Optimum Complexing of Measurements when Maintaining a Maneuvering Object in Statistically Uncertain Situations

Abstract

The problem of synthesizing optimal and quasi-optimal algorithms for complex information processing is solved using the methods of Markov theory for estimating random processes when maintaining a maneuvering object and two-channel vector observation with violations in statistically uncertain situations. The problem is solved in relation to a discrete-continuous Markov process for the case when its continuous part is a vector Markov sequence, and the discrete part is characterized by a three-component discrete Markov process, each component of which is described by a Markov chain to several positions. A block diagram of quasi-optimal complex information processing is given. Using a simple example, simulation modeling shows the performance of a quasi-optimal algorithm in statistically uncertain situations.

APPENDIX

Let us introduce into consideration the posterior distribution of the mixed state vector ${{\left[ {{\mathbf{X}}_{k}^{{\text{T}}},{\mathbf{S}}_{k}^{{\text{T}}}} \right]}^{{\,{\text{T}}}}} = {{\left[ {{\mathbf{X}}_{k}^{{\text{T}}},{{A}_{k}},{{B}_{k}},{{C}_{k}}} \right]}^{{\,{\text{T}}}}}$:

$$f\left( {{\mathbf{x}}_{k}^{{}},{{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}\,{\text{|}}\,{\mathbf{y}}_{1}^{k}} \right) = f\left( {{\mathbf{x}}_{k}^{{}},{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{k}} \right) = w_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right).$$

According to the definition of conditional distributions, we have

$$f\left( {{\mathbf{x}}_{k}^{{}},{{{\mathbf{s}}}_{k}},{\mathbf{y}}_{1}^{k}} \right) = f\left( {{\mathbf{y}}_{1}^{k}} \right){\kern 1pt} {\kern 1pt} f\left( {{\mathbf{x}}_{k}^{{}},{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{k}} \right) = f\left( {{\mathbf{y}}_{1}^{k}} \right){\kern 1pt} {\kern 1pt} w_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right).$$

(A.1)

Using (A.1) and the main properties of the Markov processes, we write expressions for the distribution law of the posterior distribution of the mixed state vector ${{\left[ {{\mathbf{X}}_{k}^{{\text{T}}},{\mathbf{S}}_{k}^{{\text{T}}}} \right]}^{{\,{\text{T}}}}}$

$$\begin{gathered} w_{{jmd}}^{*}({\mathbf{x}}_{k}^{{}}) = \frac{1}{{f({\mathbf{y}}_{1}^{k})}}f({\mathbf{x}}_{k}^{{}},{{{\mathbf{s}}}_{k}},{\mathbf{y}}_{1}^{k}) = \frac{1}{{f({\mathbf{y}}_{1}^{k})}}\sum\limits_{{{{\mathbf{s}}}_{{k - 1}}}} { \cdots \sum\limits_{{{{\mathbf{s}}}_{1}}} {\int { \cdots \int {f\left( {{{{\mathbf{x}}}_{k}},{{{\mathbf{s}}}_{k}},{{{\mathbf{y}}}_{k}},...,{{{\mathbf{x}}}_{1}},{{{\mathbf{s}}}_{1}},{{{\mathbf{y}}}_{1}}} \right)\prod\limits_{g = 1}^{k - 1} d } } } } {{{\mathbf{x}}}_{g}} \\ \, = \frac{{f({\mathbf{y}}_{1}^{{k - 1}})}}{{f({\mathbf{y}}_{1}^{k})}}\sum\limits_{{{{\mathbf{s}}}_{{k - 1}}}} {\int {f\left( {{{{\mathbf{x}}}_{k}},{{{\mathbf{s}}}_{k}},{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}},{{{\mathbf{y}}}_{{k - 1}}}} \right)f({{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}})\,d{{{\mathbf{x}}}_{{k - 1}}}} } \\ \, = \frac{{f({\mathbf{y}}_{1}^{{k - 1}})}}{{f({\mathbf{y}}_{1}^{k})}}\sum\limits_{{{{\mathbf{s}}}_{{k - 1}}}} {\int {f\left( {{{{\mathbf{x}}}_{k}},{{{\mathbf{s}}}_{k}},{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}},{{{\mathbf{y}}}_{{k - 1}}}} \right){\kern 1pt} {\kern 1pt} w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right){\kern 1pt} {\kern 1pt} d{{{\mathbf{x}}}_{{k - 1}}}} } , \\ \end{gathered} $$

(A.2)

where integration over variable x is carried out in the area ${{\Re }^{{{{n}_{x}}}}}$.

