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 \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 Hybrid Automotive Systems\\[-2mm] {\large (Draft paper)}}

\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 hybrid automotive systems. Some basic
properties of the POG technique are firstly given. An extended
dynamic model of a planetary gear with internal elasticity is
presented. Then  a POG congruent state space transformation is used
to transform and reduce the system when the elasticities go to zero.
The obtained reduced model is used as central element of an hybrid
automotive power structure (endothermic engine, multi-phase
synchronous motor and vehicle dynamics). Simulation results will be
reported in the final full paper.
\end{abstract}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% no \IEEEPARstart

Nowadays, the planetary gears are key elements for the design of new
hybrid power structures in the automotive area. In this paper, some
different dynamic models for these systems are obtained using the
the POG modeling technique.

(References and a detailed description of the work done in the
literature in this field will be given in the final full paper).

The paper is organized as follows: Sec.~\ref{POGproperties}
describes the basic properties of the POG modeling technique.
Sec.~\ref{Planetary_Gears} and Sec.~\ref{Hybrid_Structure} show,
respectively, the POG dynamic models of full and reduced  planetary
gears, and of an hybrid automotive power structure composed by an
endothermic engine, a multi-phase synchronous motor and the vehicle
dynamics. 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 correspondence 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}.
 %.


\section{POG modeling of a planetary gear} \label{Planetary_Gears}

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*}[t]
  \centering
  \schemaPOG{(-1.5,-3)(66,13)}{2.9mm}{
 \thicklines
 \bloin{\tau_{s}}{\omega_{s}}
 \flowrightsu{}
 \nrsezxsu{0.75}{\footnotesize $3$}{1}
 \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 $4$}{1}
 \flowrightsu{}
 }
  \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
\[
 \footnotesize
 \underbrace{\mat{@{}c@{}c@{\;}c@{}c@{\,}c@{\,}c@{}}{
 J_{s} &  0 &  0 &  0 &  0 &  0 \\
 0 &  \frac{1}{K_{sc}} &  0 &  0 &  0 &  0 \\
 0 &  0 &  J_{p} &  0 &  0 &  0 \\
 0 &  0 &  0 &  J_{c} &  0 &  0 \\
 0 &  0 &  0 &  0 &  \frac{1}{K_{cr}} &  0 \\
 0 &  0 &  0 &  0 & 0 &  J_{r}}}_{\ds \ooL}\!
 \underbrace{\mat{@{}c@{}}{\dot\omega_{s}\\ \dot F_{sc} \\ \dot \omega_{p}\\ \dot \omega_{c}\\ \dot F_{cr} \\ \dot \omega_{r}}}_{\ds \dot\oox}\!
 =
 -\underbrace{\mat{@{}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{}}{
 b_{s}\!+\!r_{s}^2d_{sc} & r_{s} & r_{s} d_{sc}r_{p} & \!-r_{s}^2 d_{sc} & 0 & 0\\
 \!-r_{s}         &   0  &    \!-r_{p}       &   r_{s}     & 0 & 0\\
