1
ADRIANO PEDRO RODRIGUES
ID: UB18207SSO26040
COURSE TITLE: CIVIL ENGINEERING TECHNOLOGY
Index
1- Introduction…………………………………………………………………………………..pag. 6
2- Civil Engineering Technology…………………………….…..………………………..pag. 7
3-Urban Planing………………………………………………………………………………….pag. 7
3.1-Planing and Reconstruction…………………………………………..………………pag. 8
3.2- Planing and Transport…….……………………………………………………………pag. 8
3.3-Planing and the Environment………………………………………..………………pag. 9
3.4-Actors in the Planing Process……………………..………………………..………pag.10
3.5-Land use Planing……………………………………………………………………………pag.10
3.6-Functions of “Land use Planing”..…………………………………………………….pag.10
3.7-Regional Planing……………………………………………………………………………..pag.10
3.8-Principles of Regional Planing……………………………………………….………..pag.11
4-Design Engineer…………..……………………………….………………………………..….pag.11
5-Municipality……………………………………………………………………...……………….pag.11
6-Building Engineering…….….…………………………………..……………………………pag.11
7-Thin-Shell Structure..………………………………………………………………………....pag.12
8-Construction Worker……………………………………………………..….……...……...pag.13
9-Architectural Engineering………..……………….………………..…..………………..pag.14
10—Vectors and Scalars…..………………………………………………………………....pag.14
10.1 Vector Spaces………………..…………………………….……………………………….pag.14
10.2 Vector Basics………………………………………………………….……………………pag.15
11-Norm………….…………………………………………….…………………………………….pag.15
12-Orthogonality………………………………………………………………………………….pag.16
13-Metric (Distance)……………………………….…………………………………………………pag.16
2
14-Linear Independence an Basis…….…………………………………………………………..pag.16
15-Non-Square Matrix V………………………………………………………………………………pag.16
16-Rank……………………………………………………………..………………………………..……...pag.16
17-Span………..………………………………………………..…………………………………………...pag.17
18-Basis……………………………………………………………………………………………………….pag.17
18.1-Basis Expansion…………………………………………………………………………………...pag.17
18.2-Change of Basis……………………………………………………………………………………pag.17
18.3-Reciprocal Basis…………………………………………………………………………………..pag.17
19-Linear Transformations……………..……………………………………………………………pag.17
20-Null Spaces……………………………………………………………………………………………..pag.17
21-Linear Equations…………………………………………………………………………………….pag.18
22-Complete Solution………………………………………………………………………………….pag.18
22.1-Minimum Norm Solution…………………………………………………………………….pag.18
23-Minimization…………………………………………………………………………………………pag.18
23.1-Projection…………………………………………………………………………………………..pag.19
23.2-Distance between V and W…………………………………………………………………pag.19
23.3-Intersections……………………………………………………………………………………….pag.19
23.4-Matrices…………………………………………………………………………………………….pag.20
24-Matrix Forms……………………………………………………………………………………...…pag.20
24.1-Diagonal Matrix………………………………………………………………………………….pag.20
24.2-Companion Form Matrix…………………………………………………………………….pag.20
24.3-Quadratic Forms…………………………………………………………………………………pag.21
25-Eigenvalues and Eigenvectors………………………………………………………………..pag.21
25.1-Caracteristic Equations……………………………………………………………………….pag.21
25.2-Eigenvalues…………………………………………………………………………………………pag.21
25.3-Eigenvectors……………………………………………………………………………………….pag.22
25.4-Right and Left Eigenvectors…………………………………………………………………pag.22
25.5-Matrix Diagonalization………………………………………………………………………..pag.22
3
25.6-Complex Eigenvalues…………………………………………………………………………..pag.22
25.7-Generalized Eigenvvectors…………………………………………………………………..pag.22
25.8-Spectral Decomposition……………………………………………………………………….pag.23
26-Matrix Exponential………………………………………………………………………………….pag.23
26.1-Derivative……………………………………………………………………………………………..pag.23
26.2-Differential Equations……………………………………………………………………………pag.23
27-Lyapunov Equation……………………………………………………………………………………pag.23
27.1-L2 Space…………………………………………………………………………………………………pag.23
27.2-L2 Functions…………………………………………………………………………………………..pag.24
27.3-Null Function………………………………………………………………………………………….pag.24
27.4-Norm……………………………………………………………………………………………………..pag.24
27.5-Scalar Product………………………………………………………………………………………..pag.24
27.6-Metric…………………………………………………………………………………………………….pag.24
27.7-Hilbert Space…………………………………………………………………………………………..pag.24
27.8-Fourier Series………………………………………………………………………………………….pag.25
28-a0: The Constant Term……………………………………………………………………………….pag.25
28.1-Sin Coefficients…………………………………………………………………………………………pag.26
28.2-Cos Coefficients……………………………………………………………………………………….pag.26
29-Arbitrary Basics Expansion…………………………………………………………………………..pag.26
30-Bessel Equation and Parseval Theorem……………………………………………………….pag.26
31-Multi Dimensional Fourier Series…………………………………………………………………pag.27
32-Wevelets……………………………………………………………………………………………………..pag.27
32.1-Mother Wavelet……………………………………………………………………………………….pag.27
32.2-Wevelet Series………………………………………………………………………………………….pag.27
33-Scaling Function……………………………………………………………………………………………pag.27
33.1-Probability Density Function………………………………………………………………………pag.27
33.2-Fair Coin……………………………………………………………………………………………………pag.28
4
33.3-Fair Dice……………………………………………………………………………………………………pag.28
33.4-Probability Density Function………………………………………………………………………pag.28
34-Cumulative Distribution Function…………………………………………………………………pag.28
34.1-X Between Two Bounds……………………………………………………………………………..pag.29
35-Uniform Distribution……………………………………………………………………………………pag.29
35.1-Gaussian Distribution………………………………………………………………………………..pag.29
35.2-Moments…………………………………………………………………………………………………..pag.29
35.3-Mean………………………………………………………………………………………………………..pag.29
35.4-Central Moments………………………………………………………………………………………pag.30
35.5-Variance……………………………………………………………………………………………………pag.30
36-Siso Transformations……………………………………………………………………………………pag.30
36.1-Miso Transformations……………………………………………………………………………….pag.30
37-Optimization…………………………………………………………………………………………………pag.30
37.1-Minimization……………………………………………………………………………………………..pag.31
37.2-Unconstrained Minimization………………………………………………………………………pag.31
37.3-Hessian Matrix…………………………………………………………………………………………..pag.31
37.4-Newton-Raphson method………………………………………………………………………….pag.31
37.5-Steepest Descent Method………………………………………………………………………….pag.32
37.6-Constrained Minimization………………………………………………………………………….pag.32
37.7-Equality Constraints……………………………………………………………………………………pag.32
37.8-Inequality Constraints…………………………………………………………………………………pag.32
37.9-Equality and Inequality Constraints…………………………………………………………….pag.32
38-Infinit Dimension Minimization………………………………………………………………………pag.32
38.1-Gateaux Derivative…………………………………………………………………………………….pag.33
38.2-Euler-Lagrange Equation……………………………………………………………………………pag.33
39-Normal Distribution………………………………………………………………………………………pag.33
39.1-Probability Content From −∞ 𝑡𝑜 𝑍 (𝑍 ≤ 0)………………………………………………pag.34
5
40-Systems Engineering Modeling and Design…………………………………………………..pag.34
40.1-Systems Engineering…………………………………………………………………………………pag.35
40.2-The Scope of System………………………………………………………………………………….pag.35
41-Introduction to System Dynamics…………………………………………………………………pag.35
41.1-System Dynamics as Simulation Modeling…………………………………………………pag.35
41.2-System Simulation Analysis and Design……………………………………………………..pag.35
42.3-Solving Problems……………………………………………………………………………………….pag.36
41.4-Systems Thinking and Systems Dynamics…………………………………………………..pag.37
41.6-Cause and Effects………………………………………………………………………………………pag.37
41.7-Feedback……………………………………………………………………………………………………pag.38
41.8-Building Blocks in Systems…………………………………………………………………………pag.38
42-Levels and Flows………………………………………………………………………………………….pag.38
43-Auxiliaries……………………………………………………………………………………………………pag.39
43.1-Constants…………………………………………………………………………………………………pag.39
43.2-Informations Links……………………………………………………………………………………pag.40
43.3-Decisions and Policies………………………………………………………………………………pag.40
43.4-Decision Making Process…………………………………………………………………………pag.40
43.5-Problem Definition……………………………………………………………………………………pag.41
43.6-Identification of Variables…………………………………………………………………………pag.41
44-Software Markting Research and analysis……………………………………………………pag.41
44.1-Software Markting Strategy and Implement Planing.………..…………………….pag.42
45-General Analysis…………………………………………………………………………………………pag.43
46-Conclusion………………………………………………………………………………………………….pag.44
47-Bibliography…………………………………………………………………………………….…………pag.45
6
CIVIL ENGINEERING TECHNOLOGY
1- INTRODUCTION
Civil engineering is one of the fastest growing areas currently, and their
accompanying these growth technologies.