We transform the first conditional probability density in the integrand:

$$f\left( {{{{\mathbf{x}}}_{k}},{{{\mathbf{s}}}_{k}},{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}},{{{\mathbf{y}}}_{{k - 1}}}} \right) = f\left( {{{{\mathbf{x}}}_{k}},{{{\mathbf{s}}}_{k}},{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}}} \right).$$

Here the absence ${{{\mathbf{y}}}_{{k - 1}}}$ does not matter for a given ${{{\mathbf{x}}}_{{k - 1}}}$. Using the property of conditional probability densities, we write

$$f\left( {{{{\mathbf{x}}}_{k}},{{{\mathbf{s}}}_{k}},{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}}} \right) = f\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}},{{{\mathbf{s}}}_{k}}} \right){\kern 1pt} {\kern 1pt} {\text{P}}\left( {{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{{{\mathbf{s}}}_{{k - 1}}},{{{\mathbf{x}}}_{{k - 1}}}} \right)f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{{\mathbf{S}}}_{k}},{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{S}}}_{{k - 1}}}} \right).$$

(A.3)

Taking into account the identities

$${\text{P}}\left( {{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{{{\mathbf{s}}}_{{k - 1}}},{{{\mathbf{x}}}_{{k - 1}}}} \right) \equiv {\text{P}}\left( {{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{{{\mathbf{s}}}_{{k - 1}}}} \right),$$

$$f\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}},{{{\mathbf{s}}}_{k}}} \right) \equiv f\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{A}_{k}} = {{a}_{j}}} \right) = {{f}_{j}}\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}}} \right),$$

$$f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{{\mathbf{S}}}_{k}},{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{S}}}_{{k - 1}}}} \right) \equiv f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}} \right),$$

which are satisfied according to the conditions of the problem statement, we rewrite (A.3) in the form

$$f\left( {{{{\mathbf{x}}}_{k}},{{{\mathbf{s}}}_{k}},{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}},{{{\mathbf{s}}}_{{k - 1}}}} \right) = {\text{P}}\left( {{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{{{\mathbf{s}}}_{{k - 1}}}} \right){{f}_{j}}\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}}} \right)f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}} \right).$$

(A.4)

After substituting (A.3) into (A.1), we have

$$w_{{jmd}}^{*}({\mathbf{x}}_{k}^{{}}) = \frac{{f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}},{{C}_{k}}} \right)}}{{f({{{\mathbf{y}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}})}}\sum\limits_{{{{\mathbf{S}}}_{{k - 1}}}} {{\text{P}}\left( {{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{{{\mathbf{s}}}_{{k - 1}}}} \right)\int {{{f}_{j}}\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}}} \right){\kern 1pt} {\kern 1pt} w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right){\kern 1pt} {\kern 1pt} d{{{\mathbf{x}}}_{{k - 1}}}} } .$$

(A.5)

Taking into account the fact that ${\text{P}}\left( {{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{{{\mathbf{s}}}_{{k - 1}}}} \right) = \pi _{{ij}}^{a}\pi _{{nm}}^{b}\pi _{{ed}}^{c}$, we rewrite (A.5):

$$w_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right) = \frac{{f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}} \right)}}{{f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}}} \right)}}\sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}\pi _{{nm}}^{b}} } \pi _{{de}}^{c}} \int {{{f}_{j}}\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}}} \right)\,w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right)\,d{{{\mathbf{x}}}_{{k - 1}}}} ,$$

(A.6)

where ${{f}_{j}}\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}}} \right)$ is the conditional PD, determined on the basis of equation (1.3); $w_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right)$ is the posterior distribution of the mixed state vector ${{\left[ {{\mathbf{X}}_{k}^{{\text{T}}},{\mathbf{S}}_{k}^{{\text{T}}}} \right]}^{{\,{\text{T}}}}}$obtained at the previous k – 1^st step in the measurement sequence y₁, y₂, …, ${{{\mathbf{y}}}_{{k - 1}}}$; $f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}} \right)$ is the one-step likelihood function, and $f({{{\mathbf{y}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}})$ is the normalizing factor.

