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Control Systems

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

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What are Control Systems?

Wikipedia has related information at Control system.

Wikipedia has related information at Control engineering.

The study and design of automatic Control Systems, a field known as control engineering, is a large and expansive area of study. Control systems, and control engineering techniques have become a pervasive part of modern technical society. From devices as simple as a toaster, to complex machines like space shuttles and rockets, control engineering is a part of our everyday life. This book will introduce the field of control engineering, and will build upon those foundations to explore some of the more advanced topics in the field. Note, however, that control engineering is a very large field, and it would be foolhardy of any author to think that they could include all the information into a single book. Therefore, we will be content here to provide the foundations of control engineering, and then describe some of the more advanced topics in the field. 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. Let's start off with an immediate 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.

Now, we have a little bit of a problem that, while simplistic, can be a point of concern to people who are both designing this motor system, and to the people who might potentially buy it. It would seem obvious that we should start the motor at a higher voltage, so that the motor accelerates faster, and then we can reduce the supply back down to 10 volts once it reaches speed.

Now this is clearly a simplistic example, but it illustrates one important point: That we can add special "Controller units" to preexisting systems, to increase performance, and to meet new system specifications. There are essentially two methods to approach the problem of designing a new control system: the Classical Approach, and the Modern Approach.

It will do us good to formally define the term "Control System", and some other terms that are 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.

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.

Classical and Modern

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

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

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

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

Who is This Book For?

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

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

What are the Prerequisites?

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

Algebra
Calculus

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

State-space system representation draws heavily on linear algebra techniques. Students should know how to operate on matrices. Students should understand basic matrix operations (addition, multiplication, determinant, inverse, transpose). Students would also benefit from a prior understanding of Eigenvalues and Eigenvectors, but those subjects are covered in this text.
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 this book, and this book will refer to the relevant sections in the Engineering Analysis book for further information on some subjects. This is essentially a math book, but with a focus on various engineering applications. It relies on a previous knowledge of the other math books in this list.
Signals and Systems

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

How is this Book Organized?

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

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

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

Versions

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

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

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

Differential Equations Review

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

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:

$\frac{dP}{dt} = rP$

Where $\frac{dP}{dt}$ 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:

$\frac{dP(t)}{dt} = rP(t)$

To solve this basic, first-order 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:

$\frac{dP(t)}{P(t)} = r\ dt$

And integrating both sides gives us:

log | P (t) | = r t + C

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:

P (t) = e r t + C

Where we can separate out the constant as such:

D = e C

P (t) = D e r t

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

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

History


Pierre-Simon Laplace 1749-1827	Joseph Fourier 1768-1840

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

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

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

Oliver Heaviside

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

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

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

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

Branches of Control Engineering

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

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

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

MATLAB

Information about using MATLAB for control systems can be found in
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.

For more information about properly referencing MATLAB, see:
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]

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

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

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

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

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

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

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

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

Text Conventions

Information which is tangent or auxiliary to the main text will be placed in these "sidebox" templates.

Examples will appear in TextBox templates, which show up as large grey boxes filled with text and equations.

Important Definitions: Will appear in TextBox templates as well, except we will use this formatting to show that it is a definition.

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

We will begin our study by talking about systems. Systems, in the barest sense, are devices that take input, and produce an output. The output is related to the input by a certain 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.

There are many different types of systems, and the process of classifying systems in these ways is called system identification. Different types of systems have certain properties that are useful in analysis. By properly identifying a system, we can determine which analysis tools can be used with the system, and ultimately how to analyze and manipulate those systems. In other words, the first step is system identification.

System Identification

Physical Systems can be divided up into a number of different catagories, depending on particular properties that the system exhibits. Some of these system classifications are very easy to work with and have a large theory base for analysis. Some system classifications are very complex and have still not been investigated with any degree of success.

The early sections of this book will focus primarily on linear time-invariant (LTI) systems. LTI systems are the easiest class of system to work with, and have a number of properties that make them ideal to study. In this chapter we will discuss some properties of systems and we will define exactly what an LTI system is.

Later chapters in this book will look at time variant systems and later still we will discuss 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 real-world systems are time-variant, or nonlinear or both.

Initial Time

The initial time of a system is the time before which there is no input. Typically, we define the initial time of a system 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 t₀.

