signals & system-ans - Distance Education

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30 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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SIGNALS AND SYSTEMS



5 marks

(1)

Signal processing

Signal transmission using electronic signal processing.

Transducers

convert signals from other physical

waveforms

to
electrical

current

or

voltage
waveforms, which then ar
e processed, transmitted as

electromagnetic waves
, received
and converted by another transducer to final form.

Signal processing

is an area of

systems engineering
,

electrical engineering

and

applied
mathematics

that deals with operations on or analysis of

signals
, or measurements of ti
me
-
varying or
spatially
-
varying physical quantities. Signals of interest can include

sound
,

images
, and

sensor

data, for
example biological data such as

electrocardiograms
,

control system

signals,
telecommunication

transmission

signals, a
nd many others.

Digital signal processing is the processing of digitised discrete time sampled signals. Processing is done
by general
-
purpose

computers

or by digital circuits such as

ASICs
,

field
-
programma
ble gate arrays

or
specialized

digital signal processors

(DSP chips). Typical arithmetical operations include

fixed
-
point

and

floating
-
point
, real
-
valued and complex
-
valued, multiplication and addition. Other typical
operations support
ed by the hardware are

circular buffers

and

look
-
up tables
. Examples of algorithms are
the

Fast Fourier transform

(FFT),

finite impulse response

(FIR) filter,

Infinite impulse response

(IIR) filter,
and

adaptive filters

such as th
e

Wiener

and

Kalman filters
.


(2)

Systems engineering

Systems engineering techniques are used in com
plex projects: spacecraft design, computer chip design, robotics,
software integration, and bridge building. Systems engineering uses a host of tools that include modeling and
simulation, requirements analysis and scheduling to manage complexity.

Systems e
ngineering

is an

interdisciplinary

field of

engineering

focusing on how complex engineering
proj
ects should be designed and managed over their

life cycles
. Issues such as

logistics
, the co
ordination
of different teams, and automatic control of machinery become more difficult when dealing with large,
complex projects. Systems engineering deals with work
-
processes and tools to manage

risks

on such
projects, and it overlaps with both technical and human
-
centered disciplines such as

control
engineering
,

industrial engineering
,

organizational studies
, and

project management
.

may greatly differ from the sum of the parts' properties, motivated the

Department of Defense
,
NASA
, and
other industries to apply the he term

systems engineering

can be traced back to

Bell

Telephone
Laboratories

in the 1940s.
[1]

The need to identify and manipulate the properties of a system as a whole,
which in complex engineering projects discipline.
[2]

When it was no longer possible to rely on design evolution to improve upon a system and the existing
tools were not sufficient to meet growing demands, new methods began to be de
veloped that addressed
the complexity directly.
[3]

The continuing evolution of systems engineering comprises the development
and identification of new methods and modeling te
chniques. These methods aid in better comprehension
of engineering systems as they grow more complex. Popular tools that are often used in the systems
engineering context were developed during these times, including

USL
,

UML
,
QFD
, and

IDEF0
.


(3)

Scope of system engineering.


One way to understand the motivation behind systems engineering is to see it as a method, or practice, to
identify and i
mprove common rules that exist within a wide variety of systems.
[
citation needed
]

Keeping this in
mind, the principles of systems engineering


holism, em
ergent behavior, boundary, et al.


can be
applied to any system, complex or otherwise, provided

systems thinking

is employed at all
levels.
[19]

Besides defense and aerospace, many information and technology based companies, software
development firms, and industries in the field of electronics & communications require systems engineers
as par
t of their team.
[20]

An analysis by the INCOSE Systems Engineering center of excellence (SECOE) indicates that optimal
effort spent on systems engineering is about 15
-
20% of

the total project effort.
[21]

At the same time,
studies have shown that systems engineering essentially leads to reduction in costs among other
benefits.
[21]

However, no quantitative survey at a larger scale encompassing a wide variety of industries
has been conducted until recently. Such studies are underway to determine the e
ffectiveness and
quantify the benefits of systems engineering.
[22]
[23]

Systems engineering

encourages the use of modeling and simulation to validate assumptions or theories
on systems and the interactions within them.
[24]
[25]

