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% Verbal description for system PDe (PDe_desc.tex)
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The acausal bond graph of system \textbf{PDe} is
displayed in Figure \Ref{fig:PDe_abg.ps} and its label
file is listed in Section \Ref{sec:PDe_lbl}.
The subsystems are listed in Section \Ref{sec:PDe_sub}.
This is a proportional + derivative (PD) controller for a
collocated sutuation where the control signal is an effort and the
measured signal is a (collocated) flow.
The controller can be thought of as controlling \emph{integated
flow}, and it is with respect to this that the P and D terms are defined.
The setpoint is a \emph{flow}; and must be generated to give the
desired \emph{integrated} flow.
Physically, the controller is a \textbf{C} and an \textbf{R}
component - for mechanical systems a mass and a spring.
Mathematically, in integral causality, the equations are:
%file: pde_{dae}.tex
%differential-algebraic equations
\begin{equation}
\begin{aligned}
\dot x_{1} &=
{
f_d - f
}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
u &=
{
- k_{d} f + k_{p} x_{1}
}
\end{aligned}
\end{equation}
The state $x_1$ is the the integrated difference between
\emph{desired} flow $f_d$ and the actual flow $f$. Thus the control
signal $u$ is $k_p$ multiplied by the position error minus $k_d$ time
the flow.