Commit 10acdf5b authored by Leif Andersson's avatar Leif Andersson
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Förkortat beamerExample.tex för att det skall passa med eightbeamer

parent f7786014
......@@ -17,9 +17,9 @@
*.xdv
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# *.ps
# *.eps
# *.pdf
*.ps
*.eps
*.pdf
## Generated if empty string is given at "Please type another file name for output:"
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% ----------------------------------------------------------------------
\begin{frame}{\hspace*{10pt}Alexandr Mihailovich Lyapunov (1857--1918)}
\begin{frame}{Alexandr Mihailovich Lyapunov (1857--1918)}
\begin{columns}
\begin{column}{0.3\textwidth}
\includegraphics[width=\textwidth]{figures/lyapunov_stamp}
......@@ -334,270 +334,4 @@ then $x=0$ is unstable.
\end{itemize}
\end{frame}
% ----------------------------------------------------------------------
\begin{frame}
%
\frametitle{\large Proof of (1) in Lyapunov's Linearization Method}
Lyapunov function candidate $V(x)=x^TPx$. $V(0)=0$, $V(x)>0$ for
$x\neq 0$, and
\begin{align*}
\dot{V}(x)&=x^TPf(x)+f^T(x)Px\\
&=x^TP[Ax+g(x)]+[x^TA+g^T(x)]Px\\
&=x^T(PA+A^TP)x+2x^TPg(x)=-x^TQx+2x^TPg(x)
\end{align*}
%
\begin{equation*}
x^TQx\geq\lambda_{\min}(Q)\Vert x\Vert^2
\end{equation*}
%
and for all $\gamma>0$ there exists $r>0$ such that
%
\begin{equation*}
\Vert g(x)\Vert<\gamma\Vert x\Vert, \qquad \forall\Vert x\Vert<r
\end{equation*}
%
Thus, choosing $\gamma$ sufficiently small gives
%
\begin{equation*}
\dot{V}(x)\leq-\big(\lambda_{\min}(Q)-2\gamma\lambda_{\max}(P)\big)\Vert x\Vert^2<0
\end{equation*}
%
\end{frame}
% ----------------------------------------------------------------------
\end{document}
\begin{frame}
%
\frametitle{Invariant Sets}
\textbf{Definition} A set $M$ is called \textbf{invariant} if for
the system
$$
\dot{x}=f(x),
$$
$x(0)\in M$ implies that $x(t)\in M$ for all $t\geq 0$.
\medskip
\begin{center}
\psfrag{x0}[][]{$x(0)$}
\psfrag{xt}[][]{$x(t)$}
\psfrag{M}[][]{$M$}
\includegraphics[width=0.5\hsize]{figures/invariant_set.eps}
\end{center}
\end{frame}
% ----------------------------------------------------------------------
\begin{frame}
%
\frametitle{Invariant Set Theorem}
\textbf{Theorem} Let $\Omega\in\mathbf{R}^n$ be a bounded and closed set
that is invariant with respect to
$$
\dot{x}=f(x).
$$
Let $V:\mathbf{R}^n\rightarrow\mathbf{R}$ be a radially unbounded
$C^1$ function such that
$\dot{V}(x)\leq 0$ for $x\in\Omega$. Let $E$ be the set of points
in $\Omega$ where $\dot{V}(x)=0$. If $M$ is the largest invariant set in
$E$, then every solution with $x(0)\in\Omega$ approaches $M$ as
$t\rightarrow\infty$ (proof on p.~73)
\begin{center}
\psfrag{Om}[][]{$\Omega$}
\psfrag{E}[][]{$E$}
\psfrag{M}[][]{$M$}
\includegraphics[width=0.4\hsize]{figures/lyap_invariant.eps}
\end{center}
\end{frame}
%----------------------------------------------------------------------
\begin{frame}
%
\frametitle{Example---Stable Limit Cycle}
Show that $M=\{x:\,\Vert x\Vert= 1\}$ is a globally stable limit cycle for
\begin{align*}
\dot{x}_1&=x_1-x_2-x_1(x_1^2+x_2^2)\\
\dot{x}_2&=x_1+x_2-x_2(x_1^2+x_2^2)
\end{align*}
Let $V(x)=(x_1^2+x_2^2-1)^2$.
\begin{align*}
\frac{dV}{dt}&=2(x_1^2+x_2^2-1)\frac{d}{dt}(x_1^2+x_2^2-1)\\
&=-2(x_1^2+x_2^2-1)^2(x_1^2+x_2^2)\leq 0\quad\text{for } x\in\Omega\\
\end{align*}
$\Omega=\{0<\Vert x\Vert\leq R\}$ is invariant for $R=1$.
\end{frame}
%----------------------------------------------------------------------
\begin{frame}
%
\frametitle{Example---Stable Limit Cycle}
\begin{equation*}
E=\{x\in\Omega:\,\dot{V}(x)=0\}=\{x:\,\Vert x\Vert= 1\}
\end{equation*}
$M=E$ is an invariant set, because
\begin{equation*}
\frac{d}{dt}V=-2(x_1^2+x_2^2-1)(x_1^2+x_2^2)=0\quad\text{for
} x\in M
\end{equation*}
We have shown that $M$ is a stable limit cycle (globally stable in
$R-\{0\}$)
\begin{center}
\includegraphics[height=0.3\hsize]{figures/guck}
\end{center}
\end{frame}
% ----------------------------------------------------------------------
\begin{frame}
%
\frametitle{A Motivating Example (cont'd)}
$$
m\ddot{x}=-b\dot{x}\vert\dot{x}\vert-k_0x-k_1x^3
$$
$$
V(x,\dot{x})=(2m\dot{x}^2+2k_0x^2+k_1x^4)/4>0, \qquad V(0,0)=0
$$
$\dot{V}(x,\dot{x})=-b\vert\dot{x}\vert^3$ gives
$E=\{(x,\dot{x}):\,\dot{x}=0\}$.
