Changeset 918 for applications/dual

Timestamp:

04/28/10 00:16:27 (15 years ago)

Author:

zimamiro

Message:

 

Location:

applications/dual/SIDP/text

Files:

• baksimple.pdf (modified) (previous)
• ch1.tex (modified) (1 diff)
• ch2.tex (modified) (2 diffs)
• ch3.tex (modified) (1 diff)
• ch4.tex (modified) (1 diff)
• znaceni.tex (modified) (1 diff)

applications/dual/SIDP/text/ch1.tex ¶

@@ -46 +46 @@
 \end{equation}
 
-Takto specifikovan�loha se d�e�it pou�it�dynamick� programov� []. Dynamick�rogramov� je p�p k ��ptimaliza�ch � na kter�e m� d�t jako na posloupnost rozhodnut�pro kter�lat�zv. princip optimality.  Ten � �e optim��osloupnost rozhodnut��u vlastnost, �e pro libovoln�te� stav a rozhudnut�us��chna n�eduj� rozhodnut�ptim��zhledem k v��zhodnut�rvn�. D� �e pro ztr� tvaru \eqref{adi} plat�rincip optimality je snadn�e ho nal� nap�d v [].
+Takto specifikovan�loha se d�e�it pou�it�dynamick� programov� []. Dynamick�rogramov� je p�p k ��ptimaliza�ch � na kter�e m� d�t jako na posloupnost rozhodnut�pro kter�lat�zv. princip optimality.  Ten � �e optim��osloupnost rozhodnut��u vlastnost, �e pro libovoln�te� stav a rozhudnut�us��chna n�eduj� rozhodnut�ptim��zhledem k v��zhodnut�rvn�. D� �e pro ztr� tvaru \eqref{adi} plat�rincip optimality je snadn�e ho nal� nap�d v [ref].
 
 P��en�lohy stochastick� �� aditivn�tr�u je tedy mo�n�ostupovat, jak je u ���moc�ynamick� programov� zvykem. Minim��odnotu st� ztr� od okam�iku $t$ do $N$ v z�slosti na $x_t$ ozna�e $J_t(x_t)$. M� pro ni ps�

applications/dual/SIDP/text/ch2.tex ¶

@@ -44 +44 @@
 
 O��nou ztr� nyn�� ps�ve tvaru
-\begin{equation}
-J_N(I_N)=\tilde{g}_N(I_N)
-\end{equation}
-\begin{equation}
+\begin{gather}
+J_N(I_N)=\tilde{g}_N(I_N)\\
 J_t(I_t)=\min_{u_t \in U_t}\E_{w_t,y_{t+1}}\left\{\tilde{g}_t(I_t,u_t,w_t)+J_{t+1}((I_t,u_t,y_{t+1}))|I_t,u_t\right\} \qquad t=0,\ldots,N-1
-\end{equation}
+\end{gather}
 
 Tato � ji� m��ena pomoc�ynamick� programov�. P��en�udeme postupovat od konce �� horizontu a postupn�ledat $J_t(I_t)$. Potom libovoln�\pi=\{\mu_0,\ldots,\mu_{N-1}\}$, kter�ab�nim����n�tr� $J_0(y_0)$ je optim��osloupnost rozhodnut�
@@ -159 +157 @@
 P_{t+1}=(I-K_tA_t)P_t
 \end{equation}
-Celkov�edy od p�� odhadu parametru $N(\hat{\theta}_t,P_t)$ k nov� $N(\hat{\theta}_{t+1},P_{t+1})$ p�me pomoc�\begin{equation}
-K_t=P_tA_t(A_t^TP_tA_t+Q_t)^{-1}
-\end{equation}
-\begin{equation}
-\hat{\theta}_{t+1}=\hat{\theta}_t+K_t(x_{t+1}-f_t(x_t,u_t)-A_t\hat{\theta}_t)
-\end{equation}
-\begin{equation}
-P_{t+1}=(I-K_tA_t)P_t
-\end{equation}
+Celkov�edy od p�� odhadu parametru $N(\hat{\theta}_t,P_t)$ k nov� $N(\hat{\theta}_{t+1},P_{t+1})$ p�me pomoc�\begin{gather}
+K_t=P_tA_t(A_t^TP_tA_t+Q_t)^{-1}\\
+\hat{\theta}_{t+1}=\hat{\theta}_t+K_t(y_{t+1}-\tilde{h}_t(I_t,u_t)-A_t\hat{\theta}_t),\\
+P_{t+1}=(I-K_tA_t)P_t.
+\end{gather}
 
