes particular constant in linear model

i
K Denotes particular constant in linear model
2j
-i i i
K j Constant as a function of K j and K2j in li

Sl i i
K Constant as a function of K and K in li
2j 1n 2l
1
P Represents linear-constant multiplier in op


near model


near model


near model


timal control


law

2
P Represent state variable multiplier in optimal control law

P Represents control variable multiplier in optimal control
law

4
P. Represents load variable multipliers for present, past, and
future values

X. State variable
1


Greek Letters


a Constant to insure convergence in dynamic program solution

A Represents a deviation quantity. For time this is t2-tl.

For variables this is actual value steady-state value.

T Time constant in linear model

Y Summation











Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy



EXPERIMENTAL STUDIES IN OPTIMAL COMPUTER CONTROL
OF A CONTINUOUS DISTILLATION COLUMN


By


Robert Carl Fschenbacher

August, 1970

Chairman: Dr. Frank P. May

Major Department: Chemical Engineering


A continuous distillation column and an analog-digital inter-
r-- e o cere
face for connecting process variables to an IBM 360/65 computer were

built, and a general software pac!agc for data logging and real-time

computer control was writte; and debugged.

A dynamic programming problem with a quadratic performance

index and a constraint cst consisting of a linear first order distilla-

tion column model was proposed and solved by an inductive method, by

hand, to obtain a solution. The analytic solution was run on the

IEM 3'60/65 C-'uter. eff-14ia to Jcbt;in a set of s'a.e irvrii-7 t

steady-state matrix rtipli.ir fror eich of four cost coefficient cases.

When on-line control was instituted, these multipliers were multiplied

by the present stai:e nd co ntro!l C -.i 1 abls and by the present and

futui.e :psets to give lan op.',,~I.a output at e~ach stage.

i -luine optimal conmpuLter coT:rtrol w;-s investigated for the four

sets oif cost L oeffi.ci .its ;in the p rfo i ranca index, int3i xa ng fo:'i high

state vari .ble c'::- i high 'cl'oitro] vaJriable co-:ts. An iJpsiet pTL ttcrn










consisting of pulses in feed rate and feed composition constituted the

load variable changes on the system. The system state variables were

top and bottom product composition and flow rates, and the control

variables were steam and reflux flow rates.

The same upset pattern was made for column control configurations

of (a) no control (uncontrolled column response), (b) dual composition

control where top and bottom product compositions were maintained

constant by manipulating reflux and steam rates, respectively, and

(c) optimal computer control.

Noise in the system proved to be a major problem which made it

difficult to interpret some results and make reproducible runs. This

problem must be dealt with more effectively in any future work.

Despite the noise problem, it was demonstrated that the optimal

control scheme, when used with the known upset pattern of pulses,

produced bjit.er control than either composition control or uncontrolled

operation when i;easared by the specified quadratic performance index.

It was found that by manipulatingg cost coefficients in the performance

index, one could cause the control to 'rc s nd in a predictable manner.

As the cost of control was decreased in comparison to the cost

of state variable deviation, control action became more a nd more

orenouncoe until a ratio of th- se relati'. costs was r.ach-d at which

the controls tended to stu~:ate (ratio of about 100 for system lrder

consideration).

The. General Int-erface for Process Systems Instrumentation (GIP1 I)

was designed and built to be used as a .cr -nral research and teaching

tool and has proven its usefilnes thi:ougjl".ut this work.


X iii











CHAPTER I

PROBLEM DEFINITION, LITERATURE DISCUSSION


The possibility of the control of continuous processes in

an optimal fashion has been apparent since the advent of computers

and the development of a substantial body of optimization theory.

However, past efforts have been highly theoretical in nature with

little potential for implementation in practice. The work reported

here had as its goal the development of a practical scheme of optimiza-

tion for the continuous operation of a distillation column and the

experimental investigation of its effectiveness in on-line operation.

The work described in this dissertation consists of three

parts: (a) the design and construction of an analog-digital inter-

face for linking together a process and a digital computer, (b)

the writing .nd i:plerr. station oE a software package for general

usage of the analog-.digital interface, and (c) the investigation of

real-time optimal computer control of a binary distillation colur.n.

Optimal. compuzr control is .i,.pliien-ted by solving a stage-

wise dynamic programming problem of four state variables and two

.coUL~.)l variables off-line on an IEi 5t60/65 comipuzro. This produces

a set ou ste..i y-Lstate matrices which, when multiplied by i the state. of

the system, the current controls, and the present .nd future upsets,

gives the optimal. control law a:t erSh st age. Tiis con-trol law is

then played out to minipi'late the s.:t---yoints o5 i the control variables.

This type of control. is .thus sai' to .u uperviscry control.

Sit-gewi'se d;:iyniic progra;ming w.s chosen as the solution












algorithm for several reasons: (a) it allows the system model to be

written in discrete form, (b) stages in time can represent dynamic

programming stages, (c) it provides both feedback and feedforward

capability in the optimal control law, and (d) it permits an off-

line solution of the problem, giving a steady-state control law which

is fast and easy to implement zomputationally in the on-line control

at each stage.

The state variables in this system were top and bottom composi-

tions and flow rates. The control variables were steam and reflux

rates, and the load vari ables were fe d rate and composition.

By definition-. -tai-e vari hblas -re things vnrinhblAs which

describe the state of the system control variables are those variables

which when manipulated change the state of the system, and load

variables are those variables -,,hich change the state of the system

but which one has no control over. One must realize, however, that in

an experimental research proj-ct, such as this one, load variables are

controlled in a predetermined manner to follow some a prior pattern

designed by the researcher.

An atmospheric-biarcy distill ti .- column with a oartilal

reboiler and -ot: cu -mens:c has 5i.x process dC pajc s of freedio. In

C 3S 0 J U.. I 1. .l C aiLe, iL C O 1.1iOC L i 'X atlu

feed temperature. .. er. the six d-egree uf f.E'edocn. T'ro of these

variables, reflo'%_ -tad fee-d tePrperatui. es, .were cocsi'dc.-ed to be

variables which did nmt cha.nge. F':d tc '..perac-.rc ',as s iply assumed to

b) constant at the a: mient laboratory ltmpernatLuret nd reflux te:ipf'atire












was controlled at a constant value by manipulating cooling water to

the condenser.

A quadratic performance index was minimized in the optimization

problem. The quadratic form was used to represent a deviation in

operating costs of the distillation column about a particular steady-

state operating point for upsets in load variables. Chapter 3

discusses in more detail the performance index and the associated cost

coefficients. Since the performance index represents a deviation in

operating costs about a particular steady-state operating point, it

was, therefore, convenient to represent all variables as deviations from

their steady-rtate values throughout th8s nwtork,

The literature contains some very interesting articles on

optimization of processes; however, little information is available on

the optimal computer control of distillation columns. Some of this

literature is discussed below.

Zahradnik et al;. (1) considered the c1ti-,cl control of a

distillation column using conventional calculus, the calculus of

variations, the maximum principle, and dynamic programming for a two-

control-va'r-i.b.le, one-.state- -,,riable prob!enm. Their control variables

wcre s.eam rate and top product rate, wi'- top product compo-iti.ion

b.ing tl.i' Lstate .':i-ib T. iod vas- aia e in cnhcir rsytcn. was

feed composition.

They used a 1 ineeri :led modell for th ate tvariablei a

function of load and control vriab.les. One model they usod was steady-

state in nature and assumed instantaneousus column d'arics, while












another model included dynamics in the form of first order lags, similar

to the model used in this project. Because their control law was not

a function of future upsets, it was only feedback in nature giving

only an optimal feedback control. (Compare this to the feedback-

feedforward control law developed in this work.)

They made no comparison to either the uncontrolled case or the

conventional control case as far as control or performance index

values were concerned. They solved the optimal problem of maximum

profit using an integral dynamic programming algorithm as opposed to

the method of stagewise solution presented in this work. No actual

on-line control was performed in their problem and all computations were

performed (column simulation) on a digital computer. Another point of

interest was their so-called steady-state optimization which yielded an

optimal control law giving a new optimal steady-state for a load

variable upset.

Paradis and Perlmutter (2) deviate from the usual choice of

performance indi:2x and choose instead one which is a quadratic function

of the statue variables, and then solve the problem of minimizing the

time derivative of this performance index. The resulting control law

is bang-bang in nature. This form of perfer~;ance ;idex was chosen for

ccmputtional expediCency and, cC.3syquantly, doeos rt naccssarily -yield

useful operating cost iuforrmetion. TLeir coni:ol law moves their

system from somn i:i.[ -. stat-e to some li'nal state -!.iile minimizing the

performance index ;a:d is, ca-:equen lly, a prede.tecLined control law

based solely on the given systein ..ndel for a given upset. It is rot

corrective in either the feedback or EeedfoL.iard sense and 7ould not be











applicable to on-line transient control for load variable upsets. No

comparison is made to any form of conventional control which might in

reality cause at least as good a system performance as their optimal

control. No on-line control was tested.

Rafal and Stevens (3) solved the problem of a quadratic

performance index and linearized system dynamics by using a suitable

quadratic programming algorithm. They, thereby, generated a step-by-

step optimal control which, of course, was not necessarily optimal in

the overall sense. Their control law was feedback in nature and was

updated at each stage of optimization. Here again, no actual on-line

control was accomplini ed nd nnly the lineari7.d system dVynmics were

available to represent the real system. A comparison was made to the

uncontrolled response of the lineari'ed system but no conventional

control was tested.

The state variables in their system were tray compositions in

a distillation column; the control variables were feed rate, reflux

ratio, and reboiler heat load, and the load variables were feed composi-

tion and temperature. No values of the performance index were given for

comparison.

Brosi low and rlaidley (L) jIokcd ;t th:e problem of optim.a feed-

back coiLrol of a fiCLeeu-LLay rectifying column. LThy used a quadratic

performance index with negligible cost on controls ard solved the

problem using a modified matrix Riccati equation for the control law.

