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The Wikibook of automatic
And Control Systems Engineering
With
Classical and Modern Techniques
And
Advanced Concepts
Introduction to Control Systems
What are control systems? Why do we study them? How do we identify them? The chapters in this section should answer these questions and more.
Introduction
This Wikibook
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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 as a foundation of control engineering and 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 to meet a set of design criteria. For example:
We have a particular electric motor that is supposed to turn at a rate of 40 RPM. To achieve this speed, we must supply 10 Volts to the motor terminals. However, with 10 volts supplied to the motor at rest, it takes 30 seconds for our motor to get up to speed. This is valuable time lost.
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:
 Control System
 A Control System is a device, or a collection of devices that manage the behavior of other devices. Some devices are not controllable. A control system is an interconnection of components connected or related in such a manner as to command, direct, or regulate itself or another system.
 Control System is a conceptual framework for designing systems with capabilities of regulation and/or tracking to give a desired performance. For this there must be a set of signals measurable to know the performance, another set of signals measurable to influence the evolution of the system in time and a third set which is not measurable but disturb the evolution.
 Controller
 A controller is a control system that manages the behavior of another device or system.
 Compensator
 A Compensator is a control system that regulates another system, usually by conditioning the input or the output to that system. Compensators are typically employed to correct a single design flaw, with the intention of affecting other aspects of the design in a minimal manner.
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 socalled "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, timedomain models for systems are frequently modeled using highorder 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 timedomain ODE mathematics, converts the differential equations into a system of lowerorder 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 ZTransform, 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 undergraduate 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 graduatelevel 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 corequisite instead of a prerequisite.
 Linear Algebra
 Statespace 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 statespace method of system description and control. After section 3, topics in the book will use statespace 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 Laplacedomain stability analysis techniques (RouthHurwitz, rootlocus), 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 continuoustime systems that we will be looking at are described in the time domain by ordinary differential equations (ODE), some of which are relatively highorder.
Let's review some differential equation basics. Consider the topic of interest from a bank. The amount of interest accrued on a given principal balance (the amount of money you put into the bank) P, is given by:
Where is the interest (rate of change of the principal), and r is the interest rate. Notice in this case that P is a function of time (t), and can be rewritten to reflect that:
To solve this basic, firstorder equation, we can use a technique called "separation of variables", where we move all instances of the letter P to one side, and all instances of t to the other:
And integrating both sides gives us:
This is all fine and good, but generally, we like to get rid of the logarithm, by raising both sides to a power of e:
Where we can separate out the constant as such:
D is a constant that represents the initial conditions of the system, in this case the starting principal.
Differential equations are particularly difficult to manipulate, especially once we get to higherorders 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 loworder 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 PierreSimon 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).
PierreSimon Laplace 17491827 
Joseph Fourier 17681840 
The "golden age" of control engineering occurred between 19101945, 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 closedloop 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 highorder differential equations are broken into a system of firstorder 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
the Appendix
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.
Resources
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 {{eqn}} 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:
[Inverse Laplace Transform]
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 nonEnglish 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 timedomain functions and variables will be denoted with lowercase letters, along with a t in parenthesis, such as: x(t), y(t), and h(t). Discretetime 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 timevariant 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
Examples will appear in TextBox templates, which show up as large grey boxes filled with text and equations.
 Important Definitions
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Notes of interest will appear in "infobox" templates. These notes will often be used to explain some nuances of a mathematical derivation or proof. 
Warnings will appear in these "warning" boxes. These boxes will point out common mistakes, or other items to be careful of. 
System Identification
Systems
Systems, in one sense, are devices that take input and produce an output. A system can be thought to operate on the input to produce the output. The output is related to the input by a certain relationship known as the system response. The system response usually can be modeled with a mathematical relationship between the system input and the system output.
System Properties
Physical systems can be divided up into a number of different categories, 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 analysis. Some system classifications are very complex and have still not been investigated with any degree of success. By properly identifying the properties of a system, certain analysis and design tools can be selected for use with the system.
The early sections of this book will focus primarily on linear timeinvariant (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. This chapter discusses some properties of systems.
Later chapters in this book will look at time variant systems and nonlinear systems. Both time variant and nonlinear systems are very complex areas of current research, and both can be difficult to analyze properly. Unfortunately, most physical realworld systems are timevariant, nonlinear, or both.
An introduction to system identification and least squares techniques can be found here. An introduction to parameter identification techniques can be found here.
Initial Time
The initial time of a system is the time before which there is no input. Typically, the initial time of a system is defined to be zero, which will simplify the analysis significantly. Some techniques, such as the Laplace Transform require that the initial time of the system be zero. The initial time of a system is typically denoted by t_{0}.
The value of any variable at the initial time t_{0} will be denoted with a 0 subscript. For instance, the value of variable x at time t_{0} is given by:
Likewise, any time t with a positive subscript are points in time after t_{0}, in ascending order:
So t_{1} occurs after t_{0}, and t_{2} occurs after both points. In a similar fashion above, a variable with a positive subscript (unless specifying an index into a vector) also occurs at that point in time:
This is valid for all points in time t.
Additivity
A system satisfies the property of additivity if a sum of inputs results in a sum of outputs. By definition: an input of results in an output of . To determine whether a system is additive, use the following test:
Given a system f that takes an input x and outputs a value y, assume two inputs (x_{1} and x_{2}) produce two outputs:
Now, create a composite input that is the sum of the previous inputs:
Then the system is additive if the following equation is true:
Systems that satisfy this property are called additive. Additive systems are useful because a sum of simple inputs can be used to analyze the system response to a more complex input.
Example: Sinusoids
Given the following equation:
Create a sum of inputs as:
and construct the expected sum of outputs:
Now, substituting these values into our equation, test for equality:
The equality is not satisfied, and therefore the sine operation is not additive.
Homogeneity
A system satisfies the condition of homogeneity if an input scaled by a certain factor produces an output scaled by that same factor. By definition: an input of results in an output of . In other words, to see if function f() is homogeneous, perform the following test:
Stimulate the system f with an arbitrary input x to produce an output y:
Now, create a second input x_{1}, scale it by a multiplicative factor C (C is an arbitrary constant value), and produce a corresponding output y_{1}:
Now, assign x to be equal to x_{1}:
Then, for the system to be homogeneous, the following equation must be true:
Systems that are homogeneous are useful in many applications, especially applications with gain or amplification.
Example: StraightLine
Given the equation for a straight line:
Comparing the two results, it is easy to see they are not equal:
Therefore, the equation is not homogeneous.
Exercise:
Prove that additivity implies homogeneity, but that homogeneity does not imply additivity.
Linearity
A system is considered linear if it satisfies the conditions of Additivity and Homogeneity. In short, a system is linear if the following is true:
Take two arbitrary inputs, and produce two arbitrary outputs:
Now, a linear combination of the inputs should produce a linear combination of the outputs:
This condition of additivity and homogeneity is called superposition. A system is linear if it satisfies the condition of superposition.
Example: Linear Differential Equations
Is the following equation linear:
To determine whether this system is linear, construct a new composite input:
Now, create the expected composite output:
Substituting the two into our original equation:
Factor out the derivative operator, as such:
Finally, convert the various composite terms into the respective variables, to prove that this system is linear:
For the record, derivatives and integrals are linear operators, and ordinary differential equations typically are linear equations.
Memory
A system is said to have memory if the output from the system is dependent on past inputs (or future inputs!) to the system. A system is called memoryless if the output is only dependent on the current input. Memoryless systems are easier to work with, but systems with memory are more common in digital signal processing applications.
Systems that have memory are called dynamic systems, and systems that do not have memory are static systems.
Causality
Causality is a property that is very similar to memory. A system is called causal if it is only dependent on past and/or current inputs. A system is called anticausal if the output of the system is dependent only on future inputs. A system is called noncausal if the output depends on past and/or current and future inputs.
A system design that is not causal cannot be physically implemented (to operate in a realtime). If the system can't be built, the design is generally worthless. However, there are applications of noncausal systems, e.g. when a system does not need to operate in a realtime and already has signals stored in its memory (sound and image compression). 
TimeInvariance
A system is called timeinvariant if the system relationship between the input and output signals is not dependent on the passage of time. If the input signal produces an output then any time shifted input, , results in a timeshifted output This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output. If a system is timeinvariant then the system block is commutative with an arbitrary delay. This facet of timeinvariant systems will be discussed later.
To determine if a system f is timeinvariant, perform the following test:
Apply an arbitrary input x to a system and produce an arbitrary output y:
Apply a second input x_{1} to the system, and produce a second output:
Now, assign x_{1} to be equal to the first input x, timeshifted by a given constant value δ:
Finally, a system is timeinvariant if y_{1} is equal to y shifted by the same value δ:
LTI Systems
A system is considered to be a Linear TimeInvariant (LTI) system if it satisfies the requirements of timeinvariance and linearity. LTI systems are one of the most important types of systems, and they will be considered almost exclusively in the beginning chapters of this book.
Systems which are not LTI are more common in practice, but are much more difficult to analyze.
Lumpedness
A system is said to be lumped if one of the two following conditions are satisfied:
 There are a finite number of states that the system can be in.
 There are a finite number of state variables.
The concept of "states" and "state variables" are relatively advanced, and they will be discussed in more detail in the discussion about modern controls.
Systems which are not lumped are called distributed. A simple example of a distributed system is a system with delay, that is, , which has an infinite number of state variables (Here we use to denote the Laplace variable). However, although distributed systems are quite common, they are very difficult to analyze in practice, and there are few tools available to work with such systems. Fortunately, in most cases, a delay can be sufficiently modeled with the Pade approximation. This book will not discuss distributed systems much.
Relaxed
A system is said to be relaxed if the system is causal, and at the initial time t_{0 } the output of the system is zero, i.e., there is no stored energy in the system.
In terms of differential equations, a relaxed system is said to have "zero initial state". Systems without an initial state are easier to work with, but systems that are not relaxed can frequently be modified to approximate relaxed systems.
Stability
Stability is a very important concept in systems, but it is also one of the hardest function properties to prove. There are several different criteria for system stability, but the most common requirement is that the system must produce a finite output when subjected to a finite input. For instance, if 5 volts is applied to the input terminals of a given circuit, it would be best if the circuit output didn't approach infinity, and the circuit itself didn't melt or explode. This type of stability is often known as "Bounded Input, Bounded Output" stability, or BIBO.
There are a number of other types of stability, most of which are based off the concept of BIBO stability. Because stability is such an important and complicated topic, an entire section of this text is devoted to its study.
Inputs and Outputs
Systems can also be categorized by the number of inputs and the number of outputs the system has. Consider a television as a system, for instance. The system has two inputs: the power wire and the signal cable. It has one output: the video display. A system with one input and one output is called singleinput, single output, or SISO. a system with multiple inputs and multiple outputs is called multiinput, multioutput, or MIMO.
These systems will be discussed in more detail later.
Exercise:
Based on the definitions of SISO and MIMO, above, determine what the acronyms SIMO and MISO mean.
Digital and Analog
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.
Continuous Time
A signal is called continuoustime if it is defined at every time t.