 r_{s}d_{sc}r_{p}   & r_{p} & b_{p}\!+\!d_{sc}r_{p}^2\!+\!d_{cr}r_{p}^2 & \!-r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} &
 r_{p} & \!-d_{cr}r_{p}r_{r}\\
 \!-r_{s}^2d_{sc} & \!-r_{s} & -r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} &
 b_{c}\!+\!r_{s}^2d_{sc}\!+\!d_{cr}r_{r}^2 & r_{r} & \!-d_{cr}r_{r}^2\\
 0 & 0 & \!-r_{p} & \!-r_{r} & 0 & r_{r}\\
 0 & 0 & \!-d_{cr}r_{p}r_{r} & \!-d_{cr}r_{r}^2 & \!-r_{r} & b_r\!+\!d_{cr}r_{r}^2
 }}_{\ds\ooA}\!
 \underbrace{\mat{@{}c@{}}{\omega_{s}\\ F_{sc} \\ \omega_{p}\\ \omega_{c}\\  F_{cr} \\ \omega_{r}}}_{\ds\oox}\!
 +
 \underbrace{\mat{@{}c@{\,}c@{\,}c@{}}{
 1 &  \,0 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,1 &  \,0 \\
 0 &  \,0 &  \,0 \\
 0 &  \,0 &  \,1 }}_{\ds\ooB}\!
  \underbrace{\mat{@{}c@{}}{\tau_{s}\\ \tau_{c}\\ -\tau_{r}}}_{\ds\u}
\]
 \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 the two elastic elements
$K_{cr}$ and $K_{sc}$. 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 system (\ref{expanded_POG_model}) satisfies the
following properties:
 \\[1mm]
 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 $\ooA_{s}$:
 \[
 E_{s}=\frac{1}{2}\,\oox\tras\,\ooL\,\oox,
 \hspace{15mm}
 P_{d}   =\oox\tras\,\ooA_{s}\,\oox,
 \]
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 the power
redistribution within the system. One can easily verify that all the
dissipating parameters of the system appear only within matrix
$\ooA_{s}$, while matrix $\ooA_{w}$ is completely defined by all the
connection parameters;
 \\[1mm]
2)  all the loops of any POG scheme which graphically represents
system (\ref{expanded_POG_model}) contain always an ``odd'' number
of minus signs ``$-$'' (i.e. the black spots in the summation
blocks);
 \\[1mm]
 3)
the direction of the power flowing through a section of the POG
schemes (i.e. the dashed lines) is positive if an ``even'' number of
signs ``$-$'' is present along any path which goes from the input to
the output of the section. Note, for example, that in
Fig.~\ref{POG_epicicloidale} the power is entering the system in the
power section \nrsec{3} and it is going out in the power section
\nrsec{4}.
\\[1mm]
 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:
\[
\footnotesize
 \x_{1}\!=\!\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}}\!\!,
 \hspace{1mm}
 \x_{2}\!=\!\!\mat{@{}c@{}}{\omega_{p}\\ \omega_{r}}\!\!,
 \hspace{1mm}
 \x_{3}\!=\!\!\mat{@{}c@{}}{F_{cr} \\ F_{sc}}\!\!,
 \hspace{1mm}
 \J_{1}\!=\!\mat{@{}c@{}c@{}}{
 J_{s} &  0 \\
 0 &  J_{c} }\!\!,
 \hspace{1mm}
 \J_{2}\!=\!\mat{@{}c@{}c@{}}{
 J_{p}  &  0 \\
0 &  J_{r} }\!\!,
\]
\[
\footnotesize
\B_{1}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 1 &  0 &  0 \\
 0 &  1 &  0 }\!,\!
 \hspace{1mm}
\B_{2}\!=\!\!\mat{@{}c@{\,}c@{\,}c@{}}{
 0 &  0 &  0 \\
 0 &  0 &  1 }\!,\!
 \hspace{1mm}
 \A_{11}\!=\!\mat{@{}c@{}c@{}}{
 b_{s}\!+\!r_{s}^2d_{sc} & \!-\!r_{s}^2d_{sc}                          \\
