1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
|
+
+
+
|
% -*-latex-*- Put EMACS into LaTeX-mode
% Verbal description for system ImplicitRC (ImplicitRC_desc.tex)
% Generated by MTT on Wednesday June 24 09:50:17 BST 1998.
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %% Version control history
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %% $Id$
% %% $Log$
% %% Revision 1.1 2000/12/28 18:06:50 peterg
% %% To RCS
% %%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This report describes the \emph{implicit} integration methods
available in MTT. They are introduced to provide simulation
of systems within the following context:
\begin{enumerate}
\item The system may be stiff with a mixture of slow and fast
|
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
|
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
|
-
-
+
+
-
-
+
+
-
-
+
+
|
required.
In contrast, the implicit method is stable.
\subsubsection{Example}
The acausal bond graph of system \textbf{ImplicitRC} is
displayed in Figure \Ref{ImplicitRC_abg} and its label
file is listed in Section \Ref{sec:ImplicitRC_lbl}.
displayed in Figure \Ref{fig:ImplicitRC_abg.ps} and its label
file is listed in Section \Ref{sec:ImplicitRC_lbl}
The subsystems are listed in Section \Ref{sec:ImplicitRC_sub}.
The system represents two simple RC circuits in series with
differential equations as given in Section \Ref{sec:ImplicitRC_ode.tex} and
transfer function as given in Section \Ref{sec:ImplicitRC_tf.tex}.
differential equations as given in Section \Ref{sec:ImplicitRC_ode-noargs.tex} and
transfer function as given in Section \Ref{sec:ImplicitRC_tf-noargs.tex}.
For the purposes of this example the two time constants are $1$ and
$\epsilon=10^{-3}$ -- this is a stiff system. All of the simulations
use a sample interval of $\Delta t = 0.1$ ang the input is a unit
step. Section \Ref{sec:ImplicitRC_sro} shows the exact (computed from
the matrix exponential) solution, and Section \Ref{sec:ImplicitRC_odeso}
step. Section \Ref{sec:ImplicitRC_sro-noargs.ps} shows the exact (computed from
the matrix exponential) solution, and Section {sec:ImplicitRC_odeso-cc.ps}
shows the solution by implicit integration.
The explicit solution is not shown, but was found to be unstable for
$\Delta t > 0.002$ as predicted.
\subsection{Implicit integration - the nonlinear case}}
\label{sec:nonlinear}
|