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\title{COMPARATIVE STUDY OF CONVENTIONAL AND GENETIC ALGORITHMS IN
ADAPTIVE SIGNAL PROCESSING AND CONTROL\thanks{This work was supported 
in part by the Russian Ministry of Education
(grant No.~T02-03.2-3427).}}

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\begin{document}
\begin{slide} {\Large
\begin{center} \ \
\vspace{0.5cm}
%%%%%%%%%%%%%%%%%%%%%%Title list%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\bf {COMPARATIVE STUDY OF CONVENTIONAL AND GENETIC ALGORITHMS IN
ADAPTIVE SIGNAL PROCESSING AND CONTROL
\footnote{This work was supported by research grant 
No.~T02-03.2-3427 from the Russian Ministry of Education and Science.}
}} \\ \vspace{0.5cm}

{\bf Innokenti Semoushin$^\ast$ and Maria A. Fedorova$^\ast$
\\ }
{\normalsize $^\ast${\em School of Mathematics and Mechanics, Ulyanovsk State University\\
42 Leo Tolstoy Street, Ulyanovsk 432970, Russia\\
\em fax: +7 8422 41 2340 \quad e-mail: {\tt SemushinIV@ulsu.ru}}}
\vspace{0.2cm}

\begin{abstract}\vspace{0.2cm}
{\large A comparative study is made of genetic algorithms (GA) and
conventional numeric algorithms (NA) for the purpose of filter
optimization in a class of adaptive stochastic systems known as
trackers. The need for using NA or GA is associated with uncertainty in
covariances of the noises driving a reference signal model (RSM)
(a shaping filter) and a controlled plant (CP).
}
\end{abstract}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%End of Title list%%%%%%%%%%%%%%%%%%%%%
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\newslide

\section{INTRODUCTION}
The increasing interest in synthetic systems capable to maintain
high performance under uncertainty and  unpredictability has
heightened the need for borrowing the mechanisms of adaptation
from Biology, Evolution and Genetics suitable to improve the
behavior of the system under synthesis. That such is the case is
suggested by the fact that the standard mathematical programming
models, that are synthetic as well, are not able sometimes to
guarantee good results: the swiftness of response, short settling
time and high steady-state accuracy for low computational cost.

To an increasing degree many non-standard approaches such as Fuzzy
Technology, Neural Networks and Evolutionary Computing are merged
in the area of ``Intelligent Techniques'' or ``Computational
Intelligence'' (CI).  The last three decades have witnessed a very
strong growth of CI. Since the early seventies of the last century
these techniques have also been applied to a large variety of
problems to deliver efficient solutions and benefits to the users.
Thus, the joint usage of expert system technology and fuzzy logic
enables data base size to be cut in tens \cite{Zhang}. Another
approach to intelligent system optimization and learning is
closely associated with developing the combined technologies of
fuzzy neural structures \cite{Buckley, Perneel}. Modern special
purpose program-instrumental tools make possible not only
simulate, in all details, the system under construction, but
evaluate efficiency of the adopted design solutions based on one
or other intelligent technology, as well. For example, the
application soft package \emph{WinFACT} (Windows Fuzzy and Control
Tools) provides a means for going from a classic design to a
neural network version of the fuzzy control being constructed
through it \cite{Kahlert}.

At the same time, some distrust as  to the novel, non-standard
approaches to system optimization remains a fact within applied
and computational mathematics community. This fact stimulates a
comparative study of  conventional and genetic approaches to the
problem of eliminating uncertainty during the system life cycle.
The genetic methods are of special interest because of their
ability to treat the complex optimization problems using a simple
and relatively inexpensive model.

In this paper, a problem of  discrete time filter optimization
both in open and close loop is considered as applied to trackers
\cite{Mb}. Two approaches to filter optimization within a
stochastic control system operating under uncertainty are
experimentally compared:
\begin{description}
    \item{1)} The auxiliary performance index approach based on
classic numerical optimization methods \cite{Caines} and
    \item{2)} Non-numerical filter optimization using genetic
algorithms \cite{GA1,GA2}.
\end{description}

The outline of the paper  is as follows. Section~\ref{problem}
describes the problem including the monitored system, Kalman
filter and adaptive filter. A general framework for Kalman filter
identification is given in Section~\ref{framework}.
Section~\ref{genetic} contains a short description of genetic
algorithms. Some computational experiment results are shown in
Section~\ref{simul}.  Finally, Section~\ref{conclusion} concludes
the paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newslide

