\documentclass[conference]{IEEEtran}

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\input{pst-plot}
\input{pst-node}
\input{pst-coil}

\input{pogbase_new_par}
\input{BG_base.tex}

\input{mathbase}
\input{VPPC09_POG_Automotive_Modelling_Base}

 \newrgbcolor{mygreen}{0 0.6 0}
 \newrgbcolor{vgreen}{0.1 0.5 0.1}
 \newrgbcolor{vred}{0.5 0.1 0.1}
 %\newrgbcolor{green}{0 0 0}
 %\newrgbcolor{red}{0 0 0}

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\begin{document}

\title{The POG Technique for Modeling Planetary Gears and Automotive Hybrid Systems}

\author{\IEEEauthorblockN{Roberto Zanasi}
\IEEEauthorblockA{Information Engineering Department\\
University of Modena e Reggio Emilia\\
Via Vignolese 905\\ 41100
Modena, Italy\\
roberto.zanasi@unimore.it} \and
\IEEEauthorblockN{Federica Grossi}
\IEEEauthorblockA{Information Engineering Department\\
University of Modena e Reggio Emilia\\
Via Vignolese 905\\ 41100 Modena, Italy\\
federica.grossi@unimore.it} }

\maketitle


\begin{abstract}
In this paper the Power-Oriented Graphs (POG) technique is used for
modeling planetary gears and CVT structures within automotive hybrid
systems. Some basic properties of the POG technique are given. In
particular, the use of congruent state space transformations for
reducing the system is analyzed.  Some simulation results are
finally reported.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% no \IEEEPARstart
Nowadays, the planetary gears and the CVT structures are key
elements for the design of new hybrid power structures in the
automotive area. In this paper the full dynamic models of these
systems are obtained using the the POG modeling technique. The
obtained dynamic model shows a very compact graphical representation
that clearly puts in evidence the power internal structure of the
motor. The paper is organized as follows: Sec.~\ref{POGproperties}
describes the basic properties of the POG modeling technique.
Sec.~\ref{ElectricalMotors} shows how the POG technique can be used
for modeling  the planetary gears and the CVT structures. Finally,
in Sec.~\ref{Simulations} some simulation results are reported.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Power-Oriented Graphs basic principles}\label{POGproperties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Power-Oriented Graphs technique, see \cite{Zana1} and
\cite{bodypap93}, is suitable for modeling physical systems. The POG
block schemes are normal block diagrams combined with a particular
modular structure essentially based on the use of the two blocks
shown in Fig.~\ref{figflow}.a and Fig.~\ref{figflow}.b: the {\em
elaboration block} (e.b.) stores and/or dissipates energy (i.e.
springs, masses, dampers, capacities, inductances, resistances,
etc.); the {\em connection block} (c.b.) redistributes the power
within the system without storing nor dissipating energy (i.e. any
type of gear reduction, transformers, etc.).
 % The c.b. transforms the power variables with the constraint $\x_{1}\tras\y_{1}=\x_{2}\tras\y_{2}$.
The e.b.~and the c.b.~are suitable for representing both scalar and
vectorial systems. In the vectorial case, $\G(s)$ and $\K$ are
matrices: $\G(s)$ is always a square matrix composed by positive
real transfer functions; matrix $\K$ can also be rectangular. The
circle present in the e.b. is a summation element and the black spot
represents a minus sign that multiplies the entering variable. The
main feature of the Power-Oriented Graphs is to keep a direct
correspondence between the dashed sections of the graphs and real
power sections of the modeled systems: the scalar product
$\x\tras\y$ of the two {\em power vectors} $\x$ and $\y$ involved in
each dashed line of a power-oriented graph, see Fig.~\ref{figflow},
has the physical meaning of {\em the power flowing through that
particular section}. The Bond Graphs technique, see \cite{karnopp},
is based on the same idea, but it uses a different and specific
graphical representation.