Let us introduce into consideration the extrapolated distribution of the mixed state vector ${{\left[ {{\mathbf{X}}_{k}^{{\text{T}}},{\mathbf{S}}_{k}^{{\text{T}}}} \right]}^{{\,{\text{T}}}}}$

$$\tilde {w}_{{jmd}}^{{}}\left( {{{{\mathbf{x}}}_{k}}} \right) = f({\mathbf{x}}_{k}^{{}},{{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}}) = f({\mathbf{x}}_{k}^{{}},{{{\mathbf{s}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}}).$$

(A.7)

According to (A.7), we rewrite (A.6) as a system of two recurrent equations:

$$\tilde {w}_{{jmd}}^{{}}\left( {{{{\mathbf{x}}}_{k}}} \right) = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}\pi _{{nm}}^{b}} } \pi _{{de}}^{c}} \int {{{f}_{j}}\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}}} \right)w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right)d{{{\mathbf{x}}}_{{k - 1}}}} ,$$

(A.8)

$$w_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right) = \frac{{f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}} \right)}}{{f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}}} \right)}}\tilde {w}_{{jmd}}^{{}}\left( {{{{\mathbf{x}}}_{k}}} \right).$$

(A.9)

Equation (A.7) describes the evolution of the extrapolated distribution ${{\tilde {w}}_{{jmd}}}\left( {{{{\mathbf{x}}}_{k}}} \right)$ of the mixed state vector ${{\left[ {{\mathbf{X}}_{k}^{{\text{T}}},{\mathbf{S}}_{k}^{{\text{T}}}} \right]}^{{\,{\text{T}}}}}$. Using relation (A.9), the extrapolated distribution is refined based on the obtained measurement y_k and the posterior distribution is determined $w_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right)$.

Further transformation of expressions (A.8) and (A.9) can be performed taking into account the properties of conditional distributions. In this case, equation (A.6) can be represented by the method in [19] as the following system of recursive equations:

$${{{{\tilde {P}}}}_{{jmd}}}\left( k \right) = {\text{P}}\left( {{{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}}} \right) = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}{\text{P}}_{{ine}}^{*}\left( {k - 1} \right),$$

(A.10)

$${\text{P}}_{{jmd}}^{*}\left( k \right) = {\text{P}}({{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}}) = \frac{{f({{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}},{\mathbf{y}}_{1}^{{k - 1}})}}{{f({{{\mathbf{y}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}})}}{{{{\tilde {P}}}}_{{jmd}}}\left( k \right),$$

(A.11)

$$\begin{gathered} \tilde {f}_{{jmd}}^{{}}\left( {{{{\mathbf{x}}}_{k}}} \right) = f({{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}},{\mathbf{y}}_{1}^{{k - 1}}) \\ \, = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}\pi _{{nm}}^{b}} } \pi _{{de}}^{c}} P_{{ine}}^{*}(k - 1)\int {{{f}_{j}}\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{{k - 1}}}} \right)w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right)d{{{\mathbf{x}}}_{{k - 1}}}} , \\ \end{gathered} $$

(A.12)

$$f_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right) = f\left( {{{{\mathbf{x}}}_{k}}\,{\text{|}}\,{{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}},{\mathbf{y}}_{1}^{k}} \right) = \frac{{f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}} \right)}}{{f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}},{\mathbf{y}}_{1}^{{k - 1}}} \right)}}{{\tilde {f}}_{{jmd}}}\left( {{{{\mathbf{x}}}_{k}}} \right),$$

(A.13)

$$f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}},{{C}_{k}},{\mathbf{y}}_{1}^{{k - 1}}} \right) = \int {f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}} \right)\tilde {f}_{{jmd}}^{{}}\left( {{{{\mathbf{x}}}_{k}}} \right)d{{{\mathbf{x}}}_{k}}} ,$$

(A.14)

$$f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{\mathbf{y}}_{1}^{{k - 1}}} \right) = \sum\limits_i {\sum\limits_n {\sum\limits_j {\sum\limits_m {\sum\limits_e {\sum\limits_d {\pi _{{ij}}^{a}} } } } } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}f\left( {{{{\mathbf{y}}}_{k}}\,{\text{|}}\,{{{\mathbf{x}}}_{k}},{{B}_{k}},{{C}_{k}},{\mathbf{y}}_{1}^{{k - 1}}} \right){\text{P}}_{{jm}}^{*}\left( k \right),$$

(A.15)

where $\tilde {f}_{{jmd}}^{{}}\left( {{{{\mathbf{x}}}_{k}}} \right)$ is the conditional extrapolated PD of the vector x_k on the condition ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$; $f_{{jmd}}^{*}\left( {{{{\mathbf{x}}}_{k}}} \right)$ is the conditional posterior PD of the vector x_k on the condition ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$; $P_{{jmd}}^{*}\left( k \right)$ is the posterior probability ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$, calculated by integrating (A.6) over the variables x_k, ${{{\mathbf{x}}}_{{k - 1}}}$; ${{\tilde {P}}_{{jmd}}}\left( k \right)$ is the extrapolated probability ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$.