\section{PROBLEM DESCRIPTION}\label{problem}

To construct and study the adaptive  tracker, first express the
problem in terms of the augmented state process \footnote{$^{)}$
The notation ``$\doteq$'' instead of ``$=$'' is meant in this
paper to indicate ``equality by definition''.}$^{)}$
\[%begin{equation}
   x_{\mathrm{a}}(\cdot)\doteq\left[\begin{array}{c} x(\cdot)\\
x_{\mathrm{r}}(\cdot)\\ \end{array}\right]\in\Rset^n
\]%end{equation}
as generated in discrete time by
\begin{equation}\label{AS1}
  \begin{array}{l}
     \left[\begin{array}{c}
             x(t_{i+1})\\
             x_{\mathrm{r}}(t_{i+1})\\
           \end{array}\right]=
           \left[\begin{array}{cc}
             \Phi&0\\
             0&\Phi_{\mathrm{r}}\\
           \end{array}\right]
           \left[\begin{array}{c}
             x(t_i)\\
             x_{\mathrm{r}}(t_i)\\
           \end{array}\right]+
           \left[\begin{array}{c}
             \Psi\\
             0\\
           \end{array}\right]u(t_i)
     +
           \left[\begin{array}{c}
             w(t_i)\\
             w_{\mathrm{r}}(t_i)\\
           \end{array}\right]
  \end{array}
\end{equation}
%
\begin{equation}\label{Q}
   \E{
   \left[\begin{array}{c} w(t_i)\\ w_{\mathrm{r}}(t_j)\\
\end{array}\right]
   \left[\begin{array}{cc} w^T(t_i)& {w_{\mathrm{r}}}^T(t_j)\\
\end{array}\right]
     }=
         \left[\begin{array}{cc}
             Q&0\\
             0&Q_{\mathrm{r}}\\
            \end{array}
      \right]\delta_{i,j}
\end{equation}
with a control input $u(\cdot)$ and zero mean white noises
$w(\cdot)$, $w_{\mathrm{r}}(\cdot)$, and observed by
\begin{equation}\label{AS2}
  z_{\mathrm{a}}(\cdot)\doteq\left[\begin{array}{c} z(\cdot)\\
z_{\mathrm{r}}(\cdot)\\ \end{array}\right]\in\Rset^m, \quad
z_{\mathrm{a}}(t_{i})=\left[\begin{array}{c}
         z(t_i)\\
         z_{\mathrm{r}}(t_i)\\
       \end{array}\right]=
       \left[\begin{array}{cc}
         H&0\\
         0&C_{\mathrm{r}}\\
       \end{array}\right]  x_{\mathrm{a}}(t_i)  +
       \left[\begin{array}{c}
         v(t_i)\\
         v_{\mathrm{r}}(t_i)\\
       \end{array}\right]
\end{equation}
%
\begin{equation}\label{R}
   \E {
   \left[\begin{array}{c} v(t_i)\\ v_{\mathrm{r}}(t_j)\\
\end{array}\right]
   \left[\begin{array}{cc} v^T(t_i)& {v_{\mathrm{r}}}^T(t_j)\\
\end{array}\right]
      }=
         \left[\begin{array}{cc}
             R&0\\
             0&R_{\mathrm{r}}\\
            \end{array}
      \right]\delta_{i,j}
\end{equation}
with  zero mean white noises $v(\cdot)$, $v_{\mathrm{r}}(\cdot)$,
where $\delta_{i,j}$ stands for Kroneker's delta function.

The tracker is designed to regulate the tracking error
\begin{equation}\label{e}
   e(t_i)\doteq C_{\mathrm{a}}
   \left[\begin{array}{c} x(t_i)\\ x_{\mathrm{r}}(t_i)\\ \end{array}\right]  \quad  \mbox{with}  \quad C_{\mathrm{a}}\doteq \left[\begin{array}{cc} C&\vdots -C_{\mathrm{r}}\\ \end{array}\right]
\end{equation}
to zero which is  to be accomplished by minimization of the mean
quadratic cost
\begin{equation}\label{J}
   \begin{array}{ll}
     J_{\mathrm{control}}\doteq\lim\limits_{t_0\to-\infty}  & \E {\frac{1}{2}  \sum\limits_{j=0}^{i}
            \left[\begin{array}{c} x_{\mathrm{a}}(t_j)\\ u(t_j)\\
\end{array}\right]^T
                     \left[\begin{array}{cc}
                     X_{\mathrm{a}}& S_{\mathrm{a}}\\
                     S_{\mathrm{a}}^T& U\\
                     \end{array}\right]
            \left[\begin{array}{c} x_{\mathrm{a}}(t_j)\\ u(t_j)\\
\end{array}\right]
      }\\
   \end{array}
\end{equation}
where $X_a\doteq C_{\mathrm{a}}^{T}YC_{\mathrm{a}}$ and $S_a\doteq
C_{\mathrm{a}}^{T}S$ with $S$ allowed to be nonzero, and some
constant $Y> 0$ and $U> 0$, so that the composite matrix in
(\ref{J}) is positive semidefinite. Thus,  the tracker must cause
a controlled variable $y_{\mathrm{c}}(\cdot)=Cx(\cdot)$ to track
some target (\emph{reference}) variable
$y_{\mathrm{r}}(\cdot)=C_{\mathrm{r}}x_{\mathrm{r}}(\cdot)$.

Restricting our  attention to time-invariant system, we can seek
the steady state (constant) gains $G_{\mathrm{c}1}$ and
$G_{\mathrm{c}2}$ in the full-state feedback law of
\begin{equation}\label{U}
  u(t_i)=-\left[\begin{array}{cc} G_{\mathrm{c}1} &\vdots \hphantom{-} G_{\mathrm{c}2}\\ \end{array}\right]
              \left[\begin{array}{c} x(t_i)\\ x_{\mathrm{r}}(t_i)\\
\end{array}\right]=-G_{\mathrm{c}1} x(t_i)-G_{\mathrm{c}2} x_{\mathrm{r}}(t_i) \ .
\end{equation}
This law \cite{Mb} could be used if we had perfect knowledge of
both $x(t_i)$ and $x_{\mathrm{r}}(t_i)$. As it is not the case,
one employs the Kalman filter synthesis to generate estimates
$\hat x(t_i^+)$ and $\hat x_{\mathrm{r}}(t_i^+)$ and substitute
them in (\ref{U}) for $x(t_i)$ and $x_{\mathrm{r}}(t_i)$,
correspondingly. To obtain the filter decoupled into two totally
independent filters, we require $x(t_0)$ and $x_{\mathrm{r}}(t_0)$
to be uncorrelated; the same requirement is adopted with respect
to $w(\cdot)$ and $w_{\mathrm{r}}(\cdot)$ in (\ref{AS1}) and
$v(\cdot)$ and $v_{\mathrm{r}}(\cdot)$ in (\ref{AS2}). Further, to
guarantee existence  of the steady state filters, we assume that
$\Phi_{\mathrm{r}}$ is \emph{asymptotically stable} (has all of
its eigenvalues strictly within the unit circle on the complex
plane), the pair $(\Phi, Q^{1/2})$ is \emph{stabilizable}, the
pair $(\Phi, H)$ is \emph{observable}, and the pair $(\Phi, \Psi)$
is \emph{controllable}. Also, to exploit the Auxiliary Performance
Index (API) approach for the optimal filters identification
\cite{API-filt, API-cont}, we assume that both subsystems in
(\ref{AS1}), (\ref{AS2}) are given in the \emph{standard
observable form}, SOM.

