\section{Problem statement}\label{problem}\vspace{1ex}%\noindent

Let a given system (\ref{x})--(\ref{y}) be parameterized by a
vector $\theta \in {\Rset}^{p}$ and available data $z=\left[y\atop
u\right]$ be a vector composed of two parts: the control input $u
\in {\Rset}^q$ and the measurement output $y\in {\Rset}^m$. The
control system is described for $i\in \Zset$ by the equations:
\begin{eqnarray}
x(t_{i+1})&=&\Phi_{\theta} x(t_i)+\Psi_{\theta} u(t_i)+ w(t_i)%,
\quad
%x\in \Rset^n
\label{x}
\\
y(t_i) &=&
H_{\theta} x(t_i) + v(t_i), \; x\in \Rset^n  \label{y}\\
\hat x_{0}(t_{i+1}^-)&=&{\Phi_{0}} \hat x_{0}(t_i^+)+ {\Psi_{0}}
u(t_i),\; \hat x_{0}\in \Rset^n
\label{hatx-}\\
\hat x_{0}(t_i^+) &=& \hat x_{0}(t_i^-)+ {K_{0}}\nu (t_i) \label{hatx+}\\
\nu (t_i)&=& y(t_i) - {H_{0}}\hat x_{0}(t_i^-)\label{nu} \\
u(t_i) &=& f_{\mathrm{R}}[\hat x_{0}(t_i^+)] = - {G_{0}^{\star}}
\hat x_{0}(t_i^+) \quad %u\in \Rset^q
\label{u}
\end{eqnarray}
with $\{w(\cdot)\}$, $\{v(\cdot)\}$ being  zero mean i.i.d.
sequences of covariance $Q_{\theta}$ and $R_{\theta}$
respectively. The state difference equation (\ref{x}) is
propagated forward from the initial condition $x(t_{-s})$ of the
finite expectation $\E{\|x(t_{-s})\|^2 }<\infty$. The initial
state is placed at some $t_{-s}\in \Rset^1$ where $s>0$ determines
what is termed settling time $T_{s}=t_{0}-t_{-s}$ needed to think
of all processes in (\ref{x})--(\ref{u}) as wide-sense stationary
for $i\ge 0$. As usual, equation (\ref{x}) represents a given
plant, equation (\ref{y}) a given sensor, and
(\ref{hatx-})--(\ref{u}) a feedback to be iteratively designed.
The initial feedback design is to be composed of Kalman Filter
($\mathrm{KF}_{0}$) with the Kalman gain $K_{0}$,
(\ref{hatx-})--(\ref{nu}), cascaded with Regulator
($\mathrm{R}_{0}$), (\ref{u}), described by a given function
$f_{\mathrm{R}}[\cdot]$ of the measurement updated estimate $\hat
x_{0}(t_i^+)$ or chosen according to the second equality in
(\ref{u}) with a given (or initially designed) regulator gain
matrix ${G_{0}^{\star}}$.

Matrices  defining the feedback (\ref{hatx-})--(\ref{u}) are
assumed to be initially given as $\Phi_{0}$, $\Psi_{0}$, $Q_{0}$,
$H_{0}$ and $R_{0}$ for \emph{a nominal mode of operation}, i.e.,
for \emph{a nominal value} $\theta_{0}\in\Theta$ of the
uncertainty parameter $\theta\in\Theta$ taken from a set $\Theta$
of possible modes.

However, parameter $\theta$ is  subject to abrupt changes. Each
single change occurs at an unknown time point $t_{\mathrm{c_{1}}}>
t_{0}$. This can be viewed as a switch of $\theta$ from
$\theta_{0}$ to some other unknown value $\theta_{1}\in\Theta$. To
keep the feedback close to optimal for the newly emerged mode
(defined by $\theta_{1}$), it is necessary to correspondingly
re-design the feedback. The optimal re-design would be Kalman
Filter number 1 ($\mathrm{KF}_{1}$) corresponding to
$\theta_{1}$, with Kalman gain $K_{1}$ and outputs $\hat
x_{1}(t_{i+1}^{-})$ and $\hat x_{1}(t_i^+)$, cascaded with
Regulator number 1 ($\mathrm{R}_{1}$) described by the same
function $f_{\mathrm{R}}[\cdot]$ of $\hat x_{1}(t_i^+)$ or chosen
similarly to the second equality in (\ref{u}) as $ -
{G_{1}^{\star}}$ with the correspondingly re-designed regulator
gain matrix ${G_{1}^{\star}}$. The feedback so re-designed (and
marked by subscript $_{1})$ would have to be substituted for the
above described initial feedback (marked by subscript $_{0}$).
This can be figuratively expressed by the following assignment
operators:
\begin{equation}\label{effector}
\mathrm{KF}_{0}:=\mathrm{KF}_{1},\quad \mathrm{R}_{0}:=\mathrm{R}_{1},\quad t_{0}:=t_{\mathrm{c_{1}}}
\end{equation}

