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\begin{document}

\title{Jointly Performed Computational Tasks in the Multi-Mode System Identification
\thanks{This work was supported in part by the Russian Ministry of Education
(grant No.~T02-03.2-3427).}}

\author{Innokenti Semoushin}

\institute{Ulyanovsk State University,
42 Leo~Tolstoy Str., 432970 Ulyanovsk, Russia\\
\email{SemoushinIV@ulsu.ru $\quad$
http://staff.ulsu.ru/semoushin/} }

\maketitle

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

\begin{abstract}
In this paper, we propose a new closed-loop multi-mode system identification
setup  that allows the jointly performed
change point detection, fault diagnosis, supervised training of a bank of
signature generators, unsupervised identification of the optimal filter,
and monitored system accommodation on the basis of incomplete noisy data.
We validate the setup by simulating numerical examples.
\end{abstract}

\thispagestyle{empty}

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

\section{Introduction}
\label{section:intro} In the literature on  modern control
systems, more emphasis is placed on detection \cite{BassNik93} and
identification \cite{Landau01} as separate problems. At the same
time, reality provides evidence that in many cases fault
detection-diagnosis must be performed together with system
identification and accommodation. In adaptive multi-mode or hybrid
stochastic systems with an uncertain mode switching mechanism,
identification is to be launched repeatedly every time when a mode
switch is detected. In the equivalent manner, identification must
be stopped when the identification algorithm provides the highest 
(or tolerable) pitch
of agreement with the observed data. Thus, FDD-SIA is a composite
challenging problem needing not only a theoretical but also
computational research.

In this work, we propose a unified
approach to perform FDD-SIA
computational tasks simultaneously in a single system. 
The outline
of the paper is as follows.  Section~\ref{section:System}
describes the monitored system (MS).  In Sect.~\ref{section:Stat}
we formulate the Statistical Orthogonality Principle ({SOP}) and
refer to the Accessible Indirect Performance Index ({AIPI}). The
{SOP}-based FDD is described in Sect.~\ref{section:FDD}.
Section~\ref{section:Optin} shows one of identification algorithms
(IA) intended to identify the optimal Steady-State Kalman Filter
({SSKF}) as a substitute for the Feedback Filter (FF). The {MS}
accommodation is briefly described in Sect.~\ref{section:MSA}.
Some experimental results are shown in Sect.~\ref{section:Valid}.
Finally, Sect.~\ref{section:Conclus} concludes the paper.

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

\section{Monitored System}
\label{section:System} For the monitored system we consider an
errors-in-variables (EIV) control system with the observed data
$z(t_i)=\left[y(t_i)\atop u(t_{i-1})\right]$, $i=1,2,\dots$,
composed of the control input $u \in{\bbbr}^q$ and the measurement
output $y \in {\bbbr}^m$. The system is parameterized by an
uncertainty vector $\theta \in{\bbbr}^l$ and consists of a plant
(P),  a sensor (S) and a stabilizing feedback controller(FC)
(Fig.\ref{fig1}($a$)).
\begin{figure}
\input picture1.txt
\caption{$(a)$ Monitored System (MS), and $(b)$ Statistical
orthogonality. Legend: P~--~plant, S~--~sensor, FC~--~feedback
controller, RG~--~residual generator, SM~--~sensitivity model,
SG~--~signature generator} \label{fig1}
\end{figure}
The P state equation is
\begin{eqnarray}
\label{eq:1}
x(t_{i+1}) & = & \Phi_{\theta} x(t_i)+\Psi_{\theta} u(t_i)+ w(t_i), \quad i=0,1,\dots %\\
\end{eqnarray}
with  $x \in {\bbbr}^n$ and a noise  $w(t_i)$ where the random
initial $x(t_0)$ at some $t_0$ has a finite mean $\bar x_0$ and a
finite covariance $P_0\ge 0$ and $\{u(t_i)\}$ is wide-sense
stationary and so $\E{||u(t_i)||^2}<\infty$ for all $t_i$. The S
equation is
\begin{eqnarray}
\label{eq:2}
y(t_i) & = & H x(t_i) + v(t_i), \quad i=1,2,\dots %\\
\end{eqnarray}
with ${\rm rank}(H)<n$ and a noise   $v(t_i)$. Both $\{w(t_i)\}$
and $\{v(t_i)\}$  are i.i.d. zero mean mutually independent
wide-sense stationary  sequences whose covariances are
$\E{w(t_i)w(t_i)^T}=Q_{\theta}\ge 0$ and
$\E{v(t_i)v(t_i)^T}=R_{\theta}>0$ for all $t_i$. The FC, in
compliance with the well-known separation property, is formed by
a feedback filter (FF) cascaded with a feedback regulator (FR).
The FF equations are
\begin{equation}
\label{eq:1.10}
\begin{array}{rcl}
\tilde x(t_{i+1}^-) & = &{\bar \Phi}\tilde x(t_i^+)+ \Psi u(t_i), \quad
\tilde x(t_i^+)=\tilde x(t_i^-)+ \bar K[y(t_i) - H\tilde x(t_i^-)]
\end{array}
\end{equation}
with some initials $\tilde x(t_0^+)$,  $u(t_0)$.  Constant
transition matrix $\bar \Phi$ and constant gain $\bar K$ are
chosen in a sense arbitrarily or designed to be optimal in a
steady state (as $t_0 \to -\infty$) w.r.t. a {\it fixed}
(fault-free) system mode specified by a  nominal value
$\theta_{0}$ of $\; \theta$. The FR equation is
\begin{equation}
\label{eq:1.9} u(t_i)=f_{\mathrm{FR}}[\tilde x(t_i^+)] = - \bar
{G}_{\mathrm{FR}}\tilde x(t_i^+)
\end{equation}
with  a function $f_{\mathrm{FR}}[\cdot]$ which can be chosen to
satisfy the second equality in (\ref{eq:1.9}) with a constant gain
matrix $\bar {G}_{\mathrm{FR}}$. By (\ref{eq:1.10}),
(\ref{eq:1.9}), it is predetermined that $u(t_i)$
%%%%in the left case of (\ref{eq:1.9})
depends  on all available data
$z(t_1^i)=[z(t_{1}),\cdots,z(t_{i})]$, the latter is a notation
for a composite (stackable) column vector.
%%%%In the right case of (\ref{eq:1.9}), it depends on
%%%%$z_{1}^{i-1}=[z(t_{1}),\cdots,z(t_{i-1})]$.
Note that
$\left[w(t_i)\atop v(t_j)\right]$ is independent of $u(t_k)$
and $x(t_k)$ for all $t_i \ge t_k$,  $t_j > t_k$.