Note that this gives only feedback control since this form of control

law is a function of the present state of the systci. The control law

must be a function of tha future upsets to actually have a feedforward











effect.

The load variable used by Brosilow was vapor feed to the bottom

of the column. The top product composition, as state variable, was

controlled by adjusting reflux rate as the control variable. This

then represents a one-state-variable, one-control-variable problem.

Their model was a linear model which considered the effect of each

tray and, in fact, they considered the temperature on each tray to be

representative of the composition on that tray. They compared their

results to the uncontrolled response of the column in the form of the

behavior of the top product composition and plate temperatures.

To show bhat hie rePult was truly optiin-mal they should have

compared performance index values for their optimal control experiments

with the performance index values generated in the uncontrolled case.

In fact, since their control law is feedback in nature,the question

arises as to whether their system would actually generate better results

than a similar form of conventional control.

Lapidus (5) has investigated analytically the general case of

optimal control of linear systems with quadratic performance criteria.

His problem treats Lte optimal control of a process between two s;'cady-

st:atc conditi-Lns, or, in effect, he solves an initial-final value problem.

lie sihow how u .n- uijtaid. the '.iimpldr.-v ita optimal control law and

continuous optimal control .lw and applies these, on a digital computer,

to a six-plate absorption column.

Lapidus discusses the fedback--feedforward prop-erties of optimal

control. His results indicate that for the quadcatic perform. dance











index, the magnitude of control is dependent on the cost coefficients

in the performance index. Ile also points out that by manipulating

these cost coefficients, one can obtain almost any optimal control

pattern one might desire for the sampled-data control case.

The performance index considered in this work is a function of

the state of the system and the control variables; consequently, cost

coefficients for the state and control variables both play important

parts in the control law.

Four cases of cost coefficients were investigated for the optimal

control problem: (a) large state variable costs relative to control

variable csts, (b) 1gare controll vari sb cocts relative to state

variable costs, (c) relatively equal costs for state and control

variables, and (d) large state variable costs on two of the state

variables with low control variable costs.

The performance index values for optim:al control were compared

to those for conventional control of the process 'or an upset pattern

in load variables for each of the various cost coefficient cases. Guide-

lines for implemerrt.ting and using optimal control for each of the four

cost cacss are give and tb~h effect of cost coefficierts zom control Is

shown.

Pi-e (6) discusses the aspects of using dynamic programming and

further points out that two cases can be treated: (a) the case of

determiniistic systems where all upsets over all time are known, and

(b) the case of stochastic systems where cn.y the statistical properties

of upsets are known or c-n be estate d. !n this aithor's work the











deterministic case was treated and,as was mentioned earlier, full

control over load variables was exercised to give the previously

determined upset pattern.

In order to solve the dynamic programming problem a system

model was required. This :model is presented in Chapter III.

It was mentioned earlier that optimal control would be compared

to conventional control. Conventional control in this case means dual

composition control, i.e., control of both top and bottom product

compositions, simul taneousliy. Chapter II gives a complete discussion

of the column and its control schemes. Chapter II also discu:ses the

analoq-digital interface us.-d to connect thap 11(tl]ti on Column to

the IBM 360/65 computer as \:ell as the associated software.











CHAPTER II

HARDWARE-SOFTWARE SYSTEM, DISTILLATION COLUMN,
AND COMPUTER LINKUP


As was stated in the first chapter, there were three definite

aims to this problem. Tw.o of those, the hardware and software that

make up the General Interface for Process Systems Instrumentation

(GIPSI) and the distillation column will be discussed in this chapter

and the third in the following chapters.


A. General Interface for Process Systems
Instrumental ion


GIPSI in its present design state can be used for digital

input ond cutput and two levels of analog to digital conversion. In

the present state, GIPSI can be used for on-line, real-time direct

digital control, supervisory control, or .data logging. Several pro-

ceaies or individuals or both can usehelhe hardwSare system simu!l:'eou.sly.

The d.i Ji:al c:4ip. rer used vwith the interface is an IEM 360/55.

This gives the capability of extremely fast computation time as we31

as large core availability. Data tracs:fer in ei th-r direction occurs

over d. l .ied tel .nphne li r. :Te u20 system -op-re ;3a o a tjiae-

Zihacred basis and a priority system,. The time--shai ng s1-yst.em allows

the uJse of one ccmpipLting Fprtition of the 360's core for long periods

of t.i with little actual comi;pitation ti.me occurri.Lg. The 360 has a

F'-a ratioT,. wait tate cap.biliLy uht allow t-he C. ?U (central

proc-essin u.nit) to be a.-a':ne thait the GI';S is in use but is not in a

comiputati orl ..t-. While in this -.ia-' a st..te, te 360 processes











other programming. When the GIPSI signals the 360 that computation is

necessary, the 360 stops whatever work is in process and returns to the

GIPSI programming. When finished with the GIPSI, the previous program-

ming is picked up again. Since the input-output time of the 360 is

very fast, small CPU times are logged and computational costs are small

provided control computations, material and energy balances, kinetic

calculations, or in general any computation done with GIPSI software are

not extensive. For real-time control and data logging, the CPU time

is proportional to the number of stages, i.e., the frequency of input-

output operations or sample time. Indications are that about 0.5 minutes

of CPU time would be logged per hour of oprat-ion on GIPSI.

The basic operating equipment of GIPSI is an IBM 1070 Analog-

Digital Conversion system. The central terminal unit is a 1071 unit

which handles all multiplexing, analog conversion, and input-output

communication with the 360 computer. The actual multiplexer points

(terminal connections for digital inputs-outputs and analog input signals)

reside in two 1072 multiplexer units giving a total availability of

100 points. Twenty of the points are allocated to analog input signals

of -1 to +5 vdc. Tventy of the points are allocated to analog inputs

of -10 to +50 millivolt de. Thirty points are allocated to digital

iuLpu and 30 points to digital output.

Another part of the interface is an IBM 1073 Model-3 digital

pulse converter. This converter takes a digital n-umber generated by

user software and converts it to tlhe equivalent number of up-down pulses

which can be used to drive any device which uses up-down pulses. This











device is used as the feedback from the I~~ 360 and drives, in this

work, pulse-set-point controllers which control the control variables

(steam rate and reflux rate) for the optimal control problem.

An IBM 1075 decimal display completes the IBM hardware in GIPSI.

This device is used to display error codes and variable values. In

this work the 1075 was used to display the stage number so changes in

load variables could be made easily on tire.

When a user program is entered into the central processing

unit of the IBM 360 computer, the computer becomes aware of GIPSI

through a process alert bus located in the 1070 and attached to the
process alert ch-.anc- s i -1P -. '"-en t.h-. l is


activated by a contact closure, the computer senses this closure and

activates the software to find out hat caused the closure. The devices

which handle thece contact closures are process alerts. The contacts

on the process alert bus can be closed Mraaislly for random operation,

by external devices such as pressure sGitches, or with an automatic

polling device for accurate periodic operation. GIPSI has available on

one of the six process alert ch.nnela a polling device called a hardware

pollcr. This policr cycles on a periodic basis and closes the process

alert contacts every XXX secouds. The physical limits on the poller's

cycle time are twvo to 59 seconds. The software can be used to increase

this cycle time to as long as the user pleases. Sinca process alerts

directly activate the GIPSI software sy~ st, any critical devices which

have contact closing capability and represent alarms, pressure sumitches

or any safety device, should connect directly to the process alerts

for mediatee action.










GIPSI has 15 digital input sense pairs which allow the user to

sense the state of toggle switches, sense the state of two independent

digital inputs or sense the state of one-two position process digital

input. These devices can be used to activate computer programs at the

user's desire (manual operation with toggle switches) or at the closure

of any contact closing device. The digital input sense pairs do not

activate the process alert bus and have to be scanned by the software;

consequently, the computer can respond to their commands only when the

scanner program is activated.

GIPSI has 12 digital output pairs operated manually or by the

computer. These digital output pairs give on-off capability in the

normally-open-normally-closed sense for 110 vac or user powered

devices. These devices could be solenoid valves for blending operations,

pumps for liquid movement, fans for air purging, or motors for opening

and closing control valves for direct digital control.

There are ten pulsed output channels for feedback from the

computer to any user device which can be operated by up-down pulses.

The 1073 Model-3 puts out both up and down pulses. The pulsed output

channels along with Lh,1 software package are available to the user to

choose e either the up or the down pulsed output to drive his equipment.

GIPST conta;iis A :sx-d-g decimal injrtpt device for inp-uttin

data to the computer either randomly or periodically (with the automatic

hardware poller) to change values of variables during computer opera-

tion, update constants, cause certain variables to display, or activate

other programming. This demr-n] nput device gives the system almost

unlimited demand function caj:)- A; .lity.











The problem at hand uses seven of the -10 to +50 millivolt dc

analog input points for thermocouple inputs, and 13 of the -1 to +5 vdc

analog input points for other column parameters such as flow rates

and compositions. The 1073 Model-3 digital-pulse converter and two of

the pulsed output channels are used to drive two pulse-set-point

controllers on the control variables steam rate and reflux rate. The

automatic hardware poller was connected across process alert number one

for periodic contact closure of the process alert bus allowing the

software package to operate the system under real-time-supervisory

control.

The k;jlJi housing consists of three standard 19 inch rack panel

cabinets tied together. The system is moveable to any location and

needs only to have a telephone line to the IBM 360 for direct use. ,-

GIPSI is layed-out with simple plugs and switches to f. .i litate easy

operation by an inexperienced student r.erely by plIgging into the

analog inputs, digital inputs, digital outputs, or pulsed output

channels. Figure 2.1 gives th: fron-t panel lay-out for The three-

cabinet system. A detailed docucbimentation package of the hardware

system incljidjng uipm lljnri t r:, quiremiients and wiring diagrams has been

written for the general user (8).