Use of methods that allow early detection of possible failures, in

safety engineering
, are integrated into
the design proc
ess. At the same time, decisions made at the beginning of a project whose consequences
are not clearly understood can have enormous implications later in the life of a system, and it is the task
of the modern systems engineer to explore these issues and ma
ke critical decisions. There is no method
which guarantees that decisions made today will still be valid when a system goes into service years or
decades after it is first conceived but there are techniques to support the process of systems engineering.
Ex
amples include the use of soft systems methodology,

Jay Wright Forrester
's

System d
ynamics

method
and the

Unified Modeling Language

(UML), each of which are currently being explored, evaluated and
developed to support the engineering de
cision making process.

Education

Education in systems engineering is often seen as an extension to the regular engineering
courses,
[26]

reflecting the industry attitude that

engineering students need a foundational background in
one of the traditional engineering disciplines (e.g.

automotive engineering
,

mechanical
engineering
,

industrial engineering
,

computer engineering
,

electrical engineering
) plus practical, real
-
world experience in order to be effective as

systems engineers. Undergraduate university programs in
systems engineering are rare. Typically, systems engineering is offered at the graduate level in
combination with interdisciplinary study.

INCOSE

maintains a continuously updated Directory of Systems Engineering Academic Programs
worldwide.
[5]

As of 2009, there are about 80 institutions in United Sta
tes that offer 165 undergraduate and
graduate programs in systems engineering. Education in systems engineering can be taken as

Systems
-
centric

or

Domain
-
centric
.



Systems
-
centric

programs treat systems engineering as a separate discipline and most of the
c
ourses are taught focusing on systems engineering principles and practice.



Domain
-
centric

programs offer systems engineering as an option that can be exercised with
another major field in engineering.

Both of these patterns strive to educate the systems en
gineer who is able to oversee interdisciplinary
projects with the depth required of a core
-
engineer.
[27]


20 marks

(1)

Types of system engineering


Cognitive sy
stems engineering

Cognitive systems engineering (CSE) is a specific approach to the description and analysis of
human
-
machine systems or

sociotechnical systems
.
[39]

The three main themes of CSE are how
humans cope with complexity, how work is accomplished by the use of artifacts, and how human
-
machine systems and socio
-
technical sy
stems can be described as joint cognitive systems. CSE
has since its beginning become a recognised scientific discipline, sometimes also referred to
as

cognitive e
ngineering
. The concept of a Joint Cognitive System (JCS) has in particular
become widely used as a way of understanding how complex socio
-
technical systems can be
described with varying degrees of resolution. The more than 20 years of experience with CSE

has been described extensively.
[40]
[41]

Configuration Management

Like systems engineering
,

configuration management

as practiced in the

defense

and

aerospace
industry

is a broad systems
-
level practice. The field parallels the taskings of systems engineering;
where systems engineering deals with requirements development, allocat
ion to development
items and verification, Configuration Management deals with requirements capture, traceability to
the development item, and audit of development item to ensure that it has achieved the desired
functionality that systems engineering and/o
r Test and Verification Engineering have proven out
through objective testing.

Control engineering

Control engineering

and its design and implementation of

control systems
, used extensively in
nearly every industry, is a large sub
-
field of systems engineering. The cruise control on an
automobile and the guidance system for a ballistic mis
sile are two examples. Control systems
theory is an active field of applied mathematics involving the investigation of solution spaces and
the development of new methods for the analysis of the control process.

Industrial engineering

Industrial engineering

is a branch of

engineering

that concerns the development, improvement,
implementation

and evaluation of

integrated systems

of people, money, knowledge, information,
equipment, energy, material and

process. Industrial engineering draws upon the principles and
methods of engineering analysis and synthesis, as well as mathematical, physical and social
sciences together with the principles and methods of engineering analysis and design to specify,
pred
ict and evaluate the results to be obtained from such systems.

Interface design

Interface design

and its specification are concerned with assuring that the pieces of a syst
em
connect and inter
-
operate with other parts of the system and with external systems as necessary.
Interface design also includes assuring that system interfaces be able to accept new features,
including mechanical, electrical and logical interfaces, incl
uding reserved wires, plug
-
space,
command codes and bits in communication protocols. This is known as

extensibility
.

