Mathematical Theory of Systems Engineering

365 Pages·1967·20.586 MB·English

by  A.Wayne Wymore

A Mathematical Theory
of Systems Engineering
—The Elements
A. Wayne Wymore
The University of Arizona
John Wiley and Sons, Inc. new york . London . Sydney
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Copyright © 1967 by John Wiley & Sons, Inc. All Rights
Reserved. This book or any part thereof must not be reproduced
in any form without the written permission of the publisher.
Library of Congress Catalog Card Number: 67-22416
Printed in the United States of America
Preface
Every author has several motivations for writing, and authors of technical
books always have, as one motivation, the personal need to understand;
that is, they write because they want to learn, or to understand a phenom¬
enon, or to think through a set of ideas. In my case motivation is abetted
by fascination with a set of ideas, each one of which I regard now as being
a forerunner of a general theory of systems underlying systems engineering.
This set of ideas includes information theory (Shannon 1949); the statis¬
tical theory of communication (Wiener 1949, Lee); some of the ideas to
be found in statistical mechanics (Gibbs) and topological dynamics
(Birkhoff); the engineering work on control mechanisms (Chestnut 1961,
Graham) and optimal processes (Chang, Pontryagin); the theory of
finite state machines (Gill) and more general automata (Ginsburg, Shannon
1956); some of the work, in particular, on computability and Turing
machines (Turing, Davis); the mathematical models of operations
research (Churchman); the digital computer technology including both
hardware (Ware) and software (Flores) aspects; analog computer tech¬
nology (Korn); decision theory (Fishburn); even reliability (Roberts)
and quality control (Grant); and, most significantly, some of the work
labeled cybernetics (Wiener 1948, Ashby). It seems that each of these
ideas represents an aspect of a general theory of systems; each has a
bearing on general systems theory; each is individually useful in systems
engineering. What has been lacking is a general theoretical framework
within which all these ideas, in concert, might be brought to bear on the
engineering problems of design and analysis, on theoretical considerations
of existence, uniqueness, characterization, and limitations of solutions to
problems in systems engineering. The objective here is to provide such a
framework in order that a place may be found for each of the theories
mentioned above within this framework.
This book, in particular, contributes to two principal lines of thought:
(a) toward defining systems engineering and providing mathematical tools
for the practice of systems engineering, and (b) toward the development
v
VI PREFACE
of a general theory of systems. The first line of thought is represented by the
books of Goode and Machol, Hall, Chestnut (1965), and Wilson; the
second line of thought is represented by the writings of Bartalanffy, Ashby,
Kalman, and Mesarovic. Two books that have especially contributed to
the writing of this book, at least in spirit, are Ellis and Ludwig and Zadeh
and Desoer.
Specifically, then, a mathematical framework is here provided in which
theoretical problems of systems engineering might at least be precisely
formulated and solutions methodically sought with some hope of rigor,
consistency, and mathematical proof: assertions about system theory (if
not systems engineering) should be proved mathematically or disproved
by formal counterexample.
Through working within such a mathematical framework it will be
possible, eventually, to obtain insight into the problem of ordering and
classification of systems, into the nature of complexity and organization,
into the necessary mathematical properties of simulation, into system
characteristics such as “intelligence” (artificial or otherwise), “adaptive,”
and “goal seeking,” and into the processes of system design and
analysis.
It is because I have included models of the processes of system design
and analysis that the preposition “of” appears in the title of the book
rather than “for” and qualifies the work to be titled A Mathematical Theory
of Systems Engineering rather than just A Mathematical Theory of Systems.
It is as important to understand the processes of systems engineering as it
is to understand systems.
I want particularly (eventually) to provide a scientific basis for the design
of large-scale, complex, man/machine systems. I believe the development
of a general theory of systems and of systems engineering is the first step
toward these goals.
The problem of teaching systems engineering has provided further
motivation for the research the results of which are described in this book.
Students who are to be trained in systems engineering cannot stay in
school long enough to study in detail each of the theories listed in the first
paragraph of this preface and to study all the supporting disciplines clearly
necessary to a concept of systems engineering as general as that to be
enunciated in Chapter 1. There is a question of pedagogical efficiency: if
all the theories pertinent to systems engineering could be discussed within
a common framework by means of a standard set of nomenclature and
definitions, many separate courses might not be necessary; the total body
of required knowledge might be packaged in a reasonable, manageable
curriculum. Many hours of discussion with the faculty of the Systems
Engineering Department at the University of Arizona with respect to the
PREFACE Vll
problem of curriculum design convinced me that such a common frame¬
work for the discussion of system theory is the only solution to the
problem.