B. ,G-IPSI Software


A general purpose softwa e pack]g was developed so anyone with

a knowledge of Fortran programming could ,:-e t{he GIPSI system. Sub-

routines were written io h'-'ndl process, arts, digital input sense













S IBM 1070 DISPLAY ,
i4


RELALY BANJ K






PROCESS ALERTS


IBM 1073 MODEL 3
(DIGITAL-PULSE
CONVERTER):


IBM 1072
(MULTIPLEXOR UNIT)


DIGITAL INPUT SENSE
PAIRS


DIGITAL OUTPUT




SIPULSEL OUTPUTS









POWER SUPPLY

CABINET 1


IBM 1072


THERMOCOUPLE INPUTS


THERMOCOUPLE TEST
SIGNALS


ANALOG TEST
SIGNALS ;


ANALOG INPUTS


TRANSDUCERS
(PRESSURE-VOLTAGE)


PRESSURE RECORDER







ii


IBM 1071
(CENTRAL TERMINAL
UNIT)


S POWER SUPPLY



CABINET 2 CABINET 3


Figure 2.1. GIPSI: Genaral Interface for Process Systems Instrumentation,
Front Layout.


___ __I~~ ~


I -I----~-llP i.


t '











pairs, digital output pairs, digital-pulse conversion and pulsed

output, the decimal device, and to read and decode the analog inputs

for both voltage ranges. It is required of the user to write his

computational subroutines and one transfer subroutine called GOTO.

Available are a series of subroutines associated with the decimal

input device. These subroutines make up twelve demand functions which

will allow many operations such as data transfer or change, or data

display as well as starting and stopping certain system programs and

user programs. The user, of course, can write many more demand

functions of his own.

An tnsive ar'oi' code systiii ws LvejL itVL U thie sLfi ware to

announce when errors occur either in data transfer or if the user makes

an error while using a demand function or if the software system commits

or develops a problem. Three levels of errors are considered: (1) a

fatal error occurs whenever some operation causes a problem which the

software cannot overcome, (2) semi-fatal errors occur whenever problems

arise which require user response such as ignoring the error and pro-

ceeding, stopping the machine, or restarting, (3) a non-fatal error

occurs whenever the '.c er rakeh a mistake in using a demand function.

Nu software or user action is t'ken for the last type of error; the

user merely tries again. Fatel errors always stop the computer. A

bell sounds and the display shows che arror code ,whenever one ccurs.

The real-time computer operation functions through a series of

control programs starting with the process alert handler PROCPA.

Briefly, when a process alert occurs P;'.CCPA deter .Ines .which channel











caused the alert and stacks the subroutine associated with that channel

on an execution list according to a priority level. When all the

necessary channels (those causing process alert contact closures) have

been checked and stacked for execution subroutinee STACK does the

stacking), a subroutine called EXECUT begins to execute the members of

the execution stack by calling subroutine GOTO, which has been written

by the user to include any and all channels he is interested in

executing and is composed of a series of Fortran calls to the proper

channels. Any channel called may introduce new members to the execution

list at any priority level. When all programs on the execution list

have been executed, EXECUT returns to PROCPA to wait for the next

process alert contact closure.

Periodic operation is handled by a subroutine called CLOCK

which is placed on the execution list whenever process alert number

one is activated either manually or by the hardware poller or by some

external device and the user has specified the poller option. CLOCK

looks at a list of subroutines, called the delay list, and determines

whether or not it is time to execute any of the subroutines on the list.

If a subroutine is scheduled to run, CLOCK calls STACK and places the

subroutine on the execution list and deletes it from the delay list.

A subroutine can only get back on the delay list by calling subroutine

DELAY. This subroutine checks the existing delay list and if the

calling subroutine is not already on the list, DELAY puts it on the list.

The CLOCK-DELAY combination givesthe user the time-delay capability of

running programs at time intervals longer than the stage length of











the hardware poller. CLOCK works by checking a delay time parameter,

specified by the user against the computer clock and determines if it

is time to run that program.

Figure 2.2 gives a simplified flow diagram of the execution

and delay phases of the software system. A thorough documentation

package of the software has been written for the general user (9).


C. Distillation Column: Design and Control


The distillation system used in this work is typical of pilot-

plant equipment. The basic system consists of a 10-inch, 12-plate-

bubble cap column with a 30-gallon reboiler. The rilu mn has ins.trumn.ts

to measure the variables listed in Table 2.1.


TABLE 2.1

COLuI4N VARILR,'iES READ BY COfP.TTER


V;r iabl e

Feed Flow Rate
Feed Comnposition
Top Product Flow Rate
Top Composition
Bottom Product Flow Rate
Bot i.toa Composition
Steai ,'low Rate
Reflux Flo'w Rate
Cooling V'ater Flow Rate
Cooling Water Tc :imprature, IN
Cooling Tai.ter T'f;,p:eLrature, OUT
Top Colutm n Vapor Temyp:r.ture
Feed Temperature
Steam Temperature
ReboilJer Temcper-: nature
Reflux T'ro :p- nature


Ori jinal Signal Form

Pneumatic
Electric
Pneumatic
Electric
Pneumatic


PeuiuCa t:lc
Electric
Pneuma Lic
P .eumi aLic
P, ieumriatic
Electric
Electric

Electric

Electric
CL t :-1















STACK CLOCK





I -EXECUTABLE
PROCPA EXECUT GOTO -----' PROGRAMS




DELAY




Figure 2.2. Execution and Delay Phases of Software Package.










All flow rates, with the exception of steam, are measured with

Fischer-Porter transmitting rotameters whose output signals are

standard 3 to 15 psi pneumatic signals. Steam flow rate is measured by

an orifice meter whose output signal is also a 3 to 15 psi pneumatic

signal. All temperatures are measured with copper-Constantan thermo-

couples, each of which is plugged directly into a -10 to +50 nmvdc input

point on the GIPSI hardware system.

The pneumatic signals generated by the flow measuring devices

are transduced by Taylor pressure-voltage transducers from 3 to 15 psi

pneumatic signals to 0 to 5 vdc signals for use with GIPSI hardware.

Table 2.2 shows the maximum operating conditions for column

variables.


TABLE 2.2

tM/LIMPI COLUMN VARIABLE OPE_"_ATING CAPABILITY

Variable Main um Cp ab ility

Feed Flow Rate 80 gaIl/hr spgr 0.75
Top Product Flew Rate 50 gal/hr spgr 0.75
Bottom Peoduct Flow "ate 50 -al/hr upgr 0.8
Reflux: Flow, Rate 00 gal/hr spgr 0.8
Cool-'...g Wate:r Rate 30 gal/min spgr 1.0
Stcan Rate 250 lbs/hr



5e aerialss of d.sLtillauiun fur thi: work are methanol and

tertiary butyl alcohol chc; a because of th' -v-ail]abil.ity of therrmo-

dynamic ard effiicncy data.

To per foj:I c:i-li'ine da1a l;gg n 1 and control of the system r, it

is necessary to ha.7ve cutii ~.cus~ .i s for ai1 Ce .arables i onitored











by the computer. The pneumatic signals from the flow meters as well

as the electric signals from the thermocouples are continuous analog

signals and present no problems. Since the system uses a binary

mixture and since the refractive index of the binary is a function of

composition, Electron Machine Corporation refractive index meters were

used to generate the necessary analog signals proportional to composi-

tion, 0 to 5 vdc for computer purposes and 4 to 20 milliamp dc for driv-

ing Fisher-Governor amp-pressure transducers. Three such meters were

used, one on each composition stream.

The distillation ccluItn --a.s designed to be operated under

several different itypv n F rci-trol: () no control l herTe 'team rate,

reflux rate, fed rate, ai-.d feed composition are controlled at constant

values and Lop composition and flow a'n l-ottom composition and flow are

allowed to vary; (b) -du-l c.opiiosition control, where the top composition

is controlled by manipulating reflux rate and the bottom composition is

controlled by manipulating steam rate; (c) co'r puter control, where

configuration (a) above is followed, .and the cutroller set-points on

steam and reflux are manipulated by the ; cpu'.r. The computer control

by manipulation of controller set-points is known as au.rvii. o:y control.

Figure 2.3 slhnus a typical ccTnp!.~?r control. loop. ihe three 'ypCes of

control da:-r i:stiti ted by o ptJin-ug and clTsinrg the a.pprop-'Jarie hand

valves to supply input variables to the various coitrollers. The

computer-operated controllers are Fischer-Porter pulse--:-et-pojnt

controllers.

The feed rate and composition control loops are shown in























Ref ux Lirzo


CC

S-i ... .
S* ........I C C
ITR

H CV Column
L.I cv


Reflux Flow-Top Composition Control


Legend

CM Composition Monitor
CC Composition Controller
CV Control Valve
FC Flow Controller
TR Transmitting Rotameter
Liquid Flow
*- Air Line


Figure 2.3. Typical Computer Control Loop Showing
Alternate Configuration.











Figure 2.4. Because feed rate and composition are load variables and

need to be manipulated to specified values, Fischer-Porter pneumatic-

set-point controllers are used to enable one to generate set-point

changes in the form of pneumatic signals from some external device

such as an analog computer or optical line follower, etc.

Figure 2.5 shows the overall column control scheme, while

Figure 2.6 shows the column-computer hookup through the GIPSI system.

Computer progra;is were written to read the column variables

in Table 2.1 every ?2 seconds and institute control if one was

operating under the computer control scheme. Appendix B gives a

listing of the user software. The 72-second sample and control time

was arrived at as stage length by experience.














--

SCV C I



SoI Column


CV c

f,.- L-- .~


Legend


Composition Controller
Composition Monitor
Control Valve
Flow Controller
Current-Pressure Transducer


P Pump
T Tank
TR Transmitting Rotameter
- Air Line

Liquid Flow


Figure 2.4. Control Scheme, Feed System.


CC
CM
CV
FC
CPT
































Legend


C Controller

CM Composition Monitor

CV Control Valve

DP Differential Pressure Transducer

LC Lever Controller

0 Orifice

TR Transmitting Rotameter

T Steam Trap

-.-*- Air Lines

Flow Lines


Figure 2.5. Total Column Control Scheme.

