A system is a continuoustime system if it takes a continuoustime input signal, and outputs a continuoustime output signal. Here is an example of an analog waveform:
Discrete Time
A signal is called discretetime if it is only defined for particular points in time. A discretetime system takes discretetime input signals, and produces discretetime output signals. The following image shows the difference between an analog waveform and the sampled discrete time equivalent:
Quantized
A signal is called Quantized if it can only be certain values, and cannot be other values. This concept is best illustrated with examples:
 Students with a strong background in physics will recognize this concept as being the root word in "Quantum Mechanics". In quantum mechanics, it is known that energy comes only in discrete packets. An electron bound to an atom, for example, may occupy one of several discrete energy levels, but not intermediate levels.
 Another common example is population statistics. For instance, a common statistic is that a household in a particular country may have an average of "3.5 children", or some other fractional number. Actual households may have 3 children, or they may have 4 children, but no household has 3.5 children.
 People with a computer science background will recognize that integer variables are quantized because they can only hold certain integer values, not fractions or decimal points.
The last example concerning computers is the most relevant, because quantized systems are frequently computerbased. Systems that are implemented with computer software and hardware will typically be quantized.
Here is an example waveform of a quantized signal. Notice how the magnitude of the wave can only take certain values, and that creates a steplike appearance. This image is discrete in magnitude, but is continuous in time:
Analog
By definition:
 Analog
 A signal is considered analog if it is defined for all points in time and if it can take any real magnitude value within its range.
An analog system is a system that represents data using a direct conversion from one form to another. In other words, an analog system is a system that is continuous in both time and magnitude.
Example: Motor
If we have a given motor, we can show that the output of the motor (rotation in units of radians per second, for instance) is a function of the voltage that is input to the motor. We can show the relationship as such:
Where is the output in terms of Rad/sec, and f(v) is the motor's conversion function between the input voltage (v) and the output. For any value of v we can calculate out specifically what the rotational speed of the motor should be.
Example: Analog Clock
Consider a standard analog clock, which represents the passage of time though the angular position of the clock hands. We can denote the angular position of the hands of the clock with the system of equations:
Where φ_{h} is the angular position of the hour hand, φ_{m} is the angular position of the minute hand, and φ_{s} is the angular position of the second hand. The positions of all the different hands of the clock are dependent on functions of time.
Different positions on a clock face correspond directly to different times of the day.
Digital
Digital data is represented by discrete number values. By definition:
 Digital
 A signal or system is considered digital if it is both discretetime and quantized.
Digital data always have a certain granularity, and therefore there will almost always be an error associated with using such data, especially if we want to account for all real numbers. The tradeoff, of course, to using a digital system is that our powerful computers with our powerful, Moore's law microprocessor units, can be instructed to operate on digital data only. This benefit more than makes up for the shortcomings of a digital representation system.
Discrete systems will be denoted inside square brackets, as is a common notation in texts that deal with discrete values. For instance, we can denote a discrete data set of ascending numbers, starting at 1, with the following notation:
 x[n] = [1 2 3 4 5 6 ...]
n, or other letters from the central area of the alphabet (m, i, j, k, l, for instance) are commonly used to denote discrete time values. Analog, or "nondiscrete" values are denoted in regular expression syntax, using parenthesis. Here is an example of an analog waveform and the digital equivalent. Notice that the digital waveform is discrete in both time and magnitude:
Example: Digital Clock
As a common example, let's consider a digital clock: The digital clock represents time with binary electrical data signals of 1 and 0. The 1's are usually represented by a positive voltage, and a 0 is generally represented by zero voltage. Counting in binary, we can show that any given time can be represented by a base2 numbering system:
Minute Binary Representation 1 1 10 1010 30 11110 59 111011
But what happens if we want to display a fraction of a minute, or a fraction of a second? A typical digital clock has a certain amount of precision, and it cannot express fractional values smaller than that precision.
Hybrid Systems
Hybrid Systems are systems that have both analog and digital components. Devices called samplers are used to convert analog signals into digital signals, and Devices called reconstructors are used to convert digital signals into analog signals. Because of the use of samplers, hybrid systems are frequently called sampleddata systems.
Example: Automobile Computer
Most modern automobiles today have integrated computer systems that monitor certain aspects of the car, and actually help to control the performance of the car. The speed of the car, and the rotational speed of the transmission are analog values, but a sampler converts them into digital values so the car computer can monitor them. The digital computer will then output control signals to other parts of the car, to alter analog systems such as the engine timing, the suspension, the brakes, and other parts. Because the car has both digital and analog components, it is a hybrid system.
Continuous and Discrete
We are not using the word "continuous" here in the sense of continuously differentiable, as is common in math texts.
A system is considered continuoustime if the signal exists for all time. Frequently, the terms "analog" and "continuous" will be used interchangeably, although they are not strictly the same.
Discrete systems can come in three flavors:
 Discrete time (sampled)
 Discrete magnitude (quantized)
 Discrete time and magnitude (digital)
Discrete magnitude systems are systems where the signal value can only have certain values.Discrete time systems are systems where signals are only available (or valid) at particular times. Computer systems are discrete in the sense of (3), in that data is only read at specific discrete time intervals, and the data can have only a limited number of discrete values.
A discretetime system has a sampling time value associated with it, such that each discrete value occurs at multiples of the given sampling time. We will denote the sampling time of a system as T. We can equate the squarebrackets notation of a system with the continuous definition of the system as follows:
Notice that the two notations show the same thing, but the first one is typically easier to write, and it shows that the system in question is a discrete system. This book will use the square brackets to denote discrete systems by the sample number n, and parenthesis to denote continuous time functions.
Sampling and Reconstruction
The process of converting analog information into digital data is called "Sampling". The process of converting digital data into an analog signal is called "Reconstruction". We will talk about both processes in a later chapter. For more information on the topic than is available in this book, see the Analog and Digital Conversion wikibook. Here is an example of a reconstructed waveform. Notice that the reconstructed waveform here is quantized because it is constructed from a digital signal:
System Metrics
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.
Standard Inputs
All of the standard inputs are zero before time zero. All the standard inputs are causal.
There are a number of standard inputs that are considered simple enough and universal enough that they are considered when designing a system. These inputs are known as a unit step, a ramp, and a parabolic input.
 Unit Step
 A unit step function is defined piecewise as such:
[Unit Step Function]
 The unit step function is a highly important function, not only in control systems engineering, but also in signal processing, systems analysis, and all branches of engineering. If the unit step function is input to a system, the output of the system is known as the step response. The step response of a system is an important tool, and we will study step responses in detail in later chapters.
 Ramp
 A unit ramp is defined in terms of the unit step function, as such:
[Unit Ramp Function]
 It is important to note that the unit step function is simply the differential of the unit ramp function:
 This definition will come in handy when we learn about the Laplace Transform.
 Parabolic
 A unit parabolic input is similar to a ramp input:
[Unit Parabolic Function]
 Notice also that the unit parabolic input is equal to the integral of the ramp function:
 Again, this result will become important when we learn about the Laplace Transform.
Also, sinusoidal and exponential functions are considered basic, but they are too difficult to use in initial analysis of a system.
Steady State
To be more precise, we should have taken the limit as t approaches infinity. However, as a shorthand notation, we will typically say "t equals infinity", and assume the reader understands the shortcut that is being used.
When a unitstep function is input to a system, the steadystate value of that system is the output value at time . Since it is impractical (if not completely impossible) to wait till infinity to observe the system, approximations and mathematical calculations are used to determine the steadystate value of the system. Most system responses are asymptotic, that is that the response approaches a particular value. Systems that are asymptotic are typically obvious from viewing the graph of that response.
Step Response
The step response of a system is most frequently used to analyze systems, and there is a large amount of terminology involved with step responses. When exposed to the step input, the system will initially have an undesirable output period known as the transient response. The transient response occurs because a system is approaching its final output value. The steadystate response of the system is the response after the transient response has ended.
The amount of time it takes for the system output to reach the desired value (before the transient response has ended, typically) is known as the rise time. The amount of time it takes for the transient response to end and the steadystate response to begin is known as the settling time.
It is common for a systems engineer to try and improve the step response of a system. In general, it is desired for the transient response to be reduced, the rise and settling times to be shorter, and the steadystate to approach a particular desired "reference" output.
Target Value
The target output value is the value that our system attempts to obtain for a given input. This is not the same as the steadystate value, which is the actual value that the system does obtain. The target value is frequently referred to as the reference value, or the "reference function" of the system. In essence, this is the value that we want the system to produce. When we input a "5" into an elevator, we want the output (the final position of the elevator) to be the fifth floor. Pressing the "5" button is the reference input, and is the expected value that we want to obtain. If we press the "5" button, and the elevator goes to the third floor, then our elevator is poorly designed.
Rise Time
Rise time is the amount of time that it takes for the system response to reach the target value from an initial state of zero. Many texts on the subject define the rise time as being the time it takes to rise between the initial position and 80% of the target value. This is because some systems never rise to 100% of the expected, target value, and therefore they would have an infinite risetime. This book will specify which convention to use for each individual problem. Rise time is typically denoted t_{r}, or t_{rise}.
Rise time is not the amount of time it takes to achieve steadystate, only the amount of time it takes to reach the desired target value for the first time. 
Percent Overshoot
Underdamped systems frequently overshoot their target value initially. This initial surge is known as the "overshoot value". The ratio of the amount of overshoot to the target steadystate value of the system is known as the percent overshoot. Percent overshoot represents an overcompensation of the system, and can output dangerously large output signals that can damage a system. Percent overshoot is typically denoted with the term PO.
Example: Refrigerator
Consider an ordinary household refrigerator. The refrigerator has cycles where it is on and when it is off. When the refrigerator is on, the coolant pump is running, and the temperature inside the refrigerator decreases. The temperature decreases to a much lower level than is required, and then the pump turns off.
When the pump is off, the temperature slowly increases again as heat is absorbed into the refrigerator. When the temperature gets high enough, the pump turns back on. Because the pump cools down the refrigerator more than it needs to initially, we can say that it "overshoots" the target value by a certain specified amount.
Example: Refrigerator
Another example concerning a refrigerator concerns the electrical demand of the heat pump when it first turns on. The pump is an inductive mechanical motor, and when the motor first activates, a special counteracting force known as "back EMF" resists the motion of the motor, and causes the pump to draw more electricity until the motor reaches its final speed. During the startup time for the pump, lights on the same electrical circuit as the refrigerator may dim slightly, as electricity is drawn away from the lamps, and into the pump. This initial draw of electricity is a good example of overshoot.
SteadyState Error
Sometimes a system might never achieve the desired steadystate value, but instead will settle on an output value that is not desired. The difference between the steadystate output value to the reference input value at steady state is called the steadystate error of the system. We will use the variable e_{ss} to denote the steadystate error of the system.
Settling Time
After the initial rise time of the system, some systems will oscillate and vibrate for an amount of time before the system output settles on the final value. The amount of time it takes to reach steady state after the initial rise time is known as the settling time. Notice that damped oscillating systems may never settle completely, so we will define settling time as being the amount of time for the system to reach, and stay in, a certain acceptable range. The acceptable range for settling time is typically determined on a perproblem basis, although common values are 20%, 10%, or 5% of the target value. The settling time will be denoted as t_{s}.