The  overall closed-loop control system (\ref{eq:1}),
(\ref{eq:2}), (\ref{eq:1.10}), (\ref{eq:1.9}) is assumed to be
asymptotically stable in all modes of operating referred to by the
subscript $\theta$, and so all the processes within the system are
wide-sense stationary at every $t_i$ as $t_0 \to -\infty$ ({\em
the main assumption}). The system is designed to operate with a
minimum expected control cost
\begin{eqnarray}
\label{equa:1.3}
J_{\mathrm{c}}  =  \lim\limits_{j\to-\infty}{\rm E}\{ x^T(t_{N+1})X_f x(t_{N+1})
+\sum\limits_{i=j}^N[x^{T}(t_i)X x(t_i)+u^T(t_i)Uu(t_i)]\}
\end{eqnarray}
for $N>0$, $X_f\ge 0$, $U > 0$, $X \ge 0$,
and in all modes holds the {\em main properties}:
\begin{description}
\item[$\star \hphantom{a}$]  $({\Phi_{\theta}}, Q_{\theta}^{1/2})$ is
stabilizable, $({\Phi_{\theta}}, X^{1/2})$ is detectable,
$({\Phi_{\theta}}, H)$ is completely observable, and
$({\Phi_{\theta}}, {\Psi}_{\theta})$ is completely controllable.
\end{description}
to guarantee the existence of the optimal steady-state parameters for
(\ref{eq:1.10}), (\ref{eq:1.9}).

The  system itself can operate in several ({\em finitely} or {\em
infinitely} many) modes. This brings the problem to the realm of
hybrid or multi-mode systems. The mode characterized by
$\theta_{0}$ can be treated as the main one. The other modes
characterized by some $\theta_{1}$, $\theta_{2}$, $\dots$, $\theta_{\cal M}$ -- for the case of {\em finitely} many modes, --
can be viewed as some alternative modes not obligatory (albeit
possibly) faulty. In this case we have to test the ${\cal M}$
pairs of hypotheses, in each pair between
\begin{eqnarray}
\label{eq:22} {\rm\tt H}_{0} =
\{\theta^{\dagger}=\theta_{0}\}\rm{\quad and\quad }{\rm\tt
H}_{\mu} = \{\theta^{\dagger}=\theta_{\mu}=\theta_{0}+ {\it \Upsilon}_{\mu}\}
\end{eqnarray}
where $\theta^{\dagger}$ is the {\it true} value of $\theta$,
$\mu=1,2,\ldots,{\cal M}$, and $\Upsilon_{\mu}$ is the $\mu$-th
change on $\theta$ (the $\mu$-th alternative mode of operation).
Allowing for arbitrary faults, which are modelled by any sizable
change $\Upsilon$ on $\theta$ not violating the above main
assumption, we come to the case of {\em infinitely} many modes. By
this, we generalize the problem as we have to test the continuum
set of pairs of hypotheses
\begin{eqnarray}
\label{eq:4} {\rm\tt H}_{0} =
\{\theta^{\dagger}=\theta_{0}\}\rm{\quad and\quad }{\rm\tt H}_{\it
\Upsilon} = \{\theta^{\dagger}=\theta_{\Upsilon}=\theta_{0}+{\it
\Upsilon}\} \enspace .
\end{eqnarray}
In both cases (\ref{eq:22}) and (\ref{eq:4})  we do not assume a
specific mode switching behavior (for example, a Markovian one)
and view it as deterministic (albeit unknown to the observer) like
controlled by an independent actor.

