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\section{Thesis Acknowledgement}
\paragraph{} I guess I wont write this thesis, if i couldnt get comp392 from my school and i wouldnt see the importantce of Control Theory.
\paragraph{} I owe special thanks to a lot of people. Firstly Elburus Caferov who lectured comp392 
and Ayşe Karaca who was asistant of comp392. But i have to thank Kotshiko Ogata who writes very nice
and understandle book. Thanks to this book, i realized how significant all issue on that book for 
academic life and normal life.
\paragraph{} Then as next step, i saw all of method on that book was placed under a lot of mathematical 
approaches. I want to thank my supervisor Elif Pınar Hacıbeyoğlu for giving such idea, to make thesis
about mathematical approaches for Control Theory and its system.I hope i met a lot of people, see a lot of mathematical approaches and realized its huges history.
\paragraph{} Finally, i want to thank Alparslan Parlakçı for listenin my idea, motivate and convince me  to go further.
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\section{Abstract}
\paragraph{}Since antiqiuty, there has been a lot of changes and developments have been made by mathematician people that met people necessity in life. These changes and developments has been placed by such mathematician into Control theory with some specificc title called mathematical approach. As people produce or build new systems, needs of time period has been risen up. Such needs brings new method and terminology to control theory. Such method and terminology examined under title of 'mathematical approaches for control system'. 

\paragraph{}That thesis will be like a documentary that aims to show cotrol theory to people with huge history, its application and theories. This will  make by not also showing these theories and application but also making historical rewiew. On the other hands, their theories and history that makes such theories will be in one place.

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\section{Özet}

\paragraph{} Antik çağlardan beri, matematik insanları tarafından insan hayatındaki gereksinimleri karşılayan bir çok değişim ve geliştirme yapıldı.
Matematik insanları tarafından bu değişim ve geliştirmeler kontrol teorisinin içine yerleştirildi. Bu değişim ve gelişmeler insanlar belli başlıkların altında inceledi.. Bu başlıkların toplandığı ortak alan ise Kontrol teorisindeki matematiksel yaklaşımlar oldu. İnsanlar yeni bir sistem geliştirdikçe, yeni metod ve terimler hayatımıza girdi.

\paragraph{} Bu tez insanlara kontrol teorinin büyük tarihi gelişimini ve bu tarihte yapılmış olan uygulamaları, teorileriyle birlikte, birbirleriyle ilişkilerindererek gösterme amacındadır. 




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\section{Introduction}

\paragraph{} I want to examine mathematical approaches for control system that covers a lot of theory that has huge history.
(Statemet of Purpose)

\paragraph{} Last year i took comp392. In this course, there are two many subject that gets my attention highly. But we make all subjects superficially. We accept a lot of formulas, theory and so on. And we never wonder about background of this property (control theory). In that semester, Elif Pınar Hacıbeyoğlu gave me an idea 'you wonder background so you can make a study about mathematics of control theory. Then, we called this thesis as mathematical approaches for control system. And we start to make.

\paragraph{} Mathematical modelling can be thought as landmark studies in Control Theory and its related systems. My belief about 'mathematic al modelling', that is a fact that helps Mathematician people to compose mathematical approach for control theory. My several google research indicates this facts with some books and articles. Firstly i want to talk about stability. Stability is very important issue and it has mathematical models see Lyapunov and Nyquist stability criertion. 

\paragraph{}  non-exhaustive list of areas: mathematical modelling in engineering, industrial mathematics, control theory. (Ref absolute stability of non linear control system.) Other referrences are stockhastic Control and mathematical modelling,http://www.ctit.utwente.nl/research/groups/eemcs/msct.doc/, 

\paragraph{} This thesis actually is aimed to present mathematical approaches not only professor who check my thesis, but also this thesis is for student who will take Control courses in Bilgi University. So i have some method to write this thesis and some hyphothesis about this thesis. Anyhow these hyphothesis states my methodology. In adittion I have some other methodology for education in Bilgi.

\paragraph{} During the history of Control theory, we can see this hiearchy. Firstly we see the system then we can see the theory that purposes to explain, examine these systems. What i am saying is, making mathematical approaches for control theory is to cover theorotical part and practical part of Control theory. I stated that in my proposal.
My hyphothesis about that If we/Student is able to cover two sides of Control theory, we can understand several things. We can use this method/hyphothesis instead of summarizing some graph or formulas or whatever.s

\paragraph{} I have new areas that i think to make. One of them is historical review. My observation is while we examine mathematical approaches fo control systems, we have to know the history a little. Because all things in control theory has a background on the other hands historical developments. If we dont know that it is like something is missing. My theory about that If we/student want to know Control theory, we have to know history of that.

