root/applications/dual/texts/simple_system_LIDP.lyx @ 1130

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\begin_body

\begin_layout Title
System pro simulaci
\end_layout

\begin_layout Section*
Puvodni zadani:
\end_layout

\begin_layout Standard
Vychazime ze zadani 
\begin_inset CommandInset citation
LatexCommand cite
key "ThoClu:05"

\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray}
SYSTEM:\,\, y_{t+1} & = & y_{t}+bu_{t}+\sigma e_{t},\,\,\, e_{t}\sim\mathcal{N}(0,1),\label{eq:sys}\\
ZTRATA:\,\,\,\,\, L_{t} & = & (y_{t+1}-r_{t+1})^{2}\label{eq:los}\end{eqnarray}

\end_inset

Reseni schematicky:
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{equation}
V_{t}=\min_{u_{t}}\mathsf{E}_{e_{t},b}\left\{ L_{t}+V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \label{eq:dp}\end{equation}

\end_inset

kde stredni hodnota se pocita pres neurcitost v 
\begin_inset Formula $e_{t}$
\end_inset

 a pres neurcitost v 
\begin_inset Formula $b$
\end_inset

.
\end_layout

\begin_layout Standard
Pro linearni a Gausovsky system (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:sys"

\end_inset

) je k dispozici konjugovana hustota ve forme Normalniho rozlozeni pravdepodobno
sti 
\begin_inset Formula $f(b_{t})=\mathcal{N}(\hat{b}_{t},P_{t}),$
\end_inset

 jejiz parametry se vyvijeji rekurzivne, rovnice (28) v 
\begin_inset CommandInset citation
LatexCommand cite
key "ThoClu:05"

\end_inset

.
 Tim padem je mozne vycislit ocekavanou hodnotu pres 
\begin_inset Formula $b$
\end_inset

 v (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:dp"

\end_inset

) analyticky:
\begin_inset Formula \begin{eqnarray*}
V_{t} & = & \min_{u_{t}}\mathsf{E}_{e_{t},b}\left\{ (y_{t}+bu_{t}+\sigma e_{t}-r_{t+1})^{2}+V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \\
 & = & \min_{u_{t}}\mathsf{E}_{e_{t}}\left\{ (y_{t}+\hat{b}u_{t}+\sigma e_{t}-r_{t+1})^{2}+P_{t}u_{t}^{2}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} +\\
 &  & +\mathsf{E}_{e_{t},b}\left\{ V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \end{eqnarray*}

\end_inset

Muzeme provest preznaceni
\begin_inset Formula \[
V_{t+1}(H_{t})=\mathsf{E}_{e_{t},b}\left\{ V_{t+1}|y_{t},u_{t-1},y_{t-1},u_{t-2},\ldots\right\} \]

\end_inset

kde 
\begin_inset Formula $H_{t}=[y_{t},\hat{b}_{t},P_{t}]$
\end_inset

.
 Vysledna uloha je ekvivalentni tomu, kdyby zadani bylo:
\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray}
SYSTEM:\,\, H_{t+1} & = & \left[\begin{array}{c}
y_{t+1}\\
\hat{b}_{t+1}\\
P_{t+1}\end{array}\right]=\left[\begin{array}{c}
y_{t}+\hat{b}_{t}u_{t}\\
(28)\\
(28)\end{array}\right]+\left[\begin{array}{c}
\sigma e_{t}\\
0\\
0\end{array}\right]\label{eq:sys2}\\
ZTRATA:\,\,\,\,\, L_{t} & = & (y_{t+1}-r_{t+1})^{2}+P_{t}u_{t}^{2}.\label{eq:los2}\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
(28) je opet rovnice (28) z 
\begin_inset CommandInset citation
LatexCommand cite
key "ThoClu:05"

\end_inset

.
 Takto upravenou ulohu muzeme resit pomoci algoritmu 
\begin_inset CommandInset citation
LatexCommand cite
key "TodTas:09"

\end_inset

.
\end_layout

\begin_layout Subsection*
LQ rizeni
\end_layout

\begin_layout Standard
Algoritmus LQ rizeni je aplikovatelny v pripade, ze 
\begin_inset Formula $b$
\end_inset

 v (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:sys"

\end_inset

) je zname.
 V pripade, ze 
\begin_inset Formula $b$
\end_inset

 nezmame je mozne optimalni rizeni aproximovat tzv.
 receding horizon strategii.
 Tato strategie spociva v nahrazeni 
\begin_inset Formula $b\equiv\hat{b}_{t}$
\end_inset

, spocteni optimalniho zasahu, provedeni 
\begin_inset Formula $u_{t}$
\end_inset

 , oprava 
\begin_inset Formula $b_{t}$
\end_inset

 a opetovne prepocteni strategie.
\end_layout

\begin_layout Standard
Tomuto postupu se rika certainty equivalence.
 Nevyhodou tohoto pristupu je, ze chyba rizeni pro chybny odhad 
\begin_inset Formula $\hat{b}$
\end_inset

 je znacna.
\end_layout

\begin_layout Standard
Druhou moznosti aproximace je pouziti systemu (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:sys2"

\end_inset

) s nahradou 
\begin_inset Formula $\hat{b}_{t+1}=\hat{b}_{t}$
\end_inset

, 
\begin_inset Formula $P_{t+1}=P_{t}$
\end_inset

.
 Vysledek je velmi podobny jako u CE strategie, avsak do ztratove funkce
 pribyl penalizacni clen 
\begin_inset Formula $P_{t}u_{t}^{2}$
\end_inset

, ktery penalizuje velke hodnoty 
\begin_inset Formula $u_{t}$
\end_inset

.
 Pro velke hodnoty 
\begin_inset Formula $P_{t}$
\end_inset

 tak vznika preference pro male hodnoty 
\begin_inset Formula $u_{t}$
\end_inset

.
 Vysledne strategii rizeni se proto rika cautious, tedy opatrna.
 Nevyhodou teto strategie je prilisna 
\begin_inset Quotes eld
\end_inset

opatrnost
\begin_inset Quotes erd
\end_inset

, ktera vychazi z predpokladu konstantnosti 
\begin_inset Formula $P_{t}$
\end_inset

, tedy velke penalizace 
\begin_inset Formula $u_{t}$
\end_inset

 na celem horizontu.
 Kvuli aproximaci neni ve strategii zohlednen vliv 
\begin_inset Formula $u_{t}$
\end_inset

 na 
\begin_inset Formula $P_{t}$
\end_inset

, a tim i fakt, ze vhodne zvolene 
\begin_inset Formula $u_{t}$
\end_inset

 muze hodnoty 
\begin_inset Formula $P_{t}$
\end_inset

 snizit.
\end_layout

\begin_layout Standard
Tento efekt se da kompenzovat tim, ze predpokladame, ze 
\begin_inset Formula $P_{t}$
\end_inset

 bude s casem klesat, napr:
\begin_inset Formula \[
P_{t+1}=\frac{1}{2}P_{t}.\]

\end_inset

pripadne az do krajnosti:
\begin_inset Formula \[
P_{t+1}=0.\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "vs-world,world_classics,mk,world"
options "plain"

\end_inset


\end_layout

\end_body
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