The  challenge is that the above optimal re-design cannot be
performed because $\theta_{1}\in\Theta$ and $t_{\mathrm{c_{1}}}$
are unknown and so in reality can be only replaced by estimates
$\hat \theta_{1}\in\Theta$ and  $\hat t_{\mathrm{c_{1}}}$ produced
by  an applicable practical parameter estimation algorithm for
re-design (APPEAR). The problem is to construct the APPEAR such
that it could be applied iteratively when the next change occurs.
Undeniably, the next change is supposed to occur at a next time
point $t_{\mathrm{c_{1}}}\gg t_{0}$ such that the time interval
$t_{\mathrm{c_{1}}} - t_{0}$ would be enough for completing the
operation of the APPEAR.

In fact, one has two problems to be solved:

{1.} \textsl{Detection}. It  is necessary to detect the every
change point $t_{\mathrm{c}_{1}}$ for low cost and with a
reasonable delay and required reliability, i.e.,  a \emph{decision
generator}, $\mathscr{DG}$, is needed. It makes its decision at
the alarm time $t_{\mathrm{a}}$ in support of an abrupt change
thus assigning $\hat t_{\mathrm{c_{1}}}:=t_{\mathrm{a}}$.


{2.} \textsl{Selection}. After  the above decision has been made,
a system population, $\mathscr{SP}$, should be assumed to
approximately reflect the set $\Theta$ of possible (after-change)
modes of operation, so that the corresponding Kalman filter
population, $\mathscr{FP}$, could be built.  Then it will be
necessary to select from $\mathscr{FP}$ the only one filter,
$\mathscr{F}_{\kappa}$,  that could be recognized as the best
performer in the sense that it most closely matches the newly
occurred mode of system operation, according to Figure~\ref{fig2}.

\begin{figure*}
\input pict2.txt
\caption{The general framework for  change point detection and
filter selection. {\sl Legend:} $\mathscr{SP}$ stands  for system
population, each member of $\mathscr{SP}$ being in reality the
system (\ref{x})--(\ref{y}) with a particular value
$\theta_{k}\in\Theta$, $k\in\{0,1,\dots, K\}$, $K\in\Nset$. \,
$\mathscr{FP}$ stands for filter population of size $K$; each
member $\mathscr{F}_{k}$ of $\mathscr{FP}$ is the Kalman filter
$\mathrm{KF}_{\theta_{k}}$ being the optimal one with respect to
system (\ref{x})--(\ref{y}) with a particular value
$\theta_{k}\in\Theta$. \, For each single
$\mathrm{KF}_{\theta_{k}}$, decision $d_{k}=0$ is made in support
of  $\mathrm{KF}_{\theta_{k}}$ or decision $d_{k}=1$ is made in
refusal of  $\mathrm{KF}_{\theta_{k}}$. $\mathscr{FS}$  stands for
filter selection based on the all partial decisions $\{d_{k}\}$.
\, The stopping rule generated by the $\mathscr{DG}$ is defined by
the alarm time $t_{\mathrm{a}}=\min \Bigl\{ t_{k}: \; \bigl(
\exists \kappa : \ d_{\kappa}(t_{k})=0 \;\: \&
\mathop{\forall}\limits_{j\ne\kappa} d_{j}(t_{k})=1 \bigr)
\Bigr\}$.\,  $\mathscr{FE}$ stands for feedback effector to
implement the desired action (\ref{effector}) in  the system
feedback $\mathscr{SF}$. The only filter selected and uploaded to
$\mathscr{SF}$ is $\mathscr{F}_{\kappa}$. \label{fig2}}
\end{figure*}

These problems can be solved by one of  many existing methods
\cite{basseville:1993}, for example, by using the weighted
square innovation as a fitness function (FF)
\cite{semoushin:2004-2}. In this paper we assume that these
problems must be solved using another type of FF proposed in
\cite{semoushin:2003} in the context of the so called
``Statistical Orthogonality Principle''.