Drain
Cooling Water


Cooling Water


Product
Stream


Feed
Tanks


Column


Vapor Line


-~I


Bottom
Stream


Vapor


---~7---


--t~---










.- -- .Cooling Water
CONDENSER




Reflux Product


Feed


FEED COMPOSITION

FEED TE.;P

REFLUX TEMP

-.STEAM- TEMP. .-... -.....- ..---.,
BOTTOM TEMP

TOP COLUMN TEMP

COOLING WATER IN TEMP

COOLING WATER OUT TEMP

STEA M RATE

PRODUCT RATE

REFLUX RATE

,,EEED ..RATF ... ...... ... __

.BOTIONMATEF._
PRODUCT COMPOSITION

BOTTOM COMPOSITION

COOLING WATER RATE


GIPSI


--- REBOILER


Steam .i



B ottom


Figure 2.6. Computer--Column Connections.


COLUMN


COMPUTER

- --=










CHAPTER III

THE OPTIMAL CONTROL PROBLEM


As was stated in Chapter I, the optimal control law is derived

by the stagewise dynamic programming algorithm. A general statement

of the problem is (7)

N
Min r.(X(j),U(j)) (3.1.a)
U j=0

subject to


X(j-l) = G(X(j),U(j),F(j)) (3.l.b)


where stages represent time intervals. In implementing this general

mathematical statement, several tasks arise: (a) find a suitable

performance index for the system, (b) derive a suitable set of

constraints, and (c) obtain the optimal control law.


A. Derivation of the Performance Index


A general quadratic performance index was proposed with the

following form:


TAX + UTBU + 2CTX + 2DTU (3.2)


where A is a matrix of cost coefficients relating costs due to

quadratic effects in state variables, B is a matrix of cost coefficients


T
relating costs due to quadratic effects in control variables, and CT

and D respectively, are vectors of cost coefficients for linear

effects in state and control variables.











In order to maintain control of the system about the steady

state operating point for all variables such that the unconstrained

dynamic programming problem would always give a bounded solution, the

linear terms in equation (3.2) were made zero giving the performance

index a strictly quadratic nature, as


SAX + U BU (3.3)


The state variables X., i=1,4, the control variables U.,

i=1,2, and the load variables F., i=1,2, represent deviations from

steady state, or


X = XD XD = AXD
1 ss
X2 = XB XB = AXB
2 ss

X = D D = AD
3 ss
X = B B = AB
4 ss (3.4)
U = S S = AS
1 ss
U = R R = AR
2 ss
F = F F = AF
1 ss
F2 = XF XF = AXF
2 ss


Therefore, the performance index represents a deviation from some

constant operating cost level for load variables as well as control and

state variables. Only diagonal terms are considered in the A and B

matrices to emphasize the individual variable deviations from steady

state without considering interaction between variables.

Various cases of cost coefficients (aii, i=1,4, b.., i=1,2)













are possible but only four cases were considered: (a) relatively large

control variable costs, (b) relatively equal costs for both state and

control variables, (c) relatively large state variable costs, and (d)

large state variable costs on AXD and AXB and small control costs.

(These four cases are referred to as high costs, equal costs, low

costs, and extra low costs.) Equality in this instance implies that

if a one-unit change in the steam rate adds 1.5 units to the performance

index, the corresponding effect caused by that change in each state

variable will add one unit to the performance index.

Table 3.1 gives the values of a.. and b.. for the four cost
11 11

cases considered.



S.BJTE 3.1

PNFORN UCE 1\DE7 COk COEFFICIENTS,





a. = a 4- 1 40.0 b = 15.0
J-1

1, = a 1.0 h 10.0
a,! '22

-- j ?. 0, i f j




,,b,, 1.5
.L .L L4.. J1-.
a 1.0 b2 1.0
4 4 22

a.. = *, ; j b.. = 1: i 1 j
3 31
ij C' i











TABLE 3.1 (Continued)


Low Cost Case

all = a22 = 144440.0 bll = 0.15

a33 = a44 = 1.0 b22 = 0.10

a. = 0, i j b.. = 0, i # j


Extra Low Cost Case

all = a22 = 1444400.0 bl = 0.15

a33 = a4 = 1.0 b22 = 0.10

a.. 0, i j b.. =0, i i j




E. Derivation of a Suitable Constraint Set

Since equation (3.1.b) represents the stage transformations in

the dynamic problem, and since the stages are periods in time, it is

obvious that the stage transformations represent the relationships

between the state variables an the output of a stage and the input to

the stage, the controls (decision variabl.-s) at the stage, and load

variables at the stage. These rclaticiahips are, in fact, a model for

the distil ;tion c olun.

There are several .ays to dertve a i.modll for the system which

satisfies ithe oeeds of the constraint set given by equation (3.1.b).

The imdel itself mrnus he dcscribable in a difference form allowing the

new state of the syste;,- to be written s; a function of the old state

of the system, the current and p-.vios. c controls and the current and











previous upsets. For computational efficiency it is desirable to

restrict the model to a first order difference form.

Two forms of a linear first order model were considered:

(a) a first order lag, and (b) a proportional response plus a first

order lag. Past experience of the column response to changes in load

and control variables led to the selection of the proportional response

plus a first order lag as the column model.

After the model had been formulated, as shown below, it was

decided to evaluate the constants in the -odel experimentally. Small

step changes in load and control variables were nade on the column

itself to determine the nu-rerLcar values of th:se constants.

Appendix A gives the response curves for t.e upsets made in load and

control variables. The magnitudes of these ,p-ets were such as to try

to stay -.'ithin the region of liner column response.

One can see from th: response curves in Appendix A that a second

order model would give a better fit to the data. 'This fact as gi-ven

coinsiEdJratioc Lut it as als-c seen that the second order model would

yield a set of difference equations vhich would significantly increase

their d(ificult" y of a h-.nd solution of the d~oilic Cro.n-rammin r porblemr.

The gererl fori. of the transfer function for the proporrional

response plus first order lag ;model, as given in Lsplacian transform

notation for an input variable, is



X.(s) 4.
.... (3.5)
(*p *1 s + 1
1 s -
]1











where i = AXD, AXB, AD, AB, state variables, and

j = AS, AR, AF, AXF, load and control variables.

The proportional response is seen only for two state variables,

top and bottom product flow rates, and then only for changes in steam

rate and reflux rate; therefore, the constants K in equation (3.5)
2j
will be zero for all cases except those just mentioned.

The constant K in equation (3.5) gives the change of state

variable i per unit change of variable j or

X X
Ki Xiss2 issl .
= (3.6)
Kj Ijss Ij
jss2 jssl

where ssi is the initial starting steady state and ss2 is the final

steady state attained in the state variables for some upset in variable

I..
J
The constant K represents the step in stata variable i

caused by a unit change in variable j or


X. X
I -Lstep issl
2j 1 I
-jss2 jssl


should rci:cmber that X. and I. represent deviations about
1 j
the '--it.. .l c.on<;tio. acnd, c.iCuqueiiLly, in L-he derivation no constants

appear. d- voting the Laiti'l conditions.

Consider, fr .xajuple, tbe top product flow rate. The Lheorem

of supecC 'i tion stiles h:at for a i~Iear system the responses of a

variable to other vri..'ables sha-gcd singly may be added together to

give the total rcsp;'se of that variable i-.n all the independent

variables are ch.:;ng(d si.-'ii.n.ouy. I!Thu from this thnocr.m, and












from equation (3.5), one would obtain the total response for the top

flow rate as

AD
K AF(s)
AD(s) = X (s) A + K AF(s) +
3 AD 2AF
TAF s + 1
AF

AD AD (s)
K AXF(s) K AR(s)
XF + K D AXF(s) + +
AD + K21XF-- AD
TAXF s + 1 TAR s + 1


AD S(s)
K AR(s) + S + K AS(s) (3.8)
2AR AD 2AS
AS s + 1
AS

Similar eq-uations are obtained f-r the other st~L-- e .Ia--iar es

The time constants T. were obtained from the response curves

shown in Appendix A by finding the time at which 63.2 percent of the

final value of the state variable was obtained. Table 3.2 gives the

values of these time constants. Table 3.3 gives the values of the
i i
constants Klj and K2j for equations of the form of equation (3.8).
Ij 2j
In order to maintain the column material balance, the bottom

product fl%,w rate was~ obtained by difference, i.e.,


AB L' AD (3.9)


This equation is effected by taking ;,.:e n-gative of the constants K !
i
and K,. for AD a.id ,,aking Ki corresponding to the bottom flow rate

for feed rate change equal Lo one. Tab.i 3,3 illustrates the effect of

equation (3.9) on the ii.e.ir model.

The normal method for solJving equ-:tion (3.8) is to multiply













TABLE 3.2
i
TIME CONSTANTS T. FOR EQUATION (3.8)

FOR ALL STATE VARIABLES


Load and Control Variables
F XF R S

0.632 0.319 0.320 0.486
0.577 0.638 0.861 0.556
0.553 0.375 0.340 0.205
AB = AF AD


Time constants are in hours.



TABLE 3.3
i i
CONSTANTS K AND K2 FOR EQUATION (3.8)
FOR ALL STATE VARIABLES
FOR ALL STATE VARIABLES


State
Variables (i)
XD
XB
D
B


K


Load and Control Variables
F XF R
0.00276 1.043 0.0023
0.00156 0.403 0.00261
-0.947 -176.1 0
0.947 176.1 0


(j)
S
-0.0035
-0.00315
0.15
-0.15


State
Variables (i)
XD
XB
D
B


Load
F
0
0
0
1.0


2j
and Control
XF
0
0
0
0


Variables
R
0
0
-1.739
1.739


State
Variables

XD)
XB
D
B


(j)
S
0
0
3.42
-3.42











through by a common denominator. When this is done, a fourth order

transfer function is obtained which when inverted yields a linear

fourth order differential equation. It was pointed out earlier that

a first order differential equation was desired to give a first order

difference equation to meet the specifications of the constraint set.
i
This problem was overcome by averaging the time constants T. to obtain

a single time constant T.. When this is done, one can factor T.s + 1

in each equation and obtain the desired first order differential

equation. Equation (3.10) gives the new time constants


T = ( TXD)/4 = 0.438 hours
j J


TAXB = 0.658 hours (3.10)

TlD = 0.446 hours


Since all upset variables carry the same weight for column behavior, a

simple arithmetic average was used.

