02-07-2014, 02:08 PM
Control Systems
[attachment=65555]
Introduction
What are Control Systems?
The study and design of automatic Control Systems, a field known as control engineering, is a large and
expansive area of study. Control systems, and control engineering techniques have become a pervasive part of
modern technical society. From devices as simple as a toaster, to complex machines like space shuttles and
rockets, control engineering is a part of our everyday life. This book will introduce the field of control
engineering, and will build upon those foundations to explore some of the more advanced topics in the field. Note,
however, that control engineering is a very large field, and it would be foolhardy of any author to think that they
could include all the information into a single book. Therefore, we will be content here to provide the foundations
of control engineering, and then describe some of the more advanced topics in the field.
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 then the techniques labled "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 it's
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 and simply, 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 (matrices). This book will
consider Modern Methods second
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:
? Calculus
? Algebra
? Linear Algebra
? Differential Equations
? Engineering Analysis
The last book in the list, Engineering Analysis is especially recommended, because it analyzes a number of
mathematical topics from the perspective of engineering. However the subject matter in that book relies on the 4
previous books.
Also, an understanding of the material presented in the following wikibooks will be helpful, but is not required:
? Signals and Systems
The Signals and Systems book will provide a basis in the field of systems theory, of which control systems is a
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 interchangably (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 contributers
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 it's 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
MATLAB
field of control engineering. We will not consider MATLAB in the
main narrative of this book, but we will provide an appendix that
will show how MATLAB is used to solve control problems, and
design and model control systems. This appendix can be found 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
System Identification
Systems
We will begin our study by talking about systems. Systems, in the barest sense, are devices that take input, and
produce an output. The output is related to the input by a certain relation known as the system response. The
system response usually can be modeled with a mathematical relationship between the system input and the
system output.
There are many different types of systems, and the process of classifying systems in these ways is called system
identification
System Identification
Physical Systems can be divided up into a number of different catagories, depending on particular properties that
the system exhibits. Some of these system classifications are very easy to work with, and have a large theory base
for studying. Some system classifications are very complex, and have still not been investigated with any degree
of success. This book will focus primarily on linear time-invariant (LTI) systems. LTI systems are the easiest
class of system to work with, and have a number of properties that make them ideal to study. In this chapter, we
will discuss some properties of systems, and we will define exactly what an LTI system is
Digital and Analog
There is a significant distinction between an analog system and a digital system, in the same way that there is a
significant difference between analog and digital data. This book is going to consider both analog and digital
topics, so it is worth taking some time to discuss the differences, and to display the different notations that will be
used with each.
System Metrics
When a system is being designed and analyzed, it doesn't make any sense to test the system with all manner of
strange input functions, or to measure all sorts of arbitrary performance metrics. Instead, it is in everybody's best
interest to test the system with a set of standard, simple, reference functions. Once the system is tested with the
reference functions, there are a number of different metrics that we can use to determine the system performance.
It is worth noting that the metrics presented in this chapter represent only a small number of possible metrics that
can be used to evaluate a given system. This wikibook will present other useful metrics along the way, as their
need becomes apparent.
Analysis
Once a system is modeled using one of the representations listed above, the system needs to be analyszed. We can
determine the system metrics, and then we can compare those metrics to our specification. If our system meets the
specifications, you are finished (congratulations). If the system does not meet the specifications (as is typically the
case), then suitable controllers and compensators need to be designed and added to the system.
Once the controllers and compensators have been designed, the job isn't finished: we need to analyze the new
composite system to ensure that the controllers work properly. Also, we need to ensure that the systems are stable:
unstable systems can be dangerous.
Classical Controls
The classical method of controls involves
analysis and manipulation of systems in the
complex frequency domain. This domain,
entered into by applying the Laplace or
Fourier Transforms, is useful in examining
the characteristics of the system, and
determining the system response.
Transforms
There are a number of transforms that we will be discussing throughout this book, and the reader is assumed to
have at least a small prior knowledge of them. It is not the intention of this book to teach the topic of transforms
to an audience that has had no previous exposure to them. However, we will include a brief refresher here to
refamiliarize people who maybe cannot remember the topic perfectly. If you do not know what the Laplace
Transform or the Fourier Transform are yet, it is highly recommended that you use this page as a simple guide,
and look the information up on other sources. Specifically, Wikipedia has lots of information on these subjects
Stability
System stability is an important topic,
because unstable systems may not perform
correctly, and may actually be harmful to
people. There are a number of different
methods and tools that can be used to
determine system stability, depending on
whether you are in the state-space, or the
complex domain
BIBO Stability
When a system becomes unstable, the output of the system approaches infinity (or negative infinity), which often
poses a security problem for people in the immediate vicinity. Also, systems which become unstable often incur a
certain amount of physical damage, which can become costly. This chapter will talk about system stability, what
it is, and why it matters.