System Order
The order of the system is defined by the number of independent energy storage elements in the system, and intuitively by the highest order of the linear differential equation that describes the system. In a transfer function representation, the order is the highest exponent in the transfer function. In a proper system, the system order is defined as the degree of the denominator polynomial. In a statespace equation, the system order is the number of statevariables used in the system. The order of a system will frequently be denoted with an n or N, although these variables are also used for other purposes. This book will make clear distinction on the use of these variables.
Proper Systems
A proper system is a system where the degree of the denominator is larger than or equal to the degree of the numerator polynomial. A strictly proper system is a system where the degree of the denominator polynomial is larger than (but never equal to) the degree of the numerator polynomial. A biproper system is a system where the degree of the denominator polynomial equals the degree of the numerator polynomial.
It is important to note that only proper systems can be physically realized. In other words, a system that is not proper cannot be built. It makes no sense to spend a lot of time designing and analyzing imaginary systems.
Example: System Order
1=Find the order of this system:
The highest exponent in the denominator is s^{2}, so the system is order 2. Also, since the denominator is a higher degree than the numerator, this system is strictly proper.}}
In the above example, G(s) is a secondorder transfer function because in the denominator one of the s variables has an exponent of 2. Secondorder functions are the easiest to work with.
System Type
Let's say that we have a process transfer function (or combination of functions, such as a controller feeding in to a process), all in the forward branch of a unity feedback loop. Say that the overall forward branch transfer function is in the following generalized form (known as polezero form):
[PoleZero Form]
we call the parameter M the system type. Note that increased system type number correspond to larger numbers of poles at s = 0. More poles at the origin generally have a beneficial effect on the system, but they increase the order of the system, and make it increasingly difficult to implement physically. System type will generally be denoted with a letter like N, M, or m. Because these variables are typically reused for other purposes, this book will make clear distinction when they are employed.
Now, we will define a few terms that are commonly used when discussing system type. These new terms are Position Error, Velocity Error, and Acceleration Error. These names are throwbacks to physics terms where acceleration is the derivative of velocity, and velocity is the derivative of position. Note that none of these terms are meant to deal with movement, however.
 Position Error
 The position error, denoted by the position error constant . This is the amount of steadystate error of the system when stimulated by a unit step input. We define the position error constant as follows:
[Position Error Constant]
 Where G(s) is the transfer function of our system.
 Velocity Error
 The velocity error is the amount of steadystate error when the system is stimulated with a ramp input. We define the velocity error constant as such:
[Velocity Error Constant]
 Acceleration Error
 The acceleration error is the amount of steadystate error when the system is stimulated with a parabolic input. We define the acceleration error constant to be:
[Acceleration Error Constant]
Now, this table will show briefly the relationship between the system type, the kind of input (step, ramp, parabolic), and the steadystate error of the system:
Unit System Input Type, M Au(t) Ar(t) Ap(t) 0 1 2 > 2
ZDomain Type
Likewise, we can show that the system order can be found from the following generalized transfer function in the Z domain:
Where the constant M is the type of the digital system. Now, we will show how to find the various error constants in the ZDomain:
[ZDomain Error Constants]
Error Constant Equation Kp Kv Ka
Visually
Here is an image of the various system metrics, acting on a system in response to a step input:
The target value is the value of the input step response. The rise time is the time at which the waveform first reaches the target value. The overshoot is the amount by which the waveform exceeds the target value. The settling time is the time it takes for the system to settle into a particular bounded region. This bounded region is denoted with two short dotted lines above and below the target value.
System Modeling
The Control Process
It is the job of a control engineer to analyze existing systems, and to design new systems to meet specific needs. Sometimes new systems need to be designed, but more frequently a controller unit needs to be designed to improve the performance of existing systems. When designing a system, or implementing a controller to augment an existing system, we need to follow some basic steps:
 Model the system mathematically
 Analyze the mathematical model
 Design system/controller
 Implement system/controller and test
The vast majority of this book is going to be focused on (2), the analysis of the mathematical systems. This chapter alone will be devoted to a discussion of the mathematical modeling of the systems.
External Description
An external description of a system relates the system input to the system output without explicitly taking into account the internal workings of the system. The external description of a system is sometimes also referred to as the InputOutput Description of the system, because it only deals with the inputs and the outputs to the system.
If the system can be represented by a mathematical function h(t, r), where t is the time that the output is observed, and r is the time that the input is applied. We can relate the system function h(t, r) to the input x and the output y through the use of an integral:
[General System Description]
This integral form holds for all linear systems, and every linear system can be described by such an equation.
If a system is causal (i.e. an input at t=r affects system behaviour only for ) and there is no input of the system before t=0, we can change the limits of the integration:
TimeInvariant Systems
If furthermore a system is timeinvariant, we can rewrite the system description equation as follows:
This equation is known as the convolution integral, and we will discuss it more in the next chapter.
Every Linear TimeInvariant (LTI) system can be used with the Laplace Transform, a powerful tool that allows us to convert an equation from the time domain into the SDomain, where many calculations are easier. Timevariant systems cannot be used with the Laplace Transform.
Internal Description
If a system is linear and lumped, it can also be described using a system of equations known as statespace equations. In statespace equations, we use the variable x to represent the internal state of the system. We then use u as the system input, and we continue to use y as the system output. We can write the statespace equations as such:
We will discuss the statespace equations more when we get to the section on modern controls.
Complex Descriptions
Systems which are LTI and Lumped can also be described using a combination of the statespace equations, and the Laplace Transform. If we take the Laplace Transform of the state equations that we listed above, we can get a set of functions known as the Transfer Matrix Functions. We will discuss these functions in a later chapter.
Representations
To recap, we will prepare a table with the various system properties, and the available methods for describing the system:
Properties StateSpace
EquationsLaplace
TransformTransfer
MatrixLinear, TimeVariant, Distributed no no no Linear, TimeVariant, Lumped yes no no Linear, TimeInvariant, Distributed no yes no Linear, TimeInvariant, Lumped yes yes yes
We will discuss all these different types of system representation later in the book.
Analysis
Once a system is modeled using one of the representations listed above, the system needs to be analyzed. We can determine the system metrics and then we can compare those metrics to our specification. If our system meets the specifications we are finished with the design process. However 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.
Frequency Domain
For proposals, early stage designs, and quick turn around analyses a frequency domain model is often superior to a time domain model. Frequency domain models take disturbance PSDs (Power Spectral Densities) directly, use transfer functions directly, and produce output or residual PSDs directly. The answer is a steadystate response. Oftentimes the controller is shooting for 0 so the steadystate response is also the residual error that will be the analysis output or metric for report.
Input  Model  Output 

PSD  Transfer Function  PSD 
Brief Overview of the Math
Frequency domain modeling is a matter of determining the impulse response of a system to a random process.
 ^{[1]}
where
 is the onesided input PSD in
 is the frequency response function of the system and
 is the onesided output PSD or auto power spectral density function.
The frequency response function, , is related to the impulse response function (transfer function) by
Note some texts will state that this is only valid for random processes which are stationary. Other texts suggest stationary and ergodic while still others state weakly stationary processes. Some texts do not distinguish between strictly stationary and weakly stationary. From practice, the rule of thumb is if the PSD of the input process is the same from hour to hour and day to day then the input PSD can be used and the above equation is valid.
Notes
 ↑ Sun, JianQiao (2006). Stochastic Dynamics and Control, Volume 4. Amsterdam: Elsevier Science. ISBN 0444522301.
See a full explanation with example at ControlTheoryPro.com
Modeling Examples
Modeling in Control Systems is oftentimes a matter of judgement. This judgement is developed by creating models and learning from other people's models. ControlTheoryPro.com is a site with a lot of examples. Here are links to a few of them
 Hovering Helicopter Example
 Reaction Torque Cancellation Example
 List of all examples at ControlTheoryPro.com
Manufacture
Once the system has been properly designed we can prototype our system and test it. Assuming our analysis was correct and our design is good, the prototype should work as expected. Now we can move on to manufacture and distribute our completed systems.
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
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.
Transform Basics
A transform is a mathematical tool that converts an equation from one variable (or one set of variables) into a new variable (or a new set of variables). To do this, the transform must remove all instances of the first variable, the "Domain Variable", and add a new "Range Variable". Integrals are excellent choices for transforms, because the limits of the definite integral will be substituted into the domain variable, and all instances of that variable will be removed from the equation. An integral transform that converts from a domain variable a to a range variable b will typically be formatted as such:
Where the function f(a) is the function being transformed, and g(a,b) is known as the kernel of the transform. Typically, the only difference between the various integral transforms is the kernel.
Laplace Transform
The Laplace Transform converts an equation from the timedomain into the socalled "Sdomain", or the Laplace domain, or even the "Complex domain". These are all different names for the same mathematical space and they all may be used interchangeably in this book and in other texts on the subject. The Transform can only be applied under the following conditions:
 The system or signal in question is analog.
 The system or signal in question is Linear.
 The system or signal in question is TimeInvariant.
 The system or signal in question is causal.
The transform is defined as such:
[Laplace Transform]
Laplace transform results have been tabulated extensively. More information on the Laplace transform, including a transform table can be found in the Appendix.
If we have a linear differential equation in the time domain:
With zero initial conditions, we can take the Laplace transform of the equation as such:
And separating, we get:
Inverse Laplace Transform
The inverse Laplace Transform is defined as such:
[Inverse Laplace Transform]
The inverse transform converts a function from the Laplace domain back into the time domain.
Matrices and Vectors
The Laplace Transform can be used on systems of linear equations in an intuitive way. Let's say that we have a system of linear equations:
We can arrange these equations into matrix form, as shown:
And write this symbolically as:
We can take the Laplace transform of both sides:
Which is the same as taking the transform of each individual equation in the system of equations.
Example: RL Circuit
Circuit Theory
Here, we are going to show a common example of a firstorder system, an RL Circuit. In an inductor, the relationship between the current, I, and the voltage, V, in the time domain is expressed as a derivative:
Where L is a special quantity called the "Inductance" that is a property of inductors.
Let's say that we have a 1st order RL series electric circuit. The resistor has resistance R, the inductor has inductance L, and the voltage source has input voltage V_{in}. The system output of our circuit is the voltage over the inductor, V_{out}. In the time domain, we have the following firstorder differential equations to describe the circuit:
However, since the circuit is essentially acting as a voltage divider, we can put the output in terms of the input as follows:
This is a very complicated equation, and will be difficult to solve unless we employ the Laplace transform:
We can divide top and bottom by L, and move V_{in} to the other side:
And using a simple table lookup, we can solve this for the timedomain relationship between the circuit input and the circuit output:
Partial Fraction Expansion
Calculus
Laplace transform pairs are extensively tabulated, but frequently we have transfer functions and other equations that do not have a tabulated inverse transform. If our equation is a fraction, we can often utilize Partial Fraction Expansion (PFE) to create a set of simpler terms that will have readily available inverse transforms. This section is going to give a brief reminder about PFE, for those who have already learned the topic. This refresher will be in the form of several examples of the process, as it relates to the Laplace Transform. People who are unfamiliar with PFE are encouraged to read more about it in Calculus.
