Table Of ContentAn Introduction to
Engineering Systems
Samuel Seely
University of Connecticut
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Thomas F. Irvine, Jr.
State University of New York at Stony Brook
James P. Hartnett
University of Illinois at Chicago Circle
EDITORS
William F. Hughes
Carnegie-Mellon University
Arthur T. Murphy
PMC Colleges
William H. Davenport
Harvey Mudd College
Daniel Rosenthal
University of California, Los Angeles
SECTIONS
Continuous Media Section
Engineering Design Section
Engineering Systems Section
Humanities and Social Sciences Section
Information Dynamics Section
Materials Engineering Section
Engineering Laboratory Section
Preface
Most existing introductory systems engineering texts are concerned with
the "hand solution" of systems response problems. With the advent of the
digital computer with its great power and versatility, it is no longer
necessary to restrict the classes of problems that may be studied, and
many of the constraints imposed by the hand-solution requirement can be
relaxed. It is desirable, therefore, that the methods of analysis appropriate
to hand solution be supplemented by numerical methods that are better
adapted to the computer.
This book is designed to provide the student with an introduction to
many important aspects of systems analysis. It seeks to provide a suffi
cient background to both the classical methods, which, at least in simple
cases, are appropriate to hand-solution and the matrix approach that is
more appropriate to computer solutions. Because numerical methods are
essential to an understanding of computer oriented discussions, and be
cause such methods can accomodate nonlinear as well as linear systems,
the general scope of the study has been broadened accordingly. Thus the
Kirchhoffian and the state variable formulations are considered in about
equal detail and emphasis, and numerical as well as analytic methods are
essential parts of the study. Furthermore, owing to their importance in
system studies, discrete time systems, and continuous time systems under
sampled conditions, also receive attention. The book will provide the
student with an understanding of the presently important methods of
systems study, and it likewise seeks to show the correlation that often
exists in alternate formulations.
The book is divided into four parts of unequal length; Part I, Models
vii
viii Preface
and Modeling; Part II, Interconnected Systems; Part III, Systems
Response a. Time Domain, b. Frequency Domain; Part IV, Special
Topics. Chapter 1 is an introduction to models and modeling. It contains
a discussion of the means for describing the elements which constitute the
systems under study.
Chapters 2,3,4 contain the methods for describing the dynamics of the
interconnected system. Chapter 2 details the means for describing the
system dynamics by a Kirchhoff description relating the input-output
variables. It also discusses in considerable detail an equilibrium descrip
tion that involves input, output and state. Both the Kirchhoff and the state
formulations receive general treatment. Also considered are discrete time
systems, since these are in a form particularly appropriate for digital
computer study. Chapter 3 introduces the signal flow graph, a graphical-
analytical representation that not only has interest in itself as a tool in
analysis, but also serves as a convenient step in the preparation of analog
computer programs. It is also found to be of considerable aid in relating
the Kirchhoff to the state formulation. Chapter 4 is a somewhat more
formal approach to equilibrium equation formulation than in Chapter 2.
It introduces the idea that an interconnected group of elements is conven
iently studied by a two step process involving the topological description
of the interconnection and the algebraic description of the elements in the
interconnection.
Chapters 5 and 6 are concerned with system response and behavior in
the time domain. Chapter 5 contains a discussion of classical methods of
finding the response of simple circuits to different excitation functions.
This chapter introduces certain important systems concepts, but it also
serves to show the rather poor match to numerical problem solving,
except in the simplest cases. Also included is consideration of a number
of important features of numerical methods of solution, plus a discussion
of the important aspects of machine methods in the solution of the
dynamic equations of interconnected systems. Chapter 6. extends this
study to systems which have been described in state form, with particular
emphasis on the digital computer considerations, and the time-constant
barrier and sampling time problem.