 \!-\!r_{s}^2d_{sc}    & b_{c}\!+\!r_{s}^2d_{sc}\!+\!d_{cr}r_{r}^2 }\!\!,
 \]
 \[
 \footnotesize
 \!\A_{12}\!\!=\!\!\mat{@{}c@{\;}c@{}}{
 r_{s}d_{sc}r_{p}                          & 0                      \\
 \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r} & \!-\!d_{cr}r_{r}^2    }\!\!,
 \hspace{0mm}
 \A_{13}\!\!=\!\!\mat{@{}c@{\;}c@{}}{
 0            & r_{s}     \\
 r_{r}       & \!-\!r_{s}}\!\!,
 \hspace{0mm}
 \A_{31}\!\!=\!\!\mat{@{}c@{}c@{}}{
 0                   & \!-\!r_{r} \\
 \!-\!r_{s}            & r_{s}         }\!\!,
\]
\[
 \footnotesize
\!\A_{21}\!=\!\mat{@{}c@{\;}c@{}}{
r_{s}d_{sc}r_{p}        & \!-\!r_{s}d_{sc}r_{p}\!+\!d_{cr}r_{p}r_{r}  \\
0                   & \!-\!d_{cr}r_{r}^2 }\!\!,
 \hspace{8mm}
 \A_{23}\!=\!\mat{@{}c@{}c@{}}{
 r_{p}          & r_{p}  \\
 \!-\!r_{r}  & 0    }\!\!,
\]
 \[
 \footnotesize
 \A_{22}\!=\!\mat{@{}c@{\;}c@{}}{
  b_{p}\!+\!d_{sc}r_{p}^2\!+\!d_{cr}r_{p}^2   & \!-\!d_{cr}r_{p}r_{r}   \\
  \!-\!d_{cr}r_{p}r_{r}                  & b_r\!+\!d_{cr}r_{r}^2 }\!\!,
 \hspace{6mm} \scriptsize
 \A_{32}\!=\!\mat{@{}c@{\;}c@{}}{
 \!-\!r_{p}                              & r_{r}  \\
 \!-\!r_{p}                              & 0     }\!\!,
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure*}[t]
        \begin{center}
 \schemaPOG{(0,-4)(58,14)}{3mm}{
 \blosuretor{\begin{minipage}[c]{1.1cm} Control \end{minipage}}{\V}{\I}
 \nrsezxgiu{0.5}{$\scr 1$}{-1.75}
  \scrivigiu{8}{2}{\scriptsize Synchronous machine}
 \bloxboxgiu{5}{\begin{minipage}[c]{1cm} PMSM \end{minipage}}{}{\omega_{m}}
 \nrsezxgiu{0.5}{$\scr 2$}{-1.75}
  \scrivigiu{8}{2}{\scriptsize Elasticity}
 \blosuattdue{}{K_{1}}{b_{1}}{}{\tau_{s}}{\dst}
 \nrsezxgiu{0.5}{$\scr 3$}{-1.75}
  \scrivigiu{8}{2}{\scriptsize Planetary Gear}
\put(2,10.5){\vector(0,1){1.5}} \put(6,12){\vector(0,-1){1.5}}
\psline[linestyle=dashed, linewidth=0.2pt](1,11.25)(7,11.25)
\put(2,12.5){\makebox(0,0)[c]{$\omega_{sc}$}}
\put(6,12.5){\makebox(0,0)[c]{$\tau_{sc}$}}
%\put(7.6,11){\makebox(0,0)[c]{$\nrsec{7}$}}
 \bloxboxgiu{5}{\begin{minipage}[l]{0.7cm} Plan. \\ Gear \end{minipage}}{\omega_{c}}{\omega_{r}}
 \nrsezxgiu{0.5}{$\scr 4$}{-1.75}
 \put(-0.5,0.25){\nrsezxsu{0.5}{$\scr 7$}{-0.2}}
  \scrivigiu{8}{2}{\scriptsize Elasticity}
 \blosuattdue{}{K_{2}}{b_{2}}{}{\tau_{r}}{\dst}
 \nrsezxgiu{0.5}{$\scr 5$}{-1.75}
  \scrivigiu{10}{2}{\scriptsize Vehicle}
 \bloxboxgiu{7}{ \begin{picture}(5,2) \includegraphics[width=1.5cm]{icon_car.eps} \end{picture}}{\omega_{1}}{}
 \nrsezxgiu{0.5}{$\scr 6$}{-1.75}
  \scrivigiu{6}{2}{\scriptsize Air Resistance}
 \blosuretao{\begin{minipage}[l]{0.7cm} Air \\ Res. \end{minipage}}{\F_{res}}{\dot{\X}}
 }
  \end{center}
  %\vspace{4.6cm}
 \caption{\label{POG_hybrid} POG graphical representation of the considered hybrid vehicle.}
 \end{figure*}
%
The last equation of system (\ref{tran_sys_pog_t}) is 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 follows:
