1


UNITED STATES DEPARTMENT OF COMMERCE . John T. Connor, Secretary NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Systems Engineering in Ceramics

Proceedings of a Symposium

April 19, 1964

Held under the auspices of the Ceramic Educational Council and National Institute of Ceramic

Engineers of the American Ceramic Society, with the cooperation of the National Bureau of Standards, and under the sponsorship of the Office of Naval Research. The symposium took place

at the 66th Annual Meeting of the American Ceramic Society in Chicago, Illinois.

National Bureau of Standards Miscellaneous Publication 267

Issued May 1, 1965

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402

Price $2.00
Library of Congress Catalog Card Number: 65-60036
Contents

Introduction

1. Systems Engineering and Modern Engineering Design

Allen B. Rosenstein

University of California

Los Angeles, California

2. Basic Concepts of Systems Engineering

H. E. Koenig and C. M. Bacon

Dept. of Electrical Engineering

Michigan State University

East Lansing, Michigan

3. Process Control in the Ceramic Industry

Ernest Hucke

Honeywell Industrial Products Group

Philadelphia, Pennsylvania

4. Application of Computers in Design and Control of Systems

Joseph L. Hammond, Jr. School of Electrical Engineering

Georgia Institute of Technology

Atlanta, Georgia

5. Statistical Methods in Systems Design

W. C. Hackler

North Carolina State of the

University of North Carolina

Raleigh, North Carolina

3. Systems of Interest to Ceramic Engineers

A. H. Hurst and E. R. Bullock

Instrument Dept.

Minnesota Mining & Manufacturing Co. Hawthorne, California

Page

iv

1

11

39

81

109

119

ili
Introduction

The last decade has witnessed the increased interest of Ceramic Engineers in the

development of a number of engineering techniques for the analysis and synthesis of

groups of interconnected elements or components. These techniques, which have been

developed by many engineering disciplines, have been found to be applicable and useful

for studying overall system behavior independently of the disciplinary origin of the

components. It is not surprising that an emerging area called "Systems Engineering

has begun to encompass these techniques and methods of attack, and to emphasize their

usefulness when attention is focussed on the interaction of two or more elements.

Although the average Ceramic Engineer is unlikely to shift his professional

activities wholly into this new area, Systems Engineering overlaps the interest of

Ceramic Engineers in several important areas. Where this overlap occurs, Ceramic

Engineers can benefit from well established techniques and methods. The design and use of process control systems is possibly the most important single area where such

overlap takes place. Other areas of overlap include the design of manufacturing operations, pilot plants, and systems for experimental testing and evaluation.

The Symposium on Systems Engineering was designed to present an overall concept

of this emerging discipline with emphasis on topics of interest to Ceramic Engineers.

In Chapter 1 Dr. Rosenstein introduces the concept of systems engineering and discusses

its boundaries and its relation to engineering design in other disciplines. In Chapter 2 Drs . Koenig and Bacon present a general discussion of some of the basic con- cepts used in any systems engineering problem.

Chapter 3, by Mr . Hucke, presents a discussion of the important class of process

control systems. In the next two chapters, Drs. Hammond and Hackler consider two of

the more important system engineering techniques, namely: computer methods and statistical methods. The final chapter, by Mr. Hurst and Mr. Bullock, ties together the

material of the first five chapters by considering in detail several examples of systems of interest to Ceramic Engineers.

W. E. Moody, Chairman

Georgia Institute of Technology

H. F. McMurdie, Editor

National Bureau of Standards

W. E . Coffeen

M & T Chemicals, Inc.

Winston Duckworth

Battelle Memorial Institute

J. L. Hammond

Georgia Institute of Technology

E. C. Henry

General Electric Co.

G. E. Lorey

Missouri School of Mines

G. C. Robinson

Clemson College

iv
Systems Engineering and Modern Engineering Design

Allen B. Rosenstein

1. Introduction

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technology to furnish the "good life" ifa raSalv^ ^?*Ppeared' man ^

turned to interdependence upon one another ha! i rapidly shrinking, crowded world. Our

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lllustrate the above. 4 They note

and 2400 million in 1950? Maximum tra^nn ? T ^ " 175 °' t0 l2 °° million in "50.

100 mph in 1900, to coTM rciaTtran Z "ortaf

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indication of increasing complexitv V S I 1 \

communication systems are a good

55 million in 1955. comP^xity. U.S. telephones jumped from 350,000 in 1900, to
The return on investment in machinery has increased productivity with the attendant opportunity for more leisure time and more education. As a good example of this, one farm worker supported six people in 1900, and fifteen in 1950. The penalty has

been increased complexity, more sophisticated and difficult design problems, and

greater interdependence. The telephone industry in 80 years has gone from simple

manual switchboards with direct connection to world wide automatic switching wherein

machines select the "best" of alternate routes, place the call, record the charges,

and mail you the bill.

2. History

Hall of the Bell Laboratories has attempted to trace the historical development

of Systems Engineering. 1 R.C.A. is credited with recognizing a need for a systems

approach in the 1930' s for the development of a television broadcasting service.

World War II witnessed a quantum jump in the complexity of systems which were met with

tools and techniques ranging from Operations Research to more sophisticated control

theory. Although the Bell Telephone Laboratories Incorporated is credited with using

the term "Systems Engineering" in the early 1940's, widespread usage of the terms did

not appear until the 1950' s.

Hall points out that, "It is hard to say whether increasing complexity is the cause or effect of man's effort to cope with his environment. In either case a cen- tral feature of the trend has been the development of large and very complex systems

which tie together modern society. These systems include abstract or non-physical

systems, such as government and the economic system. They also include large physical

systems like pipe line and power distribution systems, transportation and electrical

communication systems." In either event the results have been very clear. The need

for communication systems, military systems, transportation systems, and managerial

systems of increasing complexity has called for the development of models, analytical

techniques, and design methods of increasing sophistication and power. In the past

decade this call has been met with increasing success. The knowledge and experience

gained has opened still more doors. Process control systems, for example, can be

analyzed in greater detail and regulated with increasing precision. Our insight into

the modeling, organization and mechanization of man-made systems has now prepared us for a massive assault upon the secrets of the biological systems.

One important aspect of today's technology that is often overlooked lies in the

increasing importance of engineering system decisions. Because of the size of our systems and the extent of their interaction with society, the opportunities for catastrophic (or near catastrophic) engineering failures are increasing. While engineers still are called upon to design simple two lane highways, some engineers are expected to design complete transportation systems. These systems in turn will control the growth

of entire regions for many generations. The author is particularly sensitive on this

point since he lives in a city whose "solution" to the mass transportation problem has

created an equally serious hazard called smog. The total dependence of our national

military position upon approximately one dozen highly complex weapons systems xs a further example which should give us cause for concern.

3. "Definition" of Systems Engineering

As a term becomes popular it is customary to seek a concise definition which clearly illustrates its meaning. If the term is applicable to a process - and a very genera]

process at that - the lucid definition more often than not becomes elusive. For a unique definition of such processes, it usually becomes necessary to look to a description of the elements which comprise the process. On the other hand, there have been

those useful activities that have resisted precise categorization in their formative

years, but which have grown up into a very respectable, well-defined middle age.

It is too early to make categoric statements about the ultimate acceptance and

usage of the term System Engineering. There is not yet a consensus of opinion about

the activites encompassed. However, large areas of agreement are beginning to appear

as each author in turn takes a crack at defining the boundaries of the subject.

2
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3
4. The Content of Systems Engineering

"Engineering" is the name of a profession, and at the same time the description

of the activity in which engineers engage. The foregoing definitions of Systems

Engineering can be given more body by examining the important ingredients of that

activity. Examination of three of the more penetrating studies of the subject reveal

a remarkable unanimity as to what constitutes the body of Systems Engineering. 2 ' 3 ' 5 With apologies to the respective authors, if their works have been misinterpreted,

summaries of the particular parts of their publications in which they have listed

the essential elements of Systems Engineering follows:

Phases of Systems Engineering - Arthur D. Hall

I. Systems Studies:

1. To investigate the Environment

2. To create background of Information

II. Exploratory Planning

1. Problem Definition - data gathering , customer requirements , inputs , outputs

2. Obj ective selection - search for alternatives , criteria

for optimum

3. Synthes es of system - alternative systems to satisfy

objective with data for decision.

4. Analyses of system - consequences of alternative systems

5. Selecting of the Best system

6. Communicating results

III. Development Planning

1. Recycling of above in more detail

2. Experimentation may be conducted

IV. Studies During Development

1. Development and manufacture of first models

2. Final evaluation of system in its working environment

V. Current Engineering

1. Continues as long as system is used

2. Feedback to develop "know how" for future systems

Approach to Systems Engineering Problems - Harold Chesnut6

I. Problem Formulation - customer requirements or needs ; obj ectives ; restraints including inputs , environment,

human factors; weighting functions .

II. Synthesize System - alternate solutions

III. Way to Make System Synthesized

IV. Measure and Compare with Objectives

V. Refine

The Design of Engineering Systems - W. Gosling

I. Completion of Specifications - Inputs , Outputs , Environment , Measures of Value

II. Search for Alternative Solutions

III. Feasibility Check - Physical laws , Environment Compatibility including

human laws and customs, cost , company capability

4
IV. Modeling

V. Reliability Study

VI. Selection of a Best System

VII. Evaluation by Trial

VIII. Collection of system performance statistics

The ideas embodied in these operational statements of Systems Engineering are quite familiar to engineers. What does perhaps appear new or perhaps only reemphasized

on a new scale are certain clearly stated concerns. Three of these that stand out

are:

1. A concern with the overall problem ranging from the initial conception to the recording of the operating life performance.

2. A concern with optimization of system performance in terms of formal measures

of value.

3. A concern with the needs of the customer that are determined on several

levels

In addition to the structure for Engineering System studies, a number of tools of especial value to the Systems Engineer have been given. High on most lists are

probability, statistics, logic, and computing. Often included are queueing theory,

game theory, linear programming, dynamic programming, simulation, information

theory, automatic control theory, topology, philosophy, economics, psychology and language in addition to the more traditional topics of engineering, mathematics and

science.

The list contains enough items that have come from the field of Operations

Research to warrant a comparison. Since the Operations Research proponents have had even greater difficulty in defining their bailiwick than the Systems Engineers,

it is not surprising that a ready distinction is not available. The writings of

O.R. experts do not necessarily aid us. Churchman et al. 7 list the following phases

of operations research which compare rather closely with those of Systems Engineering.

1. Formulating the problem

2. Constructing a mathematical model

3. Deriving a solution from the model

4. Testing the model and the solutions derived from it

5. Establishing controls over the solution

6. Putting the solution to work: Implementation

Hall believes that Operations Research is usually concerned with the operation of

an existing system while System Engineering emphasizes the planning and design of

new systems. Goode and Machal state that the operations research is primarily interested in making procedural changes while the systems engineer is primarily interested

in making equipment changes. The line may be seen although it is drawn rather thin.

5 . Component Engineering

Those readers who have been patient enough to remain with us this long have certainly begun to ask themselves, "When is a component not a component, but a system, and who decides when to apply the new Systems Engineering in preference to the old

components engineering?" An air transportation complex is complicated enough to warrant the designation of engineering system. The jet transport would be a component

in the transportation system, and yet its design would exercise all of the elements we have attributed to Engineering Systems. The aeroplane's radar and communication

electronics are components of the craft but are substantial systems in their own

right. We can continue down to a radar component such as a pulse transformer. The

5
transformer in turn possesses a magnetic circuit, a thermal circuit, a mechanical

structure, electric circuit, and insulating system all of which are functions of their geometries and materials.

It is generally agreed that the system-component distinction is an individual decision to be made from the point of view of the engineer who is faced with the decision. While the system-component contrast is admittedly made on an apparently arbitrary basis, the distinction in the final analysis will be found to be one of engineering survival. As the number of individual items in a project increase, the

amount of information that can be economically processed per item must decrease. When

the description of some items have been reduced to either a statistic or hardly more than an input-output relationship, we are inclined to classify these items as system

components

The procedures, methods, models, and tools of system design and of component design are essentially the same. The items and information content of complex, large scale systems have spurred the development of models and techniques for dealing with

what may be called high density traffic in matter, energy, and/or information. But

as is usually the case, these techniques are finding increasing application in the design of components which in turn have become more sophisticated,,

6 . Modern Engineering Design

As the differences between Systems Engineering, Component Engineering, and even Operations Research become increasingly difficult to discern, we should re-examine our ordering of engineering and of engineering design. Such a study was begun several

years ago by the U.C.L.A. Department of Engineering. A committee spent several years exploring the role of engineering, the content of modern design, and its place in the

engineering of the future.

The committee concluded that design taken in its broadest sense is the essential

activity of engineering. Design was defined as an iterative decision making process

to develop systems to optimize the value of resources. In this context, design is

intended to encompass the entire process from the appreciation of a need of society

to the ultimate disposal of the systems created to satisfy the need. The research

required to provide knowledge necessary to implement the design is considered an element of the process

Studies of the activities of prominent engineers convinced the committee that the

design process could be described (and therefore taught) . A horizontal and vertical

structure was observed to exist in the design process. The vertical structure which

has been called the "Morphology of Design" gives the complete step by step design

procedure. 3 The horizontal structure consists of those elements and ingredients of

the process which are repeated over and over again throughout the design. These are collected into what has been called the "Anatomy of Design". The elements of the

Design Anatomy, listed below are the essential ingredients of the design process which are applicable to all design, whether system or component. A knowledge of these

elements, their interrelationships, and the techniques associated with each element

provides a foundation for all modern engineering design.

6
ANATOMY OF DESIGN

1. IDENTIFICATION OF THE NEED .

As analysis of whether or not a need exists is required. Often a need, as originally stated, is not what is really needed to satisfy the final objective .

2. INFORMATION COLLECTION AND ORGANIZATION . All factors which relate to the system need to be considered. Where necessary, experiments must be devised to obtain data otherwise unavailable.

3. IDENTIFICATION, STATEMENT, AND MODELING OF SYSTEM VARIABLES . All factors influencing the system - the so-called "boundary conditions".

Engineering systems, subsystems and components can be analyzed into basic

elements which, when described or prescribed in appropriate detail and when

properly synthesized, will constitute the design of the system.

a. INPUTS: Those resources and other environmental factors which are converted (or modified) by the system in question.

b. OUTPUTS: That which is produced by the system.

c. TRANSFORMING MEANS: Used to obtain the relationship between

inputs and outputs

d. CONSTRAINTS: All elements and factors which express limitation

and/or need to be accounted for in the design.

4. CRITERIA DEVELOPMENT FOR OPTIMUM DESIGN. The rules

for judging relative merit.

a. Development of Value Systems

b. Criteria relationship among values

5. SYNTHESIS AND ANALYSIS . Evolving of systems to convert the inputs into the desired outputs. At this

step only the requirement of realizability is usually met. Analysis forms

the feedback loop around Synthesis.

a. Physical Realizability d. Producibility

b. Economic Worthwhileness (realizability) e. Maintain ability

c. Financial Feasibility (realizability) f. Reliability

6. TEST EVALUATION, AND PREDICTION .

7. OPTIMIZING . Optimizing (Maximizing the performance. Reduction to "best" system with

available knowledge.)

8. DECISION.

7
9. ITERATION.

It is recognized that the above operations are found throughout the design

process. Many iterations will be taken around several or all of them In particular the engineer continually re-examines his previous findings and decisions in the light of new information.

10. COMMUNICATION, IMPLEMENTATION, AND PRESENTATION . The similarities between the content of Systems Engineering and the Modern Design Process are too great to be ignored. From the pedagogical standpoint an organized view of the general design process and its important constituents offer an excellent opportunity to provide the student with insight into and capability for undertaking the essential functions of engineering regardless of whether he is employed at a sys- tems or component level.

7 . Conclusion

Labels are, of course, not too important. In the long run it does not matter whether or not the titles Systems Engineering or Component Engineering become fully accepted general terms. It is, however, important that their practitioners are pre- pared to exploit to the utmost all of the tools and experience that modern engineering

can provide. Many, if not most of the present day significant engineering undertakings have become too large for a single individual to encompass. Divisions of responsibil- ity and labor cannot be avoided. Yet regardless of how the distinctions are made there is a greater need than ever for a full appreciation of the functions and respon- sibilities of all facets of the design process.

I would in particular like to stress the work "responsibility". No matter wheth we are working at a system level or a component level, every engineering project is

part of some larger organism or system. It consequently becomes a subsystem to this larger system. All of us therefore in this sense, design components which ultimately

serve a greater system. As professional engineers there exists a responsibility to continually question and re-examine the inputs to our subsystem, the weighting func- tions we use to optimize value, and to check the ultimate performance of our products. There are the individuals who are willing to blindly accept specifications, perform their computations, and submit their designs to other departments for implementation

with little further interraction . Indeed, large engineering projects tend to organize

themselves along these lines. However, the man who does not participate in any of the value judgments is not functioning as a professional engineer. He is serving as a technician, a technologist or perhaps as a human computer. In any event, it is a sub- professional activity that possesses a real possibility for future relegation to machine operation.

Within every large engineering organization there is an increasing need for

specialists in breadth as well as the continuing need for the specialists in depth . It is, of course, this need for specialization in breadth that has forced the development of "Systems Engineering". Yet we have seen that the Design Process is the same

for both the specialists in depth and in breadth, the systems and the components

engineer. The basic tools, models, techniques are also very much the same. We need only to concede that at this time that the specialist in breadth may have a greater

need for traffic handling techniques. Since the differences in the specialists do not

appear in the general design process, or in their procedures or models, we have to look further for significant variances. The most important will be the respective

storehouses of the characteristics of the real world, that each will accumulate to perform their given tasks. If we can allow ourselves the luxury of a rather broad

generalization, we could observe that the specialist in depth will orient toward microscopic phenomena while the specialist in breadth will be concerned with the

macroscopic

The clearest distinction lies in the kind of information processed rather than

the quality. If we assume that men of equal intelligence are capable of processing information at equal rates, then the specialist in breadth dealing with a large

er
number of items is forced to reduce the amount of information associated with each item. Conversely, the specialist in depth who restricts the number of items involved m hxs designs, will be free to increase the degrees of freedom allowed each item with an attendant increase in information per item. To this end, it is interesting

to note that the complexity of important organic molecules often exceeds that of our most intricate processing or communcation systems.

References :

1. "A Methodology for Systems Engineering," Arthur D. Hall, Van Nostrand 1962.

2. "What is System Theory and Where Is It Going?", A Panel Discussion, IEEE Trans. on Circuit Theory, June 1963.

3. "Integrating of Systems Engineering with Component Development", J. A. Morton, Electrical Manufacturing, August 1959.

4. "System Engineering, An Introduction to the Design of Large-Scale Systems", Harry H. Goode, Robert E. Machal, McGraw-Hill, 1957.

5. "The Design of Engineering Systems," W. Gosling, Wiley, 1962.

6. "Preliminary Survey of Systems Engineering Problems", H. Chesnut, General Electric Laboratory Report No. 60 GL 5, January, 1960.

7. "Introduction to Operations Research", C. W. Churchman, R. L. Ackoff, Wiley, 1957.

8. "Design as a Basis for a Unified Engineering Curriculum", UCLA 1962,

A. B. Rosenstein and J. M. English.

9. "Introduction to Design", Morris Asimow, Prentice-Hall, 1961.

10. "Operations Research and Systems Engineering", C. D. Flagle, W. H. Huggins,

R. H. Roy, Johns Hopkins Press 1960.

9

Basic Concepts of Systems Engineering

Herman E. Koenig and Charles M. Bacon

1. Introduction

The appeal of system theory to scientific workers in many disciplines has increased tremendously in the last decade. The work "system" is currently applied in

a host of scientific endeavors including many fields of study not directly related

to the Physical Sciences and Engineering, such as Economics, Sociology, Biology,

Transportation and Supply, etc.

In an effort to apply "systems concepts" to their own particular problems,

workers in these latter areas have turned to the engineer and mathematician for a clear concise exposition of system theory. This responsibility has caused the engi- neer to re-examine the entire area of systems analysis, with a view toward defining

the discipline more clearly, generalizing the concepts and broadening the base of

application.

The authors of this paper assume that the primary application of system theory

in ceramic engineering involves feedback control problems, as they arise in materials

handling and process control. If a satisfactory process control system is already

in operation, system theory has little to offer in the way of providing new and useful information. However, if a process control system is to be introduced, improved

or replaced, the performance characteristics of the proposed system can be completely

analyzed to see if it satisfies specifications. If it does, the process engineer

can be confident that the system, when built, will meet those specifications within

an acceptable tolerance level. Such a procedure is called system analysis . If on

the other hand, the analysis shows that a proposed system does not meet specifications, then system theory is used to alter certain components of the system so that

the altered version does meet specifications. This procedure is called system

design .

All quantitative system analysis and design procedures are based on what is

called a mathematical model of the system. Mathematical models of a given system

may vary widely depending on the performance characteristics to be modeled, the ac- curacy to be realized and the analytical techniques to be used in solving the model.

Prior to the second World War, quantitative system theory existed only in a crude

form; being concerned largely with steady-state performance characteristics. During

the war, an intense research effort, centered about the dynamic analysis and design

of radar and gun control systems, brought forth transform methods as an efficient

means for analyzing systems characterized by linear differential and difference

equations. The transfer function became, and still remains to a large extent, the

standard mathematical model for linear systems. Many excellent texts exist which

deal with system analysis and design using the transfer function approach.

Since about 1958, the profound need for more effective methods for analyzing

the nonlinear systems encountered in missile and launch vehicle systems, coupled with

the availability of computing machines, has shifted the emphasis in system theory

from transform methods to what are called state- space models. These models are ideally suited for system simulation on either analogue or digital computers and serve as the basis for modern system theory.

In a tutorial development such as this, space does not permit the development

of a working knowledge of modern system theory. Entire books are devoted to this

subject. This paper, rather, attempts to develop, mostly through the use of ex- amples, an appreciation for the main stream of what modern system theory is about

so that the reader might evaluate its potential applications to the process control

systems involved in ceramic engineering. To this end, two aspects of system theory

are delineated; 1) the problem of developing mathematical models of the system and

2) the "processing" or solution of the model to ascertain the performance characteristics of the system. The first aspect is sometimes referred to as "system simulation" and in the case of complex process control systems containing mechanical.

1
hydraulic and electrical equipments, frequently represents the most difficult aspect

of system analysis. Fortunately, systematic modeling procedures have recently been

defined. A discussion of these procedures is presented in Section 3 after developing

in Section 2 some of the prerequisite system concepts pertaining to measurements and

component models. Section 4 considers several methods of solution of the system model

as well as the derivation of the system transfer function. Finally, the important

concepts of feedback and stability are illustrated by example in Section 5.

2. Variables, Measurements and Component Models

A system is the sum of its parts. The performance characteristics of a system

are, therefore, a function of the performance characteristics of the individual com- ponents or parts which comprise the system. It follows that a mathematical model

which describes the characteristics of a system, can only be obtained when the mathematical models of the individual components are available. And yet while the models

of the individual components are necessary to the formulation of the system model,

such information is insufficient. Three given components interconnected in two alternate configurations may give entirely different system performance. Thus, in

general, two pieces of information are required before a mathematical model of a system can be derived; 1) a model of the performance characteristics of the individual

components and 2) a mathematical statement or model of their interconnection . It is implied that the mathematical model of each component is characteristic

of the component itself and independent of the way in which the component is connected

with other components in the system. That is, like the name plate and serial number,

the mathematical model of the component characteristics goes with the component, to all "customers".

To illustrate this point and show more explicitly what is meant by a model of

a component, consider the hydraulic lift mechanism in Fig. 2.1. To develop a mathematical model of this system, we first attempt to separate the system into components

or subassemblies for which mathematical models can be more easily obtained. The system has been divided into three subassemblies as indicated by the dashed lines. These

subassemblies can be regarded as "black boxes" interconnected as shown in Fig. 2.2,

where the letters A, B, H. and J indicate points of union between the components

points at which the performance characteristics of each component as a. separate entity

are to be measured and characterized mathematically.

Assume that only vertical forces and displacements with respect to the fixed

reference G are of interest. The control level assembly merely translates a given

force and displacement "input" at point A into another force and displacement at

point B. At point B, the hydraulic subassemb y receives this mechanical input and

translates it into a much larger force and displacement at point H. The level G-H-J

transmits this force and displacement to the mass at J. Each subassembly model must

relate, mathematically, the mechanical quantities of force and displacement at the

terminals where connections are made to other components. To develop these models

each subassembly can be considered in greater detail.

The hydraulic subassembly, for example, is divided into the four basic components

shown in Figure 2.3. It is assumed that the characteristics of the hydraulic pump in

Fig. 2.1 and the fittings between the pump and the control valve can be included in

the mathematical model of the control valve. From Fig. 2.3, it is evident that the

control valve translates the mechanical force and displacement at point B into hydraulic fluid flow and pressure at points C and D. Each hydraulic line exhibits a pres- sure differential over its length which is a function of the fluid flow through the

line. Finally, the hydraulic ram translates a difference in pressure and flow rate

at points E and F into mechanical force and displacement at point H. Thus, the hydraulic subassembly is considered as a system having four components each having

"terminal" characteristics that can be modeled mathematically. From the models of

these components we can hopefully derive a model of the hydraulic subassembly, then

a model of the entire system.