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

\section{Statistical Orthogonality Principle}
\label{section:Stat}
\begin{definition}
\label{def1} Given  $l$ pairs $\{r, s_j\}$,  $j=1,2,\dots,L$, each
formed by column vectors $r=r(t_i)$ and $s_j=s_j(t_k)$. Then
$r(t_i)$ and $s_j(t_k)$ are said to be {\em statistically
orthogonal} at distance $\Delta t= t_k-t_i$, that is denoted as
$r(t_{i}) \perp s_j(t_k)$,  if
\begin{eqnarray}
\label{eq:12} \forall t_{i}, t_{k}, \Delta t= t_k-t_i=\mathrm{const}: \quad
\E{s_j^{T}(t_{k})r(t_{i})} = 0 %%%\enspace .
\end{eqnarray}
and $r(t_i)$ is said to be  statistically orthogonal to
$S(t_k)=[s_1(t_k)\,|\,\cdots\,|\,s_L(t_k)]$ at $\Delta t= t_k-t_i \,$
if (\ref{eq:12}) holds for all given pairs, that is denoted as
$r(t_i) \perp S(t_k)$.
\end{definition}
Let $r$ and $S$ in  Definition~\ref{def1} be computed by $L$ pairs
of parallel blocks RG and SM as shown  in Fig.\ref{fig1}($b$). By
RG and SM are meant {\em Residual Generator} and {\em Sensitivity
Model} both parameterized by a design parameter $\hat \theta
\in~{\bbbr}^L $ not fully but partly being an exact replica of
$\theta$, the MS parameter, in its elements' numbering and sense.
Thus,   $r(t_i)= r(t_i, {\hat \theta})$ and $S(t_k)=S(t_k, {\hat
\theta})$.
\begin{definition}
\label{def2} Let {\em RG} and {\em SM} be designed in such a way
that {\em in case (\ref{eq:22})} for each $\mu=0,1,2,\ldots,{\cal
M}$, parameter $\hat \theta$ takes a specified value $\hat
\theta_\mu$ such that
\begin{eqnarray}
\label{eq:61} \forall \mu: \quad
\{\hat \theta=\hat \theta_\mu\} \;  \equiv \;
\{ \: r(t_i, {\hat \theta}) \perp S(t_i, {\hat \theta})
\: \; \getsto \: \; \theta^{\dagger}=\theta_{\mu}\, \}
\end{eqnarray}
or {\em in case (\ref{eq:4})} for each $\Upsilon$, parameter $\hat
\theta$ takes a specified value $\hat \theta_{\Upsilon}$ such that
\begin{eqnarray}
\label{eq:62} \forall \Upsilon: \quad
\{\hat \theta=\hat \theta_{\Upsilon}\} \;  \equiv \;
\{ \:  r(t_i, {\hat \theta}) \perp S(t_i, {\hat \theta}) \: \; \getsto \: \; \theta^{\dagger}=\theta_{\Upsilon} \, \}
\end{eqnarray}
where $\getsto$ stands for ``iff''  (if and only if). Then
(\ref{eq:61}) and (\ref{eq:62}) is termed  {\em Statistical
Orthogonality Principle}.
\end{definition}

The parallel blocks in Fig.\ref{fig1}($b$), RG and SM,   may be
variously designed to satisfy Definitions \ref{def1} and
\ref{def2}. However as system (\ref{eq:1}), (\ref{eq:2}),
(\ref{eq:1.10}), (\ref{eq:1.9}) is stochastic, the presence of an
{\em (adaptive) estimator} (AE), denote it by $g(t_{i|i-1}, {\hat
\theta})$, is requisite for obtaining a residual $r(t_i, \hat
\theta)$. With it,  writing $r(t_i, \hat \theta) \perp S(t_i, \hat
\theta)$ is straightforward as a necessary (and on occasion,
sufficient) condition for minimum of
\begin{eqnarray}
\label{eq:6-16}
J({\hat \theta})  = 1/2 \lim \limits_{t_j \to -\infty}
 \E{\Vert r(t_i, \hat \theta)\Vert^{2} \; \mid \; y(t_j^{i-1})} \enspace .
\end{eqnarray}
Thus, SOP implementation boils down to the  construction of
(\ref{eq:6-16}) so as to obtain
\begin{eqnarray}
\label{eq:6-17}
S(t_i, \hat \theta)  =  \left[
\frac{\partial r(t_i, \hat \theta)}{\partial \hat \theta_1} \right\vert \cdots
\left\vert \frac{\partial r(t_i, \hat \theta)}{\partial \hat \theta_L } \right] \enspace .
\end{eqnarray}

