\documentclass[12pt]{article}
\usepackage{latexsym}
\renewcommand{\baselinestretch}{1.4}
\setlength{\topmargin}{-0.25in}
\setlength{\textheight}{8.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}
\newcommand{\mb}{\mbox}
\newcommand{\bm}{\boldmath}
\def\b#1{\mb{\bm$#1$}}
\newfont{\Bb}{msbm10 scaled\magstep1}
\input setbmp
\input{psfig.sty}
\usepackage{epsfig}
%\pagestyle{empty}
\begin{document}
\section{Fuzzy Controller Design Using Evolutionary Algorithm}

In this section, we show how an evolutionary algorithm can be used to fine tune parameters of a fuzzy system to approximate a nonlinear function. We then use this method to design fuzzy controllers. The essential element of approximating a function using fuzzy system or fuzzy controller design is a fuzzy base whose rules are used to approximate a given function or to design a fuzzy controller. Suppose that we are faced with approximating a vector valued  function $\b f:\mb{\Bb R}^p \to \mb{\Bb R}^m$ of $p$ variables, $x_1, x_2,\ldots, x_p$. An example of the form the $i$-th rule that we will use to constaruct a fuzzy approximator of the the function $f$ is:
\begin{eqnarray*}
&{}&\mb{IF  } x_1 \mb{  IS  } F_1^i \mb{  WITH  } \alpha_1^i \mb{  AND  } x_2  \mb{  IS  } F_2^i \mb{  WITH  } \alpha_2^i \mb{  AND  }\cdots \mb{  AND  } x_p \mb{  IS  } F_p^i \mb{  WITH  } \alpha_p^i\\
&{}&\qquad\qquad\qquad\qquad \mb{  THEN  } y_1\mb{  IS  } G_1^i\mb{  AND  }\cdots\mb{  AND  } y_m \mb{  IS  } G_m^i
\end{eqnarray*}
In the above rule, $\alpha_i^j$ is a non-negative real number reflecting the weight attached to the $j$-th fuzzy set $F_i^j$ in the $i$-th rule, and $G_j^i$ is the output singelton function. The fuzzy sets used by us are characterized by a trapezoidal-type memebership functions whose typical graph is depicted in Figure~\ref{fig:trapez}.
\begin{figure}
\vspace{2in}
\hskip.5in\special{eps:c:/mybook/trapez.eps y=2in}
%\centering\epsfig{file=trapez.eps,height=2in}
\label{fig:trapez}
\caption{Trapezoidal-type fuzy set for the fuzzy controller design using evolutionary algorithm.}
\end{figure}
The memebership function shown in Figure~\ref{fig:trapez} can be described analytically as follows:
\begin{equation}
F_j^i(x)=\left\{\begin{array}{lll}
\frac{1}{d_{ij}^l}\left(x+d_{ij}+d_{ij}^l-c_{ij} \right) &\mb{if}& c_{ij} -d_{ij} - d_{ij}^l\le x < c_{ij} - d_{ij}\\
1 &\mb{if}& c_{ij} - d_{ij}\le x  <  c_{ij} + d_{ij}\\
\frac{1}{d_{ij}^r}\left(-x + d_{ij}+d_{ij}^r + c_{ij} \right) &\mb{if}& c_{ij} + d_{ij}\le x < c_{ij} + d_{ij} +  d_{ij}^r\\
0 &{}& \mb{otherwise}
\end{array}\right.
\label{eq:analytdescrip}
\end{equation}
Careful analysis of the above trapezoidal fuzzy set leads us to the conclusion that the memebership function characterizing this set can be uniquely described with four non-negative parameters, $c_{ij}$, $d_{ij}$, $d_{ij}^j$, and $d_{ij}^r$. We employ this fact in the construction of the evolutionary algorithm for fine tuning parameters of the fuzzy system approximating a given nonlinear function. To proceed, let
\begin{equation}
w_i=F^i_1(x_1)\times \alpha_1^i\times F^i_2(x_2) \times \alpha_2^i \times\cdots \times F^i_p(x_p)\times \alpha^i_p.
\label{eq:weightdefn}
\end{equation}
Then, with $r$ rules in the rule base, the $q$-th output of the fuzzy system is
\begin{equation}
y_q=\frac{\sum_{i=1}^r w_i \times G^i_q}{\sum_{i=1}^r w_i}.
\label{eq:qthfuzout}
\end{equation}

\end{document}