Theory of Control Stochastic Systems with Unsolved Derivatives

Abstract

Various types of stochastic differential systems with unsolved derivatives (SDS USD) arise in problems of analytical modeling and estimation (filtering, extrapolation, etc.) for control stochastic systems, when it is possible to neglect higher-order time derivatives. Methodological and algorithmic support of analytical modeling, filtering, and extrapolation for SDS USD is developed. The methodology is based on the reduction of SDS USD to SDS by means of linear and nonlinear regression models. Two examples that are illustrating stochastic aspects of methodology are presented. Special attention is paid to SDS USD with multiplicative (parametric) noises.

1. Introduction

Approximate methods of analytical modeling (MAM) of the wideband stochastic processes (StP) in stochastic differential systems with unsolved derivatives (SDS USD) based on normal approximate method (NAM), orthogonal expansions method, and quasi moment methods are developed in [1, 2]. For stochastic integrodifferential systems with unsolved derivatives reducible to SDS corresponding equations for MAM are given in [3, 4]. In [3, 4], problems of mean square (m.s.) synthesis of normal (Gaussian) estimators (filters, extrapolators, etc.) were firstly stated and solved in [1, 2, 3, 4]. Results presented in [1, 2, 3, 4] are valid for smooth (in m.s. sense) functions in SDS USD. For unsmooth functions in SDS USD theory of normal filtering and extrapolation is developed in [5].

Let us present an overview and generalization of [1, 2, 3, 4, 5] for linear and nonlinear regression models. Section 2 is developed to normal analytical modeling algorithms. Normal linear filtering and extrapolation algorithms are given in Sections 3 and 4. Linear modeling and estimation algorithms for SDS USD with multiplicated (parametric) noises are presented in Section 5. Normal nonlinear algorithms for filtering and extrapolation are described in Section 6. Section 7 contains two illustrative examples. In Section 8, main conclusions and some generalizations are given.

2. Normal modeling

Different types of SDS USD arise in problems of analytical modeling and estimator design for stochastic nonlinear dynamical systems when it is possible to neglect higher-order time derivatives [1, 2, 3].

First-order SDS USD is described by the following scalar equation:

$$\phi =\phi \left(t,X_t,{\dot{X}}_t,U_t\right)=0,$$E1

where $X_t$ and ${\dot{X}}_t$ are scalar state variable and its time derivative; $U_t$ is noise vector StP $\left(dim{U}_t=n^U\right)$; nonlinear function $\phi$ admits regression approximation [6, 7, 8].

For vector SDS USD, we have the following vector equation:

$$\overline{\phi }=\overline{\phi }\left(t,X_t,{\overline{X}}_t,U_t\right)=0.$$E2

Here ${\overline{X}}_t$ being vector of derivatives till $l$ order

$${\overline{X}}_t={\left[{\dot{X}}_t^T\dots X_t^{\left(l-1\right)T}\right]}^T;$$E3

$U_t$ being autocorrelated noise vector defined by linear vector equation:

$${\dot{U}}_t=a_{0t}^U+a_{1t}{U}_t+b_t^U{V}_t,$$E4

where $dim{X}_t=n^X;dim{U}_t=n^U$; $V_t$ is white noise, $dim{V}_t=n^V;$ $dim{a}_{0t}^U=n^U\times 1;$ $dim{a}_t^U=n^U\times n^U;$ $dim{b}_t^U=n^U\times n^V.$ Further, we consider the Wiener white noise $W_{0t}$ with matrix intensity $v_0=v_0\left(t\right)$ and the mixed Wiener-Poisson white noise [9, 10, 11, 12, 13]:

$$V_t={\dot{W}}_t,W_t=W_{0t}+\underset{R_0^q}{\int }c\left(\rho \right)P^0\left(t,\mathit{d\rho }\right),$$E5

$$v_t=v_{0t}^W+\underset{R_0^q}{\int }c\left(\rho \right){\left[c\left(\rho \right)\right]}^T{v}_P\left(t,\rho \right)\mathit{d\rho }.$$E6

Here, $dimc\left(\rho \right)=dim{W}_{0t}=n^V$; stochastic Ito integrals are taken in $R_0^q$ ($R_0^q$ with pricked origin).