 \begin{figure}[t]
 \centering
 \schemaPOG{(-2.0,-1.0)(40,12)}{2.1mm}{
 \thicklines
 \bloin{\x_{1}}{\y}
 \rput(3,-2.9){\mbox{\footnotesize a) {\em e. b.}}}
 \blovigiu{\ds \G(s)}{}{\dst}
 \bloout{\x_{2}}{\y}
 \addspace{6}
 \bloin{\x_{1}}{\y_{1}}
 \rput(2.25,-2.9){\mbox{\footnotesize b) {\em c. b.}}}
 \blokxyor{2.5}{2.5}{$\K$}{$\K$\tras}{}{}
 \bloout{\x_{2}}{\y_{2}}
 %
 \rput[l](5.5,-2.9){\mbox{\footnotesize c) {\em Across and through variables}}}
 \rput[l](4.5,5){\scriptsize
 \begin{array}{|@{\;}l@{\;}|@{\;}l@{\;}|@{\;}l@{\;}|}\hline
 & \multicolumn{2}{c|}{\mbox{\bf POG Variables}}  \\ \hline
   \mbox{\bf Domains:} & \mbox{\em Across: }     \vgreen v_e
                       &   \mbox{\em Through: }  \vred v_f
 \\ \hline
 \mbox{Electrical}&  \mbox{Voltage } \vgreen V  &  \mbox{Current } \vred I \\ \hline
 \mbox{Mec. Tra.} & \mbox{Velocity }   \vgreen v & \mbox{Force } \vred  F \\ \hline
 \mbox{Mec. Rot.} & \mbox{Ang. Vel. }  \vgreen \omega   & \mbox{Torque } \vred \tau \\ \hline
 \mbox{Hydraulic} & \mbox{Pressure }  \vgreen P  & \mbox{Flow rate }  \vred    Q  \\  \hline
\end{array}
 } }
 %%
%
% %%%%%%%%%%%%%%%%
\caption{\label{figflow} POG basic blocks and variables:  a) {\em
elaboration block}; b) {\em connection block}; c) {\em across and
through variables}.}
 \end{figure}
 %
\begin{figure}[t]
 \begin{center}
 %
 %
 \schemaPOG{(-3.0,-3)(9.0,22)}{2.2mm}{
 \put(0,19){
 \setlength{\unitlength}{1.6mm}
 \psset{unit=1.0\unitlength}
 \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](0,0){0.5}
 \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](8,0){0.5}
 \rput(0,0){\punto}
 \rput(8,0){\punto}
 \psline[linewidth=1.2pt]{-}(0,0)(2,0)
 \psline[linewidth=1.2pt]{-}(6,0)(8,0)
 \psline[linewidth=0.4pt]{-}(0,-8)(8,-8)
 \rput[l](2,0){\framebox(4,3){PE}}
 \rput(0,-8){\punto}
 \psline{->}(0,-7.5)(0,-0.5)
 \rput[r](-0.5,-4){$\vgreen v_{e1}$}
 \rput[r](-0.5,-8){\footnotesize $0$}
 \psline{->}(-3,0)(-0.5,0)
 \rput[b](-2,0.5){$\vred v_{f}$} %%%
 \rput(8,-8){\punto}
 \psline{->}(8,-7.5)(8,-0.5)
 \rput[l](8.5,-4){$\vgreen v_{e2}$}
 \rput[l](8.5,-8){\footnotesize $0$}
 \psline{->}(8.5,0)(11,0)
 \rput[b](10,0.5){$\vred v_{f}$}
 %
 \rput[b](0,2.25){\footnotesize $1$}
 \rput[b](8,2.25){\footnotesize $2$}
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(0,-7.5)(0,1.75)
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(8,-7.5)(8,1.75)
  }
 \thicklines %
%   \psellipse[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt](3,10)(2.2,1.2)
%   \psellipticarc[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{->}(-9.75,4.0)(2.1,2.1){-70}{250}
%   \psline[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{<-}(-8.0,5.0)(1,9)
%   \rput(-9.75,4.0){\tiny \mbox{\shortstack{Kirchhoff's \\voltage law}}}
 \bloin{\vgreen v_{e1}}{\vred v_{f}}
 \rput[t](3,-2.5){\mbox{\footnotesize a) Element in series}.}
 \POGvigiu{\scr f_e(\!{\vgreen v_{e}}\!)}{\vgreen v_{e}}{\vred v_{f}}{\dst}
 \bloout{\vgreen v_{e2}}{\vred v_{f}}
 }
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hspace{10mm}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
 \schemaPOG{(-3.0,-3)(9.0,22)}{2.2mm}{
 \put(0,19){
 \setlength{\unitlength}{1.6mm}
 \psset{unit=1.0\unitlength}
  \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](4,0){0.5}
 \pscircle[fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](4,-8){0.5}
 \rput(0,0){\punto}
 \psline[linewidth=1.2pt]{-}(0,0)(4,0)(4,-2)
 \rput[t](4,-2){\framebox(3,4){PE}}
 \psline[linewidth=1.2pt]{-}(4,-6)(4,-8)(0,-8)
 \rput(0,-8){\punto}
 \psline{->}(0,-7.5)(0,-0.5)
 \rput[r](-0.5,-4){$\vgreen v_{e}$}
 \psline{->}(-3,0)(-0.5,0)
 \rput[b](-2,0.5){$\vred v_{f1}$} %
 %
 \rput(4,0){\punto}
 \rput(4,-8){\punto}
 \rput(8,0){\punto}
 \rput(8,-8){\punto}
 \psline[linewidth=1.2pt]{-}(4,0)(8,0)
 \psline[linewidth=1.2pt]{-}(4,-8)(8,-8)
 \psline{->}(8,-7.5)(8,-0.5)
 \rput[l](8.5,-4){$\vgreen v_{e}$}
 \psline{->}(8.5,0)(11,0)
 \rput[b](10.0,0.5){$\vred v_{f2}$}
 %
 \rput[b](4,1.0){\footnotesize$1$}
 \rput[t](4,-9){\footnotesize$2$}
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(0,-7.5)(0,1.75)
 \psline[linewidth=0.4pt,linestyle=dashed]{-}(8,-7.5)(8,1.75)
% \rput[t](4,-11.5){\footnotesize a) Elemento parallelo}
  }
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \thicklines %
 %   \psellipse[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt](3,10)(2.2,1.2)
 %   \psellipticarc[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{->}(-9.75,4.0)(2.1,2.1){-70}{250}
 %   \psline[linestyle=dashed,linecolor=mygreen,linewidth=0.3pt]{<-}(-8.0,5.0)(1,9)
 %   \rput(-9.75,4.0){\tiny \mbox{\shortstack{Kirchhoff's \\voltage law}}}
 \bloin{\vgreen v_{e}}{\vred v_{f1}}
 \rput[t](3,-2.5){\mbox{\footnotesize b) Element in parallel}.}
 \POGvisu{\scr f_f(\!{\vred v_{f}}\!)}{\vgreen v_{e}}{\vred v_{f}}{\dst}
 \bloout{\vgreen v_{e}}{\vred v_{f2}}
 }
 %
 \end{center}
\caption{POG representations of  Physical Elements (PE): a)
connected in series (inputs  {\vgreen $v_{e1}$}, {\vgreen
$v_{e2}$}); b) connected in parallel (inputs  {\vred $v_{f1}$},
{\vred $v_{f2}$}).}\label{PE_connected_in_series}
\end{figure}
 %
The main energetic domains encountered in modeling physical systems
are the electrical, the mechanical (translational and rotational)
and the hydraulic, see Fig.~\ref{figflow}.c. Each energetic domain
is characterized by two {\em power variables}: an {\em
across-variable} ${\vgreen v_e}$ defined between two points (i.e.
the voltage $V$), and a {\em through-variable} ${\vred v_f}$ defined
in each point of the space (i.e. the current $I$). Each Physical
Element (PE) interacts with the external world through the power
sections associated to its terminals. A Physical Element is
connected {\em in series} when its terminals share the same
through-variable ${\vred v_f}$: see the physical element and the
corresponding POG scheme in Fig.~\ref{PE_connected_in_series}.a. A
Physical Element is connected {\em in parallel} when its terminals
share the same across-variable ${\vgreen v_e}$: see the physical
element and the POG scheme in Fig.~\ref{PE_connected_in_series}.b.
An example of POG modeling where a C parallel element is connected
with an R series element is shown in Fig.~\ref{Esempio_CP_RS}.
 % The $C$ and $R$ physical elements are described by two POG elaboration blocks.
There is a direct corrispondenze between physical power
sections and dashed sections in the POG model. Note:  the summation
elements present in the elaboration blocks are a mathematical
description of the current and voltage Kirchhoff's laws applied to
the considered electrical system.
%
%
% \begin{figure}[t]
% \centering
% \schemaPOG{(-2.0,-1.0)(20,13)}{2.8mm}{
% \thicklines
% \bloin{\x_{1}}{\y}
% \rput(3,12.5){\footnotesize \mbox{a) {\em elaboration block}}}
% \blovigiu{\ds \G(s)}{}{\dst}
% \bloout{\x_{2}}{\y}
% \addspace{7}
% \bloin{\x_{1}}{\y_{1}}
% \rput(2.25,12.5){\footnotesize \mbox{b) {\em connection block}}}
% \blokxyor{2.5}{2.5}{$\K$}{$\K$\tras}{}{}
% \bloout{\x_{2}}{\y_{2}}
% }
% %%
%\caption{\label{figflow} POG basic blocks:  a) {\em elaboration
%block}; b) {\em connection block}.}
% \end{figure}
% %
% %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %
 \begin{figure}[t]
 \centering
\begin{tabular}{@{}cc}
  % \hline  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \psfrag{V1}[r][r]{$\scr V_{1}$}
  \psfrag{V2}[l][l]{$\scr V_{2}$}
  \psfrag{V3}[r][r]{$\scr $}
  \psfrag{V4}[t][t]{$\scr V_{4}$}
  \psfrag{I1}[b][b]{$\scr I_{1}$}
  \psfrag{I2}[b][b]{$\scr I_{2}$}
  \psfrag{I3}[r][r]{$\scr I_{3}$}
  \psfrag{I4}[t][t]{$\scr I_{2}$}
  \psfrag{C}[l][l]{$\scr C$}
  \psfrag{R}[b][b]{$\scr R$}
  \includegraphics[width=3.8cm]{eC_0_R_1.eps}
   &
 \schemaPOG{(-2,-0.5)(13,13)}{2.1mm}{
 \thicklines
   \rput(-20,0){\nrsezgiu{$\scr 1$}{5.25}}
   \rput(-14.5,0){\nrsezgiu{$\scr 2$}{5.25}}
   \rput(-14.5,0){\psline[linestyle=dashed,linewidth=0.2pt](0,-0.75)(0,8.25)}
   \rput(-7.5,0){\nrsezgiu{$\scr 3$}{5.25}}
 \bloin{\scr I_1}{\scr V_1}
   \nrsezgiu{$\scr 1$}{-0.0}
   \rput[l](3.5,8.5){\scr I_{3}}
   \psellipse[linestyle=dashed,linecolor=red,linewidth=0.3pt](3,10)(2.2,1.2)
   \psellipse[linestyle=dashed,linecolor=red,linewidth=0.3pt](-16.5,10)(2.2,1.2)
   \pscurve[linestyle=dashed,linecolor=red,linewidth=0.3pt]{<-}(-15,11)(-11,12.5)(-2,12.5)(2,11)
   \rput[b](-16.75,11.5){\tiny \mbox{\shortstack{Kirchhoff's \\current law}}}
 \blovigiu{\ds \frac{1}{C\,s}}{\scr }{\dst}
   \nrsezgiu{$\scr 2$}{-0.0}
   \rput[l](3.5,1.5){\scr V_{4}}
   \psellipse[linestyle=dashed,linecolor=vgreen,linewidth=0.3pt](3,0)(2.2,1.2)
   \psellipticarc[linestyle=dashed,linecolor=vgreen,linewidth=0.3pt]{->}(-17,3.5)(3,3){-70}{250}
   \pscurve[linestyle=dashed,linecolor=vgreen,linewidth=0.3pt]{<-}(-14.5,1.5)(-10,-0.25)(-2,-1.25)(1,-1)
   \rput[b](-17.0,2.5){\tiny \mbox{\shortstack{Kirchhoff's \\voltage law}}}
 \blovisu{\ds \frac{1}{R}}{\scr }{\dst}
   \nrsezgiu{$\scr 3$}{-0.0}
 \bloout{\scr I_{2}}{\scr V_2}
 }
  \\
   &\\
  % \hline  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tabular}
%
 %%
\caption{\label{Esempio_CP_RS} POG modeling of an electrical RC
circuit.}
 \end{figure}
 %