Thus, expression (A.6) corresponds to formula (2.1); expressions (A.7)–(A.8) to formulas (2.2)–(2.3), expressions (A.10)–(A.11) to formulas (2.4) and (2.5).

To derive quasi-optimal filtering algorithms for the DNMP components, we use the assumption that the posterior distribution is normal $w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right)$ at the previous k – 1-th filtering step [15]:

$$\begin{gathered} w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right) = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}}}}}\det \{ {\mathbf{R}}_{{ine}}^{*}\left( {k - 1} \right)\} } }} \\ \, \times \exp \left\{ { - \frac{1}{2}{{{({{{\mathbf{x}}}_{{k - 1}}} - {\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right))}}^{{\text{T}}}}{{{({\mathbf{R}}_{{ine}}^{*}\left( {k - 1} \right))}}^{{ - 1}}}({{{\mathbf{x}}}_{{k - 1}}} - {\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right))} \right\}, \\ \end{gathered} $$

(A.16)

where ${\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) \equiv {\mathbf{M}}\left\{ {{{{\mathbf{X}}}_{{k - 1}}},{{A}_{{k - 1}}} = {{a}_{i}},{{B}_{{k - 1}}} = {{b}_{n}},{{C}_{{k - 1}}} = {{c}_{e}}\,{\text{|}}\,{\mathbf{Y}}_{1}^{{k - 1}}} \right\}$ is the mathematical expectation and ${\mathbf{R}}_{{ine}}^{*}\left( {k - 1} \right) \equiv {\mathbf{cov}}\left\{ {{{{\mathbf{X}}}_{{k - 1}}},{{{\mathbf{X}}}_{{k - 1}}},{{A}_{{k - 1}}} = {{a}_{i}},{{B}_{{k - 1}}} = {{b}_{n}},{{C}_{{k - 1}}} = {{c}_{e}}\,{\text{|}}\,{\mathbf{Y}}_{1}^{{k - 1}}} \right\}$ is the covariance matrix of the posterior distribution $w_{{ine}}^{*}\left( {{{{\mathbf{x}}}_{{k - 1}}}} \right)$, and det{R} is the matrix determinant R.

From (A.16) based on the linear transformation (1.3) of the conditionally Gaussian random variable ${{{\mathbf{X}}}_{{k - 1}}}$ we rewrite (A.12)

$$\begin{gathered} {{{\tilde {f}}}_{{jmd}}}\left( {{{{\mathbf{x}}}_{k}}} \right) = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}}}}}} }}\sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}{\text{P}}_{{in}}^{*}(k - 1)\frac{1}{{\sqrt {\det \left\{ {{{{{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{R} }}}}_{{jmd}}}\left( k \right)} \right\}} }} \\ \, \times \exp \left\{ {\frac{1}{2}{{{\left( {{{{\mathbf{x}}}_{k}} - {{{{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}}}_{{jmd}}}\left( k \right)} \right)}}^{{\text{T}}}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{R} }}_{{jmd}}^{{ - 1}}\left( k \right)\left( {{{{\mathbf{x}}}_{k}} - {{{{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}}}_{{jmd}}}\left( k \right)} \right)} \right\}, \\ \end{gathered} $$

(A.17)

where

$${{{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{X} }}}_{{jmd}}}\left( k \right) = {\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right),$$

$${{{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{R} }}}_{{jmd}}}\left( k \right) = {\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{R}}_{{ine}}^{*}\left( {k - 1} \right){\mathbf{\Phi }}_{{}}^{{\text{T}}}({{A}_{k}} = {{a}_{j}}) + {\mathbf{\Gamma }}({{A}_{k}} = {{a}_{j}}){\mathbf{\Gamma }}_{{}}^{{\text{T}}}({{A}_{k}} = {{a}_{j}}).$$

As a result of the two-moment Gaussian approximation of the extrapolated distribution (2.3), we write

$${{\tilde {f}}_{{jmd}}}\left( {{{{\mathbf{x}}}_{k}}} \right) = \frac{1}{{\sqrt {{{{\left( {2\pi } \right)}}^{{{{n}_{x}}}}}\det \left\{ {{{{{\mathbf{\tilde {R}}}}}_{{jmd}}}\left( k \right)} \right\}} }}\exp \left\{ { - \frac{1}{2}{{{\left( {{{{\mathbf{x}}}_{k}} - {{{{\mathbf{\tilde {X}}}}}_{{jmd}}}\left( k \right)} \right)}}^{{\text{T}}}}{\mathbf{\tilde {R}}}_{{jmd}}^{{ - 1}}\left( k \right)\left( {{{{\mathbf{x}}}_{k}} - {{{{\mathbf{\tilde {X}}}}}_{{jmd}}}\left( k \right)} \right)} \right\},$$