To guarantee existence of the steady-state Kalman gain
$K_{\theta}$ for each single mode of operation (uniquely
predetermined by a $\theta\in\Theta$), we assume the pair
$(\Phi_{\theta}, Q_{\theta}^{1/2})$ to be \emph{stabilizable}, the
pair $(\Phi_{\theta}, H_{\theta})$ \emph{observable}, and the pair
$(\Phi_{\theta}, \Psi_{\theta})$
\emph{controllable}~\cite{grewal:2001}. Then the matrix
$G_{\theta}^{\star}$ can be designed to be LQG optimal for each
single mode of operation, too, (i.e., optimal for each
$\theta\in\Theta$) \cite{maybeck:1982}.

\section{Decision generator for detection}\label{detection} \vspace{1ex}%\noindent

Given  the $k$-th mode of operation ($\theta=\theta_{k}$), for a
filter $\mathrm{KF}_{\hat\theta}$ with the design parameter $\hat
\theta\in\Rset^{p}$, to be optimal, i.e., equal to
$\mathrm{KF}_{\theta_{k}}$ minimizing a mean square criterion
\begin{equation}\label{API}
J(\hat \theta)\triangleq \half \E {\varepsilon(t_i, \hat \theta)^T \varepsilon(t_i, \hat \theta)}
\end{equation}
with a residual $\varepsilon(t_i, \hat \theta)$, it is
necessary  (and, in most practical cases, sufficient) that
all the expectations
\begin{equation}
\label{zeta} \zeta_{j}^{k}(t_{i}, \hat \theta)  =  \E {\left (\frac {\partial \varepsilon(t_{i}, \hat \theta)}{\partial {\hat \theta_j}}, \varepsilon(t_i, \hat \theta)
\right)\, \biggm| \, {\theta=\theta}_{k}}
\end{equation}
of  the dot product  $(\cdot \, , \cdot)$ lie in the immediate
vicinity of zero ($\forall k\in\{0,1,\dots, K\}, \,
j\in\{1,2,\dots, p\}$). Consequently, it is possible to base the
$\mathscr{DG}$ (Fig.~\ref{fig2}) on the following partial
decisions:
\begin{equation}
\label{d} d_{k}=\left\{
\begin{array}{ll}
0& \textrm{if  } \sum_{j=1}^{p}
\left|  0.5- \frac{n_{j}^{k}}{M}
\right|< \gamma_{0}\\
1& \textrm{otherwise}
\end{array}
\right.
\end{equation}
with  a threshold $\gamma_{0}$ where $n_{j}^{k}$ is the number of
negative outliers of the process $\hat \zeta_{j}^{k}(t_{i})$, an
estimator of $\zeta_{j}^{k}(t_{i})$,  in the current sample of
size $M$.

Thus,  the  $\mathscr{DG}$ implementation boils down to
construction of a workable residual $\varepsilon(t_i, {\hat
\theta})$, -- ``workable'' means available and such that the above
necessary and sufficient property of the performance index
(\ref{API}) concerning the process (\ref{zeta}) holds. The useful
solution of this problem known as ``\textsl{Auxiliary Performance
Index}'' (API) approach can be seen in \cite{semoushin:1985,
semoushin:1994} or in \cite{semoushin:2000}. Using this
approach, we construct the desired ``signal functions''
(\ref{zeta}) in the form of their exponentially smoothed
estimators
\begin{equation}
\label{hatzeta} {\hat \zeta_{j}^{k}(t_{i}, \hat \theta)}  =
\alpha {\hat \zeta_{j}^{k}(t_{i-1}, \hat \theta)} +
\beta \left (\frac {\partial \varepsilon(t_{i}, \hat \theta)}{\partial {\hat \theta_j}}, \varepsilon(t_i, \hat \theta)
\right)
\end{equation}
with the smoothing factor $\alpha$, $\alpha+\beta=1$.

\textsl{Remark 1}.\, In the avove, $\hat \theta$ stands for any particular value $\theta_{k}$ in the $\mathscr{FP}$.

\textsl{Remark 2}.\, The  same  signal functions (\ref{hatzeta}) we
use in the following conventional numerical methods chosen to
compare with them the GAs as a concurrent engineering tool
for the control systems iterated design optimization: %\\
(i) Simple Stochastic Approximation (SSA), %\\
(ii) Simplified Least Squares (SSA) \cite{caines:1988}.

%**********************************************************