Another important property of the POG technique is the direct
correspondence between the POG block schemes and the corresponding
state space dynamic equations. For example, the POG scheme shown in
Fig.~\ref{basesysgraph} can be represented by the state space
equations (\ref{generic}) where the {\em energy matrix} $\L$ is
symmetric and positive definite: $\L=\L\tras>0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t]
 \centering
 \schemaPOG{(-2,0)(28,12)}{3.0mm}{
 \thicklines
  \bloin{\u}{\y}
  \bloko{\B}{\B\tras}{}{}
  \bloivgiuintm{\L\muno}{\x}{\dst}
  \bloseiretaor{\A}{}
  \rput(4,5){
  \begin{minipage}{4.25cm}
 \begin{equation}
  \left\{
 \begin{array}{@{}r@{\;}c@{\;}l@{}}
 \L\,\dot{\x}&=&-\A \x + \B \u \\
 \y &=& \B\tras\x
 \end{array}
 \right.
 \label{generic}
 \end{equation}
 \vspace{0mm}
 \[
 \Downarrow
 (\x=\T\z)
 \]
 \vspace{0mm}
 \begin{equation}
 \left\{
 \begin{array}{@{\,}r@{\;}c@{\;}l@{\,}}
 \ooL\dot{\z}&=&-\ooA \z\! +\! \ooB \u \\[1mm]
 \y &=& \ooB\tras\z
 \end{array}
 \right.
 \label{genericTrans}
 \end{equation}
  \end{minipage}}
 }
 \caption{\label{basesysgraph} POG block scheme of a generic
dynamic system.}
 \end{figure}
%For such a system, the stored energy $E_{s}$ and the dissipating
%power $P_{d}$ can be expressed as follows:
% $E_{s}=\frac{1}{2}\x\tras\L\,\x$,
%   $P_{d}   =\x\tras\A\,\x$.
% where $\A_{s}=(\A+\A\tras)/2$ is the symmetric part of the {\em
% power matrix} $\A$.
% The skew-symmetric part
%$\A_{w}=(\A-\A\tras)/2$ of matrix  $\A$ represents power
%redistribution within the system.
 When an eigenvalue of matrix $\L$ tends to zero (or to infinity),
system (\ref{generic}) degenerates towards a lower dimension dynamic
system.  In this case, the dynamic model of the ``reduced'' system,
see (\ref{genericTrans}), can be directly obtained from
(\ref{generic}) using a simple ``congruent'' transformation
$\x=\T\z$ ($\T$ is constant) where $\ooL=\T\tras\L\T$,
$\ooA=\T\tras\A\T$ and $\ooB=\T\tras\B$.
% If matrix $\T$ is time-varying, an additional term $\T\tras\L\dot{\T}\z$ appears in
% the transformed system. When matrix $\T$ is rectangular, the system
% is transformed and reduced at the same time.
 %  More details on Power-Oriented Graphs are reported in \cite{lille},
 % \cite{mitmemo}, \cite{mim93}.
 %.