Since man came out of the caves and was looking for a safer and more comfortably,
the need to improve construction techniques have become increasingly larger. Since
man came out of the caves and was looking for a safer and more comfortably, the
need to improve construction techniques have become increasingly larger.
The great transformations in the use of materials, techniques in the construction of
buildings followed the needs of modern man, on the requirements of a new way of
life.
New technologies emerge at all times, replacing old ways of building. However when it
comes to restoration of cultural and historical knowledge of specific techniques of past
periods is essential to guarantee their preservation without distortion. To do this,
courses are offered to engineers, architects in the intention that these not-so-modern
techniques, help in the restoration of numerous estates.
The civil engineer plans, designs, implements, supervises and monitors the work
related to the construction, operation and maintenance of buildings, roads, railways,
dams ports tunnels airports, ports etc. The civil engineer is responsible for the
calculations relating to the structures of the work.
The various topics that will be addressed in this paper reflect precisely the range of
knowledge needed to fully exercise the civil engineer and building technologies serving
as pillars of innovation in this area.
Adriano Rodrigues
2013-03-26
7
2-CIVIL ENGINEERING TECHNOLOGY
In modern usage, civil engineering is a broad field of engineering that deals with the
planning, construction, and maintenance of fixed structures, or public works, as they
are related to earth, water, or civilization and their processes. Most civil engineering
today deals with power plants, bridges, roads, railways, structures, water supply,
irrigation, environment, sewer, flood control, transportation and traffic. In essence,
civil engineering may be regarded as the profession that makes the world a more
agreeable place in which to live.
Engineering has developed from observations of the ways natural and constructed
systems react and from the development of empirical equations that provide bases for
design. Civil engineering is the broadest of the engineering fields, partly because it is
the oldest of all engineering fields. In fact, engineering was once divided into only two
fields - military and civil. Civil engineering is still an umbrella term, comprised of many
related specialities.
General civil engineering is concerned with the overall interface of human created
fixed projects with the greater world. General civil engineers work closely with
surveyors and specialized civil engineers to fit and serve fixed projects within their
given site, community and terrain by designing grading, drainage (flood control),
pavement, water supply, sewer service, electric and communications supply and land
(real property) divisions.
3-Urban planning
Urban, city, or town planning is the discipline of land use planning which deals with the
physical, social, and economic development of metropolitan regions, municipalities
and neighborhoods.
Historically, urban development was more often a haphazard, incremental event than
a deliberate planned process. In the nineteenth century, urban planning became
influenced by the newly formalized disciplines of architecture and civil engineering,
which began to codify both rational and stylistic approaches to solving city problems
through physical design.
During the last two centuries in the Western world (Western Europe, North America,
Japan and Australasia) planning and architecture can be said to have gone through
various stages of general consensus.
Around the turn of the 20th Century there began to be a movement for providing
people, and factory workers in particular, with healthier environments. The concept of
garden cities arose and some model towns were built, such as Letchworth and
Welwyn Garden City the world's first garden cities, in Hertfordshire, UK. However,
these were principally small scale in size, typically dealing with only a few thousand
residents.
There were plans for large scale rebuilding of cities, such as Paris in France, though
nothing major happened until the devastation caused by the Second World War. After
this, some modernist buildings and communities were built. Successful urban planning
considers character, of "home" and "sense of place", local identity, respect for natural,
artistic and historic heritage, an understanding of the "urban grain" or "townscape,"
pedestrians and other modes of traffic, utilities and natural hazards, such as flood
zones.
8
Many conventional planning techniques are being repackaged using the contemporary
term, smart growth.
Some of the most successful planned cities consist of cells that include park-space,
commerce and housing, and then repeat the cell. Usually cells are separated by
streets.
Often each cell has unique monuments and gardening in the park, and unique gates or
boundary markers for the edges of the cell. City planning tries to control criminality
with structures designed from theories such as socio-architecture or environmental
determinism. These theories say that an urban environment can influence individuals'
obedience to social rules.
Oscar Newman’s defensible space theory cites the modernist housing projects of the
1960s as an example environmental determinism, where large blocks of flats are
surrounded by shared and disassociated public areas which is harder to identify with
for the residents. Jane Jacobs is another notable environmental determinist and is
associated with the "eyes on the street" concept.
3.1 Planning and reconstruction
Areas devastated by war or invasion represent a unique challenge to urban planners:
the area of development is not one for simple modification, nor is it a "blank slate".
Buildings, roads, services and basic infrastructure like power, water and sewerage are
often severely compromised and need to be evaluated to determine what, if anything,
can be salvaged for re-incorporation. Development, needs to be sensitive to the needs
of this community and its existing culture, businesses and needs. Urban Reconstruction
Development plans must also work with government agencies as well as private
interests to develop workable designs.
3.2 Planning and transport
9
Very densely built-up areas require high capacity urban transit, urban planners must
consider these factors in long term plans.
There is a direct, well-researched connection between the density of an urban
environment, and the need to travel within it . Good quality transport is often followed
by development. Development beyond a certain density can quickly overcrowd
transport. Good planning attempts to place higher densities of jobs or residents near
high-volume transportation. Densities can be measured in several ways. A common
method, used is the Floor area ratio, using the floor area of buildings divided by the
land area. Ratios below 1.5 could be considered low density, and plot ratios above five
very high density.
Problems can often occur at residential densities between about two and five. These
densities can cause traffic jams for automobiles, yet are too low to be commercially
served by trains or light rail systems. The conventional solution is to use buses, but
these and light rail systems may fail where automobiles and excess road network
capacity are both available, achieving less than 1% ridership.
Some theoreticians speculate that personal rapid transit (PRT) might coax people from
their automobiles, and yet effectively serve intermediate densities, but this has not
been demonstrated.
3.3 Planning and the environment
Environmental protection and conservation are of upmost importance to many
planning systems across the world. Not only are the specific effects of development to
be mitigated, but attempts are made to minimize the overall effect of development on
the local and global environment.
In most advanced urban or village planning models, local context is critical. In many,
gardening assumes a central role not only in agriculture but in the daily life of citizens.
The modern theory of natural capital emphasizes this as the primary difference
between natural and infrastructural capital, and seeks an economic basis for
rationalizing a move back towards smaller village units.
An urban planner is likely to use a number of Quantitative tools to forecast impacts of
development on a variety of environmental concerns including roadway air dispersion
models to predict air quality impacts of urban highways and roadway noise models to
predict noise pollution effects of urban highways.
10
3.4 Actors in the planning process
The traditional planning process focussed on top-down processes where the town
planner created the plans. Engineering or architecture, bringing to the town planning
process ideals based around these disciplines. They typically worked for national or
local governments.
Many recent developments were results of large and small-scale developers who
purchased land, designed the district and constructed the development from scratch.
The Melbourne Docklands, for example, was largely an intiative pushed by private
developers who sought to redevelop the waterfront into a high-end residential and
commercial district.
3.5 Land use planning
Land Use Planning is the term used for a branch of public policy which encompasses
various disciplines which seek to order and regulate the use of land in an efficient way.
The Canadian Planners Association offers a definition that; "[Land Use] Planning means
the scientific, aesthetic, and orderly disposition of land, resources, facilities and services
with a view to securing the physical, economic and social efficiency, health and wellbeing of urban and rural communities.
3.6 Functions of ‘Land Use Planning’
At its most basic level land use planning is likely to involve zoning and transport
infrastructure planning. In most developed countries, land use planning is an
important part of social policy, ensuring that land is used efficiently for the benefit of
the wider economy and population as well as to protect the environment.
Land Use Planning encompasses the following disciplines:
• Architecture
• Environmental planning
• Landscape architecture
• Regional Planning
• Spatial planning
• Sustainable Development
• Transportation Planning
• Urban design
• Urban planning
• Urban Renaissance
• Urban renewal
3.7 Regional planning
Regional planning is a branch of land use planning and deals with the efficient
placement of land use activities, infrastructure and settlement growth across a
significantly larger area of land than an individual city or town.
Although the term ‘Regional planning’ is nearly universal in English speaking countries
the areas covered and specific administrative set ups vary widely Regions require
various land uses; protection of farmland, cities, industrial space, transportation hubs
and infrastructure, military bases, and wilderness. Regional planning is the science of
efficient placement of infrastructure and zoning for the sustainable growth of a region.
A ‘region’ in planning terms can be administrative or at least partially functional, and is
likely to include a network of settlements and character areas.
11
3.8 Principles of regional planning
Specific interventions and solutions will depend entirely on the needs of each region in
each country, but generally speaking, regional planning at the macro level will seek to:
• Resist development in flood plains or along an earthquake faults. These areas may
be utilised as parks, or unimproved farmland.
• Designate transportation corridors using hubs and spokes and considering major
new infrastructure
• Some thought into the various ‘role’s settlements in the region may play, for
example some may be administrative, with others based upon manufacturing or
transport.
• Consider designating essential nuisance land uses locations, including waste
disposal.