 Tato odhadovac�rocedura se naz�lman�ltr [ref].

applications/dual/SIDP/text/ch3.tex ¶

@@ -28 +28 @@
 P�u�it�etody Certainty equivalent control (CEC) [ref] se v rovnici pro o��nou ztr� nahrad��dn�eli�y sv��mi hodnotami. O��n�tr� tak p� v
 \begin{gather}
+\label{CE}
 J_N(I_N, \theta_N)=g_N(y_N),\\
 J_t(I_t, \theta_t)=\min_{u_t \in U_t}\left\{g_t(y_t,u_t,\hat{v}_t) +J_{t+1}(I_t,\theta_{t+1},u_t,\hat{y}_{t+1}))|I_t,\theta_t,u_t\right\}, \\ \qquad t=0,\ldots,N-1,

applications/dual/SIDP/text/ch4.tex ¶

@@ +1 @@
+V t� kapitole je pops�jednoduch��popsan�ef]. Na n�jsou porovn� ��lgoritmy uveden� p�l�apitole.
+
 \section{Popis syst�}
-\section{Transformace syst�}
+V�syst� je pops�jako
+\begin{gather}
+\label{simple}
+y_{t+1}=y_t+\theta_tu_t+v_{t+1} \qquad t=0,\ldots,N-1,\\
+v_t\sim N(0,\sigma^2).\\
+\theta_t\sim N(\hat{\theta},P_t),\\
+\cov(v_{t+1},\theta)=0.
+\end{gather}
+
+Ztr�vou funkci vol� kvadratickou, tedy
+\begin{equation}
+g(y_{0:N},u_{0:N-1},v_{0:N-1})=\sum_{t=0}^{N-1}y_{t+1}^2.
+\end{equation}
+
+Za odhadovac�roceduru pro parametr $\theta$ vezmeme Kalman�ltr. Pro syst�\eqref{simple} bude m�tvar
+\begin{gather}
+\label{kal}
+K_t=\frac{u_tP_t}{u_t^2P_t+\sigma^2}\\
+\hat{\theta}_{t+1}=\hat{\theta}_t+K_t(y_{t+1}-u_t\hat{\theta}_t),\\
+P_{t+1}=(1-K_tu_t)P_t. 
+\end{gather}
+
+O��n�tr� je
+\begin{equation}
+J_t(y_t,\theta_t)=\min_{u_t \in U_t}\E_{y_{t+1},v_t}\left\{y_{t+1}^2+J_{t+1}(y_{t+1},\theta_{t+1})|y_t,\theta_t,u_t\right\}, \qquad t=0,\ldots,N-1.
+\end{equation}
+
+Ta po dosazen� \eqref{simple} a �te�m proveden�t� hodnoty p� na tvar
+\begin{gather}
+\label{dos}
+J_t(y_t,\theta_t)=\min_{u_t \in U_t}\left\{(y_t+\hat{\theta}_tu_t)^2+u_t^2P_t+\sigma^2+\E_{y_{t+1},v_t}(J_{t+1}(y_{t+1},\theta_{t+1}))|y_t,\theta_t,u_t\right\}.
+\end{gather}
+
+\section{Specifika jednotliv��up� tomto odd� jsou pops� n�er�spekty algoritm�er�udeme srovn�t, p�likaci na syst�\eqref{simple}.
+
+\subsection{Certainty equivalent control}
+O��n�tr� \eqref{CE} prejde v
+\begin{gather}
+J_t(y_t, \theta_t)=\min_{u_t \in U_t}\left\{\hat{y}_{t+1}^2 +J_{t+1}(y_{t+1},\theta_{t+1})|I_t,\theta_t,u_t\right\}.
+\end{gather}
+St� hodnota v� je 
+\begin{equation}
+\hat{y}_{t+1}=y_t+\hat{\theta}_tu_t     
+\end{equation}
+a rozhodnut�ude tedy
+\begin{equation}
+\mu_t(y_t,\hat{\theta}_t)=-\frac{y_t}{\hat{\theta}_t}.
+\end{equation}
+
+\subsection{Metoda separace}
+V prvn�� metody separace polo�� ���h 
+\begin{equation}
+u_0=\sqrt{C-\frac{1}{P_0}}.
+\end{equation}