\subsection{POG modeling of a planetary gear}

Let us consider the planetary gear shown in Fig.~\ref{epicicloida}
together with the main parameters of the system.
 %
\begin{figure} [!htbp]
 \centering
 \includegraphics[angle=0,width=0.8\columnwidth]{{epicicloida3.eps}}
 \caption{Planetary gear and related parameters.}
 \label{epicicloida}
\end{figure}
%Dove:
% \begin{itemize}
%  \item $r_{s}$  raggio del solare;
%  \item $r_{p}$  raggi dei satelliti;
%  \item $r_{re}$ raggio esterno del ring;
%  \item $r_{r}$ raggio interno del ring;
%  \item $J_{s}$, $b_{s}$  inerzia e attrito del solare;
%  \item $J_{p}$, $b_{p}$  inerzia e attrito dei satelliti;
%  \item $J_{r}$, $b_{r}$  inerzia a attrito del ring;
%  \item $K_{sc}$, $b_{sc}$ elasticit\`a e attrito tra solare e satelliti;
%  \item $K_{cr}$, $b_{cr}$ elasticit\`a e attrito tra satelliti e
%  ring.
%\end{itemize}
%\begin{figure}[h] \hspace{-1.8cm}
% \centering
% \psfrag{A}[0][0][1.5][0]{$\omega_r$}
% \psfrag{B}[0][0][1.5][0]{$\omega_{s}$}
%\psfrag{C}[0][0][1.5][0]{$\omega_{c}$}
% \includegraphics[width=7.5cm, angle=0]{{Gruppo_epicicloidale33.eps}}
% \caption{Gruppo epicicloidale del sistema CVT}
% \label{gruppo epicicloidale}
%\end{figure}
%
%
 \begin{figure*}[t]
  \centering
  \schemaPOG{(-1.5,-3)(66,13)}{2.9mm}{
 \thicklines
 \bloin{\tau_{s}}{\omega_{s}}
 \nrsezxsu{0.75}{\footnotesize $3$}{0}
 \scrivigiu{9.5}{3}{\footnotesize Carrier}
 \blovigiu{\ds \frac{1}{J_{s}s\!+\!b_{s}}}{\omega_{s}}{\dst}
 \blokxyaor{1.5}{1.5}{r_{s}}{r_{s}}{}{}
 \scrivigiu{8}{3}{\footnotesize Elastic element}
 \put(1,8.3){\makebox(0,0){$F_{sc}$}}
 \blosuattdue{}{K_{sc}}{d_{sc}}{}{}{\dst}
 \scrivigiu{24}{3}{\footnotesize Planets: two-dimensional dynamics}
 \blokxyor{4.0}{4.0}{\scr\mat{@{}c@{}}{-r_{p} \\ r_{s}} }{\scr\mat{@{}c@{\,}c@{}}{-r_{p} & r_{s}}}{}{}
 \put(3,12.5){\vector(0,-1){2}}
 \put(3,12){$\mat{@{}c@{}}{ \tau_{p} \\ \tau_{c}}$}
 \blovigiumxm{\scr\mat{@{}c@{\;}c@{}}{\frac{1}{J_{p}} & 0 \\ 0 & \frac{1}{J_{c}}}}{\mat{@{}c@{}}{\\[-5mm]\omega_{p} \\[-1mm] \omega_{c}\\[-0.3mm]} }{\dst
 }{\frac{1}{s}}{}
 \blovisuaor{\mat{@{}c@{\;}c@{}}{b_{p} & 0 \\ 0 & b_{c}}}{}{}
 \blokxyaor{4.0}{4.0}{\mat{@{}c@{}}{r_{p} \\ r_{r}}}{\mat{@{}c@{\;}c@{}}{r_{p} & r_{r}}}{}{}
 \scrivigiu{8}{3}{\footnotesize Elastic element}
 \put(1,8.3){\makebox(0,0){$F_{cr}$}}
 \blosuattdue{}{K_{cr}}{d_{cr}}{}{}{\dst}
 \scrivigiu{10}{3}{\footnotesize Ring}
 \blokxyor{2}{2}{r_{r}}{r_{r}}{}{}
 \blovigiu{\ds \frac{1}{J_{r}s\!+\!b_r}}{\omega_r}{\dst}
 \bloout{\tau_{r}}{\omega_{r}}
 \scrivigiu{10}{3}{}
 \nrsezxsu{0.75}{\footnotesize $4$}{0}
 }
 %\[
% \footnotesize
% \mat{@{}c@{}c@{\;}c@{}c@{\,}c@{\,}c@{}}{
% J_{s} &  0 &  0 &  0 &  0 &  0 \\
% 0 &  \frac{1}{K_{sc}} &  0 &  0 &  0 &  0 \\
% 0 &  0 &  J_{p} &  0 &  0 &  0 \\
% 0 &  0 &  0 &  J_{c} &  0 &  0 \\
% 0 &  0 &  0 &  0 &  \frac{1}{K_{cr}} &  0 \\
% 0 &  0 &  0 &  0 & 0 &  J_{r}
% }
% \mat{@{}c@{}}{\dot\omega_{s}\\ \dot F_{sc} \\ \dot \omega_{p}\\ \dot \omega_{c}\\ \dot F_{cr} \\ \dot \omega_{r}}
% =
% \mat{@{}c@{}c@{\;}c@{}c@{\,}c@{\,}c@{}}{
% \!-\!b_{s}\!-\!r_{s}^2d_{sc} & \!-\!r_{s} & \!-\!r_{s}d_{sc}r_{p} & r_{s}^2d_{sc} & 0 & 0\\
% r_{s}         &   0  &    r_{p}       &   \!-\!r_{s}     & 0 & 0\\
% \!-\!r_{s}d_{sc}r_{p}   & \!-\!r_{p} & \!-\!b_{p}\!-\!d_{sc}r_{p}^2\!-\!d_{cr}r_{p}^2 & r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r} &
% \!-\!r_{p} & d_{cr}r_{p}r_{r}\\
% r_{s}^2d_{sc} & r_{s} & r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r} &
% \!-\!b_{c}\!-\!r_{s}^2d_{sc}\!-\!d_{cr}r_{r}^2 & \!-\!r_{r} & d_{cr}r_{r}^2\\
% 0 & 0 & r_{p} & r_{r} & 0 & \!-\!r_{r}\\
% 0 & 0 & d_{cr}r_{p}r_{r} & d_{cr}r_{r}^2 & r_{r} & \!-\!b_r\!-\!d_{cr}r_{r}^2
% }
% \mat{@{}c@{}}{\omega_{s}\\ F_{sc} \\ \omega_{p}\\ \omega_{c}\\  F_{cr} \\ \omega_{r}}
% +
% \mat{@{}c@{\,}c@{\,}c@{}}{
% 1 &  \,0 &  \,0 \\
% 0 &  \,0 &  \,0 \\
% 0 &  \,0 &  \,0 \\
% 0 &  \,1 &  \,0 \\
% 0 &  \,0 &  \,0 \\
% 0 &  \,0 &  \,1 }
%  \mat{@{}c@{}}{\tau_{s}\\ \tau_{c}\\ \tau_{r}}
%\]
  \caption{The POG model of the considered planetary gear: the carrier, the planets and the ring interact through elastic elements.}
  \label{POG_epicicloidale}
\end{figure*}
 %
 %
 \begin{figure*}[t]