A system is defined to be BIBO Stable if every bounded input to the system results in a bounded output. This
means that so long as we don't input infinity to our system, we won't get infinity output.
State-Space and Stability
Determining whether a state-space system is stable is a little bit more tricky, but there are some tests that we can
perform to show whether a system is stable. There are methods that use the eigenvalues of the system matrix to
show whether the system is stable, and then there is the Lyapunov Method that determines whether a system
matrix is stable or not. We will learn about these methods in the upcoming chapters
Marginal Stablity
When the poles of the system in the complex S-Domain exist on the complex frequency axis (the horizontal axis),
the system exhibits oscillatory characteristics, and is said to be marginally stable. A marginally stable system may
become unstable under certain circumstances, and may be perfectly stable under other circumstances. It is
impossible to tell by inspection whether a marginally stable system will become unstable or not
Physical Models
This page will serve as a refresher for various different engineering disciplines on how physical devices are
modeled. Models will be displayed in both time-domain and Laplace-domain input/output characteristics. The
only information that is going to be displayed here will be the ones that are contributed by knowledgable
contributors.
Z Transform Mappings
There are a number of different mappings that can be used to convert a system from the complex Laplace domain
into the Z-Domain. None of these mappings are perfect, and every mapping requires a specific starting condition,
and focuses on a specific aspect to reproduce faithfully. One such mapping that has already been discussed is the
bilinear transform, which, along with prewarping, can faithfully map the various regions in the s-plane into the
corresponding regions in the z-plane. We will discuss some other potential mappings in this chapter, and we will
discuss the pros and cons of each
COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other documents released under this License, and
replace the individual copies of this License in the various documents with a single copy that is included in the
collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all
other respects.
You may extract a single document from such a collection, and distribute it individually under this License,
provided you insert a copy of this License into the extracted document, and follow this License in all other
respects regarding verbatim copying of that document
[attachment=65555]
Introduction
What are Control Systems?
The study and design of automatic Control Systems, a field known as control engineering, is a large and
expansive area of study. Control systems, and control engineering techniques have become a pervasive part of
modern technical society. From devices as simple as a toaster, to complex machines like space shuttles and
rockets, control engineering is a part of our everyday life. This book will introduce the field of control
engineering, and will build upon those foundations to explore some of the more advanced topics in the field. Note,
however, that control engineering is a very large field, and it would be foolhardy of any author to think that they
could include all the information into a single book. Therefore, we will be content here to provide the foundations
of control engineering, and then describe some of the more advanced topics in the field.
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 then the techniques labled "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 it's
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 and simply, 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 (matrices). This book will
consider Modern Methods second
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:
? Calculus
? Algebra
? Linear Algebra
? Differential Equations
? Engineering Analysis
The last book in the list, Engineering Analysis is especially recommended, because it analyzes a number of
mathematical topics from the perspective of engineering. However the subject matter in that book relies on the 4
previous books.
Also, an understanding of the material presented in the following wikibooks will be helpful, but is not required:
? Signals and Systems
The Signals and Systems book will provide a basis in the field of systems theory, of which control systems is a
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 interchangably (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 contributers
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 it's 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
MATLAB
field of control engineering. We will not consider MATLAB in the
main narrative of this book, but we will provide an appendix that
will show how MATLAB is used to solve control problems, and
design and model control systems. This appendix can be found 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
System Identification
Systems
We will begin our study by talking about systems. Systems, in the barest sense, are devices that take input, and
produce an output. The output is related to the input by a certain relation known as the system response. The
system response usually can be modeled with a mathematical relationship between the system input and the
system output.
There are many different types of systems, and the process of classifying systems in these ways is called system
identification
System Identification
Physical Systems can be divided up into a number of different catagories, depending on particular properties that
the system exhibits. Some of these system classifications are very easy to work with, and have a large theory base
for studying. Some system classifications are very complex, and have still not been investigated with any degree
of success. This book will focus primarily on linear time-invariant (LTI) systems. LTI systems are the easiest
class of system to work with, and have a number of properties that make them ideal to study. In this chapter, we
will discuss some properties of systems, and we will define exactly what an LTI system is
Digital and Analog
There is a significant distinction between an analog system and a digital system, in the same way that there is a
significant difference between analog and digital data. This book is going to consider both analog and digital
topics, so it is worth taking some time to discuss the differences, and to display the different notations that will be
used with each.