As it is known [6, 7, 8], a deterministic model for real StP defined by $Y=\phi \left(Z\right)$ at $Z={\left[X^T{\overline{X}}^T{U}^T\right]}^T$ in (2) is given by the formula

$$\widehat{y}\left(z\right)=E\left[Y|z\right],\widehat{y}\left(z\right)\in \mathrm{\Psi }$$E7

at accuracy criterion

$$\epsilon \left(z\right)=\sum _{p=1}^{n^Y}E\left[{\left|,{\widehat{y}}_p-Y_p\right|}^2|z\right],p=\left[1,\dots ,n^Y\right].$$E8

Class of functions $\psi \in \mathrm{\Psi }$ represents linear functional space satisfying the following necessary and sufficient conditions:

$$\mathrm{tr}E\left\{[||\widehat{y}\left(z\right)-Y|]\psi {\left(z\right)}^T\right\}=0.$$E9

For linear shifted and unshifted regression models, we have two known models:

$$\widehat{y}\left(z\right)=g^{B}z,g^B={\mathrm{\Gamma }}_{\mathit{yz}}{\mathrm{\Gamma }}_z^{-1}$$E10

(Booton [6, 7, 8]),

$$\widehat{y}\left(z\right)=a+g^K{z}^0,g^K=K_{\mathit{yz}}{K}_z^{-1},a=E^Y-g^K{E}^z$$E11

(Kazakov [6, 7, 8]),

where $E^z,{\mathrm{\Gamma }}_z,K_z$ being first and second moments for given one-dimensional distribution.

For Eq. (2), linear regression model takes the Booton form

$$\widehat{\phi }={\widehat{\phi }}_0+k_1^{\phi }{X}_t+k_2^{\phi }{\overline{X}}_t+k_3^{\phi }{U}_t=0,$$E12

where ${\widehat{\phi }}_0$, $k_{1,2,3}^{\phi }$ being regressors depending on $\phi$ and joint distribution of StP $X_t,{\overline{X}}_t,U_t$. After Eq. (12) differentiation till the $\left(l-1\right)$ order, we get the following set of $\left(l-1\right)$ equations:

$${\dot{\widehat{\phi }}}_t=0,\dots ,{\widehat{\phi }}_t^{\left(l-1\right)}=0.$$E13

At algebraic solvability condition of linear Eqs. (12) and (13), we reduce SDS USD to SDS of the following form:

$${\dot{X}}_t=A_0+A_1{X}_t+A_2{U}_t,$$E14

where $A_0,A_1,A_2$ are expressed in terms ${\widehat{\phi }}_0$, $k_{1,2,3}^{\phi }$ $\left(det{\left(k_2^{\phi }\right)}^{-1}\ne 0\right)$ and indirectly depends on statistical characteristics of $X_t$, its derivatives and noise $U_t$. For combined vector $\left[X_t^T{U}_t^T\right]={\tilde{Y}}_t$ we have equation:

$${\dot{\tilde{Y}}}_t=B_0+B_1{\tilde{Y}}_t+B_2{V}_t,Y_{t0}=Y_0,$$E15

Its one and second probabilistic moments satisfy the following equations [12, 13, 14]:

$${\dot{\tilde{Y}}}_t=B_0+B_1{\tilde{Y}}_t+B_2{V}_t,Y_{t0}=Y_0,$$E16

$${\dot{E}}_t^{\tilde{Y}}=B_0+B_1{E}_t^{\tilde{Y}},E_{t0}^{\tilde{Y}}=E_0^{\tilde{Y}},$$E17

$${\dot{K}}_t^{\tilde{Y}}=B_1{K}_t^{\tilde{Y}}+K_t^{\tilde{Y}}{B}_1^T+B_2{\mathit{vB}}_2^T,{\dot{K}}_{t0}^{\tilde{Y}}=K_0^{\tilde{Y}},$$E18

$$\frac{\partial K^{\tilde{Y}}\left(t_1,t_2\right)}{\partial t_2}=K^{\tilde{Y}}\left(t_1,t_2\right)B_{1t_2}^T,K^{\tilde{Y}}\left(t_1,t_1\right)=K_{t_1}^{\tilde{Y}}$$E19

where $E_t^{\tilde{Y}}=E\left[{\tilde{Y}}_t\right]$, $K_t^{\tilde{Y}}=E\left[\left({\tilde{Y}}_t-{\tilde{E}}_t^Y\right){\left({\tilde{Y}}_t-{\tilde{E}}_t^Y\right)}^T\right]$, $\left(t_1>t_2\right)$. So, we get two proposals.

