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\begin{document}
\title{gBSB Snapshots}
\author{Stanislaw H. \.{Z}ak \thanks{School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907}}
\maketitle
\begin{abstract}
Adaptive fuzzy tracking controller for a class of dynamical systems is analyzed. It is shown that the closed-loop system driven by this fuzzy adaptive controller stable and the adaptation parameters are bounded.
\end{abstract}
\section{Introduction}

\begin{example}\label{ex:gbsbsnaps}\rm In this example, we use a gBSB neural net with the state in the hypercube $\{-1, 1\}^{25}$. We construct the weight matrix to store the four letters of alphabet whose bit maps are shown in Figure~\ref{fig:bitmaps}.
\begin{figure}
\centerline{\epsfig{figure=sn_stor.eps,height=1in}}
\caption{Bit maps of the characters $A$, $B$, $C$, and $D$ stored in the gBSB neural net of Example~\ref{ex:gbsbsnaps}.}
\label{fig:bitmaps}
\end{figure}
The bit maps are $5\times 5$ with elements from the set $\{-1,1\}$ with $-1$ corresponding to black and 1 to white. The pattern vectors are obtained from the bit maps by stacking their columns. For example, if $A=\left[\begin{array}{ccccc}
\b a_1,&\b a_2,&\b a_3,&\b a_4,&\b a_5
\end{array}\right]$, where $\b a_i$, $i=1,2,\ldots,5$ is the $i$-th column of $A$, then the corresponding pattern vector is obtained from $A$ by applying the stacking operator $\b s:\mb{\Bb R}^{5\times 5} \to \mb{\Bb R}^{25}$  to get
\[
\b s(A)=A(:)=\left[\begin{array}{c}
\b a_1\\
\vdots\\
\b a_5
\end{array}\right] \in \{-1,1\}^{25}.
\]
The weight matrix, $\b W$, of the gBSB net was constructed using the weight matrix construction algorithm, where
\[
\b V=\left[\begin{array}{cccc}
A(:) & B(:) & C(:) &D(:)
\end{array}\right].
\]
The values of other parameters chosen in this example were: $\b D=1.1\b I_{25}$, $\b{\Lambda}=-2 \b I_{25}$, $\b b=0.3 A(:)+0.1 B(:)$, and $\alpha=0.5$. Snapshots of the network iterates for different initial patterns are shown in Figures~\ref{fig:snap_at}--\ref{fig:snap_dt}.
\begin{figure}
\centerline{\epsfig{figure=snap_at.eps,height=2.5in}}
\caption{Snapshots of the iterations of the gBSB net of Example~\ref{ex:gbsbsnaps} converging towards the letter $A$.}
\label{fig:snap_at}
\end{figure}
\begin{figure}
\centerline{\epsfig{figure=snap_bt.eps,height=2.5in}}
\caption{Snapshots of the iterations of the gBSB net of Example~\ref{ex:gbsbsnaps} converging towards the letter $B$.}
\label{fig:snap_bt}
\end{figure}
\begin{figure}
\centerline{\epsfig{figure=snap_ct.eps,height=2.5in}}
\caption{Snapshots of the iterations of the gBSB net of Example~\ref{ex:gbsbsnaps} converging towards the letter $C$.}
\label{fig:snap_ct}
\end{figure}
\begin{figure}
\centerline{\epsfig{figure=snap_dt.eps,height=2.5in}}
\caption{Snapshots of the iterations of the gBSB net of Example~\ref{ex:gbsbsnaps} converging towards the letter $D$.}
\label{fig:snap_dt}
\end{figure}

\end{example}\markex
\end{document}