Chapters 7 through 11 contain a discussion of the analysis of systems
response in the frequency domain. The material in these chapters is
largely limited to linear systems, since transformation techniques are
essential to most of these studies. Chapter 7 introduces the essential
features of the Laplace transform, and a general introduction to problem
solving. For linear systems, these techniques provide a better match
Preface ix
between problem and problem solver, since the transformation allows
many of the steps in the solution to be accomplished algebraically. Chap
ter 8 is a detailed study of features of systems in the transformed domain.
Chapter 9 examines the response of systems in the steady-state to sinusoi
dal excitation functions, a form that is of particular importance in many
areas of endeavor. Chapter 10 considers many special theorems and
forms that are useful in problem analysis and solution. Chapter 11 extends
considerations to the periodic nonsinusoidal function, and shows the
place of the Fourier series in systems studies. These considerations are
extended to the case of nonperiodic excitations, with an introduction of
the Fourier integral and the features of the Fourier transform. Attention
is given to the numerical solution of Fourier series components, and also
to the features of the Fast Fourier transform.
Chapters 12 and 13 are interested in certain time-domain problems.
Chapter 12 directs attention to the sampled data problem and to the use
of the Z-transform in sampled data systems. The relation between the
difference equation and the sampled data Z-transform techniques are
examined. Chapter 13 considers the question of system stability from a
number of viewpoints, a matter of particular interest in feedback system
studies.
There are many important topics that are not discussed, including time
varying systems, and systems with stochastic inputs. It is expected that
the content will provide the student with a background in modern systems
analysis on which he can build further in his subsequent studies. Parts of
the content are rather sophisticated, but have been included for the
sake of completeness. An instructor can eliminate that material which he
considers to be outside of the objectives of his course.
This book is a direct outgrowth of discussions with Professor E. Folke
Bolinder of Chalmers University of Technology, Gothenburg, Sweden,
concerning an introductory course in systems analysis. These discussions
have set the technical level and content of the book. Acknowledgement is
also made of ideas that come from the reports published under the Discrete
System Concepts Project funded by the National Science Foundation
under Grant GE-2546. Particular note is made of Monograph No. 6,
"Discrete System Techniques at the Sophomore Level," by John G.
Truxal.
University of Connecticut SAMUEL SEELY
1
Modeling of System Elements
In undertaking a study of the behavior of an engineering system, or when
undertaking the design of a new system, it is usually necessary to employ
a suitable manipulable representation of the system, often referred to as a
model of the system. This chapter will be concerned with the modeling
process.
1-1 INTRODUCTION
System models are of three basic types —physical, analytical, and
descriptive. The traditional and perhaps most familar type of model is a
physical representation of the system. Such models have the advantage
that experiments can be conducted on them; they have the disadvantage
that they may be very costly. Analytical models are mathematical expres
sions of the characteristics of the system, usually in the form of sets of
simultaneous equations. Descriptive models, and these may bridge the
entire scale of abstraction, may be simple word pictures of the processes
that make up a system. The description will cover the interactions among
the variables, the attributes of the variables, and the constraints which
limit the performance. Such models can be prepared in a minimum of
time, are relatively low in cost, and their development requires few special
skills. Descriptive models can be applied to solve a wide variety of prob
lem situations requiring decisions. Models of business systems, for
example, are usually most easily expressed in descriptive terms.
The need for a wide variety of model types is evident when one con
siders that the description of an electronic circuit that carries out a
3
4 Modeling of System Elements
specified function, and which is made up of many resistors, capacitors,
inductors, transistors, diodes, etc., would be undertaken in a manner that
is quite different from that of a chemical processing plant, with its oxida
tion reactors, heat exchangers, distillation columns, and compressors.
Both of these would have rather different descriptions from a highway
network with its exits, access points, interchanges, and the random intro
duction and removal of vehicles, or from a business concern that manu
factures rubber products. But in spite of their differences, to undertake a
study of these or almost any other system requires that some convenient
model be developed. Further, it may not be possible to represent fully all
quantitative aspects of the physical system. Clearly, therefore, the
validity of a study can certainly be no more accurate than the quality
of the model that is used.