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

\section{Identification Performance Indices}
\label{section:Perf} Minimum Prediction Error methods
\cite{Caines88} offer to predict by $g(t_{i|i-1}, {\hat \theta})$
the system output $y(t_i)$ and use  $r(t_i, {\hat \theta})=y(t_i)
- g(t_{i|i-1}, {\hat \theta})$ as a residual. Minimizing
(\ref{eq:6-16}) with such residual leads to the biased parameter
estimates for the given EIV system \cite{Soderstrom81},
\cite{Ansay99}. Had $r(t_i, {\hat \theta})$ in (\ref{eq:6-16})
instead been in one of the following forms
\begin{eqnarray}
\label{eq:6}
r(t_i, {\hat \theta})= \{\; x(t_{i}) -
g(t_{i|i-1},{\hat \theta})  & \; {\rm or} \;  &
\hat{x}_{\mathrm{OPT}}(t_{i|i-1}) - g(t_{i|i-1},{\hat \theta}) \;  \}
\end{eqnarray}
where $\hat{x}_{\mathrm{OPT}}(t_{i|i-1})$ is the  one-step SSKF
predictor for $x(t_{i})$, minimizing (\ref{eq:6-16}) would have
helped to avoid the bias. Obviously, (\ref{eq:6}) provides two
clear examples of {\em Direct {\rm (but)} Inaccessible Residual}
(DIR) and in this case (\ref{eq:6-16}) is an example of  
{\em Direct {\rm (but)} Inaccessible Performance Index} (DIPI).
\begin{remark}
In the limit  as $t_0 \to -\infty$, FF equations   (\ref{eq:1.10})
assure coincidence
\begin{eqnarray}
\label{eq:6-13}
\tilde{x}(t_{i}^{+}) = \hat{x}_{\mathrm{OPT}}(t_{i|i})  & \quad {\rm and} \quad  &
\tilde{x}(t_{i}^{-}) = \hat{x}_{\mathrm{OPT}}(t_{i|i-1})
\end{eqnarray}
if optimally designed for the  fault-free system mode specified by
$\theta^{\dagger}=\theta_{0}$.
\end{remark}
\begin{remark}
If a set of AEs $g(t_{i|i-1}, {\hat \theta})$ in  (\ref{eq:6})
contains SSKF, then $g(t_{i|i-1}, {\hat \theta})$ minimizing
(\ref{eq:6-16}) coincides with
$\hat{x}_{\mathrm{OPT}}(t_{i|i-1})$. If a fault occurs, we
consider SSKF a {\em hidden} object to be identified  to
substitute FF and be used to modify FR.
\end{remark}
\begin{problem}
To implement both SOP-based FDD and SSKF identification, we need
to construct the {\em Accessible {\em albeit} Indirect Residual}
(AIR) as a substitute for the DIR in order to change from the DIPI
to the {\em Accessible {\em albeit} Indirect Performance Index}
(AIPI) in (\ref{eq:6-16}), the latter having the same minimizing
estimator as the DIPI.
\end{problem}
\begin{solution}
Being the system completely observable enables the solution to be found
in the form:
\begin{eqnarray}
\label{eq:6-14}
{\mathrm{AIR}(t_i, \hat \theta)}&=&{\mathrm{DIR}(t_i, \hat \theta)}+\mathrm{SP}(t_i), \quad
\mathrm{SP}(t_i) \perp {\mathrm{DIR}(t_i, \hat \theta)}, \; \mathrm{SP}(t_i) \ne f(\hat \theta)\\
\label{eq:6-15} {\mathrm{AIPI}(\hat \theta)}&=&{\mathrm{DIPI}(\hat
\theta)}+\mathrm{Const}, \quad \mathrm{Const}= 1/2 \E{\Vert
\mathrm{SP}(t_i)\Vert^{2} \; \mid \; y(t_j^{i-1})}
\end{eqnarray}
where  $\mathrm{SP}(t_i)$ is a {\em slack process}  fully
independent of $\hat \theta\,$ [$\mathrm{SP}(t_i) \ne f(\hat
\theta)\,$] and (\ref{eq:6-15}) is the half mean squared Euclidean
norm of  (\ref{eq:6-14}) in the limit as $t_j \to - \infty$.
\end{solution}
\begin{remark}
\label{rem3} The solution is given in \cite{Semoushin85},
\cite{Semoushin94}, \cite{Semoushin00} and thoroughly tested in
\cite{Semoushin02-2} for identification purposes only. In what
follows, we denote the $\mathrm{AIR}(t_i, \hat \theta)$ by
$\varepsilon(t_i, \hat \theta)$ and use it here for the jointly
performed FDD-SIA computational tasks. So from now on,
$\varepsilon(t_i, \hat \theta)$ serves instead of $r(t_i, \hat
\theta)$ in Formulas (\ref{eq:12}) to (\ref{eq:6-17}).
\end{remark}