The value of any variable at the initial time t₀ will be denoted with a 0 subscript. For instance, the value of variable x at time t₀ is given by:

x (t 0) = x 0

Likewise, any time t with a positive subscript are points in time after t₀, in ascending order:

$t_0 \le t_1 \le t_2 \le \cdots \le t_n$

So t₁ occurs after t₀, and t₂ occurs after both points. In a similar fashion above, a variable with a positive subscript (unless we are specifying an index into a vector) also occurs at that point in time:

x (t 1) = x 1

x (t 2) = x 2

And so on 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 $x 3 (t) = x 1 (t) + x 2 (t)$ results in an output of $y 3 (t) = y 1 (t) + y 2 (t)$ . To determine whether a system is additive, we can use the following test:

Given a system f that takes an input x and outputs a value y, we use two inputs (x₁ and x₂) to produce two outputs:

y 1 = f (x 1)

y 2 = f (x 2)

Now, we create a composite input that is the sum of our previous inputs:

x 3 = x 1 + x 2

Then the system is additive if the following equation is true:

y 3 = f (x 3) = f (x 1 + x 2) = f (x 1) + f (x 2) = y 1 + y 2

Systems that satisfy this property are called additive. Additive systems are useful because we can use a sum of simple inputs to analyze the system response to a more complex input.

Example: Sinusoids

Given the following equation:

y (t) = sin(3 x (t))

We can create a sum of inputs as:

x (t) = x 1 (t) + x 2 (t)

and we can construct our expected sum of outputs:

y (t) = y 1 (t) + y 2 (t)

Now, plugging these values into our equation, we can test for equality:

y 1 (t) + y 2 (t) = sin(3[x 1 (t) + x 2 (t)])

We can see from this that our equality is not satisfied, and the equation 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 $a x 1$ results in an output of $a y 1$ . In other words, to see if function f() is homogenous, we can perform the following test:

We stimulate the system f with an arbitrary input x to produce an output y:

y = f (x)

Now, we create a second input x₁, scale it by a multiplicative factor C (C is an arbitrary constant value), and produce a corresponding output y₁:

y 1 = f (C x 1)

Now, we assign x to be equal to x₁:

x 1 = x

Then, for the system to be homogenous, the following equation must be true:

y 1 = f (C x) = C f (x) = C y

Systems that are homogenious are useful in many applications, especially applications with gain or amplification.

Example: Straight-Line

Given the equation for a straight line:

y = f (x) = 2 x + 3

y 1 = f (C x 1) = 2(C x 1) + 3 = C 2 X 1 + 3

x 1 = x

And comparing the two results, we see they are not equal:

$y_1 = C2x + 3 \ne Cy = C(2x + 3) = C2x + C3$

Therefore, the equation is not homogenous.

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:

We take two arbitrary inputs, and produce two arbitrary outputs:

y 1 = f (x 1)

y 2 = f (x 2)

Now, a linear combination of the inputs should produce a linear combination of the outputs:

f (A x + B y) = f (A x) + f (B y) = A f (x) + B f (y)

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:

$\frac{dy(t)}{dt} + y(t) = x(t)$

To determine whether this system is linear, we construct a new composite input:

x (t) = A x 1 (t) + B x 2 (t)

And we create the expected composite output:

y (t) = A y 1 (t) + B y 2 (t)

And plug the two into our original equation:

$\frac{d[Ay_1(t) + By_2(t)]}{dt} + [Ay_1(t) + By_2(t)] = Ax_1(t) + Bx_2(t)$

We can factor out the derivative operator, as such:

$\frac{d}{dt}[Ay_1(t) + By_2(t)] + [Ay_1(t) + By_2(t)] = Ax_1(t) + Bx_2(t)$

And we can convert the various composite terms into the respective variables, to prove that this system is linear:

$\frac{dy(t)}{dt} + y(t) = x(t)$

For the record, derivatives and integrals are linear operators, and ordinary differentialy 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 instantaneous systems.

Causality

Causality is a property that is very similar to memory. A system is called causal if it is only dependent on past or current inputs. A system is called non-causal if the output of the system is dependent on future inputs.

A system design that is not causal cannot be physically implemented. If you can't build your system, the design is generally worthless.

Time-Invariance

A system is called time-invariant if the system relationship between the input and output signals is not dependant on the passage of time. If the input signal $x (t)$ produces an output $y (t)$ then any time shifted input, $x (t + δ)$ , results in a time-shifted output $y (t + δ)$ 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 time-invariant then the system block is commutative with an arbitrary delay. We will discuss this facet of time-invariant systems later.