12
Note that exactly two physical quantities or variables have been cited in

describing the effect of one component on another mechanical force and displacement or pressure differential and fluid flow. In the ensuing development, it is as- sumed that, in general, two such complementary variables are required to describe the

effect of one component on another at their common interface. The mathematical model

of the control valve, for example, is complete when the complementary pair of variables at the "output" have been expressed as a function of the complementary pair of

variables at the "input". This functional dependence can be established by removing

the component from the system and subjecting it to certain performance tests. During

such performance tests, the vertical displacement at point B in Fig. 2.3 must be

measured. But this measurement must be taken with respect to an appropriately selected reference point. If point G is used as a fixed reference, the vertical displacement of B with respect to G at any instant of time can be described by a real number.

Similarly a pressure measurement at point C has no meaning unless a reference is

specified. The selection of point D as reference implies a pressure differential

between points C and D on the valve. If gage pressure at C is desired, the local atmospheric pressure is taken as reference. Thus, a well defined reference is required.

A similar statement can be made about the measurement of other physical quantities

such as voltage, rotational displacement and temperature difference. This reference

requirement implies a "two-point" property of physical measurements.

Component mathematical models are conveniently classified with respect to the

number of points (or terminals) on the component at which physical measurements are

made. A model of the control valve in Fig. 2.3 is designated as a four-terminal model

when it is based on measurements between points B and G and between points C and D. Similarly, the hydraulic line CE can be modeled as a two-terminal component by con- sidering measurements between points C and E. Fig. 2.4a shows schematically a two- terminal physical device subjected to a performance test. A "driver" excites the

component in order to simulate its behavior in a system while two complementary

physical quantities (pressure difference and flow rate, for example) are measured

with respect to terminals A and B. One of these quantities exhibits a

" difference " property with respect to A and B, i.e., it assumes different values at A and B. For

this reason, it is identified as the across variable x(t); in general, a time-varying

quantity. The second quantity exhibits a

" common " property with respect to A and B. That is, its values at A and B are equal. Thus, it is identified as the through var- iable y(t) ; also time varying „ The meters are to be regarded as conceptual since the

component may be electrical, hydraulic or mechanical and the physical quantities to be measured will vary accordingly. The polarity signs on the meters indicate that

they furnish both positive and negative values. Thus, a meter indication of opposite

sign is obtained if the meter connections are momentarily reversed.

The mathematical model of the component in Fig. 2.4a describes two distinct

features of the conceptual measurement apparatus. First, the orientation of the

meters and their manner of connection to the component are represented by the

oriented line segment of Fig. 2.4b. Second, the time variation of one measurement,

as a function of the second measurement is approximated by an equation; one possible

form appearing in Fig. 2.4c. The line segment, hereafter called the terminal graph , along with the equation, hereafter called the terminal equation , constitutes a mathematical model of the two terminal component.

If, for example, the component being subjected to performance test is the length

of hydraulic line in Fig. 2.5a, the through and across variables become hydraulic

fluid flow and pressure difference. The conceptual measurement apparatus of Fig. 2.4a

is represented in the mathematical model by identifying the ends A and B of the hose

by the pair of points a and b on the terminal graph shown in Fig. 2.5b. The orientation of the graph signifies that p(t) represents the pressure differential of a with

respect to b, and that g(t) represents positive fluid flow into terminal a and out

of b. If the orientation of the terminal graph is reversed, then both pressure and

flow measurements are reversed. A typical measured static pressure-flow characteristic appears in Fig. 2.5c together with a terminal equation of a straight-line approximation of the recorded characteristic. The model is incomplete unless both the

terminal graph and terminal equation are given. One is useless without the other!

f3
The terminal graph alone tells nothing of the functional relation between p(t) and

g(t) and the terminal equation alone furnishes no information about the measurements

represented by p(t) and g(t).

In each mathematical model, one across variable and one through variable are used to describe the performance characteristics of each type of physical device

(electrical, mechanical, hydraulic, etc.). Table 2.6 lists some of the important physical quantities and their classification as through or across variables. Many

practical devices can be modeled on the basis of only two measurements related by a single terminal equation and are thus classified as two-terminal components. 'Table

2.7 lists the most common two-terminal components together with their terminal

equations

Depending upon the approximation which can be tolerated in the analysis of a system, terminal equations can range in complexity from simple linear equations to nonlinear differential equations. The analysis of a system may become exceedingly difficult if it is necessary to use nonlinear terminal equations to adequately relate

the recorded performance characteristics. Consequently, every attempt is made to ap- proximate performance characteristics by linear algebraic or linear differential

equations

In many instances the terminal equation can be established from manufacturer's

specifications or derived from geometry (length, diameter, etc.) of the component.

It is not our purpose, however, to discuss such procedures here.

The mechanical lever in Fig. 2.8a is considered as a three terminal component. Only vertical displacements and forces are to be considered with positive displacement

6(t) indicated by the arrow. The terminal graph in Fig. 2.8b indicates that two sets

of variables are to be related by terminal equations. 6 X (t) is the instantaneous

vertical displacement of point A with respect to the fulcrum point G and f1 (t) the

corresponding vertical force applied between the fixed reference G and point A. The

variables 6 2 (t) and f3 (t) are similarly defined except they are measured relative to points B and G. When the lever characteristics are approximated by linear differential equations, the terminal equations may take the form

6

i (t) = F

ii<zr>- f

i (t) + F

i2<ar>-

(2. i;

where F (—) , (i, j = 1, 2) represent differential operators applied to the indicated

variables. A more compact matrix notation for (2.1) is

f,(t) (2.2)

5 2

(t)

The differential operators appearing in the coefficient matrix of (2.2) can be established by considering the conceptual measurement apparatus of Fig. 2.9. The operators

Fii<^) and F2i(^) are obtained by setting 6 3 (t) = 0 in (2.1) or (2.2) and allowing

fj (tf to vary. The equivalent physical operation is shown in Fig. 2.9(a) where the

lever is held in a horizontal position with end B fixed by a force meter measuring

f3 (t) . The lever is excited by a force driver as indicated through the force meter

measuring fx (t) . Under these conditions b 1 (t) will also be zero, provided that the

lever can be considered rigid. It is evident that f1 (t) and f2 (t) are related by

fa (t) = n f1 (t) li where n = —. Thus we find that

< 2

5^t)

f 2(t)

F n£>

21 v dt '

F 12^)

F

22<ST>

14
Fll (^) = 0 and F31 (^) = n

To evaluate Fia (—) and F22 (~j~) . set fx (t) = 0 (allow A to swing freely) and

drive end B with a displacement driver as shown in Fig. 2.9b. For a rigid beam with

rotational mass M and negligible friction we have

and

6j. (t) = -n6 2 (t)

fa (t) = M ~j^s~ 6 3 (t)

Thus, the terminal equations modeling the performance characteristics of the lever are

5 (tj = -n6

2

(t)

2

f2(t) = nf^t) + M —2" 6

2

(t) dt

or in matrix form

f 2

(t) M

dt

f

l

(t »

5 2

(t) (2.3)

When the mass of the lever is negligible, M = 0 in (2.3) and the component is

described as an "ideal" lever. The terminal equations in (2.3) with M = 0 are representative of a class of so-called "perfect coupler" components. Other examples of

perfect couplers are electrical transformers with negligible losses and gear trains

with negligible bearing friction and rotational mass.

Since the hydraulic control valve in Fig. 2.1 translates the mechanical variables of force and displacement measured between B and G to hydraulic variables of

pressure and flow rate between C and D, it is referred to as a two-port transducer . Figure 2.10a shows the control valve together with a hydraulic pump where the two

devices can be considered as a single two-port component. The terminal graph in

Fig. 2.10b identifies each of the ports and the associated pairs of complementary

variables. The terminal graph is separated into two parts since two different types

of variables are identified.

The terminal equations for the valve depend upon those characteristics of the

valve which must be taken into consideration. A reasonable linear model results when

the variables are interrelated by equations of the form

£

x (t)

g,(t)

(B

i ir + Mi 4-

dt K

21 22 P,(t)

(2.4)

These equations include the mass and friction characteristics of the spool but assume

that the pressure difference at points C and D does not affect the motion of the

spool. The constants and Kg a depend upon the valve dimensions and the pressure

applied to the valve by the hydraulic pump; the pump being of constant pressure type.

15
In a similar manner, it can be shown that the linear differential equations approximating the performance of the hydraulic ram in Figs. 2.1 and 2.11a are of the

form

d

dt sW <- K65dT>

L 56 6

6

(t)

P

5

(t) (2.5)

where the variables in the model, of course, are identified by the terminal graph of

two parts in Fig. 2.11b.

In general, a mathematical model of an n-terminal electrical, mechanical or hydraulic component consists of: 1) a terminal graph of £ parts having n vertices cor- responding to the n terminals of the component and (n-p) oriented line segments

connecting the n vertices, and 2) a set of (n-p) equations relating the 2 (n-p)

variables identified by the terminal graph.

3. Basic Graph Theory and Formulation of the State Model

The process of deriving or formulating a mathematical model of a system of interconnected components can best be described as a uniting of two sets of equations; this

union producing a third set of equations called the system model . The two sets of

equations from which the final model is evolved are derived from two separate properties of the system. One set, called the component equations, is simply a list of all

terminal equations of the components in the system. They represent the characteristics

of the individual system components and do not depend on the manner in which the com- ponents are connected together. The interconnection of the components (the system

topology) is described by the second set of equations called interconnection or constraint equations . This set is always algebraic in form and relates the through

and across variables at interfaces where the components are connected together. It

is generally difficult to write the interconnection equations directly from a picture

or schematic of the system. An examination of Fig. 2.1, for example, does not immediately reveal how many independent relationships are available and which variables

should be related. This information is more efficiently obtained from a collection

of oriented line segments called the system graph .

The system graph is formed by simply connecting together the terminal graphs

from the mathematical models of the components in a one-to-one correspondence to the

actual physical connections of the components in the system. For example, the system

graph for the hydraulic subassembly shown in Figs. 2.1 and 2.3 is obtained by uniting

the terminal graph of the control valve and pump assembly in Fig. 2.10b with a terminal

graph for each of the hydraulic lines given in Fig. 2.5b and the terminal graph of the

hydraulic ram given in Fig. 2.11b. The result is the collection of line segments shown

in Fig. 3.1.

The oriented line segments in the system graph are called edges . The end points

of the edges are called vertices and a set of edges forming a closed path is identified

as a circuit . In this particular example, the graph contains 8 vertices, 6 edges, and

only one circuit. In this respect it is a very simple system graph.

A simple graph having more than one circuit is shown in Fig. 3.2. In this case, three distinct circuits are formed by the sets of edges (3,4,5), (5,6,7) and (3,4,6,7).

The set of edges (1,2) forms a path between vertices a_ and b while three distinct

paths exist between vertices _f and .e; (5), (3,4) and (7.6). The subgraph (1,2,3,4)

is not a path. A graph is said to be connected if there exists at least one path

between every pair of vertices in the graph. All other graphs are separated graphs.

The graph of Fig. 3.2 is a separated graph. It contains two connected subgraphs

(1,2) and (3,4... 7). The subgraph (4,5,7) is connected while the subgraph (4,7) is not.

16
The two fundamental postulates of system theory state that 1) the summation of

across variables associated with the edges of a connected graph vanish around any circuit and 2) the summation of the through variables vanish at the vertices. Thus, in

the connected subgraph (3, 4,... 7) of Fig. 3.2 we have for the three circuits

x + x - x. - x = 0

3 7 6 4

x + x - x = 0 5 6 7 and for the four vertices

y 3 + y4

= 0

y4

+ y5

" y6

= 0

y7 + y6

= 0

y 3 y5

- y7 = 0

(3.2)

The reader can easily verify that one of the three circuit equations in (3.1) can be obtained as a linear combination of the other two and one of the four vertex

equations in (3.2) can be obtained as a linear combination of the remaining three.

The equations are, therefore, not independent.

A basis for selecting independent circuit and vertex equations is provided in

the concept of a tree. A tree of a connected graph is defined as a connected subset of edges that contains all vertices of the graph but contains no circuits. Each

connected subgraph of a graph contains at least one tree. For example, the connected

subgraph (1,2) in Fig. 3.2 contains only one tree and it is the subgraph itself.

On the other hand, the connected subgraph (3,4,5,6,7) contains several trees; (3,4,7),

(3,7,6), (4,6,7), (3,4,6), (4,5,7), (3,5,7), (4,5,6) and (3,5,6). Subgraph (3,4,5,7)

is not a tree because it contains a circuit. Likewise, subgraphs (3,6) and (3,5)

cannot be trees because the former is not connected and the latter does not contain

all of the vertices of the subgraph.

A collection of trees is appropriately called a forest , and a forest of a graph

contains exactly one tree from each connected subgraph. The subgraph (1,2,3,4,7) comprises one forest of the graph in Fig. 3.2 and it is possible to define seven other

distinct forests in the graph. After leading the reader this far into the woods,

there should be no argument if the edges of a tree are called branches . For every tree defined in a connected graph, there is a corresponding subgraph

called a co-tree which contains those edges of the graph not contained in the tree. The edges of a co-tree are called chords . As a usual order of events, a tree of the

graph is selected first and the corresponding co-tree is thus defined. In the con- nected subgraph (3,4,5,6,7) of Fig. 3.2 for example, the tree (4,5,6) defines a co-tree with chords 3 and 7. If the tree of a graph is the graph itself, as in the

subgraph (1,2), then the co-tree is an empty set.

If a connected graph has e_ edges and v vertices, then as a result of the definitions above, any tree of the graph always has v-1 branches and e-v+1 chords . The connected subgraph (3,4,5,6,7) has 5 edges and 4 vertices. Thus every tree of

this graph must have (v-1) = 3 branches and (e-v+1) = 2 chords.

One important property of a tree is that the addition of a chord between any two

vertices of the tree establishes a circuit. This is evident when the edges of a tree

are identified by heavy lines as in Fig. 3.3a. The tree indicated defines a co-tree

with chords 5 and 7. Each of these chords, when added to the tree defines a unique

circuit composed of the defining chord and a subset of the branches of the tree.

17
The circuits defined by the chords of a tree are called fundamental circuits of

a graph. Since a defining chord appears in only one fundamental circuit they are unique and form a basis for writing independent circuit equations from the graph. For example, the two fundamental circuit equations for the graph of Fig. 3.3a are

x

5

- x

4

+ x

3 = 0

(3-3) X

7

" X

6

" X

4

+ X

3 = °

an and we see that the two equations are independent since each equation includes across variable not contained in the other. The two circuit equations in (3.3) can also be written as

5 4 3 x

7

= x

6 + x

4

" X

3

(3.4)

It can be shown that, as in the above example, the circuit equations for the

system graph can always be written so as to show the across variables of the chords

as. an explicit function of the across variables of the branches . Such circuit equa- tions are called the fundamental circuit equations and are written after a tree has

been selected in the system graph. The number of such independent circuit equations,

of course, is equal to the number of chords in the graph, i.e., (e-v+1) if the graph

is connected and has e edges and v vertices

A similar criterion for obtaining independent relationships among the through variables follows from another interesting property of the tree. Let S be a set of

edges of a connected system graph which has the following properties: (1) removal

of the set S leaves the graph in exactly two parts (an isolated vertex qualifies as

a part) and (2) the set S contains exactly one branch of a tree plus the minimum

number of chords necessary to satisfy property (1) . Such a set of edges is called

a fundamental cut- set or simply a cut-set. Referring to the graph of Fig. 3.3b, the

three cut-sets defined by the branches of the tree are indicated by the dashed lines

I, II, and III and consist, respectively, of edges (3,5,7), (4,5,7) and (6,7). It

is evident that each cut-set is unique in that each branch of the tree appears in

exactly one cut-set.

An almost obvious generalization of the vertex equations is that the summation

of the through variables associated with a cut-set of edges vanishes identically.

The three cut-set equations for the graph of Fig. 3.3b, for example, are

y 3 - y

5

- y7

= o

y4

+ y 5 + y 7

= ° (3-5)

y6 + y7 =o

These equations are clearly independent since each one contains a variable not con- tained in either of the others. This implies also that the three cut-set equations

in (3.5) can also be written as

y 3 = y 5 + y7

Y4

=

~ Y 5~ y

7 (3.6)

y

6

=

- y

7

18
Note that these equations show the branch throu gh variables as an explicit function

of the chord through variables . Such a system of equations is called the fundamental

cut-set equations. It is not difficult to see that since each cut-set contains

exactly one branch of the tree, the cut-set equations for a graph can always be written in this form and that the number of such equations is exactly equal to the

number of branches in the graph, i.e., (v-1) if the graph is connected and contains

v vertices.

It might appear that the circuit equations given in (3.4) and the cut-set

equations given in (3.6) are entirely unrelated by virtue of the separate methods

used to derive them. To the contrary, they are very much related. But the relationship between them is only evident when (3.4) and (3.6) are written in the matrix

forms

and

• 1 1 0

• 1 1 1

1 1

-1 -1

0 -1

(3.7)

(3.8)

The arrays of ± 1 and 0 coefficients to the right of the equality signs are called

the fundamental circuit and cut-set matrices, respectively. Note that one coefficient

matrix is the negative transpose of the other, i.e., one matrix has two rows and

three columns while the other has three rows and two columns and the entries in the

rows of one matrix are identical to the negative of the entries in the columns of

the other. The coefficient matrices of the fundamental circuit and cut-set equations

of any system graph are related in this manner. Thus, if the vectors X^ and X

represent the across variables associated, respectively, with the branches and chords

of a system graph, and if Y and are the corresponding vectors of through variables,

then (3.7) and (3.8) can be regarded as specific examples of the general system of

fundamental circuit and cut-set equations.

3Xb X = (3.9)

and

Y, = AY b c (3.10)

where A and B are rectangular matrices and A = -B , i.e. A is the negative transpose

Of B.

Among other things, (3.9) and (3.10) imply conservation of energy in any closed

system. To show this, note that from (3.9) and (3.10) it follows that

X = U

e X

c B

(3.11)

19
Y =

e

Y

b A ~-B

T

"

Y

Y U

c U

c (3.12)

aid whi n

e liters containing all across and through variables, respectively, and where U represents the unit or identity matrix. The total power (or energy) in the system is given by the scalar product, or dot product of the vectors X and Y written from 3.11 and 3.12 as e e

T T T X Y = X [U ; B ] -B

U

T T T Y = X^f-B 1 + B

X CD )Y

c (3. 13)

For the circuit and cut-set equations in (3.7) and (3.8) the expression symbolized in (3.13) is specifically

LX

5 X

7 > "3 ~4 ^6 J X

4 X

6

] "l 0 0 -1 -1"

0 1 0 1 1

0 0 1 0 1

1 1

-1 -1

0 -1

= [-3 x

4 XJ

= 0

0 0

0 0

0 0

In summary, if a connected graph contains v vertices and e edges, there are always e interconnection equations; (v-1) cut-set equations expressing the branch through variables and (e-v+1) circuit equations expressing the chord across variables in terms of the branch across variables. Thus, the interconnection equations of a system provide e independent equations in 2e unknowns. These e equations combine with the e component equations, (in the same variables) characterizing the performance characteristics of the components, to provide 2e equations in 2e variables. This set of equations constitutes the basis for all mathematical models of the performance characteristics of the system.

The final explicit form of the mathematical model of a system varies greatly, depending on the form of the equations used to model the characteristics of the components, the type of information to be gained from the analysis and the "machinery" available for solving the model. The so-called state model is easily derived for most systems and is exceptionally well suited for machine solution by analog or digital

computer.

20
Considering the hydraulic lift system of Fig. 2.1, the state model of the entire

system can be formulated in one operation but it is simpler to consider the system as

a collection of three subassemblies as in Fig. 2.2. In this way, the complete system

state model is derived from the three individual models; again considering the system

as the sum of its parts.

Consider first the hydraulic subassembly shown in Fig. 2.3. We desire a mathematical model of the input-output performance characteristics of the assembly as a two-port component. The models of the various components in this subassembly and the

resulting system graph have already been discussed in detail with the system graph

shown explicitly in Fig. 3.1. This graph is repeated in Fig. 3.4a with a tree indicated by heavy lines. The required mathematical model is obtained by using the fundamental circuit and cut-set equations of the system graph to derive a set of equations

showing f1 (t) and 6 S (t) as a function of b 1 (t) and fs (t) only. Note that 6 X (t) and

6 8 (t) are not in general measured with respect to the same fixed reference G. Since

the relative time variation between 6 X (t) and 6 6 (t) is of primary interest, the derived

mathematical model will not account for the constant distance between the two fixed

references. In the interest of simplifying the algebra in this example development,

the mass and friction of the control valve are neglected and the hydraulic lines are modeled as pure resistors. With these simplifications, the component terminal equations (2.4) and (2.5) together with the terminal equations for the hydraulic lines can be listed as follows:

d_

dt

(-B

6 /M

6

)66

- (K

56/M

6

)p 5 + (VM

6

)f6

(3. 14)

' (-K

21/K22) 6j + (1/K 22 ) g 2 P 3 R

3 h

P 4 R

4

g4

" K

6 5 *6

(3.15)

Note that the component equations have been divided into two groups; the differential

equations and the algebraic equations. Note further that the differential equations have been written in a form explicit in the first derivative of the variable. In addition, the algebraic equations are explicit in the across variables if, in the

system graph of Fig. 3.4a, the corresponding edges are branches of the tree and

explicit in the through variable if the edge is a chord.

The second step in developing the required model of the subassembly is to use

the fundamental circuit and cut-set equations of the system graph in Fig. 3.4a to eliminate all branch through variables and all chord across variables from the component equations in (3.14) and (3.15). In this particular example, the interconnects

equations are simply

P 5 =

- P

3 + P 2 + P 4

(3. 16)

g

3 = g

5

. g4

= - g5

, g2

= - g

5

21
substituting (3.16) into (3.14) and (3.15) gives

d

dt

( -VM6 ,6

6

- (K

56 /M

6

)( P 2

- P

3 + P4 > + < 1/m

6 >V (3: 17)

1 0 0 1/K 22~ P 2

( - K

21/K 22> 6

l

0 1 0 " R

3 P

3 0

0 0 1 R

4 P

4

0 (3.1

0 0 0 1 %

" K

6 5 5

6

The third, and last, step in developing the required model of the subassembly is

to solve the algebraic system of equations in (3.18) for p2 , p3 and p4 and substitute

the result into the differential equations in (3.17). The solution in this particular

case is very simple and the resulting model of the system is

_d_ dt

_d_ dt

K

56 K65

M6

0

0

1

1 • KqA K — ?1 — + R + R )6, +

1 6 + K

22

3 4 6 M6 K22 1 M6 (3.19)

< K

56 K 21> M6 K22

1 M,

where

\ - - < B6 /M6

J " (K

56 K

65 /M6 )(i/K 22

+ R

3 + V

(3. 20)

The final model in (3.19) has the form of two simultaneous differential equations

in the unknowns 6 6 and 6 S . The equations are first order nonhomogenous and are linear

under the assumption that the coefficient matrices have constant entries. For any

specified time variation of the terminal variables (drivers) 6 X and f6 , the solution

to (3.19) yields 6 S and 6 6 as functions of time and the specified drivers. The "state"

of the system is dependent upon 6 S and 6 S since the remaining variables in the subassembly are determined by the appropriate relationships in (3.18) and/or the interconnection equations. For this reason, the variables 6 S and 6 S are called state

variables and the differential equations the state equations . These equations together

with the trivial terminal equation f1 = 0 comprise the state model of the subassembly.

Combining the terminal graph of Fig. 3.4b and the state model, we have a complete

mathematical model of the component ; thus allowing the hydraulic subassembly to be

regarded as one two-port component of the larger system.

In a similar manner, it can be shown that the state model of the lever and mass

load in Fig. 3.5a, considered as a three-terminal component, is

22
_d_ dt

(-1/ MT K ) L s

£ 1= ( - n

e /Ks) 5

8 +(n

e /Ks )6 7

(n / M K ) (l/M, ) e L, s L 1

(3.21)

where the variables in the state and terminal equations refer to the graph of Fig. 3.5b.

Also, n = UaAj ) represents the ratio of lever arms, K represents the spring con- stant of the lever and represents the load mass. TheS mass of the lever has been

neglected.

Assuming that the control lever AB in Fig. 2.1 is "ideal" its terminal equations

and terminal graph are taken, respectively from (2.3) with M = 0 and Fig. 2.8b. A

state model of the complete hydraulic lift system is obtained by first combining the

terminal graph of the control lever with the terminal graphs in Figs. 3.4b and 3.5b, consistent with the union of the components, to establish the system graph of Fig. 3.6.

The unoriented line is simply an expanded vertex, edge 11 identifies the independent

input variable 6 11( edges 9 and 10 identify, respectively, the input and output variables of the control lever, edges 1 and 6 identify the input and output variables of

the hydraulic subassembly and 7 and 8 identify the input and output of the mass-lever

subassembly. The respective component graphs in this case are said to be connected

in cascade.

A list of the terminal equations for the components in the system is

_d_ dt MT K

L s

K

56 K21 M6 K22

M, K

L s

M.

(3. 22)

and

10 \

6 9 — 6

8

s

-n f c 10

s (3.23)

where n represents the lever arm ratio of

the system graph has been selected so that

explicit in branch displacements and chord

the control lever. Note that the tree in

the algebraic equations can be written

forces or their time derivatives.

23
As in the development of the state model of the hydraulic subassembly, the state

model of the entire hydraulic lift system is obtained by: 1) eliminating all branch

forces and chord displacements from (3.22) and (3.23) by applying the fundamental circuit and cut-set equations for the tree indicated in the system graph, 2) solve the

resulting algebraic system of equations in (3.23) for 6 10 and f7 and 3) substitute

this solution into the differential equations. The result is the set of first-order

differential equations characterizing the performance of the system.

dt

M. K

6 s

MT K

L s

M,K

6 s

-1 MT K

L s

n

c K

2lK

56 M6 K

22

0

'11

(3.24)

11

The differential equations are the state equations of the system since all variables in the system are obtainable as a linear combination of the variables appearing

in the state vector. In this particular case, the two input variables are the control

lever position b xl and the externally applied load force f8

. If no additional load

force is applied then fe = 0 . The output variables of primary interest is the load

position 6 8 and the force fxl required to move the control lever. The latter force is

identically zero under the assumptions used to model the control valve and

explicitly as one of the variables in the state vector.