When equation (3.8) is combined with equation (3.10), the

general form becomes

K I.(s)
j T. s + j1 2j j


Multiplying through by T. s+1 gives


(T.s + 1)X.(s) = 1.I(s) + (Ts + 1) Ki T (s) (3.12)
J j

Taking the inverse of equation (3.12) to obtain the differential

equation form gives










dX.
T. + X
i dt i


= K.I(t) +
13j


i
S 2j
J


dI.(t)
dt


+ Kji I(t)
j


(3.13)


Approximating the derivative by a first order backward difference

equation gives


dX.
1
dt


X.(m) X.(m 1)
At


(3.14)


where t for this problem is the stage length. Substituting equations

(3.14) and (3.13) gives


Ti(Xi(m) X.(m 1))
1 1 3


+ X.(m) = l (m)
1 j3


+ ~ I .(m) I.( )) + K .2jIj
j j


At
Collecting terms and multiplying through by + At
T. + Att


gives


X.(m 1) +
J


At
T + At


i
(K j
13


i K Ki
+ K + i 2j
2j T + At
1


Jj


SK.I.(m 1)
. + At 2j
1


i i
(K + K ) +
lj 2j


(3.15)


X (m) =
1


T .
1 +
t. + At
1


Define


(3.16)


At
. + At
1


i
K2
2j


1


(3.17)











i = i
2K i
2j t. + At 2j
1


(3.18)


Substitute equations (3.17) and (3.18) into (3.16) to obtain


X.(m) = X.(m 1) + K I.(m) + KI(m- 1)
S. + At 1 2j
1 J ]


(3.19)


Table 3.4 gives the computed values of T./T. + At and the
1 1
-i i
constants Kj and K2j for At = 0.02 hours or 50 stages per hour.



TABLE 3.4
-i -i
COMPUTED VALUES OF T./T. + At, Kl. AND K
1 1 lj 2j


i

Ti/Ti + At
-i
K1j










-i
K2
2j


j
i
AXI)



AD
AXB







j





AXB
i



AD

AXB

AB


AXD

0.9564

AF

0.0001202

0.000046

-0.0407

1.0407


AF


AXB

0.9705

AXF

0.0455

0.01189

-7.57

7.57


AXF


AD

0.957

AS

-0.0001525

-0.0000929

3.426

-3.426


AS


0

0

-3.278

3.278


AB

-0.957

AR

0.0001001

0.000077

-1.739

1.739


AR


0

0

1.662

-1.662










Note that I. represents both load and control variables.

Breaking I. into F. for load variables and into U. for control variables

and writing equation (3.19) in matrix form gives


X(m) = EX(m 1) + GF(m) + HF(m 1) + JU(m) + KU(m 1)

(3.20)

where U = (AS,AR) F= (AF,AXF), and



AXD
E0 0 0
STXD + At

0 AX
T AXB

0 0 rA 0
AD
0 0 0
TAD + At


0 0 0 AD
TAD + At





S-AXD -AXD
= lAF 1AXF

-AXB -AXB
1 AF 1AXF

-AD -AD
K K
1AF 1AXF

-AB -AB
K K
A1F 1AXF


S= [0]













J


-AXD
IAS

-AXB
KAS
1AS

-AD
K1AS


-AB
KAS




0

0
-AD
K2AS


-AD
-K
2AS


-AXD
KAR


-AXB
IAR

-AD
K1AR


-AB
KAR


0

0

-AD
K2AR


AD
-2A
2aR


Equation (3.20) gives the linear model and constraint set in

time stage form. The convenient dynamic programming stage form numbers

backwards to that of time, thus, stage m=0 in time represents stage

n=N in dynamic programming notation (7). Consider Figure 3.1


Figure 3.1. Time-Dynamic Programming Stage
Relationships.











One can show that the two stages in Figure 3.1 are equivalent and that

one can obtain the dynamic programming stage notation by merely trans-

posing the m and m-1 on the mth time stage. This can be shown as

follows: On the mth time stage reverse the input and output stream

names aLLn call them, respectively, m and m-l. Then, if m = N-m+1,

m-1 = N-m.

Combining equations (3.3) and (3.20) after switching the

indexes in equation (3.20) gives the dynamic programming problem as


N
Min W X (n)AX(n) + UT(n)BU(n) (3.21.a)
U n=0

subject to



X(n-l) = EX(n) + GF(n-l) + IF(n) + JU(n-l) + KU(n) (3.21.b)



C. Obtaining the Optimal Control Law


The solution to the problem stated above in equation (3.21)

gives the optimal control law. As was stated in Chapter I, this

control law was obtained off-line. When on-line, it combines the

state, control, and load variables to give controller settings at each

stage of a run such that a minimum performance index occurs about the

steady state over that run. Pike (6) points out that this performance

index value, in fact, is a deviation from an initial value and that

this resultant delta value results from using deviations in state and

control variables in the problem definition.











To insure convergence to a solution when the hand solution of

the dynamic programming problem is solved off-line, equation (3.21.a)
N-n
is multiplied by a constant a giving equation (3.21) the following

form:


Min aN (X(n)_X(n) + U(n) U(n)) (3.22.a)
U n=0


subject to


X(n-l) = EX(n) + GF(n--l) + HF(n) + JU(n-l) + KU(n) (3.22.b)


X(N) = Constant (3.22.c)



The normal recursive solution to this problem generates the

hand solution mentioned above which gives the necessary set of optimal

control vectors. Pike (6) shows the derivation of this optimal control

law and gives it form. Pike further points out that in using the

analytic form of the control law vast amounts of computer storage as

well as large computation times are rneccessary to compute the optimal

control law at c-.ch stage.

Eor.pr of this probl-em. it is ct feasible to -nlve thl~e

dynamic program on-luine at asch stage- h;'ever, one can, in this

situation, take advantage of one of the characteristics of the solution

to the infinite stage dylJinic prograr'ming problem. ?emhauser (7)

points out that time stage probl!-rs wre frequently treated as infinite

stage problems. The characteristic of interest is the fact that in

problems with large uuinaers of stages there Iappears a point in the











solution of the problem where the optimal control output at a stage

differs only slightly from that of the next stage. In fact, the

control law has a tendency to approach, asymtotically, a so-called

steady-state or optimal stage invariant solution. The hand solution

of the dynamic programming problem is then solved off-line to compute

the matrix multipliers in the optimal control law and computation is

terminated when the matrix multipliers at one stage differ by a

specified amount from those at the next stage (normal convergence

criteria). Once this steady-state value of the control is determined

(note that this control law is a function of the state of the system

and all future upsets) it is only necessary to supply the present state

of the system, the upset pattern and the mathematics to multiply the

state and upsets by their proper matrices at each stage. The hand

solution of the dynamic programming problem was solved off-line and the

steady-state solution determined for the several cases cf cost

coefficients discussed earlier. Equation (3.23) gives the form of the

optimal control law as it applies to the problem defined in equation

(3.22).

1. 2
u k.1-"-L) := 1 n- ) .n-_.d -,nj + v ,n--..L) U i.n;

n
l V -- l \- .
j l ] -3
j=1


Appendix C gives the multiplier iatri.ces obtained for the cost coefficient

cases considered.

It must be pointed out that it is necessary to assume a large











number of stages or a very long time horizon for the finite stage

problem in order to approximate it by the infinite stage problem to

obtain the steady state solution. This assumption was valid in this

problem since one could assume a time horizon of say 1000 stages, obtain

the optimal control convergence in 100 stages and have 900 stages in

which the optional control law changes negligibly from stage to stage.

If then, the run which one makes is only 200 stages long, one is

assured of operating in the steady-state range.

One must realize that in the above discussion, time stages

run from zero time to the final time, whereas, the dynamic programming

problem runs from stage 1000, equivalent to time zero, to stage one,

equivalent to the final time, thus, convergence is obtained at stage

100 or nearly the final time on the time horizon, and the computer

control run of 200 stages runs from zero time to 200 times the stage

length or at the opposite end of the time spectrum from the conver-

gence of the control law.

Constraints physically exist on the state, control, and load

variables but were not considered in the problem solution because of

the difficulty in dealing with constraints in solving the dynamic

programming problem. This was handled in the following manner: The

on-line control problem occurs about a steady state approximately mid-

range to all variables. The upset patterns were chosen along with

cost coefficients to maintain close proximity to the operating steady

state over the range of upsets to insure linear column response and to

insure that none of the physical constraints would be violated.












CHAPTER IV

EXPERIMENTAL TECHNIQUE


The experimental work covered four distinct areas: (a)

building and debugging the distillation column, the analog-digital

interface, and the general software package for real-time computer

control and data logging, (b) collecting data for the linear distilla-

tion model, (c) obtaining the optimal control law, and (d) making on-

line optimal control runs, composition control runs, and uncontrolled

runs.


A. Building and Debugging the System:
Distillation Column, Analog-Digital Interface
Software Package


After the column, interface and software had been constructed,

a thorough debugging procedure was required to test each phase. The

column feed tanks were loaded and the column brought to steady state.

The control loops on feed rate and composition, reflux rate, steam rate,

and reflux temperature were tuned to give fast, stable response to

step changes in controller setpoints. Reflux and steam rate control-

lers were simple loops to tune since they are independent control

loops. Proportional band settings of 100 percent and integral times

of 0.05 minutes were used on each of the proportional-integral

controllers. These same two control loops were used as the control

variables in the computer control scheme.

Feed rate and feed composition control loops were not

independent and required a somewhat more elaborate search to find











satisfactory controller settings. The two controllers are also

proportional-integral types and the settings obtained were 50 to 200

percent proportional bands and 0.05 and 0.3 minutes integral time for

feed rate and composition, respectively.