\paragraph{} My other new area is to make case-study part. I have some idea in my mind. As i meantioned before, i try to find some articles and i try to express that very efficiently or I use some specific topic from subject based book or i use different example that covers real area of that. For example Controller cononical representation. In this hyphothesis we as student knows how theory can be related to real world problem and we student understands the topic more neatly.

\paragraph{} Last a rea that is not in my proposal is suggestion part.
I try to advise students advanced suggestion that provides more way to student so as to understand mathematics and control theory. Actually my thesis is about mathematics. We cover mathematical suggestion.

\paragraph{} We have also matlab part. It will be easier to tell what is going on, what parameter is in spesific subject.

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\section{Mathematical models for Control System}
\paragraph{} The mathematical model of a system signifies mathematical relation for input, system, and the output. This relation guarantee at least one system can determine system's output from given input. This relation is not ordinary relation but it is very special relation that facilitates system's examining. \cite{Ref2} \footnote{Mathematical model is not an ordinary relation. That is very special relation that has capacity to analyze, observe systems' behaivour. \cite{Ref2}}
\begin{center}

\includegraphics[scale=0.25]{controlpicture-1.png}
\end{center}
Figure : 1 System, input and output
 

\subsection{General aspect of mathematical models}
\paragraph{} The problem of determining a system’s mathematical model is essentially a problem
of approximating the behavior of a physical system with an ideal mathematical
expression. \cite{Ref2}

\subsection{Type of Mathematical Models} \footnote{The important thing that has to be made is to understand terms that are listed above.}
\paragraph{}There are several mathematical models. We care four of them which are as fallows; \footnote{All method that we will tell later on, has an own advantages and disadvantages relatively than other. The most important and good ones we have, we can manually select one of them and taking help from its own flexibility \cite{Ref2}}

\begin{itemize}
\item The differential equation.
\item The transfer function. \footnote{Transfer function is the most important ones}
\item The impulse response.
\item The state space equation.
\end{itemize}
\footnote{The important thing that has to be made is to understand terms that are listed above.}



\paragraph{} There are also other way to understand and analyze system. This schematic representation that works with mathematical relations. \footnote{There are two types of system that is open loop and close loop. We can draw two different block diagram and we also make two different transfer function. Differential equation is always helper to obtain transfer function. See connections ?}
\begin{itemize}
\item Block diagram.
\item The signal flow diagrams.
\end{itemize}



\subsection{Differential equation}
\paragraph{} Differential equation is one of the most oldes and most widely used method for describing systems. This system description contain includes all the linearly independent equations of the system, as well as the appro-
priate initial conditions.

\paragraph{} Example 1: 'Consider the network shown in Figure 3.3. Derive the network’s differential equa-
tion mathematical model.'

Apply Kirchhoff’s voltage law \footnote{'Kirchhoff’s voltage law. The algebraic sum of the voltages in a loop is equal
to zero.} , So 
'
$ L di/dt int_0^t \! idt + Ri = v(t)$

\paragraph{}'The above integrodifferential equation constitutes a mathematical description
of the network. To complete this description, two appropriate initial conditions must
be given, since the above mathematical model is essentially a second-order differen-
tial equation. As initial conditions, we usually consider the inductor’s current iL ðtÞ
and the capacitor’s voltage vC ðtÞ at the moment the switch S closes, which is usually
at t 1/4 0. Therefore, the initial conditions are'

$i_L(0) = I_0$ and $v_c(0) = V_0$ where $I_0$ and $V_0$ are given constant.

