Changeset 1244 for applications/dual/texts/pmsm_system.lyx

Timestamp:

11/02/10 21:20:17 (14 years ago)

Author:

smidl

Message:

Doc for PI for PMSM

Files:

• applications/dual/texts/pmsm_system.lyx (modified) (2 diffs)

applications/dual/texts/pmsm_system.lyx

@@ -1 @@
-#LyX 1.6.4 created this file. For more info see http://www.lyx.org/
+#LyX 1.6.7 created this file. For more info see http://www.lyx.org/
 \lyxformat 345
 \begin_document
@@ -625 +625 @@
 
 \begin_layout Subsection
+PI control
+\end_layout
+
+\begin_layout Standard
+The classical control is based on transformation to 
+\begin_inset Formula $dq$
+\end_inset
+
+ reference frame:
+\begin_inset Formula \begin{eqnarray*}
+i_{d} & = & i_{\alpha}\cos(\vartheta)+i_{\beta}\sin(\vartheta),\\
+i_{q} & = & i_{\beta}\cos(\vartheta)-i_{\beta}\sin(\vartheta).\end{eqnarray*}
+
+\end_inset
+
+Desired 
+\begin_inset Formula $i_{q}$
+\end_inset
+
+ current, 
+\begin_inset Formula $\overline{i}_{q}$
+\end_inset
+
+, is derived using PI controller
+\begin_inset Formula \[
+\overline{i}_{q}=PI(\overline{\omega}-\omega,P_{i},I_{i}).\]
+
+\end_inset
+
+This current needs to be achieved through voltages 
+\begin_inset Formula $u_{d},u_{q}$
+\end_inset
+
+ which are again obtained from a PI controller
+\begin_inset Formula \begin{eqnarray*}
+u_{d} & = & PI(-i_{d},P_{u},I_{u}),\\
+u_{q} & = & PI(\overline{i}_{q}-i_{q},P_{u},I_{u}).\end{eqnarray*}
+
+\end_inset
+
+These are compensated (for some reason) as follows:
+\begin_inset Formula \begin{eqnarray*}
+u_{d} & = & u_{d}-L_{S}\omega\overline{i}_{q},\\
+u_{q} & = & u_{q}+\Psi_{pm}\omega.\end{eqnarray*}
+
+\end_inset
+
+Conversion to 
+\begin_inset Formula $u_{\alpha},u_{\beta}$
+\end_inset
+
+ is 
+\begin_inset Formula \begin{align*}
+u_{\alpha} & =|U|\cos(\phi), & u_{\beta} & =|U|\sin(\phi)\\
+|U| & =\sqrt{u_{d}^{2}+u_{q}^{2}}, & \phi & =\begin{cases}
+\arctan(\frac{u_{q}}{u_{d}})+\vartheta & u_{d}\ge0\\
+\arctan(\frac{u_{q}}{u_{d}})+\pi+\vartheta & u_{d}<0\end{cases}\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+PI controller is defined as follows:
+\begin_inset Formula \begin{eqnarray*}
+x & = & PI(\epsilon,P,I)\\
+ & = & P\epsilon+I(S_{t-1}+\epsilon)\\
+S_{t} & = & S_{t-1}+\epsilon\end{eqnarray*}
+
+\end_inset
+
+Constants for the system:
+\begin_inset Formula \[
+P_{i}=3,\,\, I_{i}=0.00375,\,\, P_{u}=20,\,\, I_{u}=0.5.\]
+
+\end_inset
+
+The requested values for 
+\begin_inset Formula $\omega$
+\end_inset
+
+ should be kept in interval 
+\begin_inset Formula $<-30,30>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
 Poor-man's dual LQG control
 \end_layout