In these assumptions,  the Steady-State Kalman Filter (SSKF) is
presented by
\begin{eqnarray}
\hat x_{\mathrm{a}}(t_{i+1}^-) & = & \Phi_{\mathrm{a}} \hat x_{\mathrm{a}}(t_i^+)+ \Psi_{\mathrm{a}} u(t_i), \label{Kal1}\\
\hat x_{\mathrm{a}}(t_i^+) & = & \hat x_{\mathrm{a}}(t_i^-)+ K_{\mathrm{a}}[z_{\mathrm{a}}(t_i) - H_{\mathrm{a}}\hat x_{\mathrm{a}}(t_i^-)] \label{Kal2}
\end{eqnarray}
with $\Phi_{\mathrm{a}}$ and $\Psi_{\mathrm{a}}$ readily seen from
(\ref{AS1}), $H_{\mathrm{a}}$ from (\ref{AS2}) and
$K_{\mathrm{a}}$ computed through the  forward Riccati recursion
\cite{Mb} for use $\hat x_{\mathrm{a}}(t_i^+)=[\hat x(t_i^+)^T \
\vdots \ \hat x_{\mathrm{r}}(t_i^+)^T]^T$  instead of
$x_{\mathrm{a}}(t_i)$ in (\ref{U}).

The problem we consider is  caused by the fact of parameter
uncertainty inherent to (\ref{AS1}), (\ref{Q}), (\ref{AS2}) and
(\ref{R}). In this work we concentrate on one level of
uncertainty:
\begin{description}
  \item[Case 1:]
    Four matrices in the  system description, namely $Q$, $R$, $Q_{\mathrm{r}}$ and $R_{\mathrm{r}}$ in (\ref{Q}) and (\ref{R}),
     allow for  the dependence on an uncertainty vector
$\theta\in\Theta$ \footnote{$^{)}$ $\Theta$ is a compact set where the SSKF (\ref{Kal1}), (\ref{Kal2}) exists.}$^{)}$.
    Each particular value of $\theta$ specifies a \emph{mode}.
    \end{description}
Estimates of the Kalman gains $K$ and $K_{\mathrm{r}}$ in $K_{\mathrm{a}}=\diag{K}{K_{\mathrm{r}}}$ will be denoted correspondingly as  $\bar K$ and $\bar K_{\mathrm{r}}$ in $\bar K_{\mathrm{a}}=\diag{\bar K}{\bar K_{\mathrm{r}}}$ within a suboptimal filter (SOF)
\begin{eqnarray}
\bar x_{\mathrm{a}}(t_{i+1}^-) & = & \Phi_{\mathrm{a}} \bar x_{\mathrm{a}}(t_i^+)+ \Psi_{\mathrm{a}} u(t_i), \label{SOF1}\\
\bar x_{\mathrm{a}}(t_i^+) & = & \bar x_{\mathrm{a}}(t_i^-)+ \bar K_{\mathrm{a}}[z_{\mathrm{a}}(t_i) - H_{\mathrm{a}}\bar x_{\mathrm{a}}(t_i^-)] \label{SOF2}
\end{eqnarray}
and as $\tilde K$ and $\tilde K_{\mathrm{r}}$ in $\tilde K_{\mathrm{a}}=\diag{\tilde K}{\tilde K_{\mathrm{r}}}$ within an adaptive filter (AF)
\begin{eqnarray}
\tilde x_{\mathrm{a}}(t_{i+1}^-) & = & \Phi_{\mathrm{a}} \tilde x_{\mathrm{a}}(t_i^+)+ \Psi_{\mathrm{a}} u(t_i), \label{AF1}\\
\tilde x_{\mathrm{a}}(t_i^+) & = & \tilde x_{\mathrm{a}}(t_i^-)+ \tilde K_{\mathrm{a}}[z_{\mathrm{a}}(t_i) - H_{\mathrm{a}}\tilde x_{\mathrm{a}}(t_i^-)] \label{AF2}
\end{eqnarray}

All the AF defined  by (\ref{AF1}), (\ref{AF2}) vary in the
algorithm of adaptation (selection) of $\tilde K_{\mathrm{a}}$.
Two algorithms for this are compared: conventional  numeric (NA)
and genetic (GA).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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\newslide

\section{GENERAL FRAMEWORK FOR IDENTIFICATION}\label{framework}

Conventional  identification methods such as stochastic
approximation (or Robbins-Monroe  procedure), least-squares method
and many others are widely presented in the literature
\cite{Caines}. For comparison study, we select one of them known
as Auxiliary Performance Index (API) method developed for adaptive
filtering \cite{API-filt} and extended for control problems
\cite{API-cont}.