Aside from such personal and local motivations for the writing of this
book, there are groups to whom this book is addressed. In addition to
mathematicians and engineers, there will be many operations researchers,
management scientists, economists, computer scientists, information
scientists, physicists, physicians, sociologists, psychologists, biologists,
and ecologists (to name a few), who will profit from and enjoy this book:
The systems I discuss and model are not necessarily specialized to the
engineering type. I have done nothing, consciously, to exclude biological
and business systems (for examples) from the discussion, and it is to be
hoped that what I have done here will eventually be applied to the study
of such systems.
The discussion in this book, however, is primarily addressed to two
groups: mathematicians and engineers; mathematicians who may be at
least a little bit applied and engineers who aspire to deal with systems. I
want to interest mathematicians in general system theory as mathematics,
for there are some very subtle and very deep mathematical problems
involved in general system theory; the mathematical public must accept
general system theory as mathematics before these problems will be solved.
I want to interest engineers in general system theory as a part of the scientific
basis for systems engineering.
Reflecting somewhat the dual composition of the invited audience, the
exposition is distinctly in two separate parts: the formal mathematical
theory and the informal discussions, interpretations, motivations, and
explanations. The parts of the formal mathematical theory are labeled, as
usual, “Definition,” “Theorem,” “Lemma,” “Corollary,” “Scholium,”
“Proof.” The parts of the exposition that are not so rigorous have been
labeled “Discussion;” it is under this label that the mathematical state¬
ments are interpreted in terms of “reality,” in terms of the practical,
applied problems of system design and analysis; it is under the Discussion
label that explanations and motivation are provided. Every part of the
exposition will have one or the other of these labels: one of the labels from
formal theory or the informal label.
Exercises are also of two kinds: formal and informal. The formal
exercises are unmistakable; they invariably begin with the imperative,
“Prove or find a counterexample.” Following this is a mathematical
proposition stated in terms of the formal mathematical theory. The
informal exercises begin, “Discuss . . . ,” and following is some sort of
question about hardware, economics, human factors, or other aspects I
have chosen not (or have been unable) to formalize at that point.
Vlll PREFACE
The book is completely self-contained from the mathematical point of
view, at least as far as the formal development is concerned. Some
elementary mathematical constructs not developed in the book are used
in the discussion of examples but all the mathematics used in the formal
development is derived completely.
It is noteworthy—because it is the exception in these times—that it is
unnecessary for me to acknowledge direct Federal support in any form of
the writing of this book. In spite of this it is difficult to make my acknowl¬
edgments inclusive enough. My indebtedness to a rich heritage of
technical work is indicated by the bibliographical references. My indebted¬
ness to secretaries, Mrs. Felker and Mrs. Endebrock, who bravely typed
many, many versions of this book—almost all of which were neonatal
fatalities—cannot be repaid in any coin commensurate with the value of
the service rendered. My indebtedness to my wife Muriel; even to my
children Farrell, Darcy, Melanie, and Leslie; to my colleagues Duckstein,
Perry, Sanders, Titt, Tucker, Weldon; and, of course to outstanding
teachers Vinograde, Eberlein, Hammer, is without limit. Help and en¬
couragement are hereby acknowledged and thanks sincerely given. Special
recognition in this respect is owed to Dr. T. L. Martin under whom I was
working when this whole thing was begun.
I have spent a great deal of time, however, as an administrator at the
departmental and research activity level at the University of Arizona, and
I am thereby qualified to assert that only the academic administrator can
appreciate the damage to research activity caused by interruptions for
routine administrative matters. And it is only the father of a growing
family who can appreciate the amount of time consumed by such a family
at the expense of his book-writing activity. Nonetheless, it is to these:
my family, my friends, my colleagues, my staff, my superiors, in spite of all
of whom this book got itself written, this book is affectionately dedicated.