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

\section{Fault Detection and Diagnosis}
\label{section:FDD} In accordance with Definition~\ref{def2} and
Remark~\ref{rem3},  being the $k$-th expectation
\begin{eqnarray}
\label{eq:15} \zeta_{j}(t_{i}, \hat \theta)  =  \E {\left (\frac {\partial
\varepsilon(t_{i}, \hat \theta)}{\partial {\hat \theta_j}}, \varepsilon(t_i, \hat \theta)
\right)\, \mid \, {\hat \theta={\hat \theta}_{\mu}}\, \mathrm{or}\;{\hat \theta}_{\Upsilon}},\;
\; j=1,2,\ldots,L
\end{eqnarray}
of the dot product  $(\cdot \, , \cdot)$ in the immediate vicinity
of zero is equipollent to the fact that
$\theta^{\dagger}=\theta_{\mu}\,
\mathrm{or}\;{\theta}_{\Upsilon}$. On this basis we use it as a
signature of the fact as it is shown schematically in
Fig.~\ref{fig1}($b$) where $\zeta$ is composed of (\ref{eq:15}),
$\zeta \in \bbbr^L$.

In FDD, when we test  between (\ref{eq:22}), the theoretical
signature
\begin{eqnarray}
\label{eq:23}
\forall \, \mu \; \mathrm{and}\; j=1,2,\ldots,L: \quad \zeta_{j}(t_{i}, \hat \theta_\mu) = 0
\end{eqnarray}
is checked by the  following approximate decision rule:
\begin{eqnarray}
\label{eq:24} \forall \, \mu =1,2,\ldots, {\cal M} \;
\mathrm{and}\; j=1,2,\ldots,L:  \quad \left| \, 0.5 -
\frac{n_{j}^{\mu}}{N}\right| {{{\rm\tt H}_{\mu} \atop >} \atop {<
\atop {\rm\tt H}_{0}}} \gamma_{j}
\end{eqnarray}
where $n_{j}^{\mu}$  is the number of negative outliers of the
process $\hat{\zeta}_{j}(t_{i}, \hat \theta_\mu)$ in the current
sample of size $N$. Here $\hat{\zeta}_{j}(t_{i}, \hat \theta_\mu)$
is an estimator (for which we recommend the exponential smoothing)
of ${\zeta}_{j}(t_{i}, \hat \theta_\mu)$ defined for (\ref{eq:23})
according to (\ref{eq:15}) for each $\mu$-th {\em Signature
Generator} (SG) within the {\em Bank of Signature Generators}
(BSG), $\mu=1,2,\ldots,{\cal M}$ (Fig.~\ref{fig2}($a$)). Signature
Evaluator and Selector (SES) in this setup operates according to
(\ref{eq:24}), where the sample size $N$ and thresholds
$\gamma_{j}$ should be determined experimentally to guarantee the
maximum decision power with the limited error probability. If
several hypotheses among ${\rm\tt H}_{\mu}$, $\mu=1,2,\ldots,{\cal
M}$, have been selected, we discriminate between them by the
additional rule: select the ${\rm\tt H}_{\mu_{*}}$ with
\begin{eqnarray}
\label{eq:25} \mu_{*} & = & {\rm {arg}} \min\limits_{\mu}
\sum_{j=1}^{L}\left\vert \, 0.5 - \frac{n_{j}^{\mu}}{N}\right\vert \enspace .
\end{eqnarray}
If none ${\rm\tt H}_{\mu}$, $\mu=1,2,\ldots,{\cal M}$, has been
selected, we adopt a decision that ${\rm\tt H}_{0}$  is true.

\begin{figure}
\input picture2.txt
\caption{$(a)$ Fault  detection and diagnosis (discrimination
between ${\cal M}+1$ hypotheses). $(b)$ Fault diagnosis and system
accommodation using TBSG. $(c)$ Fault point detection to switch IA
and AM. Legend: MS~--~monitored system, BSG~--~Bank of Signature
Generators, TBSG~--~Trained BSG, SES~--~Signature Evaluator and
Selector, AM~--~Accommodation Mechanism, IA~--~Identification
Algorithm } \label{fig2}
\end{figure}

