root/applications/dual/SIDP/text/ch4.tex @ 919

V t� kapitole je pops�jednoduch��diskutovan�ite{astrom1986dual}. Na n�jsou porovn� ��lgoritmy uveden� p�l�apitole.

\section{Popis 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 \cite{astrom1986dual} 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