# Control Systems/Introduction

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## What are Control Systems?

The study and design of automatic **Control Systems**, a field known as
**control engineering,** has become important in modern technical
society. From devices as simple as a toaster or a toilet, to complex
machines like space shuttles and power steering, control engineering is
a part of our everyday life. This book introduces the field of control
engineering and explores some of the more advanced topics in the field.
Note, however, that control engineering is a very large field and this
book serves only as a foundation of control engineering and an
introduction to selected advanced topics in the field. Topics in this
book are added at the discretion of the authors and represent the
available expertise of our contributors.

Control systems are components that are added to other components to
increase functionality or meet a set of design criteria. For example:


This simple example can be complex to both users and designers of the
motor system. It may seem obvious that the motor should start at a
higher voltage so that it accelerates faster. Then we can reduce the
supply back down to 10 volts once it reaches ideal speed.

This is clearly a simplistic example but it illustrates an important
point: we can add special \"Controller units\" to preexisting systems to
improve performance and meet new system specifications.

Here are some formal definitions of terms used throughout this book:

There are essentially two methods to approach the problem of designing a
new control system: the **Classical Approach** and the **Modern
Approach**.

## Classical and Modern

**Classical** and **Modern** control methodologies are named in a
misleading way, because the group of techniques called \"Classical\"
were actually developed later than the techniques labeled \"Modern\".
However, in terms of developing control systems, Modern methods have
been used to great effect more recently, while the Classical methods
have been gradually falling out of favor. Most recently, it has been
shown that Classical and Modern methods can be combined to highlight
their respective strengths and weaknesses.

Classical Methods, which this book will consider first, are methods
involving the **Laplace Transform domain**. Physical systems are modeled
in the so-called \"time domain\", where the response of a given system
is a function of the various inputs, the previous system values, and
time. As time progresses, the state of the system and its response
change. However, time-domain models for systems are frequently modeled
using high-order differential equations which can become impossibly
difficult for humans to solve and some of which can even become
impossible for modern computer systems to solve efficiently. To
counteract this problem, integral transforms, such as the **Laplace
Transform** and the **Fourier Transform**, can be employed to change an
Ordinary Differential Equation (ODE) in the time domain into a regular
algebraic polynomial in the transform domain. Once a given system has
been converted into the transform domain it can be manipulated with
greater ease and analyzed quickly by humans and computers alike.

Modern Control Methods, instead of changing domains to avoid the
complexities of time-domain ODE mathematics, converts the differential
equations into a system of lower-order time domain equations called
**State Equations**, which can then be manipulated using techniques from
linear algebra. This book will consider Modern Methods second.

A third distinction that is frequently made in the realm of control
systems is to divide analog methods (classical and modern, described
above) from digital methods. Digital Control Methods were designed to
try and incorporate the emerging power of computer systems into previous
control methodologies. A special transform, known as the
**Z-Transform**, was developed that can adequately describe digital
systems, but at the same time can be converted (with some effort) into
the Laplace domain. Once in the Laplace domain, the digital system can
be manipulated and analyzed in a very similar manner to Classical analog
systems. For this reason, this book will not make a hard and fast
distinction between Analog and Digital systems, and instead will attempt
to study both paradigms in parallel.

## Who is This Book For?

This book is intended to accompany a course of study in under-graduate
and graduate engineering. As has been mentioned previously, this book is
not focused on any particular discipline within engineering, however any
person who wants to make use of this material should have some basic
background in the Laplace transform (if not other transforms), calculus,
etc. The material in this book may be used to accompany several
semesters of study, depending on the program of your particular college
or university. The study of control systems is generally a topic that is
reserved for students in their 3rd or 4th year of a 4 year undergraduate
program, because it requires so much previous information. Some of the
more advanced topics may not be covered until later in a graduate
program.