• Designate Green belt land or similar to resist settlement amalgamation and protect
the environment.
• Set regional level ‘policy’ and zoning which encourages a mix of housing values
and communities.
• Consider building codes, zoning laws and policies that encourage the best use of
the land.
4-Design engineer
A design engineer is an engineer whose job is to produce a detailed design from a
conceptual design, thereby bringing the real from the abstract on a day-to-day basis.
The output of a design engineer is usually a set of drawings and specifications that
should produce a working product with very little final adjustment needed.
in the conceptual design that go uncaught by the design engineer or others will not
surface until production or construction, so there is a potential for a design engineer to
become a scapegoat for those problems.
A design engineer would usually work on a newer and less proven design than a
designer, since once the engineering principles are proven to be correct in a similar
design, further engineering knowledge and skill is not generally needed to produce
similar designs.
5-Municipality
A municipality is an administrative entity composed of a clearly defined territory and
its population and commonly referring to a city, town, or village, or a small grouping of
them. A municipality is typically governed by a mayor and a city council or municipal
council.
A municipality is a generalpurpose district, as opposed to a special-purpose district. In
most countries, a municipality is the smallest administrative subdivision to have its
own democratically elected representative leadership.
In some countries, municipalities are referred to as "communes" (for example, French
commune or Spanish comuna). The term derives from the medieval commune. Note
that the word has absolutely no implication of communism.
6-Building engineering
Building engineering, commonly known in the US as architectural engineering, is an
emerging engineering discipline that concerns with the planning, design, construction,
operation, renovation, and maintenance of buildings, as well as with their impacts on
the surrounding environment.
12
As building construction projects are increasingly large and complex, the discipline
requires pertinent knowledge integrated from traditional wellestablished disciplines:
• Civil engineering for building structures and foundation;
• Mechanical engineering for Heating, Ventilation and Air-Conditioning system
(HVAC), and for mechanical service systems;
• Physics for building science, lighting and acoustics.
• Electrical engineering for power distribution, control, and electrical systems;
• Chemistry and biology for indoor air quality;
• Architecture for form, function, building codes and specifications;
• Economics for project management.
Building engineers are trained in all phases of the life cycle of a building and develop
an appreciation of the building as an advanced technological system requiring close
integration of many sub-systems and their individual components. Technical problems
are identified and appropriate solutions found to improve the performance of the
building in areas such as:
• Energy efficiency, passive solar engineering, lighting and acoustics;
• Construction management;
• HVAC and control systems; Indoor air quality;
• Advanced building materials; building envelope;
• Earthquake resistance, wind effects on buildings, computer-aided design.
7-Thin-shell structure
Ex. The world's first double curvature lattice steel Shell by V.G.Shukhov (during
construction), Vyksa near Nizhny Novgorod, 1897
Thin-shell structures can be defined as curved structures capable of transmitting loads
in more than two directions to supports Loads applied to shell surfaces are carried to
the ground by the development of compressive, tensile, and shear stresses acting in
the inplane direction of the surface. Thin shell structures are uniquely suited to
carrying distributed loads and find wide application as roof structures in building.
13
Ex. Swiss Re "Gherkin", with a lattice shell by Norman Foster and Ken Shuttleworth,
London.
8-Construction worker
Construction workers are employed in the construction industry and work
predominately on construction sites as opposed to being office based and are typically
engaged in aspects of the industry other than design or finance. The term includes
general construction workers, also referred to as labourers and members of specialist
trades such as electricians, carpenters and plumbers.
After working as a journeyman for a specified period, a tradesman may go on to study
or test as a master craftsman. In some countries, such as Germany or Japan, this is a
process requiring extensive knowledge and skill to achieve master certification. In
others, it can be a loosely used term to describe a skilled carpenter.
The construction industry is the most dangerous land based civilian work sector (the
fishing industry is more dangerous). In the European Union, the fatal accident rate is
nearly 13 workers per 100,000 as against 5 per 100,000 for the all sector average
(Source: Eurostat).
Under European Union Law, there are European Union Directives in place to protect
workers, notably Directive 89/391 (the Framework Directive) and Directive 92/57 (the
Temporary and Mobile Sites Directive). This legislation is transposed into the Member
States and places requirements on employers (and others) to assess and protect
workers health and safety.
Construction workers are usually associated with wearing a hard hat, this along with
steel-toe boots are the most common personal protective equipment worn. The
standard use of high visibility jackets is also widespread. Additional personal protective
equipment is required on the basis of a risk assessment, for example when dealing
with situations involving hazardous substances, protective gloves and googles would
be specified.
14
Ex. Construction worker donning a high-visibility fluorescent vest.
9-Architectural engineering
An architectural engineer applies the skills of many engineering disciplines to the
design, construction, operation, maintenance, and renovation of buildings while
paying attention to their impacts on the surrounding environment.
An architectural engineer applies the skills of many engineering disciplines to the
design, construction, operation, maintenance, and renovation of buildings while
paying attention to their impacts on the surrounding environment.
In countries such as Canada, the UK and Australia, architectural engineering is more
commonly known as Building engineering, building systems engineering, or building
services engineering.
Many practicing 'architectural engineers' hold degrees or registration in civil,
mechanical, electrical, or another engineering field and become architectural
engineers via experience.
Architectural engineers' roles can overlap with that of the architect and other project
engineers. Like architects, they seek to achieve optimal designs within the overall
constraints, except using primarily the tools of engineering rather than architecture. In
most parts of the world, architectural engineers are not entitled to practice
architecture unless they are also licensed as architects.
10-Vectors and Scalars
A scalar is a single number value, such as 3, 5, or 10. A vector is an ordered set of
scalars. A vector is typically described as a matrix with a row or column size of 1. A
vector with a column size of 1 is a row vector, and a vector with a row size of 1 is a
column vector.
[Column Vector]
[Row Vector]
10.1 Vector Spaces
A vector space is a set of vectors and two operations (addition and multiplication,
typically) that follow a number of specific rules.
15
A space V is a vector space if all the following requirements are met. We will be using x
and y as being arbitrary vectors in V. We will also use c and d as arbitrary scalar values.
There are 10 requirements in all:
Given:
1. There is an operation called "Addition" (signified with a "+" sign) between two
vectors, x + y, such that if both the operands are in V, then the result is also in V.
2. The addition operation is commutative for all elements in V.
3. The addition operation is associative for all elements in V.
4. There is a neutral element, φ, in V, such that x + φ = x. This is also called a one
element.
5. For every x in V, then there is a negative element -x in V.
6. cx ∈ 𝑉
7. c(x + y) = cx + cy
8. (c + d)x = cx + dx
9. c(dx) = cdx
10. 1 × x = x
Some of these rules may seem obvious, but that's only because they have been
generally accepted, and have been taught to people since they were children.
10.2 Vector Basics
Scalar Product
A scalar product is a special type of operation that acts on two vectors, and returns a
scalar result. Scalar products are denoted as an ordered pair between angle-brackets:
<x,y>. A scalar product between vectors must satisify the following four rules:
1.
2.
only if x=0
3.
4.
If an operation satisfies all these requirements, then it is a scalar product.
11-Norm
The norm is an important scalar quantity that indicates the magnitude of the vector.
Norms of a vector are typically denoted as || 𝑥||To be a norm, an operation must
satisfy the following four conditions:
1.
2.
only if x=0
3.
4.
A vector is called normal if it's norm is 1. A normal vector is sometimes also referred to
as a unit vector. To make a vector normal, but keep it pointing in the same direction,
we can divide the vector by it's norm:
Examples
One of the most common norms is the cartesian norm, that is defined as the squareroot of the sum of the squares:
16
12-Orthogonality
Two vectors x and y are said to be orthogonal if the scalar product of the two is equal
to zero:
Two vectors are said to be orthonormal if their scalar product is zero, and both vectors
are unit vectors.
13-Metric (Distance)
The distance between two vectors in the vector space V, called the metric of the two
vectors, is denoted by d(x, y). A metric operation must satisfy the following four
conditions:
1. d(𝑥, 𝑦) ≥ 0
2. d(𝑥, 𝑦) = 0 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥 = 𝑦
3. d(𝑥, 𝑦) = 𝑑(𝑦, 𝑥)
4.
14-Linear Independence and Basis
A set of vectors are said to be linearly dependant on one another if any vector v from
the set can be constructed from a linear combination of the other vectors in the set.
Given the following linear equation:
The set of vectors V is linearly independant only if all the a coefficients are zero. If we
combine the v vectors together into a single row vector:
15-Non-Square Matrix V
If the matrix is not square, then the determinate can not be taken, and therefore the
matrix is not invertable. To solve this problem, we can premultiply by the transpose
matrix:
And then the square matrix
must be invertable:
16-Rank
The rank of a matrix is the largest number of linearly independant rows or columns in
the matrix.
To determine the Rank, typically the matrix is reduced to row-echelon form. From the
reduced form, the number of non-zero rows, or the number of non-zero colums
(whichever is smaller) is the rank of the matrix.