\subsection{Transfer Equation}
\paragraph{} Transfer function is also system descriptor. In contrant differential equation, transfer function is used in frequency domain.
\footnote{Differential equation is used in time domain}
\footnote{Concept of Transfer function is restricted to linear, time invariant different equational system}
\footnote{Transfer function is used to design systems also it can be used in time domain.}

\paragraph{} There are more to say about transfer function. I try to list something about transfer function that has to be kept it in your mind.
\begin{itemize}
\item Transfer function is operational method for expressing differential equation that relates the output variable of a system.
\item Transfer function is independent from magnitude of systems.
\end{itemize}
\footnote{Transfer function gives full description of dynamic characterication of system, but not give any physical structure information.}


  \em { \small {If transfer function,}} 

  \em{ \small {knows :  The output or response can be expressed with various system to understand nature of systems. }} 

  \em{ \small {unknows :  It may be found experimentally by studying with known  input and output of system.}}




\paragraph{} Example : Consider the electrical network shown Determine the transfer function
$H(sÞ) =  I_2(s) / V(s).$

\paragraph{} First we need to write two loop equation.

$[R_1 + 1 / Cs]I_1(s) - 1/Cs I_2(s) = V(s)$
$-1/Cs I_1(s) + [R_2 + Ls + 1/Cs] I_2(s) =0$

\paragraph{} Now solve the equation $I_2(s)$ . The second equation yields.

$I_1(s) = [LCs^2 + R_2Cs + 1] I_2(s)$

\paragraph{} Substituting this result with firest equation.

$(R_1Cs + 1)(LCs^2 + R_2Cs +1) I_2(s) - I_2(s) = CsV(s)$

\
$ H(s) = I_2(s)/V(s) =  Cs / (R_1Cs + 1)(LCs_2 + R_2Cs + 1) -1$
$ = 1/ R_1LCs^2 + (R1 R2 C + L)s + R_1 + R_2$

\subsection{Impulse Response}
\paragraph{} Impulse response is system descriptor that works on time domain. Impulse response works on limited area such as;
\begin{itemize}
\item the linear time invariant
\item time varying system
\end{itemize}
\footnote{ Important to know initial condition must be 0.}
\footnote{ it may be better to know what these systems are as description from wikipedia.}
\footnote {The impulse response show with $h(t)$.}

\subsubsection{ Definition :} 
\paragraph{}The impulse response function $h(t)$ of a system with zero initial condition is the system’s output when its input is the unit impulse function $\theta (t)$

\paragraph{} There  is a special case of linear, time invariant system where  $Y(s) = H(s)U(s)$ holds. Let input be unit impulse function $u(t) = \theta(t)$. Since $u(t) =1$  we have  $Y(s) = H(s)$. So 
\begin{itemize}
\item Transfer function becomes $H(s) = L(h(t))$.
\item Impulse response function $h(t) = L^-1(H(s)]$.
\end{itemize}
\footnote{ $H(s)$ is in frequency domain $h(t)$ is in time domain. $H(t)$ and $h(s)$ are same descriptor.}
\footnote{This is one tricky way to determine impulse response in linear, time invariant system provided by $H(s)$ is to use equation $h(t) = L^-1(H(s))$.}

\subsection{State space equation}
\subsubsection{General point}
\paragraph{} State space equation or simply state equation  is descriptor in time domain system. This method is used in very wide category of system. Such as;
\begin{itemize}
\item linear and non linear systems.
\item time invariand and time variant systems.
\item systems with zero initial condition and others.
\end{itemize}
\footnote{The term state of a system refers to the past, present, and
future of the system. From the mathematical point of view, the state of a system is
expressed by its state variables. Usually, a system is described by a finite state variable which is designated by $x_1(t), x_2(t),..., x_n(t)$.}

\subsubsection{Definition}
\paragraph{} The state variables $x_1(t), x_2(t); . . . ; x_n(t)$ of a system are defined as a (minimum)
number of variables such that if we know (a) their values at a certain moment t0 ,
(b) the input function applied to the system for t ! t0 , and (c) the mathematical
model which relates the input, the state variables, and the system itself, then the
determination of the system’s states for t > t0 is guaranteed.

 \[ u(t) = \begin{pmatrix}u_1(t)\\ u_2(t)\\\vdots u_n(t)\end{pmatrix} \]


\paragraph{} n is the number of input(s). Output vector $y(t)$ is indicated like below.

 \[ y(t) = \begin{pmatrix}y_1(t)\\ y_2(t)\\\vdots y_m(t)\end{pmatrix} \]

\paragraph{} m is number of output(s). The state vector $x(t)$ is indicated like below.

 \[ x(t) = \begin{pmatrix}x_1(t)\\ x_2(t)\\\vdots x_k(t)\end{pmatrix} \]

\paragraph{} m is the number of state variables.