Many colleges and universities only offer one or two classes
specifically about control systems at the undergraduate level. Some
universities, however, do offer more than that, depending on how the
material is broken up, and how much depth that is to be covered. Also,
many institutions will offer a handful of graduate-level courses on the
subject. This book will attempt to cover the topic of control systems
from both a graduate and undergraduate level, with the advanced topics
built on the basic topics in a way that is intuitive. As such, students
should be able to begin reading this book in any place that seems an
appropriate starting point, and should be able to finish reading where
further information is no longer needed.

## What are the Prerequisites?

Understanding of the material in this book will require a solid
mathematical foundation. This book does not currently explain, nor will
it ever try to fully explain most of the necessary mathematical tools
used in this text. For that reason, the reader is expected to have read
the following wikibooks, or have background knowledge comparable to
them:

Algebra\
Calculus
:   The reader should have a good understanding of differentiation and
    integration. Partial differentiation, multiple integration, and
    functions of multiple variables will be used occasionally, but the
    students are not necessarily required to know those subjects well.
    These advanced calculus topics could better be treated as a
    co-requisite instead of a pre-requisite.

Linear Algebra
:   State-space system representation draws heavily on linear algebra
    techniques. Students should know how to operate on matrices.
    Students should understand basic matrix operations (addition,
    multiplication, determinant, inverse, transpose). Students would
    also benefit from a prior understanding of Eigenvalues and
    Eigenvectors, but those subjects are covered in this text.

Ordinary Differential Equations
:   All linear systems can be described by a linear ordinary
    differential equation. It is beneficial, therefore, for students to
    understand these equations. Much of this book describes methods to
    analyze these equations. Students should know what a differential
    equation is, and they should also know how to find the general
    solutions of first and second order ODEs.

Engineering Analysis
:   This book reinforces many of the advanced mathematical concepts used
    in the Engineering Analysis book,
    and we will refer to the relevant sections in the aforementioned
    text for further information on some subjects. This is essentially a
    math book, but with a focus on various engineering applications. It
    relies on a previous knowledge of the other math books in this list.

Signals and Systems
:   The Signals and Systems book will
    provide a basis in the field of **systems theory**, of which control
    systems is a subset. Readers who have not read the Signals and
    Systems book will be at a severe
    disadvantage when reading this book.

## How is this Book Organized?

This book will be organized following a particular progression. First
this book will discuss the basics of system theory, and it will offer a
brief refresher on integral transforms. Section 2 will contain a brief
primer on digital information, for students who are not necessarily
familiar with them. This is done so that digital and analog signals can
be considered in parallel throughout the rest of the book. Next, this
book will introduce the state-space method of system description and
control. After section 3, topics in the book will use state-space and
transform methods interchangeably (and occasionally simultaneously). It
is important, therefore, that these three chapters be well read and
understood before venturing into the later parts of the book.

After the \"basic\" sections of the book, we will delve into specific
methods of analyzing and designing control systems. First we will
discuss Laplace-domain stability analysis techniques (Routh-Hurwitz,
root-locus), and then frequency methods (Nyquist Criteria, Bode Plots).
After the classical methods are discussed, this book will then discuss
Modern methods of stability analysis. Finally, a number of advanced
topics will be touched upon, depending on the knowledge level of the
various contributors.

As the subject matter of this book expands, so too will the
prerequisites. For instance, when this book is expanded to cover
**nonlinear systems**, a basic background knowledge of nonlinear
mathematics will be required.

### Versions

This wikibook has been expanded to include multiple
versions of its text,
differentiated by the material covered, and the order in which the
material is presented. Each different version is composed of the
chapters of this book, included in a different order. This book covers a
wide range of information, so if you don\'t need all the information
that this book has to offer, perhaps one of the other versions would be
right for you and your educational needs.

Each separate version has a table of contents outlining the different
chapters that are included in that version. Also, each separate version
comes complete with a printable version, and some even come with PDF
versions as well.

Take a look at the **All Versions Listing
Page** to find the version of
the book that is right for you and your needs.

## Differential Equations Review

Implicit in the study of control systems is the underlying use of
differential equations. Even if they aren\'t visible on the surface, all
of the continuous-time systems that we will be looking at are described
in the time domain by ordinary differential equations (ODE), some of
which are relatively high-order.