System Metrics
When a system is being designed and analyzed, it doesn't make any sense to test the system with all manner of
strange input functions, or to measure all sorts of arbitrary performance metrics. Instead, it is in everybody's best
interest to test the system with a set of standard, simple, reference functions. Once the system is tested with the
reference functions, there are a number of different metrics that we can use to determine the system performance.
It is worth noting that the metrics presented in this chapter represent only a small number of possible metrics that
can be used to evaluate a given system. This wikibook will present other useful metrics along the way, as their
need becomes apparent.
Analysis
Once a system is modeled using one of the representations listed above, the system needs to be analyszed. We can
determine the system metrics, and then we can compare those metrics to our specification. If our system meets the
specifications, you are finished (congratulations). If the system does not meet the specifications (as is typically the
case), then suitable controllers and compensators need to be designed and added to the system.
Once the controllers and compensators have been designed, the job isn't finished: we need to analyze the new
composite system to ensure that the controllers work properly. Also, we need to ensure that the systems are stable:
unstable systems can be dangerous.
Classical Controls
The classical method of controls involves
analysis and manipulation of systems in the
complex frequency domain. This domain,
entered into by applying the Laplace or
Fourier Transforms, is useful in examining
the characteristics of the system, and
determining the system response.
Transforms
There are a number of transforms that we will be discussing throughout this book, and the reader is assumed to
have at least a small prior knowledge of them. It is not the intention of this book to teach the topic of transforms
to an audience that has had no previous exposure to them. However, we will include a brief refresher here to
refamiliarize people who maybe cannot remember the topic perfectly. If you do not know what the Laplace
Transform or the Fourier Transform are yet, it is highly recommended that you use this page as a simple guide,
and look the information up on other sources. Specifically, Wikipedia has lots of information on these subjects
Stability
System stability is an important topic,
because unstable systems may not perform
correctly, and may actually be harmful to
people. There are a number of different
methods and tools that can be used to
determine system stability, depending on
whether you are in the state-space, or the
complex domain
BIBO Stability
When a system becomes unstable, the output of the system approaches infinity (or negative infinity), which often
poses a security problem for people in the immediate vicinity. Also, systems which become unstable often incur a
certain amount of physical damage, which can become costly. This chapter will talk about system stability, what
it is, and why it matters.
A system is defined to be BIBO Stable if every bounded input to the system results in a bounded output. This
means that so long as we don't input infinity to our system, we won't get infinity output.
State-Space and Stability
Determining whether a state-space system is stable is a little bit more tricky, but there are some tests that we can
perform to show whether a system is stable. There are methods that use the eigenvalues of the system matrix to
show whether the system is stable, and then there is the Lyapunov Method that determines whether a system
matrix is stable or not. We will learn about these methods in the upcoming chapters
Marginal Stablity
When the poles of the system in the complex S-Domain exist on the complex frequency axis (the horizontal axis),
the system exhibits oscillatory characteristics, and is said to be marginally stable. A marginally stable system may
become unstable under certain circumstances, and may be perfectly stable under other circumstances. It is
impossible to tell by inspection whether a marginally stable system will become unstable or not
Physical Models
This page will serve as a refresher for various different engineering disciplines on how physical devices are
modeled. Models will be displayed in both time-domain and Laplace-domain input/output characteristics. The
only information that is going to be displayed here will be the ones that are contributed by knowledgable
contributors.
Z Transform Mappings
There are a number of different mappings that can be used to convert a system from the complex Laplace domain
into the Z-Domain. None of these mappings are perfect, and every mapping requires a specific starting condition,
and focuses on a specific aspect to reproduce faithfully. One such mapping that has already been discussed is the
bilinear transform, which, along with prewarping, can faithfully map the various regions in the s-plane into the
corresponding regions in the z-plane. We will discuss some other potential mappings in this chapter, and we will
discuss the pros and cons of each
COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other documents released under this License, and
replace the individual copies of this License in the various documents with a single copy that is included in the
collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all
other respects.
You may extract a single document from such a collection, and distribute it individually under this License,
provided you insert a copy of this License into the extracted document, and follow this License in all other
respects regarding verbatim copying of that document