Proposal 1. Let vector non-Gaussian SDS USD (2) satisfy conditions:

1. vector functions $\phi$ in Eq. (2) admit m.s. regression of linear class $\mathrm{\Psi }$;

2. linear Eqs. (12) and (13) are solvable regards all derivatives till $\left(l-1\right)$ order.

Then SDS USD may be reduced to parametrized SDE. First and second moments of joint vector ${\tilde{Y}}_t={\left[X_t^T{U}_t^T\right]}^T$ satisfy Eqs. (16)–(19).

Proposal 2. For normal joint distribution $\mathcal{N}=\mathcal{N}\left(E_t^Y,K_t^Y\right)$ of vector variables in Eqs. (16)–(19) it is necessary in equations of Theorem 1 to put

$$\begin{array}{l}{W}_t=W_{0t},E_t^{\tilde{Y}}=E_{\mathcal{N}}^{\tilde{Y}},K_t^{\tilde{Y}}=E_{\mathcal{N}}^{\tilde{Y}}\left[\left({\tilde{Y}}_t-E_{\mathcal{N}}^{\tilde{Y}}\right){\left({\tilde{Y}}_t-E_{\mathcal{N}}^{\tilde{Y}}\right)}^T\right],\\ K^{\tilde{Y}}\left(t_1,t_2\right)=E_{\mathcal{N}}\left[\left({\tilde{Y}}_{t_1}-E_{\mathcal{N}{t}_1}^{\tilde{Y}}\right){\left({\tilde{Y}}_{t_2}-E_{\mathcal{N}{t}_2}^{\tilde{Y}}\right)}^T\right].\end{array}$$E20

For Eq. (2) using Kazakov form

$$\overline{\phi }={\overline{\phi }}_0+{\overline{\phi }}^0=0$$E21

where

$${\overline{\phi }}^0=k_1^{\overline{\phi }}{X}_t^0+k_2^{\overline{\phi }}{\overline{X}}_t^0+k_3^{\overline{\phi }}{U}_t^0,$$E22

we have two sets of equations for mathematical expectations and centered variables:

$${\dot{\overline{\phi }}}_0=0,\dots ,{\overline{\phi }}_0^{\left(l-1\right)}=0$$E23

$${\dot{\overline{\phi }}}_0=0,\dots ,{\overline{\phi }}_0^{0\left(l-1\right)}=0.$$E24

So, we reduce SDS USD to two sets of equations for $E_t^X$ and $X_t^0=X_t-E_t^X$

$${\dot{E}}_t^X={\overline{A}}_0+{\overline{A}}_1{E}_t^X+{\overline{A}}_2{E}_t^U,$$E25

$${\dot{X}}_t^0=A_1{X}_t^0+A_2{U}_t^0.$$E26

For the composed vector ${\overline{Y}}_t^0={\left[X_t^{0T}{U}_t^{0T}\right]}^T$ its probabilistic one and second moments satisfy the following equations:

$$E_t^{\overline{Y}}={\overline{B}}_0+{\overline{B}}_1{E}_t^{\overline{Y}},{\overline{Y}}_{t0}={\overline{Y}}_0,$$E27

$${\dot{K}}^{\overline{Y}}={\overline{B}}_1{K}_t^{\overline{Y}}+K^{\overline{Y}}{\overline{B}}_1^T+{\overline{B}}_{2}v{\overline{B}}_2^T,K_{t0}^{\overline{Y}}=K_0^{\overline{Y}},$$E28

$$\frac{\partial K^{\overline{Y}}\left(t_1,t_2\right)}{\partial t_2}=K^{\overline{Y}}\left(t_1,t_2\right){\overline{B}}_{1t_2}^T,K^{\overline{Y}}\left(t_1,t_1\right)=K_{t_1}^{\overline{Y}},t_2>t_1.$$E29

Here $v=v_0$ being defined by Eq. (6).

So for Kazakov regression, Eqs. (21)–(24) are the basis of Proposal 3.