$\x_{2}=-\A_{32}\muno\,\A_{31}\,\x_{1}$. 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
\[
 \L_{r}
 \!=\!
 \mat{@{}c@{\;\,}c@{}}{
 J_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2} J_{p}\!+\!\frac{r_{s}^2 }{r_{r}^2} J_r &
 \!-\!\frac{r_{s}^2 }{r_{p}^2 } J_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! J_r \\[1mm]
 \!-\frac{r_{s}^2 }{r_{p}^2 } J_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! J_r &
 J_{c}\!+\!\frac{r_{s}^2}{r_{p}^2} J_{p} \!+\! \left(1\!+\!\frac{r_{s} }{ r_{r}}\right)^2 \!\!J_r}
\]
\[
 \A_{r}
\!=\!
 \mat{@{}c@{\;\,}c@{}}{
 b_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2} b_{p}\!+\!\frac{r_{s}^2 }{r_{r}^2} b_r &
 \!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! b_r \\[1mm]
 \!-\frac{r_{s}^2 }{r_{p}^2 } b_{p}  \!-\!\frac{r_{s}}{r_{r}} \!\left(1\!+\!  \frac{r_{s} }{ r_{r}}\right)\! b_r &
 b_{c}\!+\!\frac{r_{s}^2}{r_{p}^2} b_{p} \!+\! \left(1\!+\!\frac{r_{s} }{ r_{r}}\right)^2 \!\!b_r}
%\mat{@{}c@{\;\;}c@{}}{
% b_{s}\!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p}\!+\! \frac{r_{s}^2 }{r_{r}^2} b_r &
% \!\!\!\!-\!\left(\!1\!+\!\!\frac{r_{s} }{ r_{r}}\!\right)\!\!\frac{r_{s}}{r_{r}} b_r\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} \\[1mm]
% -\!\left(\!1\!+\!\frac{r_{s} }{ r_{r}}\!\right)\!\frac{r_{s}}{r_{r}} b_r\!-\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} &
% b_{c} \!+\!\frac{r_{s}^2 }{r_{p}^2 } b_{p} \!+\!\left(\!1\!+\!  \frac{r_{s} }{ r_{r}}\!\right)^2 \!\!b_r }
\]
\[
 \B_{r}
 \!=\! \mat{@{\,}c@{\;\;}c@{\;\;}c@{\,}}{
1 & 0 & \!-\frac{r_{s}}{r_{r}}\\[1mm]
0 & 1 & 1\!+\!\frac{r_{s}}{r_{r}}
 },
 \hspace{4mm}
\x_{1}\!=\!\mat{@{}c@{}}{\omega_{s}\\\omega_{c}},
 \hspace{4mm}
 \y
 \!=\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\ \omega_{r}}
% \!=\!\mat{@{}c@{}}{\omega_{s}\\ \omega_{c}\\  }
\]
and $\omega_{r}\!=\!\omega_{c}\!+\!
\frac{r_{s}}{r_{r}}(\omega_{c}\!-\!\omega_{s})$. Note: the
possibility of using a rectangular matrix $\T_{2}$ for transforming
and reducing at the same time a dynamical system is a peculiar
characteristics of the POG technique and it holds only for system
described in the POG form (\ref{generic}). A simulative comparison
between the extended and the reduced models of the planetary gear is
shown in Fig.~\ref{Simulative_comparison} (The simulation details of
this figure will be given in the final paper).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tbp]
 \centering
 \psfrag{Velocita del solare}[c][tc]{\footnotesize Angular velocity $\omega_{s}$ of the sun}
 \psfrag{Wc [rad/s]}[c][c]{\footnotesize $\omega_s$ [rad/s]}
 \psfrag{Time [s]}[ct][c]{\footnotesize Time [s]}
 \includegraphics[width=8cm,height=4cm]{VPPC09_Sun_Figure.eps}
 \caption{Simulative comparison between the extended and the reduced model of the planetary gear.}
 \label{Simulative_comparison}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