The reflux temperature controller was found to have a very

slow response time because of the accumulator at the condenser exit.

The final controller settings arrived at were 100 percent proportional

band and 10 minutes' integral time.

The reflux and steam rate controllers were tested with the

general software package and the analog-digital interface by adjusting

the setpoints externally with the IBM 360/65 computer. Other parts

of the interface and software were tested by a logical procedure.

The composition control scheme on the column was tested by

bringing the column to total reflux and then manually bringing the

column to steady state at which point steam rate manipulation to

control bottom composition and reflux rate manipulation to control top

composition were used.

The two composition control loops (which used the same

controllers used for computer control) were tuned by individual testing

and then by an individual Ziegler-Nichols technique on each loop.

The t.wo me thods. pr.odUCd assantially the sa'e control. .lle settings.

100 percent proportional band and 10 minutes' integral time in both

cases. In all probability, since the two control loops are not

independent of one another, there exists a better set of controller

settings than those obtained for overall operation, but this was not

investigated.











B. Collecting Data for the Linear Model


As was described in Chapter III, the linear model was derived

from data obtained by making step upsets in control and load variables,

one at a time, holding the others constant, and monitoring the state

variables. (This technique is known as the response curve technique.)

Small upsets were made in the control and load variables so column

response would remain linear. Table 4.1 lists the upsets made as

well as the steady-state about which the upsets were made.



TABLE 4.1

CONTROL AND LOAD VARIABLE


Upsets for Linear Model Determination

Feed Rate 25 lbs/hr
Feed Composition + 0.046
Reflux Rate 23 lbs/hr
Steam Rate + 10 lbs/hr

Steady State Operating Conditionts for Upsets
Feed Rate 275 10 lbs/hr
F.ed Composition 0.5 + 0.01
Steam Rate 185 + 3 lbs/hr
Reflux R ite20 + 8 Ibs/hr
Feed T.p era .i.ure 82 5OF
Reflux Temperature 115F












TABLE 4.1 (Continued)


State Variable Values

Top Composition 0.9 + 0.02
Bottom Composition 0.1 + 0.02
Top Flow Rate 130 5 Ibs/hr
Bottom Flow Rate 145 + 5 lbs/hr




The uncontrolled column configuration was used in making these

runs. To perform one of these runs, the distillation column was

brought to steady state. The upset was made and strip chart recorders

were used to track the state variable response until a new steady-state

was reached. Data for the linear model were then obtained from the

strip charts. The time required to bring the column to total reflux

from a cold state was approximately 45 minutes, the time to attain the

initial steady-state was approximately one hour and 15 minutes, and the

time to attain the new steady-state was about one hour and 15 minutes

giving a total run time of approximately three to three and one-half

hours.


C. Obtaining the Optimal Control Law


The hand-obtained solution of the dynamic program was batched

into the IBM 360/65 computer and solved off-line until convergence

(time invariant steady-state solution) was obtained for each of the

four cost coefficient cases listed in Chapter III. The output of each

of these computer runs generated the matrix multipliers for the optimal












control runs. Appendix C gives the multiplier matrices for the four

cost cases as well as the cost coefficients and the number of stages

for convergence.


D. On-Line Computer Control Runs


In order to accomplish the computer control and data logging

runs, it was first necessary to write and debug the subroutines

required to handle these problems. Appendix B lists the software

used to do these runs.

An optimal computer control run was made for each of the four

cost coefficient cases by bringing the column to the initial steady

state and then instituting computer control of the column when

disturbed by the following upset pattern:


TABLE

LOAD VARIABLE


Feed Rate
F-F
ss

0
-20.25

9.53
9.59

8.02
-13.05
0


4.2

UPSET PATTERN


Stage


1-9
10
11-14
15-22
23-24
25-30
31-


Feed Composition
XF-XF
ss

0.046
-0.0332

0.0033
0.0355
-0.011
-0.0206
0


Stage


1
2-6
7-10
11-15
16-20
21-30
31-











Duplication runs were made to test reproducibility in some

cases and composition control and uncontrolled runs were made for the

purpose of comparison. For every run made on the distillation column,

the same initial steady state was used, as well as the same upset

pattern; those given in Tables 4.1 and 4.2 above.

One computer control run was made with no upset pattern in

order to obtain information on noise effects on feedback through the

computer to the control variable setpoints. The information gained

through this run is discussed in Chapter V.













CHAPTER V

RESULTS, CONCLUSIONS, RECOMMENDATIONS


A. Results


The obvious question at this point is: "What was accomplished

and what does it mean?" When one formulates and solves an optimiza-

tion problem, in the final analysis, the most important result is

whether or not the optimal solution is in fact optimal.

The solution of the off-line dynamic programming problem

insures that, for a system identical to the given linear model, the

solution is optimal in terms of the specified performance index.

However, the response of the linear model, as seen in Appendix A,

shows that the model and the column do not always agree for simul-

taneous variable upsets. While individual upsets of magnitudes in

the neighborhood of 5 percent of the operating levels of variables

considered gave linear response of the state variables, simultaneous

upsets of similar magnitude in each of the several variables gave a

linear response different from that predicted by the superposition of

the individual responses.

Because of the difference in model and real column response,

one could expect the action of the optimal control as simulated on

the computer to be somewhat different from the response of the real

system. The results of this research demonstrate this. Because of

the accuracy of the computer and the inherent properties of an

optimization scheme, one knows that optimal control will give better











results than any other type of control in terms of the given

performance index and, in fact, this has been demonstrated (1,2,3,4,5).

The real system does not have the accuracy of the computer nor does

it respond as the computer sees it in Lhe model; therefore, one would

expect to see, and does see limitations on the system.

Figures 5.1, 5.2, 5.3 and 5.4 show the performance index

values for the optimal control, the composition control, and the

uncontrolled column response to the particular upset pattern given in

Table 4.2. The results clearly show that the optimal control scheme

generates lower performance index values than composition control for

all sats of cosiL iufficleint iLvestigated. The results also show

that the optimal control gives lower performance index values than the

uncontrolled case. However, in the case of Figures 5.2 and 5.3, the

uncontrolled column response performance index value is close enough

to the optimal control to require some comment on system noise.

Noise levels about steady state for all the column variables

(state, control, and load) were determined by monitoring the input

signal of each variable to the analog-digital interface while the

system was at steady state. Table 5.1 gives the results of the noise

level tests. The noise on the bottom composition was lowered to

+ .003 by damping the sensitivity of the controller flow out of the

reboiler. This implies that the bottom composition monitor was flow

sensitive which was confirmed by independent experiments.

There are several possible methods of handling the noise

problem. Three of them considered were: (a) do nothing about noise













TABLE 5.1

NOISE LEVELS OF COLUTNN VARIABLES


Variable

Reflux Rate

Steam Rate

Feed Rate

Top Rate
Bottom Rate

Feed Comp.

Top Comp.
Bottom Comp.


Noise Level About Steady State


4.3 lbs/hr

2.9 lbs/hr

1.9 lbs/hr

3.1 lbs/hr
5.0 lbs/hr
.004

.001
.012





















(3)


* (1)
(2)
(3)


Optimal Control
Composition Control
Uncontrolled


0 5 10 15 20 25 30 35 40 45
Stage
Figure 5.1. Performance Index Values for Various Control
Cases for Cost Case One.


Cost



















Cost


(1) Optimal Control
(2) Composition Control

(3) Uncontrolled











r I I I I


104












10













102
0
O


30 35 40 45 50



es for Various Control
o .


5 10 15 20 25

Stage


Figure 5.2. Performance Index Valu
Cases for Cost Case Tw



















Cost


(1)


Optimal Control

Composition Control

Uncontrolled


I

I

-II

il
-I


0 5 10 15 20 25 30 35 40 45 50

Stage


Figure 5.3.


Performance Index Values for
Cases for Cost Case Three.


Various Control

















Cost


(3)
I~
I *"


(1) Optimal Control

(2) Composition Control

(3) Uncontrolled


5 10 15 20 25 30 35 40 45 50

Stage


Figure 5.4.


Performance Index Values for Various Control
Cases for Cost Case Four.


102
0


M












and live with it, (b) place noise filters on the variables, (c)

place noise bands on the variables and control bands on the control

variables.

A noise filter implies a digital filter in the form of a one-

or two-stage exponential filter written into the software. This type

of filter dampens out rapid changes in variable values and has a

smoothing level dependent on the damping coefficient a. Equation (5.1)

illustrates the single stage exponential filter.


X = caX + (l-a)*X (5.1)
new old act


This equation states that the new value of the variable is a times

the old variable value plus 1-a times the actual-present value of the

variable. Limits on a are given as 0 < a < 1. Noise bands imply

that if the present variable value is within the noise level of that

variable, the present variable value should equal the old variable

value or


IF X a < Noise Level
act -


X =X
new old

IFX act > Noise Level
acc

X = X
new act


A control band simply means that if the change in a control variable is

not greater than a specified amount, do not institute any change in

control.












The method chosen here was to place two-stage exponential

filters on the state and load variables with a = 0.7 and no filters or

control bands on the control variables. Optimal control runs were

made for cost cases one, two and four in Table 3.4 with this noise

control scheme. An attempt was made to duplicate the run for extra

low cost of control (case four) but the controls were set to zero by

the computer. A re-evaluation of the noise control was made and it

was decided to dampen the bottoms flow rate, increase a to 0.9 on the

bottoms composition filter, and remove the filtering from the load

variables. Table C.4 shows in P., j=1,25 the extreme effect on the

control. law the load vai~i bd iave cfor the case four cost coefficients.

Since the upsets made in those variables were pulses, the filters did

not allow the load variable values to be read exact at each stage and

it was felt this contributed to the system shut-down.

Another duplication run was made with the new noise control

conditions. In this run the computer also set the controls to zero.

After an investigation of the optimal control law for this cost case,

it was decided that, due to system noise and model inaccuracy, an

upper limit had been reached in the separation of state and control

variable cost coefficients, i.e.,


Limit tatc Variable Costs 1
limit = .--- 100
Control Variable Costs

for this upset pattern and this linear model.