The regression $E^y\left(z\right)$ and its m.s. estimator $\widehat{y}\left(z\right)$ represent deterministic regression model. So to obtain a stochastic regression model, it is sufficient to represent $Y$ in the form $Y=E^y\left(z\right)+Y\text{'}$ or $Y=\widehat{y}\left(z\right)+Y\text{'}\text{'}$, where $Y\text{'},Y\text{'}\text{'}$ being some random variables. For finding a deterministic linear regression model, it is sufficient to know the mathematical expectations $E^z,E^y$ and covariance matrices $K^z,K^{\mathit{yz}}$. In the case of a stochastic linear regression model, it is necessary to know the distribution of $Y$ for any $z$ or at list its regression $\widehat{y}\left(z\right)$ and covariance matrix $K^y\left(z\right)$ (coinciding with the covariance matrices $K^{Y\text{'}}\left(z\right)$ or $K^{Y\text{'}\text{'}}\left(z\right)$). A more general problem of the best m.s. approximation of the regression by a finite linear combination of given functions ${\chi }_1\left(z\right),\dots ,{\chi }_N\left(z\right)$ is reduced to the problem of the best approximation to the regression, as any linear combination of the functions ${\chi }_1\left(z\right),\dots ,{\chi }_N\left(z\right)$ represents a linear function of variables $z_1={\chi }_1\left(z\right),\dots ,z_N\left(z\right)={\chi }_N\left(z\right)$. Corresponding models based on m.s. optimal regression are given in [7].

In the general case, we have the following vector equation:

$${\dot{Z}}_t=a^z\left(Z_t,t\right)+b^z\left(Z_t,t\right)V_t,$$E30

where $V_t$ being defined by Eqs. (5) and (6). Functions $a^z=a^z\left(Z_t,t\right)$ and $b^z=b^z\left(Z_t,t\right)$ are composed on the basis of Eq. (2) after nonlinear regression approximation ${\widehat{\phi }}_t={\sum }_j{c}_j\chi \left(Z_t\right)$ and Eq. (13).

According to normal approximation method (NAM), we have for Eq. (30) the following equations for normal modeling [9, 10, 11, 12]:

$${\dot{E}}_t^z=F_1\left(E_t^z,K_t^z,t\right),$$E31

$${\dot{K}}_t^z=F_2\left(E_t^z,K_t^z,t\right),$$E32

$$\frac{\partial K_t^z\left(t_1,t_2\right)}{\partial t_2}=F_3\left(E_t^z,K^z\left(t_2\right),K^z\left(t_1,t_2\right),t_1,t_2\right).$$E33

Here

$$F_1\left(E_t^z,K_t^z,t\right)=E_N{a}^z\left(Z_t,t\right),$$E34

$$F_2\left(E_t^z,K_t^z,t\right)=F_{21}\left(E_t^z,K_t^z,t\right)+F_{21}{\left(E_t^z,K_t^z,t\right)}^T+F_{22}\left(E_t^z,K_t^z,t\right),$$E35

$$F_{21}\left(E_t^z,K_t^z,t\right)=E_N{a}^z\left(Z_t,t\right)\left(Z_t-E_t^z\right),$$E36

$$F_{22}\left(E_t^z,K_t^z,t\right)=E_N{b}^z\left(Z_t,t\right){\mathit{vb}}^z{\left(Z_t,t\right)}^T,$$E37

$$\begin{array}{l}{F}_3\left(E_{t_2}^z,K_{t_2}^z,K^z\left(t_1,t_2\right),t\right)=K^z\left(t_1,t_2\right){\left(K_{t_2}^z\right)}^{-1}{F}_{21}{\left(E_{t_2}^z,K_{t_2}^z,t_2\right)}^T,\\ E_{t_0}^z=E^Z\left(t_0\right),K_{t_0}^z=K^Z\left(t_0\right),K^z\left(t_1,t_2\right)=K_{t_1}^z\end{array}$$E38

where $E_N$ being symbol of normal mathematical expectation.

3. Normal linear filtering

In filtering SDS USD problems, we use two types of equations: reduced SDE USD for vector state variables $X_t$ and equation for vector observation variables $Y_t$ and ${\dot{Y}}_t\equiv Z_t$.