If we multiply two matrices A and B, and the result is C:
17
AB=C
Then the rank of C is the minimum value between the ranks A and B:
17-Span
A Span of a set of vectors V is the set of all vectors that can be created by a linear
combination of the vectors.
18-Basis
A basis is a set of linearly-independant vectors that span the entire vector space.
18.1 Basis Expansion
If we have a vector 𝑦 ∈ 𝑉and V has basis vectors 𝑣1 𝑣2 … 𝑣𝑛 by definition, we can
write y in terms of a linear combination of the basis vectors:
Or
18.2 Change of Basis
Frequently, it is useful to change the basis vectors to a different set of vectors that
span the set, but have different properties. If we have a space V, with basis vectors and
a vector in V called x, we can use the new basis vectors W to represent x:
or,
18.3 Reciprocal Basis
A Reciprocal basis is a special type of basis that is related to the original basis. The
reciprocal basis W can be defined as:
19-Linear Transformations
A linear transformation is a matrix M that operates on a vector in space V, and results
in a vector in a different space W. We can define a transformation as such:
In the above equation, we say that V is the domain space of the transformation, and W
is the range space of the transformation. Also, we can use a "function notation" for the
transformation, and write it as:
M(x) = Mx = y
Where x is a vector in V, and y is a vector in W. To be a linear transformation, the
principle of superposition must hold for the transformation:
M(av1 + bv2) = aM(v1) + bM(v2)
Where a and b are arbitary scalars.
20-Null Space
The Nullspace of an equation is the set of all vectors x for which the following
18
relationship holds:
Mx = 0
Where M is a linear transformation matrix. Depending on the size and rank of M, there
may be zero or more vectors in the nullspace. Here are a few rules to remember:
1. If the matrix M is invertable, then there is no nullspace.
2. The number of vectors in the nullspace (N) is the difference between the rank(R)
of the matrix and the number of columns(C) of the matrix:
N=R–C
If the matrix is in row-eschelon form, the number of vectors in the nullspace is given by
the number of rows without a leading 1 on the diagonal. For every column where
there is not a leading one on the diagonal, the nullspace vectors can be obtained by
placing a negative one in the leading position for that column vector.
We denote the nullspace of a matrix A as:
21-Linear Equations
If we have a set of linear equations in terms of variables x, scalar coefficients a, and a
scalar result b, we can write the system in matrix notation as such:
Ax = b
Where x is a m × 1 vector, b is an n &times 1 vector, and A is an n × m matrix.
Therefore, this is a system of n equations with m unknown variables. There are 3
possibilities:
1. If Rank(A) is not equal to Rank([A b]), there is no solution
2. If Rank(A) = Rank([A b]) = n, there is exactly one solution
3. If Rank(A) = Rank([A b]) < n, there are infinately many solutions.
22-Complete Solution
The complete solution of a linear equation is given by the sum of the homogeneous
solution, and the particular solution. The homogeneous solution is the nullspace of
the transformation, and the particular solution is the values for x that satisfy the
equation:
A(x) = b
A(xh + xp) = b
Where
xh is the homogeneous solution, and is the nullspace of A that satisfies the equation
A(xh) = 0 xp is the particular solution that satisfies the equation A(xp) = b
22.1Minimum Norm Solution
If Rank(A) = Rank([A b]) < n, then there are infinately many solutions to the linear
equation. In this situation, the solution called the minimum norm solution must be
found. This solution represents the "best" solution to the problem. To find the
minimum norm solution, we must minimize the norm of x subject to the constraint of:
Ax − b = 0
23-Minimization
Khun-Tucker Theorem
19
The Khun-Tucker Theorem is a method for minimizing a function f(x) under the
constraint g(x). We can define the theorem as follows:
Where Λ is the lagrangian vector, and < , > denotes the scalar product operation. We
will discuss scalar products more later. If we differentiate this equation with respect to
x first, and then with respect to Λ, we get the following two equations: Where Λ is the
lagrangian vector, and < , > denotes the scalar product operation. We will discuss
scalar products more later. If we differentiate this equation with respect to x first, and
then with respect to Λ, we get the following two equations:
23.1 Projection
The projection of a vector 𝑣 ∈ 𝑉onto the vector space 𝑊 ∈ 𝑉 is the minimum distance
between v and the space W. In other words, we need to minimize the distance
between vector v, and an arbitrary vector :𝜔 ∈ 𝑊
[Projection onto space W]
23.2 Distance between v and W
The distance between 𝑣 ∈ 𝑉 and the space W is given as the minimum distance
between v and an arbitrary 𝜔 ∈ 𝑊:
23.3 Intersections
Given two vector spaces V and W, what is the overlapping area between the two? We
define an arbitrary vector z that is a component of both V, and W:
Where N is the nullspace.
20
23.4 Matrices
Derivatives
Consider the following set of linear equations:
a = bx1 + cx2
d = ex1 + fx2
We can define the matrix A to represent the coefficients, the vector B as the results,
and the vector x as the variables:
And rewriting the equation in terms of the matrices, we get:
B = Ax
24-Matrix Forms
Matrices that follow certain predefined formats are useful in a number of
computations.
We will discuss some of the common matrix formats here. Later chapters will show
how these formats are used in calculations and analysis.
24.1 Diagonal Matrix
A diagonal matrix is a matrix such that:
In otherwords, all the elements off the main diagonal are zero, and the diagonal
elements may be (but don't need to be) non-zero.
24.2 Companion Form Matrix
If we have the following characteristic polynomial for a matrix:
We can create a companion form matrix in one of two ways:
Or, we can also write it as:
21
24.3 Quadratic Forms
If we have an n × 1 vector x, and an n × n symmetric matrix M, we can write:
xTMx = a Where a is a scalar value. Equations of this form are called quadratic forms.
25-Eigenvalues and Eigenvectors
The Eigen Problem
In this page I’m going to talk about the concept of Eigenvectors and Eigenvalues,
which are important tools in linear algebra, and which play an important role in StateSpace control systems. The "Eigen Problem" stated simply, is that given a square
matrix A which is n × n, there exists a set of n scalar values λ and n corresponding nontrivial vectors v such that:
Av = λv
We call λ the eigenvalues of A, and we call v the corresponding eigenvectors of A. We
can rearrange this equation as:
(A − λI)v = 0
For this equation to be satisfied so that v is non-trivial, the matrix (A - λI) must be
singular. That is:
| A − λI | = 0
25.1 Characteristic Equation
The characteristic equation of a square matrix A is given by:
[Characteristic Equation]
| A − λI | = 0
Where I is the identity matrix, and λ is the set of eigenvalues of matrix A. From this
equation we can solve for the eigenvalues of A, and then using the equations discussed
above, we can calculate the corresponding eigenvectors.
In general, we can expand the characteristic equation as:
[Characteristic Polynomial]
25.2 Eigenvalues
The solutions, λ, of the characteristic equation for matrix X are known as the
eigenvalues of the matrix X.
Eigenvalues satisfy the following properties:
1. If λ is an eigenvalue of A, λn is an eigenvalue of An.
2. If λ is a complex eigenvalue of A, then λ* (the complex conjugate) is also an
eigenvalue of A.
22
3. If any of the eigenvalues of A are zero, then A is singular. If A is non-singular, all
the eigenvalues of A are nonzero.
25.3 Eigenvectors
The characteristic equation can be rewritten as such:
Xv = λv
Where X is the matrix under consideration, and λ are the eigenvalues for matrix X. For
every unique eigenvalue, there is a solution vector v to the above equation, known as
an eigenvector. The above equation can also be rewritten as:
| X − λI | v = 0
Where the resulting values of v for each eigenvalue λ is an eigenvector of X. There is a
unique eigenvector for each unique eigenvalue of X. From this equation, we can see
that the eigenvectors of A form the nullspace:
Eigenvectors satisfy the following properties:
1. If v is a complex eigenvector of A, then v* (the complex conjugate) is also an
eigenvector of A.
2. Distinct eigenvectors of A are linearly independant.
3. If A is n × n, and if there are n distinct eigenvectors, then the eigenvectors of A
form a complete basis set for 𝑅 𝑛
25.4 Right and Left Eigenvectors
The equation for determining eigenvectors is:
(A − λI)v = 0 And because the eigenvector v is on the right, these are more
appropriately called "right eigenvectors". However, if we rewrite the equation as
follows:
u(A − λI) = 0
The vectors u are called the "left eigenvectors" of matrix A.
25.5 Matrix Diagonalization
Some matricies are similar to diagonal matrices using a transition matrix, T. We will
say that matrix A is diagonalizable if the following equation can be satisfied:
T − 1AT = D
Where D is a diagonal matrix. An n × n square matrix is diagonalizable if and only if it
has n linearly independant eigenvectors.
25.6 Complex Eigenvalues
Consider the situation where a matrix A has 1 or more complex conjugate eigenvalue
pairs. The eigenvectors of A will also be complex.
In engineering situations, it is often not a good idea to deal with complex matrices, so
other matrix transformations can be used to create matrices that are "nearly
diagonal".
