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\documentclass[a4paper, oneside, 12pt, final, hidelinks]{article}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{siunitx}
\usepackage[czech]{babel}
\usepackage{csquotes}
\usepackage{fontspec}
\usepackage[final]{pdfpages}
\usepackage{outlines}
\usepackage{minted}
\usepackage{physics}
\usepackage{amsmath}
\usepackage{nicematrix}
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\usepackage[normalem]{ulem}
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%\usepackage{showframe}

\sisetup{range-phrase = \text{--}, group-separator = \text{\,}, group-four-digits = true, output-decimal-marker = {,}}
\MakeOuterQuote{"}

\setminted
{
frame=lines,
framesep=2mm,
baselinestretch=1.2,
fontsize=\footnotesize,
linenos=true,
}

\setmintedinline[c++]
{
frame=none,
baselinestretch=1.2,
fontsize=\footnotesize,
linenos=false,
}

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\newcommand{\msout}[1]{\text{\sout{\ensuremath{#1}}}}

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\begin{document}

\title{Model zvyšujícího měniče digitronových hodin}
%\author{Martin Vítek}
\maketitle
%
%\tableofcontents
%\newpage


\section{Zvyšující měnič}
\begin{figure}[!h]
    \centering
    \includegraphics[width=\textwidth]{../BP/pictures/zvysujici_menic}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{Q24 v uzavřeném stavu}
\begin{figure}[!ht]
    \centering
    \includegraphics[width=.9\textwidth]{boost_model/boost_model_closed.pdf}
\end{figure}

\begin{align}
U_{in} &= (R_{62} + R_{Q25} + R_{L3})i_l + L_3\derivative{i_l}{t} + R_{Q24}i_l \\
-C_{out}\derivative{u_c}{t}  &= \frac{u_c}{R_{63}+R_{load}}
\end{align}

\vspace{1em}
\centerline{}

\begin{align}
\derivative{i_l}{t} &= \frac{1}{L_3}U_{in}-\frac{R_{62}+R_{Q25}+R_{L3}+R_{Q24}}{L_3}i_l \\
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\derivative{u_c}{t} &= -\frac{1}{C_{out}(R_{63}+R_{load})}u_c \\
\derivative{U_{in}}{t} &= 0
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\end{align}

\vspace{1em}
\centerline{}

\begin{align}
\derivative{}{t}
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\begin{bNiceMatrix} i_l \\ u_c \\ U_{in} \end{bNiceMatrix}
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&=
\begin{bNiceMatrix}
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-\frac{R_{62}+R_{Q25}+R_{L3}+R_{Q24}}{L_3} & 0 & \frac{1}{L_3} \\
0 & -\frac{1}{C_{out}(R_{63}+R_{load})} & 0 \\
0 & 0 & 0
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\end{bNiceMatrix}
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\begin{bNiceMatrix}i_l \\ u_c \\ U_{in}\end{bNiceMatrix}
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\end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{Q24 v otevřeném stavu}
\begin{figure}[!ht]
    \centering
    \includegraphics[width=.9\textwidth]{boost_model/boost_model_opened.pdf}
\end{figure}

\begin{align}
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U_{in} &= (R_{62} + R_{Q25} + R_{L3})i_l + L_3\derivative{i_l}{t} + R_{s_{D6}}i_l + u_c \\
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C_{out}\derivative{u_c}{t} &= i_l - \frac{u_c}{R_{63}+R_{load}}
\end{align}

\vspace{1em}
\centerline{}

\begin{align}
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\derivative{i_l}{t} &= \frac{1}{L_3}(U_{in}-u_c)-\frac{R_{62}+R_{Q25}+R_{L3}+R_{s_{D6}}}{L_3}i_l \\
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\derivative{u_c}{t} &= \frac{1}{C_{out}}i_l-\frac{1}{C_{out}(R_{63}+R_{load})}u_c \\
\derivative{U_{in}}{t} &= 0
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\end{align}