Consider SDS USD Eq. (2) reducible to SDE Eq. (3.9) at conditions of Theorem 1. We introduce new variables putting $X_t\equiv {\tilde{Y}}_t$,

$${\dot{X}}_t=A_{0t}+A_{1t}{X}_t+A_{2t}{V}_{1t}.$$E39

Let the observation vector variable $Y_t$ satisfy the following linear equations:

$$Z_t={\dot{Y}}_t=B_{0t}+B_{1t}{X}_t+B_{2t}{V}_{2t}.$$E40

where $V_{1t}$ and $V_{2t}$ are normal white noises with matrix $v_{1t}=v_{01}$ and $v_{2t}=v_{02}$ intensities.

Equations of Kalman-Bucy filter in case of Eqs. (39) and (40) for the Gaussian white noises are as follows [12, 13, 14]:

$${\dot{\widehat{X}}}_t=A_0+A_1{\widehat{X}}_t+{\beta }_t\left[Z_t-\left(B_0+B_1{\widehat{X}}_t\right)\right].$$E41

$${\beta }_t=R_t{B}_{1t}^T{v}_{2t}^{-1},det{v}_{2t}\ne 0.$$E42

$${\dot{R}}_t=A_{1t}{R}_t+R_t{A}_{1t}^T+v_{1t}-{\beta }_t{v}_{2t}{\beta }_t^T$$E43

at corresponding initial conditions. $R_t$ being m.s. covariance matrix error, ${\beta }_t$ being gain coefficient. So, we have the following result.

Proposal 4. Let:

1. USD are reducible to SDS according to Proposal 2 or Proposal 3;

2. observations are performed according to Eq. (40).

Then equations for m.s. normal filtering have the generalized Kalman-Bucy filter of the form (41)–(43).

4. Normal linear extrapolation

Using equations of linear m.s. extrapolation for time interval $\mathrm{\Delta }$ [12, 13, 14] we get the following equations for the generalized Kalman–Bucy extrapolator:

$${\dot{\widehat{X}}}_{t+\mathrm{\Delta }t\mid t}=A_1{\widehat{X}}_{t+\mathrm{\Delta }t\mid t}\left(\mathrm{\Delta }>0\right)$$E44

with initial condition

$${\left[{\widehat{X}}_{t+\mathrm{\Delta }t\mid t}\right]}_{\mathrm{\Delta }=0}={\widehat{X}}_t.$$E45

For the initial time moment $t$ and for the final time moment $t+\mathrm{\Delta }$ according to Eq. (44), we get

$$X_{t+\mathrm{\Delta }t\mid t}=u\left(t+\mathrm{\Delta },t\right)X_t+\underset{t}{\overset{t+\mathrm{\Delta }}{\int }}u\left(t+\mathrm{\Delta },\tau \right)a_0\left(\tau \right)\mathit{d\tau }+\underset{t}{\overset{t+\mathrm{\Delta }}{\int }}u\left(t+\mathrm{\Delta },\tau \right)\psi \left(\tau \right)\mathit{dW}\left(\tau \right).$$E46

where $u\left(t,\tau \right)$ being the fundamental solution of equation ${\dot{u}}_t=A_{1t}{u}_t$ at condition $u\left(t,t\right)=I$. For conditional mathematical expectation relatively $Y_{t_0}^t$ in Eq. (46), we get m.s. estimate future state $X_{t+\mathrm{\Delta }}$

$${\widehat{X}}_{t+\mathrm{\Delta }t\mid t}=\mathrm{E}\left[X_{t+\mathrm{\Delta }\mid t}|Y_{t_0}^t\right]=u\left(t+\mathrm{\Delta },t\right){\widehat{X}}_{t\mid t}+\underset{t}{\overset{t+\mathrm{\Delta }}{\int }}u\left(t+\mathrm{\Delta },\tau \right)a_0\left(\tau \right)\mathit{d\tau }.$$E47

In this case, error covariance matrix $R_{t+\mathrm{\Delta }t\mid t}$ satisfies the following equation:

$${\dot{R}}_{t+\mathrm{\Delta }t\mid t}=a_1{R}_{t+\mathrm{\Delta }\mid t}+R_{t+\mathrm{\Delta }\mid t}{a}_1^T+\psi v_0{\psi }^T.$$E48

At initial condition

$${\left[R_{t+\mathrm{\Delta }\mid t}\right]}_{\mathrm{\Delta }=0}=R_t.$$E49

Hence, the error matrix $R_t$ is known from Proposal 4. So, we have the following proposition.