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

\section{Optimal Filter Identification and BSG Training}
\label{section:Optin} To reason what change  (or what fault) has
occurred, one of two strategies may be used: parallel or
sequential. The first one is adequate for Case (\ref{eq:22}),
while the latter for Case (\ref{eq:4}). The parallel strategy
implies the existence of the BSG as discussed in
Sect.~\ref{section:FDD}. Each SG can be trained beforehand to
match the corresponding system mode, i.e. to satisfy
(\ref{eq:23}). Training algorithms for a single ($\mu$-th) SG are
standard identification algorithms for
$\hat{\theta}=\hat{\theta}_\mu$. For example, one can use the
following SLS:
\begin{enumerate}
\item[] {\it Simplified Least Squares}.  For $k=1,2,\ldots$ and
$\Lambda={\rm{diag}}[\lambda^{(j)}]$, $j=1,2,\ldots,L$, with
$\Lambda_{1}=I$ do  (\ref{eq:17}) and then (\ref{eq:18}) where $k$
is the step number:
\begin{eqnarray}
\label{eq:17} \lambda_{k+1}^{(j)} & = & \lambda_{k}^{(j)}  +
\left\Vert \frac{\partial\varepsilon(t_{i}, \hat{\theta}_k)}{\partial
 \hat{\theta}_{j}}\right\Vert^{2}\\
\label{eq:18} \hat{\theta}_{k+1} & = & \hat{\theta}_{k}  -
\Lambda_{k+1}^{-1} {\cal E} \{S^T(t_i, \hat{\theta}_{k}) \varepsilon(t_{i}, \hat{\theta}_k)  \}
\end{eqnarray}
\end{enumerate}
where  ${\cal E}\{ \cdot \}$ is a smoothing algorithm. In the case
of exponential smoothing we obtain $\hat \zeta (t_i, \hat
\theta)={\cal E}\{ \zeta (t_i, \hat \theta) \}$ from the signature
vector $\zeta (t_i, \hat \theta)=S^T(t_i, \hat{\theta})
\varepsilon(t_{i}, \hat{\theta})$ by
\begin{eqnarray}
\label{eq17}
\hat \zeta (t_i, \hat \theta)=\beta \hat \zeta (t_{i-1}, \hat \theta)+ (1-\beta)\zeta (t_i, \hat \theta),
\qquad 0 \le \beta<1 \enspace .
\end{eqnarray}

In Case  (\ref{eq:4}) we have to use the sequential strategy of
Fig.~\ref{fig2}($c$) with one SG, one SES and one identification
algorithm (IA). In this setup,  SG  is intended to detect and
track all possible abrupt changes as they occur in time in the
monitored system. The SG performs this by means of IA, which may
be of (\ref{eq:17})--(\ref{eq:18}) type or another. The SES tests
between hypotheses (\ref{eq:4}) so that the rule (\ref{eq:24}) is
implemented with $\mu=1$. Decision to adopt ${\rm\tt H}_{1}$
starts up the IA; decision to adopt ${\rm\tt H}_{0}$ stops it and
starts the accommodation mechanism (AM) in order to bring the
feedback of MS and the real situation after change into better
agreement and, in doing so, to adapt the system feedback to suit
the new (after-change) conditions. Figure~\ref{fig4} shows how it
is implemented except AM. Accommodation mechanism is discussed
briefly in Sect.~\ref{section:MSA}. \nopagebreak
\begin{figure}
\begin{center}
\begin{tabular}{|p{0.5cm}p{10.1cm}|}
\hline & \begin{tabbing} {\tt begin} \=\tt while \=\tt begin \=\tt
begin \=\tt ififi \=\tt begin \=\tt if \=\tt begin \kill
\tt begin\>$i:=0;$\+\\
\tt for $j:=1$ \tt to $L$ \tt do\+\\
{\tt begin} $n_{j}:=0;$ $s_{j}:=${\em off}; $\lambda_{j}:=1$ \tt end\rm;\-\\
\tt while $i\le imax$ \tt do\+\\
\tt for $j:=1$ \tt to $l$ \tt do\+\\
\tt begin $i:=i+1;$\+\\
\tt if $\hat \zeta_j(t_i, \hat{\theta})<0$ \tt then $n_{j}:=n_{j}+1;$\\
$f_{j}:=0.5-n_{j}/N;$\\
\tt if $i$ mod $N=0$ then\+\\
\tt begin $n_{j}:=0;$\+\\
\tt if $abs(f_{j})<\gamma_j$ then\+\\
$s_{j}:=${\em off} $\,$ \tt else $s_{j}:=on$\-\-\\
\tt end\rm; \-\\
\tt if $s_j=on$ then\+\\
\tt begin\>\tt         \+\\
$\lambda^{(j)}  :=  \lambda^{(j)}  +
\left\Vert \frac{\partial\varepsilon(t_{i}, \hat{\theta})}{\partial
 \hat{\theta}_{j}}\right\Vert^{2}$;\\[4pt]
$\hat{\theta}^{(j)}:=\hat{\theta}^{(j)} -
\hat{\zeta_j}(t_{i}, \hat{\theta})/\lambda^{(j)}$\-\\
\tt end  else\\
\tt if $i$ mod $N=0$ then $\lambda^{(j)}:=1$\-\-\\
\tt end\-\-\-\\
\tt end\rm.
\end{tabbing}\\
\hline
\end{tabular}
\end{center}
\caption{Pseudocode  sketching the inter-independent parameter
identification algorithm} \label{fig4}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Monitored System Accommodation}
\label{section:MSA} To accommodate the  MS to the newly developed
(maybe, faulty) mode, the accommodation mechanism (AM) is expected
to be capable to replace parameters $\bar \Phi$ and $\bar K$ of
(\ref{eq:1.10}) with the  newly identified matrix $\Phi$ and
optimal SSKF gain $K$, and $\bar G_{\mathrm{FR}}$ of
(\ref{eq:1.9}) with the $G_{\mathrm{FR}}$ computed by solving the
{\it Discrete Algebraic Riccati Equation}, DARE, in compliance
with the well known LQG control law using matrices of
(\ref{equa:1.3}).