)@ appears

4. Solution of the State Model

All performance characteristics of the hydraulic lift system are implied in the

solution of the equations of state given in (3.24). Linear state models of this type can be solved analytically to obtain the output variables. In addition, they may be

solved numerically by either analog or digital computers to obtain a plot of the output for any particular specified time variations in the input variables. Thus, the

solution for a state variable z x (t) might typically be given in any one of the three

forms shown in Fig. 4.1.

It is evident that the analytic solution is the most desirable since the other

two solutions can be written directly from it. On the other hand, the exact analytic

solution can never be found from a numerical solution although under certain conditions

it can be approximated. As might be expected the analytic solution, as the most

desirable type, is frequently the most difficult to obtain. In fact, an analytic

solution may not be obtainable in a usable form for some state models, particularly

nonlinear ones.

The use of analog computers in the solution of state models is considered by

Hammond. The emphasis here is, therefore, on analytical procedures. These procedures

fall into two basic classes; Laplace transform' methods and functions of matrices. Both

of these procedures are considered briefly together with a short discussion of a digital computer method based on functions of matrices.

A widely-used analytical method for solving ordinary linear differential equations

is based on the Laplace transform. Many useful design concepts and procedures employ

the so-called complex frequency models. These models can be obtained by directly

applying the Laplace transform method to the state equations. The reader unfamiliar

2 A
with the Laplace transform will find it instructive to supplement the discussion of

the subject given here with reading in almost any modern book on feedback systems.

Our purpose here is to show the application of the Laplace transform to the solution

of the state model and derivation of the complex frequency model from the state model,

If F(t) is a vector of time functions

F(t)

f

x (t)

f 2

(t)

fn(t > n

(4.1)

then the Laplace transform of F(t), written L[F(t)], is defined to be the vector

L[F(t)] = F(s) =

Lt^ft)]

L[f

2

(t)]

\(s)

f2

(s)

_

fn

(S »

(4.2)

where

L[f.(t)] = f.(s) = / f.(t)e"

Stdt, i = l,2, ...,n (4.3)

and _s is a complex variable s = o + jiu sometime called the complex frequency . It is

assumed, of course that f.(t) in

integral exists. A short table c transforms is given in Table 4.2.

assumed, of course that f.(t) in the definition of (4.3) is such that the indicated

integral exists. A short table of time functions and their corresponding Laplace

It is easy to show through integration by parts that if the Laplace transform of

a vector Z(t) is Z(s) then the Laplace transform of — Z(t) is given by

dt

L[ — Z(t)] = sZ(s) - Z(0+) (4.4)

where Z(0+) = Z = lim Z(t) with t assuming only positive values.

° t >0

Thus, Z represents the initial value of the vector,

o

25
Consider now the state model

-j-jT Z(t) = AZ(t) + BF(t) (4>5)

E(t) = CZ(t) + DF(t) (4-6)

Where F(t) and E(t) are vectors representing the input and output variables respective ly, Z(t) is an n-dimensional state vector and A, B, c and D are constant matrices of appropriate dimensions. Note that the state model developed in the previous section is a degenerate from of the general model in (4.5) and (4.6).

Taking the Laplace transform of (4.5) and (4.6) and applying (4.4) gives the so-called complex frequency or s-domain model .

sZ(s) - Z

q

= AZ(s) + BF(s) (4.7)

E(s) = CZ(s) + DF(s) -(4.8)

Note that the transformation has converted the differential equations in (4 5) to an algebraic system of equations. Consequently, it is now possible to solve (4.7) for Z(s). Collecting the coefficients of Z(s) in (4.7) yields

(sU - A) Z(s) = Z + BF{s) (4.9)

o where U is an nxn unit matrix. The coefficient matrix (sU - A) is called the charact eristic matrix and its inverse can be written as ^-..iay*^.^ (4. 10)

P(s) is the adjoint matrix of (sU - A) and has as its entries polynomials in s of degree (n-1) or less. The polynomial d(s) is called the characteristic polynomia l of A and represents the determinant of (sU - A). Solving (4.9) for Z(s) by multiply- ing both sides by (sU - A) 1 defined in (4.10) we obtain

Z{S)= WT P(s)Z

o

+ dTIT P < S > BF< S

> (4.H)

For any given F(t), F(s) is a function of s and the vector Z(s) as given by (4.11) represents the Laplace transform of some vector function of time. The only question

is what function of time. If this can be determined then the solution to the differ- ential equations in the state model has been determined. The process of going back from a known function of s to the corresponding function of time is called the in- verse Laplace transform and is best achieved by expanding the ratios of polynomials on the right-hand side of (4.11) in partial fractions. But to realize this expansion, the zeros of d(s) must be known. Let d(s) be written in the factored foi>rm

d(s) = (s - Sl ) (s - s 2

) --- (s - s ) (4.12)

Assuming that the zeros s x , ss , — s are distinct, Z(s) can be expanded in partial fractions as

R, R 2 R F, F,

Z(s)=- i + + . + 5 +

1

, 2

, (s - s ) (s - s ) +

(s - s ) +

(s - B ) +

(s - sj +

1 *" n l L

26
where the constant vectors and F. are given by

where the constant vectors R. and F. are given by

(s-s.)

R

i = Hm "dfif P(slZ

0

' (i=:U-'=- 11) s —>s

1

F. = lim

s —>s

1

(s-s.)

d(sf~ p( s ) BF( s )-(i = 1. 2 n,a,b..k)

and V %

S k are distinct Poles contributed by the vector F(s). By definition

a vector F(s) has a pole at s = s if for at least one entry f . (s) in F(s) we have hr X lim f.(s) = oo (4 14) s—>sp Each factor in the partial fraction expansion in (4.13) corresponds to the Laplace transform pair:

F(t) = Keat =>F(s,,

II^

r

(4 . 15)

where K is a constant matrix (or vector) and a is a constant scalar. Applying this equivalence to (4.13), it is evident that the state vector Z(t) which has Z(s) as its Laplace transform is

(4. 16)

Z(t) =Gxe + G2

e +... + Ge n

+ F e a

+ ... + F, e

^

where the vector = R. +F., i = l,2,...,n.

Thus, (4.16) represents one general form of the solution to the state equations of (4.5). The purely exponential character normally occurs only when the zeros of d(s) are distinct. When repeated zeros occur, the solution may have terms of the

s . t

p i- *- m i torm G

± t e , m = 1,2 n. In general, the zeros of d(s) can be real or complex and when complex they occur as conjugate pairs. A stable solution Z(t) is one having bounded (finite) state variables as time increases without bound (assuming the input variables m F(t) are bounded). An examination of (4.16) reveals that Z(t) is a stable solution if all zeros of d(s), s llSs ,..s^ have negative real parts . This follows

immediately from the fact that e

j - 0 as t - - when the real part of

s < 0, j - 1,2, ...n. Further discussion of stability and a detailed example of a sblution by Laplace transform methods is given in Section 5. The complex- frequency or s-domain model of the system is, by definition, obtained by substituting the solution of (4.7) for z(s) into (4.8) with Z =0. The result is o

E(s) =[C(sU - A)

_1 B + D]F(s) = G(s)F(s) (4.17)

where G(s) = [C (sU - A)

1 B + D] is a matrix whose entries are ratios of polynomials. Note that the initial conditions are taken as zero. This is always implied in a given s-domain model. Thus, the s-domain mathematical model of a typical two-port component takes the form

27
Gn

(s)

x

2

(s)

G

12

(s, Xl (s)

y 2

(s)

(4. 18)

where G (s) , (1, j =1,2), is a ratio of polynomials in s If nnrf i •„ as the input port and 2 as the output port then G if f " ldentlfie<2 quency input admittance, GS2 ( s ) is 1 ^15' 13 Called the COmPlex

G31 (s) the forward transfer function and G LfTf ^equency output impedance,

unction. Th e entire coefficientTtrS £^"^•LTS^TSl,.

tern a^d ^^00^ th6

, f0rWard f—ion of a sys . (4.10) in the .efinition^f^LS^UrSLTu?!^!": IST

^

G(s) L d(s) J

tracer L^"or(4°SrS SfoTalj' Tl^'f "tl0 G

ii "> * «»

Therefore, the foJro

~

G21

(S) - d(T)

n(s; (s-s )(s-s ) ... (s-s ) 1 * n

Sg

mP

2ei

4

;« oSain^T" f

nSf" characteristics °f the hydraulic subassembly in c.s

g

; M'lBrh^lne^pa'ti^arfLr5 Pr°"dUre deS"ibed "TM '° - this

6

6

(s; J61' J66

s) f6

(s)

(4. 19)

where G. , (s) =

fa 1 K

56 K

21 M6 K2^ A

h

s) (4. 20)

G

66

(s >= 7—T M6

(s -Ah

s)

(4. 21)

28
Note that the characteristic polynomial of the coefficient matrix in (3.19),

d(s) = det (sU - A) = s

2 - s appears in the denominator of GB1 (s) and G66 (s)

Also, in this particular case both the s-domain input admittance and return transfer

function are identically zero. Such a component is said to be a unilateral component

with zero input admittance—the output variable fs (s) has no effect on the input

variable fx (s) . In a special application where the load connected to the assembly

_is negligible , f6 (s) is taken as zero and only the no - load transfer function Gs 1 (s) is of interest, and we write

66

(s) = G6l

(s) 6j(*) (4.22)

Consequently, if two unilateral components are cascaded to form the system in

Fig. 4.3a, then under the assumption of no loading, the system transfer function is

given as the product of the component no-load transfer functions. The block diagram

shown in Fig. 4.3b is used to indicate this very special but important result. The

blocks and arrows imply that

x

4

(s) = G2

(s) x

3 (s) = G2

(s) G^s) x^s) (4. 23)

and therefore the system transfer function is, by definition

G{S)= 1^T = G2 (S)G1< S)

< 4 - 24 )

The transfer function and the associated block diagram are valuable tools in

treating complex systems where no "loading" occurs between components, i.e., when

the through variable measured at the interface between each pair of compoents is zero

or can be assumed so. More discussion relating the transfer function to the concepts

of feedback and stability appears in Section 5. Another technique for obtaining the analytical solution to a linear state model

is based on functions of matrices . A function of a matrix is a matrix istelf and

can be evaluated in the same way as the scalar function h = f (x) . For example, let

the matrix Y be the following polynomial function of the matrix X

Y = f(X) = U + 3X + 8X

2

- 15X

3

(4. 25)

Upon substituting a square matrix X into the right side of (4.25), the matrix Y can

be evaluated.

To illustrate briefly the use of matrix functions in the solution of linear differential equations, consider the nohomogenous state equations

Z(t) = A Z(t) (4. 26)

with initial conditions Z (0+) = Z . Suppose the nxn matrix eAt

is defined so that o

and

^(eAt

) = AeAt

(4.27)

At lim e = U (4. 28)

where U is the nxn unit matrix. Under these conditions the solution of the state

equations in (4.26) can be expressed as

At

Z(t) = e Z (4.29)

o

29
By substituting (4.29) into (4.26), Z(t) can be verified as a solution with initial conditions Z . o At The matrix e is a function of the matrix A; this function being defined by the infinite matrix series

e =U+At+ A t At

2 2 A

3

t 3 + ~JT~ + • • • (4. 30)

Notice that the properties given in (4.27) and (4.28) follow directly from (4.30).

The series in (4.30) can be evaluated analytically, and in closed form in terms of constant matrices and exponentials, i.e.

At V V X t e = C e + C e + . . . + C e n

1 ° n

(4.31)

where the constant matrices C, , Ca , — are called constituent matrices and the constants X 1 , \ s — X are the eigenvalues of the matrix A; assumed here to be dis- tinct. Substituting (4.31) into (4.29) yields the final analytic solution to the homogeneous state model as

X t x

2

t X t Z(t) = R e + Re +...+R e n

(4.32)

1 L n

where the vector R. = C Z i = 1,2 n. While the procedure for obtaining the constituent matrices cx , C8 . . . C is not difficult, a more detailed discussion of the method is beyond the scope of this paper. The interested reader should consult the attached list of references.

It should be noted that the method outlined above applies only to the homogeneous state model given by (4.26). However, the non-homogeneous model in (4.5) can be trans- formed into a homogeneous form when certain conditions on the input variables in F(t) are satisfied. These conditions are not stringent; being roughly equivalent to the requirement that the Laplace transform of each variable exist.

In view of the fact that a later paper discusses the application of computers

to the solution of differential equations, a detailed development is not given here

However, one particular method, based on the series representation of the matrix eA* m (4.30) deserves some comment. This method utilizes the digital computer and pro- vides a discrete numerical solution of the homogeneous state model of (4.26). Although the method provides an approximate solution, the approximation error can be made as small as desired.

Let the interval of time over which a solution is desired be divided into small increments of length h as shown in Fig. 4.4. Let z.(t) be a typical state variable

of the state vector Z(t). With the time increment £ specified, we approximate the matrix eAn by evaluating the first few terms of the series

2 2 3 3 e Ah Ah Ah = U + Ah + + ... (4.33)

The number of terms considered naturally depends upon the allowable solution error. From (4.29) we can write

Z(l) = Z(h) = e An

Z (4.34)

o * '

30
Z(l) approximates the value of the state vector Z(t) at time t = h. Letting Z(l) be

the new initial condition we can similarly write from (4.29) that

Ah

Z(2) = Z(2h) = e Z(l)

• Ah 2

= (e ) Z (4. 35)

Continuing in this manner, the value of the state vector at any discrete time t = nh can be approximated by

Ah

Z(n) = e Z(n-l)

Ah n

Z(n) = (e ) Z (4.36)

The approximation can obviously be improved by reducing the size of the increment h.

The recursion formula given in (4.36) is easily programmed on the digital com- puter as is the original evaluation of e^11 from (4.33). At each iteration, the matrix

eAh is raised to the next highest power and multiplied into the given initial vector

Z . This process yields a sequence of numbers for each of the state variables.

Fig. 4.4 shows the resulting solution for a typical variable.

5. Feedback Systems

The concept of "feedback" is basic to Hie study of almost all control systems and

is centered about a particular pattern of interconnection between components. Although deceptively simple to define, the effect of feedback on a given system may be

complex. Suppose it is desired to accurately control the value of the output across variable x (t) by varying the input x. (t) . Theoretically, if the exact numerical cor- respondence between input and output values is known, the required input signal for a desired output can be calculated and the control problem is trivially solved on paper

by an open-loop system . However, in practical systems, no exact relationship between

the input and output is available. Even an approximation to the relationship constantly changes since it is dependent upon load conditions at the output and variations in

the system itself due to wear, age and change in environmental conditions. The objective is to design a system that is insensitive to such changes.

To this end, "feedback" is applied to the open-loop system by sensing the value

of the output and returning this information, possibly with modification, to the input

for comparison with the input signal. Such a scheme is indicated in Fig. 5.1a where

a feedback component is used to sense, modify and return the output signal of the open- loop system to the input terminals. Considering the open- loop system and the feedback

component as two two-port devices, it is evident that ports 1 and 4 are in series with

the input port 5 while ports 2 and 3 are in parallel with the output port 6. Such an interconnection is called the series -parallel feedback connection and this description

is further substantiated by the form of the system graph in Fig. 5.1b. Ideally, the

components should be unilateral with zero input admittance. However, this restriction

is not necessary.

Given the state models of the open-loop system and the feedback component in

Fig. 5.1a, the state model of the corresponding closed- loop system can be derived

directly by combining the two state models and the interconnection equations from the

indicated system graph.

31
As a practical example of feedback control, consider the electro-hydraulic system

in Fig. 5.2. The output variable is the vertical displacement of the hydraulic ram

piston. The position of the piston is to be precisely controlled by an input voltage

v. at the potentiometer FE in

The components enclosed by the dashed lines in Fig. 5.2 correspond to the openloop system block in Fig. 5.1a„ The amplifier-solenoid forms an ideal linear transducer. That is, the vertical displacement of point P is proportional to the voltage

v1 and further it is assumed that P undergoes a positive displacement (up) when vx is

positive (A positive with respect to B) „ Under these assumptions, the state model of

the open-loop system is obtained by combining the state model of the hydraulic subassembly given in (3.19) and the linear relationship for the amplifier-solenoid

assembly: 6, = K v, v

1 The result is

d

dt

6

2 A

h

0 6

2

1 0 6 2j

K • K

h v

1M

(5.1)

K

56 K

21 where K. = —

— , M = M, and A, is defined by (3. 20). h M. K-.-. 6 h

1 ' O 22 The feedback component in Fig. 5.1 consists of the potentiometer DE in Fig. 5.2

which furnishes a voltage v4 proportional to the vertical position of the piston of

the hydraulic ram. Its simple algebraic model is

V

4

= K

f 63

f 3 =°

(5.2)

Finally, two independent variables are specified at the input and output terminals

of the feedback system. These are

V = V. 5 m

6 L

(5.3)

Writing the interconnection equations from the system graph in Fig. 5.1b and combining

them appropriately with (5.1-3), the state model of the feedback system takes the form

_d_ dt

25=6

-K K K

v h f v h

(5.4)

In general, it is desirable to obtain from the mathematical model as much information as possible concerning the performance of the system without actually having

to obtain a detailed solution. Assuming for the moment that the system is stable, the

steady-state performance can be evaluated directly from the state model. Let a con- stant voltage input of magnitude V1 be applied to the system simultaneously with a

32
constant force loading at the output with magnitude F . Under steady-state conditions

and constant excitation, the time derivatives of 6 S vanish. Hence, from the first

state equation of (5.4), we can write

0 = - Kvhf2 K K 5 j + KvhlM K V - i F

L

or V F

If the input potentiometer is calibrated so that numerically, Vx /K equals the

desired value of 6 , then f 6=6 ^

L, L K K KM v h f (5.6)

The effect of loading the piston shaft is clearly indicated by (5.6); the steady-state

position error being zero only when F =0.

L The stability of the system can be obtained from the poles of the no-load transfer function. For simplicity, let the various constants in the state equations of

(5.4) take on numerical values so as to give

_d_ dt

" 6

2~ -5 -6 6

2 +

6 -2 V

in

5 2

1 0 _

6

2 0 0 fL

(5.7)

Following the procedure outlined in Section 4 for obtaining the transfer function from

the state model, we set the through variable f = 0 in (5.7), assume zero initial con- ditions for the state variables 6 2 and 6 2 and fake the Laplace transform of both sides

to obtain

6

2 (s) - 5 -6 6 2 CJ 6

s +

6

2

(s) 1 0 6

2

<s) 0

s + 5 S 6 2(s) 6

-1 5 _6 2

(s) 0

V. (s in

Solving for the state vector gives

6

2

(s)

6 2

(s)

1 s(s+5)+6

-6

s+5

(5.8)

33
Combining the last equation of (5.8) with the s-domain terminal relation 6 (s) = 6 S (s)

,

we obtain the no-load transfer function L

m in s + 5s + 6

6 L

(s) =

(s + 2Hs + 3) ' V

in

(S)

< 51 °)

The poles of G(s) are s: = -2 and s 3 = -3. Thus, if the input variable v. is

bounded, the time function 6 L

(t) corresponding to 6 (s) in (5.10) is a stafile time

function and the feedback system is stable. Note that we have assumed no force loading at the output.

When the components are modeled as ideal transducers, a very useful relationship

exists between the no-load transfer function G(s) of a closed-loop system and the

no-load transfer functions of the open-loop system G (s) and the feedback component

G

f (s). With regard to the system graph of Fig. 5.1b? we have the following relations

holding between the s-domain across variables.

By definition

x

1 (s) = x

5 (s) - x4

(s)

x

6

( s ) = x

3

( s ) = x

2

(s)

x

2

( s

) x

4

(s;

(5. 11)

G (s) = —— , GJs) = —— (5.12) O x^s) f x

3 (s) v

'

Combining (5. 11) and (5. 12) we obtain

x

6 (s) G (s) G(S)=

^ =

l + G

o

(s) Gf(s) < 5 - 13 >

Thus, for unilateral components, the transfer function of the closed-loop system is

easily expressed in terms of the transfer functions of the open-loop system and the

feedback component. Since the poles of G(s) determine stability and transient response

characteristics of a closed-loop system, much emphasis is placed on the zeros of the

expression 1 + G

q (s)G (s) . Essentially, all linear feedback system design techniques

such as root-locus ana Nyquist plots are based on this expression.

The complete transient response of a feedback system for a given input can only

be obtained by a full solution of the state model. Such a solution can b e accomplished

either by computer using a numerical technique or by a hand calculation using Laplace

transforms or functions of matrices. For simple problems, hand calculation suffices.

As a final example, we consider a complete solution to the state model of (5.7) with

Laplace transforms.

Let the initial conditions on the state vector be taken as

5

2

0) 0 z = o

5

2

0) -2

34
and let

'in = Vl u(t)

(5. 15)

define the independent variables where u(t) is the unit step function

u(t) = 1 for t > 0

= 0 otherwise

(5. 16)

with Laplace transform

L[u(t)] (5. 17)

Taking the Laplace transform of both sides of (5.7) and applying (5.14-17), we obtain

6 2

(s) -5 -6 62

{s) +0 +

6 -2 Vj/s

6 2

(s) 1 0 6

2

fs) -2 0 0 F

L

/ s

(5. 18)

Solving for the s-domain state variables gives

6 2

(s)

5 2

(s) (s+2)(s+3)

(s+2)(s+3)

1

-6

s+5

s -6

1 s+5

0

-2

6 -2

0 0

Vj/s

F / s J

(5. 19) The form of (5.19) can be compared directly with (4.11). A partial fraction expansion

of (5.19) has the form

5 2

(s)

6 2

(s)

R. R, F F. F 1 2 1 2 a

(s- Sl ) (s-s

2 ) (s-Sj) (s-s

2

)

" (s-s

a

)

1 s+2

12 -12

1 1 + 7TI

-6 4

6V, -2F

L

3V

1 +F

L

s+3

-bV

1 + 2FL

2 2V, --F 1 3 L

1 + —

s V F

1 3 L

3 5 (5. 20)
where, for example, the vector

(s+3)

F.. = lim

2

„

' * (s+2)(s+3)

-6

s+5 0 0

lim

< S+2 H S )

-2s

V /s

-6V, + 2F

2V,

Finally, applying the transform pair

2

3 F

L

, r , at. A

L[ A e ] s + a

we obtain as the inverse Laplace transform of (5.20) the time solution

-2t

e +F.

6

2

(t) 6L

(t) 12 +6V

X

B 2

(t) _6L

(t)_ -6 " 3V

1

12 -6V +2F

4 +2V, -|f

1 3 L

-3t

e + V F

1 3 L

(5. 21)

Assuming specific magnitudes of the input voltage and output load force Vx =0.5 and

F^ = 3, the time solution becomes

(5. 22)

6L

(t) 9

-2t -9

-3t 0

e + e +

6L

(t)^ -4. 5 3 -0. 5

Notice that the initial conditions (5.14) are satisfied and the solution is stable.

As t -> co, 6 - -0.5, resulting in a steady-state error of 1 unit. This error is

caused by the output load force. With F = 0, the solution (5.21) reduces to

6L(t)

6 L

(t)

15

-7. 5

2t

15

5

3t 0

0. 5 (5.23)

Thus, as predicted by (5.6), the steady-state error between input and output is zero when the load force is zero.

36
REFERENCES

Feedback Control Systems:

Chestnut, H. and Mayer, R. W. , Servomechanisms and Regulating System Design , Volume 1, John Wiley and Sons, Inc., 1951.

Kuo, B. C, Automatic Control Systems , Prentice-Hall Inc., 1962.

Thaler, G. J. and Brown, R. G. , Analysis and Design of Feedback Control Systems , second edition, McGraw-Hill Book Co., Inc., 1960.

Truxal, J. G., Automatic Feedback Control System Synthesis . McGraw-Hill Book Co., Inc. 1955.

Linear Graph Theory:

Seshu, S. and Reed, M. B., Linear Graphs and Electrical Networks , Addison-Wesley Pub.

Co., Inc., 1961.

Koenig, H. E. and Frame, J. S., An Introduction to Functions of Matrices with

Applications, Part III, IEEE Spectrum, to appear, Spring, 1964.

Mathematical Models of Physical Systems:

Koenig, H. E. and Blackwell, W. A., Electromechanical System Theory , McGraw-Hill Book

Co., Inc., 1961.

Koenig, H. E., Tokad, Y. and Kesavan, H. K. , Analysis of Discrete Physical Systems , McGraw-Hill Book Co., Inc., in publication.

Matrix Theory:

Basic Theory:

Hohn, F. E., Elementary Matrix Algebra , MacMillan Co., 1958.

Perlis, S., Theory of Matrices , Addison-Wesley Pub. Co., Inc., 1952.

Advanced Theory and Functions of Matrices:

Frame, J. S., An Introduction to Functions of Matrices with Applications, Parts I and

II i IEEE Spectrum, to appear, Spring, 1964.

Gantmacher, F. R., The Theory of Matrices , Vol. I, Chelsea Pub. Co., 1960.

Transform Methods and Complex Variables

Aseltine, J. A. , Transform Method in Linear System Analysis , McGraw-Hill Book Co.,

Inc., 1958.

Churchill, R. V., Complex Variables and Applications , second edition, McGraw-Hill Book

Co., Inc., 1960.

