Changeset 754 for applications/dual/texts

Timestamp:

12/10/09 23:59:34 (15 years ago)

Author:

smidl

Message:

fixes + LQ for dual control

Location:

applications/dual/texts

Files:

• simple_system_LIDP.lyx (modified) (4 diffs)
• simple_system_LIDP.pdf (modified) (previous)

applications/dual/texts/simple_system_LIDP.lyx

@@ -66 +66 @@
 \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}\nonumber \end{eqnarray}
+ZTRATA:\,\,\,\,\, L_{t} & = & (y_{t+1}-r_{t+1})^{2}\label{eq:los}\end{eqnarray}
 
 \end_inset
@@ -145 +145 @@
 
 \begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
+\begin_inset Formula \begin{eqnarray}
 SYSTEM:\,\, H_{t+1} & = & \left[\begin{array}{c}
 y_{t+1}\\
@@ -155 +155 @@
 \sigma e_{t}\\
 0\\
-0\end{array}\right]\\
-ZTRATA:\,\,\,\,\, L_{t} & = & (y_{t+1}-r_{t+1})^{2}+P_{t}u_{t}^{2}.\end{eqnarray*}
+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
@@ -180 +180 @@
 
 .
-\begin_inset CommandInset citation
-LatexCommand cite
-key "Ber:01"
-
-\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