Table 5.2 illustrates the performance index value expected per

stage for each of the four cost cases bout the steady-state operating











point for a no-upset situation using the noise levels given in

Table 5.1.



TABLE 5.2

MAXIL-IM VALUES OF PERFORMANCE IIDEX PER STAGE

Performance Index
Cost Case Value Per Stage

1 366.4
2 86.5
3 58.6
4 246.1



Because expected values of the random noise are less than the

maximum values shown in Table 5.1 and because of filters on state

variables, the per stage value of the performance index which the

computer actually sees for each of the cost cases is less thnn that

given in Table 5.2.

More realistic average values of the actual noise would be

about 190, 50, 40, ]50, respectively, for the four cost cases. Thus,

for a 50-stage run at steady state one would expect 9400, 2500, 2000,

7500, respectively, to be the final performance index value For the

four cost cases. The Eirst and fourth values are large compared to

the middle two because of the extreme cost coefficients in these two

cases.

As was mentioned in Chapter V, one run was made to determine

the effect of noise on the computer feedback control loop. The per-

formance index value generated at 40 stages in this run was 14448.0











units. This run was made for high control costs or cost case one.

The matrix multipliers in the optimal control law for this case are

much smaller for all variables than in cost case three. The steady-

state noise level generated only 7600 units of cost in 40 stages

leaving 6800 units-as that contributed by the feedback through the

computer. Since the feedback effect in the control law is larger

for cost cases two and three, one would expect more noise contribution

than that shown for cost case one.

In this light, one can visualize the closeness of the un-

controlled cost case and the optimal control cost case curves in

Figures 5.2 and 5.3 with somewhat more perspective in that the same

random noise is apparent in all the control runs but only the optimal

control runs have the increased noise effect due to computer feedback.

While Figures 5.1 to 5.4 illustrate the fact that the perform-

ance index for optimal control is smaller than for the other types of

control considered, they do not make clear some equally interesting

results which occur in the control and state variables. Figures 5.5,

5.6, and 5.7 show the control variables for the four cost cases and

composition control. It is apparent that control action becomes

larger as the cost of control decreases. This is to be expected

since these movements contribute less to the performance index.

Control variables for the uncontrolled column response about the upset

pattern are not plotted since they were maintained at constant values.

While sub-saturation in control could be assured only for


State Costs
o o< 100
Control Costs





61




AS
-5
A. Cost Case One
0 5 10 15 20 25 30 35 40 45 50





AS a5 J1
B. Cost Case Tvo _
-10 1 -------- J
0 5 10 15 20 25 30 35 40 45 50





AS -F.

-10 -
C. Cost Case Three
-15 -- --' .: __ ____ .t .,. j
0 5 10 15 20 25 30 35 40 45 50



0 n
-5
AS
-10 -
D. Cost
-15 Case
-20 Er--j ----- I -
0 5 10 15 20 25 30 35 40 45



5 -
E. Composition Control

AS 0


-5
0 5 10 15 20 25 30 35 40 45 50
Stage
Figure 5.5. Control Variable, Steam, for Composition
Control and All Optimal Control Cost Cases.









5 5




AR 0 [
-5 -
A. Cost Case
One
-10 ---- L---_ ____
0 5 10 15 20 25 30 35 40 45 50

5





AR L
-5

B. Cost Case Two
-10 -J.

0 5 10 15 20 25 30 35 40 45 50




5 -
C. Cost Case Three




ARr
_5


-10




0 5 10 15 20 25 30 35 40 45 50

Stage

Figure 5.6. Control Variable, Reflux, for Cost Cases
One, Two, Three.






























0 5 10 15 20 25 30 35 40 45


-20


-30


-40











5


0


0 5 10 15 20 25 30 35 40 45 50
Stage


Figure 5.7.


Control Variable, Reflux, for
Four and Composition Control.


B. Composition Control


-10


-15


Cost Case











no upper limit on control costs was expected, i.e.,



State Costs
0
Control Costs


since it was found that control action diminished for increasing

control variable costs. The lower limit of control action, of course,

is no action at all, which gives, obviously, a stable system, since

the distillation column is completely stable in the uncontrolled case

for this upset pattern.

Lapidus (5) pointed out that one could, by adjusting cost

coefficients in the performance index, obtain whatever form of control

he desired. Of course, Lapidus performed his calculations on a computer

and had not only noiseless variables but the speed and accuracy of the

computer to generate exact results.

Appendix D contains the results, in the form of graphs, of

all the runs made for optimal control, composition control, and the

uncontrolled column case.

Figures 5.5.A, 5.6.A, and D.2 represent cost case one with

high control costs relative to state variable costs. When these

figures are compared with Figure D.7 the absense of movement in the

control variables as well as the similarity of state variable response

leaves little doubt that as control variable costs are increased the

system response tends toward the uncontrolled case.

Based on the above results one would expect that for low

control variable costs and high state variable costs on two state

variables, say top and bottom composition, the optimal control would












respond in a similar fashion to the composition control. When

Figures 5.5.D, 5.7.A, and D.5 (optimal control, cost case four) are

compared with Figures 5.5.E, 5.7.B, and D.6 (composition control) one

sees that, while the optimal control on steam and reflux rates differ

somewhat from those of composition control, the state variable

response under optimal control actually performs better in the sense

of constant top and bottom composition control than does the composi-

tion control. The fact that attempts to duplicate this run in two

additional tries, both of which ran into saturation problems, should

not discredit the results obtained from the run but does emphasize

the effect of noise and model inaccuracy when coupled with the tendency

toward saturation of control action when no cost of control is

considered.

Figure D.4 (cost case three) shows that when state variable

costs are approximately equal tha control does not selectively choose

any one state variable over another in Lrying to maintain minimum

objective function value even when control costs are low and control

action is large.

Figures 5.5.B, 5.6.B and D.3 illustrate the fact that ior the

case of equal costs (cost case two), the optimal control allows both

state and control variables to vary. One must realize that bottoms

Flow is not plotted. Since it was calculated as a difference between

feed flow and top product flow, it can be obtained merely by taking

AF-AD on any set of figures.

The r usme upset pattern was used for all runs. An attempt was











made for every run to implement the upset pattern in the same way,

but the computer did not always see exactly the same pattern.

Figure D.1, while typical of the upset pattern, shows two cases where

the computer saw the pattern differently.

The upset pattern was made by adjusting the set points of the

feed rate and feed composition controllers. If one observes closely

the upset curves one can see the dependency of the two control loops

on one another. In several cases, when the upset pattern was returned

to zero, steady-state load variable values, the computer and the system

saw off-set values for feed composition and flow rate. This was a

result of not being able to set the controllers exactly to their steady-

state values.

Table 5.3 gives a set of control and load variable deviations

based on controller settings 1 percent of the steady-state set point

value.

In certain of the optimal control runs, Figures 5.5.C, 5.5.D,

and 5.6.C, 5.7.A, after the upset pattern was set to zero, the control

variables approached zero deviation but actually never attained zero.

It appears that noise was responsible for this effect and a look at the

optimal control law and the matrix multipliers for cost cases three

and four shows that, in fact, noise could have significant effect on

the control variables even at steady state.

There is a difference in graphs of control variables for

optimal control runs and the other cases given in Figures 5.5, 5.6,

and 5.7. Control variables for optimal control runs are plotted as a














TABLE 5.3

CONTROL AND LOAD VARIABLE DEVIATIONS FOR
1 PERCENT CONTROLLER SET POINT DEVIATIONS


Variable

Steam Rate


Steady-State
Controller Set Point


Variable
Deviation

1.0
0
-1.5


Reflux Rate


Feed Rate



Feed Composition


37.5
36.5
35.5


7.0
0
-7.0

5.0
0
-6.0

-0.01
0
+0.01











sequence of pulses which most closely resembles what the computer

actually saw from stage to stage. Control variables for the conven-

tional and uncontrolled cases are plotted as a sequence of straight

lines since control in both these cases was continuous in nature.

For certain of the cost coefficient cases (cases three and

four), reprcducibility was not possible in the few attempts made.

Figures D.8 and D.9 show the control variable response for two

reproduction runs made for cost case four. Compare those figures to

Figures 5.5.D and 5.7.A.

For cost case two reproducibility was successful, although

nut exact, as seea iu Figures D.12 to D.14. No reproduction runs were

attempted for cost case one.


B. Conclusions


A large part of this research effort is represented by the

finished physical system and the operating computer software. The

time required to obtain each successful run on the average was a

matter of days because of the need to have the equipment running

properly at the same time that the Computer Center was coordinated to

operate with the system. Therefore, the number of successful runs

reported here is rather lir.ited.

Since only one upset pattern w rs observed, and since only a

few cost coefficient cases were examined, certainly no very exact

guidelines can be drawn from the results presented. However, one

can speak in generali.ties by saying that the results appear to show











that optimal control is both feasible and profitable.

With the results at hand one seems to be limited to cost

coefficient cases where


State Variable Costs
< 100
Control Variable Costs


In this work only quadratic cost coefficients for variable

deviations about some steady state were considered. Such a performance

index should be clearly understood. Costs for a quadratic performance

index only imply cost per pound. Lapidus (5) points out that the

quadratic cost coefficients are suitably chosen values to meet such

requirements as accuracy and the amount of control desired. lHe

considers state variable cost coefficients in the quadratic performance

index to be determined in order to specify the criterion on amount of

deviation from steady state one is willing to accept. Similarly, one

chooses control variable cost coefficients based on the amount of

control one is willing to put up with. From this analysis it is

immediately apparent that these cost coefficients are relative only

to one another and consequently any value can be chosen for any one

coefficient as long as all otLars are chosen i.lativc to that one,

with the restriction that the matrices of coefficients be positive

definite and synmmetric.

When sccn in this light, and noting the control obtainable

with cost coefficient separations of 100, it becomes clear that the

restriction

State Variable Cost
Control Variable Cost
Control Variable Cost













is not very binding.