  \centering
\[
 \footnotesize
 \underbrace{\mat{@{}c@{}c@{\;}c@{}c@{\,}c@{\,}c@{}}{
 J_{s} &  0 &  0 &  0 &  0 &  0 \\
 0 &  \frac{1}{K_{sc}} &  0 &  0 &  0 &  0 \\
 0 &  0 &  J_{p} &  0 &  0 &  0 \\
 0 &  0 &  0 &  J_{c} &  0 &  0 \\
 0 &  0 &  0 &  0 &  \frac{1}{K_{cr}} &  0 \\
 0 &  0 &  0 &  0 & 0 &  J_{r}}}_{\ds \ooL}\!
 \underbrace{\mat{@{}c@{}}{\dot\omega_{s}\\ \dot F_{sc} \\ \dot \omega_{p}\\ \dot \omega_{c}\\ \dot F_{cr} \\ \dot \omega_{r}}}_{\ds \dot\oox}\!
 =
 -\underbrace{\mat{@{}c@{\;}c@{\;}c@{\;\;}c@{\,}c@{\;}c@{}}{
 b_{s}\!+\!r_{s}^2d_{sc} & r_{s} & r_{s} d_{sc}r_{p} & \!-\!r_{s}^2 d_{sc} & 0 & 0\\
 \!-\!r_{s}         &   0  &    \!-\!r_{p}       &   r_{s}     & 0 & 0\\
 r_{s}d_{sc}r_{p}   & r_{p} & b_{p}\!+\!d_{sc}r_{p}^2\!+\!d_{cr}r_{p}^2 & \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} &
 r_{p} & \!-\!d_{cr}r_{p}r_{r}\\
 \!-\!r_{s}^2d_{sc} & \!-\!r_{s} & r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} &
 b_{c}\!+\!r_{s}^2d_{sc}\!+\!d_{cr}r_{r}^2 & r_{r} & \!-\!d_{cr}r_{r}^2\\
 0 & 0 & \!-\!r_{p} & r_{r} & 0 & r_{r}\\
 0 & 0 & \!-\!d_{cr}r_{p}r_{r} & \!-\!d_{cr}r_{r}^2 & \!-\!r_{r} & b_r\!+\!d_{cr}r_{r}^2
%% b_{s}\!+\!r_{s}^2d_{sc} & \!-\!r_{s}^2d_{sc}                          & r_{s}d_{sc}r_{p}                          & 0                      & 0            & r_{s}     \\
%% \!-\!r_{s}^2d_{sc}    & b_{c}\!+\!r_{s}^2d_{sc}\!+\!d_{cr}r_{r}^2 & \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} & \!-\!d_{cr}r_{r}^2    & r_{r}       & \!-\!r_{s}\\
%% r_{s}d_{sc}r_{p}        & \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r}     & b_{p}\!+\!d_{sc}r_{p}^2\!+\!d_{cr}r_{p}^2   & \!-\!d_{cr}r_{p}r_{r}   & r_{p}          & r_{p}     \\
%% 0                   & \!-\!d_{cr}r_{r}^2                         & \!-\!d_{cr}r_{p}r_{r}                  & b_r\!+\!d_{cr}r_{r}^2 & \!-\!r_{r}  & 0       \\
%% 0                   & \!-\!r_{r}                               & \!-\!r_{p}                              & r_{r}                   & 0            & 0       \\
%% \!-\!r_{s}            & r_{s}                                       & \!-\!r_{p}                              & 0                      & 0            & 0
 }}_{\ds\ooA}\!
 \underbrace{\mat{@{}c@{}}{\omega_{s}\\ F_{sc} \\ \omega_{p}\\ \omega_{c}\\  F_{cr} \\ \omega_{r}}}_{\ds\oox}\!
 +
 \underbrace{\mat{@{}c@{\,}c@{\,}c@{}}{
 1 &  \,0 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,1 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,0 &  \,1 }}_{\ds\ooB}\!
  \underbrace{\mat{@{}c@{}}{\tau_{s}\\ \tau_{c}\\ -\tau_{r}}}_{\ds\u}
\]
 \caption{
The POG state space dynamic model of the considered planetary gear:
only the inputs $\tau_{s}$,  $\tau_{c}$ and $\tau_{r}$ have been
considered.}
  \label{POG_epicicloidale_POG}
\end{figure*}
 %
The extended POG dynamic model of the considered planetary gear is
shown in Fig.~\ref{POG_epicicloidale}: the carrier, the planets and
the ring interact each other through two elastic elements. The
corresponding state space dynamic equations are shown in
Fig.~\ref{POG_epicicloidale_POG}:
\begin{equation}
\label{expanded_POG_model}
 \ooL\,\dot\oox
 =
 -\ooA\,\oox+\ooB\,\u,
 \hspace{12mm}
 \y=\ooB\tras\oox
\end{equation}
The POG linear systems described in the form
(\ref{expanded_POG_model}) satisfy the following properties: 1) the
energy $E_{s}$ stored in the system and the dissipating power
$P_{d}$ can be expressed as quadratic functions of matrices $\ooL$
and $\ooA_{s}$ as follows:
 \[
 E_{s}=\frac{1}{2}\,\oox\tras\,\ooL\,\oox,
 \hspace{15mm}
 P_{d}   =\oox\tras\,\ooA_{s}\,\oox
 \]
where $\ooA_{s}=(\ooA+\ooA\tras)/2$ is the symmetric part of the
{\em power matrix} $\ooA$. The skew-symmetric part
$\ooA_{w}=(\ooA-\ooA\tras)/2$ of matrix  $\ooA$ represents power
redistribution within the system.
 A POG scheme always satisfies the following two rules:
 1) {\it along all the loops of the scheme must
 be present an ``odd'' number of signs ``$-$''} (the black spots in the summation blocks);
 2) {\it the direction of the power flowing through a section is
positive if an ``even'' number of signs ``$-$'' is present along all
the paths which link the input to the output}.