Human
-
Computer Interaction

(HCI) or Human
-
Machine Interface (HMI) is another aspect of interface
design, and is a critical aspect of modern systems engineering. Systems engineering principles
are applied in the

design of network protocols

for

local
-
area networ
ks

and

wide
-
area networks
.

Mechatronic engineering

Mechatronic enginee
ring
, like Systems engineering, is a multidisciplinary field of engineering that
uses dynamical systems modeling to express tangible constructs. In that regard it is almost
indistinguishable from Systems Engineering, but what sets it apart is the focus on

smaller details
rather than larger generalizations and relationships. As such, both fields are distinguished by the
scope of their projects rather than the methodology of their practice.

Operations research

Operations research

supports systems engineering. The tools of operations research are used in
systems analysis, decision making, and trade studies. Several

schools teach SE courses
within
the

operations research

or

industrial engineering

department
[
citation needed
]
, highlighting the role
systems eng
ineering plays in complex projects.

Operations research
, briefly, is concerned with
the optimization of a process under multiple constraints.
[42]

Performance engineering

Performance engineering

is the discipline of ensuring a system will mee
t the customer's
expectations for performance throughout its life. Performance is usually defined as the speed with
which a certain operation is executed or the capability of executing a number of such operations
in a unit of time. Performance may be degra
ded when an operations queue to be executed is
throttled when the capacity is of the system is limited. For example, the performance of a

packet
-
switched netwo
rk

would be characterised by the end
-
to
-
end packet transit delay or the number of
packets switched within an hour. The design of high
-
performance systems makes use of
analytical or simulation modeling, whereas the delivery of high
-
performance implementati
on
involves thorough performance testing. Performance engineering relies heavily
on

statistics
,

queuein
g theory

and

probability theory

for its tools and processes.

Program management and project management.

Program management

(or programme management) has many similarities with systems
engineering, but has broader
-
based origins than the engineering ones of systems
engineering.

Project management

is also closely related to both program management and
systems engineering.

Proposal engineering

Proposal engineering is the application of scientific and mathematical principles to design,
construct, and op
erate a cost
-
effective proposal development system. Basically, proposal
engineering uses the "
systems engineering process
" to create a cost effective
proposal and
increase the odds of a successful proposal.

Reliability engineering

Reliability engineering

is the discipline of ensuring a system will meet the
customer's expectations
for reliability throughout its life; i.e. it will not fail more frequently than expected. Reliability
engineering applies to all aspects of the system. It is closely associated
with

maintainability
,

availability

and

logistics engineering
.

Reliability engineering is always a
critical component of safety engineering, as in

failure modes and effects analysis

(FMEA)
and

hazard fault tree

analysis, and of

security engineering
. Reliability engineering relies heavily
on

statistics
,

probability theory

and

reliability theory

for its tools and processes.

Safety engineering

The techniques of

safety engineering

may be applied by non
-
specialist engi
neers in designing
complex systems to minimize the probability of safety
-
critical failures. The "System Safety
Engineering" function helps to identify "safety hazards" in emerging designs, and may assist with
techniques to "mitigate" the effects of (potent
ially) hazardous conditions that cannot be designed
out of systems.

Security engineering

Security engineering

can be viewed as an

interdisciplinary

field that integrates the

community of
practice

for control systems design, reliability, saf
ety and systems engineering. It may involve
such sub
-
specialties as

authentication

of system users, system targets and others: people,
objects and processes.

Software engineeri
ng

From its beginnings,

software engineering

has helped shape modern systems engineering
practice. The techniques used in the handling of complexes of large softwar
e
-
intensive systems
has had a major effect on the shaping and reshaping of the tools, methods and processes of SE.


(2)

Systems theory

Systems theory

is the

interdisciplin
ary

study of

systems

in general, with the goal of elucidating principles that
can be applied to all types of systems at all nesting levels in all fields of research.