25.7 Generalized Eigenvectors
If the matrix A does not have a complete set of eigenvectors, that is, that they have d
eigenvectors and n - d generalized eigenvectors, then the matrix A is not
diagonalizable.
Each set of generalized eigenvectors that are formed from a single eigenvector basis
will create a jordan block.
All the distinct eigenvectors that do not spawn any generalized eigenvectors will form
a diagonal block in the Jordan matrix.
23
25.8 Spectral Decomposition
If λi are are the n distinct eigenvalues of matrix A, and vi are the corresponding n
distinct eigenvectors, and if wi are the n distinct left-eigenvectors, then the matrix A
can be represented as a sum: this is known as the spectral decomposition of A.
26-Matrix Exponential
If we have a matrix A, we can raise that matrix to a power of e as follows:
eA
It is important to note that this is not necessarily (not usually) equal to each individual
element of A being raised to a power of e. Using taylor-series expansion of
exponentials, we can show that:
In other words, the matrix exponential can be reducted to a sum of powers of the
matrix.
This follows from both the taylor series expansion of the exponential function, and the
cayley-hamilton theorem discussed previously.
26.1 Derivative
The derivative of a matrix exponential is:
Notice that the exponential matrix is commutative with the matrix A. This is not the
case with other functions, necessarily.
26.2 Differential Equations
If we have a first-degree differential equation of the following form:
x'(t) = Ax(t) + f(x) With initial conditions x(t0) = c Then the solution to that equation is
given in terms of the matrix exponential:
This equation shows up frequently in control engineering.
27-Lyapunov Equation
[Lyapunov's Equation]
AM + MB = C
Where A, B and C are constant square matrices, and M is the solution that we are
trying to find. If A, B, and C are of the same order, and if A and B have no eigenvalues
in common, then the solution can be given in terms of matrix exponentials:
27.1 L2 Space
The L2 space is very important to engineers, because functions in this space do not
need to be continuous. Many discontinuous engineering functions, such as the delta
(impulse) function, the unit step function, and other discontinuous finctions are part of
this space.
27.2 L2 Functions
A large number of functions qualify as L2 functions, including uncommon,
discontinuous, piece-wise, and other functions.
For example, a unit step and an impulse function are both L2 functions.
24
Also, other functions useful in signal analysis, such as square waves, triangle waves,
wavelets, and other functions are L2 functions. In practice, most physical systems have
a finite amount of noise associated with them.
27.3 Null Function
The null functions of L2 are the set of all functions φ in L2 that satisfy the equation:
for all a and b.
27.4 Norm
The L2 norm is defined as follows:
[L2 Norm]
If the norm of the function is 1, the function is normal.
27.5 Scalar Product
The scalar product in L2 space is defined as follows:
[L2 Scalar Product]
If the scalar product of two functions is zero, the functions are orthogonal.
27.6 Metric
The metric of two functions (we will not call it the "distance" here, because that word
has no meaning in a function space) will be denoted with ρ(x,y). We can define the
metric of an L2 function as follows:
[L2 Metric]
2.7 Hilbert Space
A Hilbert Space is a Banach Space with respect to a norm induced by the scalar
product.
That is, if there is a scalar product in the space X, then we can say the norm is induced
by the scalar product if we can write:
That is, that the norm can be written as a function of the scalar product. In the L2
space, we can define the norm as:
In a Hilbert Space, the Parallelogram rule holds for all members f and g in the function
space:
The L2 space is a Hilbert Space. The C space, however, is not.
27.8 Fourier Series
The L2 space is an infinite function space, and therefore a linear combination of any
25
infinite set of orthogonal functions can be used to represent any single member of the
L2
space. The decomposition of an L2 function in terms of an infinite basis set is a
technique known as the Fourier Decomposition of the function, and produces a result
called the Fourier Series.
Fourier Basis
Let's consider a set of L2 functions, φ as follows:
φ = 1,sin(nπx),cos(nπx),n = 1,2,...
We can prove that over a range [a, b] = [0, 2\pi], all of these functions are orthogonal:
The sinusoidal functions are orthogonal with the function φ(x) = 1. Because this
serves as an infinite orthogonal set in L2, this is also a valid basis set in that space.
Therefore, we can decompose any function in L2 as the following sum:
[Classical Fourier Series]
However, the difficulty occurs when we need to calculate the a and b coefficients. We
will show the method to do this below:
28- a0: The Constant Term
Calculation of a0 is the easiest, and therefore we will show how to calculate it first. We
first create an error function, E, that is equal to the squared norm of the difference
between the function f(x) and the infinite sum above:
We will write all the basis functions as the set φ, described above:
Combining the last two functions together, and writing the norm as an integral, we can
say:
We attempt to minimize this error function with respect to the constant term. To do
this, we differentiate both sides with respect to a0, and set the result to zero:
And solving for a0, we get our final result:
28.1 Sin Coefficients
Using the above method, we can solve for the an coefficients of the sin terms:
26
28.2 Cos Coefficients
Also using the above method, we can solve for the bn terms of the cos term.
29-Arbitrary Basis Expansion
The classical Fourier series uses the following basis:
φ(x) = 1,sin(nπx),cos(nπx),n = 1,2,...
However, we can generalize this concept to extend to any orthogonal basis set from
the L2 space.
We can say that if we have our orthogonal basis set that is composed of an infinite set
of arbitrary, orthogonal L2 functions:
We can define any L2 function f(x) in terms of this basis set:
[Generalized Fourier Series]
30- Bessel Equation and Parseval Theorem
Bessel's equation relates the original function to the fourier coefficients an:
[Bessel's Equation]
If the basis set is infinitely orthogonal, and if an infinite sum of the basis functions
perfectly reproduces the function f(x), then the above equation will be an equality,
known as Parseval's Theorem:
[Parseval's Theorem]
Engineers may recognize this as a relationship between the energy of the signal, as
represented in the time and frequency domains. However, parseval's rule applies not
only to the classical Fourier series coefficients, but also to the generalized series
coefficients as well.
31- Multi-Dimensional Fourier Series
The concept of the fourier series can be expanded to include 2-dimensional and
ndimensional function decomposition as well. Let's say that we have a function in
terms of independant variables x and y. We can decompose that function as a doublesummation as follows:
27
Where φij is a 2-dimensional set of orthogonal basis functions. We can define the
coefficients as:
This same concept can be expanded to include series with n-dimensions.
32-Wavelets
Wavelets are orthogonal basis functions that only exist for certain windows in time.
This is in contrast to sinusoidal waves, which exist for all times t. A wavelet, because it
is dependant on time, can be used as a basis function. A wavelet basis set gives rise to
wavelet decomposition, which is a 2-variable decomposition of a 1-variable function.
Wavelet analysis allows us to decompose a function in terms of time and frequency,
while fourier decomposition only allows us to decompose a function in terms of
frequency.
32.1 Mother Wavelet
If we have a basic wavelet function ψ(t), we can write a 2-dimensional function known
as the mother wavelet function as such:
ψjk = 2j / 2ψ(2jt − k)
32.2 Wavelet Series
If we have our mother wavelet function, we can write out a fourier-style series as a
double-sum of all the wavelets:
33-Scaling Function
Sometimes, we can add in an additional function, known as a scaling function:
The idea is that the scaling function is larger then the wavelet functions, and occupies
more time. In this case, the scaling function will show long-term changes in the signal,
and the wavelet functions will show short-term changes in the signal.
33.1 Probability Function
The probability function, P[], will denote the probability of a particular occurance
happening. Here are some examples:
• P[X < x], the probability that the random variable X has a value less then some
variable x.
• P[X = x], the probability that the random variable X has a value equal to some
variable x.
• P[X < x,Y > y], the probability that the random variable X has a value equal to x,
and the random variable Y has a value equal to y.
33.2 Fair Coin
If we consider the example that a fair coin is flipped. We will define X to be the random
28
variable, and we will define "head" to be 1, and "tail" to be 0. What is the probability
that the coin is a head?
P[X = 1] = 50%
33.3 Fair Dice
Considering now a fair 6-sided dice. X is the r.v., and the numerical value on the face
of the die is the value that X can take. What is the probability that when the dice is
rolled, the value is less then 4?
P[X < 4] = 50%
What is the probability that the value will be even?
P[X is even] = 50%
33.4 Probability Density Function
The probability density function, or pdf of a random variable is the function defined
by:
fX(x) = P[X = x]
X is the random variable, and x is a related variable (but is not
random). The subscript X on fX denotes that this is the pdf for the X variable.
pdf's follow a few simple rules:
1. The pdf is always non-negative.
2. The area under the pdf curve is 1.
34- Cumulative Distribution Function
The cumulative distribution function, (CDF), is also known as the Probability
Distribution Function, (PDF). to reduce confusion with the pdf of a random variable,
we will use the acronym CDF to denote this function. The CDF of a random variable is
the function defined by:
The CDF and the pdf of a random variable are related:
The CDF is the function corresponding to the probability that a given value x is less
then the value of the random variable X. The CDF is a non-decreasing function, and is
always non-negative.
