Changeset 1309 for applications/dual/texts/pmsm_system.lyx

Timestamp:

03/24/11 18:32:27 (14 years ago)

Author:

smidl

Message:

description of reduced order system

Files:

• applications/dual/texts/pmsm_system.lyx (modified) (1 diff)

applications/dual/texts/pmsm_system.lyx

@@ -520 +520 @@
 \end_layout
 
+\begin_layout Subsubsection
+Reduced order version
+\end_layout
+
+\begin_layout Standard
+Equations (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:model"
+
+\end_inset
+
+) ca be restructured by considering 
+\begin_inset Formula $i_{s\alpha}$
+\end_inset
+
+ and 
+\begin_inset Formula $i_{s\beta}$
+\end_inset
+
+ as external observations.
+ Then the state variables are 
+\begin_inset Formula $x_{t}=[\omega_{t},\vartheta_{t}]$
+\end_inset
+
+ and as follows:
+\begin_inset Formula \begin{align}
+\om_{t+1} & =d\om_{t}+e\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right),\label{eq:rord-state}\\
+\vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\nonumber \end{align}
+
+\end_inset
+
+and the onbservation equations are
+\begin_inset Formula \begin{align}
+\isa{,t+1} & =a\,\isa{,t}+b\omega_{t}\sin\vartheta_{t}+c\usa{,t},\nonumber \\
+\isb{,t+1} & =a\,\isb{,t}-b\omega_{t}\cos\vartheta_{t}+c\usb{,t},\label{eq:rord-obs}\end{align}
+
+\end_inset
+
+These equations are used by the EKF to update estimates of mean values.
+ The new matrices 
+\begin_inset Formula $A$
+\end_inset
+
+ and 
+\begin_inset Formula $C$
+\end_inset
+
+ are
+\begin_inset Formula \[
+A=\left[\begin{array}{cc}
+d & -e(\isb{}\sin\th+\isa{}\cos\th)\\
+\Dt & 1\end{array}\right],\quad C=\left[\begin{array}{cc}
+b\sin\th & b\om\cos\th\\
+-b\cos\th & b\om\sin\th\end{array}\right].\]
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Subsection
 Test system