Lepage, W. R., Complex Variables and the Laplace Transform for Engineers , McGraw-Hill

Book Co. , Inc. , 1961.

37

Process Control in the Ceramic Industry

Ernest Hucke

ABSTRACT

The basic components in a block diagram depicting a general closed loop process

control system are covered. Emphasis is given to pneumatic and electronic sensors, transducers, transmitters, as well as controllers. Typical control systems, using

some of these components are shown in block diagram, and their characteristics are discussed briefly.

I . INTRODUCTION

The importance of representing each unit of control equipment as a block and to

show all units inthe correct relative position for easier evaluation of system performance has been discussed. In order to properly construct a complete system, however, one must know something about the components which will make up the control

system.

A simple closed loop control system is shown in Fig. 1.

PROCESS

PRIMARY

DETECTING

ELEMENT

[transmitter;

j transducer!

i i

MEASURING

ELEMENT

PROCESS

VARIABLE

SIGNAL O

SET

POINT

FINAL CONTROL

ELEMENT

COMMAND

SIGNAL CONTROLLEF

ERROR

SIGNAL

Fig. 1 Block Diagram of Closed Loop Control System

The first blocks that are encountered and discussed in the basic control loop

are the Primary Detecting Element and the Measuring Element. It is generally accepted

that the primary detecting element is that portion of the feedback that either utilizes or transforms energy from the controlled medium to produce a signal which is a function of the value of the directly controlled variable. The measuring element is

that portion of the feedback elements which converts the signal from the primary detecting element to a form comparable with the set point reference input. This chapter

will be limited to hardware that is being used in the Ceramic Industry and which can

form part of a control loop.

II. Temperature Detecting Elements

Filled Thermal Systems

Filled-system thermometers are generally classified as vapor, gas or mercury

filled, depending uponthe fluid in tie sealed measuring unit. The measuring element

is connected by capillary tubing to the primary detecting element, in this case, a bulb thermally coupled to the process. The measuring element converts the motion

proportional to temperature to a form compatible with the reference input, and, in

most cases, is also mechanically linked to a pen or pointer or to a control element.

39
The response speed of a typical thermometer-bulb is indicated in Figure 2, and a ther- mal system is shown in Figure 3„

MEASURING SPIRAL

Fig. 3 Thermal System with Case Compensation

Thermocouples

Two dissimilar metals welded together at one end and heated at this junction will develop a voltage at the free ends. This voltage is proportional to the temperature difference between the welded and the free end. Such a device can be used with a voltage measuring galvanometer to construct a pyrometer as shown in Figure 4.

v

1

i 1 r

thermocouple

compensating

resistor

Fig 4. Pyrometer Using Simple Deflection Galvanometer

40
TIME, MINUTES

Fig. 2 Speed of Response of Thermometer Bulb

By proper selection of the two metals, temperature ranges of from -300°F to +4000°F

can be covered by thermocouples. Figure 5 shows typical calibration curves.

A number of small thermocouples arranged in series is called a Thermopile. In normal use infrared radiation from a heated body is focused by a lens onto the measuring junctions of these thermocouples. The radiation detector is not immersed in the

heated source as with other temperature measuring devices. The thermopile can be used

for temperature ranges of from ambient to 7000 °F.

Radiation

Fig. 6 Radiation Pyrometer-Thermopile

41
Resistance Thermometers

The property of metals to increase in electrical resistance as temperatures rises

provides a temperature measurement known as resistance thermometry. The primary detecting element is a wire wound resistor, called a resistance thermometer bulb. The bulb

is connected to a measuring instrument incorporating a Wheatstone bridge. The speed

of response of a typical resistance thermometer is given in Figure 7. Response of Temperature Sensors

The dynamic response of most temperature sensors is determined by a thermal resist- ance and a thermal capacitance in the signal flow path. A bare thermocouple has a significant thermal resistance at the boundary between the environment and the junction.

Such an arrangement is analogous to an electrical R-C filter circuit as shown in

Figure 8. Its transfer function is a first order linear lag, described by the equation

[1] •

R

AAAA/WV

e, c

E, [']

Fig. 8 An Electrical R-C Filter Circuit

42
THERMOCOUPLE CALIBRATIONS 70

Reference Junction

At 32 °F © Shromel-Constantan §lron-Constantan (Type J) Copper- Constantan(TypeT) @Chromel- Alumenel (Type K) _(f)Platinum-Platinum-Rhodium(Type R)(I3% Rh) ©Platinum-Platinum-Rhodium(Type S)(IO% Rh)

12 14 16 IS 22 24 26 28 32 34 36 38 42 1000 2000 3000 4000

Temperature. °F —»- Tenative Curves: ® Rhenium-Molybdenum

® Rhenium - Tungsten © Iridium- Iridium- Rhodium

100 —

n Moving Liquid

6 8

Time, Minutes

In Moving Air

10 12 14

Fig. 7 Speed of Response of Resistance Thermometer Bulb

G(s

R = thermal (film) resistance

c = thermal capacitance

RC = T = time constant

8j_ = temperature input

EQ

= EMF output

s = Laplace operator

) = transfer function

43
A protective well surrounding the thermocouple will add another section in cas- cade to the network described above. Since there is a loading or interacting effect the overall transfer function will be of the following form:

G(s)

R1R

2

C1C2

S +

( R

1

C1

+ R

2

C

2 + : 7Js + 1 DO

The equation is factorable into the fo]

G(s)

(T s + 1 a

Where Rx ,C1 and Rg ,CS refer, respectively, to the well and to the junction; the time constants T

a ,Tb

are the roots of the denominator quadratic (2).

Radiation pyrometers, bare-bulb resistance thermometers and bare-bulb filled sys- tems exhibit an approximate first order response, however, their time constants vary considerably with the various media and arrangements in question (1), (3), (5). Figures 8, 9, and 10 give time responses of typical temperature measuring ' devices

Pyrometric Cones

They are cones made on the form of slender, truncated, trigonal pyramids from raw materials used in the manufacture of many ceramic products. Exposed to the same firing and furnace conditions as the product, the physical and chemical changes in the cones are indicative as to the changes occurring in the product as a function of temperature and time (8)

III. Pressure Detecting Elements

BRONZE SP/PAL

EZE

BOURDO/V TUBE X I

STEEL SPIRAL

J~l 1 ~ > SPKm £BELLOWS INSIDE C/IS£

J

1 ABSOLUTE I II

PfESSURS UNIT

L IMP DIAPHRAGM III I INVERT£D BELL III I LOW PRESSURE CONTROLLER ATMOSPHERIC PRESSURE

J L GAOGE -/4.7(±) Q /S 30 100 400 I00O 4000 IO.00O ABSOLUTE O l<f-.7(±)SO Fig. 11 Application Chart of Pressure Measuring Devices

44
10Q:

1.2 1.6 2.0

Time in Minutes

FIG. 9 SPEED OF RESPONSE OF THERMOCOUPLE, WITH & WITHOUT;

PROTECTING WELLS (IN AIR)

FFFfFF

2.8

1.2 1.8 2.4

Time, Seconds

'FIG. 10 SPEED OF RESPONSE OF A RADIATION DETECTOR

45
There are 3 main types of elastic pressure elements; diaphragms, bellows and Bourdon

tubes. All follow "Hooks Law", which states that stress is proportional to strain, or deflection is proportional to the pressure applied. Mathematically

and

[4]

F = P A

c

[5]

where x = output deflection

F = force

k = spring gradient

P = inside pressure

X = effective area of the sensor

In all three devices, the pressure built-up in the enclosure is determined by the capillary restriction and the volume of the capsule. For small changes in pressure it

is possible to define a pneumatic resistance R and a pneumatic capacitance C. Then

the resulting transfer function relating capsule pressure AP to input pressure AP. is given by 1

G(s) - wjrf RCs + 1 DO

Combining equations 4, 5, and 6 yields

G(s) = A X(s)

"A P (s)"

i

A/k

RCs + 1

Equation 7 can be represented by the block diagram shown below

AP<(s) RCs + 1 AP (s)

c AF(s) AX(s)

In some instances, the attached mass will become significant; this would lead to a third order transfer function.

Diaphragm - Non-Metallic

SCftLB

Fig. 12 Non Metallic Diaphragm

46
One type of diaphragm is made of flexible leather, (oiled to assure flexibility over

a long period of time) , colon leather, or of noeprene. The diaphragm is generally

opposed by a spring, which is deflected by an amount directly proportional to the applied pressure. A linkage connects the diaphragm to the pen and/or measuring element.

A typical structure is shown in Figure 12.

Metallic Diaphragm

Fig. 13 Metallic Diaphragm

Flexible metal diaphragms are also used and their action is similar to that previously

described. The diaphragm, however, is of thin brass or other flexible metal chosen to be compatible with the corrosive action of the process.

Bellows Elements

The bellows element is axially flexible and is a one piece expansible and collapsible unit. It is usually formed of thin seamless tubing made into a corrugated unit

either hydraulically or mechanically. These elements are made out of phosphor bronze,

brass, stainless steel, beryllium copper and other non corrosive materials as required

by the corrosive condition of the process. They are calibrated for various pressure

ranges by a bellows spring, which is external, and therefore not subject to corrosion

by the fluid under measurement. For vacuum and compound ranges an internal spring is

also required as the bellows materials itself is not suitable for accurate calibration.

47
Fig. 16 Simple Application of Bourdon Tube

The Bourdon tube is probably the oldest pressure detecting element and is the most

widely used pressure measuring device in industry. The tube itself is flattened on diametrically opposite sides to produce an approximately elliptical or oval section

and is then bent into an arc of a circle. Pressure applied to the inside of the tube

will distend the flattened positions and tend to restore the original circular cross

section. This change in shape tends to straighten the tube and in so doing moves the

tip a sufficient distance to actuate the linkage. The most commonly used Bourdon

tubes are made out of phosphor bronze, alloy steel, stainless steel and beryllium

copper. Their ranges cover from vacuum to 10,000 psi. Bored out of solid steel forgings they will detect pressure up to 80,000 psi.

The Spiral

Fig. 17 A Spiral Gauge
Another useful gauge is the spiral gauge which is formed out of a metal tube,

flattened in cross section and wound into a spiral shape. The open inner end of the

spiral is connected to the pressure source while the linkage is attached to the closed

outer end. When pressure is applied, the spiral, which is firmly fastened at its

inner end, unwinds, causing the outer end to rotate in a counter-clockwise rotation.

The Inverted Bells

V'

Fig. 18 An Inverted Bell Pressure Gauge

Another pressure measuring device is the inverted bell. Two inverted bells are shown

detecting the differential pressure. The pressures are being piped to the bells on opposite ends of a balanced beam from which the bells are suspended. Static pressure

can be detected by a single bell with a counterweight suspended from the other end of

the beam. The dynamic response of the inverted bell is second order.

Electrical Pressure Transducers

LOA/lT

Fig. 19 An Electrical Pressure Transducer

The strain gage is a common pressure-to-electrical voltage transducer. The strain

gage itself consists of wire grid bonded to the surface of a small slip of rag bond

or bakelite impregnated paper. When the gages are cemented to a surface which is

subjected to load, the wires, constituting the grids are stretched, thereby changing

their diameter and length and consequently their electrical resistance. Through

49
measurement of the change in resistance of the wire grid with the gage under stress, an evaluation of the force producing the load is obtained. The strain gages are electrically connected to form two legs of a Wheatstone bridge circuit. The stress to voltage response is instantaneous.

IV. FLOW METERS

Of the several types of flowmeters, the most frequently used is the differential

pressure or heat meter, which measures the differential pressure produced by a restriction in the flow line. This resitriction, which is the primary element, can be produced in a number of ways. Common devices for measuring flow in this way are orifice plate, the venturi tube, the flow nozzle and the pitot tube shown respectively

in Figures 20 through 23

CONCENTRIC ecC£MT/PiC S£$M£NTM

Fig. 20 Orifice Plates

HP C0AWecT/ON THROAT i£> C0#HECT,OK

Fig. 21 A Venturi Tube
HP CONNECTION LP CONNECTION

OB7

ENTeANCe CONS THROAT

Fig. 22 A Flow Nozzle

ONNEC TION

TO QA&E

GUIOS VANS FOR

POSIT/ON fiND DIRECTION OF

IMPACT OPENING ^stuffing box

^T/9T/C

p/zEssuee Taae

Fig. 23 A Pitot Tube

51
Principle of Operation

There are many types of measuring elements, which measure the relationship of

flow rate and differential pressure where the flow varies as the square root of the

differential pressure of Q = K H

Fig. 24 Differential Pressure Across an Orifice

The Mercury Float Type Flow Meter

Fig. 25 Schematic of Mechanical Flow Meter

This device basically comprises a mercury filled manometer with a large chamber con- taining the float and the range tube in a smaller chamber.

52
The Bell Type

COVER FLANGE FILLING PLUG

CHARACTERIZED BELL

LOW PRESSURE

CONNECTION

3-WIRE CABLE

TO INSTRUMENT TRANSMITTER

COIL

Fig. 26 Ledoux Bell Meter Body

The bell type flow meter is essentially a pressure chamber, which contains a characterized bell floating in a pool of mercury. The low pressure is applied to the inside of the bell and the high pressure to the outside. This type sensor is basical- ly of the 2nd order (no specific data are available)

The Bellows Type

rcee ARM

Fig. 27 Differential Pressure Converter, Bellows Type

lu

B ?

i

ff

erential Pressure acts against metal bellows, with a torque tube to transmit the bellows motion. The liquid filled damping system provides essentially first order response as the typical curves of Figure 28 indicate.

53
54
The Force-Balance Type

transmitted air zero adjustment

Fig. 29 Differential Pressure Converts, Force-Balance Type

In this type device the pressure differential is applied across a diaphragm. By

mechanical and pneumatic means the force on the diaphragm is balanced so that the output pressure is proportional to the applied differential. The cascading of mechanical

and pneumatic systems results in a high order transfer function, as shown for a typical device in Figure 30.

The Area Type Meter

Fig. 31 Area Meter Body

In the area type meter, the area of the orifice is varied to maintain a constant

differential pressure. The flow enters the meter body horizontally, is deflected

upwards against a floating piston, and passes downstream through an orifice on the

side opposite the inlet. A passageway is provided from the low-pressure side of the

meter to the top of the piston, so that the downstream pressure always exists on top

of the piston. As the piston moves up on an increase of flow, it opens the orifice

just enough to maintain a constant pressure differential across the high and lowpressure sides of the orifice. Referring to Basic formula, Q = A H, since the area

is varied and not the differential pressure, the flow will vary directly as the change

in area. The flow is indicated by the position of the piston. Electric, electronic

or pneumatic position transmission is available. The response is of the second order

or higher.

55
56
The Rotameter

METERING

FLOAT TUBE

OUTLET SCALE jjlet Fig. 32 Rotameter

The rotameter consists of an upright tapered tube, large end up, containing a weighted plummet, which is lifted to the position of equilibrium between the downward

force of the plummet and the upward force of the fluid flowing past it through an annular orifice. The transfer function is of the second order since mass, spring and

damping are present.

Magnetic Flow Sensor

Based on Faraday's law of electro-magnetic induction, the poles of a permanent

or electro magnet are arranged so that the flux lines between North and South pole

transverse the flow at right angles. The fluid flowing acts as a conductor moving

through the magnetic field and causes an emf to be induced in the fluid which is

proportional to velocity. Electrodes flush with the inner pipe surface in contact

with the fluid measure this EMF. The response is practically instantaneous with the

change of flow.

Fig. 33 Principle of Magnetic Flow Meter

57
Mass Flow-Dry Solids

The radioactive density meter frequently used to measure % solids in slurries or % concentration of acids has been found to be quite accurate in measuring mass flow

per unit time of dry solids. The response is instantaneous.

Pioneering applications are in the cement and gypsum industries. The important

conditions are:

1) Constant velocity

This is most of ten obtained by using free fall of the material at a point within 1-1/2 ft. of the start of free fall.

2) No large changes in composition or moisture

Use of steam below point of measurement may change moisture content of

material

3) Sufficient quantity of material flow to absorb minimum amount of radiation

for instrument response. (l#/ft3 apparent density across pipe.)

Fig. 34 Nuclear Density Gage Measuring Mass Flow of Solids in Free Fall

58
V. LIQUID LEVEL

Primary Detectors

The primary detectors are pressure gauges and the diaphragm box.

maximum I eve I

minimum level

seal required

for corrosive

or viscous fluids V

pressure gage.

this run of pipe should have a slope of approx

I'per foot.

Fig. 35 Liquid Level Measurement by Pressure Gage Method

to pressure gage level recorder

open type closed type

Fig. 36 Liquid Level Measurement by Diaphragm Method

Static Pressure Method

The static pressure measurement is used in measuring the level in "open" or

"closed" vessels. This method is based upon the fact that the static pressure exerted

by any liquid is directly proportional to the height of liquid above the point of measurement, irrespective of volume. Thus any instrument which measures pressure can be

calibrated in terms of the height of a given liquid and used to measure liquid level

in vessels under atmospheric pressure. The water columns will add a quadratic lag

term to the transfer function of the detector.

In closed vessels, however, the pressure existing above the liquid is added to the liquid head and must therefore be compensated for.

59
The Air Purge Method

regulator (c-clamp) rotameter

pressure gage

as level recorder

Fig. 39 Liquid Level Measurement by Air Purge Method

60
An air line is immersed in the liquid to the minimum level as shown with the

pressure and volume of the air supply controlled by a regulator so that slow bubbling

occurs when the vessel is filled. The pressure in the air line is then equal to the

back pressure exerted by the head of the liquid. Measurement of the air pressure is

equivalent to measurement of the static pressure of the liquid, i.e. the liquid level

The Dynamic Response is essentially that of the pressure transmitter used.

The Gamma Ray Method

This method is based on the change in the number of gamma rays that penetrate a layer of liquid. As the thickness of the layer increases, the number decreases.

A radiation source unit is used, and its output is measured by a high-pressure,

high-efficiency ionization chamber. The ionization current developed is directly

proportional to the detected radiation and is amplified for measuring purposes.

This is an extremely accurate method and is frequently used on liquids difficult to measure since the source and detector can be located on the outside of the vessel.

(See Fig. 34)

Liquid Level by Means of Noise Level Detection

Fig. 40 Liquid Level Detection by Noise Level Measurement

Used frequently on grinding mills to maintain an optimum feed rate. A microphone located near the mill is actuated by the sound, extraneous noises are filtered out, and

the transient noise pattern is integrated and converted to a usable current or voltage. The response is essentially instantaneous.

Liquid X Solid Level by Electrical Capacitance

Fig. 41 Liquid or Solid Level Detection by Electrical Capacitance Measurement

61
The sensing probe assembly consists of a tensioned insulated wire fixed secur

at the top and bottom of a container. It operates on the principle of a change in

electrical capacitance, the material in the container forming the dielectric betwe

two plates of a capacitor as represented by the probe and the container wall. The

response is essentially instantaneous.

VI. SPEED MEASUREMENT

Electric Generator Tachometers

The most commonly used primary elements to measure speed are the AC or DC Tachometer generators. Coupled to a shaft, each will develop an emf which is proportional

to shaft speed.

Proximity Probe Method

41(53 | MAWttTE* j

-8VDC

Pig. 42 Speed Measurement Using Proximity Probe and Pulse-Current Convertor

A magnetic proximity pickup senses the rotation of a gear and produces a small

signal. This signal, alternating at a frequency dependent on the number of teeth on

the gear and its speed of rotation is converted by a frequency discriminator to a proportional DC output.

VII. DENSITY OF LIQUIDS AND SLURRIES IN PIPES

The gamma ray method, used most frequently has been previously described.

VIII. DEWPOINT DETECTING ELEMENTS

The Dew Probe consists of bifilar wire electrodes wound on a cloth sleeve, which

covers a hollow tube or bobbin. The sleeve is impregnated with a lithium chloride

solution which is allowed to dry. The bifilar electrodes are not interconnected but

depend on the conductivity of the atmospherically moistened lithium chloride for

current flow.

62
At values of 11% RH and below, the lithium chloride becomes conductive and an electric current passes through and heats it. Some of the moisture is thereby evaporated until an equilibrium temperature is reached on the bobbin. At this temperature

the R.H. of the air next to the heated bobbin is 11%. The equilibrium or bobbin temperature is thus related to the dewpoint temperature of the air. A filled thermometer

bulb or a resistance thermometer is mounted inside the bobbin to measure the cavity temperature.

IX. MEASURING ELEMENTS AND TRANSMITTERS

Having studied a few of the primary detecting elements we are now ready to discuss briefly the next box on our basic control system, which is the measuring element.

This is the portion of the feedback elements, which converts the signal from the primary detecting element to a form comparable with the reference input, usually the set

point located in the indicating or recording device.

It is quite difficult to say, where we will find the measuring element. In some

cases it is incorporated into the indicating and recording device, in others we will

find, that due to the remote location of the comparator the measuring element is built

into the same housing with the primary detector.

f Bourdon A Connecting Beom

Oteilloior Fig. 44 Method of Operation, PP/1 transmitter

OscillOtor

Fig. 45 Method of Operation P/l Transducer

See Fig. 46 for response of P/l transducer. Many times it becomes necessary to change the signal from one form to another, such as pneumatic 3-15 psi to electric

4-20 MA in order to make it suitable as an input to an electronic comparator. Transducers and converters have been designed for this purpose.

There are a number of devices on the market which will permit you to pneumatically

divide, multiply, ratio, add and subtract. The response for ratioing is given by

Fig. 47.

63


P2 = KplP3 K = Constant

of Mechanical

Prop.

Fig. 48 Multiplication by Pneumatic Means

KP1 Ratio Adjustment

LEGEND

1. Manual Ratio Adjustment (P3)

2. Ratio Adjustment Scale

3. Fulcrum Shaft Bushing

Fulcrum

5. Input Capsule (PI)

6. Input Weighbeom

7. Bias Adjustment (factory-set)

8. Rebalancing Capsule (P2)

9. Zero Adjustment

10. Rebal oncing Weighbeom

11. Span Adjustment

12. Flapper Beam

13. Nozzle

14. Pilot Reloy OUTPUT FAS

Fig. 49 Ratioing by Pneumatic Computer

66
ZERO

SCREW

VENTS

Fig. 50 Adding and Subtracting RelayThere are many electronic solid state auxiliary devices which allow you to:

Add

Subtract

Compare voltages to detect alarm and limit points

Maintain system control at the last operating point upon loss of input

Extract square root - (Fig. 51)

Ratio - (Fig. 52)

Integrate - (Fig. 53)

Compensate for residence time in grinding loop - (Fig. 54)

Make up feed losses

Multiply

Automatically select control

Delay signal - (Fig. 55)

Override set point signal

67
SQUARE ROOT EXTRACTOR RATIO/CASCADE AUXILIARY

Fig. 51 Fig. 52

SIGNAL DELAY RELAY

Fig. 54 Fig. 55

68
X . CONTROLLERS

The basic characteristic of an automatic controller is the manner in which it

acts to restore the controlled variable to the desired value. This is called the mode of control. The common modes of control are:

1. Two Position

2. Floating or Single Speed Floating Control

3. Proportional

4. Proportional + Reset

5. Proportional + Reset + Rate

Two Position Control

45 49.5 50 505 55 Controlled Vonoble

Percent of Instrument Ronge

Fig. 56 Two Position Control

In two position control, whenever the controlled variable deviates a predetermined

amount from the set point, the controller moves the final control element to either of two extreme positions.

Because of its simplicity, two position control is very popular and often adequate

for process regulation. In general, it functions satisfactorily if the process has a slow reaction rate, minimum transfer lag or dead time, and the two extreme positions

are adjusted to permit an input just slightly above and below requirements for normal operation

Floating or Single Speed Floating Control

Floating or single speed floating control moves the final control element at a constant speed, in either direction, whenever the controlled variable deviates a pre determined amount from the set point. The floating controller changes its speed to and off, while the two position controller changes the position of the final control

element. from on to off.

The outstanding advantage of floating control over two-position is that gradual

load changes can be counteracted by gradual shifting of the valve position, thus min- imizing cycling.

Floating control cannot be used where there is any significant lag, or where load changes even though small, occur rapidly.

6 9
Proportional Control

In proportional position, mostly called proportional or single-mode control, th

is a fixed linear relationship between the value of the controlled variable and the

position of the final control element. The proportional controller moves the final

control element to a definite position for each value of the controlled variable.

Set Point

50 (00 125 150 175 ZOO 250 Controlled Vorioble

Fig. 57 Proportional-position Control

The proportional controller provides smoother control than the more simple modes

because it can move the control valve to intermediate positions that are proportional

to the deviation from set point.

The proportional controller, however, can move the control valve to one, and only

one, position for any given value of the controlled variable, and therefore is only

applicable where the load does not change. It is unable to change valve position for

a change of load.

This characteristic of proportional-control results in offset, which is a sustained

deviation of the controlled variable from the set point as the result of a sustained

process load change. Most proportional controllers have a manual reset adjustment that

eliminates offset by shifting the proportional band about the set point.

Pen Position, % Full Scale

Fig. 58 Effect of Manual Reset

70
Proportional control is used where process capacitance is relatively large, process reaction rate relatively slow and process lag and dead time relatively small. All of these characteristics promote stability and permit the use of a narrow proportional

band, which leads to faster corrective action and less cycling. Under these conditions proportional control can even overcome small load changes.

The wider the proportional band, the greater the offset when there is a load

change.

Proportional + Reset Control

. A:Proporlionol Action

.8 Reset Action

. C:Prop.- PlusReset Action Time Fig. 59 Proportional + Reset Control

In the proportional + reset mode of control, reset is automatic. As soon as the controlled variable deviates above and below the set point, there is a gradual and automatic shift of the proportional band to bring the variable back to the set point causing changes in valve position to accommodate for load changes.