(A.18)

where ${{{\mathbf{\tilde {X}}}}_{{jmd}}}\left( k \right) \equiv {\mathbf{M}}\left\{ {{{{\mathbf{X}}}_{k}},{{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}\,{\text{|}}\,{\mathbf{Y}}_{1}^{{k - 1}}} \right\}$ is the conditionally predictive mathematical expectation at ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$, which, taking into account the weighted sum of conditional distributions, has the form:

$${{{\mathbf{\tilde {X}}}}_{{jmd}}}\left( k \right) = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{{\text{P}}_{{ine}}^{*}\left( {k - 1} \right)}}{{{{{{{\tilde {P}}}}}_{{jmd}}}\left( k \right)}}{\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right),$$

(A.19)

${{{\mathbf{\tilde {R}}}}_{{jmd}}}\left( k \right) \equiv {\mathbf{cov}}\left\{ {{{{\mathbf{X}}}_{k}},{{{\mathbf{X}}}_{k}},{{A}_{k}} = {{a}_{j}},{{B}_{k}} = {{b}_{m}},{{C}_{k}} = {{c}_{d}}\,{\text{|}}\,{\mathbf{Y}}_{1}^{{k - 1}}} \right\}$ is the covariance matrix of prediction errors for ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$, which can be calculated by the formula [14]

$${{{\mathbf{\tilde {R}}}}_{{jmd}}}\left( k \right) = \int\limits_{ - \infty }^\infty {\left( {{{{\mathbf{x}}}_{k}} - {{{{\mathbf{\tilde {X}}}}}_{{jmd}}}\left( k \right)} \right)} {{\left( {{{{\mathbf{x}}}_{k}} - {{{{\mathbf{\tilde {X}}}}}_{{jmd}}}\left( k \right)} \right)}^{{\text{T}}}}{{\tilde {f}}_{{jmd}}}\left( {{{{\mathbf{x}}}_{k}}} \right)d{{{\mathbf{x}}}_{k}}.$$

Using the extrapolation probability density representation ${{\tilde {f}}_{{jmd}}}\left( {{{{\mathbf{x}}}_{k}}} \right)$ as a weighted sum of conditional probabilities, and also taking into account the fact that

$${{{\mathbf{x}}}_{k}} - {{{\mathbf{\tilde {X}}}}_{{jmd}}}\left( k \right) \equiv ({{{\mathbf{x}}}_{k}} - {\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right)) + ({\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {{{\mathbf{\tilde {X}}}}_{{jmd}}}\left( k \right)),$$

it can be shown that the covariance matrix of prediction errors ${{{\mathbf{\tilde {R}}}}_{{jmd}}}\left( k \right)$ has the form

$$\begin{gathered} {{{{\mathbf{\tilde {R}}}}}_{{jmd}}}\left( k \right) = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{P_{{in}}^{*}(k - 1)}}{{{{{\tilde {P}}}_{{jm}}}(k)}} \\ \, \times \left\{ {{\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{R}}_{{ine}}^{*}\left( {k - 1} \right){\mathbf{\Phi }}_{{}}^{{\text{T}}}({{A}_{k}} = {{a}_{j}}) + {\mathbf{\Gamma }}({{A}_{k}} = {{a}_{j}}){\mathbf{\Gamma }}_{{}}^{{\text{T}}}({{A}_{k}} = {{a}_{j}})\mathop {}\limits_{_{{_{{_{{_{{}}}}}}}}} } \right. \\ \, + ({\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {{{{\mathbf{\tilde {X}}}}}_{{jmd}}}\left( k \right))\left. {({\mathbf{\Phi }}({{A}_{k}} = {{a}_{j}}){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {{{{\mathbf{\tilde {X}}}}}_{{jmd}}}\left( k \right))_{{}}^{{\text{T}}}} \right\}. \\ \end{gathered} $$

(A.20)

Based on the linear measurement equation (1.4) from (A.19) and (A.20), it is easy to obtain expressions for conditionally predictive values of the measurement vectors ${\mathbf{\tilde {Y}}}_{{jmd}}^{{(1)}}\left( k \right)$ and ${\mathbf{\tilde {Y}}}_{{jmd}}^{{(2)}}\left( k \right)$ and conditional covariance matrix ${{{\mathbf{\tilde {W}}}}_{{jmd}}}\left( k \right)$ of measurement-prediction errors, respectively:

$${\mathbf{\tilde {Y}}}_{{jmd}}^{{(1)}}\left( k \right) = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{{\text{P}}{{{_{{ine}}^{*}}}_{{}}}(k - 1)}}{{{{{{{\tilde {P}}}}}_{{jmd}}}(k)}}{{{\mathbf{H}}}_{1}}\left( {{{b}_{m}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right),$$