34.1 X between two bounds
To determine whether our random variable X lies between two bounds, [a, b], we can
take the CDF functions:
29
35-Uniform Distribution
The uniform distribution is one of the easiest distributions to analyze. Also, uniform
distributions of random numbers are easy to generate on computers, so they are
typically used in computer software.
35.1 Gaussian Distribution
The gaussian distribution, or the "normal distribution" is one of the most common
random distributions. A gaussian random variable is typically called a "normal" random
variable.
Where μ is the mean of the function, and σ2 is the variance of the function. we will
discuss both these terms later.
This operator is very useful, and we can use it to derive the moments of the random
variable.
35.2 Moments
A moment is a value that contains some information about the random variable. The n
moment of a random variable is defined as:
35.3 Mean
The mean value, or the "average value" of a random variable is defined as the first
moment of the random variable:
We will use the greek letter μ to denote the mean of a random variable.
35.4 Central Moments
A central moment is similar to a moment, but it is also dependant on the mean of the
random variable:
The first central moment is always zero.
30
35.5 Variance
The variance of a random variable is defined as the second central moment:
E[(x − μX)2] = σ2
36-SISO Transformations
Let's say that we have a random variable X that is the input into a given system. The
system output, Y is then also a random variable that is related to the input X by the
response of the system. In other words, we can say that:
Y = g(X)
Where g is the mathematical relationship between the system input and the system
output.
To discover information about Y, we can use the information we know about the r.v. X,
and the relationship g:
Where xi are the roots of g.
36.1 MISO Transformations
Consider now a system with two inputs, both of which are random (or pseudorandom,
in the case of non-deterministic data). For instance, let's consider a system with the
following inputs and outputs:
• X: non-deterministic data input
• Y: disruptive noise
• Z: System output
Our system satisfies the following mathematical relationship:
Z = g(X,Y)
Where g is the mathematical relationship between the system input, the disruptive
noise, and the system output. By knowing information about the distributions of X and
Y, we can determine the distribution of Z.
37-Optimization
Optimization is an important concept in engineering. Finding any solution to a
problem is not nearly as good as finding the one "optimal solution" to the problem.
Optimization problems are typically reformatted so they become minimization
problems, which are well-studied problems in the field of mathematics.
Typically, when optimizing a system, the costs and benefits of that system are
arranged into a cost function. It is the engineers job then to minimize this cost
function (and thereby minimize the cost of the system) Optimization typically becomes
a mathematical minimization problem.
37.1 Minimization
Minimization is the act of finding the numerically lowest point in a given function, or in
a particular range of a given function.
If we have a function f(x), we can find the maxima, minima, or saddle-points (points
where the function has zero slope, but is not a maxima or minima) by solving for x in
the following equation:
31
we can test those points to see if they are relatively high (maxima), or relatively low
(minima). Some other words to remember are:
Global Minima: A global minima of a function is the lowest value of that function
anywhere.
Local Minima: A local minima of a function is the lowest value of that function within a
given range A < x < B. If the function derivative has no roots in that range, then the
minima occurs at either A, or B.
37.2 Unconstrained Minimization
Unconstrained Minimization refers to the minimization of the given function without
having to worry about any other rules or caveats. On the other hand, refers to
minimization problems where there are other factors or constraints that must be
satisfied.
37.3 Hessian Matrix
The function has a local minima at a point x if the Hessian matrix H(x) is positive
definite:
Where x is a vector of all the independant variables of the function. If x is a scalar
variable, the hessian matrix reduces to the second derivative of the function f.
37.4 Newton-Raphson Method
The Newton-Raphson Method of computing the minima of a function, f uses an
iterative computation. We can define the scheme:
Where
The above equation, plugging in consecutive values for n, our solution will
converge on the true solution. However, this process will take infinitely many
iterations to converge, so oftentimes an approximation of the true solution will suffice.
37.5 Steepest Descent Method
The Newton-Raphson method can be tricky because it relies on the second derivative
of the function f, and this can oftentimes be difficult (if not impossible) to accurately
calculate. The Steepest Descent Method, however, does not require the second
derivative, but it does require the selection of an appropriate scalar quantity ε, which
cannot be chosen arbitrarily (but which can also not be calculated using a set formula).
37.6 Constrained Minimization
Constrained Minimization' is the process of finding the minimum value of a function
32
under a certain number of additional rules or constraints.
If we say "Find the minium value of f(x), but g(x) must equal 10". These kinds of
problems are difficult, but fortunately we can utilize the Khun-Tucker theorem, and
also the Karush=Khun-Tucker theorem to solve for them.
There are two different types of constraints: equality constraints and inequality
constraints. We will consider them individually, and then we will consider them
together.
37.7 Equality Constraints
The Khun-Tucker Theorem is a method for minimizing a function f(x) under the
equality constraint g(x). We can define the theorem as follows:
If we have a function f, and an equality constraint g in the following form:
g(x) = 0,
Then we can convert this problem into an unconstrained minimization problem by
constructing the Lagrangian function of f and g:
Where Λ is the lagrangian vector, and < , > denotes the scalar product operation of the
Rn vector space (where n is the number of equality constraints).
37.8 Inequality Constraints
Similar to the method above, let's say that we have a cost function f, and an inequality
constraint in the following form:
Then we can take the Lagrangian of this again:
37.9 Equality and Inequality Constraints
If we have a set of inequality and equality constraints:
g(x) = 0
38-Infinite Dimension Minimization
The above methods work well if the variables involved in the analysis are finitedimensional vectors, especially those in the RN space.
If we consider the L2 space, we have an infinite-dimensional space where the members
of that space are all functions. We will define the term functional as follows:
Functional A functional is a function that takes one or more functions as arguments,
and which returns a scalar value.
Let's say that we have a function x of time t. We can define the functional f as:
f(x(t))
With that function, we can associate a cost function J:
Where we are explicitly taking account of t in the definition of f
To minimize this function, like all minimization problems, we need to take the
derivative of the function, and set the derivative to zero. However, we are not able to
take a standard derivative of J with respect to x, because x is a function that varies with
time. However, we can define a new type of derivative, the Gateaux Derivative that
can handle this special case.
38.1 Gateaux Derivative
33
We can define the Gateaux Derivative in terms of the following limit:
Which is similar to the classical definition of the derivative, except with the inclusion of
the term ε.
38.2 Euler-Lagrange Equation
The Euler-Lagrange Equation uses the Gateaux derivative, discussed above, to find the
minimization of the following types of function:
the solutions to this problem:
δJ(x) = 0
And the solution is:
The partial derivatives can be done in an ordinary way ignoring the fact that x is a
function of t. Solutions to this equation are either the maxima or minima of the cost
function J.
Example: Shortest Distance
We've heard colloquially that the shortest distance between two points is a straight
line.
If we have two points in R2 space, a, and b, we would like to find the minimum function
that joins these two points.
We can define the differential ds as the differential along the function that joins points
a and b:
Our function that we are trying to minimize then is defined as:
or:
39-Normal Distribution
The normal distibution is an extremely important family of continuous probability
distributions. It has applications in every engineering discipline. Each member of the
family may be defined by two parameters, location and scale: the mean ("average", μ)
and variance (standard deviation squared, σ2) respectively.
The probability density function, of the distribution is given by:
The cumulative distibution function, or cdf, of the normal distribution is:
34
Given a normal distibution ,
distribution, Z, is:
the standardised normal
Due to this relationship, all tables refer to the standardised distibution, Z.
39.1-Probability Content from –∞ to Z (Z≤0)
Table of Probability Content between –∞ and z in the
Standardised Normal Distribution Z~N(0,1) for z≤0
Z
0.0
0.02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
40-Systems Engineering Modeling
and Design
System theories, analysis and design have been deployed within every corporate
function and within a broad section of businesses and markets. In systems thinking,
analysis and design we look for interrelationships among the elements of a system.
The chapter focuses on: the real-world goals for, services provided by, and constraints
on systems; the precise specification of system structure and behavior, and the
implementation of specifications; the activities required in order to develop an
assurance that the specifications and real-world goals It is also concerned with the
processes, methods and tools for the development of systems in an economic and
timely manner.
A system is usually made up of many smaller systems, or subsystems. For example, an
organization is made up of many administrative and management functions, products,
services, groups and individuals.
40.1 Systems Engineering
A management technology involving the interactions of science, an organization, and
its environment as well as the information and knowledge bases that support each.
Systems engineering has triple bases: a physical (natural) science basis, an
organizational and social science basis, and an information science and knowledge
basis.
35
The organizational and social science basis involves human, behavioral, economic, and
enterprise concerns.
40.2 The Scope of System
Engineering Activities
One way to understand the motivation behind systems engineering is to see it as a method, or
practice, to identify and improve common rules that exist within a wide variety of systems.
At times a systems engineer must assess the existence of feasible solutions, and rarely will
customer inputs arrive at only one. Various modeling methods can be used to solve the
problem including constraints and a cost function.
Systems engineering encourages the use of modeling and simulation to validate assumptions
or theories on systems and the interactions within them.