The illustration shows the changes of a control valve position during a sustained

change in process load under:

A - Proportional Action

B - Reset Action

C - Proportional + Reset Action

The primary advantage of proportional-plus-reset control action is the reduction

of offset. It can be used even when process capacitance is small, reaction rate if

fast and load changes are large.

Rate

A: Proportlonol Action

B;Rote Action

C: Prop- Plus- Rote Action Time

Fig. 60 Proportional-plus-Rate Control

71
Rate action provides a continuous relationship between the rate of change of the

controlled variable and the position of the final control element. Rate action never

exists alone, but in combination with proportional or proportional + reset actions.

It is a temporary overcorrection proportional to the amount of deviation.

The illustration shows how a proportional + rate controller changes the position

of the control valve when the controlled variable deviates from set point.

The diagram breaks the control action into its two components: There is no change in the proportional response A.

Rate response B causes an overcorrection first in one direction then in the

other.

C shows that the return of the controlled variable to the set point is sooner

than it would without rate action.

Rate action is introduced on processes with large dead time, 2 minutes or more, or large transfer lag that are difficult to control with the proportional + reset

mode. Especially on those, which are controlled with a wide proportional band and

very slow reset rate. With load change, there is a rather wide deviation and it

takes a long time for the controlled variable to return it to the set point. The

addition of rate will often solve this problem. Rate time is the time interval by

which rate action advances the effect of proportional action upon the final control

element, Fig. 61, 62, and 63.

72
73
I

74

XI. COMPLETE CONTROL LOOPS

It is not the scope of this paper to discuss the last box of the basic diagram,

the final control element. However, examples of two control loops employing some of

the components discussed are given.

Example 1:

It is desired on certain coolers to control the Fan Damper from secondary air

temperature, and cooler grate speed from Undergraduate pressure. Should secondary air

temperature show a consistent deviation from setpoint, the setpoint of the grate speed

controller should be automatically changed to correct this condition. This action

should not be erratic.

TYP£ T/C

MV/I

DP/iPP^

co/iP^e/jTojs

coA/r/eoiL£-/e

INPUT

(See Fig. 6k for response)

DEL/9Y

•S/AIQLE MODE

COOLER

DAWPc/e

-SIQNPC

DEiP Y ACTION

x/?T£-p.

cor?PAR#TOg

SP£<PO

>CONTgOL

CONTEOLtETye

4- 2o

TO G&ATE

T/ME

Fig. 65 Typical Closed Loop Control System

Characteristics of this system:

1. Since an ordinary thermocouple in the throat of the cooler would be affected

by heat radiation from the clinker, an aspirating type thermocouple is used.

2. If the error signal caused by the deviation of the temperature measured and

the setpoint in the comparator are allowed to affect the setpoint of the

speed controller directly, the control action would be unstable.

Once the controllable limit of the Fan Damper has been reached, the system would

tend to cycle around this limit. Therefore, a time delay unit has been employed, the

action of which is very much like an RC network, forward and backward.

76
77
The single-mode controller is the biasing means of the electrical setpoint and

acts otherwise like a controller with proportional band.

Example 2:

A system to blend 3 materials is required. The material should be fed to a 3-

compartment ball mill in a closed circuit. Thru-put measured by a sonic device shou

control feed and primary water additions. Slurry should be held at 35% water by

weight ahead of a DSM screen.

System should be of highest accuracy possible.

Flow Diagram of System:

a - coA/reouees

ei£creic motok posst/oa/sizs

e/)T/0 UNITS

F- zecoeoee oNty

/V- TOTAL/Zf/Z

L - CCOS£-£> 10OP

Fig. 66 Typical Blending System

Basic characteristics of system:

1. Weight inaccuracy due to inherent moisture in the materials.

2. Variation in the material size.

3. A lag produced by the material residence time in the grinding mill.

4. Variation in circulating load returns.

78
The worst condition would exist if the materials to be blended have a different

inherent moisture content. The solution for this problem would be to preheat the

materials or take the average of separate moisture measurements. Neither solution is

satisfactory in this process because of the high cost of moisture control. Frequently

it is decided to not control moisture and it demands an increase in the accuracy of

the weighing equipment which furnishes the feedback signal to the control loop. If

this signal is of poor quality, the entire system becomes affected by it. Variation

in material sizes results in considering only the weighing method. Residence time in

the Ball Mill results in considering the circulating load returns, and adds an unknown

amount to the primary water. This will have to be considered if it is necessary to

control the required percentage of water by weight of the slurry.

Bibliography

1. D. P. Eckman: Automatic Process Control (John Wiley & Sons)

2. J. G. Truxal: Control Engineer's Handbook (McGraw-Hill)

3. Fundamentals of Industrial Instrumentation (Honeywell)

4. G. K. Tucker & D. M. Wills: A Simplified Technique of Control System

Engineering (Honeywell)

5. T. C. Linahan: The Dynamic Response of Industrial Thermometers in Wells

(ASME Trans. 1956 Vol. 78, No. 4)

6. D. P. Eckman: Industrial Instrumentation (John Wiley & Sons)

7. D. M. Considine: Process Instruments & Controls Handbook (McGraw-Hill)

8. Orton Sfd Pyrometric Cones: The Properties and Uses of Pyrometric Cones

79

Application of Computers in Design and

Control of Systems

Joseph L. Hammond, Jr.

1 . Introduction

The inherent complexity of most problems which are of interest to ceramic engineers makes tractable mathematical models almost impossible, and thus analytical results

have historically been of considerably less interest than practical experience. The

rapidly advancing fields of automatic control and automated design, however, depend

to a large extent on the use of mathematical models for the problems of interest. Perhaps the key to making automatic control and automated design possible in the ceramic

industry can be the modern computer which has the ability to cope with mathematical

models far more complex than can be handled efficiently with hand calculation.

To use a computer, some sort of mathematical formulation of the problem must be

available. However, the present state of the art is such that many formulations such

as nonlinear equations, large sets of simultaneous equations, equations with random

parameters etc., can be handled numerically with a computer even though such formulations are not analytically tractable. The problem of choosing a mathematical model

is frequently difficult and of course has important significance on the overall

results. In order to approach the use of computers intelligently, the engineer must

be cognizant of the properties of the chosen computer when he formulates the system

equations. Only in this way can full use be made of the capabilities of the machine.

In later sections of this paper this point will be illustrated by specific examples.

The role of the computer in systems work has two distinct parts: (1) the com- puter can be used as an element in a control system, or (2) it can be used as a tool

in system design. The use of computers in system design can be further divided into

several catagories, namely (a) their use in analyzing systems and system components,

(b) their use in simulating systems, and (c) their use in automated design procedures.

In the discussion below each distinct use of the computer will be given separate

attention. Before discussing the uses of computers, however, the two computer types - analog and digital - will be described, and the essential features of methods for

solving ordinary differential equations will be reviewed as an illustration of how

computers are programmed.

2 . Characteristics of Analog Computers

. An analog computer is distinguished by the fact that every variable in a problem

to be solved is represented in the computer by the magnitude of some physical quantity.

Electrical voltage is currently the most popular quantity and this will be assumed

here for definiteness . In almost all cases the independent problem variable is represented by time in the computer. Thus the result of a computation is one or more

voltages which vary with time in a manner approximating the solution to the given

problem. These voltages can be recorded to give a permanent graph of the solution to the problem.

Since the dependent problem variables are represented by voltages, the analog

computer must have available elements to operate on these voltages in the manner dictated by the equation. Thus analog elements perform the following operations:

1. Integration 5. Generation of appropriate functions of dependent variables.

2. Addition (subtraction)

3. Multiplication of a variable by a constant

4. Multiplication of two variables together

81
Note that differentiation is not normally included since it is a noise accentuating

operation. A true analog computer with no digital accessories is handicapped by being

able to perform only a limited number of logical operations such as "when voltage A

exceeds voltage B then do operation C, otherwise do operation D.

The accuracy of an analog computer is inherently limited by the fact that problem

variables are represented by physical quantities. Experimentalists in all areas realize that measurement accuracies cannot be made arbitrarily high, and that devices

responding to physical variables are limited in operating range. Good general purpose

analog computing equipment will have a typical component accuracy of from 0.01 to 0.1%

of full scale, and a linear range on the order of 1000 to 1. This limitation on ac- curacy makes an analog computer best suited to solving ordinary differential equations

in problems where "engineering" accuracy is adequate.

As stated above, the independent variable in an analog computer is time. Fortunately, however, actual time in the analog computer can be proportional to the

independent problem variable and does not necessarily have to be equal to it. The

same comment can be made relative to the voltages representing dependent problem variables. The constants of proportionality for the independent and dependent variables

are called scale factors, and their use extends greatly the useful range of the computer. Applying the scale factors to the problem equation is essentially a change of

variables which results in a "machine equation" for use in the computer.

The basic steps in using an analog computer are shown in Figure 1. First scale

factors are chosen and applied to the problem equation. The result of this change

of variables is a machine equation which is solved on the computer with voltages representing the dependent machine variables and time representing the independent

machine variable. The computer outputs are plotted vs time on an appropriate recorder

and finally these plots are calibrated in problem variables by use of an inverse

scale factoring procedure.

3 . Characteristics of Digital Computers

As contrasted to analog computers, the dependent variables at any time in a digital computer are represented as binary numbers by a collection of computer elements

which have only two stable states. The nature of the digital computer is such that

the variables cannot change continuously, and thus differential equations, for ex- ample, must be approximated by difference equations for which the independent variables

progress in steps of a nonzero length.

A digital computer performs five basic operations, (1) input from tape, cards,

etc., (2) arithmetic operations on data in the machine, (3) storage, (4) output to

typewriters, etc., and (5) logical control of the sequence of operations which are performed. The actual computation is done by proper sequencing of arithmetical

operations on the input and stored data. The basic arithmetic operation is addition,

but this can be combined with storage to perform multiplication, division, integration

and other operations. A block diagram showing these operations is given in Figure 2.

The versatility of the modern digital computer is a result of the speed with

which the basic operations of addition, storage, and shifting of information can be

accomplished. For present day machines the time required for a single operation is on the order of 10~ 6 sec. To make proper use of this type of computer, however, involved programming is required to convert the required calculations into the proper

sequences of the basic machine operations. Fortunately much of the detailed instruction can be preprogrammed once and for all into the machine memory or stored on paper

tape. A number of so called interpretive programs such as Fortan are available which

preprogram the machine to receive directly instructions to perform operations such

as multiplication integration etc., which are ultimately carried out as sequence of

the simpler operations.

82
As pointed out above the digital computer works with numbers, usually in the

form of binary digits. Thus the dependent variables at a particular multiple of the

step size of the independent variable are represented by binary numbers at appropriate

locations in the machine. The high accuracy of the digital computer results from the

fact that each digit of the binary number is represented by one of two possible states.

For example, 1 could be represented by +100 volts and 0 by -100 volts. An error due

to noise is seldom so large that +100 volts is interepreted as -100. Thus the binary

numbers can be processed almost without error and the primary source of error is rounding off to a fixed number of binary digits. Even here the digital machine has the

flexibility of being able to swap computing time for accuracy by using a large number

of binary digits „ The accuracy of simple operations such as addition, multiplication

etc., can thus be made almost arbitrarily high. However, as in the case of the analog

computer, it is a difficult matter to determine overall accuracy for a non trivial

problem in terms of the accuracy of the elementary operations, and of course, there

are many problems which cannot be solved accurately on a digital computer.

In addition to high accuracy the digital computer has great flexibility in performing logical instructions. This fact makes automated design feasible since, for

example, a digital computer has the capability of solving a set of equations for a system, comparing the result to a performance criterion, and then changing the system

parameters and repeating the calculation.

4. Computer Solution of Differential Equations

Many systems of interest to ceramic engineers can be described by ordinary differential equations. Those systems which cannot be exactly described in this way can frequently be approximated by ordinary differential equations. Thus this very important class of equations is chosen to illustrate the technique of machine computation.

Most systems of ordinary differential equations which can be solved for the

highest derivatives can be reduced to a system of the form

dx

1 /dt f^t, x

1 (t), -, x (t), y^t), -

(i)

dx

n

/dt = fn[t, x

1 (t), , x

n

(t), y

1 (t), , ym

(t)]

with appropriate initial conditions. In Equation (1) the f. are real valued function,

of the indicated arguments, the y. (t) are known forcing functions, and n is the order

of the given system of equations. For a physical system the variables x1

, , x are just the state variables discussed in an earlier paper, and Equation (1) can be

thought of as the description of a physical system in state variable form.

To illustrate the procedure for obtaining a numerical solution to a set of equations such as Equation (1), consider the case of only one state variable. For this case the equation is simply

dx/dt = f[ t , x(t), y (t) ] (2) with an initial condition given by

x(t = t ) = A. o

83
A numerical solution to equation (2) is obtained when x is specified for values of t over the range of interest. It is instructive to note that the given equation pre- scribes the derivative or slope of the desired curve at every value of t Thus in effect the differential equation prescribes the locus of tangents to the desired curve This fact can be used in obtaining a numerical solution to the equation. To begin with, x is known to be A at t = t . At this point Equation (2) gives the slope of the curve as

dt

The problem is then to obtain the values of x at values of t other than t A conven- ient approximation is to assume that the actual x(t) vs t curve coincides°with the tangent to the curve at t = t for t values near zero. This results in the expression

* t x Ct) = / f [ t Q

, A, y(t

Q

) ] dt + A (4)

which is a good approximation to x(t) for t "close" to t , say in the range

t0

£ fc £ t + e . Now the approximation

x (t) = /

t Q+ E , x (t

Q+ e), y(t

o+ £ ) ]dt + x* (t

Q+ e) (5)

can be used for t in the range t + e £ t £ t + 2e, and the process can be continued

to determine x*(t) over the whole range of interest.

Approximations of the general type of x*(t) are used in both analog and digital computation. Questions of convergence of x*(t) to x(t) of course arise, and such considerations lead to more involved approximations but the basic idea is preserved.

The circuit of Figure 3 is used to generate x*(t) with an analog computer. The circuit operates as follows. At t = t the integrator output A, a voltage correspond- ing to tQ

, and a voltage corresponding to y(t ) are the inputs to a collection of elements which produce f [

Q

, A, y (t ) ] . As time progresses, the inputs to this

"function generator" change so that f [ t , x* ( t) , y ( t) ] is produced and integrated to produce x*(t). The "closed loop" feature of this circuit, namely the fact that the integrator output, x*(t), is one input to the function generator which produces the integrator input to correct x*(t), is characteristic of any machine solution of a dif- ferential equation.

In a digital computer the computation of x*(t) is accomplished by essentially the same logic as shown in Figure 3. For this computer, however, the integral in

Equations (4) and (5) is also approximated, and the independent variable is made to progress in discrete steps, say of value e. Thus Equation (4) becomes

**(t + e ) = e f[t A, y(t )] + A, (6)

and the general expression for any discrete t value is given by

x*[t

Q + (i+l) e ] = e f[t

o + i e , x*(t

Q + i e ), y(t

Q + i E )] + x*(t

Q + ie) (7)

84
Equation (7) is a difference equation which can be solved by the computer circuit of

Figure 4. In this circuit starting at i = 0, the values of x*(t + ie), y(t + ie),

and t + ie are stored in the memory and used at each step to compute first

f[t ¥ is, x*(t + ie), y(t + ie)] and then the corrected approximation x*[t +(i + l)e].

o o o o The quantity x*(t + ie) can be printed out as the machine progresses through the

calculation.

The fact that x* is a continuous function of t for the analog computer, and a discrete function of t for the digital computer is misleading in comparing the two

machines. The analog computer in actuality has an inherent smoothing or averaging

effect which causes the continuous output. It is well known, however, that due to limited bandwidth the continuous analog signal contains no more, and usually less,

information than the discrete digital output.

The general rr^- order ordinary differential equation, expressed in state variable

form by Equation (1) , can be solved in the same way as the single variable case. For

the n variable case, n circuits of the type shown in Figure 3 or 4 must operate simultaneously. Furthermore each of these circuits is coupled together through the fact

that in the general case the value of each function f. depends not on just x1 *(t),

but on all n state variables as well as additional inputs. Of course convergence

problems are more severe in the general case so that practical circuits frequently

contain modifications of the basic idea.

An interesting logical simplification of the procedure for the general case

results from writing Equation (1) in vector form as

dx/dt = f[t, x(t), y(t)] (8)

where

x

n

(t)

; y(t)

y L (t)

y (0

and dx/dt

dx/dt

dx /dt n

A circuit showing the logic for solving Equation (8) is given in Figure 5. In this figure the double lines denote the flow of vector quantities, and the complete

circuit is a short hand representation for n interconnected subloops of the type shown

in either Figure 3 or Figure 4. It is to be understood that x(t), the output of the

circuit of Figure 5, will be n plots representing each x.*(t) vs t for the analog computer, and n sets of data tabulating the x *(t + ie) at values of i = 0, 1, 2, - - - for the digital computer.

5 . Operating Characteristics of Computers

In using a computing machine it is important to take into account the obvious

fact that best results are obtained on equations which are matched to the capabilities

of the computer. Thus use of a computer is itself a system problem for which the

physical process, the mathematical model for the physical process, and the computer

must all be considered to obtain best results. For example, a mathematical model

should be chosen to not only represent the physical process, but to also be compatible

with the computer which will be used to give the numerical result. It should be noted

that classical mathematical models in almost all fields are "matched" to analytical

computation, and may or may not be appropriate to machine computation.

The important problem of choosing a mathematical model which is matched to both

the computer and the physical process is unfortunately still an art not a science.

However, some of the characteristics of computing machines which have bearing on the

choice of mathematical model can be listed and used as a guide in choosing mathematical models

85
First, both analog and digital computers have the following common characteristics:

1. Computers can operate only on numerical equations for which all parameter

values are specified. While it is possible to generate a family of solutions for different parameter values, a computing machine cannot produce a general solution in the sense of giving an answer as a function of parameter

values.

2. The operation of differentiation accentuates noise in both types of computers

and should be avoided.

3. Multipoint boundary value problems cannot be solved directly in most cases.

4. Only continuous functions can be produced as solutions to equations.

5. Partial differential equations must be approximated by sets of ordinary

differential equations.

Digital computers have the following additional operating characteristics worthy

of noting.

1. This type of machine is well suited to a large number of simple operations.

2. Answers are produced for discrete increments in the independent variable,

thus finite, as opposed to continuous, state models are appropriate to this

type of computation.

3. Memory is limited to a small fraction of the number of operations which can be performed in a short period of time.

4. Tabular data is easily handled by digital computers.

5. Logical decisions of the type "compare A and B, if A is greater than B do

operation C, if not do operation D" can be made easily.

6. The method of operation of a digital computer is seldom related to the

physical process being analyzed so that computing accuracy has nothing to do

with accuracy in the physical process.

7. Complexity in the sense of the total number of operations does not necessarily decrease accuracy. Such complexity can be had at the expense of com- puting time with little change in the number of components used.

Analog computers are distinguished by the following characteristics:

1. The linear operations of addition, subtraction, multiplication by a constant,

and integration can be performed with high accuracy.

2. Multiplication of two dependent variables, and generation of functions of

dependent variables are less accurate operations than the linear ones.

3. Logical operations are not easily performed.

4. Continuous state models are appropriate for this type of computer.

5. Complexity, in the sense of the total number of operations, is obtained by a proportional increase in the number of computing elements. Accuracy is almost

always a decreasing function of complexity.

86
6. For many problems there is a direct correspondence between the analog computing elements and elements of the physical process being analyzed. In

such cases, the accuracy with which a given mathematical variable can be

computed is comparable with the accuracy possible in generating and measuring

the corresponding physical variable. Problems of this type are well suited

to the analog computer.

6 - Use of Computers in Analysis of Systems and System Components

This application is perhaps the oldest and most direct use of computing machines. The type of problems which arise in this area are as many and as varied as the

branches and aspects of engineering. Thus subject to the comments made in section 5, the choice of computer method depends on the particular problem, and little more of

a general nature can be said. The remainder of this section will be devoted to a dis- cussion of a specific problem as an illustration of typical procedures.

Consider the problem of diffusion of liquids or gases. The concentration,

C(x, t) of diffused substance at a point x and time t is usually assumed to satisfy

Fick's law which for a one dimensional problem can be written as

8C(x. t) 3 3t 3x D[C(x, t)] j)C(x, t) 3x

(9)

with appropriate boundary conditions. In this equation the parameter D(C) is the diffusion coefficient.

Equation (9) can be solved analytically if the diffusion coefficient is independent of concentration or if the equation has certain simple boundary values. From a practical point of view, if a given physical process does not have these simple boundary values, D(C) must be assumed to be constant in order to obtain an analytical

solution. For a constant D(C), a general solution to Equation (9) can be obtained. The problem then is to choose an appropriate function of the general solution to match

the boundary values, and hence solve a specific problem. The final solution for spe- cific boundary conditions is frequently a complicated function of the general solution

to Equation (9) and it is sometimes expedient to employ a computer to evaluate this

function at specific values of x and t or to expedite other steps in the procedure. Such a use of the computer implements, and makes practical laborious computation.

Subject to certain restrictions. Equation (9) can also be solved numerically on

an analog or digital computer for either constant or nonconstant diffusion coefficient.

A machine solution of this type of course involves approximations and imposes restric-'

tions which are different from the corresponding ones for an analytic procedure. In

the case being considered the mathematical models which can be handled by machine com- putation can frequently yield better approximations to the physical process. In

particular, these are practical problems with nonconstant diffusion coefficients which

can be worked on a computer (see reference 40). Although this class of problem is obviously more involved than the simpler constant diffusion coefficient case, use of the computer in such problems extends the analytical methods.

Direct numerical solution of Equation (9) on either an analog or digital computer involves approximating at least one of the derivatives by a finite difference. On the analog computer the partial derivatives with respect to the space variable x are ap- proximated by differences to reduce the partial differential equation to a system of ordinary equations with time as the independent variable. Physically this is

equivalent to separating the axis along which diffusion takes place into a number of cells within which the concentration is assumed to be constant. The simplest approxi- mation of this sort, assuming a constant diffusion coefficient, results in the equation

87
dc

i =

DLc i+i " 2c

i + c i-i ]

(Ax)

i = 0, 1, 2, , N

where

Ax = x . - x . , l l-l

(10)

C. = c( Xi,t) N = number of divisions of x-axis

A similar but more involved expression could be obtained in the case of a nonconstant

diffusion coefficient. If a large number of subdivisions of the x-axis is required,

it is frequently impractical to solve the resulting large number of equations on an analog computer and a digital computer must be used.

For digital computation the derivative of C with respect to t must also be approximated by a finite difference with the effect that the time axis is also subdivided

into cells. This results in the set of equations

c i.m - c

i.i = D[c i+i,r 2c

i,i + c i-i,i ]

- (ii)

(Ax)

2

i = 0, 1, 2, , N

where t = t - t

J J

-1

C. . = C(x. , t.)

and other symbols have the significance assigned above. Equation (11) represents the

simplest type of difference equation equivalent to Equation (9) assuming a constant

diffusion coefficient. Many times in practice, stability of the solution becomes a problem and more sophisticated approximations must be used. Some questions of this

sort are discussed in the references cited at the end of the paper.

Note that a mathematical model of the form of Equation (11) could be developed

for a nonconstant diffusion coefficient. On the other hand for numerical computation

using such an equation both the x-axis and the t-axis must be subdivided into cells

in which the concentration is constant. Usually the lack of restriction on D(C) more

than overrides the necessity to subdivide into cells; and thus if a properly convergent solution can be obtained, the finite number of data points present in the solution is adequate in normal engineering work.

Example 1 : As an illustration of the solution of a diffusion problem on a digital

computer, consider a one dimensional problem discussed by Crank (Ref . 39, p. 207)

The problem involves a plane sheet of material of thickness 2L in the x-dimension,

and infinite extent in the y - and z - dimensions. The plane sheet is immersed in a bath of vapour which diffuses into the material in such a way that the equation

3C(x.t) = _3 |D[C(x,t)] |£ (x,t)

3t 3x L (12)

is satisfied. The diffusion coefficient varies with concentration as prescribed by

88
D(C) = De2 ' 303C/C

o

o (13)

where D

q

is a constant, C is the concentration at any x and t, and C is the concentration at each surface of the sheet. It will be assumed that the concentration

x = ±L remains at C

q for all t, and that C(s,t) = 0 for -L < x < L and t = 0. The

physical configuration is shown in Figure 6. To solve this problem, Equation (12) was reduced to a finite difference equation,

analogous to Equation 11, by using a method due to Crank. The nondimensional variables

C = C/C , X = x/L, T = D

Q t/L

2

, D* = D/D

q

were introduced to make the results as general as possible. The solutions obtained

are given in Figure 7 as a plot of C vs X with T as a parameter. The equations used

on the computer and some further details of the solution are discussed in Appendix 1.

1 Use of Computers in the Design of Systems

a. Design by Successive Analysis : The most direct system design procedures are based

on the successive analysis of trial designs. The procedure is initiated by choosing

a particular set of parameter values and a particular configuration of the system

elements. The trial design is then analyzed to see if it meets specifications. If

it does not, the system is modified, and a new analysis and performance evaluation is

made. The procedure is continued until a satisfactory system is obtained.

For some systems the interaction between elements is so complex that it is necessary to deal almost entirely with the overall system equations for a given trial

design. In such cases the calculations can be implemented by a computer which is

simply used to solve a sequence of analysis problems.