As noted, a system study is usually done by developing an equivalent
model of the actual system and then studying the model. This is especially
true in carrying out a design, since no actual system exists, but is ulti
mately to be developed. The system is often represented by the intercon
nection of a collection of appropriate components, each of which is
individually modeled. For carrying out mathematical studies, a mathe
matical model is usually necessary; for physical studies, physical models
would be used. Thus, for example, to ascertain the performance character
istics of a chemical plant, a pilot plant is built as a physical model for test
and study. With the advent of the digital computer, it is quite common to
model the chemical process by mathematical relationships that fully
describe the interconnected components of the system, and then ascertain
the performance characteristics from computer studies.
In carrying out the modeling, the procedure may be rigorously analytical
and may be accomplished from the basic laws that describe the compon
ents. This is often possible for some physical devices, and much of our
subsequent studies will be of systems composed of such elements. Often
the components are so involved that the models are approximations that
involve empirical data correlations.
We shall be concerned in this chapter with a study of the mathematical
modeling process of the elements which may comprise the components of
an extensive physical system. We shall discuss a variety of elements, but
the systems that we shall study will be composed largely of electrical or
mechanical elements.
Model Characteristics 5
1-2 MODEL CHARACTERISTICS
Our study will be concerned only with lumped elements. A lumped
element is one that can be isolated and treated as one that possesses the
features of idealized elements. Actually in many cases the real elements
are not lumped. Clearly, it might appear quite unreasonable to consider
that a long section of pipe or a long electrical transmission line could be
considered as being lumped, yet in many cases this is a valid approxima
tion. Also, clearly discernable springs exist, and if these are not too
large, it seems reasonable that they would well satisfy any reasonable
criterion of lumpiness. However, a solid rod also possesses compliance
or spring qualities, and such a rod under loading does stretch, and with the
removal of the load, the rod will return to its original length, provided
that it had not been stressed beyond its elastic limit. An easily stated
criterion for lumpiness does not appear possible. Thus one must be care
ful that in any critical case that the assumption of lumpiness, if it is made,
is indeed valid.
Often it is assumed that the elements are linear. This assumes that there
is a linear relation between cause and effect, or in engineering terms,
between excitation and response. A casual inspection will not indicate
linearity —both lumped or distributed elements can be linear. We shall
find that systems analysis can often be carried out completely in closed
mathematical form for systems composed of linear elements. This is
rarely possible for nonlinear systems, and nonanalytic methods (graphical
or numerical) must be employed in studying such systems. In an effort
to deduce a closed-form solution, one often may assume linearity for a
nonlinear system. Extreme care is necessary in such cases, since the
assumption of linearity may completely negate the important features of
the system. The use of a digital computer in analysis often makes un
necessary such linear approximations, since the computer can handle
nonlinear systems with about the same ease as a linearized description.
Most elements that are considered linear are, in fact, nonlinear.
However, there is usually a reasonable range over which their response-
excitation characteristics are linear, to a reasonable engineering approxi
mation. Where the approximation is not valid, then the behavior of the
real system will probably not agree with that predicted for the modeled
system. The discussion is clarified somewhat by reference to Fig. 1-1
which shows graphically a response-excitation relationship. If the excita
tion were limited to the small range a-b (this is often referred to as the
small-signal or incremental operation), then the assumption of linearity of
6 Modeling of System Elements
Response
0 . Excitation
Fig. 1-1. A representative nonlinear response-excitation relationship.
response over the associated range a'-b' is reasonably valid. If the excita
tion were to extend over the range O-b, then, of course, the assumption
of a linear response would not be valid.
There is an implication in the foregoing that must be discussed. It is
that upon reversing the sense of the excitation that the response-excitation
relation will remain unchanged. Elements for which this is true are known
as bilateral elements. There is a wide range of elements which are not
bilateral. Figure 1-2 illustrates a nonbilateral characteristic. In the extreme
Response
i 1 y^00^
Excitati<
Fig. 1-2. A nonbilateral response-excitation characteristic.