With different choices of matrix $\T_{1}$ it is possible to obtain
different but equivalent reduced systems, similar to
(\ref{modello_epiciclo_ridotto}), with different state variables in
vector $\x_{1}$, for example
$\x_{1}=\mat{@{\,}c@{\;}c@{\,}}{\omega_{s}&\omega_{r}}\tras$,
$\x_{1}=\mat{@{\,}c@{\;}c@{\,}}{\omega_{r} & \omega_{c}}\tras$, etc.
Using the POG reduction technique it is also possible to obtain from
(\ref{modello_epiciclo_ridotto}) the reduced static model of the
planetary gear when all the inertias $J_{s}$, $J_{c}$, $J_{p}$ and
$J_{r}$ go to zero. The details of this last transformation will be
given in the final version of this paper.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}[t]
 \centering
 \includegraphics[width=5cm, angle=-90]{architetture_epi.eps}
 \caption{Simulink block scheme of the considered hybrid power structure.}
 \label{architetture_epi}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%




\section{POG modeling of an hybrid automotive system}
\label{Hybrid_Structure}

The planetary gear is the key element for modeling the hybrid
powertrain structure shown in Fig.~\ref{schema_epicicloidale}.
\begin{figure}[htbp]
 \centering
 \psfrag{r}[c]{R}
 \psfrag{c}[c]{C}
 \psfrag{s}[c]{S}
 \psfrag{i}[l]{ICE}
 \psfrag{p}[c]{PMSM}
 \psfrag{k}[b]{$K_{2}$}
 \psfrag{h}[b]{$K_{1}$}
 \psfrag{a}[b]{\nrsec{7}}
 \psfrag{g}[b]{\nrsec{1}}
 \psfrag{d}[b]{\nrsec{2}}
 \psfrag{b}[b]{\nrsec{3}}
 \psfrag{e}[b]{\nrsec{4}}
 \psfrag{f}[b]{\nrsec{5}}
 \includegraphics[width=7.5cm, angle=0]{epicicloidale.eps}
 \caption{Scheme of the considered power structure of the vehicle}
 \label{schema_epicicloidale}
\end{figure}
The considered structure includes an internal combustion engine
(ICE) and a multi-phase permanent magnet synchronous machine (PMSM).
The planetary gear is the element connecting the two motors and the
driving wheels. The ICE is rigidly connected to the Carrier (C), the
PMSM is connected to the Sun (S) by means of an elastic element
$K_{1}$ and the vehicle driving axle is connected to the Ring (R) by
means of an elastic element $K_{2}$. This hybrid system can be
dynamically described by the  ``high level'' POG block scheme shown
in Fig.~\ref{POG_hybrid}. Note that the POG power sections
\nrsec{1}-\nrsec{7} shown in Fig.~\ref{POG_hybrid} correspond to the
physical power sections indicated  in
Fig.~\ref{schema_epicicloidale}.
 The first POG block of Fig.~\ref{POG_hybrid} represents the electric control
of the PMSM. The POG model of the synchronous machine present
between sections \nrsec{1} and \nrsec{2} is  given in
Fig.~\ref{POG_PMSM}, see \cite{ElectrIMACS_2008_Multi_phase} and
\cite{EPE}. The elaboration block between sections \nrsec{2} and
\nrsec{3} represents the elasticity of the shaft connecting the
electric motor to the Planetary Gear Sun. The POG model of the
Planetary Gear present between sections \nrsec{3} and \nrsec{4} is
given in Fig.~\ref{POG_epicicloidale}. The corresponding POG reduced
model proposed in Sec.~\ref{Planetary_Gears} can also be used. The
planetary gear is connected to the driving shaft of the vehicle
through the elastic element present between sections \nrsec{4} and
\nrsec{5}. The dynamics of the vehicle is described by the POG block
present between sections \nrsec{5} and \nrsec{6}, see the POG model
given in \cite{2009_ELECTROMOTION}. Finally the resistance of the
air is described by the last POG block.