The term does not yet hav
e a
well
-
established, precise meaning, but systems theory can reasonably be considered a specialization
of

systems thinking
, a generalization of

systems science
, a systems approach. The term originates
from

Bertalanffy
's

general system theory

(GST) and is used in later efforts in other fields, such as the

action
theor
y

of

Talcott Parsons

and the system
-
theory of

Niklas Luhmann

In this context the word

systems

is used to refer specifically to

self
-
regulating systems
, i.e. that are self
-
correcting through

feedback
. Self
-
re
gulating systems are found in nature, including the physiological systems of
our body, in local and global ecosystems, and in climate

and in human learning processes.

Contemporary ideas from systems theory have grown with diversified areas, exemplified by
the work
of

Béla H. Bánáthy
, ecological systems with

Howard T. Odum
,

Eugene Odum

and

Fritjof
Capra
,

organizational theory

and

management

with individuals such as

Peter Senge
, interdisciplinary
study
with areas like
Human Resource Development

from the work of

Richar
d A. Swanson
, and insights
from educators such as

Debora Hammond

and

Alfonso Montuori
. As a transdisciplinary, interdisciplinary
and multiperspectival domain, the area brings together principles and concepts from

ontology
,

philosophy
of science
,
physics
,

computer science
,

biology
, and

engineering

as well as

geography
,

sociology
,

political
science
,

psychotherapy

(within

family systems therapy
) and
economics

among others. Systems theory
thus serves as a bridge for interdisciplinary dialogue between autonomous areas of study as well as
within the area of

systems sci
ence

itself.

In this respect, with the possibility of misinterpretations, von Bertalanffy
[1]

believed a general theory of
systems "should be an important regulative device in sci
ence," to guard against superficial analogies that
"are useless in science and harmful in their practical consequences." Others remain closer to the direct
systems concepts developed by the original theorists. For example,

Ilya Prigogine
, of

the Center for
Complex Quantum Systems

at the University of Texas, Austi
n, has studied

emergent properties
,
suggesting that they offer

analogues

for

living systems
. The theories of

autopoiesis

of

Francisco
Varela

and
Humberto Maturana

are a further development in this field. Important names in contemporary
systems science include

Russell Ackoff
,

Béla H. Bánáthy
,

Anthony Stafford Beer
,

Peter
Checkland
,
Robert L. Flood
,

Fritjof Capra
,

Michael C. Jackson
,

Edgar Morin

and

Werner Ulrich
, among
others.

With the modern foundations for a general theory of systems following the World Wars,

Ervin Lasz
lo
, in
the preface for Bertalanffy's book

Perspectives on General System Theory
, maintains that
the

translation

of "general system theory" from German into English has "wrought a ce
rtain amount of
havoc".
[2]

The preface explains that the original concept of a general system theory was "
Allgemeine
Systemtheorie

(or

Lehre
)", pointing out the fact t
hat "Theorie" (or "Lehre") just as "Wissenschaft"
(translated Scholarship), "has a much broader meaning in German than the closest English words ‘theory’
and ‘science'".
[2]

With these ideas referring to an organized body of knowledge and "any systematically
presented set of concepts, whether they are

empirical
,

axiomatic
, or
philosophical
, "Lehre" is associated
with theory and science in the etymology of general systems, but also does not translate from the
German very well; "teaching" is the "closest equivalent", but "sounds dogmatic and off the mark".
[2]

While
many of the root meanings for the idea of a "general systems

theory" might have been lost in the
translation and many
[
who?
]

were led to believe that the systems theorists had articulated nothing but
a

pseudoscience
, systems theory became a

nomenclature

that early investigators used to describe
the

interdependence

of relationships in
organization

by defining a new way of thinking about science
and

scientific paradigms
.