(A.21)

$${\mathbf{\tilde {Y}}}_{{jmd}}^{{(2)}}\left( k \right) = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{{\text{P}}_{{ine}}^{*}(k - 1)}}{{{{{{{\tilde {P}}}}}_{{jmd}}}(k)}}{{{\mathbf{H}}}_{2}}\left( {{{c}_{d}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right),$$

(A.22)

$${{{\mathbf{\tilde {W}}}}_{{jmd}}}\left( k \right) = {\mathbf{H}}{{{\mathbf{\tilde {R}}}}_{{jmd}}}\left( k \right){{{\mathbf{H}}}^{{\text{T}}}} + {\mathbf{V}}\left( {{{b}_{m}},{{c}_{d}}} \right){{{\mathbf{V}}}^{{\text{T}}}}\left( {{{b}_{m}},{{c}_{d}}} \right).$$

(A.23)

Taking into account the fact that ${{{\mathbf{Y}}}_{k}} = {{\left[ {{\mathbf{Y}}_{k}^{{(1){\text{T}}}},{\mathbf{Y}}_{k}^{{(2){\text{T}}}}} \right]}^{{\,{\text{T}}}}}$, the conditional covariance matrix ${{{\mathbf{\tilde {W}}}}_{{jmd}}}\left( k \right)$ of measurement-prediction errors has a block form

$${{{\mathbf{\tilde {W}}}}_{{jmd}}}\left( k \right) = \left[ {\begin{array}{*{20}{c}} {{\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}} \\ {{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \end{array}} \right].$$

(A.24)

Using (A.20) and (A.23), it is easy to obtain expressions for the elements of the block matrix ${{{\mathbf{\tilde {W}}}}_{{jmd}}}\left( k \right)$:

$$\begin{gathered} {\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}} = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{{\text{P}}_{{ine}}^{*}(k - 1)}}{{{{{{{\tilde {P}}}}}_{{jmd}}}(k)}} \\ \, \times \left\{ {{{{\mathbf{H}}}_{1}}\left( {{{b}_{m}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right){{{\mathbf{\Phi }}}^{{\text{T}}}}\left( {{{a}_{j}}} \right){\mathbf{H}}_{1}^{{\text{T}}}\left( {{{b}_{m}}} \right) + {{{\mathbf{V}}}_{1}}\left( {{{b}_{m}}} \right){\mathbf{V}}_{1}^{{\text{T}}}\left( {{{b}_{m}}} \right)\mathop {}\limits_{_{{_{{_{{_{{}}}}}}}}} } \right. \\ \, + \left\lfloor {{{{\mathbf{H}}}_{1}}\left( {{{B}_{k}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(1)}}\left( k \right)} \right\rfloor \times \left. {{{{\left\lfloor {{{{\mathbf{H}}}_{1}}\left( {{{b}_{m}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(1)}}\left( k \right)} \right\rfloor }}^{{\text{T}}}}} \right\}, \\ \end{gathered} $$

(A.25)

$$\begin{gathered} {\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}} = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{{\text{P}}_{{ine}}^{*}(k - 1)}}{{{{{{{\tilde {P}}}}}_{{jmd}}}(k)}}\left\{ {{{{\mathbf{H}}}_{1}}\left( {{{b}_{m}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right){{{\mathbf{\Phi }}}^{{\text{T}}}}\left( {{{a}_{j}}} \right){\mathbf{H}}_{2}^{{\text{T}}}\left( {{{c}_{d}}} \right)\mathop {}\limits_{_{{_{{_{{_{{}}}}}}}}} } \right. \\ \, + \left\lfloor {{{{\mathbf{H}}}_{1}}\left( {{{b}_{m}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(1)}}\left( k \right)} \right\rfloor \times \left. {{{{\left\lfloor {{{{\mathbf{H}}}_{2}}\left( {{{c}_{d}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(2)}}\left( k \right)} \right\rfloor }}^{{\text{T}}}}} \right\}, \\ \end{gathered} $$

