multiphysicsL17.tex

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% Copyright © 2014 Peeter Joot.  All Rights Reserved.
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%\generatetitle{ECE1254H Modeling of Multiphysics Systems.  Lecture 17: Stability.  Taught by Prof.\ Piero Triverio}
%%\chapter{Stability}
%\label{chap:multiphysicsL17}
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%\section{Disclaimer}
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%Peeter's lecture notes from class.  These may be incoherent and rough.
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\section{Stability (continued).}
\index{stability}
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Continuing with the simple continuous time test system
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\begin{equation}\label{eqn:multiphysicsL17:20}
\dot{x}(t) = \lambda x(t)
\end{equation}
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With the application of a trial solution \( x(t) = e^{s t} \), the resulting characteristic equation is
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\begin{equation}\label{eqn:multiphysicsL17:40}
s = \lambda,
\end{equation}
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and stability follows, provided that \( \Real(\lambda) < 0 \) as sketched in \cref{fig:lecture17:lecture17Fig1}.
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\imageFigure{../figures/ece1254-multiphysics/lecture17Fig1}{Stability region.}{fig:lecture17:lecture17Fig1}{0.2}
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Utilizing LMS methods, for example TR, the discrete time system results, such as
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\begin{equation}\label{eqn:multiphysicsL17:60}
\gamma_{-1} x_{n+1} +
\gamma_{0} x_{n} +
\gamma_1 x_{n-1}
+ \cdots
= 0.
\end{equation}
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% n+1 = k
% n = k-1
% n + j = k - 1 + j
With a substitution \( x_{n + j} = z^{k - 1 + j} \), a characteristic polynomial results
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\begin{equation}\label{eqn:multiphysicsL17:80}
\gamma_{-1} z^{k} +
\gamma_{0} z^{k-1} +
\gamma_1 z^{k-2} + \cdots = 0.
\end{equation}
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This is stable provided \( \Abs{z_l} < 1 \), as illustrated in \cref{fig:lecture17:lecture17Fig2}.
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\imageFigure{../figures/ece1254-multiphysics/lecture17Fig2}{Stability region in z-domain.}{fig:lecture17:lecture17Fig2}{0.3}
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... SWITCHED TO SLIDES.
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\makedefinition{Stiff.}{dfn:multiphysicsL17:100}{
A \textAndIndex{stiff} system is one that has multiple timescales, as characterized by a significant range of eigenvalues.
}
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\makedefinition{Lossless system.}{dfn:multiphysicsL17:120}{
Lossless \index{pole!lossless} poles are those that reside strictly on the imaginary axis.
}
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\makedefinition{Lossly system.}{dfn:multiphysicsL17:140}{
Lossly poles \index{pole!lossly} are those that reside strictly to the left of the imaginary axis.
}
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\makedefinition{Active system.}{dfn:multiphysicsL17:160}{
Active poles \index{pole!active} are those that reside strictly to the right of the imaginary axis.
}
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\makedefinition{A-stable.}{dfn:multiphysicsL17:170}{
See slides.
}
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\index{Dahlquist's theorem}
\maketheorem{Dahlquist's theorem.}{thm:multiphysicsL17:180}{
There are no LMS methods of order greater than 2 that are A-stable.  Also known as the Dahlquist barrier.  The TR method has the lowest error of the A-stable LMS methods.
}
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\paragraph{Some recent developments}
\index{backward differentiation formula}
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The backward differentiation formulas (BDF) are of \( \text{order} > 2 \).  They are not A-stable, but can be sufficient with stability sketched in \cref{fig:lecture17:lecture17Fig3}.
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\imageFigure{../figures/ece1254-multiphysics/lecture17Fig3}{BDF stability region.}{fig:lecture17:lecture17Fig3}{0.2}
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Also available are the Obreshkov formulas \citep{butcherOrder}, which are both A-stable and of \( \text{order} > 2 \).  There is a cost to both of these methods: computation of the derivatives at each step.
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