Assume there exists $(\bar{x},\dot{\bar{x}})\in M$ such that
$\bar{x}(t_0)\neq 0$. Then
$$
m\ddot{\bar{x}}(t_0)=-k_0\bar{x}(t_0)-k_1\bar{x}^3(t_0)\neq 0
$$
so $\dot{\bar{x}}(t_0+)\neq 0$ so the trajectory will immediately
leave $M$. A contradiction to that $M$ is invariant.
Hence, $M=\{(0,0)\}$ so the origin is asymptotically stable.
\end{frame}
%----------------------------------------------------------------------
\begin{frame}
%
\frametitle{Adaptive Noise Cancellation by Lyapunov Design}
\begin{center}
\psfrag{u}[][]{$u$}
\psfrag{g1}[][][1.4]{$\frac{b}{s+a}$}
\psfrag{g2}[][][1.4]{$\frac{\widehat b}{s+\widehat a}$}
\psfrag{x}[][]{$x$}
\psfrag{xh}[][]{$\widehat x$}
\psfrag{xt}[][]{$\widetilde x$}
\psfrag{+}[][]{$+$}
\psfrag{-}[][]{$-$}
\includegraphics[width=0.4\hsize]{figures/noise.eps}
\end{center}
\begin{align*}
\dot x +a x &= bu\\
\dot{\widehat x} + \widehat a \widehat x &= \widehat b u
\end{align*}
Introduce $\widetilde x = x-\widehat x, \enskip\widetilde a =
a-\widehat a, \enskip \widetilde b = b-\widehat b$.
Want to design adaptation law so that $\widetilde x\to 0$
\end{frame}
\begin{frame}
Let us try the Lyapunov function
\begin{align*}
V&=\frac{1}{2}(\widetilde x^2+\gamma_a\widetilde a^2+\gamma_b\widetilde
b^2)\\
\dot V &= \widetilde x \dot{\widetilde x} + \gamma_a \widetilde a
\dot{\widetilde a} + \gamma_b \widetilde b
\dot{\widetilde b} = \\
&=\widetilde x(-a\widetilde x-\widetilde a \widehat x + \widetilde b
u) + \gamma_a \widetilde a
\dot{\widetilde a} + \gamma_b \widetilde b
\dot{\widetilde b} = -a \widetilde x^2
\end{align*}
where the last equality follows if we choose
\begin{align*}
\dot{\widetilde a} = -\dot{\widehat a} = \frac{1}{\gamma_a} \widetilde{x} \widehat x \qquad
\dot{\widetilde b} = -\dot{\widehat b} = -\frac{1}{\gamma_b} \widetilde{x} u
\end{align*}
Invariant set: $\widetilde x =0$.
This proves that $\widetilde x \to 0$.
(The parameters $\widetilde a$ and $\widetilde b$
do not necessarily converge: $u\equiv 0$.)
\fbox{Demonstration if time permits}
\end{frame}
%%------------------------------
\begin{frame}
\frametitle{Results}
\begin{center}
\psfrag{p1}[][][1.3]{$\hat{a}$}
\psfrag{p2}[][][1.3]{$\hat{b}$}
\includegraphics[width=0.7\hsize]{musik/parameters.eps} \\
Estimation of parameters starts at t=10 s.
\end{center}
\end{frame}
%----------------------------------------------------------------------
\begin{frame}
\frametitle{Results}
\begin{center}
\psfrag{x-xhat}[][][1.3]{$x-\hat{x}$}
\includegraphics[width=0.6\hsize,angle=0]{musik/adap.ps} \\
\includegraphics[width=0.3\hsize]{musik/xt.eps}
\end{center}
\begin{center}
\begin{small}
Estimation of parameters starts at t=10 s.
\end{small}
\end{center}
\end{frame}
%----------------------------------------------------------------------
\begin{frame}
%
\frametitle{Next Lecture}
\begin{center}
Stability analysis using input-output (frequency) methods
\end{center}
\begin{itemize}
item Stability analysis using input-output (frequency) methods
\end{itemize}
\vfill
\end{frame}
%----------------------------------------------------------------------
\begin{frame}
\frametitle{title}
Let us try the Lyapunov function
\begin{align*}
V&=\frac{1}{2}(\widetilde x^2+\gamma_a\widetilde a^2+\gamma_b\widetilde
b^2)\\
\dot V &= \widetilde x \dot{\widetilde x} + \gamma_a \widetilde a
\dot{\widetilde a} + \gamma_b \widetilde b
\dot{\widetilde b} = \\
&=\widetilde x(-a\widetilde x-\widetilde a \widehat x + \widetilde b
u) + \gamma_a \widetilde a
\dot{\widetilde a} + \gamma_b \widetilde b
\dot{\widetilde b} = -a \widetilde x^2
\end{align*}
where the last equality follows if we choose
\begin{align*}
\dot{\widetilde a} = -\dot{\widehat a} = \frac{1}{\gamma_a} \widetilde{x} \widehat x \qquad
\dot{\widetilde b} = -\dot{\widehat b} = -\frac{1}{\gamma_b} \widetilde{x} u
\end{align*}
Invariant set: $\widetilde x =0$.
This proves that $\widetilde x \to 0$.
(The parameters $\widetilde a$ and $\widetilde b$
do not necessarily converge: $u\equiv 0$.)
\fbox{Demonstration if time permits}
\end{frame}
\ No newline at end of file
\end{document}
\ No newline at end of file
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