Example: SecondOrder System
If we have a given equation in the Sdomain:
We can expand it into several smaller fractions as such:
This looks impossible, because we have a single equation with 3 unknowns (s, A, B), but in reality s can take any arbitrary value, and we can "plug in" values for s to solve for A and B, without needing other equations. For instance, in the above equation, we can multiply through by the denominator, and cancel terms:
Now, when we set s → 2, the A term disappears, and we are left with B → 3. When we set s → 1, we can solve for A → 1. Putting these values back into our original equation, we have:
Remember, since the Laplace transform is a linear operator, the following relationship holds true:
Finding the inverse transform of these smaller terms should be an easier process then finding the inverse transform of the whole function. Partial fraction expansion is a useful, and oftentimes necessary tool for finding the inverse of an Sdomain equation.
Example: FourthOrder System
If we have a given equation in the Sdomain:
We can expand it into several smaller fractions as such:
Canceling terms wouldn't be enough here, we will open the brackets (separated onto multiple lines):
Let's compare coefficients:
 A + D = 0
 30A + C + 20D = 79
 300A + B + 10C + 100D = 916
 1000A = 1000
And solving gives us:
 A = 1
 B = 26
 C = 69
 D = 1
We know from the Laplace Transform table that the following relation holds:
We can plug in our values for A, B, C, and D into our expansion, and try to convert it into the form above.
Example: Complex Roots
Given the following transfer function:
When the solution of the denominator is a complex number, we use a complex representation A + iB, like 3+i4 as opposed to the use of a single letter (e.g. D)  which is for real numbers:
 As + B = 7s + 26
 A = 7
 B = 26
We will need to reform it into two fractions that look like this (without changing its value):
 →
 →
Let's start with the denominator (for both fractions):
The roots of s^{2}  80s + 1681 are 40 + j9 and 40  j9.
 →
And now the numerators:
Inverse Laplace Transform:
Example: SixthOrder System
Given the following transfer function:
We multiply through by the denominators to make the equation rational:
And then we combine terms:
Comparing coefficients:
 A + B + C = 0
 15A  12B  3C + D = 90
 73A + 37B  3D = 0
 111A = 1110
Now, we can solve for A, B, C and D:
 A = 10
 B = 10
 C = 0
 D = 120
And now for the "fitting":
The roots of s^{2}  12s + 37 are 6 + j and 6  j
No need to fit the fraction of D, because it is complete; no need to bother fitting the fraction of C, because C is equal to zero.
Final Value Theorem
The Final Value Theorem allows us to determine the value of the time domain equation, as the time approaches infinity, from the S domain equation. In Control Engineering, the Final Value Theorem is used most frequently to determine the steadystate value of a system. The real part of the poles of the function must be <0.
[Final Value Theorem (Laplace)]
From our chapter on system metrics, you may recognize the value of the system at time infinity as the steadystate time of the system. The difference between the steady state value and the expected output value we remember as being the steadystate error of the system. Using the Final Value Theorem, we can find the steadystate value and the steadystate error of the system in the Complex S domain.
Example: Final Value Theorem
Find the final value of the following polynomial:
We can apply the Final Value Theorem:
We obtain the value:
Initial Value Theorem
Akin to the final value theorem, the Initial Value Theorem allows us to determine the initial value of the system (the value at time zero) from the SDomain Equation. The initial value theorem is used most frequently to determine the starting conditions, or the "initial conditions" of a system.
[Initial Value Theorem (Laplace)]
Common Transforms
We will now show you the transforms of the three functions we have already learned about: The unit step, the unit ramp, and the unit parabola. The transform of the unit step function is given by:
And since the unit ramp is the integral of the unit step, we can multiply the above result times 1/s to get the transform of the unit ramp:
Again, we can multiply by 1/s to get the transform of the unit parabola:
Fourier Transform
The Fourier Transform is very similar to the Laplace transform. The fourier transform uses the assumption that any finite timedomain signal can be broken into an infinite sum of sinusoidal (sine and cosine waves) signals. Under this assumption, the Fourier Transform converts a timedomain signal into its frequencydomain representation, as a function of the radial frequency, ω, The Fourier Transform is defined as such:
[Fourier Transform]
We can now show that the Fourier Transform is equivalent to the Laplace transform, when the following condition is true:
Because the Laplace and Fourier Transforms are so closely related, it does not make much sense to use both transforms for all problems. This book, therefore, will concentrate on the Laplace transform for nearly all subjects, except those problems that deal directly with frequency values. For frequency problems, it makes life much easier to use the Fourier Transform representation.
Like the Laplace Transform, the Fourier Transform has been extensively tabulated. Properties of the Fourier transform, in addition to a table of common transforms is available in the Appendix.
Inverse Fourier Transform
The inverse Fourier Transform is defined as follows:
[Inverse Fourier Transform]
This transform is nearly identical to the Fourier Transform.
Complex Plane
Using the above equivalence, we can show that the Laplace transform is always equal to the Fourier Transform, if the variable s is an imaginary number. However, the Laplace transform is different if s is a real or a complex variable. As such, we generally define s to have both a real part and an imaginary part, as such:
And we can show that s = jω if σ = 0.
Since the variable s can be broken down into 2 independent values, it is frequently of some value to graph the variable s on its own special "Splane". The Splane graphs the variable σ on the horizontal axis, and the value of jω on the vertical axis. This axis arrangement is shown at right.
Euler's Formula
There is an important result from calculus that is known as Euler's Formula, or "Euler's Relation". This important formula relates the important values of e, j, π, 1 and 0:
However, this result is derived from the following equation, setting ω to π:
[Euler's Formula]
This formula will be used extensively in some of the chapters of this book, so it is important to become familiar with it now.
MATLAB
The MATLAB symbolic toolbox contains functions to compute the Laplace and Fourier transforms automatically. The function laplace, and the function fourier can be used to calculate the Laplace and Fourier transforms of the input functions, respectively. For instance, the code:
t = sym('t'); fx = 30*t^2 + 20*t; laplace(fx)
produces the output:
ans = 60/s^3+20/s^2
We will discuss these functions more in The Appendix.
Further reading
 Digital Signal Processing/ContinuousTime Fourier Transform
 Signals and Systems/Aperiodic Signals
 Circuit Theory/Laplace Transform
Transfer Functions
Transfer Functions
A Transfer Function is the ratio of the output of a system to the input of a system, in the Laplace domain considering its initial conditions and equilibrium point to be zero. This assumption is relaxed for systems observing transience. If we have an input function of X(s), and an output function Y(s), we define the transfer function H(s) to be:
[Transfer Function]
Readers who have read the Circuit Theory book will recognize the transfer function as being the impedance, admittance, impedance ratio of a voltage divider or the admittance ratio of a current divider.
Impulse Response
Time domain variables are generally written with lowercase letters. LaplaceDomain, and other transform domain variables are generally written using uppercase letters.
For comparison, we will consider the timedomain equivalent to the above input/output relationship. In the time domain, we generally denote the input to a system as x(t), and the output of the system as y(t). The relationship between the input and the output is denoted as the impulse response, h(t).
We define the impulse response as being the relationship between the system output to its input. We can use the following equation to define the impulse response:
Impulse Function
It would be handy at this point to define precisely what an "impulse" is. The Impulse Function, denoted with δ(t) is a special function defined piecewise as follows:
[Impulse Function]
The impulse function is also known as the delta function because it's denoted with the Greek lowercase letter δ. The delta function is typically graphed as an arrow towards infinity, as shown below:
It is drawn as an arrow because it is difficult to show a single point at infinity in any other graphing method. Notice how the arrow only exists at location 0, and does not exist for any other time t. The delta function works with regular time shifts just like any other function. For instance, we can graph the function δ(t  N) by shifting the function δ(t) to the right, as such:
An examination of the impulse function will show that it is related to the unitstep function as follows:
and
The impulse function is not defined at point t = 0, but the impulse must always satisfy the following condition, or else it is not a true impulse function:
The response of a system to an impulse input is called the impulse response. Now, to get the Laplace Transform of the impulse function, we take the derivative of the unit step function, which means we multiply the transform of the unit step function by s:
This result can be verified in the transform tables in The Appendix.
Step Response
Similar to the impulse response, the step response of a system is the output of the system when a unit step function is used as the input. The step response is a common analysis tool used to determine certain metrics about a system. Typically, when a new system is designed, the step response of the system is the first characteristic of the system to be analyzed.
Convolution
However, the impulse response cannot be used to find the system output from the system input in the same manner as the transfer function. If we have the system input and the impulse response of the system, we can calculate the system output using the convolution operation as such:
Where " * " (asterisk) denotes the convolution operation. Convolution is a complicated combination of multiplication, integration and timeshifting. We can define the convolution between two functions, a(t) and b(t) as the following:
[Convolution]
(The variable τ (Greek tau) is a dummy variable for integration). This operation can be difficult to perform. Therefore, many people prefer to use the Laplace Transform (or another transform) to convert the convolution operation into a multiplication operation, through the Convolution Theorem.
TimeInvariant System Response
If the system in question is timeinvariant, then the general description of the system can be replaced by a convolution integral of the system's impulse response and the system input. We can call this the convolution description of a system, and define it below:
[Convolution Description]
Convolution Theorem
This method of solving for the output of a system is quite tedious, and in fact it can waste a large amount of time if you want to solve a system for a variety of input signals. Luckily, the Laplace transform has a special property, called the Convolution Theorem, that makes the operation of convolution easier:
 Convolution Theorem
 Convolution in the time domain becomes multiplication in the complex Laplace domain. Multiplication in the time domain becomes convolution in the complex Laplace domain.
The Convolution Theorem can be expressed using the following equations:
[Convolution Theorem]
This also serves as a good example of the property of Duality.
Using the Transfer Function
The Transfer Function fully describes a control system. The Order, Type and Frequency response can all be taken from this specific function. Nyquist and Bode plots can be drawn from the open loop Transfer Function. These plots show the stability of the system when the loop is closed. Using the denominator of the transfer function, called the characteristic equation, roots of the system can be derived.
For all these reasons and more, the Transfer function is an important aspect of classical control systems. Let's start out with the definition:
 Transfer Function
 The Transfer function of a system is the relationship of the system's output to its input, represented in the complex Laplace domain.
If the complex Laplace variable is s, then we generally denote the transfer function of a system as either G(s) or H(s). If the system input is X(s), and the system output is Y(s), then the transfer function can be defined as such:
If we know the input to a given system, and we have the transfer function of the system, we can solve for the system output by multiplying:
[Transfer Function Description]
Example: Impulse Response
From a Laplace transform table, we know that the Laplace transform of the impulse function, δ(t) is:
So, when we plug this result into our relationship between the input, output, and transfer function, we get:
In other words, the "impulse response" is the output of the system when we input an impulse function.
Example: Step Response
From the Laplace Transform table, we can also see that the transform of the unit step function, u(t) is given by:
Plugging that result into our relation for the transfer function gives us:
And we can see that the step response is simply the impulse response divided by s.
Example: MATLAB Step Response
Use MATLAB to find the step response of the following transfer function:
We can separate out our numerator and denominator polynomials as such:
num = [79 916 1000]; den = [1 30 300 1000 0]; sys = tf(num, den);
% if you are using the System Identification Toolbox instead of the Control System Tooolbox: sys = idtf(num, den);
Now, we can get our step response from the step function, and plot it for time from 1 to 10 seconds:
T = 1:0.001:10; step(sys, T);
Frequency Response
The Frequency Response is similar to the Transfer function, except that it is the relationship between the system output and input in the complex Fourier Domain, not the Laplace domain. We can obtain the frequency response from the transfer function, by using the following change of variables:
 Frequency Response
 The frequency response of a system is the relationship of the system's output to its input, represented in the Fourier Domain.