A system from this frame of reference is composed of regularly interacting or interrelating groups of
activities. For example, in noting the influence in organizational psycho
logy as the field evolved from "an
individually oriented

industrial psychology

to a systems and developmentally oriented

organizational
psychology
," it was recognized that organizations are complex social systems; reducing the parts from
the whole reduces the overall effectiveness of organizations.
[3]

This is different from conventional models
that center on individuals, structures, departments and units separate in part from the whole instead of
recognizing the interdependence between grou
ps of individuals, structures and processes that enable an
organization to function. Laszlo
[4]

explains that the new systems view of organized complexity went "one
step beyond the

Newtonian view of organized simplicity" in reducing the parts from the whole, or in
understanding the whole without relation to the parts. The relationship between organizations and
their

environments

became recognized as the foremost source of complexity and interdependence. In
most cases the whole has properties that cannot be known from analysis of the constituent elements in
isolation.

Béla H. Bánáthy
, who argued

along with the founders of the systems society

that "the benefit
of humankind" is the purpose of science, has made significant and far
-
reaching contributio
ns to the area
of systems theory. For the Primer Group at ISSS, Bánáthy defines a perspective that iterates this view:

The systems view is a world
-
view that is based on the discipline of SYSTEM INQUIRY. Central to
systems inquiry is the concept of SYSTEM.
In the most general sense, system means a configuration of
parts connected and joined together by a web of relationships. The Primer group defines system as a
family of relationships among the members acting as a whole. Von Bertalanffy defined system as
"e
lements in standing relationship".


[5]

Similar ideas are found in learning theories that developed from the same fundamental concepts,
emphasizing how understanding results from
knowing concepts both in part and as a whole. In fact,
Bertalanffy’s organismic psychology paralleled the learning theory of

Jean Piaget
.
[6]

Interdisciplinary
perspectives are critical in breaking away from

industrial age

models and thinking where history is history
and math i
s math, the arts and sciences

specialized

and separate, and where teaching is treated
as

behaviorist

conditioning.
[7]

The influential contemporary work of

Peter Senge
[8]

provides detailed
discussion of the commonplace critique of educational systems grounded in conventional assumptions
about learning, including the problems with fragmented knowledge and lack of holi
stic learning from the
"machine
-
age thinking" that became a "model of school separated from daily life." It is in this way that
systems theorists attempted to provide alternatives and an evolved ideation from orthodox theories with
individuals such as

Max Weber
,

Émile Durkheim

in sociology and

Frederick Winslow Taylor

in

scientific
management
, which were grounded in classical assumptions.
[9]

The theorists sought holistic methods by
developing systems concepts that could be integrated with different areas.

The contradiction of

reductionism

in conventional theory (which has as its subject a single part) is simply
an example of changing assumptions. The emphasis with systems theory shifts from parts to the
organization of parts, recognizing interactions of the parts are
not "static" and constant but "dynamic"
processes. Conventional

closed systems

were questioned with the development of

open
systems

perspectives. The shift was from

absolute

and universal authoritative principles and knowledge
to

relative

and

general

conceptual

and

perceptual

knowledge,
[10]

still in the tradition of theorists that
sought to provide means in organizing human life. Meanin
g, the

history of ideas

that preceded were
rethought not lost. Mechanistic thinking was particularly critiqued, especially the industrial
-
age
mechanistic

metaphor

of the mind from

interpretations

of

Newtonian
mechanics

by

Enlightenment

philosophers and later psychologists that laid the foundations of modern
organizational theory and management by the

late 19th century.
[11]

Classical science had not been
overthrown, but questions arose over core assumptions that historically influenced organized systems,
within both social an
d technical sciences.
[
citation needed
]

(3)

Developments

General systems research and systems inquiry

Many early systems theorists aimed at finding a gener
al systems theory that could explain all systems in
all fields of science. The term goes back to Bertalanffy's book titled "
General System theory: Foundations,
Development, Applications
" from 1968.
[6]

According to Von Bertalanffy, he developed the "allgemeine
Systemlehre" (general systems teachings) first via lectures beginning in 1937 and then via publications
beginning in 1946.
[21]

Von Bertalanffy's objective was to bring together under one heading the organismic science that he had
observed in his work as a biologist. His desire was to use the word "system" to describe those principles
which are com
mon to systems in general. In GST, he writes:

...there exist models, principles, and laws that apply to generalized systems or their subclasses,
irrespective of their particular kind, the nature of their component elements, and the relationships or
"forces
" between them. It seems legitimate to ask for a theory, not of systems of a more or less special
kind, but of universal principles applying to systems in general.