Differential equations are particularly difficult to manipulate,
especially once we get to higher-orders of equations. Luckily, several
methods of abstraction have been created that allow us to work with
ODEs, but at the same time, not have to worry about the complexities of
them. The classical method, as described above, uses the Laplace,
Fourier, and Z Transforms to convert ODEs in the time domain into
polynomials in a complex domain. These complex polynomials are
significantly easier to solve than the ODE counterparts. The Modern
method instead breaks differential equations into systems of low-order
equations, and expresses this system in terms of matrices. It is a
common precept in ODE theory that an ODE of order N can be broken down
into N equations of order 1.

Readers who are unfamiliar with differential equations might be able to
read and understand the material in this book reasonably well. However,
all readers are encouraged to read the related sections in
**Calculus**.

## History

The field of control systems started essentially in the ancient world.
Early civilizations, notably the Greeks and the Arabs were heavily
preoccupied with the accurate measurement of time, the result of which
were several \"water clocks\" that were designed and implemented.

However, there was very little in the way of actual progress made in the
field of engineering until the beginning of the renaissance in Europe.
Leonhard Euler (for whom **Euler\'s Formula** is named) discovered a
powerful integral transform, but Pierre-Simon Laplace used the transform
(later called the **Laplace Transform**) to solve complex problems in
probability theory.

Joseph Fourier was a court mathematician in France under Napoleon I. He
created a special function decomposition called the **Fourier Series**,
that was later generalized into an integral transform, and named in his
honor (the **Fourier Transform**). {{-}}

+----------------------------------+----------------------------------+
| ![](Pierre-Simon-Laplace_(174    | ![](Joseph_Fourier.jpg "J        |
| 9-1827).jpg "Pierre-Simon-Laplac | oseph_Fourier.jpg"){width="150"} |
| e_(1749-1827).jpg"){width="150"} |                                  |
+----------------------------------+----------------------------------+
| Pierre-Simon Laplace\            | Joseph Fourier\                  |
| 1749-1827                        | 1768-1840                        |
+----------------------------------+----------------------------------+

The \"golden age\" of control engineering occurred between 1910-1945,
where mass communication methods were being created and two world wars
were being fought. During this period, some of the most famous names in
controls engineering were doing their work: Nyquist and Bode.

**Hendrik Wade Bode** and **Harry Nyquist**, especially in the 1930\'s
while working with Bell Laboratories, created the bulk of what we now
call \"Classical Control Methods\". These methods were based off the
results of the Laplace and Fourier Transforms, which had been previously
known, but were made popular by **Oliver Heaviside** around the turn of
the century. Previous to Heaviside, the transforms were not widely used,
nor respected mathematical tools.

Bode is credited with the \"discovery\" of the closed-loop feedback
system, and the logarithmic plotting technique that still bears his name
(**bode plots**). Harry Nyquist did extensive research in the field of
system stability and information theory. He created a powerful stability
criteria that has been named for him (**The Nyquist Criteria**).

Modern control methods were introduced in the early 1950\'s, as a way to
bypass some of the shortcomings of the classical methods. **Rudolf
Kalman** is famous for his work in modern control theory, and an
adaptive controller called the **Kalman Filter** was named in his honor.
Modern control methods became increasingly popular after 1957 with the
invention of the computer, and the start of the space program. Computers
created the need for digital control methodologies, and the space
program required the creation of some \"advanced\" control techniques,
such as \"optimal control\", \"robust control\", and \"nonlinear
control\". These last subjects, and several more, are still active areas
of study among research engineers. {{-}}

## Branches of Control Engineering

Here we are going to give a brief listing of the various different
methodologies within the sphere of control engineering. Oftentimes, the
lines between these methodologies are blurred, or even erased
completely.