Because the frequency response and the transfer function are so closely related, typically only one is ever calculated, and the other is gained by simple variable substitution. However, despite the close relationship between the two representations, they are both useful individually, and are each used for different purposes.
Sampled Data Systems
Ideal Sampler
In this chapter, we are going to introduce the ideal sampler and the Star Transform. First, we need to introduce (or review) the Geometric Series infinite sum. The results of this sum will be very useful in calculating the Star Transform, later.
Consider a sampler device that operates as follows: every T seconds, the sampler reads the current value of the input signal at that exact moment. The sampler then holds that value on the output for T seconds, before taking the next sample. We have a generic input to this system, f(t), and our sampled output will be denoted f*(t). We can then show the following relationship between the two signals:
Note that the value of f^{ *} at time t = 1.5 T is the same as at time t = T. This relationship works for any fractional value.
Taking the Laplace Transform of this infinite sequence will yield us with a special result called the Star Transform. The Star Transform is also occasionally called the "Starred Transform" in some texts.
Geometric Series
Before we talk about the Star Transform or even the ZTransform, it is useful for us to review the mathematical background behind solving infinite series. Specifically, because of the nature of these transforms, we are going to look at methods to solve for the sum of a geometric series.
A geometic series is a sum of values with increasing exponents, as such:
In the equation above, notice that each term in the series has a coefficient value, a. We can optionally factor out this coefficient, if the resulting equation is easier to work with:
Once we have an infinite series in either of these formats, we can conveniently solve for the total sum of this series using the following equation:
Let's say that we start our series off at a number that isn't zero. Let's say for instance that we start our series off at n = 1 or n = 100. Let's see:
We can generalize the sum to this series as follows:
[Geometric Series]
With that result out of the way, now we need to worry about making this series converge. In the above sum, we know that n is approaching infinity (because this is an infinite sum). Therefore, any term that contains the variable n is a matter of worry when we are trying to make this series converge. If we examine the above equation, we see that there is one term in the entire result with an n in it, and from that, we can set a fundamental inequality to govern the geometric series.
To satisfy this equation, we must satisfy the following condition:
[Geometric convergence condition]
Therefore, we come to the final result: The geometric series converges if and only if the value of r is less than one.
The Star Transform
The Star Transform is defined as such:
[Star Transform]
The Star Transform depends on the sampling time T and is different for a single signal depending on the frequency at which the signal is sampled. Since the Star Transform is defined as an infinite series, it is important to note that some inputs to the Star Transform will not converge, and therefore some functions do not have a valid Star Transform. Also, it is important to note that the Star Transform may only be valid under a particular region of convergence. We will cover this topic more when we discuss the Ztransform.
Star ↔ Laplace
Complex Analysis/Residue Theory
The Laplace Transform and the Star Transform are clearly related, because we obtained the Star Transform by using the Laplace Transform on a timedomain signal. However, the method to convert between the two results can be a slightly difficult one. To find the Star Transform of a Laplace function, we must take the residues of the Laplace equation, as such:
This math is advanced for most readers, so we can also use an alternate method, as follows:
Neither one of these methods are particularly easy, however, and therefore we will not discuss the relationship between the Laplace transform and the Star Transform any more than is absolutely necessary in this book. Suffice it to say, however, that the Laplace transform and the Star Transform are related mathematically.
Star + Laplace
In some systems, we may have components that are both continuous and discrete in nature. For instance, if our feedback loop consists of an AnalogToDigital converter, followed by a computer (for processing), and then a DigitalToAnalog converter. In this case, the computer is acting on a digital signal, but the rest of the system is acting on continuous signals. Star transforms can interact with Laplace transforms in some of the following ways:
Given:
Then:
Given:
Then:
Where is the Star Transform of the product of X(s)H(s).
Convergence of the Star Transform
The Star Transform is defined as being an infinite series, so it is critically important that the series converge (not reach infinity), or else the result will be nonsensical. Since the Star Transform is a geometic series (for many input signals), we can use geometric series analysis to show whether the series converges, and even under what particular conditions the series converges. The restrictions on the star transform that allow it to converge are known as the region of convergence (ROC) of the transform. Typically a transform must be accompanied by the explicit mention of the ROC.
The ZTransform
Let us say now that we have a discrete data set that is sampled at regular intervals. We can call this set x[n]:
x[n] = [ x[0] x[1] x[2] x[3] x[4] ... ]
we can utilize a special transform, called the Ztransform, to make dealing with this set more easy:
[Z Transform]
the Appendix.
Like the Star Transform the Z Transform is defined as an infinite series and therefore we need to worry about convergence. In fact, there are a number of instances that have identical ZTransforms, but different regions of convergence (ROC). Therefore, when talking about the Z transform, you must include the ROC, or you are missing valuable information.
Z Transfer Functions
Like the Laplace Transform, in the Zdomain we can use the inputoutput relationship of the system to define a transfer function.
The transfer function in the Z domain operates exactly the same as the transfer function in the S Domain:
Similarly, the value h[n] which represents the response of the digital system is known as the impulse response of the system. It is important to note, however, that the definition of an "impulse" is different in the analog and digital domains.
Inverse Z Transform
The inverse Z Transform is defined by the following path integral:
[Inverse Z Transform]
Where C is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). The contour or path, C, must encircle all of the poles of X(z).
This math is relatively advanced compared to some other material in this book, and therefore little or no further attention will be paid to solving the inverse ZTransform in this manner. Z transform pairs are heavily tabulated in reference texts, so many readers can consider that to be the primary method of solving for inverse Z transforms. There are a number of Ztransform pairs available in table form in The Appendix.
Final Value Theorem
Like the Laplace Transform, the Z Transform also has an associated final value theorem:
[Final Value Theorem (Z)]
This equation can be used to find the steadystate response of a system, and also to calculate the steadystate error of the system.
Star ↔ Z
The Z transform is related to the Star transform though the following change of variables:
Notice that in the Z domain, we don't maintain any information on the sampling period, so converting to the Z domain from a Star Transformed signal loses that information. When converting back to the star domain however, the value for T can be reinsterted into the equation, if it is still available.
Also of some importance is the fact that the Z transform is bilinear, while the Star Transform is unilinear. This means that we can only convert between the two transforms if the sampled signal is zero for all values of n < 0.
Because the two transforms are so closely related, it can be said that the Z transform is simply a notational convenience for the Star Transform. With that said, this book could easily use the Star Transform for all problems, and ignore the added burden of Z transform notation entirely. A common example of this is Richard Hamming's book "Numerical Methods for Scientists and Engineers" which uses the Fourier Transform for all problems, considering the Laplace, Star, and ZTransforms to be merely notational conveniences. However, the Control Systems wikibook is under the impression that the correct utilization of different transforms can make problems more easy to solve, and we will therefore use a multitransform approach.
Z plane
The lowercase z is the name of the variable, and the uppercase Z is the name of the Transform and the plane.
z is a complex variable with a real part and an imaginary part. In other words, we can define z as such:
Since z can be broken down into two independent components, it often makes sense to graph the variable z on the Zplane. In the Zplane, the horizontal axis represents the real part of z, and the vertical axis represents the magnitude of the imaginary part of z.
Notice also that if we define z in terms of the startransform relation:
we can separate out s into real and imaginary parts:
We can plug this into our equation for z:
Through Euler's formula, we can separate out the complex exponential as such:
If we define two new variables, M and φ:
We can write z in terms of M and φ. Notice that it is Euler's equation:
Which is clearly a polar representation of z, with the magnitude of the polar function (M) based on the realpart of s, and the angle of the polar function (φ) is based on the imaginary part of s.
Region of Convergence
To best teach the region of convergance (ROC) for the Ztransform, we will do a quick example.
We have the following discrete series or a decaying exponential:
Now, we can plug this function into the Z transform equation:
Note that we can remove the unit step function, and change the limits of the sum:
This is because the series is 0 for all time less than n → 0. If we try to combine the n terms, we get the following result:
Once we have our series in this term, we can break this down to look like our geometric series:
And finally, we can find our final value, using the geometric series formula:
Again, we know that to make this series converge, we need to make the r value less than 1:
And finally we obtain the region of convergance for this Ztransform:
z and s are complex variables, and therefore we need to take the magnitude in our ROC calculations. The "Absolute Value symbols" are actually the "magnitude calculation", and is defined as such:

Laplace ↔ Z
There are no easy, direct ways to convert between the Laplace transform and the Z transform directly. Nearly all methods of conversions reproduce some aspects of the original equation faithfully, and incorrectly reproduce other aspects. For some of the main mapping techniques between the two, see the Z Transform Mappings Appendix.
However, there are some topics that we need to discuss. First and foremost, conversions between the Laplace domain and the Z domain are not linear, this leads to some of the following problems:
This means that when we combine two functions in one domain multiplicatively, we must find a combined transform in the other domain. Here is how we denote this combined transform:
Notice that we use a horizontal bar over top of the multiplied functions, to denote that we took the transform of the product, not of the individual pieces. However, if we have a system that incorporates a sampler, we can show a simple result. If we have the following format:
Then we can put everything in terms of the Star Transform:
and once we are in the star domain, we can do a direct change of variables to reach the Z domain:
Note that we can only make this equivalence relationship if the system incorporates an ideal sampler, and therefore one of the multiplicative terms is in the star domain.
Example
Let's say that we have the following equation in the Laplace domain:
And because we have a discrete sampler in the system, we want to analyze it in the Z domain. We can break up this equation into two separate terms, and transform each:
And
And when we add them together, we get our result:
Z ↔ Fourier
By substituting variables, we can relate the Star transform to the Fourier Transform as well:
If we assume that T = 1, we can relate the two equations together by setting the real part of s to zero. Notice that the relationship between the Laplace and Fourier transforms is mirrored here, where the Fourier transform is the Laplace transform with no realpart to the transform variable.
There are a number of discretetime variants to the Fourier transform as well, which are not discussed in this book. For more information about these variants, see Digital Signal Processing.
Reconstruction
Some of the easiest reconstruction circuits are called "Holding circuits". Once a signal has been transformed using the Star Transform (passed through an ideal sampler), the signal must be "reconstructed" using one of these hold systems (or an equivalent) before it can be analyzed in a Laplacedomain system.
If we have a sampled signal denoted by the Star Transform , we want to reconstruct that signal into a continuoustime waveform, so that we can manipulate it using Laplacetransform techniques.
Let's say that we have the sampled input signal, a reconstruction circuit denoted G(s), and an output denoted with the Laplacetransform variable Y(s). We can show the relationship as follows:
Reconstruction circuits then, are physical devices that we can use to convert a digital, sampled signal into a continuoustime domain, so that we can take the Laplace transform of the output signal.
Zero order Hold
A zeroorder hold circuit is a circuit that essentially inverts the sampling process: The value of the sampled signal at time t is held on the output for T time. The output waveform of a zeroorder hold circuit therefore looks like a staircase approximation to the original waveform.