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

\section{Approach Validation}
\label{section:Valid} {\it I. System}. Dimensions  are $n=2$,
$m=1$, $q=1$, $l=2$. Matrices are as follows:
$$
\begin{array}{lclclclclcl}
\Phi_{\theta}&=&
\left[
\begin{array}{rcl}
0     &\hphantom{a} &1 \\
\theta_{1} &\hphantom{a} &\theta_{2}
\end{array}\right],
&\hphantom{a}&\Psi_{\theta}&=&\Psi=
\left[
\begin{array}{r}
0     \\
1
\end{array}\right],
&\hphantom{a}&Q_{\theta}&=&Q=\left[
\begin{array}{lcr}
0 &\hphantom{a}& 0 \\
0 &\hphantom{a}& 1
\end{array}\right]\\
&&&&&&&&&&\\
H&=&\left[
\begin{array}{lcr}
1 &\hphantom{a}& 0
\end{array} \right],
&\hphantom{a}&R_\theta&=&R=\left[
\begin{array}{l}
0.1
\end{array} \right],&\hphantom{a}&\theta^T&=&
\left[
\begin{array}{lcr}
\theta_1 &\hphantom{a}& \theta_2
\end{array} \right]\\
&&&&&&&&&&\\
\theta^T_0&=&
\left[
\begin{array}{lcr}
0.30 &\hphantom{a}& 0.68
\end{array} \right],&\hphantom{a}&\theta^T_{\Upsilon}&=&
\left[
\begin{array}{lcr}
0.40 &\hphantom{a}& 0.10
\end{array} \right],&\hphantom{a}&\Upsilon^T&=&
\left[
\begin{array}{lcr}
0.10 &\hphantom{a}& -0.58
\end{array} \right]\\
&&&&&&&&&&\\
\bar \Phi&=&\left[
\begin{array}{rcl}
0     &\hphantom{a} &1 \\
0.2 &\hphantom{a} &0.2
\end{array}\right],&\hphantom{a}&\bar K&=&\left[
\begin{array}{r}
0.2     \\
0.2
\end{array}\right],&\hphantom{a}&\bar G_{\mathrm{FR}}&=&\left[
\begin{array}{lcr}
0.5 &\hphantom{a}& 0.5
\end{array} \right] \enspace . \\
\end{array}
$$
{\it II. Residual Generator}. According to Remark~\ref{rem3}, RG is composed of
AE and AIR:
\begin{eqnarray}
\label{eq23}
g(t_{i+1|i}, \hat \theta)&=&A[g(t_{i|i-1}, \hat \theta)+D\eta(t_i, \hat \theta)]
+ \Psi u(t_i)
\\
\label{eq24}
\eta(t_i, \hat \theta)&=&y(t_i)-Hg(t_{i|i-1}, \hat \theta)  \enspace .
\end{eqnarray}
For AE (\ref{eq23}), (\ref{eq24}), we have here $L=4$ and
\begin{eqnarray}
\label{eq28}
\begin{array}{lclclclclcl}
A&=&\left[
\begin{array}{rcl}
0     &\hphantom{a} &1 \\
a_1 &\hphantom{a} &a_2
\end{array}\right],&\hphantom{aaa}&D&=&\left[
\begin{array}{r}
d_1     \\
d_2
\end{array}\right],&\hphantom{aaa}&\hat \theta^T&=&\left[
\begin{array}{lcccccr}
a_1 & \hphantom{a} & a_2 & \hphantom{a} & d_1 & \hphantom{a} & d_2
\end{array} \right] \enspace . \\
\end{array}
\end{eqnarray}
With reference to Remark~\ref{rem3}, we construct AIR as
\begin{eqnarray}
\label{eq25}
\varepsilon(t_i, \hat \theta)&=&{\cal N}(D)\eta(t_{i-s+1}^{i}, \hat \theta), \quad
{\cal N}(D)=
\left(\begin{array}{cccc}
1&\hphantom{a}0&\hphantom{a}0&\hphantom{a}0\\
\ddots&\hphantom{a}1&\hphantom{a}0&\hphantom{a}0\\
d_{n-1}&\hphantom{a}\ddots&\hphantom{a}1&\hphantom{a}0\\
d_n&\hphantom{a}d_{n-1}&\hphantom{a}\ddots&\hphantom{a}1
\end{array}\right) \enspace .
\end{eqnarray}
In the first formula  of (\ref{eq25}), $s$ is the  greatest
partial observability index of the system and  the second formula
holds if MS with $m=1$ is taken in the standard observable form
\cite{Semoushin94}. Such is here the case and $s=2$,
\begin{eqnarray}
\label{eq26}
\eta(t_{i-s+1}^{i}, \hat \theta)=[\eta(t_{i-1}, \hat \theta)\quad\eta(t_i,
\hat \theta)]^T \enspace .
\end{eqnarray}
{\it III. Sensitivity Model}. To obtain  (\ref{eq:6-17}) from
$\varepsilon(t_i, \hat \theta)$, (\ref{eq25}), in view of
Remark~\ref{rem3}, we have to generate the sensitivity functions
having defined them by
\begin{eqnarray}
\label{eq27}
\mu^{(j)}(t_{i+1|i}, \hat \theta)=\frac{\partial }{\partial \hat \theta_j}
g(t_{i+1|i}, \hat \theta), \quad j=1,\ldots,L \enspace .
\end{eqnarray}
{\it IV. Pattern for Computational  Experiments}. The composite
algorithm we test is outlined by Figs.~\ref{fig2}($c$) and
\ref{fig4}. Besides the MS with the above matrices, it involves:
RG -- (\ref{eq23}), (\ref{eq24}), (\ref{eq28}), (\ref{eq25}) and
(\ref{eq26}); SM -- (\ref{eq:6-17}) and (\ref{eq27}); SG --
(\ref{eq:15}); SES -- (\ref{eq:24}) and (\ref{eq:25}); and IA --
(\ref{eq:17}), (\ref{eq:18}) and (\ref{eq17}). The algorithm is
started up with the non-optimal parameters $\bar \Phi$, $\bar K$
and $\bar G_{\mathrm{FR}}$. The SES is expected to  detect this
initial non-optimality and to start up the IA precisely in respect
to the parameters which require optimisation. The IA is expected
to eliminate the initial non-optimality (this is the SG training).
After that, at the middle time instant $t_{8000}$, the MS switches
from mode ${\rm\tt H}_{0}$ to ${\rm\tt H}_{\Upsilon}$, cf.
(\ref{eq:4}). The algorithm is expected to detect this mode
switch, to identify the new optimal parameters and accommodate the
MS feedback
to the after-switch conditions.\\

