\documentclass[12pt,fleqn]{article}
\usepackage{slovak}
\usepackage{amsmath}
\title{Pr�klad na matematiku}
\author{Meno Priezvisko}
\pagestyle{empty}

\def\ve#1{\mathchoice{\mbox{\boldmath$\displaystyle#1$}}
{\mbox{\boldmath$\textstyle#1$}}
{\mbox{\boldmath$\scriptstyle#1$}}
{\mbox{\boldmath$\scriptscriptstyle#1$}}}
\newtheorem{theorem}{Veta}
\newtheorem{note}{Pozn�mka}

\begin{document}
\maketitle
\thispagestyle{empty}

\section{Z�klady}
\begin{description}
\item[Statick� zosilnenie] -- je definovan� ako $b_0/a_0$ (po��ta sa pre stabiln� syst�my).
\item[Prenos] -- v�eobecn� tvar prenosu syst�mu s~1 vstupom a 1
  v�stupom je dan� vz�ahom
\begin{equation}
 G(s)=\frac{B(s)}{A(s)}=\frac{b_m s^m + \cdots + b_1 s + b_0 }{a_n s^n + \cdots + a_1 s + a_0}
\end{equation}
\end{description}
Limitn� hodnoty pri jednotkovom skoku vstupnej veli�iny $U(s)=1/s$:
  \begin{eqnarray*}
    Y(s) &=& \frac{s-1}{s^2+3s+2} \frac{1}{s} \\
    \lim_{t\rightarrow 0}y(t) &=& \lim_{s\rightarrow \infty} s Y(s) 
         =  \lim_{s\rightarrow \infty}  \frac{s-1}{s^2+3s+2} \\
       &=&  \lim_{s\rightarrow \infty}  \frac{\frac{1}{s}-\frac{1}{s^2}}{1+\frac{3}{s}+\frac{2}{s^2}} = 0 
  \end{eqnarray*}
Priemern� hodnota
\begin{equation}
\hat{y}_i  = \frac{\displaystyle \sum_{k=1}^{N}\Delta \bar{u}_k y_{ik}
  }{\displaystyle \sum_{k=1}^{N} \left(\Delta \bar{u}_k\right)^2 }
\end{equation}
Prechodov� funkcia syst�mu 1. r�du je dan� vz�ahom
\begin{equation} \label{eq:id:21y}
y(t) = \left\{
\begin{array}{ll}
0 & t< T_d \\
Z\left(1-e^{-\dfrac{t-T_d}{T}}  \right)
& t \geq T_d
\end{array}
\right.
\end{equation}

Ak si definujeme stavy $x_1 = y_f$, $x_2=\dot{y}_f$, potom rovnicu
filtra pre $y$ m��eme prep�sa� pomocou stavov�ho opisu v~tvare
\begin{eqnarray*}
  \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} &=&
  \begin{pmatrix}
    0 & 1 \\ -\frac{1}{c_0^2} & -\frac{2}{c_0}
  \end{pmatrix}
  \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} +
  \begin{pmatrix} 0 \\ \frac{1}{c_0^2} \end{pmatrix} y \\
  \begin{pmatrix} z_1 \\ z_2 \\ z_3\end{pmatrix} &=&
  \begin{pmatrix}
    -\frac{1}{c_0^2} & -\frac{2}{c_0} \\
    -1 & 0 \\
     0 & -1 
  \end{pmatrix}
  \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} +
  \begin{pmatrix} \frac{1}{c_0^2} \\ 0 \\ 0 \end{pmatrix} y
\end{eqnarray*}
kde v�stupy zo stavov�ho opisu s� $z_1=\ddot{y}_f $,
$z_2=-y_f$, $z_3=-\dot{y}_f$. 

\begin{subequations}
\begin{align}
x&=y       & X&=Y       & a&=b+c\\
x'&=y'     & X'&=Y'     & a'&=b\\
x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
\end{align}
\end{subequations}

\section{Vety a defin�cie}

\begin{theorem}[Findeisen, 2003] 
  Suppose that 
  \begin{itemize}
  \item the terminal region $\Omega \in X $ is closed with $0 \in
    \Omega$ and that $G > 0$
  \item $\forall \ve{x} \in \Omega$ there exists an admissible input
    $\ve{u}_{\Omega}(\tau)$ such that $\ve{x}(\tau) \in \Omega$ and
    \begin{multline}
    \frac{\partial G}{\partial \ve{x}} \ve{f}(\ve{x}(\tau),
    \ve{u}_{\Omega}(\tau))+ F(\ve{x}(\tau),\ve{u}_{\Omega}(\tau)) \le
    0\\ \ve{x}(t) \in X,\quad \ve{u}(t) \in U, \quad t \ge 0, \forall \tau \in [0,T]
    \end{multline}
  \end{itemize}
  Then the region of attraction $\mathcal{R}$ consists of the states
  for which an admissible input exists.
\end{theorem}
Nasleduje d�le�it� pozn�mka.
\begin{note}\label{note:prva}
  Poznamen�vame, �e t�mto dokument kon��.
\end{note}
Toto bola pozn�mka ��slo~\ref{note:prva}.
\end{document}

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