 For certain applications the POG model of Fig.~\ref{POG_epicicloidale} can be too much
detailed. In these cases it can be of interest to find the reduced
model when the stiffness coefficients $K_{cr}$ and $K_{sc}$ tend to
infinity. Applying to system (\ref{expanded_POG_model}) the
following congruent transformation:
\[
 \oox=\T_{1}\,\x,
 \hspace{8mm}
 \mbox{where}
 \hspace{8mm}
 \T_{1}=\mat{@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}}{
 1 &  0 &  0 &  0 &  0 &  0 \\
 0 &  0 &  0 &  0 &  0 &  1 \\
 0 &  0 &  1 &  0 &  0 &  0 \\
 0 &  1 &  0 &  0 &  0 &  0 \\
 0 &  0 &  0 &  0 &  1 &  0 \\
 0 &  0 &  0 &  1 &  0 &  0
 }.
\]
 one easily obtains the following transformed system:
\begin{equation}\label{tran_sys_pog_t}
 \underbrace{\mat{@{}c@{}c@{}c@{}}{
 \J_{1} &  0 &  0 \\
 0 & \J_{2} &   0 \\
 0 & 0 &    0 }}_{\ds \L}\!
 \underbrace{\mat{@{}c@{}}{\dot\x_{1}\\ \dot\x_{2} \\ \dot\x_{3}}}_{\ds \dot\x}\!
 =
-\underbrace{\mat{@{}c@{\;}c@{\;}c@{}}{
 \A_{11} & \A_{12} & \A_{13} \\
 \A_{21} & \A_{22} & \A_{23} \\
 \A_{31} & \A_{32} & 0
  }}_{\ds\A}\!
 \underbrace{\mat{@{}c@{}}{\x_{1}\\ \x_{2} \\ \x_{3}}}_{\ds \x}\!
 +
 \underbrace{\mat{@{}c@{}}{
 \B_{1} \\
 \B_{2} \\
 0
 }}_{\ds\B}\!
 \u
\end{equation}
 %
where $\x=\T_{1}\tras\,\oox$, $\L=\T_{1}\tras\ooL\,\T_{1}$,
$\B=\T_{1}\tras\,\ooB$ and $\A=\T_{1}\tras\ooA\,\T_{1}$ and where:
\[
\footnotesize
 \x_{1}\!=\!\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}}\!\!,
 \hspace{1mm}
 \x_{2}\!=\!\!\mat{@{}c@{}}{\omega_{p}\\ \omega_{r}}\!\!,
 \hspace{1mm}
 \x_{3}\!=\!\!\mat{@{}c@{}}{F_{cr} \\ F_{sc}}\!\!,
 \hspace{1mm}
 \J_{1}\!=\!\mat{@{}c@{}c@{}}{
 J_{s} &  0 \\
 0 &  J_{c} }\!\!,
 \hspace{1mm}
 \J_{2}\!=\!\mat{@{}c@{}c@{}}{
 J_{p}  &  0 \\
0 &  J_{r} }\!\!,
\]
\[
\footnotesize
\B_{1}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 1 &  0 &  0 \\
 0 &  1 &  0 }\!,\!
 \hspace{1mm}
\B_{2}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 0 &  0 &  0 \\
 0 &  0 &  1 }\!,\!
 \hspace{1mm}
 \A_{11}\!=\!\mat{@{}c@{}c@{}}{
 b_{s}\!+\!r_{s}^2d_{sc} & \!-\!r_{s}^2d_{sc}                          \\
 \!-\!r_{s}^2d_{sc}    & b_{c}\!+\!r_{s}^2d_{sc}\!+\!d_{cr}r_{r}^2 }\!\!,
 \]
 \[
 \footnotesize
 \!\A_{12}\!\!=\!\!\mat{@{}c@{\;}c@{}}{
 r_{s}d_{sc}r_{p}                          & 0                      \\
 \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} & \!-\!d_{cr}r_{r}^2    }\!\!,
 \hspace{0mm}
 \A_{13}\!\!=\!\!\mat{@{}c@{\;}c@{}}{
 0            & r_{s}     \\
 r_{r}       & \!-\!r_{s}}\!\!,
 \hspace{0mm}
 \A_{31}\!\!=\!\!\mat{@{}c@{}c@{}}{
 0                   & \!-\!r_{r} \\
 \!-\!r_{s}            & r_{s}         }\!\!,
\]
\[
 \footnotesize
\!\A_{21}\!=\!\mat{@{}c@{\;}c@{}}{
r_{s}d_{sc}r_{p}        & \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r}  \\
0                   & \!-\!d_{cr}r_{r}^2 }\!\!,
 \hspace{8mm}
 \A_{23}\!=\!\mat{@{}c@{}c@{}}{
 r_{p}          & r_{p}  \\
 \!-\!r_{r}  & 0    }\!\!,
\]
 \[
 \footnotesize
 \A_{22}\!=\!\mat{@{}c@{\;}c@{}}{
  b_{p}\!+\!d_{sc}r_{p}^2\!+\!d_{cr}r_{p}^2   & \!-\!d_{cr}r_{p}r_{r}   \\
  \!-\!d_{cr}r_{p}r_{r}                  & b_r\!+\!d_{cr}r_{r}^2 }\!\!,
 \hspace{6mm} \scriptsize
 \A_{32}\!=\!\mat{@{}c@{\;}c@{}}{
 \!-\!r_{p}                              & r_{r}  \\
 \!-\!r_{p}                              & 0     }\!\!,
\]
%\[
% \scriptsize
% \A\!\!=\!\!\mat{@{}c@{}c@{\;}c@{}c@{\,}c@{\,}c@{}}{
% b_{s}\!+\!r_{s}^2d_{sc} & \!-\!r_{s}^2d_{sc}                          & r_{s}d_{sc}r_{p}                          & 0                      & 0            & r_{s}     \\
% \!-\!r_{s}^2d_{sc}    & b_{c}\!+\!r_{s}^2d_{sc}\!+\!d_{cr}r_{r}^2 & \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} & \!-\!d_{cr}r_{r}^2    & r_{r}       & \!-\!r_{s}\\
% r_{s}d_{sc}r_{p}        & \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r}     & b_{p}\!+\!d_{sc}r_{p}^2\!+\!d_{cr}r_{p}^2   & \!-\!d_{cr}r_{p}r_{r}   & r_{p}          & r_{p}     \\
% 0                   & \!-\!d_{cr}r_{r}^2                         & \!-\!d_{cr}r_{p}r_{r}                  & b_r\!+\!d_{cr}r_{r}^2 & \!-\!r_{r}  & 0       \\