For other systems the interaction between elements is such that system behavior

can be usefully expressed in terms of component element behavior. For this type of

system a complete computer program would contain a number of element analysis programs

as subroutines. In such cases a given element analysis program becomes an entity in

itself, and can be used in the analysis of other systems using the same element.

Libraries of programs for the analysis of components can be built up and used to

advantage in the design of systems of this type. Furthermore a considerable amount

of work has been done in recent years by computer manufacturers as well as several

academic groups toward developing interpretive digital computer programs which ex- pedite the use of existing component analysis programs as subroutines in overall

system programs. The objective of this work is to make it possible for engineers with

only rudimentary knowledge of computer programming to use the machines. In a number

of cases, programs have been developed which make it possible for an engineer to read

into the computer with a system configuration and trial values for parameters expressed

in common engineering language, and obtain from the computer, system outputs and

other variables of interest which correspond to his assumptions.

Tables of analog computer programs for specific elements are also available in

the literature of several fields. Although it is more difficult to systematize the use of an analog computer than a digital computer, an analog computer program is inherently patterned after the physical system. Thus it is relatively easy to investigate various element configurations and different parameter values once the element

programs are available.

The use of computers in implementing system design procedures based on successive analysis of trial designs is well established and refined techniques are available in many fields. Present research is directed toward enlarging and extending the

89
scope of element and subsystem analysis programs, and in decreasing the time and effort required by engineers in using interpretive programs.

b. Completely Automated Design Procedures: A new approach which uses a digital com- puter to automate the entire design' procedure has been shown to be practical for

special systems particularly in the area of control systems. This type of approach is

receiving a considerable amount of research attention and is likely to grow in import- ance in the next decade.

The procedure referred to is based on machine techniques for handling optimization problems such as those which have been treated for many years through use of the

calculus of variations. One form for a problem of this type is the following: Given

(1) a set of ordinary differential equations describing certain unalterable elements

of the system, and (2) a mathematical statement of the criterion of performance, find

(1) that input to the unalterable elements which produces an extremum of the performance criterion subject to the restraint imposed by the unalterable elements, and (2) find a configuration of elements which produces this optimum input from the available

variables (including the desired system inputs) . Mathematically such problems can be

expressed as follows. The unalterable elements can be described by a vector state

equation of the form

& = f(x, y, t), x(0) = c (14)

where x is the state vector, for the unalterable elements, y is a vector of inputs to the unalterable elements, and f is a real valued function of x, y and t. The perform- ance criterion can frequently be expressed as a functional of y, J (y) , by equations

such as

T

J(y) = / q(x, y) dt (15)

0

where q is an instantaneous error function which is integrated over an interval (0, T)

to give the performance criterion. The design problem can now be stated as: (1) Determine y to minimize J(y) subject to the restraint imposed by Equation (14), and

(2) Determine a configuration of elements which will produce y from the desired system

inputs and x.

A less general, but sometimes more practical, formulation of the design problem

eliminates the task of automatically determining a configuration of elements to produce y (item (2) above) by assuming that the configuration is given. In such a case

the state equation becomes

& = f(i, x, i, t), x(0) = c (16)

where a is a vector of system parameters which are to be determined by the design, and

z is a vector of known system inputs. The performance criterion, J, now depends on

the parameter a, a finite set, rather than on the function y as in the previous formulation. The problem then becomes one of choosing the values of a to minimize J

subject to the restriction of Equation (16).

The computational difficulties associated with either of these formulations of

the design problem can be formidable. In particular the calculus of variations used

alone does not usually yield the required numerical results. In the past ten years,

however, several computational approaches have been developed which show general

promise, and which have been shown to solve specific problems. These methods are:

(1) Pontryagin's Maximum Principle (2) Bellman's Dynamic Programming, and (3) Techniques for implementing the Calculus of Variations type approach. Specific references to this work are given in the last section of the paper.

9D
Example 2 : This example is concerned with the design of an optimum control system

usxng a procedure developed by Webb (see reference 41) . An unalterable element satis- fying the equation

ax + x = y (17) is given, where x is the output, and y is the input to the unalterable element and a. is a constant. A performance criterion J(y) given by

I J(t) = / [(x-x.)

2 + X 2y

2 ]dt (18)

is also specified, where x

± is the input to the system which is to be designed.

Use of Webb's design method results in the equations

0n

= i - 0j / .v

022 =

022 ' aV " 20

12 " 2/ « *22 d»

012 = " ( 0

12

022> ' ^ + 0 11 "

012 /a

which were solved on a digital computer for the special case of a = 1 \ = 1/4 The structure of the final system is shown in Figure 8, and its responses to a step input for various choices of T are shown in Figure 9. Some further details of the solution are given in Appendix 2.

8 - Use of Computers in Simulation of Systems

The present category cannot be entirely separated from previously discussed meth- ods of using computers in system work. In particular the approach to be discussed in this section is closely related to the approach discussed in section 6 since the ob- jective in either case is to determine system outputs for certain inputs and initial conditions. The distinction between the use of a computer as aid in implementing system analysis and its use as a simulator can be best appreciated in the light of several general remarks concerning models.

In the typical application, a model is chosen to simulate certain properties of a physical system in such a way that there is a one-to-one correspondence between the variables in the model and certain of the system variables. The way in which the variables m the model change with respect to changes in inputs, system structure,

etc., can then be established from "experiments" with the model, and these results

can be used to infer corresponding properties of the physical variables. Although mathematical models have emerged in recent years as extremely effective in dealing with systems from many branches of engineering, such models are by no means the only type. The scale model, used by architects and engineers in the mechanical sciences is a well known example of an entirely different type of model.

A simulation of a system with a computer is a use of the computer to model the system. The computer elements and subroutines can usually be described in mathematic- al terms, and this mathematical description of the computer program for a system simu- lation is also a mathematical model for the system. The distinction between a computer model and the use of a computer in implementing calculations for a mathematical model is the following. In the use of a computer as a simulator available computer elements subroutines, etc., are taken as given, and elements and operations from this restrict-' ed class are used to model the system. On the other hand, when the computer is used to implement calculations, the mathematical model is usually specified. Thus in one

case, the computer approximates properties of the system directly; and in the other

91
the computer approximates the solution of equations which in turn approximate properties of the system.

Several examples of computer simulation in diverse fields will now be described,

and a simulation problem from ceramic engineering will be formulated and solved in an Appendix

Business Simulator : In the area of economics and business management, digital com- puter programs have been developed to combine the solution of supply and demand equations with a set of logical decisions to produce a model for the operation of a restricted segment of the business community. The computer inputs and outputs have

been tailored so as to make it possible to use the simulator in management "games."

In this problem the use of the computer to simulate the system rather than implement

the solution of equations is clear cut. Mathematical models of the complexity needed

to represent the system were not developed prior to stimulation experiments, since

their use was impractical with hand calculations.

Monte Carlo Studies : An important capability of the digital computer is its ability

to select numbers from specified random populations. For example, a digital computer

could be programmed so that the number at a particular location in the machine changes

randomly with time in such a way as to have a prescribed probability distribution

function. Systems from a number of fields can be simulated by a set of equations having inputs or parameters which change randomly with time according to some specified

distribution function. Such simulations, which are frequently called Monte Carlo

studies, have been useful in modeling highway traffic flow, work sampling activity,

and communication systems to mention several applications.

Dynamic System Simulation : Much effective work has been done in simulating dynamic

systems on computers. The nature of such problems lends itself well to analog computers, and dynamic system simulation is very likely the most important area in which

the analog computer is applied. The success of the analog computer in simulating

dynamic systems lies in the fact that most such systems can be adequately modeled in

terms of ideal elements, and these ideal elements in turn can be easily modeled by

computer elements.

Some of the simpler ideal physical elements, the equations which they satisfy,

and the analog computer program for their simulation are given in Table I. As an ex- ample of a direct simulation, consider the elementary system of Figure 10(a). A

model for this system using computer simulation of the ideal elements is given in

Figure 10 (b)

In this example the computer circuits simulating the spring, the mass, and the

friction were devised as the best analog computer approximation to the equations for

the ideal mechanical elements. The simulation point of view, however, is to think of

the analog computer program as a model for the physical system. Thus attention is

focused on the accuracy with which voltage in the computer approximates velocity in

the physical system rather than considering first the approximation to the physical

system afforded by the mathematical model, and then the approximation to the mathematical model afforded by the analog computer elements. A great deal of work has been

done in devising computer circuits to model physical elements. Tables and discussions

of many of these circuits are given in the references of section 10.

Example 3 : This example is concerned with the simulation of a mixing process and its

associated control equipment on an analog computer. Consider the process shown in

Figure 11. Liquid (or gases) from two sources - one at a high temperature T

fc and an- other at a low temperature T - are combined and the composite material flows into a tank containing liquid at a uniform temperature, T. Liquid at temperature T flows

out of the mixing tank. The temperature T is measured, and a controller is employed

to maintain T at constant reference value, T , by automatically adjusting valves on

the incoming hot and cold lines. The problem chosen for consideration is that of

determining the error between T and T after a sudden disturbance in the temperature

T . This error is a function of timer and will depend on the settings of the proportional, derivative, and integral gains of the controller.

92
Use of the equations given in Appendix 3 results in the computer circuit of

Figure 12. This circuit was used to plot the curves of Figures 13 and 14 which show, on a normalized basis, error vs time for various controller settings. The setting

corresponding to k^ = 0.5, k

fl = 0.5, k± = 0 gives the smallest transient error.

9 . Computers as Elements of Process Control Systems

In section 1 of this paper it was pointed out that computers have two distinct

roles in systems work, namely (1) computers can be used in system design, and (2) com- puters can be used as elements in process control systems. The use of computers in

system design was covered in sections 4-8. The purpose of this section is to define

and discuss briefly the role of computers as system elements.

In a certain sense the use of special purpose computers as control system elements

began almost concurrently with the conception of control systems. An error detector

which computes output minus input, a temperature sensor, and almost any other control

system element can be thought of as a special purpose computer in the sense that the

element output is a prescribed mathematical function of its inputs. The objective of

this section will be limited to a consideration of the use of general purpose com- puters as elements in physical systems, but of course it is difficult to make a sharp distinction between special purpose and general purpose computers.

General purpose computers are incorporated into systems in three different ways.

(1) The computer can store information which is used to compute, on the basis

of prescribed rules, the required time varying inputs to parts of a system.

This is essentially an open loop mode of operation since the computer processes only stored data without sensing the actual state of the system.

This use of the computer will thus be termed "open loop operation.

(2) The computer can also be operated from a fixed program to process both stored

data and data from sensors coupled to the physical system. In such an application feedback around the computer is present, and this type of opera- tion will be identified as "feedback operation."

(3) The most sophisticated use of a computer results when the computer program

as well as certain data are determined by the state of the physical system.

In such an application there is feedback around the computer, and the

potential to change the computer program makes the system adaptive. Thus

this type of operation will be termed "feedback-adaptive." Several characteristics of systems using computers in these three ways will now be

discussed.

.§_: General : Several problems and characteristics which are common to all, or at least two, of the three types of operation are the following.

Use of the digital computer with physical elements poses an immediate problem.

Most physical variables change continuously with time while on the other hand most

digital computers are variables which change in discrete steps. Thus almost all applications of a digital computer as an element of a physical system require analog-todigital or digital-to-analog converters or both.

Applications (2) and (3), which have feedback around the computer, can be unstable

so that design for stable operation becomes one of the more serious problems. The

same two applications raise problems of correctly identifying the current state of the

physical system, and choosing an appropriate measure of system performance.

As a final general comment, any use of a general purpose computer as a system

element requires economic justification. It is clear that in industrial applications

the savings in actual cost of the product of a system must be large in order to justify the use of expensive computing equipment.

93
b. Open Loop Operation : A system using a computer in an open loop mode is frequently

termed a computer directed system. A block diagram of this type of operation is given

in Figure 15, where it should be noted that the flow of information is entirely from

the computer memory through the computational operations to the input of the physical

components

One well publicized application of the use of a computer in the open loop mode is

in the numerical control of machine tools. For such an application the dimensions of

a desired product are stored in the computer memory and used to compute the desired

position of a cutting tool at any time. This information is then used to control the

machine.

The use of a computer for open loop operation is the most elementary of the cases being considered. The absence of feedback removes instability as one of the design

problems, but accuracy of the controlled variable cannot be as high as for the cases where feedback is used. The advantage provided by the computer is flexibility in

being able to perform complicated operations on the stored input data to produce the

necessary input for the physical system.

c. Feedback Operation : Use of a computer in this mode makes it possible to sense

the current state of a physical process and use this information along with stored

information to determine the required input to the physical elements. A block diagram for this type of operation is shown in Figure 16. The added feedback around the

computer makes stability a serious problem, but provides a more accurate means of

controlling the desired variables. Note that in this case the input to the physical

elements is computed from the stored data and from measurements of the current state

of the system. The added information provided in this mode of operation makes it possible to correct errors as they occur, rather than operating with a fixed procedure.

However, the computer program is fixed in advance so that the functional relation

between computer inputs and outputs does not change.

A flight control system for a modern space vehicle affords an illustration of

this use of a computer. In such a case an equation specifying the optimum trajectory

from any point in space to the desired target is programmed into an on-board computer.

The computer then uses a measurement of its current position along with stored target

data as inputs from which to compute the required thrust vector at all points of the

powered flight.

d. Feedback-adaptive operation : This type of operation has the greatest potential

of the three types being discussed. As indicated in Figure 17, the computer accepts

inputs from stored data, from sensors coupled to the physical process, and also from

other sources. In this case the program for the computer can be altered by its inputs so that the system can adapt to changes in inventory, economic factors such as sales price of the product, or any other input which is pertinent and which can be

quantitatively related to the desired output of the system.

Among serious design problems presented by this very general type of operation

are choice of criteria for weighting various factors which influence the desired

system outputs, and accurate means for identifying the current state of the physical

process

Examples of the use of computers in systems occurring in the ceramics industry

will be given in other papers which are a part of this symposium.

94
10 . References

This section lists references which cover in detail the areas summarized in the

paper. The references are divided into the following general categories to conform

with existing patterns in the literature:

1. General - references which treat both analog and digital computers.

2. Characteristics and Uses of Analog Computers.

3. Simulation using Analog Computers.

4. Analog computers as System Elements.

5. Numerical Methods - this is a mathematical area useful in preparing problems

for digital computers.

6. Characteristics of Digital Computers.

7. Use of Digital Computers in System Design by Successive Analysis.

8. Use of Digital Computers in Optimum Design Problems.

9. Use of Digital Computers in Simulation.

10. Digital Computers as System Elements.

11. References Pertaining to Specific Examples.

General

1. E. M. Grabbe, Editor, Automation in Business and Industry , John Wiley and Sons

1957.

2. P. von Handel, Electronic Computers: Fundamentals, Systems, Applications , Prentice Hall, 1961.

Characteristics and Uses of Analog Computers

3. A. E. Rogers and T. W. Connolly, Analog Computation in Engineering Design , McGraw Hill, 1960.

4. C. L. Johnson, Analog Computer Techniques , 2^ Edition, McGraw Hill, 1963.

5. A. S. Jackson, Analog Computation , McGraw Hill, 1960.

6. S. Fifer, Analog Computation , Vols. 1-4, McGraw Hill, 1961.

Simulation Using Analog Computers

7. V. L. Larrowe, "Direct Simulation-Bypasses Mathematics-Simplifies Analysis,"

Control Engineering , Nov. 1954.

8. J. M. Carteron, "Analog Representation of Heat Exchange. Application to the

Simulation of Heat Exchangers of Nuclear Power Plants, " Proceedings of the

Computers in Control Systems Conference , AIEE, May 1958.

9. W. J. Karplus, Analog Simulation , McGraw Hill, 1958. This book includes an extensive bibliography to 1957.

10. C. W. Worley, "Simulation of Industrial Processes with the Analog Computer, Automatic Control , p. 44, Jan. 1959.

11. D. E. N. Davies, "Electronic Simulation and Computer Techniques in the Design of

Automatic Control Systems," Journal of the British Institution of Radio

Engineers , p. 323, April, 1961.

12. F. A. Woods, "Simulation of Process Control with an Analog Computer," Ind. Eng . Chem. 50 , p. 1627-30, November 1958.

95
Analog Computers as Systems Elements

13. L. F. Kazda, "Control System Optimization Using Computers as Control System

Elements," Proceedings of the Computers in Control Systems Conference , AIEE,

May, 1958.

14. R. J. Bibbers, "Process Regulation with Analogue Controls," Automatic Control , Nov. 1954.

15. E. T. Davis, "An Electronic Control Svstem for Industrial Control Applications,"

Instruments , June 1951.

16. J. G. Truxal, "Computers in Automatic Control Systems," IRE Proc, p. 305,

Jan. 1961.

Numerical Methods

17. R. W. Hamming, Numerical Methods for Scientists and Engineers , McGraw Hill, 1962

18. F. B. Hilderbrand, Introduction to Numerical Analysis , McGraw Hill, 1956.

19. R. Bellman, Dynamic Programming , Princeton University Press, 1957.

Characteristics of Digital Computers

20. R. Nathan, E. Hanes, Computer Programming Handbook , A Guide for Beginners , Prentice-Hall, 1962.

21. I. Flores, Computer Logic , Prentice-Hall, 1960.

22. Computer Issue, Proceedings IRE , Vol. 41, No. 10, October 1953.

23. Computer Issue, Proceedings IRE , Vol. 49, No. 1, January 1961.

Use of Digital Computers in System Design by Successive Analysis

24. B. A. Bryce, "Circuit Synthesizing Techniques with the Aid of a Digital Computer

An Annotated Review of the Literature to 1962," Document 278 635, July 16, 1962.

This report is a review of 161 references pertinent to direct use of digital

computers in design, and other topics.

Use of Digital Computers in Optimum Design Problems

25. R. Bellman, Adaptive Control Processes ; A Guided Tour , Princeton University

Press, 1961. This book contains an extensive bibliography.

26. L. S. Pontryagin, et al., The Mathematical Theory of Optimal Processes,

John Wiley, 1962.

27. R. Bellman, et al., "A Computational Procedure for Optimal System Design and

Utilization.

Use of Digital Computers in Simulation

28. IBM, "General Purpose Systems Simulator," Reference Manual for a System

simulation program published by IBM Corp.

29. J. Moshman, "The Generation of Pseudo-random Numbers on a Decimal Machine," J. Assoc. Computing Mach .

, Vol. 1, p. 88-91, 1954.

30. M. Palevsky, "An Approach to Digital Simulation," Proc. National Simulation

Conference, Jan. 1956.

96
31. F. I. Musk, "A Monte Carlo Simulation of a Production Planning Problem,"

Computer J. Vol. 2, p. 80-94, July 1959.

32. C. P. Bonini, "A Simulation of a Business Firm," Proceeding of the Joint Computer

Conference, Vol. 21, 1962.

Digital Computer as System Elements

33. M. Pelegrin, et al., "Digital Computers in Automatic Control," Proceedings of

the Computers in Control Systems Conference, AIEE, May 1958.

34. E. L. Braun, "Digital Computers in Continuous Control Systems," IRE Trans . Vol.

EC-7 p. 123-128, June 1958.

35. A. K. Susskind, "Digital Information Process for Machine Tool Control," IRE Trans . Vol. EC-7, p. 136-140, June 1958.

References Pertaining to Specific Examples

Example 1

36. M. F. Berard, Diffusion in Ceramic Sys: A Selected Bibliography, Iowa State Univ.

Aimes Lab. Res. & Development Rpt. USAEC IS-448.

37. G. E. Forsythe, P. C. Rosenbloom, Numerical Analysis & Partial Diff. Eq., John

Wiley, 1958, survey of field.

38. L. W. Ehrlick, "A Numerical Method of Solving a Heat Flow Problem with Moving

Boundary," Asso. for Comp. Mach. Vol. 5, 1958, p. 161-176.

39. J. Crank, The Mathematics of Diffusion , Oxford University Press, London, 1956.

40. G. H. Bruce, et al., "Calculation s of Unsteady-State Gas Flow through a Porous

Media," Petroleum Trans. A.I.M.E . 198(1953) pp. 79-92.

Example 2

41. R. P. Webb, "Synthesis of Measurement Systems," Ph.D. Thesis, Georgia Institute

of Technology, July, 1963.

97
Appendix* 1: Further Discussion of Example 1

As discussed in the body of this paper, Frick's law given as Equation 12 is the

basic relation which determines the concentration of diffused substance as a function

of time and distance. The normalized variables defined on page 12 reduce Equation 12

to the equation

3T 8X [D iL) 3Xj

For the particular D(C) being used in this example, a variable s, defined by

C 1 s = / DdC/ / DdC,

o o

is introduced so that Equation A-l becomes

with the boundary conditions

s = 1, X = + 1 for T > 0.

s = 0, - 1 < X < 1 for T = 0.

V 1

V 2

To solve Equation A-2 on a digital computer, it must be reduced to a difference equation. The technique used in this example is known as the Crank-Nicolson method (see

reference 39, p. 189) and in the case being considered it results in a set of equations of the form

s(m,n+]) = s(m,n) + — 9s(m,n+l) + 9s(m,n) + 2 2(<SX) L J L

s(m+l,n+l) + s(nri-l,n) -2s(m,n+l) -2s(m,n) + s(m-l,n+l) + s(m-l,n)J

where m is the distance index, n is the time index, 6T is the step size in time and

6X is the step size in distance. The special equation

6 T s(o,n+l) = s(o,n) + —— «

s(l,n+l) + s(l, n) - s(o,n+l)

9s(o,n+l) + 9s(o,n) +

2^ [

-s(o,n)j

is required for m = 0. Equations Aj - 3 and Ax - 4 were solved for 8 steps in

X (i.e. m = 0, 1,2,3,4,5,6,7) making 6X = 0.125. A ratio 6T/(6X) 2 = 1/10, which

makes 6T = 0.0016, was chosen to insure convergence of the method used.

*The assistance of A. W. Bowers in programming this example is greatfully acknowledged.

98
em

*^at

t0nS ^ I3 T

d Al " equations

4 bSCame 3 Set ° f 8 simultaneous nonlinear algebraic for any fixed n. The procedure used in going from step n in T to ! !

Tne oTthfsT one of the s(m.n) in

SimUlt

every—term involving

^

ati °-

a— .ad'e linear ^ lls^^Ve JTM

equations was

product of two s terms. The set of linear

values.

then solved for the s(m,n) and the solutions compared to the assumed

was iterated

The new values were then taken as a better approximation and t^e procedure

xn the seventh

until the assumed solutions and actual solutions agreed witlin an error

answers for the

decimal

s(m,n)

place.

as This completed a solution at the staae n + 1 ^ ^

well as for concentration were priced oT.

^

The next step in time (i.e. to n + 2) was taken by assuming as an initial guess

s(m,n+l) = s(m,n+2)

and proceeding as before. Very small steps in T were required to insure convergence

wt TM«-

te"tive process. Further the initial concentration stated in EquationT 2 39) at V 01S

!7^°rY "

3 f

artin9 PO±nt - ThS ValUeS ^ iven * Crank (sefrefe^ence

IILTZ

= °- 172\"ere used to start the process and approximately 300 iterations were

tion" by hP

a

rnd?r

^

^° f ^?

' (C«* ^

trough 5 itera^*

The computing time required on the Borroughs B500 was several minutes.

Appendix 2: Further Discussion of Example 7

times the squared input to the unalterable elements Th P «- constant

energy supplied to the unalterable elements!

measures the

Tx Z « : IT': f

PUt \the °utput state -^i«. pae^i»;„\Hl "

99
Appendix 3: Further Discussion of Example 3

The components of the mixing process shown in Figure 11 are described mathematically as follows. The temperature T in the mixing tank is assumed to satisfy the

equation

dl/dt + | T = X Th + TM

cT

c

) A

3 - 1

where m is the mass flowing out of the tank at temperature T, m and m are the mass

flows at T and T , and M is the mass of liquid in the tank. The valves are assumed

to be described by an equation such that m^ and m^ are related to the controller output, V, by the equations

V dt A

3 m = m - k / V dt . c

'

The controller satisfies the equation

V(t) = k <T-Tr

) + k

d ft (T-T

r

) + k. /(T-T

r )dt. A

3 - 3

Combining Equations A^ - 1 and A. - 2, yields

dT/dt + J

T - gT

c + |(Th

-T

c

) /V(t) dt A

3 - 4

The following choice of constants and definitions were made m/M = 0.1 \ =fe( VVk

P ^ = &VVk

d h - i

( vT

c )k

i

Since the system is linear, no loss in generality is incurred by taking T^_ = 0, and

this results in the equation

Error = T(t) - T

r

= T(t) A. - 5

The response in error due to a sudden change in T was investigated, and this was produced mathematically by letting T be an impulse at t = 0 which is equivalent in this case to an initial condition on T.

The computer diagram which solves Equations A3 - 3 and A3 - 4 subject to the as- sumptions listed_is given in Figure 12. Plots of T(t)/T(t = 0) for various choices

for k , k,, and k. are given in Figures 13 and 14. p d l

100
Description of Element

Electrical resistance

Viscous friction

Thermal resistance

Electrical Inductance

Mechanical mass

Mechanical spring

Thermal mass

Table 1

Equation

e = Ri

f = Rv

6 = Rq

e = L di/dt

f = m dv/dt

Electrical capacitance e = — f idt

c >

f -\ I vdt

e = -i / qdt

Glossary of Symbols :

1, v or q

Analog Simulation

r(e»f or e

)

e voltage; f - force; 6 - temperature

i - current; v - velocity; q - heat flow

R - res is tance in all systems

L - electrical inductance m - mechanical mass

c - electrical or thermal capacitance

1/k- mechanical spring constant

PROBLEM »• SCALE MACHINE COMPUTER

VOLTAGES

EQUATION FACTOR 1 EQUATION * 1

PROBLEM INVERSE

SCALE

FACTOR

MACHINE RECORDER

SOLUTION SOLUTION <

Figure 1. Basic Steps in Using an Analog Computer.