41- INTRODUCTION TO SYSTEM
DYNAMICS
System dynamics is a computer-based simulation modeling methodology developed at
the Massachusetts Institute of Technology (MIT) in the 1950s as a tool for managers to
analyze complex problems.
Using system dynamics simulations allows us to see not just events, but also patterns
of behaviour over time. Sometimes the simulation looks backward, to historical results.
System dynamics simulations are good at communicating not just what might happen,
but also why. This is because system dynamics simulations are designed to correspond
to what is, or might be happening, in the real world.
41.1 System Dynamics as Simulation
Modeling
System dynamics is a subset of the field of simulation modeling. Simulation modeling is
widely practiced in many traditional disciplines such as engineering, economics, and
ecology.
The concept of simulating a system is too general and unstructured to be in itself a
paradigm that helps one organize questions and observations about the world.
System dynamics, however, includes not only the basic idea of simulation, but also a
set of concepts, representational techniques, and beliefs that make it into a definite
modeling paradigm. It shapes the world view of its practitioners.
41.2 System Simulation Analysis and
Design
System development can generally be thought of having two major components:
systems simulation & analysis and systems design. In System simulation & analysis
more emphasis is given to understanding the details of an existing system or a
proposed one and then deciding whether the proposed system is desirable or not and
whether the existing system needs improvements.
System analysis is the process of investigating a system, identifying problems, and
using the information to recommend improvements to the system.
System design is the process of planning a new business system or one to replace or
complement an existing system. Analysis specifies what the system should do. Design
states how to accomplish the objective. After the proposed system is analyzed and
designed, the actual implementation of the system occurs. After implementation,
working system is available and it requires timely maintenance.
36
The various stages involved in building an improved system
A system dynamicist is likely to look for explanations of recurring long-term social problems
within this internal structure rather than in external disturbances, small
maladjustments, or random events.
The central concept that system dynamicists use to understand system structure is the
idea of two-way causation or feedback. System dynamics models are made up of many
such loops linked together. When some factor is believed to influence the system from
the outside without being influenced itself, however, it is represented as an exogenous
variable in the model.
41.3 Solving Problems
When confronted with problems or new situations, we can react to them in several
possible ways. The most common approach to new problems and situations is to take
them apart and examine their pieces.
If a company is experiencing a serious threat to its survival, be it declining market
shares or disagreements with the labor union, resources are mobilized to deal with the
problem. Problems affecting the entire company are often blamed on a department,
as when a loss in market shares causes executives to target the sales department for
investigation or punishment. The reason for the problem might seem obvious.
A corporation is an example of such a system, composed of many departments that in
turn act as systems themselves. When we study the parts and the interactions
between them, we in fact study the entire system. Systems theory suggests that
knowledge of one type of system can be applied to many other types.
Systems theory expands further to include two major fields in management science:
systems thinking and system dynamics.
41.4 Systems Thinking and System
Dynamics
The ideas we have presented thus far are important in both systems thinking and
system dynamics. Seeing the interrelationships can also help us find leverage points
within a system (places where a slight change will have a tremendous effect on the
system’s behaviour). System thinking is powerful because it helps us to see our own
mental models and howthese models color our perception of the world. Problems can
occur, when a rigid mental model stands in the way of a solution that might solve a
problem. In such situations, adherence to mental models can be dangerous to the
37
health of the organization. Our minds do not contain real economic or social systems.
If everyone’s mental models are brought to light in the context of an organization, we
can begin to see where, how, and why the models diverge. As long as mental models
remain hidden, they constitute na obstacle to building shared understanding.
System dynamics is capable of creating a learning environment - a laboratory that acts
like the system in miniature. Even if building a learning organization - an organization
with a high degree of shared understanding and knowledge about how the
organization works - isn’t the goal, systems thinking can be a very valuable tool at the
outset of a system dynamics study.
Systems thinking, by helping people in an organization see what the problems are and
how their mental models contribute to the problems, set the stage for a successful
system dynamics study.
When we conduct a systems thinking or system dynamics study, we must base it on
existing information. The information we can use exists on several levels.
The human mind is a brilliant storage device, but we do have trouble relating cause
and effect, especially when they are not close in time. A systems thinking study usually
produces causal-loop diagrams to map the feedback structure of a system, and generic
structures to illustrate common behaviour. System dynamics takes the information
about a system’s structure that normally remains hidden in mental models and
formalizes it into a computer model.
41.5 The Tools and Rules of System
Dynamics
System dynamics simulations are based on the principle of cause and effect, feedback,
and delay. More sophisticated simulations will use all three to produce the kind of
behaviour we encounter in the real world.
41.6 Cause and Effect
Cause and effect is a simple idea, but some simulations based on methodologies other
than system dynamics don’t use it. The idea is that actions and decisions have
consequences. Price affects sales. The human mind is good at developing intuition
around complex problems, but poor at keeping track of dozens, hundreds, or even
thousands of interconnections and cause and effect relationships.
A simple causal-loop diagram illustrating
connections between price, sales, and
unit cost
When births increase, so does the population. This is a situation where a change leads to a
“change in the same direction”. It is shown by marking “s” or “+” on the arrow in the diagram.
The Figure shows a simple causal-loop diagram. In this diagram, which we will discuss closer in
38
the next section, price has a negative effect on sales, which in turn has a negative effect on
unit costs, which in turn has a positive effect on price.
41.7 Feedback
Feedback is a concept that most people associate with microphones and speakers. A
microphone that isn’t properly set up will pick up the sound coming from its own
speaker. This sound gets amplified further by the speaker and picked up by the
microphone again. This process keeps going until the speaker is producing the loudest
sound it can or the microphone cannot pick up any louder sound. If the microphone
and the speaker were set up correctly, the system would work linearly. The loudness of
the sound going into the microphone would only affect the loudness of the sound
coming out of the speaker. Because of the misplacement ofthe icrophone, however,
the loudness of soundcoming out of the speaker also affects the loudnessof sound
going into the microphone. Cause andeffect feed back on each other. This is the
general principle of feedback.
Epidemic is another example. Viruses spread when a member of an infected
population comes into contact with someone, who is uninfected, but susceptible. This
person then becomes part of the infected population, and can spread the virus to
others. The larger the infected population, the more contacts, the larger the infected
population.
41.8 Building Blocks in System
Simulation tools is a modeling environment based on the science of system dynamics.
Simulation tool allows us to model systems - with all their cause and effect
relationships, feedback loops, and delays - in an intuitive graphical manner. Symbols
representing levels, flows, and “helper” variables (so called auxiliaries) are used to
create graphical representations of the system in constructor diagrams. Flows and
information links representr elationships and interconnections
.
42-LEVELS AND FLOWS
Four common behaviours created by
Integrating a function measures the area
underneath the function
various feedback loops
In a system dynamics model, the structure of the system is represented
mathematically. A levelis the accumulation (or integration) of the flows that causes the
39
level to change. When creating a simulation model graphically in Simulation tool,
connecting the variable symbols generates the integral (flow) equations. Every variable
in the model is defined by an equation, in the same way as cells in a spreadsheet are
defined.
In Simulation tool, boxes represent levels. Double arrows represent the flows, and that
is controlled by a flow rate. The flow rate is defined in the same way as auxiliaries.
The cloud-like symbol to the left of the first flows and to the right of the second flow
represents source and sink of the structure, respectively. The cloud symbol indicates
infinity and marks the boundary of the model.
The model with extended model boundaries
43-AUXILIARIES
While it is possible to create an entire model with only levels and flows, Simulation
tool has a few more tools to help us to capture real-world phenomena in a model. In
Simulation tool, a circle represents auxiliaries, as shown in Figure. An auxiliary is used
to combine or reformulate information. It has no standard form; it is an algebraic
computation of any combination of levels, flow rates, or other auxiliaries. Auxiliaries
are used to model information, not the physical flow of goods, so they change with no
delay, instantaneously.
43.1 CONSTANTS
Constants are, unlike ordinary auxiliaries, constant over the time period of the
simulation.
A constant is defined by an initial value, and maintains this value throughout the
simulation, unless the user changes the value manually (by using a slider bar, for
example).
If the simulation were to expand to 20 years, however, workforce would most likely
become a level and be allowed to vary over time. We should think of the time period
of the problematic behaviour and whether or not it is reasonable to expect the
element to change over that period. We will then be in a better position to decide
what elements should be constants and what elements should be allowed to vary
during the simulation.
43.2 INFORMATION LINKS
Connections are made among constants, auxiliaries, and levels by means of
information links.
40
Information links connects various variables
Information links show how the individual elements of the system are put together. In
a sense they close the feedback loops.
Information links show how the individual Information links can transfer the value of
the level back to the flow, indicating a dependence of the flow on the level, as well as
the obvious dependence of the level on the flow.
43.3 Decisions and Policies
Many people intuitively understand the difference between decisions and policies. The
system of a swinging pendulum can be described in terms of its “decisions” in the face
of governing policies. Within corporations, the distinction between the two is
extremely important.