To tailor this  paper with \cite{API-filt, API-cont}, re-denote
the variables of (\ref{AF1}), (\ref{AF2}) as follows
\begin{equation}\label{AF3}
g(t_{i+1|i}) \doteq  \tilde x_{\mathrm{a}}(t_{i+1}^-)\ , \quad g(t_{i|i}) \doteq x_{\mathrm{a}}(t_i^+)\ , \quad \eta(t_{i|i}) \doteq z_{\mathrm{a}}(t_i) - H_{\mathrm{a}}\tilde x_{\mathrm{a}}(t_i^-)
\end{equation}
and denote the design  (adjustable) AF parameter as the column
vector
\begin{eqnarray}
\hat \theta\doteq \col {\ \underbrace{\hat
\theta_{11},\dots,\hat \theta_{1m};\cdots; \hat
\theta_{n1},\dots,\hat \theta_{nm}}_{{\mathrm{all} \ p \ \mathrm{entries \ of}} \
\tilde K_{\mathrm{a}} \ \mathrm{ordered \ by \ rows}}\ } \in \Rset^{p}\ , \quad  \hat \theta_{ij}\doteq [\tilde K_{\mathrm{a}}]_{ij}    \ , \quad
p=mn\ . \label{theta}
\end{eqnarray}

As it is known, API method boils down to MPE$\!\!$
\footnote{Minimum Prediction Error}$^{)}$ method except one
feature: it exploits the idea of equivalent replacing the
\emph{Original Performance Index} being Direct but Inaccessible
(DbI) one, by an \emph{Auxiliary Performance Index} being
Accessible albeit Indirect (AaI). The main requirement for  this
replacement is that OPI and API be \emph{equimodal}, that is, they
have one and the same minimizing argument. Thus, based on API
approach, we do the following:
\begin{enumerate}
\item Construct the  Sensitivity Model (SM) by partial differentiating
equations (\ref{AF1})--(\ref{AF2}) to obtain the recursive equations
for the sensitivity functions
\begin{eqnarray*}
\mu_{jl}(t_{i+1|i})&\doteq &\frac{\partial }{\partial \hat \theta_{jl}}g(t_{i+1|i})\ , \quad j=1,\ldots,n\ ; \; \; \;
\; l=1,\ldots,m
\end{eqnarray*}
\item For $j=1,\ldots,n\ ; \ l=1,\ldots,m$\ , compute
\begin{eqnarray}
\eta_{i-s+1|i-s}^{i|i-1}&=&[\eta^T(t_{i-s+1|i-s})\enspace|\enspace\cdots\enspace|\enspace\eta^T(t_{i|i-1})]^T \nonumber\\
\varepsilon(t_i, \hat \theta)&=&\mathscr{P}(\tilde K_{\mathrm{a}})\eta_{i-s+1|i-s}^{i|i-1}\  \label{varepsilon}\\
\frac{\partial \eta(t_{i|i-1})}{\partial \hat \theta_{jl}}&=&-H_{\mathrm{a}}\mu_{jl}(t_{i|i-1}) \nonumber\\
\frac{\partial \varepsilon(t_i, \hat \theta)}{\partial \hat \theta_{jl}}&=&
\left[\frac{\partial }{\partial \hat \theta_{jl}}
\mathscr{P}(\tilde K_{\mathrm{a}})\right]\eta_{i-s+1|i-s}^{i|i-1}+ \mathscr{P}(\tilde K_{\mathrm{a}})\frac{\partial
}{\partial \hat \theta_{jl}}\eta_{i-s+1|i-s}^{i|i-1} \label{eq:26}
\end{eqnarray}
where $\mathscr{P}(\cdot)$ is a  procedure defined in
\cite{API-filt} as follows.
\begin{defn}
\label{d:1} Let $s,p_1,\dots,p_m\in \Nset$   satisfy
\[s\doteq\max(p_1,p_2,\dots,p_m)\leq p_1+p_2+\dots+p_m=n\] and $A$ be an $ms\times
k$ matrix, $k\in \Nset$, composed of $s$ submatrices with $m$ rows
and $k$ columns each. Denote the $j$-th row of the $i$-th
submatrix by $a_i^j$ and rearrange all these $ms$ rows in order to
form a new $s\times km$ matrix
\[A_T\doteq
\left[
\begin{array}{ccc}
a^1_1 & \dots & a^m_1\\
\vdots & \ddots
& \vdots\\
a^1_s & \dots & a^m_s
\end{array}
\right]\,.
\]

Then  $\mathscr{S}(A)$, which is the $n\times k$ matrix, is called
the $\mathscr{S}$-transform of the matrix $A$, provided its $n$
rows are obtained by taking the elements $a_i^j$ from $A_T$ and
placing them into $\mathscr{S}(A)$ as rows in the following order:
$a_1^1,a_2^1,\ldots,a_{p_1}^1$, $a_1^2,a_2^2,\ldots,a^2_{p_2}$,
$\ldots$, $a_1^m,a_2^m,\ldots,a^m_{p_m}$.
\end{defn}
\begin{defn}
\label{d:2}  Let $(\Phi_{\star},H_{\star})$ be matrices in the SOM
such that  $\Phi_{\star}$ is of a block companion form whose
non-trivial (non-zero or non-unit) elements lie in the rows with
numbers $p_1$, $p_1+p_2$, \dots, $p_1+p_2+\dots+p_m=n$, where
$p_j$ are the given partial observability indices of
$(\Phi_{\star},H_{\star})$, and $H_{\star}$ is the $m\times n$
matrix whose $ij$-th element is $1$ if $j=p_1+p_2+\dots+p_{i-1}+1$
and \ $0$ otherwise.