The transfer function for a zeroorder hold circuit, in the Laplace domain, is written as such:
[Zero Order Hold]
The Zeroorder hold is the simplest reconstruction circuit, and (like the rest of the circuits on this page) assumes zero processing delay in converting between digital to analog.
First Order Hold
The zeroorder hold creates a step output waveform, but this isn't always the best way to reconstruct the circuit. Instead, the FirstOrder Hold circuit takes the derivative of the waveform at the time t, and uses that derivative to make a guess as to where the output waveform is going to be at time (t + T). The firstorder hold circuit then "draws a line" from the current position to the expected future position, as the output of the waveform.
[First Order Hold]
Keep in mind, however, that the next value of the signal will probably not be the same as the expected value of the next data point, and therefore the firstorder hold may have a number of discontinuities.
Fractional Order Hold
The ZeroOrder hold outputs the current value onto the output, and keeps it level throughout the entire bit time. The firstorder hold uses the function derivative to predict the next value, and produces a series of ramp outputs to produce a fluctuating waveform. Sometimes however, neither of these solutions are desired, and therefore we have a compromise: FractionalOrder Hold. Fractional order hold acts like a mixture of the other two holding circuits, and takes a fractional number k as an argument. Notice that k must be between 0 and 1 for this circuit to work correctly.
[Fractional Order Hold]
This circuit is more complicated than either of the other hold circuits, but sometimes added complexity is worth it if we get better performance from our reconstruction circuit.
Other Reconstruction Circuits
Another type of circuit that can be used is a linear approximation circuit.
Further reading
 Hamming, Richard. "Numerical Methods for Scientists and Engineers" ISBN 0486652416
 Digital Signal Processing/Z Transform
 Complex Analysis/Residue Theory
 Analog and Digital Conversion
System Delays
Delays
A system can be built with an inherent delay. Delays are units that cause a timeshift in the input signal, but that don't affect the signal characteristics. An ideal delay is a delay system that doesn't affect the signal characteristics at all, and that delays the signal for an exact amount of time. Some delays, like processing delays or transmission delays, are unintentional. Other delays however, such as synchronization delays, are an integral part of a system. This chapter will talk about how delays are utilized and represented in the Laplace Domain. Once we represent a delay in the Laplace domain, it is an easy matter, through change of variables, to express delays in other domains.
Ideal Delays
An ideal delay causes the input function to be shifted forward in time by a certain specified amount of time. Systems with an ideal delay cause the system output to be delayed by a finite, predetermined amount of time.
Time Shifts
Let's say that we have a function in time that is timeshifted by a certain constant time period T. For convenience, we will denote this function as x(t  T). Now, we can show that the Laplace transform of x(t  T) is the following:
What this demonstrates is that timeshifts in the timedomain become exponentials in the complex Laplace domain.
Shifts in the ZDomain
Since we know the following general relationship between the Z Transform and the Star Transform:
We can show what a time shift in a discrete time domain becomes in the Z domain:
Delays and Stability
A timeshift in the time domain becomes an exponential increase in the Laplace domain. This would seem to show that a time shift can have an effect on the stability of a system, and occasionally can cause a system to become unstable. We define a new parameter called the time margin as the amount of time that we can shift an input function before the system becomes unstable. If the system can survive any arbitrary time shift without going unstable, we say that the time margin of the system is infinite.
Delay Margin
When speaking of sinusoidal signals, it doesn't make sense to talk about "time shifts", so instead we talk about "phase shifts". Therefore, it is also common to refer to the time margin as the phase margin of the system. The phase margin denotes the amount of phase shift that we can apply to the system input before the system goes unstable.
We denote the phase margin for a system with a lowercase Greek letter φ (phi). Phase margin is defined as such for a secondorder system:
[Delay Margin]
Oftentimes, the phase margin is approximated by the following relationship:
[Delay Margin (approx)]
The Greek letter zeta (ζ) is a quantity called the damping ratio, and we discuss this quantity in more detail in the next chapter.
TransformDomain Delays
The ordinary ZTransform does not account for a system which experiences an arbitrary time delay, or a processing delay. The ZTransform can, however, be modified to account for an arbitrary delay. This new version of the Ztransform is frequently called the Modified ZTransform, although in some literature (notably in Wikipedia), it is known as the Advanced ZTransform.
Delayed Star Transform
To demonstrate the concept of an ideal delay, we will show how the star transform responds to a timeshifted input with a specified delay of time T. The function : is the delayed star transform with a delay parameter Δ. The delayed star transform is defined in terms of the star transform as such:
[Delayed Star Transform]
As we can see, in the star transform, a timedelayed signal is multiplied by a decaying exponential value in the transform domain.
Delayed ZTransform
Since we know that the Star Transform is related to the Z Transform through the following change of variables:
We can interpret the above result to show how the Z Transform responds to a delay:
This result is expected.
Now that we know how the Z transform responds to time shifts, it is often useful to generalize this behavior into a form known as the Delayed ZTransform. The Delayed ZTransform is a function of two variables, z and Δ, and is defined as such:
And finally:
[Delayed Z Transform]
Modified ZTransform
The Delayed ZTransform has some uses, but mathematicians and engineers have decided that a more useful version of the transform was needed. The new version of the ZTransform, which is similar to the Delayed Ztransform with a change of variables, is known as the Modified ZTransform. The Modified ZTransform is defined in terms of the delayed Z transform as follows:
And it is defined explicitly:
[Modified Z Transform]
Poles and Zeros
Poles and Zero
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively. The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.
Physically realizable control systems must have a number of poles greater than or equal to the number of zeros. Systems that satisfy this relationship are called Proper. We will elaborate on this below.
TimeDomain Relationships
Let's say that we have a transfer function with 3 poles:
The poles are located at s = l, m, n. Now, we can use partial fraction expansion to separate out the transfer function:
Using the inverse transform on each of these component fractions (looking up the transforms in our table), we get the following:
But, since s is a complex variable, l m and n can all potentially be complex numbers, with a real part (σ) and an imaginary part (jω). If we just look at the first term:
Using Euler's Equation on the imaginary exponent, we get:
If a complex pole is present it is always accomponied by another pole that is its complex conjugate. The imaginary parts of their time domain representations thus cancel and we are left with 2 of the same real parts. Assuming that the complex conjugate pole of the first term is present, we can take 2 times the real part of this equation and we are left with our final result:
We can see from this equation that every pole will have an exponential part, and a sinusoidal part to its response. We can also go about constructing some rules:
 if σ_{l} = 0, the response of the pole is a perfect sinusoid (an oscillator)
 if ω_{l} = 0, the response of the pole is a perfect exponential.
 if σ_{l} < 0, the exponential part of the response will decay towards zero.
 if σ_{l} > 0, the exponential part of the response will rise towards infinity.
From the last two rules, we can see that all poles of the system must have negative real parts, and therefore they must all have the form (s + l) for the system to be stable. We will discuss stability in later chapters.
What are Poles and Zeros
Let's say we have a transfer function defined as a ratio of two polynomials:
Where N(s) and D(s) are simple polynomials. Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s.
Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s. Because of our restriction above, that a transfer function must not have more zeros than poles, we can state that the polynomial order of D(s) must be greater than or equal to the polynomial order of N(s).
Example
Consider the transfer function:
We define N(s) and D(s) to be the numerator and denominator polynomials, as such:
We set N(s) to zero, and solve for s:
So we have a zero at s → 2. Now, we set D(s) to zero, and solve for s to obtain the poles of the equation:
And simplifying this gives us poles at: i/2 , +i/2. Remember, s is a complex variable, and it can therefore take imaginary and real values.
Effects of Poles and Zeros
As s approaches a zero, the numerator of the transfer function (and therefore the transfer function itself) approaches the value 0. When s approaches a pole, the denominator of the transfer function approaches zero, and the value of the transfer function approaches infinity. An output value of infinity should raise an alarm bell for people who are familiar with BIBO stability. We will discuss this later.
As we have seen above, the locations of the poles, and the values of the real and imaginary parts of the pole determine the response of the system. Real parts correspond to exponentials, and imaginary parts correspond to sinusoidal values. Addition of poles to the transfer function has the effect of pulling the root locus to the right, making the system less stable. Addition of zeros to the transfer function has the effect of pulling the root locus to the left, making the system more stable.
SecondOrder Systems
The canonical form for a second order system is as follows:
[Secondorder transfer function]
Where K is the system gain, ζ is called the damping ratio of the function, and ω is called the natural frequency of the system. ζ and ω, if exactly known for a second order system, the time responses can be easily plotted and stability can easily be checked. More information on second order systems can be found here.
Damping Ratio
The damping ratio of a secondorder system, denoted with the Greek letter zeta (ζ), is a real number that defines the damping properties of the system. More damping has the effect of less percent overshoot, and slower settling time. Damping is the inherent ability of the system to oppose the oscillatory nature of the system's transient response. Larger values of damping coefficient or damping factor produces transient responses with lesser oscillatory nature.
Natural Frequency
The natural frequency is occasionally written with a subscript:
We will omit the subscript when it is clear that we are talking about the natural frequency, but we will include the subscript when we are using other values for the variable ω. Also, when .
HigherOrder Systems
System Representation
Systems can be represented graphically in a number of ways. Block diagrams and signalflow diagrams are powerful tools that can be used to manipulate systems, and convert them easily into transfer functions or statespace equations. The chapters in this section will discuss how systems can be described visually, and will also discuss how systems can be interconnected with each other.
Gain
What is Gain?
Gain is a proportional value that shows the relationship between the magnitude of the input to the magnitude of the output signal at steady state. Many systems contain a method by which the gain can be altered, providing more or less "power" to the system. However, increasing gain or decreasing gain beyond a particular safety zone can cause the system to become unstable.
Consider the given secondorder system:
We can include an arbitrary gain term, K in this system that will represent an amplification, or a power increase:
In a statespace system, the gain term k can be inserted as follows:
The gain term can also be inserted into other places in the system, and in those cases the equations will be slightly different.
Example: Gain
Here are some good examples of arbitrary gain values being used in physical systems:
 Volume Knob
 On your stereo there is a volume knob that controls the gain of your amplifier circuit. Higher levels of volume (turning the volume "up") corresponds to higher amplification of the sound signal.
 Gas Pedal
 The gas pedal in your car is an example of gain. Pressing harder on the gas pedal causes the engine to receive more gas, and causes the engine to output higher RPMs.
 Brightness Buttons
 Most computer monitors come with brightness buttons that control how bright the screen image is. More brightness causes more power to be outputed to the screen.
Responses to Gain
As the gain to a system increases, generally the risetime decreases, the percent overshoot increases, and the settling time increases. However, these relationships are not always the same. A critically damped system, for example, may decrease in rise time while not experiencing any effects of percent overshoot or settling time.
Gain and Stability
If the gain increases to a high enough extent, some systems can become unstable. We will examine this effect in the chapter on Root Locus. But it will decrease the steady state error.
Conditional Stability
Systems that are stable for some gain values, and unstable for other values are called conditionally stable systems. The stability is conditional upon the value of the gain, and often the threshold where the system becomes unstable is important to find.
Block Diagrams
When designing or analyzing a system, often it is useful to model the system graphically. Block Diagrams are a useful and simple method for analyzing a system graphically. A "block" looks on paper exactly what it means:
Systems in Series
When two or more systems are in series, they can be combined into a single representative system, with a transfer function that is the product of the individual systems.