(A.26)

$$\begin{gathered} {\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}} = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{P_{{ine}}^{*}(k - 1)}}{{{{{\tilde {P}}}_{{jmd}}}(k)}}\left\{ {{{{\mathbf{H}}}_{2}}\left( {{{C}_{k}}} \right){\mathbf{\Phi }}\left( {{{A}_{k}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right){{{\mathbf{\Phi }}}^{{\text{T}}}}\left( {{{A}_{k}}} \right){\mathbf{H}}_{1}^{{\text{T}}}\left( {{{B}_{k}}} \right)\mathop {}\limits_{_{{_{{_{{}}}}}}} } \right. \\ \, + \left\lfloor {{{{\mathbf{H}}}_{2}}\left( {{{C}_{k}}} \right){\mathbf{\Phi }}\left( {{{A}_{k}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(2)}}\left( k \right)} \right\rfloor \times \left. {{{{\left\lfloor {{{{\mathbf{H}}}_{1}}\left( {{{B}_{k}}} \right){\mathbf{\Phi }}\left( {{{A}_{k}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(1)}}\left( k \right)} \right\rfloor }}^{{\text{T}}}}} \right\}, \\ \end{gathered} $$

(A.27)

$$\begin{gathered} {\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}} = \sum\limits_i {\sum\limits_n {\sum\limits_e {\pi _{{ij}}^{a}} } } \pi _{{nm}}^{b}\pi _{{ed}}^{c}\frac{{{\text{P}}_{{ine}}^{*}(k - 1)}}{{{{{{{\tilde {P}}}}}_{{jmd}}}(k)}} \\ \, \times \left\{ {{{{\mathbf{H}}}_{2}}\left( {{{c}_{d}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right){{{\mathbf{\Phi }}}^{{\text{T}}}}\left( {{{a}_{j}}} \right){\mathbf{H}}_{2}^{{\text{T}}}\left( {{{c}_{d}}} \right) + {{{\mathbf{V}}}_{2}}\left( {{{c}_{d}}} \right){\mathbf{V}}_{2}^{{\text{T}}}\left( {{{c}_{d}}} \right)\mathop {}\limits_{_{{_{{_{{_{{}}}}}}}}} } \right. \\ \, + \left[ {{{{\mathbf{H}}}_{2}}\left( {{{c}_{d}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(2)}}\left( k \right)} \right] \times \left. {{{{\left[ {{{{\mathbf{H}}}_{2}}\left( {{{c}_{d}}} \right){\mathbf{\Phi }}\left( {{{a}_{j}}} \right){\mathbf{X}}_{{ine}}^{*}\left( {k - 1} \right) - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(2)}}\left( k \right)} \right]}}^{{\,{\text{T}}}}}} \right\}. \\ \end{gathered} $$

(A.28)

To determine the matrix of the optimal transmission coefficient of the complex discrete filter [14]

$${{{\mathbf{K}}}_{{jmd}}}(k) = {{{\mathbf{\tilde {R}}}}_{{jmd}}}(k){{{\mathbf{H}}}^{{\text{T}}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{ - 1}}\left( k \right) = \left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{jmd}}^{{(1)}}\left( k \right)} \\ {{\mathbf{K}}_{{jmd}}^{{(2)}}\left( k \right)} \end{array}} \right] = {{{\mathbf{\tilde {R}}}}_{{jmd}}}\left[ {\begin{array}{*{20}{c}} {{\mathbf{H}}_{1}^{{\text{T}}}\left( {{{b}_{k}}} \right)} \\ {{\mathbf{H}}_{2}^{{\text{T}}}\left( {{{c}_{k}}} \right)} \end{array}} \right]{{\left[ {\begin{array}{*{20}{c}} {{\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}} \\ {{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \end{array}} \right]}^{{ - 1}}}$$

(A.29)

we represent the inverse block correlation matrix ${\mathbf{\tilde {W}}}_{{jmd}}^{{ - 1}}$, using the Frobenius formula [21]:

$${{\left[ {\begin{array}{*{20}{c}} {{\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}} \\ {{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \end{array}} \right]}^{{ - 1}}} = \left[ {\begin{array}{*{20}{c}} {{{{{\mathbf{\tilde {D}}}}}^{{ - 1}}}}&{ - {{{{\mathbf{\tilde {D}}}}}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}{{{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \right)}}^{{ - 1}}}} \\ { - {{{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \right)}}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}{{{{\mathbf{\tilde {D}}}}}^{{ - 1}}}}&{{{{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \right)}}^{{ - 1}}} + {{{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \right)}}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}{{{{\mathbf{\tilde {D}}}}}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}{{{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \right)}}^{{ - 1}}}} \end{array}} \right],$$

(A.30)

where ${\mathbf{\tilde {D}}} = {\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}} - {\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}{{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \right)}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}$.