Then  $\mathscr{P}(D)$, which is the $(n\times sm)$-matrix, is
called the $\mathscr{P}$-transform of an $(n\times m)$-matrix $D$,
provided
\[
\mathscr{P}(D)\doteq\mathscr{S}(S_{\theta}(H_{\star},\Phi_{\star},
\Phi_{\star} D))
\]
with the $(sm\times sm)$-matrix
\begin{equation}\label{eq:45}
\begin{array}{c}
S_{\theta}(H_{\star},\Phi_{\star},G)\doteq\left[
\begin{array}{llcl}
I &0 &\cdots &0 \\
H_{\star}G &I &\cdots &0 \\
\vdots &\vdots &\ddots &\vdots \\
H_{\star}\Phi_{\star}^{s-2} G &H_{\star}\Phi_{\star}^{s-3} G
&\cdots & I \end{array} \right]\ , \quad G=\Phi_{\star} D
\end{array}
\end{equation}

\end{defn}

\item Arrange the sensitivity functions  defined by (\ref{eq:26}) into
the $n\times p$ sensitivity matrix
\begin{eqnarray}
S(t_i)&\doteq& \left [  \ {
\frac{\partial \varepsilon(t_i, \hat \theta)}{\partial \hat \theta_{11}},\dots,
\frac{\partial \varepsilon(t_i, \hat \theta)}{\partial \hat \theta_{1m}},\cdots,
\frac{\partial \varepsilon(t_i, \hat \theta)}{\partial \hat \theta_{n1}},\dots,
\frac{\partial \varepsilon(t_i, \hat \theta)}{\partial \hat \theta_{nm}}}\ \right ]
\end{eqnarray}
\item Construct the Gradient  Model with an exponential smoothing factor $\beta$
\begin{eqnarray}
\mathcal{G} (t_i)&=&S^T(t_i)\varepsilon(t_i, \hat \theta) \label{Grad}\\
\hat \mathcal{G} (t_i)&=&\beta \hat \mathcal{G}
(t_{i-1})+(1-\beta)\mathcal{G} (t_i) \nonumber
\end{eqnarray}
\item Check the Stability  Condition (SC): $\rho [(I-\tilde K_{\mathrm{a}}H_{\mathrm{a}})\Phi_{\mathrm{a}}] < 1$  for AF and SM where
$\rho [\cdot]$ stands for the spectral radius of matrix
$[\cdot]$. In doing so, the designer should have the
characteristic polynomial for matrix $(I-\tilde K_{\mathrm{a}}H_{\mathrm{a}})\Phi_{\mathrm{a}}$, that is written as
\begin{eqnarray*}
q(\lambda)\doteq b_0{\lambda}^n+b_1{\lambda}^{n-1}+\cdots+b_{n-1}{\lambda}^{1}+b_n,
\quad b_0>0
\end{eqnarray*}
with  $b_0,\dots,b_n$ expressed as some functions $b_k=b_k(\hat
\theta)$ of the design parameter  (\ref{theta}), and then  use
Jury's criterion \cite[Sect.~3.3]{Antoniou}. If SC is met, we
write $SC(\hat\theta)= \mathrm{true}$, and
$SC(\hat\theta)=\mathrm{false}$ otherwise, and we call this
procedure by \texttt{checkSC}$(\hat\theta)$.
\item Use an \textbf{Adaptation Procedure} (AP) to update the parameter
(\ref{theta}) within an effective operating range of $t_i$ from
$t_{\mathrm{start}}$ to $t_{\mathrm{stop}}$. For  AP, one can use
different optimum seeking methods able to minimize a performance
index taken to characterize the quality of AF
(\ref{AF1})--(\ref{theta}). For such index, we use in this paper
the API
\begin{equation}\label{API}
J(\tilde \theta)\doteq \half \E {\varepsilon(t_i, \hat \theta)^T \varepsilon(t_i, \hat \theta)}
\end{equation}
with the generalized residual (\ref{varepsilon}). Using the
Simplified Least Squares (SLS) to minimize (\ref{API}), the AP
looks as follows:\\
\begin{myframe}{12cm}
\begin{tabbing}
\quad \=\quad \=\quad \=\quad \kill
\keyw{if} $(t_{\mathrm{start}}\le t_i < t_{\mathrm{stop}} )$  \keyw{then} \+ \\
\keyw{begin} \keyw{if} $(t_{\mathrm{start}}=t_i)$ \keyw{then} \+ \\
\keyw{begin} $k:=0$; $\Lambda=I$ \keyw{end} \keyw{else} $k:=k+1$; \\
\keyw{if} $(k>0 \ \& \ k=0 \ \mathrm{mod} \ s)$ \keyw{then} \\
\keyw{begin} $\tau:=[k-1/s]$; \keyw{for} $j=1$ \keyw{to} $p$ \keyw{do} \+ \\
$\lambda^{(j)}_{\tau+1}:=\lambda^{(j)}_{\tau}+ \left\Vert
\frac{\partial r(t_{i}, \hat{\theta}(\tau))}{\partial
\hat{\theta}_{j}}\right\Vert^{2}$; $
\Lambda_{\tau+1}=\mathrm{diag}\ \{ \lambda^{(j)}_{\tau+1} \}$;
\hphantom{aaaaaaa} $\setminus\star$ SLS $\star \setminus$ \\
$\pi:=\hat{\theta}(\tau)- \Lambda^{-1}_{\tau+1}\hat\mathcal{G}(t_i)$;
\hphantom{......aaaaaaaaaaaaaaaaaaaaaaaaaa} $\setminus\star$ Num$\star \setminus$ \\
\texttt{checkSC}($\pi$); \\
\keyw{if} $(SC(\pi))$ \keyw{then} $\hat\theta(\tau+1):=\pi$ \- \\
\keyw{end} \- \\
\keyw{end}
\end{tabbing}
\end{myframe}
\begin{remarka}
Changing the line marked  by $\setminus\star$ \emph{SLS}$\star
\setminus$, one can go to another minimum seeking method. For
instance, by using
\[
\lambda^{(j)}_{\tau+1}:=\lambda^{(j)}_{\tau}+1
\]
one  switches to the Simple Stochastic Approximation
$\setminus\star$ \emph{SSA}$\star \setminus$. Such cases, together
with the line marked by $\setminus\star$ \emph{Num}$\star
\setminus$, form a numeric method   to optimize $\hat \theta$.
\end{remarka}
\begin{remarka}
Changing only the two  marked  lines, one can go to a genetic
method.
\end{remarka}
\begin{remarka}
Due to $s-1$ one sample  delays in the framework of
Fig.~\ref{fig1}, the time pace for AP has to be set $s$  times
lower than for the system. To do this, the AP timer $\tau$ is
introduced in the above AP through the truncation function
$[\cdot]$.
\end{remarka}
\end{enumerate}