101
r

PROCESSING

' i

MEMORY

- n

OUTPUT

1

1

!

!

I ! —I I CONTROL

PROGRAM

Figure 2. Basic Steps Performed by a Digital Computer.

INTEGRATOR

x*(t)

1

INITIAL CONDITION

x (t0 ) = A

FUNCTION

GENERATOR ——*^

Figure 3 • Approximating the Solution of a Differential Equation

with an Analog Computer

102
r

MEMORY

CONTROL PROGRAM

1

tQ + l£ w

"*(t 0 + It ) fr>

y(t0

+ k

COMPUTATION

"*[t„ + (i+lje] OUTPUT

Figure k. Approxi mating the Solution of a Differential

Equation with a Digital Computer.

FUNCTION

GENERATOR :y(t)

t

Figure 5. Logical Program for Solving a Set of Simultaneous

Differential Equations.

103
Figure 6. An Illustrative Diffusion Problem. Figure 7. Curves Illustrating the Solution of a Diffusion Problem.

-1

Figure 8. System Configuration for a Control System.

104

1 Jo VALVE

MASS FLOW - m

VALVE CONTROLLER

TEMPERATURE

SENSOR

MASS FLOW -

TANK

Figure 11. A Mixing Process.

Figure 12. Computer Simulation for a Mixing Process.

106
107
Figure 15 • Block Diagram. Showing Open Loop

Computer Operation.

Figure 16. Block Diagram Showing Feedback Operation.

COMPUTATION

Figure 17 • Block Diagram Showing Feedback-Adaptive

Operation.

108
Statistical Methods in Systems Design

W. C. Hackler

1. Introduction

When a manufacturing process can be characterized by means of a mathematical equation one is m a position of greater understanding, hence greater ability to exer- cise control over the process. Too often in the ceramic industries qualitative relationships are understood but the appropriate constants associated with a mathemat- ical function of the process variables are frequently unknown. Statistical methodology provides a means of estimating the necessary coefficients to associate with each proc- ess variable thus providing the needed quantitative relationship between variables and process yield or quality. In addition it provides a level of assurance (probability) depending on the magnitude of the fluctuations or variability encountered in the proc- ess. Such fluctuations may be assignable to raw materials, equipment, or level of operational variables, temperature for example. In any system, decisions need to be made at various stages throughout the process as to whether the process is performing in a satisfactory manner or not. In order that a valid decision be made, one must have assurance that test data is a reliabile predictor of performance. Statistical methods are most useful in reaching a decision - either adjust the process or allow it to continue. In many instances it has been proved that frequent adjustment of some process variable was the most upsetting influence. Some variation about an average value is to be expected and as long as this variability is within specified limits, the system should not be adjusted for these minor fluctuations. Another contribution of statistical methods is that of evaluation of "noise" or error in the system. Tech- niques are available whereby the masking effect of one variable can be isolated so as to get a true measure of the desired variable. Error analysis and reduction of error (minimization of variability) are effectively handled by statistics as well as the determination of what is a minimum error for a particular system.

2. Estimation

A manufacturing process may be examined as a situation in which some response (yield, Purity, strength, resistance or some other property of a product) is a direct result of the operational level of many variables (raw materials, equipment and process variables.) If the variables can change along a continuous scale and the response can be measured in the same manner, then it is possible to describe the response as some functional relationship between the variables. In many instances it has been found that an expanded Taylor series1 will provide a suitable estimate of this response The order of the series may be extended as required for a satisfactory fit. Such 'a relationship for two variables of the second order might be:

When:

Y = b. + hl x1 4- b2 x3 + bjjXjXs + h11 x1 3 + b2 3 xl

Y = response

. th . , , x. = i variable

K = differential effect associated with i th

variabl*

(1)

The order of the series depends on the complexity of the true but unknown surface The magnitude of each b indicates the contribution of each variable to the response of interest. Hence, the variables with the largest b values are the ones that need closest control so as to maintain the response at the desired level

IOC
The coefficient b, a in equation (1) is of prime concern. This term x.x3 is known

as the interaction term or the failure of xx to produce the same effect on

1 th e response

at different levels of x2 .

Interaction can best be explained by examination of Figure 1. When a response

is plotted against a variable x^_ for two different levels of a second variable xa

,

two lines result, one for the low value of Xg_ and one for the higher level of xs . If these two lines are parallel then the effect of the variables on response is additive, and it may be assumed that there is no interaction between the two variables.

When there is lack of parallelism, interaction between x^ and Xg_ exists. Then the

effect of xa on the response is dependent on the level of xx. Hence interaction may

be characterized as factor dependence insofar as the response is concerned. Inspection of the set of numbers in Table I is also a helpful way of evaluating interaction.

Table I

Low High

Low 5 10

High 9 18

Increase of xx from the low to the high value increased the yield by 4 units.

An increase in Xg_ from the low to the high value increase the yield by 5 units.

Simultaneous increase of xx and xs from low to high value resulted in an increase in

yield of 13 units. Since 13 does not equal 5+4, interaction is present. Thus

interaction between two variables can be plotted or set up in a tabular form. When

the number of variables increases, the ability to "see" the interaction diminishes.

However, they can be estimated in a quantitative manner by means of the functional

relationship (equation 1) regardless of the number of variables.

A most useful outcome of equation (1) is sometimes referred to as response surface methodology since the equation describes a "surface" of some order. The plot

in figure 2 is just like a topographic map where points of equal elevation over a given area are connected by a contour line. These contours locate all combinations

of Xj^ and x2 within a prescribed region that will yield a given response. This con- tour system points out the necessity of simultaneous evaluation of the effect of both

variables on the response. A given response is not achieved in a unique manner rather

there are many combinations of variables which will yield the desired level.

Another interesting feature of the contour plot is the necessary control which

must be maintained over each variable. It is quite apparent that the solution to this

problem depends on the slope of the contours in the operational region. In figure 2 a deviation of a units of x3 would produce much more drastic results when operating

at point R as compared to point S_. The closeness of spacing of the contours has the same implication in this plot as does a topographic map. Close spacing of contours

is indicative of a steep slope or a rapid change in response. In this case it would

be much better to choose point S_ as operating levels of the variables to give a re- sponse of 80 rather than point R, since the slope is steeper at R than it is at S_.

It is quite probable that more than one property of a product must be kept within desired limits. The response surface for a second property will be a quite different relationship than that determined for the first. Such an example is seen in

figure 3. Let the contours labeled 60, 70, and 80 represent yields and the second

family of contours labeled A, B, and C represent cost per unit, increasing from A to

C. Since maximum yield is desired, anywhere within the 80 contour would meet this

requirement. However; in order to maximize yield and minimize costs, a smaller operational region results. The segment of the A curve between points M and N represents conditions where a yield of at least 80 is obtained with minimum cost.

Graphical interpretation can be extended to three dimensions by stacking planes

such as seen in figure 2 to yield figure 4. In figure 4 it can be seen that the con- tours are the intercepts of a surface with a plane at specified levels of the third

variable

110
The functional relationship of equation ( 1) may be compared to a transfer function

relating a response to the differential contribution of certain operational variables.

This equation is one which is empirically determined, by methods of least squares so

as to result in the best fit of the data. Data can come from laboratory experiments, pilot plant trials, or best of all from production runs. Often times "evolutionary

operations" techniques serve as a useful tool for collecting data from production runs

so as to minimize upsetting the process. Such information will characterize the

response surface, pointing out optimum operational combinations of controlled variables

The effect of control of each variable is readily available from this equation. Such

a relationship provides a method of comparing one design or control system against

an alternative one. Thus a quantitative method is available for evaluation of a par- ticular design.

In the estimation of the function of equation (1) there will be some variability

in response which cannot be associated with an assignable cause or variable. For

example, the failure of duplicate observations to be identical is such a source of

error. The contribution of an uncontrolled variable results in a large increase of

the error term. Any variability which cannot be accounted for is pooled in the error term - sometimes called lack of fit. As the number of uncontrolled variables increases

more specifically as the error term increases, the usefulness of the equation diminishe

One has to describe what is a tolerable error term, based on product specifications

and the economics involved. In some cases the error cannot be reduced to a desired

level without major changes in the equipment, control systems, or the processing itself. In any event a measure of the "noise" in the system is available as lack of fit

of the function relationship. The magnitude of this error serves as "yardstick" for

evaluating the effect of specified variables. If the contribution of a given variable

is of the same order of magnitude as is the error, then one cannot say that the specific variable contributes to the response. If, on the other hand, the effect of a variable is much greater than the magnitude of the error, it can be safely assumed

that the particular variable is a critical one. Thus as the magnitude of the "noise"

or error term diminishes the ability to discriminate between the effects of variables

increases. This error term provides the necessary scale for the evaluation of alternative designs.

3 . Raw Materials

Before any system can be set up in a satisfactory manner some control over the

starting materials must be established. Whether material comes from a variety of

suppliers or whether material is mined locally, there is an associated inherent variability. In order to be assured that the system has the opportunity to perform at its

optimum, then realistic material specifications (based on past performance or from

pilot plant studies) need to be maintained. Too frequently specifications are established for no good reason, leading to inadequate or too stringent requirement for the

material. It is essential that quality measurements made on the starting materials

be a reasonable predictor of the performance of this material in the process. Thus,

if measurements of sensitive properties of the material are not obtained control may

appear to be had when in actuality there is no correlation between measurements and

performance. Figure 5 portrays such a situation of low correlation between measurements and outcome in the process. Once such a plot is made it is quite apparent that

the measurement will not predict the outcome. One could just as well "fit" line A as line B to this set of data since really there is not a satisfactory linear fit to the

data

One method of evaluation of the quality of raw materials is by utilization of an X and R chart for measurements3 on one or more sensitive properties of the material.

By use of the X chart the level of a variable can be kept under surveillance. The R

chart, however, is a measure of variability within a sample, thus it is an estimate

of the variability in the lot of material from which the samples were taken. Figures

6 and 7 depict the X and R charts respectively.

Ill
The X chart will present the level of quality of a given sample of material whili

the R chart is indicative of the variability within this sample. Two points are to

be remembered here:

1. These charts as shown are prepared for averages of samples of a given size.

If the sample size changes, the upper and lower control limits will also

change. An increased sample size implies a better estimate of the lot

average of the particular property being determined, hence control limits move nearer to the average. Consequently, the control limits for the X

chart in figure 6 are not appropriate when considering individual items.

2. A second aspect of these charts is that they present estimates of the property of the population from which they were taken. As such these estimates are totally independent of desired values or specifications. In the event

that the specifications fall outside the control limits for individual items

then one readily accepts the material being evaluated. However, when the

specifications fall inside the control limits for individuals, the material

does not meet the specifications. It is quite possible that materials or parts could be prepared under controlled conditions and indicate a state

of statistical control, yet not be able to meet specifications. When this

situation arises there are at least three alternatives available: 1) find

new vendors, 2) help vendor improve his process, 3) examine specifications

to see if they are realistic.

There are many cases in which incoming material is evaluated as being either ac- ceptable or not acceptable. The absolute value of a characteristic is not measured.

All that is known is that the inspected item meets or fails to meet specifications.

It is not known how good or how bad an item may be. Also a part could be rejected

for any one of a number of reasons such as size, smoothness, glaze defects or warpage

When a sample from a lot of material or parts is inspected in such a manner, this is

known as sampling by attributes. Items such as resistors, capacitors, wall tile,

dinnerware, and enameling stock are examples of items which could be inspected in

such a manner.

To help in the classification of incoming material the United States government

has developed sampling methods published as MIL-STD-10 5-D. 3 These plans take into

consideration the lot size and acceptable quality level. Based on this information

sampling plans are specified for single, double, and multiple samples. The sample

size is specified and the number of defective items allowable in a good sample are recorded. An excellent feature of MIL-STD-105-D is the fact that the operating characteristic curve for each plan is presented.

The operating characteristic (OC) curve is a measure of the ability of

sampling plan to discriminate between acceptable and unacceptable material, curve is shown in figure 8.

If the lot size is at least ten times greater than the size of the sample, these curves can be obtained from the summation of the appropriate terms of the binomial

expansion (q + p) n when q = fraction acceptable

p = fraction defective

n = sample size.

In the case of the curve labeled N = 150, c = 8 the sample size is 150 and the c refers to the number allowable defective items in the sample. At a given fraction

defective (0.04 for example) the probability of acceptance is obtained from:

P = Cn(0.96) n_r

(.0.4)

r

r r where r_ indicates the number of defective items. The probability is obtained by sum- ming probabilities for no defectives, one defective, etc. up to eight defectives at that specified value of fraction defective. When p < 0.10 and pn < 5, a Poisson

distribution2 is easier to use to compute P. Such data can be read directly from a

112
chart for the Poisson distribution found in Dodge and Romig. 4 It can be seen that

the curve N = 150, C = 8 has a steeper slope, hence is more critical. The probability

of accepting a sample of p = 0.04 is about 0.80 while the other curve will accept

such a lot approximately 0.06. Lots of p = 0.08 will be accepted only 20% of the time

by N = 150, c = 8 as compared to 43% of the time for N = 150, c = 11. Thus the sample size and value of c may be found for any desired probability of acceptance at a specified fraction defective. As a general rule a probability of 0.95 is satisfactory for

the acceptance of a good lot. This remaining 5% is known as the producers risk (a)

the risk of producer takes of having a good lot rejected. By the same token the risk

of accepting a poor lot is known as the consumers' risk (3).

A sampling plan is characterized by any two of the three following:

a) fraction defective

b) producer's risk

c) consumer's risk

There are a variety of alternatives that may require some compromise so as to come

near to satisfying all three conditions. The manner in which a OC curve changes with

sample size is illustrated in figure 9. As the sample size increases the plan becomes

more discriminatory in evaluating good and bad material. Naturally the ideal curve

is a ~%_ shaped one. As sample size increases the curve approaches that shape.

4 . Process Control

Statistical techniques are employed throughout a process in order to detect

shifts in the process as quickly as possible so that adjustments can be made to return

the process to its desired operating conditions. The adjustments may come in the form

of manual changes to a piece of equipment by an operator or it may be in the form of

an electrical feedback from a sensing element to the piece of equipment. Some deci- sion making element such as a computer is needed in the line as seen in figure 10.

In this case two sensings are made, 5 furnace temperature and nominal core resistance.

This information is sent to a computer which solves the transfer function with the

data supplied by these sensings and feeds the correction to the furnace controls.

It is to be remembered that again a correlation between the quality of the product

and the process measurement must exist if effective control is to be maintained.

Quite frequently the sensing is made of something other than the product or semifinished product (e.g. temperature, atmosphere, or flow rate of the product). This

avoids handling or contact with the product, but a good correlation must exist.

Frequently too, it is the combination of two or more process variables (e.g. time and

temperature) which must be sensed and controlled simultaneously. The importance of

this simultaneous control is indicated by the magnitude of the coefficients of the

interaction terms of the response function (equation 1)

An interesting example of the use of control of an automatic process was re- ported by Boatwright. 6 The process was that of making carbon coated resistors

where control of the furnace temperature, rate of flow of gas, and speed of cone travel through the furnace was maintained. An X chart was plotted by a computer and

use was made of a non-parametric test to evaluate shifts in the process average. Averages of five resistance measurements were plotted on a chart with the desired

nominal resistance as the central value. The value of the average range was programmed

as an estimate of the anticipated value based on past experience. Significant trends were established by any of the following conditions:

1. any point falling outside the control limits

2. two out of three successive points in zone A or beyond

3. four out of five successive points in zone B or beyond

4. eight successive points in zone C or beyond

A control chart is shown in figure 11. The width of a zone is equivalent to one standard deviation. Corrections for average values of resistance were calculated by the computer, the correction being based on an average of 20 or 30 previous points, not on the point at which a decision was made.

113
5 . Reliability

To many people reliability of a given item is achieved by successive inspections

and measurements, discarding unsatisfactory pieces and retaining the good. Such a process is merely sorting, making the best of what is available rather than making a better product. Reliability is used in many senses but the most appropriate connotation is that of a probability7 of satisfactory performance over some specified period

of time. As such it is neither a quality nor a property of an item, rather it is a number between 0 and 1 with no units. Thus reliability may refer to a system or a single component in a complex system. It is often evaluated in a negative sense, such

as a failure rate or mean time between failures.

Since reliability is concerned with the ability of a given product to perform in

a specific environment, it is apparent that an intermediate heat duty fireclay refractory would not have the same reliability as a 70% alumina refractory if the environment was near the service limit for the 70% alumina refractory. However, reliability

of both refractories would be nearer to each other when service conditions were near

those appropriate to an intermediate heat duty fireclay. Hence, it is assumed that

material be selected wisely for a specified environment. The technique of "derating"

is often used to increase reliability. If design indicates that a one watt resistor

is an appropriate component then the use of a two watt resistor gives assurance or acts as a safety factor to increase reliability of the part selected.

It is quite possible that a given process can be producing a product under excellent conditions of statistical control yet the product might have a low reliability.

This would indicate the presence of unknown or uncontrolled variables in the process

which exerted a significant effect on the product. Apparently specification as well

as processing control need be changed in order to characterize the product more properly as pertains to performance in service. Such a situation readily indicates the

difference between the quality control function and the reliability function. Where

the responsibility of quality control is present time, the responsibility of the

reliability function is in the future.

Before an estimate of reliability of a product can be determined it is essential

that tests be made. By and large, extended periods of time are required to obtain

life test data, particularly when dealing with a product of low failure rate. Often

life test data is obtained by means of accelerated tests, or by testing a large num- ber of items for short periods of time. The equivalence of time and number of parts

tested to yield the number of hours to first failure is not always justifiable. For

a constant failure rate from the beginning of the test, this compression of time by

increasing the number of components tested is on safer ground. Accelerated tests are often performed at higher stress levels than a part may be expected to see in service.

In order to translate such test data to service conditions an appropriate factor must

be known which will relate this overload endurance to expected life.

Reliability seen in a product is a direct result of reliability in the performance

of a system which manufactures this product. The quality control function and the

product reliability function must be brought into close proximity in order to achieve

reliability in the system. System reliability is brought about by an understanding

of the effect of process variables on the performance of the end product. Thus the

design of the system - the materials used, the equipment selected, the closeness of

controls, the monitoring of the performance - yields the level of reliability seen in

the system. The major contribution of statistical methods is that of providing quantitative information relative to optimum operating levels of process variables.

Increased complexity of systems seems to be the order of the day, and statistical

design appears to be a most reasonable method to evaluate simultaneously the effect

of many variables. In most systems the effect of one variable is not independent of

another variable, hence the contribution of the interaction between two or more variables must be known. This is best accomplished by means of statistical design.
REFERENCES

1. W. G. Cochran and Gertrude M. Cox, Experimental Designs , Chapter 8A, John Wiley

and Sons, Inc. (1957)

2. A. J. Duncan, Quality Control and Industrial Statistics , Richard D. Irwin, Inc. Homewood, 111. (1959) .

3. Sampling Procedures and Tables for Inspection by Attributes , MIL-STD-105D

U. S. Government Printing Office, 29 April 1963.

4. H. F. Dodge and H. G. Romig, Sampling Inspection Tables - Single and Double

Sampling , John Wiley and Sons, Inc. (1944)

5. M. D. Clark, H. K. Jordan, and R. C. Ward, "Precision Carbon Coatings by Computer

Control," The Western Electric Engineer , _6 (2) April 21-27, (1962).

6. J. H. Boatright, "Computer-Plotted Control Charts," The Western Electric Engineer , 6 (1) January 20-21, (1962)

7. E.G.D. Paterson, "Reliability and Quality in Communications", Industrial Quality

Control 20 (9) March 15-23, 31-38 (1964)

115
' Figure 2. Contours of equal response in

a two variable plane.

Figure 1. Interaction between two

variables

Figure 3. Families of contours for two Figure 4. Response surface in X1 , X3 , different responses (60, 70, and X3 space,

and 80) indicate % yield and

A, B, and C indicate cost

increasing from A to C.

116
LU

O

O

o

Figure 5. Lack of correlation between

measured property and outcome

or quality of a product.

MEASUREMENT

Figure 6. Control chart for sample

average

X

UCL

X

LCL

SAMPLE NUMBER —UCL

R R Figure 7. Control chart for sample range

SAMPLE NUMBER

LCL

117
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

LOT FRACTION DEFECTIVE

Figure 9. Operating characteristic

curves for different sample

COMPUTER

CONTROL

JT

HEAT

CONTROL

_SPEED^

CONTROL'

_RES I STANCE

READINGS

TEMPERATURE

FURNACE

RESISTANCE

MEASUREMENT

Figure 11. Control chart used in non- parametric test (after

Boatwright)

Figure 8. Operating characteristic curves

for different acceptance

numbers of defective items.

N = I50

c=l

0 004 008 OI2 OI6

LOT FRACTION DEFECTIVE

Figure 10. Computer controlled process.

ZONE

UCLX

2 3 4 5 6 7 SAMPLE NUMBER

8 LCLX

118
Systems of Interest to Ceramic Engineers

A. H. Hurst & E. R. Bullock

1 . Introduction

Beginning with the pottery of early civilizations, the study of ceramics has

evolved into a highly scientific technology covering a broad range of endeavors. This

evolution has been caused by the demands of modern industries which has placed an ever increasing premium on a more exact composition, precise dimension, or repeatable physical characteristic. In order to keep pace with these demands, the modern ceramist

must rely on control technology to assist him in producing a product which will meet

or exceed such stringent requirements.

In this paper, the authors describe some typical systems that are of interest to ceramic engineers. The examples were chosen in order to give a good cross section of

the various types of systems available to industry. The practical aspects have been

emphasized in order to give its reader a basic understanding of the mechanisms at work, and to stimulate his imagination toward applying the concepts to solve everyday

problems. Both conventional analog control and more modern digital control systems

are explored.

2 . A Packaging System

Many small ceramic parts must be manufactured to extremely close tolerances and

therefore have a significant cost which makes an accurate accounting of each part

worth the investment of a system to count each part. This can be accomplished most

easily during the packing phase of the operation when a particular number of parts are to be packaged in a container.

Figure 1 shows a simplified version of such a system which has as its basic parts

a conveyor, a deflector bar, a chute, a photopickup, and an electronic preset counter.

As parts move along the conveyor, they pass the photo pickup sensing head. The re- sultant electrical signal generated by the interruption of the light beam causes a single count to be registered in the electronic counter. At the same time, the

deflector arm, capable of assuming two positions, is in position "A" in order that all

parts will enter the container in position beneath chute "A". A reserve container is

in position "B" so that when the container in position "A" is full, the delivery of

parts can be directed immediately to "B", the full container moved out of the loader,

and an empty container placed in position "A" ready to recieve parts when "B" becomes

full.

The counter such as that illustrated in Figure 2 is capable of being preset to a given number which represents the desired quantity of parts to fill a container. When

this number is reached, a relay closure is initiated within the counter and the

counter immediately resets to zero in preparation of another cycle. A solenoid positions the deflection bar to "A" and "B" depending upon whether it is activated or not

which in turn depends upon. the alternating condition of the counter output relay.

While this example represents a simple open-loop control system, it can be readily

adapted to other applications through variations in sensors or conveyors. Many parts are loaded by vibratory means which is less cumbersome and quite fast making it an attractive method for parts handling. Other sensors could be used depending on the

nature of the parts to be loaded. Air sensors operate on the interruption of an air

stream between a source and pressure diaphragm sensor, generating switch closures each

time that the air pressure drops below a previously adjusted limit. This method would

119
be useful with ceramic parts, but might lack the desired speed since the switch closure

time could be in the order of milliseconds while the photo pickup is in the microsecond

range.

A variable reluctance sensor would work only with magnetic materials such as ferroceramics , but has the advantage that it is low cost and quite dependable. This

pickup consists of a permanent magnet surrounded by a coil. As the magnetic field sur- rounding the coil is disturbed by the presence of a magnetic material, the coil sees

a magnetic flux change which generates an output signal in accordance with Lenz ' Law. Other suitable variations will occur to the reader when faced with a specific packaging

problem.

3 . An Automatic Testing System

Quality assurance becomes increasingly more important daily. Materials and com- ponents must meet continually increasing demands on reliability which in turn requires

greater knowledge of a larger number of components. To have a technician individually

check each part is time consuming and therefore costly. A large amount of data can be

obtained automatically with a system such as the one shown in Figure 3.

In this case, it is desired to know the temperature coefficient of ceramic dielectric materials. An oven with a large number of electrodes suitable for holding

ceramic samples comprises the test jig. Connected to this is a commutator capable of

selecting any of the samples and connecting it to a test oscillator. The test oscillator is running at a frequency dependent upon a known inductance-capacitance combination. The dielectric under test is connected in parallel with the tuned circuit capacitor, and thus causes a shift in frequency which is related to the value of capacitance

of the sample.

A frequency counter measures the frequency of the test oscillator. This is ac- complished by counting the cycles of the oscillator for an accurately known time,

usually one second or decimal multiple thereof. At the end of the counting interval

or "gate time", the counter input is closed, and the counting chain then holds a number

which represents the number of cycles counted. If the gating interval were one second,

the number would indicate cycles per second directly. The use of an electronic frequency counter allows the data to be measured, and converted to digital data at the same time. The resultant digital information can then be connected to the computer

input directly.