To determine if a system f is time-invariant, we can perform the following test:

We apply an arbitrary input x to a system and produce an arbitrary output y:

y (t) = f (x (t))

And we apply a second input x₁ to the system, and produce a second output:

y 1 (t) = f (x 1 (t))

Now, we assign x₁ to be equal to our first input x, time-shifted by a given constant value δ:

x 1 (t) = x (t - δ)

Finally, a system is time-invariant if y₁ is equal to y shifted by the same value δ:

y 1 (t) = y (t - δ)

LTI Systems

A system is considered to be a Linear Time-Invariant (LTI) system if it satisfies the requirements of time-invariance and linearity. LTI systems are one of the most important types of systems, and we will consider them 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 we will discuss them in more detail when we learn about modern controls.

Systems which are not lumped are called distributed. Distributed systems are very difficult to analyze in practice, and there are few tools available to work with such systems. 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 t₀ the output of the system is zero.

y (t 0) = f (x (t 0)) = 0

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

Control Systems engineers will frequently say that an unstable system has "exploded". Some physical systems actually can rupture or explode when they go unstable.

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 we apply 5 volts to the input terminals of a given circuit, we would like it 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, we have devoted an entire section to it's study.

Inputs and Outputs

Systems can also be categorized by the number of inputs and the number of outputs the system has. If you consider a Television, for instance, the system has two inputs: the power wire, and the signal cable. A system with one input and one output is called single-input, single output, or SISO. a system with multiple inputs and multiple outputs is called multi-input, multi-output, or MIMO.

We will discuss these systems in more detail later.

Excercise:

Based on the definitions of SISO and MIMO, above, can you determine what the acronyms SIMO and MISO stand for?

Digital and Analog

There is a significant distinction between an analogue 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

This operation can be performed using this MATLAB command:
isct

A signal is called continuous-time if it is defined at every time t.

A system is a continuous-time system if it takes a continuous-time input signal, and outputs a continuous-time output signal. Here is an example of an analog waveoform:

Discrete Time

This operation can be performed using this MATLAB command:
isdt

A signal is called discrete-time if it is only defined for particular points in time. A discrete-time system takes discrete-time input signals, and produces discrete-time 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 called "photons". A given particle may have 1 photon worth of energy, or 2 photons worth of energy, but it may not have 1.5 photons worth of energy.
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 computer-based. 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 step-like 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 amount of voltage and current that are input to the motor. We can show the relationship as such:

Θ(v) = f (v)

Where

Θ

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:

φ h = f h (t)

φ m = f m (t)

φ s = f s (t)

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 discrete-time 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 "non-discrete" 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:


Analog Waveform	Digital Waveform

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 base-2 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 sampled-data 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

Note:
We are not using the word "continuous" here in the sense of continuously differentiable, as is common in math texts.

A system is considered continuous-time if the signal exists for all time. Frequently, the terms "analog" and "continuous" will be used interchangably, although they are not strictly the same.

Discrete systems can come in three flavors:

Discrete time
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 discrete-time system has as 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 square-brackets notation of a system with the continuous definition of the system as follows:

x [n] = x (n T)

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 the reconstructed waveform here is quantized because it is constructed from a digital signal:

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

Note:
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]

$u(t) = \left\{ \begin{matrix} 0, & t < 0 \\ 1, & t \ge 0 \end{matrix}\right.$

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]

r (t) = t u (t)

It is important to note that the ramp function is simply the integral of the unit step function:

$r(t) = \int u(t)dt = tu(t)$

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]

$p(t) = \frac{1}{2}t^2 u(t)$

Notice also that the unit parabolic input is equal to the integral of the ramp function:

$p(t) = \int r(t)dt = \int t u(t)dt = \frac{1}{2}t^2u(t) = \frac{1}{2}t r(t)$

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

Note:
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 unit-step function is input to a system, the steady state value of that system is the output value at time $t = \infty$ . 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 steady-state 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 steady-state 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 steady-state 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 steady-state to approach a particular desired "reference" output.

An arbitrary step function with $x (t) = M u (t)$	A step response graph of input x(t) to an imaginary system

Target Value

The target output value is the value that our system attempts to obtain for a given output. This is not the same as the steady-state value, which is the actual value that the target 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 rise-time. 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 steady-state, 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 steady-state 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 then 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 then 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 counter-acting 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.

Steady-State Error

In general, the letter e or E will be used to denote error values.