\vspace{1em}
\centerline{}

\begin{align}
\derivative{}{t}
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\begin{bmatrix} i_l \\ u_c \\ U_{in} \end{bmatrix}
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=
\begin{bmatrix}
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-\frac{R_{62}+R_{Q25}+R_{L3}+R_{s_{D6}}}{L_3} & -\frac{1}{L_3} & \frac{1}{L_3} \\
\frac{1}{C_{out}} & -\frac{1}{C_{out}(R_{63}+R_{load})} & 0 \\
0 & 0 & 0
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\end{bmatrix}
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\begin{bmatrix}i_l \\ u_c \\ U_{in} \end{bmatrix}
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\end{align}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{"Zprůměrování" stavového modelu}
\begin{align}
\overline{A}  =A_1d + A_2(1-d) \\
\overline{B} = B_1d + B_2 (1-d)
\end{align}

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$A_1$, $B_1$ - Q24 v uzavřeném stavu, $A_2$, $B_2$ - Q24 v otevřeném stavu. Zanedbáno $R_{Q25}$ a $R_{s_{D6}}$.
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\vspace{1em}
\centerline{}

\begin{align}
\derivative{}{t}
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\begin{bmatrix} i_l \\ u_c \\ U_{in} \end{bmatrix}
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=
\begin{bmatrix}
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-\frac{R_{62}+R_{Q25}+R_{L3}+R_{s_{D6}}}{L_3} & -\frac{1}{L_3}(1-d) & \frac{1}{L_3} \\
\frac{1}{C_{out}}(1-d) & -\frac{1}{C_{out}(R_{63}+R_{load})} & 0 \\
0 & 0 & 0
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\end{bmatrix}
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\begin{bmatrix}i_l \\ u_c \\ U_{in} \end{bmatrix}
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\end{align}

\subsection{Odezva otevřené smyčky pro různé hodnoty střídy}
\begin{figure}[!ht]
    \centering
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    \includegraphics[width=1\textwidth, trim=4.9cm 1cm 4.3cm 1.8cm,clip]{figures/open_loop_response}
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\end{figure}


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\newpage
\section{Linearizace}
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\begin{align}
    D &= 1-d \\
    R_{in} &= R_{62}+R_{Q25}+R_{L3}+R_{s_{D6}} \\
    R_{out} &= R_{63}+R_{load}
\end{align}

\begin{align}
    \derivative{}{t}
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    \begin{bmatrix} i_l \\ u_c \\ U_{in} \end{bmatrix}
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    =
    \begin{bmatrix}
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        -\frac{R_{in}}{L_3} & 0 & \frac{1}{L_3} \\
        0 & -\frac{1}{C_{out}R_{out}} & 0 \\
        0 & 0 & 0
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    \end{bmatrix}
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    \begin{bmatrix}i_l \\ u_c \\ U_{in}\end{bmatrix}
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    +
    \begin{bmatrix}
        -\frac{\overline{u_c}}{L_3} \\
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        \frac{\overline{i_L}}{C_{out}} \\
        0
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    \end{bmatrix}
    D
\end{align}


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\section{Diskretizace}
\begin{align}
    \derivative{}{t}
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    \begin{bmatrix} i_l \\ u_c \\ U_{in} \end{bmatrix}
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    =
    \begin{bmatrix}
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        -\frac{R_{in}}{L_3} & 0 & \frac{1}{L_3} \\
        0 & -\frac{1}{C_{out}R_{out}} & 0 \\
        0 & 0 & 0
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    \end{bmatrix}
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    \begin{bmatrix}i_l \\ u_c \\ U_{in}\end{bmatrix}
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    +
    \begin{bmatrix}
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        -\frac{\overline{u_c}}{L_3} \\
        \frac{\overline{i_L}}{C_{out}} \\
        0
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    \end{bmatrix}
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    D
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\end{align}

\vspace{1em}
\centerline{}

\begin{align}
    \begin{bmatrix}
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        i_{l,t+1} \\ u_{c,t+1} \\ U_{in,t+1}
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    \end{bmatrix}
    =
    \boldsymbol{A}
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    \begin{bmatrix}i_{l,t} \\ u_{c,t} \\ U_{in,t}\end{bmatrix}
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    +
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    \boldsymbol{B}D
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\end{align}