A heater for the oven is controlled from the computer with on-off control through

a relay. The temperature control loop is closed with a digital temperature transducer/

encoder which converts temperature to a digital signal for direct input into the computer. The transducer/encoder such as that shown in Figure 4 consists of a mercury

filled thermal bulb system which is connected to a bourdon tube. As the temperature

increases, the pressure increases thus causing an expansion of the bourdon tube which

results in rotary motion. The rotary motion, which usually moves a pointer on a scale,

in this case rotates a shaft position encoder.

The shaft position encoder graphically shown in Figure 5 consists of a digitally

coded glass disc with patterns of black and clear such that light can pass through

some areas and not others. A light source is positioned opposite to a photocell with

the disc between. The photocell is divided into many tracks depending on the resolution required. A typical number of tracks is ten since this would allow 2

10 possible

combinations which would consist of ten photocells (one for each track) . The ten bit

output would then be amplified and connected to the computer input. Any given temperature would then be represented by a binary number between 0000000000 and 1111111111

which would correspond to 0 to 10 2 3 in decimal terminology.

The actual temperature in engineering units such as degrees F can be determined

by the computer through a scaling operation in which the binary number read by the

computer is multiplied by a factor determined by the range of the transducer. This

factor is determined only once and placed in computer memory. Each reading then is

corrected and stored in memory as a temperature value in degrees F.

120
A clock is included in the system to act as a reference for the sequence of oper- ations in changing temperatures. Data from the clock is continually available to the computer. The clock is a simple counter counting the cycles from the power line since this is quite stable over long periods of time and in any case is as accurate as any synchronous clock. For each sixty counts received, one count is recorded and the first section reset and repeats. The recorded counts are then seconds (at a line frequency of 60 cps) . The seconds counting register resets on 60 and transfers a minute count to the next section. Likewise the last section counts hours to 24 and resets In this way, data is in the proper form to connect directly to the computer and furnish time of day information.

The computer can now be programmed to operate all of the subsystems previously described. A typical program would have the following routines:

1. Bring temperature of oven to first test temperature.

2. Measure capacitances, store and log data.

3. Bring temperature of oven to second test temperature.

4. Measure capacitances, store and log data. (This sequence would continue

to the last temperature.

5. Calculate temperature coefficients and log data.

6. Calculate temperature coefficient distribution and log data.

In order to control the temperature of the oven the computer begins by comparing the desired temperature to the actual. The difference represents an error which is entered into the control equation. Solution of the control equation provides a deci- sion to do one of two things: Turn on power or turn off power to the oven. Clock data also enters at this time and provides a basis for decision as to the stability of the temperature. This is accomplished by comparing the programmed temperature to the actual measured temperature and establishing the time for which the difference remained between ± 2 degrees of zero for example. When this condition has been met for five consecutive minutes, the computer recognizes the fact and initiates a cycle of capacitance measurements.

The capacitance measurement is also made on a cyclic basis from control commands generated by the computer. After the first decision that temperature balance has been sustained for the given length of time, the computer issues a command signal to step the commutator to position 1, then after a small delay (on the order of 10 milliseconds) it commands the frequency counter to take a reading. The frequency counter gate

opens for the gate time and then closes. The closing of the gate signifies to the com- puter that the reading has been completed.

The computer then reads and stores the data from the counter. After calculating the capacitance from the frequency data, the computer then logs out the point number

temperature, and capacitance.

After having completed the routine for the first point, the computer commands the commutator to step to the next point and measure. Again, the same sequence of events would take place until all of the points had been measured, calculated, and logged. The last point is stored in memory and after each capacitance measuring cycle, the com- puter runs a comparison check for equality between the point just completed and the last point. When the two check as equal, the computer automatically moves to the next temperature.

After a check which indicates that the maximum desired temperature has been reached and the data read and logged, the computer moves into the final routine to calculate the temperature coefficient point by point and log each coefficient along with the point. After completing the coefficient calculations, the program shifts into a statistical analysis of the results and logs out the desired statistical data. Sensing that all data gathering and calculations were complete, the computer would then shut down the test.

121
The foregoing illustrates the power of a computer system. The system described

would not be a large complex of equipment. Figure 6 shows a typical small scale com- puter which can easily perform the tasks described. It would in fact be possible to package the whole system in a standard relay rack. Since the computer acts rapidly,

it can do many chores while waiting for other equipment to perform. For example, in

the system described, the computer can make at least one and perhaps many temperature

control calculations on the oven temperature while the frequency counter is measuring

frequency. The "gate closed" signal from the counter would enter an interrupt input

to the computer to indicate that data was ready for pickup by the computer. At the

next convenient point in the program, the interrupt would be checked. If it has re- ceived a signal, the program would shift into that portion written to operate in the

data, and having completed that operation, would return to the point at which it had

been interrupted. In this way, the computer is continually juggling all of the inputs

and output to gain the maximum amount of computation per unit of time.

Much larger computer systems are presently in operation controlling complete re- fineries, power generating systems, and cement plants. Such systems not only provide

complete automatic operation of entire plant and complexes, but have the added ability

to assimilate marketing and other pertinent data such as weather or availability of

raw materials data. With this data, the large computer system makes economic decisions

and schedule alterations which automatically adjust set point and controls within the. plant to case the most economic production of the most marketable item. Versatility

of this magnitude stretches the imagination to new horizons in the realm of the

possible

4. A Blending System

In the ceramics industry, the accuracy of material concentrations or mixtures can be quite important to the quality of the end product. Often materials are mixed in a slurry to gain even composition before further processing of the material. The addition of small amounts of reactants with a high degree of accuracy can be a difficult

task with conventional means. The following blending system could find use in mixing

a reactant solution with a slurry at unusually high accuracies.

Figure 7 shows a typical example of such a system using a variety of control

devices. Two feed lines are indicated "A" and "B". "A" is shown to have a magnetic

flowmeter since it will be assumed that the material flowing through this flowmeter

is abrasive and not suitable for measurement by either positive displacement or turbine

meters. "B" is assumed to be a liquid which may be used with a turbine meter. The

liquid "B" is to be added to the material "A" in a given ratio by volume which may be

set manually.

This system is divided into both analog and digital parts. Since the heart of the

control is a pulse counter, the first objective will be to convert the volumetric

flows into pulse trains which represent the volume which has passed. In line "A" this

is accomplished through the use of a recorder associated with the magnetic flowmeter.

Figure 8 pictures a typical magnetic flowmeter. The recorder is usually considered to be a part of the flowmeter due to the sensitive nature of this method of flow measurement, and will indicate the flow rate of the material passing through. An attachment

to the recorder will convert the rate reading to a resistance proportional to the

rate indicated. A voltage source across the variable resistance will then generate an output voltage proportional to the flow rate. This in turn when connected to a voltage

to frequency converter will provide a pulse train whose frequency is proportional to flow rate. Since the frequency is proportional to flow rate, then each pulse must

represent a given amount of column. Thus, we have now represented the volume by a series of pulses.

The turbine meter such as that illustrated in Figure 9, generates pulses directly.

This is accomplished by placing a small turbine directly in the path of the flowing

liquid. As the blades of the turbine pass a reference point on the circumference, a pulse is generated for each blade that passes. Calibration curves furnished with the

meter relate flow rate to pulse rate. This can then be interpreted as pulses per unit

122
volume. Since turbine meters are subject to wear both on the blades and in the bearings, the calibration curve is not valid indefinitely. For this reason, periodic

calibration checks are run to determine the accuracy of the meter.

When used in a system, one relies upon a single pulse representing a given volume

of liquid, therefore a drift in calibration causes loss of accuracy in the control

system dependent upon the meter. To compensate for meter wear, electronic scalers

have been developed. The scaler is a counting instrument which will generate a preset

number of output pulses for every 10N input pulses, where N is the number of decade

units in the scaler.

As an example of how this works, let us examine the case in which a turbine meter

normally generates 1000 pulses per gallon. Further, we will assume that a three decade

scaler is in use. Since we can never get more pulses out that we put in, we must

choose a convenient number of pulses per unit volume with which to work. In this case,

100 pulses per gallon seems to be the most convenient. To obtain this, we must set

100 on the scaler. Since we have three decades, we will get 100/10 or 100/1000 pulses

out. Expressed another way, we will get 100 scaler output pulses for every 1000 turbine meter pulses. As wear causes a drop in the number of pulses per unit volume for

the flowmeter, we can manually select a larger number (anywhere from 000 to 999) for

example 110 which would then cause the generation of more output pulses per 1000 input

pulses, but now the meter would be generating fewer pulses per gallon. The result

would be that the system would see no change in the pulse per unit volume ratio, and we have thus compensated for the meter wear.

Now data is available from both meter units in the form of pulse trains representing the volume of each that is passing. To use this data, the units must first be nor- malized to a common language, that is, each pulse must represent the same volume.

This can be done by adjusting the voltage span in the voltage to frequency converter

to correspond to the chosen pulse-to-volume ratio of the normalized turbine meter.

At this point, it is necessary to discuss the bi-directional counter, an example

of which appears in Figure 10, which compares the pulse trains by counting one train

in a positive direction while simultaneously counting the other in a negative direction.

The result is an instantaneous algebraic sum of all counts received. If both lines

"A" and "B" have identical flow rates and both normalized pulse train outputs are fed

to the BDC (bi-directional counter) , then the constant algebraic sum will be zero, but

if one of the lines has an increased flow, the resultant difference will be integrated.

Attached to the BDC is a digital to analog converter which generates an output

signal voltage proportional to the number and signal contained in the BDC. The D/A

output voltage in turn enters a controller with the set point set to zero. The con- troller in this instance has proportional plus reset action. Neglecting the Ratio

Set control for the moment, consider that the valve in line "B" is closed, and a slurry is pumped through "A". A pulse train would result immediately which would cause

the BDC to count away from zero thus causing an output from the D/A converter. The

result is that the controller now sees an error signal and acts upon it by opening the

valve in line "B". As liquid passes through "B", pulses are generated which tend to reduce the difference count in the BDC and thereby reduce the error signal to the controller. Eventually, the reset action will cause the valve to assume a position such

that both flows are equal. This, however, will be after the integrated difference in

the BDC has been made up by allowing more of "B" to pass in order to "catch up".

It can be seen that the control system will act upon the valve in the "B" line

so as to cause a net count difference in the BDC of zero. For balance, this means

that when the system is in operation, both inputs to the BDC must have the same frequency. If it is decided to blend in the ratio of two parts "A" to one part "B",

there would need to be a means to compensate the pulse trains to make it appear to the

BDC that the pulse rates were equal when the desired ratio existed.

To accomplish ratio control, another scaler is placed in the system and is designated Ratio Set. The Ratio Set control may be present to any value between 00.0% and

99.9%. As an example, assume the setting to be 10.0% which is interpreted to mean

123
that for every gallon of "A" passing through, 0.1 gallons of "B" will be added to the

stream. With a setting of 10.0%, there would be an output of 100 pulses for each 1000

input pulses. This in turn means that the command pulse rate to the BDC is 1/10 the

pulse rate of the "A" line meter. To reach balance then, the valve will assume a position such that the "B" line meter pulse rate is equal to the Ratio Set output, or 1/10

the rate of the "A"line. This means then that the flow rate of "B" is 10.0% that of

"A"

A blending system of the type described lends a greater measure of control to the

blending process than other methods due to the integrating action of the bi-directional

counter which generates the control action. Since this counter can resolve to a single

count, and generate an output signal for that single count, then it is possible to control to within one count representing quite a small volume. The precision in this appJication is due to the fact that the control action is taking place on volume

information and not rate.

5 . A Temperature Control System

In a tunnel kiln, the temperature gradient is controlled by zones of different

temperatures rather than by changing the temperature as a function of time. For the

tunnel kiln, the material is passed from zone to zone as a function of time while the

temperature of each zone is held constant.

This process will contain four zones of temperature control. Each zone will have

a gas valve or a saturable core reactor to control the amount of heat applied and a thermocouple to measure the temperature. This unit is illustrated in Figure 11. Here

is shown the four parts of the control system, the sensor or thermocouple, the signal

conditioner or transmitter, the amplifier or controller, and the actuator which may be

a valve position or signal level to a saturable core reactor.

The thermocouple is a simple temperature sensor. Its low cost and small size overcome some of its weaknesses such as: low signal level (a few millivolts), need

for a cold reference junction (ice bath is ideal but not practical), limited temperature range (depends upon the type of thermocouple used) , and the need to place the

thermocouple junction at the spot where the temperature is desired.

The signal conditioner or transmitter depends upon the type of sensor being used

and the type of system which follows. For the thermocouple example, the transmitter

must accept a millivolt input signal, add to this a cold reference junction millivolt

signal, and produce the needed output signal. For the case of an electronic analog controller, the output signal would be 1 - 5 milliamperes or some other common 1 to 5 ratio current levels. The output signal at this stage represents the span of the system. For example, if the input is 0 to 5 millivolts after the reference junction

representing 0 to 100 °C and the ouptut is 1 to 5 milliamperes, the span is set at 0 to 100° Centigrade.

The amplifier or controller is the heart and the brains of the system. An

amplifier by itself would be used in only the simplest of systems. The controller

contains an amplifier, but also contains a set point signal which is compared with the

signal from the transmitter. The signal from the comparison or error signal is amplified

and used to produce the signal from the controller. More control action can be obtained by integrating the error signal and/or by taking the derivative of the error signal and adding the result of each to produce the final output. This output for an electronic analog controller would be a 1 to 5 current ratio signal such as 1 to 5 ma.

The 1 to 5 milliampere signal could be used to control the firing angle of a saturable reactor directly, or it could be used to position a valve by possibly going

through an electric to pneumatic relay, before driving a pneumatic valve positioner

The same process control could be accomplished pneumatically.

124
6 . A Direct Digital Control Approach

The four loops of the tunnel kiln indicate the possibility of using a common control system which is time shared among the four loops. All direct digital control systems are designed in this manner in order to minimize the cost per loop of the digital

equipment. An example of this type of controller is shown in Figure 12. Figure 13 shows in block diagram form the operation of a direct digital controller

controlling the four temperature loops of Figure 9. Note that the thermocouple temperature sensors and the gas valve actuators are still used. In this case, the direct

digital controller takes the place of the transmitters and the analog controllers.

The DDC starts its cycle by selecting the proper input and output scanners. Closing the input scanner connects the thermocouple signal to the temperature reference com- pensator and to the amplifier. The gain of the amplifier sets the span of the system,

the same as the transmitter in the analog system. The next step is to set the desired

digital set point into the bi-directional counter. The D/A converter following the bidirectional counter changes the digital number to an analog voltage. This known analog

voltage is compared with the unknown voltage from the amplifier by the voltage comparator. The output of the voltage comparator indicates which voltage is larger and thus

the direction that the bi-directional counter must be stepped in order to make the known

analog voltage equal to the unknown voltage. The direction of stepping and the number

of steps are the important factors in this step. The direction represents the polarity

and the number of steps, the magnitude of the error in digital form. The number in the

bi-directional counter, after completion of a balance, is the digitized form of the temperature. This number is displayed in percent of span of the input millivolt signal.

While not directly readable in engineering units, this method is quite sufficient to

provide excellent control action.

The output scanner comes into play during the actual analog to digital conversion

by connecting the error signal to the proper output circuit. The error signal for this

output is converted to + and - voltages representing a positive error or a negative

error. Again, the numbers of pulses represent the magnitude of the error. The output

circuit integrates these error pulses and either increases or decreases the output current in order to reduce this error to zero. The integrator has a gain adjustment in

order to speed up or slow down the control action for each individual loop. The output

current is 1 to 5 milliamperes or other standard 1 to 5 ratio current values.

Finally, to complete one cycle of the DDC, the loop number is displayed and the

variable under control remains constant until another cycle. This is done by obtaining

the loop number from the loop scanner and the variable from the bi-directional counter.

The other three loops are repeats of this cycle, changing only the input and output scanner connections along with different set points. The simplest method of storing

a digital set point is to use 3 selector switches for each point. Each switch contains

a number in binary coded decimal form by having four decks wired with this code. Another method is to store the set point in twelve flip-flops. A big advantage of the

flip-flops is their ability to accept set points from a remote location or from a digital computer.

The direct digital controller has as one of its greatest advantages, the ability

to communicate with a digital computer. The A/D converter obtains the variable in digital form which is the form needed by the computer. The computer can either present

set points to the controller or can be made to present digital error information in the

form of digital pulses directly to the output circuits. This combination becomes a very

powerful control tool due to the fast speed and large memory capabilities of the digital

computer and the interface, as well as the control functions of the direct digital

controller.

Another feature of this approach is the ease with which written data may be obtained even without the use of a computer. Since the set points, variables, and control

loop designations are electrically available in a coded form, page copy of the data is

conveniently possible. This can be significant in the temperature control of a kiln,

125
especially where gradients are critical to performance and a permanent record of this

data would be helpful to the operator.

7 . A Computer Controlled System

While the computer has left an indelible mark in the data tabulation field, it

has not yet reached its peak in the process control area. Predictions of completely

automated plants by visionaries of the last decade are just now appearing, but if

present interest in such applications are any indication of their importance, the impact of computers on the field of industrial process and plant control could equal

that of the data processing revolution just now at its zenith.

The ceramic industry is using computer technology to control inventory of parts,

predict sales and types of product which should be stocked, and even to schedule production runs of products. Recently, however, continuous processes have gone over to

complete computer control. Figure 14 illustrates the appearance of a typical computer

control console for use in such systems. The first areas within the ceramics field

to automate in this respect have been the cement plants. Two processes constitute

the major portion of cement manufacture, these are the blending of raw materials, and

the kiln- firing of the mixture. Variations exist in the methods of blending materials,

however the kiln firing techniques seldom vary.

First, let us consider control methods used in the proportioning of materials.

In the case that raw materials are stored in separate hoppers, the control of blend

consists of controlling the feeder conveyors. Continuous belt weighing devices can present data to the computer to calculate feed rate, and store the integral in memory

in order to account for the total amount entering the process. Feed rates are automatically controlled to correspond to those rates necessary to maintain a mixture ra- tio of materials set by laboratory analysis of the process output. While this particular approach has the drawback of requiring a laboratory sample analysis to close

the loop, new on-line analyzers should allow continuous feedback on future systems.

A second method of proportioning makes use of the control of the stockpile con- veyor in order to build a stockpile with a constant mixture cross section which may

be crushed and homogenized at the same time. The fact that this proportioning method

is off-line allows a more accurate control since adjustment is possible by the addition of materials until the proper balance is reached. The primary drawback of this

method is that rapid changes in the product are difficult to accomplish. The scheduling capability of the computer in this case may be able to justify its use, however, it is doubtful that continuous on-line control with analyzer feedback would

benefit this method.

The kiln firing should next be considered. As mentioned before, this step in the

process seldom varies, however the computerized cement plant may use alternative methods to control the kiln. Two philosophies dominate this area of control technology.

Figure 15 illustrates in block diagram form these two methods. Direct computer control is illustrated in 15A which includes various inputs entering the analog to

digital converter and from thence into the computer. The digitized information con- sists of thermocouple signals, fuel flow rate information in the form of standard

current signals from transmitters, and perhaps contact closures indicative of various

states in the process. In direct computer control, the computer actually performs

rapid calculations on the data received from the inputs and relates these calculations

to the mathematical model of the kiln. Any adjustments called for as a result of the

analysis are generated as output signals in the form of contact closures, pulses to

stepper motor drive systems, or time duration signals all of which may be used to adjust valve positions or speed controls.

In this approach, plant shutdown could occur if the computer were to become inoperative. It would be virtually impossible for an operator to operate the control,

commands normally generated by the computer in an emergency situation. The best that

could be hoped for would be that all output commands would remain at the last command

which would allow continued operation until the computer could be brought back online. This of course presupposes that the plant can operate satisfactorily on fixed

output commands

126
The computer directed approach as outlined in 15B, shows that the control is performed in local loops under the direction of controllers and the controller set points

are commanded by the computer. This approach permits autonomous local loop control

during an emergency loss of computer direction in much the same way that present loop

control can be accomplished manually in an emergency condition. The fact that each

loop has its own control system means that computer time is saved since it is not

necessary to calculate the control equation. Typically, set point adjustment commands

are generated as computer output pulses which step a retransmitting slide-wire mechanism. The voltage output of the slidewire represents the desired set point which is

digitized and fed back into the computer for verification. Other all-electronic methods are available which eliminate the need for stepper motors and slidewires.

Figure 16 illustrates such a typical Output Control Module.

Strong arguments can be made in support of each approach, however, the specific

system problem to be solved will usually point to the most useful answer to the

problem. A few comments on computer control in general would be useful in grasping

the concept of this technology, particularly concerning the common areas of the two

philosophies discussed above. Some of the prime reasons for seeking computer control

of a plant are the possibility of optimizing plant output, performing data logging,

and performing alarm scanning.

Optimization is currently accomplished in two ways. First, and most obvious is

the purely mathematical approach in which the mathematical model of the plant is

entered, and a program generated which will optimize the control functions based upon

the analysis performed in the modeled system. This is undoubtedly the most accurate

method, however, as has been mentioned in other papers, the preparation of a mathematical model is the most formidable task in the solution of the ' optimization program.

The second approach, while empirical is more readily programmed. The basis of this

approach is the generation of small perturbations which in themselves do not disturb

the process significantly, yet result in a process response which will either increase

or decrease the plant output. Small set point changes are then made in a direction

to increase or optimize plant output. Local loops are randomly selected and adjusted

for optimum. While basically simple, this optimizing method has the inherent weakness

that the plant might assume maximum output on a combination of control set points

which would not represent the most economical input conditions.

Data logging is a natural by-product of computer control. Since all variables

and set points are digitized and usually stored in memory, they are all available to

a typewriter output. This enables the operator to quickly observe process parameters

and at the same time tapes can be prepared to furnish input data to the central tabulating department for accounting purposes. Logging is usually done on a programmed

basis, perhaps hourly, unless some emergency situation dictates a logging routine

more often.

Alarm scanning is similarly a by-product of computer control. This function nor- mally requires a significant amount of hardware, but with computer control, a simple

routine to compare the variable value to the limits stored in memory is all that is

necessary to accomplish this task. The change of limits can easily be accomplished

at the control console by merely selecting the loop and desired high or low limit,

then commanding an entry of the data. The result is that the memory location serving

that function assumes a new value, and the comparisons continue searching for deviations. At the time a condition is sensed in which a particular loop exceeded its

limits, an alarm is actuated and normally the loop number, time, and data will be

logged usually on a separate typewriter. Additionally, a priority interrupt input

to the computer will be actuated to cause the computer to quickly attend to this

particular situation. The priority interrupt will cause the computer to interrupt

its normal program at some convenient point, and switch into an emergency program

which will then examine the cause of alarm and initiate corrective action.

127
While large scale computer control of plants is now in its infancy, it nevertheless has received considerable attention from control users. The results to be seen within the next decade will most likely exceed the strides made in the last by an order of magnitude, and can be expected to greatly affect the lives of each of us.

References

1. "Handbook of Automation, Computation, and Control", Vols. 1, 2, and 3 Grabbe, Ramo, & Wooldridge Editors - Wiley, 1959

2. "Principles of Industrial Process Control", D. P. Eckman, March, 1945

3. "Standard Instrumentation Questions and Answers", Vols. 1 & 2. Steve Elonka

and Alonzo R. Parsons, McGraw-Hill, 1962

4. "Instrument for Measurement and Control", Werner G. Holzbock, Reinhold, 1962

5. "100 Process Control Systems", M. H. Aronson and F. D. Marton - Instruments

Publishing Company, 1962

6. "Advanced Instrumentation and the Control Computer", H. H. Flum and D. J. Fraade, Thompson Ramo Wooldridge Inc. ISA Preprint Number 22.2.63

7. "Instrumentation for Digital Computer Control of a Cement Plant",

Dr. D. D. Bedworth and J. R. Faillace, General Electric Industry Control

Department, ISA Preprint Number 22.4.63.

123
Container

Preset Counter

Figure 1. Parts Packaging System

Figure 2. Preset Counter

129
Digital

Clock

Heater

Computer

Output

Typewriter

Test Oven Oscillator &

Commutator

Frequency

Counter

Figure 3. Automatic Testing System

Figure 4. Temperature Transducer/Encoder

130
BOURDON

TUBE

O

PRESSURE

INPUT

DIGITAL

OUTPUT

PHOTO

CELLS

LIGHT

SOURCE

CODE

WHEEL

Figure 5. Transducer/Encoder Mechanism

Figure 6. Computer

131
Uncontrolled Stream "A"

Controlled Stream "B"

Voltage To Recorder Frequency

Converter

Magnetic

Flow Meter

Turbine

Flow Meter

Ratio

Setter

Control Valve

Meter Bi- Digital To

Scaler DirectionaJ

* Analog

Counter Converter

Figure 7. Blending

Controller

INSULATING PIPE LINER ELECTRICAL TERMINAL BLOCK CALIBRATION COMPONENTS

METER PIPE SECTION

LAMINATED IRON CORE METER

ELECT RODE

FLOW SIGNAL CABLE

Figure 8. Magnetic Flow Meter (Courtesy Fischer & Porter Company)

132

Transmitters

Analog

Controllers

134
Thermocouples

Figure 13. Temperature Control System with Digital Control

Figure 14. Computer/Controller Console

135
Process

Sensors

Set Point A

Loop A

Controller

1

Process J

Set Point B Loop B

Controller

Process

Analog Control

to Computer

Digital Converter

Process

Sensors

15A - Direct Computer Control

15B - Computer Directed Control

Analog

to Digital Converter

Control

Computer

Process Controls

Process

Set Points

Figure 15. Computer Control System

Figure 16. Typical Output Control Module

136 -4 U. S. GOVERNMENT PRINTING OFFICE : 1965 O - 765-415



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