Sometimes a system might never achieve the desired steady state value, but instead will settle on an output value that is not desired. The difference between the steady-state output value to the reference input value at steady state is called the steady state error of the system. We will use the variable e_ss to denote the steady-state 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 per-problem 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 highest degree 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 state-space equation, the system order is the number of state-variables 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.

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

Find the order of this system:

$G(s) = \frac{1 + s}{1 + s + s^2}$

The highest exponent in the denominator is s², so the system is order 2. Also, since the denominator is a higher degree then the numerator, this system is proper.

in the above example, G(s) is a second-order transfer function because in the denominator one of the s variables has an exponent of 2. Second-order functions are the easiest to work with, and this book will focus on second-order LTI systems.

System Type

Let's say that we have a transfer function that is in the following generalized form (known as pole-zero form):

[Pole-Zero Form]

$G(s) = \frac {K \prod_i (s - s_i)}{s^M \prod_j (s - s_j)}$

Poles at the origin are called integrators, because they have the effect of performing integration on the input signal.

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 $K p$ . This is the amount of steady state error of the system when multiplied by a unit step input. We define the position error constant as follows:

[Position Error Constant]

$K_p = \lim_{s \to 0} G(s)$

Where G(s) is the transfer function of our system.

Velocity Error: The velocity error is the amount of steady state error when the system is stimulated with a ramp input. We define the velocity error constant as such:

[Velocity Error Constant]

$K_v = \lim_{s \to 0} s G(s)$

Acceleration Error: The acceleration error is the amount of steady-state error when the system is stimulated with a parabolic input. We define the acceleration error constant to be:

[Acceleration Error Constant]

$K_a = \lim_{s \to 0} s^2 G(s)$

Now, this table will show briefly the relationship between the system type, the kind of input (step, ramp, parabolic), and the steady state error of the system:

	Unit System Input
Type, M	Au(t)	Ar(t)	Ap(t)
0	$e_{ss} = \frac{A}{1 + K_p}$	$e_{ss} = \infty$	$e_{ss} = \infty$
1	$e s s = 0$	$e_{ss} = \frac{A}{K_v}$	$e_{ss} = \infty$
2	$e s s = 0$	$e s s = 0$	$e_{ss} = \frac{A}{K_a}$
> 2	$e s s = 0$	$e s s = 0$	$e s s = 0$

Z-Domain Type

Likewise, we can show that the system order can be found from the following generalized transfer function in the Z domain:

$G(z) = \frac {K \prod_i (z - z_i)}{(z - 1)^M \prod_j (z - z_j)}$

Where the constant M is the order of the digital system. Now, we will show how to find the various error constants in the Z-Domain:

[Z-Domain Error Constants]

Error Constant	Equation
Kp	$K_p = \lim_{z \to 1} G(z)$
Kv	$K_v = \lim_{z \to 1} (z - 1) G(z)$
Ka	$K_a = \lim_{z \to 1} (z - 1)^2 G(z)$

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 Input-Output 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]

$y(t) = \int_{-\infty}^\infty h(t, r)x(r)dr$

This integral form holds for all linear systems, and every linear system can be described by such an equation.

If a system is causal, then there is no output of the system before time r, and we can change the limits of the integration:

$y(t) = \int_0^t h(t, r)x(r)dr$

Time-Invariant Systems

If a system is time-invariant (and causal), we can rewrite the system description equation as follows:

$y(t) = \int_0^t h(t - r)x(r)dr$

This equation is known as the convolution integral, and we will discuss it more in the next chapter.

Every Linear Time-Invariant (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 S-Domain, where many calculations are easier. Time-variant 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 state-space equations. In state space 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 state space equations as such:

x'(t) = A (t) x (t) + B (t) u (t)

y (t) = C (t) x (t) + D (t) u (t)

We will discuss the state space 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 state-space 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	State-Space Equations	Laplace Transform	Transfer Matrix
Linear, Time-Variant, Distributed	no	no	no
Linear, Time-Variant, Lumped	yes	no	no
Linear, Time-Invariant, Distributed	no	yes	no
Linear, Time-Invariant, 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, quick turn around analyses a frequency domain model is often superior to a time domain model. Frequency domain models take disturbance PSDs directly, use transfer functions directly, and produce output or residual PSDs directly. The answer is a steady-state response. Often times the controller is shooting for 0 so the steady-state reponse is also the residual error that will be the analysis output or metric for report.