% 0                   & \!-\!r_{r}                               & \!-\!r_{p}                              & r_{r}                   & 0            & 0       \\
% \!-\!r_{s}            & r_{s}                                       & \!-\!r_{p}                              & 0                      & 0            & 0
%}
%\]
%\[
% \scriptsize
% -\A\!\!=\!\!\mat{@{}c@{}c@{\;}c@{}c@{\,}c@{\,}c@{}}{
% \!-\!b_{s}\!-\!r_{s}^2d_{sc}& r_{s}^2d_{sc}                        & \!-\!r_{s}d_{sc}r_{p}                 & 0                   & 0       & \!-\!r_{s}\\
% r_{s}^2d_{sc}     & \!-\!b_{c}\!-\!r_{s}^2d_{sc}\!-\!d_{cr}r_{r}^2 & r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r}  & d_{cr}r_{r}^2      & \!-\!r_{r} & r_{s}\\
% \!-\!r_{s}d_{sc}r_{p}   & r_{s}d_{sc}r_{p}\!-\!d_{cr}r_{p}r_{r}       & \!-\!b_{p}\!-\!d_{sc}r_{p}^2\!-\!d_{cr}r_{p}^2  & d_{cr}r_{p}r_{r}     & \!-\!r_{p}    & \!-\!r_{p} \\
% 0               & d_{cr}r_{r}^2                       & d_{cr}r_{p}r_{r}               & \!-\!b_r\!-\!d_{cr}r_{r}^2 & r_{r}  & 0\\
% 0               & r_{r}                             & r_{p}                           & \!-\!r_{r}               & 0       & 0\\
% r_{s}             & \!-\!r_{s}                               & r_{p}                           & 0                   & 0       & 0
%}
%\]
The last equation of system (\ref{tran_sys_pog_t}) shows an
algebraic relation between the state variables:
$\A_{31}\,\x_{1}+\A_{32}\,\x_{2}=0$. Since matrix $\A_{32}$ is
invertible, vector $\x_{2}$ can be expressed as:
$\x_{2}=-\A_{32}\muno\,\A_{31}\,\x_{1}$. Applying  to system
(\ref{tran_sys_pog_t}) the following ``rectangular'' transformation:
\[
\x=\T_{2}\,\x_{1}
 \hspace{4mm}
 \mbox{where}
 \hspace{4mm}
 \T_{2}
 \!=\!
 \mat{@{}c@{}}{
 \I_{2} \\[1mm]
-\A_{32}\muno\,\A_{31}\\[1mm]
 0 }
 \!=\!
 \mat{@{}c@{\;}c@{}}{
  1 & 0  \\
  0 & 0 \\
  -\frac{r_{s}}{r_{p}} &  \frac{r_{s}}{r_{p}} \\
  -\frac{r_{s}}{r_{r}} &  1\!+\!\frac{r_{s}}{r_{r}} \\
  0 & 0 \\
  0 & 0 }
\]
one obtains the following second order transformed and reduced
system:
\begin{equation}
\label{modello_epiciclo_ridotto}
\L_{r}\,\dot\x_{1}=-\A_{r}\,\x_{1}+\B_{r}\,\u,
 \hspace{10mm}
 \y=\B_{r}\tras\x_{1}
\end{equation}
where
\[
 \L_{r}
 \!=\! \mat{@{}c@{\;\,}c@{}}{
 J_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2} J_{p}\!+\!\frac{r_{s}^2 }{r_{r}^2} J_r &
 \!-\!\frac{r_{s}^2 }{r_{p}^2 } J_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! J_r \\[1mm]
 \!-\!\frac{r_{s}^2 }{r_{p}^2 } J_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! J_r &
 J_{c}\!+\!\frac{r_{s}^2}{r_{p}^2} J_{p} \!+\! \left(1\!+\!\frac{r_{s} }{ r_{r}}\right)^2 \!\!J_r
}
\]
\[
 \A_{r}
\!=\! \mat{@{}c@{}c@{}}{
 b_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}\!+\! \frac{r_{s}^2 }{r_{r}^2} b_r &
 \!\!\!\!-\!\left(\!1\!+\!\!\frac{r_{s} }{ r_{r}}\!\right)\!\!\left(\!1\!+\!\frac{r_{s}}{r_{r}} b_r\!\right)\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} \\[1mm]
 \left(\!1\!+\!\frac{r_{s} }{ r_{r}}\!\right)\!\left(\!1\!-\!\frac{r_{s}}{r_{r}} b_r\!\right)\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} &
 b_{c} \!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} \!+\!\left(\!1\!+\!  \frac{r_{s} }{ r_{r}}\!\right)^2 \!\!b_r
 }
\]
\[
 \B_{r}
 \!=\! \mat{@{\,}c@{\;\;}c@{\;\;}c@{\,}}{
1 & 0 & \!-\frac{r_{s}}{r_{r}}\\[1mm]
0 & 1 & 1\!+\!\frac{r_{s}}{r_{r}}
 },
 \hspace{4mm}
\x_{1}\!=\!\mat{@{}c@{}}{\omega_{s}\\\omega_{c}},
 \hspace{4mm}
 \y
 \!=\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}
% \!=\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\  }
\]
and $\omega_{r}\!=\!\omega_{c}\!+\!
\frac{r_{s}}{r_{r}}(\omega_{c}\!-\!\omega_{s})$.
%
%\[
% \L_{r}=\mat{cc}{
%\,\frac{(J_{s}r_{p}^2r_{r}^2+r_{s}^2J_{p}r_{r}^2+r_{s}^2J_rr_{p}^2)}{r_{p}^2r_{r}^2}
%&
%\,\frac{-r_{s}(r_{s}J_{p}r_{r}^2+J_rr_{p}^2r_{s}+J_rr_{p}^2r_{r})}{r_{p}^2r_{r}^2}\\[5mm]
%\,\frac{-r_{s}(r_{s}J_{p}r_{r}^2+J_rr_{p}^2r_{s}+J_rr_{p}^2r_{r})}{r_{p}^2r_{r}^2}
%&
%\,\frac{(J_{c}r_{p}^2r_{r}^2+r_{s}^2J_{p}r_{r}^2+r_{s}^2J_rr_{p}^2+2J_rr_{p}^2r_{s}r_{r}+J_rr_{p}^2r_{r}^2)}{r_{p}^2r_{r}^2}
%}
%\]