A. Wayne Wymore
Tucson, Arizona
April 1967
Contents
1 Introduction 1
Set Theoretic Concepts, Definition 1.1 5
Function Theoretic Concepts, Definition 1.2 10
Notation for Sets of Real Numbers, Definition 1.3 13
Cartesian Products, Definition 1.4 17
Exercises 18
2 Systems Definitions 21
Admissible Sets of Input Functions, Definition 2.1 24
System Theoretic Notions, Definition 2.2 30
The Assemblage, Definition 2.3 40
The Relation between a System and an Assemblage, Corollary
2.1 40
Alternate Statement of System Consistency Requirements,
Theorem 2.1 42
Existence of Discrete Systems, Theorem 2.2 49
System Models for Complete Sequential Machines, Theorem
2.3 52
Semigroups, Definition 2.4 53
Turing Machines, Definition 2.5 55
System Models for Turing Machines, Theorem 2.4 56
Exercises 62
3 Modeling of Systems 68
A System Model of an Adder 72
A System Model of a Multiplier 73
A System Model of a Scalor 73
A System Model of an Integrator 74
A System Model of the Initial Condition Mode 75
IX
X CONTENTS
A System Model of a Differentiator 78
A System Model of a Stieltjes Integrator 78
A System Model of a Retail Sales Operation 80
A System Model of an Information Retrieval System 81
A System Model of a Catch Basin 83
Existence of Memory of Length d, Theorem 3.1 89
Memory of Length d, Definition 3.1 91
A System Model of an Open Pit Mine 91
A System Model of a Human Operator Tracking 104
A System Model of a Human Organization 117
Exercises 118
4 Comparison of Systems 121
Structural Comparison, Definition 4.1 124
System Properties of Homomorphs, Theorem 4.1 124
Relation between Sets of Transition Functions under Homo¬
morphism, Theorem 4.2 126
System Equivalence, Definition 4.2 133
Relation between Homomorph and Alternative, Theorem 4.3 134
Orderings, Definition 4.3 137
Order on a Set of Systems Generated by Homomorphism,
Theorem 4.4 139
State Equivalence, Definition 4.4 140
Reduction of S to a Distinguished Set of States, Theorem 4.5 141
Input Equivalence, Definition 4.5 149
Input Equivalence If Z Is State-Distinguished, Corollary 4.1 149
Input Equivalence If T = R and Z Is State-Distinguished,
Theorem 4.6 150
Input Bases, Definition 4.6 152
Step Functions, Corollary 4.2 153
Operational Criterion for a Reliable Input Base, Theorem 4.7 157
An F without a Reliable Input Base, Corollary 4.3 159
Zorn’s Lemma, a Postulate 162
The Axiom of Choice, Lemma 4.1 163
Existence of Input Bases, Theorem 4.8 164
Reduction of F to an Input-Distinguished Set, Theorem 4.9 166
General Reduction Theorem, Corollary 4.4 170
Reduction of F through P for Discrete Systems, Theorem 4.10 171
Technical System Specifications, Definition 4.7 176
Existence of a System Satisfying Consistent Technical System
Specifications, Theorem 4.11 179
CONTENTS Xi
Existence of a System Better Satisfying Explicit, Consistent
Technical System Specifications, Theorem 4.12 181
Specification Satisfaction Is Implied by Homomorphism,
Theorem 4.13 184
Exercises 185
5 Coupling of Systems 194
Admissible Sets of Input Functions Generated by Projections,
Lemma 5.1 201
Feedback Couples, Definition 5.1 202
A Condition Equivalent to Time Determinedness, Lemma 5.2 203
The Resultant of a Feedback Couple Is Well Defined, Lemma
5.3 206
Representation of Any Time-Determined System As a Result¬
ant, Corollary 5.1 207
Cartesian Products of Admissible Sets of Input Functions,
Lemma 5.4 208
^-System Couples, Definition 5.2 211
The Resultant of an ^-System Couple Is Well Defined,
Lemma 5.5 214
The Resultant of an ^-System Couple Is a System, Theorem 5.1 221
Synthesis of Two-System Couples from Cascade and Feedback
Couples, Theorem 5.2 226
Infinite System Couples, Definition 5.3 231
Exercises 234
6 Subsystems and Components 237
Subsystems and Decomposition, Definition 6.1 238
Decomposition of a Disjunction, Corollary 6.1 240
Existence of Maximal Decompositions, Theorem 6.1 241
Components and Resolution, Definition 6.2 243
Conjunctive Resolutions, Corollary 6.2 254
Every Conjunctive Resolution Determines a Two-Couple
Conjunctive Resolution, Theorem 6.2 255
Structure of Z Due to a Two-Couple Conjunctive Resolution
of Z, Theorem 6.3 258
Existence of Parallel Resolutions, Theorem 6.4 262
Structure of Z Due to a Conjunctive Resolution of Z of
Arbitrary Size, Theorem 6.5 264
The Homomorphic Image of an Input Base, Theorem 6.6 271
xii CONTENTS
Construction of a Complementary Conjunctive Component
through Input Bases, Theorem 6.7 272
Dual Systems, Definition 6.3 277
The Dual of a System Is a System and Every System Is a Dual,
Corollary 6.3 278
Reachability and Determination, Corollary 6.4 280
Duality of AND and OR, Theorem 6.8 286
Exercises 291
7 Discrete Systems 293
Discreteness, Definition 7.1 295
Primitive Sample Data Theorem, Corollary 7.1 295
Canonical Discreteness Is General, Corollary 7.2 298
^-Couples among Discrete Systems Are ^-System Couples,
Theorem 7.1 307
Existence of a Computation for the Maximal Decomposition,
Theorem 7.2 324
Existence of Cascade Resolutions of a Discrete System,
Theorem 7.3 334
Exercises 339
Appendix: Table of Symbols 343
Bibliography 347
Index 351