**Table 1: Frequency Domain Model Inputs and Outputs**
Input	Model	Output
PSD	Transfer Function	PSD

Brief Overview of the Math

Frequncy domain modeling is a matter of determining the impulse response of a system to a random process. Typically the theory for this is detailed in texts on stochastic processes and stochastic contol.

Figure 1: Frequency Domain System

$S_{YY}\left(\omega\right)=G^*\left(\omega\right)G\left(\omega\right)S_{XX}= \left | G\left(\omega\right)\right \vert S_{XX}$ ^[1] where

$S_{XX}\left(\omega\right)$ is the one-sided input PSD in $\frac{magnitude^2}{Hz}$

$G\left(\omega\right)$ is the frequency response function of the system and

$S_{YY}\left(\omega\right)$ is the one-sided output PSD or auto power spectral density function.

The frequency response function, $G\left(\omega\right)$ , is related to the impulse response function (transfer function) by

$g\left(\tau\right)=\frac{1}{2 \pi} \int_{-\infty}^{\infty}e^{i\omega t}H\left(\omega\right)\,d\omega$

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 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, Jian-Qiao (2006). Stochastic Dynamics and Control, Volume 4. Amsterdam: Elsevier Science. ISBN 0444522301.

See a full explanation with example 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.

System Representation

Systems can be represented graphically in a number of ways. Block diagrams and signal-flow diagrams are powerful tools that can be used to manipulate systems, and convert them easily into transfer functions or state-space 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 second-order system:

$T(s) = \frac{1}{s^2 + 2s + 1}$

We can include an arbitrary gain term, K in this system that will represent an amplification, or a power increase:

$T(s) = K\frac{1}{s^2 + 2s + 1}$

In a state-space system, the gain term k can be inserted as follows:

x'(t) = A x (t) + k B u (t)

y (t) = C x (t) + k D u (t)

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 rise-time 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.

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 the value of the gain, and often times the threshold where the system becomes unstable is important to find.

Block Diagrams

Block Diagram Representation

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 how it sounds:

If system H has a time-domain impulse response h(t), we can express y(t) as:

y (t) = u (t) * h (t)

Where the asterisk ( * ) denotes convolution. If system H has a Laplace-domain transfer function H(s), we show the relationship between the input and the output as:

Y (s) = U (s) H (s)

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 sum 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:

h (t) = x (t) * f (t)

Now, we can define the system output y(t) in terms of h(t) as:

y (t) = h (t) * g (t)

We can expand h(t):

y (t) = [x (t) * f (t)] * g (t)

But, since convolution is associative, we can re-write this as:

y (t) = x (t) * [f (t) * g (t)]

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 time domain we know that:

y (t) = x (t) * [f (t) * g (t)]

But, in the frequency domain we know that convolution becomes multiplication, so we can re-write this as:

Y (s) = X (s)[F (s) G (s)]

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 state-space equations for each individual system.

System 1:

x F' = A F x F + B F u

y F = C F x F + D F u

System 2:

X G' = A G x G + B G y F

Y G = C G x G + D G y F

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]

$\begin{bmatrix}x_G \\ x_F\end{bmatrix} = \begin{bmatrix}A_G & B_GC_F \\ 0 & A_F\end{bmatrix} \begin{bmatrix}x_G \\ x_F\end{bmatrix} + \begin{bmatrix}B_GD_F \\ B_F\end{bmatrix}u$

[Series output equation]

$\begin{bmatrix}y_G \\ y_F\end{bmatrix} = \begin{bmatrix}C_G & D_GC_F \\ 0 & C_F\end{bmatrix} \begin{bmatrix}x_G \\ x_F\end{bmatrix} + \begin{bmatrix}D_GD_F \\ D_F\end{bmatrix}u$

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:

Y (s) = X (s)[F (s) + G (s)]

Since the Laplace transform is linear, we can easily transfer this to the time domain by converting the multiplication to convolution:

y (t) = x (t) * [f (t) + g (t)]

State Space Model

The state-space equations, with non-zero A, B, C, and D matrices conceptually model the following system:

In this image, the strange-looking block in the center is either an integrator or an ideal delay, and can be represented in the transfer domain as:

$\frac{1}{s}$ or $\frac{1}{z}$

Depending on the time characteristics of the system. If we only consider continuous-time 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:

$\frac{Y(s)}{U(s)} = B(s)\left(\frac{1}{s - A(s)}\right)C(s) + D(s)$

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.

Transformation		Equation	Block Diagram	Equivalent Block Diagram
1	Cascaded Blocks	$Y=\left(P_1 P_2 \right) X$