\begin{figure}
\input picture1.txt
\caption{The general framework for using the API. $\Delta$ denotes
a one  sample memory (unit delay). \label{fig1}}
\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newslide

\section{THE SIMULATED GENETIC ALGORITHMS \cite{GA1, GA2, SysGA, Neubauer}}\label{genetic}

A  genetic algorithm (GA)  differs from other search techniques by
the use of concepts taken from natural genetics and evolution
theory.

The basic element processed  by a GA is the string formed by
concatenating a substring, each of which is a binary code of  a
parameter of search space \cite{SysGA}. Thus each string
represents a possible solution to the problem. The GA works with a
set of strings, called the population. This population then
evolves from generation to generation through the application of
genetic operators. A GA in its simplest form uses three operators:
Replication, Crossover and Mutation.

\s{Replication.} Replication is based on the principle of survival
of the fittest. A fitness, $F(i)$, is assigned to each individual
in the population where high numbers mean good fit. The fitness
function can be any nonlinear, non-differentiable, discontinuous,
positive function, because the algorithm only needs a fitness
assigned to each string. We  will first consider the construction
of the intermediate population from the current population. In the
first generation the current population is also the initial
population. After calculating $F(i)/\bar{F}$ (where $\bar{F}$ is
the average value of the fitness) for all the strings in the
current population, selection is carried out. In the canonical
genetic algorithm, the probability that strings in the current
population are copied (i.~e., replicated) and placed in the
intermediate generation is proportional to their fitness.

There are a number of ways to do selection. We might view the
population as mapping onto a roulette wheel, where each individual
is  represented by a space that proportionally corresponds to its
fitness. By repeatedly spinning the roulette wheel, individuals
are chosen using ``stochastic sampling with replacement'' to fill
the intermediate population.

A selection process that will more closely match the  expected
fitness values is a ``remainder stochastic sampling''. For each
string $i$ where $F(i)/\bar{F}$ is greater than $1.0$, the integer
portion of this number indicates how many copies of that string
are directly placed in the intermediate population. All strings
(including those with $F(i)/\bar{F}$ less than 1.0) then place
additional copies in the intermediate population with a
probability corresponding to the fractional portion of
$F(i)/\bar{F}$. For example, a string with $F(i)/\bar{F}= 1.36$
places $1$ copy in the intermediate population, and then receives
a $0.36$ chance of placing a second copy. A string with a fitness
of $F(i)/\bar{F}= 0.54$ has a $0.54$ chance of placing one string
in the intermediate population.

\s{Crossover:} Replication directs the search toward  the best
existing individuals but does not create any new individuals. In
nature, an offspring is rarely an exact clone of a parent, it
usually has two parents and inherits genes from both. The main
operator to work on the parents is crossover, which is applied
with a certain probability, called crossover rate ($p_c$).
Crossover takes two individuals from intermediate population, and
cuts their chromosome strings at some randomly chosen position, to
produce two ``head'' segments, and two ``tail'' segments. The tail
segments are then swapped over to produce two new full length
chromosomes:
$$
\left\{
    \begin{array}{cccccccc}
      0&0&0&\vdots &0&0&0&0\\
      1&1&1&\vdots &1&1&1&1
    \end{array}
  \times\Rightarrow
    \begin{array}{ccccccc}
      0&0&0&1&1&1&1\\
      1&1&1&0&0&0&0
    \end{array}
\right.
$$
This is known as single-point crossover.

A two-point crossover operator uses  two randomly chosen crossover
points. Strings exchange the segment that falls between these two
points:
$$
\left\{
    \begin{array}{ccccccccc}
      0&0&\vdots &0&0&0&\vdots &0&0\\
      1&1&\vdots &1&1&1&\vdots &1&1
    \end{array}
  \times\Rightarrow
    \begin{array}{ccccccc}
      0&0&1&1&1&0&0\\
      1&1&0&0&0&1&1
    \end{array}
\right.
$$
\s{Mutation:} Mutation is applied to  each child individually
after crossover. It randomly alters each gene with a small
probability (typically 0.001). Following example shows the third
gene of the chromosome being mutated.
$$
    \begin{array}{ccccccc}
      0&0&\vdots &0&\vdots &0&0\\
      \end{array}
  \Rightarrow
    \begin{array}{ccccccc}
      0&0&\vdots &1&\vdots&0&0\\
    \end{array}
$$
%\vspace{3pt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newslide