[22]

Ervin La
szlo
[23]

in the preface of von Bertalanffy's book

Perspectives on General System Theory
:
[24]

Thus wh
en von Bertalanffy spoke of Allgemeine Systemtheorie it was consistent with his view that he was
proposing a new perspective, a new way of doing science. It was not directly consistent with an
interpretation often put on "general system theory", to wit, th
at it is a (scientific) "theory of general
systems." To criticize it as such is to shoot at straw men. Von Bertalanffy opened up something much
broader and of much greater significance than a single theory (which, as we now know, can always be
falsified an
d has usually an ephemeral existence): he created a new paradigm for the development of
theories.

Ludwig von Bertalanffy outlines systems inquiry into three major domains: Philosophy, Science, and
Technology. In his work with the Primer Group, Béla H. Báná
thy generalized the domains into four
integratable domains of systemic inquiry:


ybernetics

The term cybernetics derives from a Greek word which meant steersman, and which is the origin of
English words such as "govern". Cybernetics is the study of

feedback

and derived concepts such
as

communication

and control in living organisms, machines and organisations. Its
focus is how anything
(digital, mechanical or biological) processes information, reacts to information, and changes or can be
changed to better accomplish the first two tasks.

The terms "systems theory" and "
cybernetics
" have been widely used as synonyms. Some authors use
the term

cybernetic

systems to denote a proper subset of the class of general systems, namely those
systems that include

feedback

loops. However

Gordon Pask
's differences of eternal interacting actor
loops (that produce finite products) makes general systems a proper subset of c
ybernetics. According to
Jackson (2000), von Bertalanffy promoted an embryonic form of general system theory (GST) as early as
the 1920s and 1930s but it was not until the early 1950s it became more widely known in scientific circles.

Threads of cybernetic
s began in the late 1800s that led toward the publishing of seminal works (e.g.,
Wiener's

Cybernetics

in 1948 and von Bertalanffy's

General Systems Theory

in 1968). Cybernetics arose
more from engineering fields and GST from biology. If anything it appears

that although the two probably
mutually influenced each other, cybernetics had the greater influence. Von Bertalanffy (1969) specifically
makes the point of distinguishing between the areas in noting the influence of cybernetics: "Systems
theory is freque
ntly identified with cybernetics and control theory. This again is incorrect. Cybernetics as
the theory of control mechanisms in technology and nature is founded on the concepts of information and
feedback, but as part of a general theory of systems;" then

reiterates: "the model is of wide application but
should not be identified with 'systems theory' in general", and that "warning is necessary against its
incautious expansion to fields for which its concepts are not made." (17
-
23). Jackson (2000) also clai
ms
von Bertalanffy was informed by

Alexander Bogdanov
's three volume

Tectology

that was published
in
Russia between 1912 and 1917, and was translated into German in 1928. He also states it is clear to
Gorelik (1975) that the "conceptual part" of general system theory (GST) had first been put in place by
Bogdanov. The similar position is held by Mattess
ich (1978) and Capra (1996). Ludwig von Bertalanffy
never even mentioned Bogdanov in his works, which Capra (1996) finds "surprising".

Cybernetics,

catastrophe theory
,

chaos theory

and

complexity theory

have the common goal to explai
n
complex systems that consist of a large number of mutually interacting and interrelated parts in terms of
those interactions. Cellular automata (CA), neural networks (NN), artificial intelligence (AI), and

artificial
life

(ALife) are related fields, but they do not try to describe general (universal) complex (singular)
systems. The best context to compare the different "C"
-
Theories about complex systems is historical,
which
emphasizes different tools and methodologies, from pure mathematics in the beginning to pure
computer science now. Since the beginning of chaos theory when

Edward Lorenz

accident
ally discovered
a

strange attractor

with his computer, computers have become an indispensable source of information.
One could not imagine the study of complex systems wi
thout the use of computers today.