\footnote{The state equation are a set of n first order differential equation which relates the input vector  $u(t)$ with the state vector x(t) and has form  $x^. = f[x(t), u (t)]$}
\footnote{f is column with n elements}
\footnote{generally f is a complex nonlinear function of $x(t(, u(t)$}

\paragraph{} Important to know about state space equation.
\begin{itemize}
\item  'State equations can describe a large category of systems, such as linear and
nonlinear systems, time-invariant and time-varying systems, systems with
time delays, systems with nonzero initial conditions, and others.'
\item 'Due to the fact that state equations are a set of first-order differential
   equations, they can be easily programmed and simulated on both digital
   and analog computers.'
\item  State equations, by their very nature, greatly facilitate both in formulating
   and subsequently in investigating a great variety of properties in system
   theory, such as stability, controllability, and observability. They also facil-
   itate the study of fundamental control problems, such as pole placement,
   optimal and stochastic control, and adaptive and robust control.

\item  State equations provide a more complete description of a system than the
   other three methods: i.e., differential equations, transfer function, and
   impulse response. This is because state equations involve very important
   additional information about the system—namely, the system’s state. This
   information is particularly revealing of the system’s structure (e.g., regard-
   ing controllability, observability, pole-zero cancellation in the transfer
   function, etc.).
\end{itemize}

\subsection{Block Diagram}


 
\section{Laplace Transform}
\subsection{Historical review}

\paragraph{}The laplace transform was builded by mathematician and astronomer Pierre Simon Laplace. Laplace transform method was used on probability theory in Pierre's works. And from 1744 Leonhard Eular applies integral with below formula.

z = $ \int_a^b X(x)e^{-ax} *a^x\,dx.$ and z = $ \int_a^b X(x)x^A\,dx.$


\subsection{Why Laplace Transform is used}
\paragraph{} Laplace transform is used to solve linear differential equation. Thanks to Laplace transform we can convert some common functions such as sinusoidal function, damped sinusoidal function, exponential function into algebraic function of complex variable s.
ref : ogata book



\subsection{How is it used}
\paragraph{} Differentiation and integration can be replaced by algebraic operation in the complex plane. \cite{Ref1}

\footnote{So we can see two thing Complex variable is important issue and Euler's transform. You can see in the quick historical review, Leonhard Euler make Laplace transform operational.}

\subsection{Complex variable and Complex function}  
    \subsubsection{Complex variable}
* Complex variable that is a variable that has a real part or imaginary part wihch both are constant. \\ 
\[alpha=\frac{\beta+i\gamma}{}\] \\
or \\
\[alpha=\frac{\beta+j\gamma}{}?] \\
 \]
\nocite{*}
Some formulas : \\
Magitude of F(s) : \\
\[\sqrt{Fx^2+jFy^2}\]  \\
Angle of teta F)s) \\
\[\\tangent^-1(F(y)/F(x)\] \\

\subsubsection{Complex Function}
* Complex function F(s) is a function of s that has real and imaginary part.\\
\includegraphics{foo-mps.pdf} \\
\footnote {graph for complex number} 
\footnote{x is rea axis} 
\footnote{y is imaginary axis}
The angel is measured counterclockwise from positive real axis.
Complex conjugate of f(s)
\[F(x)-jF(y)\]
 Complex function commonly encountered in linear control system analysis are single-valued function of s and are uniquely determined for agiven value of s.
\begin{itemize}
 \item Definition single-valued function :
    A single-valued function is function that, for each point in the domain, has
a unique value in the range. It is therefore one-to-one or many-to-one. (Ref :
Wolfram Mathworld)
\footnote{i think you learn something more (besides defition) from Wolfram,
wikipedia or something else}
\end{itemize}
 A complex function is G(s) is analytic iff g(s) and its all derivatives of G(s)
is in that region.The derivative of G(s) is found in that way.
\[ g'(z)=\lim_{h\to0}\frac{g(z+h)-g(z)}h \]


$\Delta s = \Delta \sigma + j \Delta w$ can approach a zero along
an infinite different path

* For a particular path $\Delta s = \Delta \sigma$  (path is in real axi)
\footnote{Why is it in real axi ?}

* For another particular path $\Delta s =j  \Delta \sigma$ (path is in
imaginary axis)
\footnote{Why is it in imaginary axis}
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