\noindent{\it V. Experimental Results}.  An example of the results
is shown in Figs.~\ref{grafic5} and \ref{grafic4}. This is one of
many results obtained for many different conditions
\cite{Kond2001}.  In these experiments, the following values were
chosen: $\gamma_j=0.05$, $N=2000$ and $\beta=0.1$.  The estimates
formed by the composite algorithm are thus seen to be satisfactory
enough. This lends support to the validity of the proposed
approach.

\begin{figure}
\setbox1=\vbox{
\hbox to \textwidth{\null\hfil
\psfig{figure=A_1.ps,angle=0,width=0.48\textwidth}\hfil
\psfig{figure=A_2.ps,angle=0,width=0.48\textwidth}\hfil\null
} \vspace{1ex}
        }
\noindent \copy1
\caption{Tracking the true parameters $\theta_1$ and $\theta_2$ by
$a_1$ ({\em left}) and $a_2$ ({\em right}) of (\ref{eq28})}
\label{grafic5}
\end{figure}

\begin{figure}
\setbox1=\vbox{
\hbox to \textwidth{\null\hfil
\psfig{figure=D_1.ps,angle=0,width=0.48\textwidth}\hfil
\psfig{figure=D_2.ps,angle=0,width=0.48\textwidth}\hfil\null
} \vspace{1ex}
        }
\noindent \copy1
\caption{Tracking the SSKF  parameters $k_1$ and $k_2$ by $d_1$
 ({\em left})  and $d_2$ ({\em right})  of (\ref{eq28})}
\label{grafic4}
\end{figure}

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

\section{Conclusions and Future Work}
\label{section:Conclus} We have shown that statistical
orthogonality principle and indirect identification performance
index provide the basis for the jointly performed computational
tasks in the multi-mode stochastic system identification. Further
investigations of this approach should be made to study all the
aspects, both theoretical and applied, for instance -- to evaluate
it in terms of false alarm and missed detection rates.

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

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\end{document}
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