Classical Controls:Control methodologies where the ODEs that describe a system are transformed using the Laplace, Fourier, or Z Transforms, and manipulated in the transform domain.\
Modern Controls:Methods where high-order differential equations are broken into a system of first-order equations. The input, output, and internal states of the system are described by vectors called \"state variables\".\
Robust Control:Control methodologies where arbitrary outside noise/disturbances are accounted for, as well as internal inaccuracies caused by the heat of the system itself, and the environment.\
Optimal Control:In a system, performance metrics are identified, and arranged into a \"cost function\". The cost function is minimized to create an operational system with the lowest cost.\
Adaptive Control:In adaptive control, the control changes its response characteristics over time to better control the system.\
Nonlinear Control:The youngest branch of control engineering, nonlinear control encompasses systems that cannot be described by linear equations or ODEs, and for which there is often very little supporting theory available.\
Game Theory:Game Theory is a close relative of control theory, and especially robust control and optimal control theories. In game theory, the external disturbances are not considered to be random noise processes, but instead are considered to be \"opponents\". Each player has a cost function that they attempt to minimize, and that their opponents attempt to maximize.

This book will definitely cover the first two branches, and will
hopefully be expanded to cover some of the later branches, if time
allows.

## MATLAB

**MATLAB** ® is a programming tool that is commonly used in the field of
control engineering. We will discuss MATLAB in specific sections of this
book devoted to that purpose. MATLAB will not appear in discussions
outside these specific sections, although MATLAB may be used in some
example problems. An overview of the use of MATLAB in control
engineering can be found in the **appendix** at: Control
Systems/MATLAB.

For more information on MATLAB in general, see: MATLAB
Programming.

Nearly all textbooks on the subject of control systems, linear systems,
and system analysis will use MATLAB as an integral part of the text.
Students who are learning this subject at an accredited university will
certainly have seen this material in their textbooks, and are likely to
have had MATLAB work as part of their classes. It is from this
perspective that the MATLAB appendix is written.

In the future, this book may be expanded to include information on
**Simulink** ®, as well as MATLAB.

There are a number of other software tools that are useful in the
analysis and design of control systems. Additional information can be
added in the appendix of this book, depending on the experience and
prior knowledge of contributors.

## About Formatting

This book will use some simple conventions throughout.

### Mathematical Conventions

Mathematical equations will be labeled with the 
template, to give them names. Equations that are labeled in such a
manner are important, and should be taken special note of. For instance,
notice the label to the right of this equation:


$$f(t) 
        = \mathcal{L}^{-1} \left\{F(s)\right\}
        = {1 \over {2\pi i}}\int_{c-i\infty}^{c+i\infty} e^{st} F(s)\,ds$$

Equations that are named in this manner will also be copied into the
List of Equations
Glossary in the end of
the book, for an easy reference.

Italics will be used for English variables, functions, and equations
that appear in the main text. For example *e*, *j*, *f(t)* and *X(s)*
are all italicized. Wikibooks contains a LaTeX mathematics formatting
engine, although an attempt will be made not to employ formatted
mathematical equations inline with other text because of the difference
in size and font. Greek letters, and other non-English characters will
not be italicized in the text unless they appear in the midst of
multiple variables which are italicized (as a convenience to the
editor).

Scalar time-domain functions and variables will be denoted with
lower-case letters, along with a *t* in parenthesis, such as: *x(t)*,
*y(t)*, and *h(t)*. Discrete-time functions will be written in a similar
manner, except with an *\[n\]* instead of a *(t)*.

Fourier, Laplace, Z, and Star transformed functions will be denoted with
capital letters followed by the appropriate variable in parenthesis. For
example: *F(s)*, *X(jω)*, *Y(z)*, and *F\*(s)*.

Matrices will be denoted with capital letters. Matrices which are
functions of time will be denoted with a capital letter followed by a
*t* in parenthesis. For example: *A(t)* is a matrix, *a(t)* is a scalar
function of time.

Transforms of time-variant matrices will be displayed in uppercase bold
letters, such as **H***(s)*.

Math equations rendered using LaTeX will appear on separate lines, and
will be indented from the rest of the text.

### Text Conventions