\section{SOME COMPUTATIONAL EXPERIMENT RESULTS}\label{simul}


A sample of GA behavior within the Adaptation Procedure of
Section~\ref{framework} is shown on Figure~\ref{Case1:GA}. For
simulation, the tracker described in~\cite{Mb} has been chosen.
In this example, we have the following constant quantities:
\[
\begin{array}{lllllll}
\Phi=[0.82],&\Phi_{\mathrm{r}}=[0.61],&\Psi=[0.18],&
C=[1],&C_{\mathrm{r}}=[1],&Y=[1],&U=[1]\\
\end{array}
\]
and $S_{\mathrm{a}}^T=[0 \ \vdots \ 0]$ for the LQG controller (\ref{U}) synthesis. This gives
\[
G_{\mathrm{c}1}=[0.36]\ , \quad G_{\mathrm{c}2}=[-0.19]\ .
\]
For the Kalman filter synthesis we have the following data:
\[
\begin{array}{lllllll}
H=[1],&Q=[0.084\sigma^{2}],&
Q_{\mathrm{r}}=[0.63\sigma_{\mathrm{r}}^{2}],&\sigma^{2}=\mbox{\sl var},
&\sigma_{\mathrm{r}}^{2}=\mbox{\sl var},&R=\mbox{\sl var},
&R_{\mathrm{r}}=\mbox{\sl var}\\
\end{array}
\]
where ``{\sl var}'' stands for the variable quantities. While
simulating, we specify the two phases named ``{\sl before}'' and
``{\sl after}'' (before and after  a parameter switch). The phase
``{\sl before}'' takes 300 time instants and phase ``{\sl after}''
takes 10000 time instants.

In these experiments, simulation parameters are as follows:
\begin{enumerate}
\item Number of Iterations, (NI) \dotfill 10000
\item Number of Experimental Samples  (NES) to average results, \dotfill 100
\item Simulated numeric algorithms and their parameters:
   \begin{enumerate}
     \item Robbins-Monroe procedure, \dotfill SSA
     \item Simplified Least Squares, \dotfill SLS
     \item Exponential smoothing parameter $\beta$, \dotfill 0.5   \end{enumerate}
\item Simulated genetic algorithm:
   \begin{enumerate}
     \item Chromosome Length, (CL) \dotfill 7
     \item Power (size) of population, (PP) \dotfill 50
     \item Eliticity Factor \footnote{$^{)}$ Number of chromosomes
     passing from current population into future population without any operations (selection-crossover-mutation) as being the best ones.}$^{)}$, (EF) \dotfill 2
     \item Mutation Probability Rate,  (MPR) \dotfill 0.100
     \item Selection Mode, (SM) \dotfill remainder stochastic sampling
     \item Crossover Mode, (CM) \dotfill two-point
   \end{enumerate}
\end{enumerate}
In experiments,  we consider the following two cases.
\s{Case~1.} See Figures~\ref{Case1:GA} to \ref{Case1:!Par}.\\
$\hphantom{aaaaaaa}$``{\sl before}''
$\hphantom{aaaaaaaaaaaaaaaaaaaaaa}$``{\sl after}''
\[
\begin{array}{cccccc|cccccc}
\sigma^{2}&\sigma_{\mathrm{r}}^{2}&R&R_{\mathrm{r}}&K&K_{\mathrm{r}}&
\sigma^{2}&\sigma_{\mathrm{r}}^{2}&R&R_{\mathrm{r}}&K&K_{\mathrm{r}}\\
\hline\\
10.0&4.0&4.0&1.0&0.287&0.736&
10.0&4.0&0.1&10.0&0.900&0.258\\\hline
\end{array}
\]
\s{Case~2.} See Figures~\ref{Case2:GA} to \ref{Case2:Par}.\\
$\hphantom{aaaaaaa}$``{\sl before}''
$\hphantom{aaaaaaaaaaaaaaaaaaaaaaa}$``{\sl after}''
\[
\begin{array}{cccccc|cccccc}
\sigma^{2}&\sigma_{\mathrm{r}}^{2}&R&R_{\mathrm{r}}&K&K_{\mathrm{r}}&
\sigma^{2}&\sigma_{\mathrm{r}}^{2}&R&R_{\mathrm{r}}&K&K_{\mathrm{r}}\\
\hline\\
10.0&4.0&4.0&10.0&0.287&0.258&
10.0&4.0&0.1&1.0&0.900&0.736\\\hline
\end{array}
\]

The Integral Percent Error has been  determined (with averaging
over the NESs) by
\begin{eqnarray}\label{IPE:ab}
\mbox{(a)}  { \mathrm{IPE}(t_{i})\doteq \frac {{\Vert
\tilde K_{\mathrm{a}}(t_{i})-K_{\mathrm{a}} \Vert}} {{\Vert
K_{\mathrm{a}} \Vert}}100 } \%   \hphantom{.}  \mbox{or}
 & \mbox{(b)}    { \mathrm{IPE}(t_{i})\doteq
\frac {\displaystyle{\Vert \tilde K(t_{i})-K \Vert}}
{\displaystyle{\Vert K \Vert}}100 \% }
\end{eqnarray}

In the other portion of experiments, we modify the parameters: NI
to 5000, NES to  50, CL to 5, PP to 30, and MPR to 0.200.
Corresponding results for Case~1 are shown in
Figures~\ref{Case1:!GA} to \ref{Case1:!Par}  \footnote{$^{)}$
Figures~\ref{Case1:!Sq} to \ref{Case2:Par} have been shown in
Appendix.}$^{)}$. The IPE is shown in the figures for all  three
methods: SSA by red lines, SLS by blue lines and GA by green
lines.