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\section{Integrace}
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\begin{align}
    \Delta_{u_c} &= u_c - u_c^* \\
    \int \Delta_{u_c} &= \Delta \\
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    \Delta_{t+1} &= \Delta_t + (u_{c,t} - u_{c,t}^*)\mathrm{d}t
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\end{align}

\begin{align}
    \begin{bmatrix}
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        i_{l,t+1} \\ u_{c,t+1} \\ U_{in,t+1} \\ \Delta_{t+1}
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    \end{bmatrix}
    &=
    \begin{bmatrix}
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        \boldsymbol{A}_{1,1} & \boldsymbol{A}_{1,2} & \boldsymbol{A}_{1,3} & 0 \\
        \boldsymbol{A}_{2,1} & \boldsymbol{A}_{2,2} & \boldsymbol{A}_{2,3} & 0 \\
        \boldsymbol{A}_{3,1} & \boldsymbol{A}_{3,2} & \boldsymbol{A}_{3,3} & 0 \\
        0 & \mathrm{d}t & 0 & 1 \\
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    \end{bmatrix}
    \begin{bmatrix}
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        i_{l,t} \\ u_{c,t} \\ U_{in,t} \\ \Delta_t
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    \end{bmatrix}
    +
    \begin{bmatrix}
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        \boldsymbol{B}_{1,1} \\
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        \boldsymbol{B}_{2,1} \\
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        \boldsymbol{B}_{3,1} \\
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        0
    \end{bmatrix}
    D +
    \begin{bmatrix}
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        0 \\ 0 \\ 0 \\ -\mathrm{d}t
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    \end{bmatrix}
    u_{c,t}^*
\end{align}

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\begin{align}
    \begin{bmatrix}
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        i_{l,t+1} \\ u_{c,t+1} \\ U_{in,t+1} \\ \Delta_{t+1}
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    \end{bmatrix}
    &=
    \begin{bmatrix}
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        \boldsymbol{A}_{1,1} & \boldsymbol{A}_{1,2} & \boldsymbol{A}_{1,3} & 0 \\
        \boldsymbol{A}_{2,1} & \boldsymbol{A}_{2,2} & \boldsymbol{A}_{2,3} & 0 \\
        \boldsymbol{A}_{3,1} & \boldsymbol{A}_{3,2} & \boldsymbol{A}_{3,3} & 0 \\
        0 & \mathrm{d}t & 0 & 1 \\
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    \end{bmatrix}
    \begin{bmatrix}
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        i_{l,t} \\ u_{c,t} \\ U_{in,t} \\ \Delta_t
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    \end{bmatrix}
    +
    \begin{bmatrix}
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        \boldsymbol{B}_{1,1} & 0  \\
        \boldsymbol{B}_{2,1} & 0  \\
        \boldsymbol{B}_{3,1} & 0  \\
        0 & -\mathrm{d}t
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    \end{bmatrix}
    \begin{bmatrix}
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        D \\ u_{c,t}^*
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    \end{bmatrix}
\end{align}

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\section{Regulátor}
\begin{align}
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    D &= -\boldsymbol{L}x_t - \boldsymbol{K}w \\
    D &= -\boldsymbol{L}
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    \begin{bmatrix}
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        i_{l,t} \\ u_{c,t} \\ U_{in,t} \\ \Delta_{t}
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    \end{bmatrix}
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    - K\cdot u_{c,t}^* \\
    [\boldsymbol{L},K] &= dlqr(\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{Q}, \boldsymbol{N}, \boldsymbol{R}) \\
    \msout{K} &= \{-\boldsymbol{C_w}[\boldsymbol{I}-(\boldsymbol{A}-\boldsymbol{BL})]^{-1}\boldsymbol{B}\}^{-1} \\
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    \boldsymbol{C}_w &= \begin{bmatrix}
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        0 & 1 & 0 & 0
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    \end{bmatrix}
\end{align}


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\end{document}