Complex adaptive systems

A complex adaptive systems are special cases of

complex systems
. They are

complex

in that they are
diverse and compos
ed of multiple, interconnected elements; they are

adaptive

in that they have the
capacity to change and learn from experience. The term

complex adaptive system

was coined at the
interdisciplinary

Santa Fe Institute

(SFI), by

John H. Holland
,

Murray Gell
-
Mann

and others.

Biomatrix systems theory

During the 1990s, an interdisciplinary team of PhD students at the University of Cape Town, South Africa,
integrated the key concepts of the systems and related fields, together with their unique theoretical
cont
ributions, into a coherent meta
-
theory called Biomatrix systems theory. The theory is also unique in
having a graphic alphabet with which it can be explained visually.
[26]

(4)

Fe
atures of complex systems

Complex systems may have the following features:

Cascading Failures

Due to the strong coupling between components in complex systems, a failure in one or more
components can lead to cascading failures which may have catastrophic c
onsequences on the
functioning of the system.
[7]

Difficult to determine boundaries

It can be difficult to determine the boundaries of a complex system. The decision is ultimately
made by the observer.

Complex systems may be open

Complex systems are usually

open systems



that is, they exist in a

thermodynamic

gradient
and dissipate energy. In other words, complex systems are frequently far from
energetic
equilibrium
: but despite this flux, there may be

pattern stability
, see

synergetics
.

Complex systems may have a memory

The history of a complex system may be important. Because complex systems are

dynamical

systems

they change over time, and prior states may have an influence on present states. More
formally, complex systems often exhibit

hysteresis
.

Complex systems may be nested

The co
mponents of a complex system may themselves be complex systems. For example,
an

economy

is made up of

organisat
ions
, which are made up of

people
, which are made up
of

cells
-

all of which are complex systems.

Dynamic networ
k of multiplicity

As well as coupling rules, the dynamic network of a complex system is important.

Small
-
world

or

scale
-
free

networks
[8]
[9]
[10]

which have many local interactions and a smaller number of
inter
-
area connections are often employed. Natural complex systems often exhibit such
topologies. In the human

cortex

for example, we see dense local connectivity and a few very
long

axon
projections between regions inside the cortex and to other brain regions.

May p
roduce emergent phenomena

Complex systems may exhibit behaviors that are

emergent
, which is to say that while the results
may be sufficiently determined by the activity of the systems' b
asic constituents, they may have
properties that can only be studied at a higher level. For example, the

termites

in a mound have
physiology, biochemistry and biological development that a
re at one level of analysis, but
their

social behavior

and mound building is a property that emerges from the collection of termites
and needs to be analysed at a different l
evel.

Relationships are non
-
linear

In practical terms, this means a small perturbation may cause a large effect (see

butterfly effect
),
a proportional effect, or even no ef
fect at all. In linear systems, effect is

always

directly
proportional to cause. See

nonlinearity
.

Relationships contain feedback loops

Both negative (damping) and positive (amplif
ying)

feedback

are always found in complex
systems. The effects of an element's behaviour are fed back to in such a way that the element
itself is altered.

his article largely discusses co
mplex systems as

a subject of mathematics

and the attempts to
emulate physical complex systems with emergent properties. For other scientific and professional
disciplines addressing complexity in their fields see the

complex systems

article and references.

A

complex system

is a

system

composed of interconnected parts that as a whole exhibit one or
more properties
(behavior among the possible properties) not obvious from the properties of the
individual parts.
[1]

A system’s

complexity

may be of one of two forms:

disorganized complexity and
organized complexity
.
[2]

In essence, disorganized complexity is a matter of a very large number of
parts, and organized complexity is a matter of the subject system (quite possibly with only a limited
number of parts) exhibiting

emergent

properties.

Examples of complex

systems

for which complexity models have been developed include

ant
colonies
, human

economies

and

social structures
,

climate
,

nervous systems
,

cells

and living things,
including human beings, as well as modern energy or

telecommunication

infrastructures. Indeed,
ma
ny systems of interest to humans are complex systems. Complex systems are studied by many
areas of

natural science
,

mathematics
, and

social science
. Fields that specialize in the
interdisciplinary study of complex systems include

systems theory
,

complexity theory
,

systems
ecology
, and

cybernetics
.