If we have two systems, f(t) and g(t), we can put them in series with one another so that the output of system f(t) is the input to system g(t). Now, we can analyze them depending on whether we are using our classical or modern methods.
If we define the output of the first system as h(t), we can define h(t) as:
Now, we can define the system output y(t) in terms of h(t) as:
We can expand h(t):
But, since convolution is associative, we can rewrite this as:
Our system can be simplified therefore as such:
Series Transfer Functions
If two or more systems are in series with one another, the total transfer function of the series is the product of all the individual system transfer functions.
In the timedomain we know that:
But, in the frequency domain we know that convolution becomes multiplication, so we can rewrite this as:
We can represent our system in the frequency domain as:
Series State Space
If we have two systems in series (say system F and system G), where the output of F is the input to system G, we can write out the statespace equations for each individual system.
System 1:
System 2:
And we can write substitute these equations together form the complete response of system H, that has input u, and output y_{G}:
[Series state equation]
[Series output equation]
Systems in Parallel
Blocks may not be placed in parallel without the use of an adder. Blocks connected by an adder as shown above have a total transfer function of:
Since the Laplace transform is linear, we can easily transfer this to the time domain by converting the multiplication to convolution:
State Space Model
The statespace equations, with nonzero A, B, C, and D matrices conceptually model the following system:
In this image, the strangelooking block in the center is either an integrator or an ideal delay, and can be represented in the transfer domain as:
 or
Depending on the time characteristics of the system. If we only consider continuoustime systems, we can replace the funny block in the center with an integrator:
In the Laplace Domain
The state space model of the above system, if A, B, C, and D are transfer functions A(s), B(s), C(s) and D(s) of the individual subsystems, and if U(s) and Y(s) represent a single input and output, can be written as follows:
We will explain how we got this result, and how we deal with feedforward and feedback loop structures in the next chapter.
Adders and Multipliers
Some systems may have dedicated summation or multiplication devices, that automatically add or multiply the transfer functions of multiple systems together
Simplifying Block Diagrams
Block diagrams can be systematically simplified. Note that this table is from Schaum's Outline: Feedback and Controls Systems by DiStefano et al
External links
Feedback Loops
Feedback
A feedback loop is a common and powerful tool when designing a control system. Feedback loops take the system output into consideration, which enables the system to adjust its performance to meet a desired output response.
When talking about control systems it is important to keep in mind that engineers typically are given existing systems such as actuators, sensors, motors, and other devices with set parameters, and are asked to adjust the performance of those systems. In many cases, it may not be possible to open the system (the "plant") and adjust it from the inside: modifications need to be made external to the system to force the system response to act as desired. This is performed by adding controllers, compensators, and feedback structures to the system.
Basic Feedback Structure
This is a basic feedback structure. Here, we are using the output value of the system to help us prepare the next output value. In this way, we can create systems that correct errors. Here we see a feedback loop with a value of one. We call this a unity feedback.
Here is a list of some relevant vocabulary, that will be used in the following sections:
 Plant
 The term "Plant" is a carryover term from chemical engineering to refer to the main system process. The plant is the preexisting system that does not (without the aid of a controller or a compensator) meet the given specifications. Plants are usually given "as is", and are not changeable. In the picture above, the plant is denoted with a P.
 Controller
 A controller, or a "compensator" is an additional system that is added to the plant to control the operation of the plant. The system can have multiple compensators, and they can appear anywhere in the system: Before the pickoff node, after the summer, before or after the plant, and in the feedback loop. In the picture above, our compensator is denoted with a C.
 Summer
 A summer is a symbol on a system diagram, (denoted above with parenthesis) that conceptually adds two or more input signals, and produces a single sum output signal.
 Pickoff node
 A pickoff node is simply a fancy term for a split in a wire.
 Forward Path
 The forward path in the feedback loop is the path after the summer, that travels through the plant and towards the system output.
 Reverse Path
 The reverse path is the path after the pickoff node, that loops back to the beginning of the system. This is also known as the "feedback path".
 Unity feedback
 When the multiplicative value of the feedback path is 1.
Negative vs Positive Feedback
It turns out that negative feedback is almost always the most useful type of feedback. When we subtract the value of the output from the value of the input (our desired value), we get a value called the error signal. The error signal shows us how far off our output is from our desired input.
Positive feedback has the property that signals tend to reinforce themselves, and grow larger. In a positive feedback system, noise from the system is added back to the input, and that in turn produces more noise. As an example of a positive feedback system, consider an audio amplification system with a speaker and a microphone. Placing the microphone near the speaker creates a positive feedback loop, and the result is a sound that grows louder and louder. Because the majority of noise in an electrical system is highfrequency, the sound output of the system becomes highpitched.
Example: StateSpace Equation
In the previous chapter, we showed you this picture:
Now, we will derive the I/O relationship into the statespace equations. If we examine the innermost feedback loop, we can see that the forward path has an integrator system, , and the feedback loop has the matrix value A. If we take the transfer function only of this loop, we get:
Premultiplying by the factor B, and postmultiplying by C, we get the transfer function of the entire lowerhalf of the loop:
We can see that the upper path (D) and the lowerpath T_{lower} are added together to produce the final result:
Now, for an alternate method, we can assume that x' is the value of the innerfeedback loop, right before the integrator. This makes sense, since the integral of x' should be x (which we see from the diagram that it is. Solving for x', with an input of u, we get:
This is because the value coming from the feedback branch is equal to the value x times the feedback loop matrix A, and the value coming from the left of the sumer is the input u times the matrix B.
If we keep things in terms of x and u, we can see that the system output is the sum of u times the feedforward value D, and the value of x times the value C:
These last two equations are precisely the statespace equations of our system.
Feedback Loop Transfer Function
We can solve for the output of the system by using a series of equations:
and when we solve for Y(s) we get:
[Feedback Transfer Function]
The reader is encouraged to use the above equations to derive the result by themselves.
The function E(s) is known as the error signal. The error signal is the difference between the system output (Y(s)), and the system input (X(s)). Notice that the error signal is now the direct input to the system G(s). X(s) is now called the reference input. The purpose of the negative feedback loop is to make the system output equal to the system input, by identifying large differences between X(s) and Y(s) and correcting for them.
Example: Elevator
Here is a simple example of reference inputs and feedback systems:
There is an elevator in a certain building with 5 floors. Pressing button "1" will take you to the first floor, and pressing button "5" will take you to the fifth floor, etc. For reasons of simplicity, only one button can be pressed at a time.
Pressing a particular button is the reference input of the system. Pressing "1" gives the system a reference input of 1, pressing "2" gives the system a reference input of 2, etc. The elevator system then, tries to make the output (the physical floor location of the elevator) match the reference input (the button pressed in the elevator). The error signal, e(t), represents the difference between the reference input x(t), and the physical location of the elevator at time t, y(t).
Let's say that the elevator is on the first floor, and the button "5" is pressed at time t_{0}. The reference input then becomes a step function:
Where we are measuring in units of "floors". At time t_{0}, the error signal is:
Which means that the elevator needs to travel upwards 4 more floors. At time t_{1}, when the elevator is at the second floor, the error signal is:
Which means the elevator has 3 more floors to go. Finally, at time t_{4}, when the elevator reaches the top, the error signal is:
And when the error signal is zero, the elevator stops moving. In essence, we can define three cases:
 e(t) is positive: In this case, the elevator goes up one floor, and checks again.
 e(t) is zero: The elevator stops.
 e(t) is negative: The elevator goes down one floor, and checks again.
StateSpace Feedback Loops
In the statespace representation, the plant is typically defined by the statespace equations:
The plant is considered to be preexisting, and the matrices A, B, C, and D are considered to be internal to the plant (and therefore unchangeable). Also, in a typical system, the state variables are either fictional (in the sense of dummyvariables), or are not measurable. For these reasons, we need to add external components, such as a gain element, or a feedback element to the plant to enhance performance.
Consider the addition of a gain matrix K installed at the input of the plant, and a negative feedback element F that is multiplied by the system output y, and is added to the input signal of the plant. There are two cases:
 The feedback element F is subtracted from the input before multiplication of the K gain matrix.
 The feedback element F is subtracted from the input after multiplication of the K gain matrix.
In case 1, the feedback element F is added to the input before the multiplicative gain is applied to the input. If v is the input to the entire system, then we can define u as:
In case 2, the feeback element F is subtracted from the input after the multiplicative gain is applied to the input. If v is the input to the entire system, then we can define u as:
Open Loop vs Closed Loop
Let's say that we have the generalized system shown above. The top part, Gp(s) represents all the systems and all the controllers on the forward path. The bottom part, Gb(s) represents all the feedback processing elements of the system. The letter "K" in the beginning of the system is called the Gain. We will talk about the gain more in later chapters. We can define the ClosedLoop Transfer Function as follows:
[ClosedLoop Transfer Function]
If we "open" the loop, and break the feedback node, we can define the OpenLoop Transfer Function, as:
[OpenLoop Transfer Function]
We can redefine the closedloop transfer function in terms of this openloop transfer function:
These results are important, and they will be used without further explanation or derivation throughout the rest of the book.
Placement of a Controller
There are a number of different places where we could place an additional controller.
 In front of the system, before the feedback loop.
 Inside the feedback loop, in the forward path, before the plant.
 In the forward path, after the plant.
 In the feedback loop, in the reverse path.
 After the feedback loop.
Each location has certain benefits and problems, and hopefully we will get a chance to talk about all of them.
SecondOrder Systems
The general expression of the transfer function of a second order system is given as:
where and are damping ratio and natural frequency of the system respectively.
Damping Ratio
The damping ratio is defined by way of the sign . The damping ratio gives us an idea about the nature of the transient response detailing the amount of overshoot & oscillation that the system will undergo. This is completely regardless of time scaling.
If :
 = zero, the system is undamped;
 < 1, the system is underdamped;
 = 1, the system is critically damped;
 > 1, the system is overdamped.
is used in conjunction with the natural frequency to determine system properties. To find the zeta value you must first find the natural response!
Natural Frequency
Natural Frequency, denoted by is defined as the frequency with which the system would oscillate if it were not damped and we define the damping ratio as .
System Sensitivity
Signal Flow Diagrams
Signalflow graphs
Signalflow graphs are another method for visually representing a system. Signal Flow Diagrams are especially useful, because they allow for particular methods of analysis, such as Mason's Gain Formula.
Signal flow diagrams typically use curved lines to represent wires and systems, instead of using lines at rightangles, and boxes, respectively. Every curved line is considered to have a multiplier value, which can be a constant gain value, or an entire transfer function. Signals travel from one end of a line to the other, and lines that are placed in series with one another have their total multiplier values multiplied together (just like in block diagrams).
Signal flow diagrams help us to identify structures called "loops" in a system, which can be analyzed individually to determine the complete response of the system.
Forward Paths
A forward path is a path in the signal flow diagram that connects the input to the output without touching any single node or path more than once. A single system can have multiple forward paths.
Loops
A loop is a structure in a signal flow diagram that leads back to itself. A loop does not contain the beginning and ending points, and the end of the loop is the same node as the beginning of a loop.
Loops are said to touch if they share a node or a line in common.