+T�se dle \eqref{kal} sn� rozptyl $P_0$ nezn�ho parametru $\theta$ na $\frac{1}{C}$. Konstanta $C$ by m� b�ena dostate� mal�aby odhad $\hat{\theta}$ pro druhou f� ��yl dostate� bl�o skute� hodnot�arametru $\theta$. P�ovn� jednotliv�goritm�l�me $C=100$.
+
+\subsection{SIDP}
+Dle \eqref{dos} je optim��u_t$ z�sl�a $(y_t,\hat{\theta}_t,P_t)$. P�mulaci m� tedy v ka�d��ov�okam�iku $t$ diskretizovat t�enzion��rostor nez�sle prom��le [ref] je v�ak p�amotnou simulac�hodn�� k transformaci prostoru $(y_t,\hat{\theta}_t,P_t,u_t)$ do nov�om��\eta_t,\beta_t,\zeta_t,\nu_t)$ dle
+\begin{gather}
+\eta_t=\frac{y_t}{\sigma} \\
+\beta_t=\frac{\hat{\theta}_t}{\sqrt{P_t}} \\
+\zeta_t=\frac{1}{\sqrt{P_t}} \\
+\nu_t=\frac{u_t\sqrt{P_t}}{\sigma}
+\end{gather}
+
+Sou�n�� neur�ost ve v� \eqref{simple} reprezentovat jedinou normalizovanou n�dnou veli�ou podle
+\begin{equation}
+s_t=\frac{y_{t+1}-y_t+\hat{\theta}_tu_t}{\sqrt{u_t^2P_t+\sigma^2}} \sim N(0,1).
+\end{equation}
+ 
+Rovnice pro v�\eqref{simple} a n�eduj� odhad nezn�ho parametru \eqref{kal} tak p� v
+\begin{gather}
+\eta_{t+1}=\eta_t+\beta_t\nu_t+\sqrt{1+\nu^2}s_t\\
+\beta_{t+1}=\sqrt{1+\nu^2}\beta_t+\nu_ts_t
+\end{gather}
+
+P�me-li k vhodn�praven���n�tr�, dostaneme 
+\begin{align}
+V_t(\eta_t,\beta_t,\zeta_t)&=\frac{J_t(y_t,\hat{\theta}_t,P_t)}{\sigma^2}\\
+&=\min_{\nu_t }\left\{(\eta_t+\beta_t\nu_t)^2+\nu_t^2+1+\E_{y_{t+1},v_t}(V_{t+1}(\eta_{t+1},\beta_{t+1},\zeta))\right\}.
+\end{align}
+
+Nyn�po�me o��nou ztr� pro $N-1$. 
+\begin{equation}
+V_{N-1}(\eta_{N-1},\beta_{N-1},\zeta_{N-1})=\min_{\nu_{N-1}}\left\{(\eta_{N-1}+\beta_{N-1}\nu_{N-1})^2+\nu_{N-1}^2+1\right\}.
+\end{equation}
+
+Derivac��� optim���h jako
+\begin{equation}
+\label{optcon}
+\nu_{N-1}=-\frac{\eta_{N-1}\beta_{N-1}}{1+\beta_{N-1}^2}
+\end{equation}
+a o��nou ztr�
+\begin{equation}
+V_{N-1}(\eta_{N-1},\beta_{N-1},\zeta_{N-1})= \frac{\eta_{N-1}^2+1}{\beta_{N-1}^2+1}
+\end{equation}
+
+Proto�e optim���h $\nu_{N-1}$ ani o��n�tr� $V_{N-1}$ nez�s�a $\zeta_{N-1}$, d� tvaru $V_t$ nebude rovn�optim���h $\nu_t$ a o��n�tr� $V_t$ z�set na $\zeta_t$. P�skretizaci tedy sta�uva�ovat pouze dvoudimenzion��rostor nez�sle prom��\eta_t,\beta_t)$.
+
 \section{Srovn� jednotliv��up�

applications/dual/SIDP/text/znaceni.tex ¶

@@ -7 +7 @@
 &a_{t:s}&&\text{posloupnost veli� } (a_t,a_{t+1}, \ldots, a_s)\\
 &g_{t:s}(a_{t:s})&&\text{posloupnost funk�ch hodnot } (g_t(a_t),g_{t+1}(a_{t+1}), \ldots, g_s(a_s))\\
-&|H|&&\text{po� prvk�no�in�}
+&|H|&&\text{po� prvk�no�in� H
 \end{align*}