Substituting into (A.29) the values (A.25)–(A.28) and taking into account (A.30), we obtain an expression for the optimal transmission coefficients of the complex discrete filter:

$$\begin{gathered} {\mathbf{K}}_{{jmd}}^{{(1)}}\left( k \right) = {{{{\mathbf{\tilde {R}}}}}_{{jmd}}}\left( k \right){\mathbf{H}}_{1}^{{\text{T}}}\left( {{{b}_{m}}} \right){{({\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}}\left( k \right) - {\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}\left( k \right){{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}}\left( k \right)} \right)}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}\left( k \right))}^{{ - 1}}} \\ \, - {{{{\mathbf{\tilde {R}}}}}_{{jmd}}}\left( k \right){\mathbf{H}}_{2}^{{\text{T}}}\left( {{{c}_{d}}} \right){{({\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}\left( k \right) - {\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}\left( k \right){{\left( {{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}\left( k \right)} \right)}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}\left( k \right))}^{{ - 1}}}, \\ \end{gathered} $$

(A.31)

$$\begin{gathered} {\mathbf{K}}_{{jmd}}^{{(2)}}\left( k \right) = - {{{{\mathbf{\tilde {R}}}}}_{{jmd}}}{\mathbf{H}}_{1}^{{\text{T}}}\left( {{{b}_{m}}} \right){{({\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}} - {\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}{{({\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}})}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}})}^{{ - 1}}}{{({\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}})}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}} \\ \, + {{{{\mathbf{\tilde {R}}}}}_{{jmd}}}{\mathbf{H}}_{2}^{{\text{T}}}\left( {{{c}_{d}}} \right)({{({\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}})}^{{ - 1}}} + {{({\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}})}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}{{({\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}} - {\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}{{({\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}})}^{{ - 1}}}{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}})}^{{ - 1}}}){\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}{{({\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}})}^{{ - 1}}}. \\ \end{gathered} $$

(A.32)

The current vector of the conditional estimate of the continuous component of the DNMP ${\mathbf{X}}_{{jmd}}^{*}\left( k \right)$ at ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$ is determined taking into account (A.19), (A.21), (A.22), (A.31) and (A.32) based on the measurement results ${\mathbf{Y}}_{k}^{{(1)}}$ and ${\mathbf{Y}}_{k}^{{(2)}}$ [15]:

$${\mathbf{X}}_{{jmd}}^{*}\left( k \right) = {{{\mathbf{\tilde {X}}}}_{{jmd}}}\left( k \right) + {\mathbf{K}}_{{jmd}}^{{(1)}}\left( k \right)({\mathbf{Y}}_{k}^{{(1)}} - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(1)}}\left( k \right)) + {\mathbf{K}}_{{jmd}}^{{(2)}}\left( k \right)({\mathbf{Y}}_{k}^{{(2)}} - {\mathbf{\tilde {Y}}}_{{jmd}}^{{(2)}}\left( k \right)).$$

(A.33)

The covariance matrix of errors in estimating the continuous component of DNMP ${\mathbf{R}}_{{jmd}}^{*}\left( k \right)$ on condition ${{A}_{k}} = {{a}_{j}}$, ${{B}_{k}} = {{b}_{m}}$, ${{C}_{k}} = {{c}_{d}}$ is determined taking into account (1.4), (A.20), (A.24), (A.30) [15]:

$${\mathbf{R}}_{{jmd}}^{*}\left( k \right) = {\mathbf{\tilde {R}}}_{{jmd}}^{{}}\left( k \right) - {{\left[ {\begin{array}{*{20}{c}} {{\mathbf{H}}_{1}^{{\text{T}}}\left( {{{b}_{k}}} \right){\mathbf{\tilde {R}}}_{{jmd}}^{{}}} \\ {{\mathbf{H}}_{2}^{{\text{T}}}\left( {{{c}_{k}}} \right){\mathbf{\tilde {R}}}_{{jmd}}^{{}}} \end{array}} \right]}^{{\text{T}}}}{{\left[ {\begin{array}{*{20}{c}} {{\mathbf{\tilde {W}}}_{{jmd}}^{{(11)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(12)}}} \\ {{\mathbf{\tilde {W}}}_{{jmd}}^{{(21)}}}&{{\mathbf{\tilde {W}}}_{{jmd}}^{{(22)}}} \end{array}} \right]}^{{ - 1}}}\left[ {\begin{array}{*{20}{c}} {{\mathbf{H}}_{1}^{{}}\left( {{{b}_{k}}} \right){\mathbf{\tilde {R}}}_{{jmd}}^{{}}} \\ {{\mathbf{H}}_{2}^{{}}\left( {{{c}_{k}}} \right){\mathbf{\tilde {R}}}_{{jmd}}^{{}}} \end{array}} \right].$$

(A.34)