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper the Power-Oriented Graphs (POG) technique has benn
used for modeling ...


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}


\newpage

\begin{equation}
\label{reduced_T2system}
 \underbrace{\mat{@{}c@{\;}c@{}}{
 L_{1} & L_{12} \\ L_{12} & L_{2}
 }}_{\ds\L_{r}}
 \,
 \underbrace{\mat{@{}c@{}}{ \dot\omega_{s} \\ \dot\omega_{c}
 }}_{\ds\dot\x_{1}}
 =
 -\underbrace{\mat{@{}c@{\;}c@{}}{
 a_{11} & a_{12} \\ a_{21} & a_{22}
 }}_{\ds\A_{r}}\,
 \underbrace{\mat{@{}c@{}}{ \omega_{s} \\ \omega_{c}
 }}_{\ds\x_{1}}
 +\underbrace{\mat{@{}c@{\;}c@{\;}c@{}}{
 1 & 0 & b_{1} \\
 1 & 0 & b_{2} \\
 }}_{\ds\B_{r}}
 \,\u
% \L_{r}\,\dot\x_{1}
% =
% -\A_{r}\,\x_{1}+\B_{r}\,\u,
% \hspace{12mm}
% \y=\B_{r}\tras\x_{1}
\end{equation}
where
\[
% \scr
\begin{array}{@{}l@{\,}c@{\,}l@{}}
 L_{1}
 % &=&\frac{J_{s} r_{p}^2 r_{r}^2+r_{s}^2 J_{p} r_{r}^2+r_{s}^2 J_r r_{p}^2}{r_{p}^2 r_{r}^2} \\
 &=& J_{s}+\frac{r_{s}^2 }{r_{p}^2} J_{p}+\frac{r_{s}^2 }{r_{r}^2} J_r
 \\[2mm]
 L_{12}
 % &=&\frac{-r_{s}(r_{s} J_{p} r_{r}^2+J_r r_{p}^2 r_{s}+J_r r_{p}^2 r_{r})}{r_{p}^2 r_{r}^2}\\
 % &=&  -\frac{r_{s}^2 }{r_{p}^2 } J_{p}  -\frac{r_{s}^2}{r_{r}^2} J_r  -\frac{r_{s}}{r_{r}} J_r \\
 &=&  -\frac{r_{s}^2 }{r_{p}^2 } J_{p}  -\frac{r_{s}}{r_{r}} \left(1\!+\!  \frac{r_{s} }{ r_{r}}\right) J_r
 \\[2mm]
 L_{2}
 % &=&\frac{J_{c}r_{p}^2 r_{r}^2+r_{s}^2 J_{p} r_{r}^2+r_{s}^2 J_r r_{p}^2+2 J_r r_{p}^2 r_{s} r_{r}+J_r r_{p}^2 r_{r}^2}{r_{p}^2 r_{r}^2}\\
 % &=&  J_{c}+\frac{r_{s}^2}{r_{p}^2} J_{p} + \frac{r_{s}^2 }{r_{r}^2}J_r +  \frac{2 r_{s} }{ r_{r}} J_r + J_r\\
 &=&  J_{c}+\frac{r_{s}^2}{r_{p}^2} J_{p} + \left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)^2 J_r
 \\[2mm]
 a_{11}
 %&=& \frac{(b_{s} r_{p}^2 r_{r}^2+r_{s}^2 b_{p} r_{r}^2+r_{s}^2 r_{p}^2 b_r)}{r_{p}^2 r_{r}^2}\\
 &=&
  b_{s}+ \frac{r_{s}^2 }{r_{p}^2 } b_{p}+ \frac{r_{s}^2 }{r_{r}^2} b_r
 \\[2mm]
 a_{12}
 % &=& \frac{-r_{s}^2 b_{p} r_{r}^2-r_{s} r_{p}^2 r_{r}-r_{p}^2 r_{r}^2-r_{s}^2 r_{p}^2 b_r-b_r r_{s} r_{p}^2 r_{r}}{r_{p}^2 r_{r}^2}\\
 % &=& -\frac{-r_{s}^2 }{r_{p}^2} b_{p}-\frac{r_{s} }{r_{r}}-1 -\frac{r_{s}^2 }{r_{r}^2}b_r -\frac{r_{s}  }{r_{r}}b_r\\
 &=& -1-\frac{r_{s} }{r_{r}} -\frac{r_{s}^2 }{r_{p}^2} b_{p} -\frac{r_{s}}{r_{r}} \left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)b_r
 \\[2mm]
 a_{21}
 %&=&\frac{r_{s} r_{p}^2 r_{r}+r_{p}^2 r_{r}^2-r_{s}^2 b_{p} r_{r}^2-r_{s}^2 r_{p}^2 b_r-b_r r_{s} r_{p}^2 r_{r}}{r_{p}^2 r_{r}^2}\\
 &=& 1+\frac{r_{s} }{r_{r}}  -\frac{r_{s}^2 }{r_{p}^2 } b_{p} -\frac{r_{s}}{r_{r}} \left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)b_r
 \\[2mm]
 a_{22}
 %&=& \frac{b_{c} r_{p}^2 r_{r}^2+r_{s}^2 b_{p} r_{r}^2+r_{s}^2 r_{p}^2 b_r+2b_r r_{s} r_{p}^2 r_{r}+b_r r_{p}^2 r_{r}^2}{r_{p}^2 r_{r}^2}\\
 %&=& b_{c} +\frac{r_{s}^2 }{r_{p}^2 } b_{p} +\frac{r_{s}^2  }{ r_{r}^2} b_r +\frac{2 r_{s} }{r_{r}} b_r + b_r\\
 &=& b_{c} +\frac{r_{s}^2 }{r_{p}^2 } b_{p} +\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)^2 b_r
 \\[2mm]
 b_{1}
 &=&\frac{-r_{s}}{r_{r}}
 \\[2mm]
 b_{2}
 &=&1+\frac{r_{s}}{r_{r}}
\end{array}
\]
\[
% \scr
\begin{array}{@{}l@{\,}c@{\,}l@{}}
 L_{1}
 &=& J_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2} J_{p}\!+\!\frac{r_{s}^2 }{r_{r}^2} J_r
 \\[2mm]
 L_{12}
 &=&  \!-\!\frac{r_{s}^2 }{r_{p}^2 } J_{p}  \!-\!\frac{r_{s}}{r_{r}} \left(1\!+\!  \frac{r_{s} }{ r_{r}}\right) J_r
 \\[2mm]
 L_{2}
 &=&  J_{c}\!+\!\frac{r_{s}^2}{r_{p}^2} J_{p} \!+\! \left(1\!+\!\frac{r_{s} }{ r_{r}}\right)^2 J_r
 \\[2mm]
 a_{11}
 &=&
  b_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}\!+\! \frac{r_{s}^2 }{r_{r}^2} b_r
 \\[2mm]
 a_{12}
% &=& \!-\!1\!-\!\frac{r_{s} }{r_{r}} \!-\!\frac{r_{s}^2 }{r_{p}^2} b_{p} \!-\!\frac{r_{s}}{r_{r}} \left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)b_r
 &=& \!-\!\left(\!1\!+\!  \frac{r_{s} }{ r_{r}}\!\right)\!\left(\!1\!+\!\frac{r_{s}}{r_{r}} b_r\!\right)\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}
 \\[2mm]
 a_{21}
 &=& \left(\!1\!+\!  \frac{r_{s} }{ r_{r}}\!\right)\!\left(\!1\!-\!\frac{r_{s}}{r_{r}} b_r\!\right)\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}
 \\[2mm]
 a_{22}
 &=& b_{c} \!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} \!+\!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)^2 b_r
 \\[2mm]
 b_{1}
 &=&\!-\!\frac{r_{s}}{r_{r}}
 \\[2mm]
 b_{2}
 &=&1\!+\!\frac{r_{s}}{r_{r}}
\end{array}
\]
\[
% \scr
\begin{array}{@{}l@{\,}c@{\,}l@{}}
 L_{1}
 &=& J_{s}+\rho^2 J_{p}+\delta^2 J_r
 \\[2mm]
 L_{12}
 &=&  -\rho^2 J_{p}  -\delta \left(1\!+\!  \delta\right) J_r
 \\[2mm]
 L_{2}
 &=&  J_{c}+\rho^2 J_{p} + \left(1\!+\!  \delta\right)^2 J_r
 \\[2mm]
 a_{11}
 &=&  b_{s}+ \rho^2 b_{p}+ \delta^2 b_r
 \\[2mm]
 a_{12}
 &=& -1-\delta -\rho^2 b_{p} -\delta \left(1\!+\!  \delta\right)b_r
 \\[2mm]
 a_{21}
 &=& 1+\delta  -\rho^2 b_{p} -\delta \left(1\!+\!  \delta\right)b_r
 \\[2mm]
 a_{22}
 &=& b_{c} +\rho^2 b_{p} +\left(1\!+\!  \delta\right)^2 b_r
 \\[2mm]
 b_{1}
 &=&-\delta
 \\[2mm]
 b_{2}
 &=&1+\delta
\end{array}
\]
where
\[
 \rho=\frac{r_{s}}{r_{p}}
 , \hspace{10mm}
 \delta=\frac{r_{s}}{r_{r}}
 ,\hspace{10mm}
 \gamma=1+\delta
\]
%%%%%%%%%%%%%%
\[
% \scr
\begin{array}{@{}l@{\,}c@{\,}l@{}}
 L_{1}
 &=& J_{s}+\rho_{s}^2 J_{p}+\rho_{r}^2 J_r
 \\[2mm]
 L_{12}
 &=&  -\rho_{s}^2 J_{p}  -\rho_{r} \gamma_{r} J_r
 \\[2mm]
 L_{2}
 &=&  J_{c}+\rho_{s}^2 J_{p} + \gamma_{r}^2 J_r
 \\[2mm]
 a_{11}
 &=&  b_{s}+ \rho_{s}^2 b_{p}+ \rho_{r}^2 b_r
 \\[2mm]
 a_{12}
 &=& -\gamma_{r} -\rho_{s}^2 b_{p} -\rho_{r} \gamma_{r} b_r
 \\[2mm]
 a_{21}
 &=& \gamma_{r}  -\rho_{s}^2 b_{p} -\rho_{r} \gamma_{r} b_r
 \\[2mm]
 a_{22}
 &=& b_{c} +\rho_{s}^2 b_{p} +\gamma_{r}^2 b_r
 \\[2mm]
 b_{1}
 &=&-\rho_{r}
 \\[2mm]
 b_{2}
 &=&\gamma_{r}
\end{array}
\]
where
\[
 \rho_{s}=\frac{r_{s}}{r_{p}}
 , \hspace{10mm}
 \rho_{r}=\frac{r_{s}}{r_{r}}
 ,\hspace{10mm}
 \gamma_{r}=1+\rho_{r}
\]

 dove
    \[
    \L_{r}
    = \mat{@{\,}c@{\;\;}c@{\,}}{
    J_{s}\!+\!\rho_{s}^2 J_{p}\!+\!\rho_{r}^2 J_r & \!-\!\rho_{s}^2 J_{p}  \!-\!\rho_{r} \gamma_{r} J_r \\[1mm]
    \!-\!\rho_{s}^2 J_{p}  \!-\!\rho_{r} \gamma_{r} J_r & J_{c}\!+\!\rho_{s}^2 J_{p} \!+\! \gamma_{r}^2 J_r
    }
    \]
    \[
    \A_{r}
    = \mat{@{\,}c@{\;\;}c@{\,}}{
    b_{s}\!+\! \rho_{s}^2 b_{p}\!+\! \rho_{r}^2 b_r &
    \!-\!\gamma_{r} \!-\!\rho_{s}^2 b_{p} \!-\!\rho_{r} \gamma_{r} b_r \\[1mm]
    \gamma_{r}  \!-\!\rho_{s}^2 b_{p} \!-\!\rho_{r} \gamma_{r} b_r &
    b_{c} \!+\!\rho_{s}^2 b_{p} \!+\!\gamma_{r}^2 b_r
    }
    \]
    \[
    \B_{r}
    \!=\! \mat{@{\;}c@{\;\;}c@{\;}c@{}}{
    1 & 0 & -\rho_{r}\\[1mm]
    0 & 1 & \gamma_{r}
    }
    , \hspace{3mm}
    \rho_{s}\!=\!\frac{r_{s}}{r_{p}}
    , \hspace{3mm}
    \rho_{r}\!=\!\frac{r_{s}}{r_{r}}
    ,\hspace{3mm}
    \gamma_{r}\!=\!1\!+\!\rho_{r}.
    \]



\end{document}