When decisions turn out to be wrong, they are often blamed for misinterpreting the
data. Sometimes the conclusion is that the manager simply didn’t have enough
information to make the correct choice. Unfortunately, the actual problem is usually
much deeper. Therefore, the only real changes we can make to a system are changes
to the structure. Other changes to the system will soon be canceled out through the
actions of negative feedback loops.
It is a better idea to examine the structure of the organization. This way we can gain
knowledge and insight about the policies of the company; the rules of the
organization, spoken or unspoken, implicit or explicit, that provides the setting for
decisions.
43.4 Decision-Making Process
Decisions must always be based on observable variables. In a system dynamics model,
this means that decisions must be based entirely upon levels, as flows are never
instantaneously observable and therefore can never affect instantaneous decisionmaking. The first attempt to solve a complex problem rarely succeeds. Usually,
corrections change the system and lead to a total redefinition of the problem.
Decisions are attempts to move the system toward our goals. This is an iterative
process. In the context of a corporate model, decisions could be how many orders to
submit to the supplier to replace inventory, how many workers to hire, or when to
replace capital equipment. A decision to replenish inventory should be based on the
present level of inventory (a level) and not on the rate of sales (a flow). Decisions are
governed by policies. Therefore, the way decisions control change is through policies.
Flows are defined by equations, and these equations are statements of system policy.
Policies describe how and why decisions are made.
Policies may be informal, such as a consequence of habit, intuition, personal interest,
and social pressures and power within the organization. Informal policies can be hazy,
but the system dynamics model attempts to make them explicit. In such a model,
informal policies are treated with as much concern as explicit policies.
41
43.5 Problem Definition
The modeling process begins with defining a problem. The problem definition is the
keystone of the entire activity. Although it might sound like the easiest part, it is not
enough to have a vague notion about the problem behaviour. Defining the problem is
essentially defining the purpose of the model. The problem should therefore be
defined as precisely as possible.
43.6 Identification of Variables
The problem definition helps us to structure our information, and to start generating
names and units of measurement for variables. The list of variables usually becomes
very long. From this list, we should identify primary system variables. We can throw
out the variables that are irrelevant to the purpose of the model and set aside the
variables that we are not sure of. The latter ones might become helpful later, when we
arrive at the stage of model design.
44-Software Marketing Research and
Analysis
Traditionally, software marketing analysis was structured into three areas: More
recently, it has become fashionable in some software marketing circles to divide these
further into five “Cs”: Customer analysis, Company analysis, Collaborator analysis,
Competitor analysis, and analysis of the industry Context.
The focus of customer analysis is to develop a scheme for market segmentation which
are called customer segments or market segments. Marketers also attempt to track
these segments’ perceptions of the various products in the market using tools such as
perceptual mapping. Software marketing management employs various tools from
economics and competitive strategy to analyze the industry context in which the firm
operates. These include five forces analysis of strategic groups of competitors, value
chain analysis and others. Depending on the industry, the regulatory context may also
be important to examine in detail using SWOT analysis. Software marketing managers
are able to make key strategic decisions and develop a software marketing strategy
designed to maximize the revenues and profits of the firm.
To achieve the desired objectives, marketers typically identify one or more target
customer segments which they intend to pursue. The company has the resources and
capabilities to compete for the segment’s business, can meet their needs better than
the competition, and can do so profitably.
44.1 Software Marketing Strategy and
Implement Planning
Once the company has obtained an adequate understanding of the customer base and
its own competitive position in the industry, software marketing managers are able to
make key strategic decisions and develop a software marketing strategy designed to
maximize the revenues and profits of the firm.
After the firm’s strategic objectives have been identified, the target market selected,
and the desired positioning for the company, product or brand has been determined,
software marketing managers focus on how to best implement the chosen strategy.
42
System Engineering modeling and design is a technique that aims to allow
understanding and modeling of complex systems. The models provide a holistic view of
the system.
This is done by showing causal relationships between different elements of the system
graphically, and describing the nature of the relationship through equations. Another
key element of the system dynamics approach is the time evolutionary view. This
allows the representation of the behavior of the system as it evolves through time,
giving a dynamic rather than a static view of the system.
43
45-GENERAL ANALYSIS
Civil engineering technology is one of the fastest growing areas and technologies now
come with this growth.
Since man came out of the caves and was looking for a safer and more comfortably,
the need to improve construction techniques have become increasingly larger.
The great transformations in the use of materials, techniques in the construction of
buildings followed the needs of modern man, with the demands of a new way of life.
New technologies emerge at all times, replacing old ways of building. However, when
it comes to restoration of cultural and historical knowledge of specific techniques of
past periods is essential to guarantee their preservation without distortion. Therefore,
courses are offered to engineers and architects in the intention that these not-somodern techniques, help in the restoration of numerous estates.
The civil engineer plans, designs, implements, supervises and monitors the work
related to the construction and maintenance of buildings, bridges, dams, railways,
roads, waterways, tunnels, airports, ports etc. The civil engineer is responsible for the
calculations for the structures of the work referred to above.
The various issues raised in this paper reflect precisely the range of knowledge needed
to fully exercise of civil engineering, and construction technologies serving as pillars of
innovation in this area.
Talk about civil engineering forces us to step back in time to understand the various
steps by which the man mainly in the improvement of habitat being the need to live in
the rooms more comfortable and decent one of the root causes of the emergence of
technical innovations of construction not only shelter but also on the improvement of
roads, construction of structures, bridges etc. These innovations were accompanied by
the evolution of concepts and calculations that extends to the present day.
Innovations in civil engineering can be found in developed countries where engineers
and architects on a combination of knowledge and sensitivities, have developed and
have been developing projects unstoppable and stunning solutions, able to attract the
attention of any visitor.
Innovations in civil engineering are reflected in the four corners of the world, each one
acting according to their sensitivity, knowledge and wealth, which arises through
fantastic projection and traits that are United by the universal laws of mathematics,
physics and chemistry.
In my country (Cape Verde) and which is open to the world, the buildings suffer
influences from universal traits, because we are a country of emigrants and carry
normally the influences of the countries where they live. However, in rural areas still
note the presence of buildings with traces very old and primitive, and in certain cases
the weak financial ability to sustain the existence of these dwellings, although
currently on grounds of interest the architects and engineers are designing buildings
with these characteristics in order to captivate the attention of visitors.
Of all, the civil engineering is the guarantee of a constant innovation and that drags
along other technologies aimed at comfort safety and much more, through
comprehensive and calculations that influence the construction and development in
their various fields.
44
46-CONCLUSION
In conclusion, in general civil engineering technology, I would say that it is on
innovation and this last is part of the strategy of development of the sector.
Unlike an image often conservative, the construction sector has been implementing
innovations of various nature that, in some cases, represent only incremental changes
and in other cases radical changes of product, processes, organizational or marketing.
There are two strands of character macro to the issue of innovation in construction:
1-the development and advancement of international and human needs in its socioeconomic context;
2-absorption and creation of call for innovative products and systems.
Make innovations to spread creating scale, and progressive loss of unit costs is crucial
for innovation to be viable in the industry.
The restrictive factors to be faced, singled out at this stage of the work require that the
focus is on the production agents, taking into account the reality of the companies and
their operation.
The innovation policy must be drawn with strategies for their growth and development
of companies toward greater competitiveness and innovation investment conditions.
You must make a large part of the entire production chain have access to innovation
and are innovative, condition that involves the generation of conditions for the
satisfactory performance of these companies, the integration between players of the
entire chain around innovation, integration with public agents that intervene in
construction activity and with research and development that occurs at universities
and research institutes.
The importance of Civil Engineering is so large that it becomes virtually impossible to
think of the world without your presence. But, in an exercise of imagination we could
create a city without your intervention, it certainly would be a jumble of shacks, no
communication, energy or water and sewer system. The chaos, anyway. The civil
engineer is, by far, the most important professional when it comes to structure. With
his knowledge, chooses the most appropriate places for a construction, solidity and
safety checks of the terrain and of the material used in the work, monitors the
progress of the project and also the operation and maintenance of water supply and
sewerage distribution.
ENGINEERING is to improve the living conditions of mankind, always preserving the
lives, health and property, promoting culture and quality of life, meet the needs of the
present and try to anticipate the development regarding the future.
45
BIBLIOGRAPHY
Civil Engineering Technology
By: Gray, Kevin, Edition 1st ed. Global Media 2007 and the Bibliography refered in the
book.
Systems Engineering Modeling and Design.
Whole System Design: The Integrated Approaches to Sustainable Engineering
By: Stasinopoulos, Peter, Earthsean L. 2009 and the Bibliography referered in the book
Forrester, J. W. Industrial Dynamics: A Major Breakthrough for Decision Makers.
Harvard Business Review, 38(4), 37-66.
Forrester, J. W. Industrial Dynamics. Pegasus Communications, Waltham, MA.
Wolstenholme, E. F. System Enquiry: a System Dynamics Approach. John Wiley & Sons,
New York.
Adriano Rodrigues
2013-03-30