As can be seen from these and many other experiments that were
conducted, in the majority of cases, the averaged GA behavior
provides the lower IPE levels. However it is important to note,
that the individual GA behavior is much more  fluctuating than the
individual numeric methods behavior.

The consistent difference between these types of methods, that was
to be expected, lies in the very mechanism of their behavior.
Numeric algorithms are sequential in operation while genetic
algorithms are parallel. The latter require  a set of individuals
forming the current population of adaptive filters, each
individual filter working through the mechanism of
Figure~\ref{fig1}. Large size of population causes the computation
load to rise so that real time data processing may occur to be
difficult to sustain.

\begin{center}
\begin{figure}[h]
\centering
\includegraphics[scale=0.72]{!GA.eps}
\caption{Behavior of the modified GA parameters for Case~1.}
\label{Case1:!GA}
\end{figure}
\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newslide

\section{...}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\newslide\subsection{The Sensor equation}


\newslide
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\bibitem{Semoushin et al. 2002b}
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\bibitem{Maybeck 1982}
Maybeck, P.S. 1982. {\em Stochastic models, estimation, and control} {\bf 3}.
Paris San Diego San Francisco Sao Paulo Sydney Tokyo Toronto: Academic Press.

\bibitem{Caines88}
Caines, P.:  Linear Stochastic Systems.  John Willey \& Sons, New
York Chichester Brisbane Toronto Singapore (1988)

\bibitem{Semoushin 1974}
Semoushin, I.V. 1974. On One Approach to Fault Detection in Linear Dynamical 
Systems with Possible Disturbances. {\em Automatics and Computing Techniques, 
The Latvian SSR Academy of Sciences} {\bf 4}: 24--30. 

\bibitem{Ponyrko et al. 1973}
Ponyrko, S.A. \&  Semoushin, I.V. 1973. On the Choice of Start-Stop Algorithm 
while Minimizing the Square Mean Performance Index. {\em Autometria, 
The USSR Academy of Sciences} {\bf 2}: 68--74. [{\em in Russian}]

\bibitem{Soderstrom81}
Soderstrom, T.: Identification  of stochastic linear systems in
presence of input noise. Automatica {\bfseries 17} (1981) 713--725

\bibitem{Ansay99}
Ansay, P., Gevers, M., Wertz, V.:  Closed-loop or open-loop models
in identification for control? In: Proc. of the European Control
Conference (1999) CD-ROM F544

\bibitem{Semoushin85}
Semoushin, I.V.:  Adaptive  Identification and Fault Detection
Methods in Random Signal Processing.  Saratov University
Publishers, Saratov  (1985) [{\em in Russian}]

\bibitem{Semoushin94}
Semoushin, I.V.: Adaptive control for stochastic linear plants
under conditions of uncertainty.  Nonlinear Dynamical Systems --
Qualitative Analysis and Control: Collected Papers, The Institute
for System Analysis of the Russian Academy of Sciences {\bfseries
2}  (1994) 104--110

\bibitem{Semoushin00}
Semoushin, I.V., Tsyganova, J.V.: Indirect error control for
adaptive filtering.  In: Neittaanmaki, P., Tiihonen, T.,
Tarvainen, P. (eds.): Proc. of the 3rd  European Conference on
Numerical Mathematics and Advanced Applications. World Scientific,
Singapore New Jersey London Hong Kong (2000) 333--340

\bibitem{Semoushin02-2}
Semoushin, I., Gorokhov, O.:  Computational processes in iterative
control design. In:  Sloot, P.M.A., Kenneth Tan, C.J., Dongarra,
J.J., Hoekstra, A.G. (eds.): Computational Science -- ICCS~2002.
Lecture Notes in Computer Science, Vol. 2329. Springer-Verlag,
Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris
Tokyo (2002) 186--195

\bibitem{Semoushin94a}
Semoushin, I.V.: 
The design of active type systems for the adaptive control and its 
application to Inertial Navigation Systems. Nonlinear Dynamical Systems --
Qualitative Analysis and Control: Collected Papers, The Institute
for System Analysis of the Russian Academy of Sciences {\bfseries
2}  (1994) 110--115

\bibitem{Lanc-Rodman 1995}
Lancaster, P.,  Rodman, L.: Algebraic Riccati Equations. Clarendon Press,
Oxford (1995)

\bibitem{Antoniou 1979}
Antoniou, A.: Digital filters: analysis and design. MdGraw-Hill, Inc., New
York (1979)

\bibitem{Gorokhov 2002}
Gorokhov, O. Yu.: Investigation of the jointly performed operations of detection,
identification and adaptation in discrete stochastic systems.
Candidate of Science dissertation (Advisor I.V. Semoushin). Ulyanovsk
State University, Ulyanovsk (2002)

\bibitem{Kond2001}
Kondratiev, A.E., Fatyanova, O.A.:  Start-stop algorithms for
adaptive filtering. In: Boulyarski, S.V. (ed.): Young Researchers'
Collected Papers of the Ulyanovsk State University. UlSU
Publishers, Ulyanovsk (2001) 6--7

\bibitem{Semoushin et al. 2002b}
Semoushin, I.V. \& Polosenko,  V.P. 2002. Iterative Cooperative CAD for
Avionics Navigation Systems. In  R.~Goncalves, R.~Roy, A.~Steigler-Garcao
(eds),  {\em Advances in Concurrent Engineering; Proc. intern. conf., 
Cranfield, July 27--31 2002}: 753--761. Lisse, Abington, Exton (PA), Tokyo: Balkema. 

\end{thebibliography}
}
\end{slide}

\end{document}
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