The Loop gain is the total gain of the loop, as you travel from one point, around the loop, back to the starting point.
Delta Values
The Delta value of a system, denoted with a Greek Δ is computed as follows:
Where:
 A is the sum of all individual loop gains
 B is the sum of the products of all the pairs of nontouching loops
 C is the sum of the products of all the sets of 3 nontouching loops
 D is the sum of the products of all the sets of 4 nontouching loops
 et cetera.
If the given system has no pairs of loops that do not touch, for instance, B and all additional letters after B will be zero.
Mason's Rule
Mason's rule is a rule for determining the gain of a system. Mason's rule can be used with block diagrams, but it is most commonly (and most easily) used with signal flow diagrams.
If we have computed our delta values (above), we can then use Mason's Gain Rule to find the complete gain of the system:
[Mason's Rule]
Where M is the total gain of the system, represented as the ratio of the output gain (y_{out}) to the input gain (y_{in}) of the system. M_{k} is the gain of the k^{th} forward path, and Δ_{k} is the loop gain of the k^{th} loop.
Examples
Solving a signalflow graph by systematic reduction : Two interlocking loops
This example shows how a system of five equations in five unknowns is solved using systematic reduction rules. The independent variable is . The dependent variables are , , , , . The coefficients are labeled .
Here is the starting flowgraph:
The steps for solving follow.
Removing edge c from x2 to x3
Removing node x2 and its inflows
has no outflows, and is not a node of interest.
Removing edge e from x3 to x1
Remove edge d from x3 to x4
Node has no outflows and is not a node of interest. It is deleted along with its inflows.
Removing selfloop at x1
Removing selfloop at x4
Remove edge from x4 to x1
Remove outflow from x4 to x_{out}
's outflow is then eliminated: is connected directly to using the product of the gains from the two edges replaced.
is not a variable of interest; thus, its node and its inflows are eliminated.
Eliminating selfloop at x1
Eliminating outflow from x1, then eliminating x1 and its inflows
is not a variable of interest; and its inflows are eliminated
Simplifying the gain expression
Solving a signalflow graph by systematic reduction: Three equations in three unknowns
This example shows how a system of three equations in three unknowns is solved using systematic reduction rules.
The independent variables are , , . The dependent variables are , , . The coefficients are labeled . The steps for solving follow:
Electrical engineering: Construction of a flow graph for a RC circuit
This illustration shows the physical connections of the circuit. Independent voltage source S is connected in series with a resistor R and capacitor C. The example is developed from the physical circuit equations and solved using signalflow graph techniques. Polarity is important:
 S is a source with the positive terminal at N_{1} and the negative terminal at N_{3}
 R is a resistor with the positive terminal at N_{1} and the negative terminal at N_{2}
 C is a capacitor with the positive terminal at N_{2} and the negative terminal at N_{3}.
The unknown variable of interest is the voltage across capacitor C.
Approach to the solution:
 Find the set of equations from the physical network. These equations are acausal in nature.
 Branch equations for the capacitor and resistor. The equations will be developed as transfer functions using Laplace transforms.
 Kirchhoff's voltage and current laws
 Build a signalflow graph from the equations.
 Solve the signalflow graph.
Branch equations
The branch equations are shown for R and C.
Resistor R (Branch equation )
The resistor's branch equation in the time domain is:
In the Laplacetransformed signal space:
Capacitor C (Branch equation )
The capacitor's branch equation in the time domain is:
Assuming the capacitor is initially discharged, the equation becomes:
Taking the derivative of this and multiplying by C yields the derivative form:
In the Laplacetransformed signal space:
Kirchhoff's laws equations
Kirchhoff's Voltage Law equation
This circuit has only one independent loop. Its equation in the time domain is:
In the Laplacetransformed signal space:
Kirchhoff's Current Law equations
The circuit has three nodes, thus three Kirchhoff's current equations (expresses here as the currents flowing from the nodes):
In the Laplacetransformed signal space:
A set of independent equations must be chosen. For the current laws, it is necessary to drop one of these equations. In this example, let us choose .
Building the signalflow graph
We then look at the inventory of equations, and the signals that each equation relates:
Equation  Signals 

The next step consists in assigning to each equation a signal that will be represented as a node. Each independent source signal is represented in the signalflow graph as a source node, therefore no equation is assigned to the independent source . There are many possible valid signal flow graphs from this set of equations. An equation must only be used once, and the variables of interest must be represented.
Equation  Signals  Assigned signal node 

The resulting flow graph is then drawn
The next step consists in solving the signalflow graph.
Using either Mason or systematic reduction, the resulting signal flow graph is:
Mechatronics example
Bode Plots
Bode Plots
A Bode Plot is a useful tool that shows the gain and phase response of a given LTI system for different frequencies. Bode Plots are generally used with the Fourier Transform of a given system.
The frequency of the bode plots are plotted against a logarithmic frequency axis. Every tickmark on the frequency axis represents a power of 10 times the previous value. For instance, on a standard Bode plot, the values of the markers go from (0.1, 1, 10, 100, 1000, ...) Because each tickmark is a power of 10, they are referred to as a decade. Notice that the "length" of a decade decreases as you move to the right on the graph. (note that this description doesn't match the chart above... there are 10 tickmarks per decade, not one, but since it is a log chart they are not evenly spaced).
The bode Magnitude plot measures the system Input/Output ratio in special units called decibels. The Bode phase plot measures the phase shift in degrees (typically, but radians are also used).
Decibels
A Decibel is a ratio between two numbers on a logarithmic scale. To express a ratio between two numbers (A and B) as a decibel we apply the following formula for numbers that represent amplitudes (numbers that represent a power measurement use a factor of 10 rather than 20):
Where dB is the decibel result.
Or, if we just want to take the decibels of a single number C, we could just as easily write:
Frequency Response Notations
If we have a system transfer function T(s), we can separate it into a numerator polynomial N(s) and a denominator polynomial D(s). We can write this as follows:
To get the magnitude gain plot, we must first transit the transfer function into the frequency response by using the change of variables:
From here, we can say that our frequency response is a composite of two parts, a real part R and an imaginary part X:
We will use these forms below.
StraightLine Approximations
The Bode magnitude and phase plots can be quickly and easily approximated by using a series of straight lines. These approximate graphs can be generated by following a few short, simple rules (listed below). Once the straightline graph is determined, the actual Bode plot is a smooth curve that follows the straight lines, and travels through the breakpoints.
Break Points
If the frequency response is in polezero form:
We say that the values for all z_{n} and p_{m} are called break points of the Bode plot. These are the values where the Bode plots experience the largest change in direction.
Break points are sometimes also called "break frequencies", "cutoff points", or "corner points".
Bode Gain Plots
Bode Gain Plots, or Bode Magnitude Plots display the ratio of the system gain at each input frequency.
Bode Gain Calculations
The magnitude of the transfer function T is defined as:
However, it is frequently difficult to transition a function that is in "numerator/denominator" form to "real+imaginary" form. Luckily, our decibel calculation comes in handy. Let's say we have a frequency response defined as a fraction with numerator and denominator polynomials defined as:
If we convert both sides to decibels, the logarithms from the decibel calculations convert multiplication of the arguments into additions, and the divisions into subtractions:
And calculating out the gain of each term and adding them together will give the gain of the system at that frequency.
Bode Gain Approximations
The slope of a straight line on a Bode magnitude plot is measured in units of dB/Decade, because the units on the vertical axis are dB, and the units on the horizontal axis are decades.
The value ω = 0 is infinitely far to the left of the bode plot (because a logarithmic scale never reaches zero), so finding the value of the gain at ω = 0 essentially sets that value to be the gain for the Bode plot from all the way on the left of the graph up till the first break point. The value of the slope of the line at ω = 0 is 0 dB/Decade.
From each pole break point, the slope of the line decreases by 20 dB/Decade. The line is straight until it reaches the next break point. From each zero break point the slope of the line increases by 20 dB/Decade. Double, triple, or higher amounts of repeat poles and zeros affect the gain by multiplicative amounts. Here are some examples:
 2 poles: 40 dB/Decade
 10 poles: 200 dB/Decade
 5 zeros: +100 dB/Decade
Bode Phase Plots
Bode phase plots are plots of the phase shift to an input waveform dependent on the frequency characteristics of the system input. Again, the Laplace transform does not account for the phase shift characteristics of the system, but the Fourier Transform can. The phase of a complex function, in "real+imaginary" form is given as:
Bode Procedure
Given a frequency response in polezero form:
Where A is a nonzero constant (can be negative or positive).
Here are the steps involved in sketching the approximate Bode magnitude plots:
Bode Magnitude Plots
 Step 1
 Factor the transfer function into polezero form.
 Step 2
 Find the frequency response from the Transfer function.
 Step 3
 Use logarithms to separate the frequency response into a sum of decibel terms
 Step 4
 Use to find the starting magnitude.
 Step 5
 The locations of every pole and every zero are called break points. At a zero breakpoint, the slope of the line increases by 20dB/Decade. At a pole, the slope of the line decreases by 20dB/Decade.
 Step 6
 At a zero breakpoint, the value of the actual graph differs from the value of the straightline graph by 3dB. A zero is +3dB over the straight line, and a pole is 3dB below the straight line.
 Step 7
 Sketch the actual bode plot as a smoothcurve that follows the straight lines of the previous point, and travels through the breakpoints.
Here are the steps to drawing the Bode phase plots:
Bode Phase Plots
 Step 1
 If A is positive, start your graph (with zero slope) at 0 degrees. If A is negative, start your graph with zero slope at 180 degrees (or 180 degrees, they are the same thing).
 Step 2
 For every zero, slope the line up at 45 degrees per decade when (1 decade before the break frequency). Multiple zeros means the slope is steeper.
 Step 3
 For every pole, slope the line down at 45 degrees per decade when (1 decade before the break frequency). Multiple poles means the slope is steeper.
 Step 4
 Flatten the slope again when the phase has changed by 90 degrees (for a zero) or 90 degrees (for a pole) (or larger values, for multiple poles or multiple zeros.
Examples
Example: Constant Gain
Draw the bode plot of an amplifier system, with a constant gain increase of 6dB.
Because the gain value is constant, and is not dependent on the frequency, we know that the value of the magnitude graph is constant at all places on the graph. There are no break points, so the slope of the graph never changes. We can draw the graph as a straight, horizontal line at 6dB:
Example: Integrator
Draw the bode plot of a perfect integrator system given by the transfer function:
First, we find the frequency response of the system, by a change of variables:
Then we convert our magnitude into logarithms:
Notice that the location of the break point for the pole is located at ω → 0, which is all the way to the left of the graph. Also, we notice that inserting 0 in for ω gives us an undefined value (which approaches negative infinity, as the limit). We know, because there is a single pole at zero, that the graph to the right of zero (which is everywhere) has a slope of 20dB/Decade. We can determine from our magnitude calculation by plugging in ω → 1 that the second term drops out, and the magnitude at that point is 6dB. We now have the slope of the line, and a point that it intersects, and we can draw the graph:
Example: Differentiator
Example: 1st Order, Lowpass Filter (1 Break Point)
Further reading
 Bode plot online generator (real and asymptotic)
 Circuit Theory/Bode Plots
 Bode Plots on ControlTheoryPro.com