The interesting conclusion to be drawn from the experimental

results is that not only can money be saved by using the optimal

control scheme to control about upsets in load variables, but better

control can be maintained over the state variables than in the con-

ventional control case. Thus when a situation is faced such that an

upset is known to be expected a change to optimal control can be

made for that particular upset, using cost coefficients based on

previous upset responses to give the desired control results, which

will maintain better control over the system than by remaining on

conventional control. There is no doubt that the case of deterministic

upsets places a restriction on the use of optimal control of this

nature.

Before this type of control scheme can be instituted, a

linear model is required and it must be solved by dynamic programming

off-line (6) to obtain the steady-state optimal control law. Certainly

the possibility of solving the problem on-line at every stage ought

to be investigated even in the face of the expense of large core

requirements and excessive computation times.

Even if one were to satisfy the necessary requirements of a

deterministic system and a linear model, one is faced with the problem

of on-line analysis of stream compositions in multicomponent cases.

For systems with extremely large response times, on-line gas or

vapor chromatography is possible as an analyzing technique. One must

be able to sample and control at a rate fast enough to insure system











stability and this single requirement limits considerably the use of

optimal closed-loop control.

In short the general conclusions are:

(a) Results show that cost coefficient separation for this

system should be less than 100.

(b) Results indicate that the optimal control is sensitive not

only to model accuracy but also to system noise.

(c) The optimal control is feasible and actually gives better

control than conventional control for selective cost coefficients.

(d) Optimal control gives selective control by varying cost

coefficients.


C. Recommendations


This research has merely scratched the surface of a large

area of interesting control work made available through the complete

instrumentation of the distillation column in the unit operations

laboratory at the University of Florida and through the building of

the analog-digital interface appropriately named GIPSI (general

interface for process systems instrumentation). Not only should a

continuation of the study of closed-loop optimal computer control be

made but other areas of control such as on-line adaptive and on-line

feedforward control should be investigated.

A continuation of this project should be made with several

goals in mind:

(a) Investigate enough cost coefficient cases to be able to











lay down very explicit guidelines for operation.

(b) Investigate higher order linear and possibly some non-

linear system models to determine their feasibility in the dynamic

program solution.

(c) Investigate many more upset patterns in order to cover a

large range of feasible responses.

(d) Investigate noise more closely and its effect on the

system. Attempt the optimal control with noise bands and control

bands in contrast to noise filters.

(a) Use the GIPSI system to look at on-line adaptive and feed-

forward control and compare these techniques to optimal control.

(f) There seems to be potential industrial use for on-line

optimal control, and industry would be well advised to investigate

not only the deterministic case but also the stochastic case, as

studied by Pike (6).

Even though noise was a major problem and affected reproduci-

bility of results, one should realize the problems involved in

computerizing a pilot plant size distillation column and in obtaining

the reliability of operation which was demonstrate ,- Enormrnous amounts

of time and effort were expended in building the GIPSI system and in

installing the instrumentation for the column. Every system has its

initial bugs and once these were removed the GTPSI system and the

distillation column performed very well.

To describe in detail the possible uses of the GIPSI system

and its effects on laboratory experimentation and teaching would in






73



itself require many pages. As was mentioned in Chapter II, GIPSI

is a general tool which can be used for many purposes and, in fact,

can be used to computerize several processes simultaneously. A

system of this nature is not only necessary but also essential to

real-time, on-line optimization and computer control problems. The

GIPSI system is already being readied for other computer control

projects and its accessibility and ease of use have already insured

its use in the future.





































APPENDICES











APPENDIX A

BASIC DATA FOR LINEAR MODEL DERIVATION


The figures contained in this appendix represent column state

variable response to upsets in load and control variables as well as

linear model fit to the response. Figures A.1 to A.12 represent

state variable response to individual step changes in load and control

variables. Figures A.13 to A.15 represent state variable response to

a set of simultaneous load and control variable upsets.

It is apparent that the model fits are not perfect and this

occurs for several reasons: (1) the typical column state variable

response curve shows second or higher order tendencies while the model

is first order, (2) steps occur in the column curves at some time

greater than zero while model curves show steps at time zero, and

(3) the time constan.l- used for model curves represents an average of

four time constants, one for each lc-ld a.nd control variable. One

should notice that the response curves for bottom flow rate are not

shown. The reason for this is that boctto flow rate is computed as

feed rate m-inu.s top product rate.

Th.. .oa-'tanlt for th .- J- ci u rvs (Chapte.r III) ,e Coltl'C

using 50 :tages per hour as the stnag- length, thos data points for the

model were calcula.cd every 0.02 hours but are plotted as stages.

Only ti:le valus of 14 lstagcs are ploL.ted on *hoe abbci.ssa.

The Eol lo.,oi,.g ,psets in load cnd coIntrol v,1 ioablrs were made

oa the column:











Individual Changes

AF = -25 Ibs/hr

AXF = 0.046

ASR = -10 lbs/hr

AR = 23 lbs/hr


Simultaneous Changes

AF = -20 lbs/hr

AS = -7 Ibs/hr

AR = 8 Ils/hr



Chanui s wiLe lade id lii ile individual variables, except L feeud

rate, in only one direction. This direction was one which would

force the top composition to increase and approach 100 percent methanol.

It was reasoned chat these changes would put the most severe test on

the linear model, Step change.: if equal :a2nitud. e :ere made in feed

rate to note the effect on the state variables. The same change was

noted in both cases, however, only one set of data is presented. It

was assuriv.. that Lih other up.;at variables would give similar results.

One should note that the rcdel response for individual changes

is good in approximating the column. The model curves, if drawn out

to steady-state approach the steady-state values given the column.

The model response for the simuLi':.neous cahangos fits good for

several :- ,.gcs and then starts to deviat.i for two variables. This

indico;:3s that the effect of the three sirultan, : .;us changes over-drives

the lie:ar response of the column. Only this one set of simnltanocu3

upsets was investigated on the column to sive come idea of the ].imi;cs







77



of the linear region of response.

It should be noted that compositions and flow rates were

obtained from strip chart recordings where compositions could be

read only to the nearest 0.5 percent (0.005) and flow rates to the

nearest 2 lbs/hr. Deviations between model and column within these

limits appear to give satisfactory agreement.










0.034 .
I/ Model
.032
.03 Column

.030

.028

.026

.024 -

.022

.020 /
AXD )
.018 -

.016 -

.014 A /

.012 -

.010 /

.008 0

.006 A

.004

.002 / r

0 U ..~.L ...... J I I I I
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Stage

Figure A.1. Top Composition: Column-Model Response to Step
Change in Feed Rate.


14























A Model

0 Column


..... O



2 3 4 5 6 7 8 9 10 11 12 13 14


Stage


Figure A.2.


Bottom Composition: Column-Model Response to
Step Change in Feed Rate.


.014

.012


010


.008

.006

.004

.002

0


AXB








12 A Model

0 Column
10 1- .. -


8 / O' ..-
6 1






0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14


Stage


Figure A.3.


Top Flow Rate: Column-Model
Change in Feed Rate.


Response to Step


A Model


1----A
V,^' .,.


2 3 4 5 6 7 8 9 10 11 12 13 14
Stage


Figure A.4.


Top Flow Rate: Column-Model Response to Step
Change in Feed Composition.


) 0


Column






Model


0 Column


.024

.022

.020

.018

.016

.014

.012

.010

.008

.006

.004

.002

0


2 3 4 5 6 7 8 9 10 11 12 13 14
Stage


Figure A.5. Top Composition: Column-Model Response to Step
Change in Feed Composition.


A Model


O Column


AXB i 0



-- O ~ O---O 0 0 0O

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Stage

Figure A.6. Bottom Composition: Column-Model Response to
Step Change in Feed Composition.


AXD


0 1


.008

.006

.004

.002

0






.018 A

.016 0O


.014

.012

.010

.008

.006

.004

.002

0


AXD


Model


Column _,-'




o O




21









2 3 4 5 6 7 8 9 10 11 12 13 14


Figure A.7.


Stage
Top Composition: Column-mlodel
Step Change in Steam Rate.


Response to


A Model

0 Column



AXB


0 1, 2 3 4 5
0 1 2 3 4 5


Figure A.8.


.O.-G- -... "-,. --.- --.- i -
6 7 8 9 10 11 12 13 14
Stage


Bottom Composition: Column-Model Response to
Step Change in Steam Rate.


.012

.010

.008

.006

.004

.002
















I oQ O--O-C O-- 3--C--- --o -*o 0--


-34

-32

-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10


2 3 4 5 6 7 8 9 10 11 12 13 14
Stage


Figure A.9.


Top Flow Rate: Column-Model Response to Step
Change in Steam Rate.


Model
Column










.034


.032


.030

.028


.026

.024

.022


,020

.018


.016


.014

.012

.010


.008

:006

.004


.002


0


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Stage
Figure A.10. Top Composition: Column-Model Response to
Step Change in Reflux Rate.


/7 Model

SO Column













AXD
-





















A Model

O Column







AXB


.022

.020

.018

.016

.014

.012

.010

.008

.006

.004

.002


0


Figure A.11.


Bottom Composition: Column-Model Response to
Step Change in Reflux Rate.


-O/ O







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Stage








--48 0

-46-
0
-44 0


-42 0O 0------ 0



O O O
-40- /- 7\ Y- 1Y f & -A A /\ 5..- L


-38

-36


-34

AD
-32 -


-30

-28

-26 Model
-26

-24 Column


-22

-20

-18

-16


-14


-12

-10


- 8

-:6 : .... !L ._,- ... j..... ...... -,. ..- !..! .!
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stage

Figure A.12. Top Flow Rate: Column-Model Response to Step
Change in Reflux Rate.
















.022 A Model

.020 0 Column

.018

.016

.014
.014



.010 -

.008

.006
0
.004

.002
0
.002






-.006 -~ &- A
-.004 --,A ^ -. A _-A-.


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Stage
Figure A.13. Top Composition: Column-Model Response to
Simultaneous Step Changes.




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