Further details of this hybrid system will be given in the final
version of this paper.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[tbp]
   \begin{center}
 \schemaPOG{(0,-4)(25,12)}{3.0mm}{
 \thicklines
 \bloin{\V}{\I}
 \nrsezgiu{$\scr 1$}{-1.75}
 \scrivigiu{10}{1.8}{ \footnotesize Electrical part}
 \blovigiumxm{\ds\L\muno}{}{\dst}{\frac{1}{s}}{ \tiny \boldsymbol{\Phi}}
 \bloivsuaor{\R}{}{}
 %\nrsezgiu{$\scr 2$}{-1.75}
 \scrivigiu{6}{1.8}{\footnotesize Energy \\ \footnotesize conversion}
 \blokxyaor{4}{3}{\ds \K_{\tau}(\theta)}
                 {\ds \K_{\tau}\tras(\theta)}{\E}{\tau_{m}}
 %\nrsezgiu{$\scr 3$}{-1.75}
 \scrivigiu{10}{1.8}{ \footnotesize Mechanical part}
 \blovisumxm{\ds\frac{1}{J_m}}{\omega_{m}}{\dst}{\frac{1}{s}}{}
 \bloivgiuor{b_m}{}{}
 \nrsezgiu{$\scr 2$}{-1.75}
 \bloout{}{\tau_{s}}
 \scrivigiu{}{1.8}{}
 }
  \end{center}
  \caption{POG scheme of the multi-phase synchronous motor (PMSM).}\label{POG_PMSM}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{Simulations}

The system shown in Fig.~\ref{schema_epicicloidale} has been
implemented in Matlab/Simulink as it is shown in
Fig.~\ref{architetture_epi}. All subsystems have been modeled with
POG and in particular the vehicle subsystem in the right part of the
scheme is a bicycle model of a car that includes the tire-road
elastic interaction, see \cite{2009_ELECTROMOTION}.

A full set of simulation results of the controlled hybrid system
will be given in the final version of the paper.




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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{1}
%
%\bibitem{leonhard}
%W.~Leonhard, {\em Control of Electrical Drives}, 3rd Edition 2001,
%Springer-Verlag Berlin Heidelberg NewYork, ISBN 3-540-41820-2.
%
% Zanasi
%
\bibitem{Zana1}
R.~Zanasi, ``Power Oriented Modelling of Dynamical System for
Simulation'', IMACS Symp. on Modelling and Control of Technological
System, Lille, France, May 1991.
%
%\bibitem{mitmemo}
%R.~Zanasi, K.~Salisbury,  ``Dynamic Modeling, Simulation and
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\bibitem{bodypap93}
Zanasi R., ``Dynamics of a $n$-links Manipulator by Using
Power-Oriented Graph'',  {\em SYROCO '94}, Capri, Italy, 1994.
%
% \bibitem{IECON_2006_Parigi_POG}
% R.~Morselli, R.~Zanasi, ``Modeling of Automotive Control Systems
% Using Power Oriented Graphs'', 32nd Annual Conference of the IEEE
% Industrial Electronics Society, IECON 2006,  Parigi, 7-10
% Novembre, 2006.
%
%
%
%
% \bibitem{Paynter} Paynter, H.M., {\em Analysis and Design of Engineering Systems},
% MIT-press, Camb., MA, 1961.
%
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 Interscience, ISBN 0-471-33301-8, 3rd ed. 2000.
%
%\bibitem{EUROSIM}
%R.~Zanasi, F.~Grossi ``The POG technique for modelling multi-phase
%permanent magnet synchronous motors'', 6th EUROSIM Congress on
%Modelling and Simulation, Ljubljana, 9-13 September 2007.
%

\bibitem{ElectrIMACS_2008_Multi_phase}
 R.~Zanasi, F.~Grossi, ``Multi-phase Synchronous Motors: POG
Modeling and Optimal Shaping of the Rotor Flux'', ELECTRIMACS
2008, Qu\'ebec, Canada, June 2008.

\bibitem{EPE}
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%\bibitem{2009_ECC_Asynchronous}
%R.~Zanasi, F.~Grossi, G.~Azzone, ``Dynamic Behavior of Multi-phase
%Asynchronous Motors'', submitted to 10th European Control
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%

\bibitem{2009_ELECTROMOTION}
F.~Grossi, W.~Lhomme, R.~Zanasi, A.~Bouscayrol, ``Modelling and
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\end{thebibliography}


\end{document}
