Following the previous results of [1,2], we assume that all data models 
forming a set 
are parameterized by an 
-component vector
. Each particular value of 
(which does not depend on time) specifies a
. Hence
where 
is the compact subset of
. A given physical data model (PhDM) is described by the following equations:
where 
denotes nonnegative integers, 
strictly positive integers, and 
all integers.
Every model 
(2) is assumed to be acting between adjacent switches as long as it is sufficient for accepting as correct the basic theoretical statement (BTS) that all processes related to the 
are wide-sense stationary. This statement amounts to the following assumptions. The random 
with 
is orthogonal [
and
, the zero-mean mutually orthogonal wide-sense stationary orthogonal sequences with 
and 
for all
; 
is orthogonal to 
and 
for all
; 
is a given signal; it is an “external input” when considering the open-loop case or a control strategy function
when considering the closed-loop setup.
Stackable vectors of previous values
constitute the experimental condition 
(cf. Ljung [
By assumption, 
is generated by the completely observable PhDM (2), so we can move from the physical state variables 
in (2) to another 
through the similarity transformation
. Such transformation uniquely determines a new state representation
of the standard observable data model (SODM) (cf. Semushin [
For convenience in the below we shall omit the subscript 
for all the matrices describing PhDM or SODM.
, the initial
, and
which is the well-known (not necessarily steady-state) Kalman filter with the innovation process
, the optimal state predictor
, the gain
satisfying the discrete Riccati iterations [
, which is equivalent to (6), can be used where 
is the optimal “filtered” estimator for 
based on experimental condition 
(4). When 
ranges (or switches) over 
as in (1), we obtain the set of Kalman filters
through its error 
in the Kalman filter. Thus in the basis forming the state-space, 
(9) is the model minimizing the Original Performance Index (OPI) 
(11) at any
, which is large enough for BTS to hold, so that writing 
or 
as well as any other finitely shifted time in (11) makes no difference.
- component vector 
of unknown system parameters, which needs to be identified. This means that the entries of the matrices
, 
, 
, 
, 
, 
are functions of
. However, for the sake of simplicity we will supress the corresponding notations below, i.e. instead of
, 
, 
, 
, 
, 
we will write
, 
, 
, 
, 
,
. We make the same assumptions about the SODM.
belongs to with the only difference that the unknown parameter 
in 
is replaced by 
to obtain
. In so doing, each particular value of
, an estimate of
, leads to a fixed model
. In accordance with The Active Principle of Adaptation (APA) [
ranges over 
in search of 
for the goal 
or 
as governed by a smart, unsupervised helmsman equipped by a vision of the goal in state space and able to pursue it, we obtain a model 
of active type within the set 
(12). In this case, 
will act as a self-tuned model parameter and so should be labeled by
, the time instant of model’s inner clock, in order to get thereby the emphasized notations 
and 
in describing parameter identification algorithms (PIAs) to be developed. From this point on 
becomes an adaptive estimator.
may differ from that of
. We shall need to discriminate between 
and 
later when developing a PIA.
or 
in an explicit form.
) the model
(12). Here 
is the self-tuned parameter intended to estimate (in one-to-one corresponddence) parameter θ. In parallel, reasoning from
, we build the adaptive model
) the model
does not depend on
. Matrices 
are evaluated according to (7), (8).
using (14)-(15) (or alternatively, 
using (16)-(17)) is supposed to contain a PIA to offer the prospect of convergence. For convergence in parameter space, we anticipate almost surely (a. s.) convergence, as it is the case for MPE identification methods [3,4]. It actuates either or both of the two other types of convergence. The type of convergence in state space, as well as in response space, is induced by the type of Proximity Criterion, PC (cf. [
generated by Data Source described in any appropriate form (2), (5)(6), or (9), and their estimates 
generated by the adaptive model 
(or
). For 
in
or
, thus bringing them into correlation with 
or 
(correspondingly, with 
or
) from (19). Then
needs clarification. Its sense correlates with the above concept of convergence (18). Necessary refinements will be done (in Theorem 1).
(20) be a vector-valued ncomponent function of
. If 
is defined by
of the API fixed out at any instant t is the necessary and sufficient condition for adaptive model 
to be consistent estimator of 
in mean square,
, that is True (Unbiased) m.s. System Identifiability
is a preassigned zero-mean orthogonal wide-sence stationary process orthogonal to 
but in contrast to 
and
, known and serving as a testing signal;
, and Setup 3 (Close-loop Control) with
, which does not depend on
.
of unknown parameters. The minimization of API by some PIA allows us to determine the optimal value
. Then it must be substituted into (14)-(15) (or (16)-(17))
(or
). At the same time, we obtain the optimal estimates 
(or
) according to OPI.
is given by
.
.
and 
of the noises 
and 
are equal to 0.04 and 0.06, correspondingly.
, 
, 
and 
should be identified. The true values of parameters are the same as in E1.
. Thus, the obtained results confirm applicability of the presented method.
in parameters
of Data Model 
or 
instead of parameters of Adaptive Filter 
or
. The obtained results were verified by two numerical simulation examples.