Proposal 5. At conditions of Proposal 4 m.s. normal extrapolation ${\widehat{X}}_{t+\mathrm{\Delta }t\mid t}$ is defined by Eqs. (47)–(49).

This extrapolator presents a sequel connection m.s. filter with gain $u\left(t+\mathrm{\Delta },t\right)$, summator $u\left(t+\mathrm{\Delta },t\right){\widehat{X}}_{t\mid t}$ and integral term ${\int }_t^{t+\mathrm{\Delta }}u\left(t+\mathrm{\Delta },\tau \right)A_{0\tau }\mathit{d\tau }$. The accuracy of extrapolation is estimated according to Eqs. (48) and (49).

5. Linear modeling and estimation in SDS USD with multiplicated noises

Let us consider vector Eqs. (2)–(6) for the multiplicative Gaussian noises:

$$\phi =\phi \left({\dot{X}}_t,X_t,V_t\right)={\phi }_1\left({\dot{X}}_t,t\right)+\left[{\phi }_{20}\left(t\right)\sum _{h=1}^{n^X}{\phi }_{2h}\left(t\right)X_h\right]V_t=0.$$E50

Here, $dim{X}_t=dim{\dot{X}}_t=n^X$, $dim\phi =n^X$, ${\phi }_1$ being nonlinear vector function of vector argument ${\dot{X}}_t$ admitting linear regression

$${\phi }_1\left({\dot{X}}_t,t\right)\approx {\phi }_{11}{\dot{X}}_t,{\phi }_{11}={\phi }_{11}\left(E_t^{\dot{X}},K_t^{\dot{X}},t\right).$$E51

Here, ${\phi }_{11}$ being matrix of regressors; $V_{1t}$ being vector Gaussian white noise, $dim{V}_t=n^V$ with matrix intensity $v=v_0\left(t\right)$. In this case, Eqs. (50) and (51) at condition $det{\phi }_{11}\ne 0$ may be resolved relatively ${\dot{X}}_t$

$${\dot{X}}_t=B_0+B_1{X}_t+\left(B_2+\sum _{r=1}^{n^X}{B}_{3r}{X}_{\mathit{rt}}\right)V_t,$$E52

where $B_0,B_1,B_2,B_{3r}$ depend upon regressors ${\phi }_{11}$. Using [9, 10, 11, 12], we get equations for mathematical expectations. $E_t^X$, covariance matrix $K_t^X$, and matrix of covariance functions $K^X\left(t_1,t_2\right)$:

$${\dot{E}}_t^X=B_0+B_1{E}_t^X,E_{t0}^X=E_0^X,$$E53

$$\begin{array}{l}{\dot{K}}_t^X=B_1{K}_t^X+K_t^X{B}_1^T+B_2{v}_0{B}_2^T+\sum _{r=1}^{n^X}\left(B_{3r}{v}_0{B}_2^T+B_2{v}_0{B}_{3r}^T\right)\\ E_{\mathit{rt}}^X\sum \sum _{r,s=1}^{n^X}{B}_{3r}{v}_0{B}_{3s}^T\left(E_{\mathit{rt}}^X{E}_{\mathit{st}}^X+K_{\mathit{rst}}^X\right),K_{t0}^X=K_0^X,\end{array}$$E54

$$\frac{\partial K^X\left(t_1,t_2\right)}{\partial t_2}=K^X\left(t_1,t_2\right)B_{1t_2}^T,K^X\left(t_1,t_1\right)=K_{t_1}^X,t_2>t_1.$$E55

Here $K_t^X=\left[K_{\mathit{rst}}^X\right]$; $K^X\left(t_1,t_2\right)=\left[K_{\mathit{rs}}^X\left(t_1,t_2\right)\right]$. So for MAM in nonstationary regimes, we have Eqs. (54) and (55) Proposal 6. In stationary case Eqs. (54) and (55) we get the following finite set of equations for $E_{\ast }$ and $K_{\ast }$ (Proposal 7):

$$B_0^{\ast }+B_1^{\ast }{E}_{\ast }^X=0,$$E56

$$\begin{array}{ll}{B}_1^{\ast }{K}_{\ast }^X& +K_{\ast }^X{B}_1^{\ast }+B_2^{\ast }{v}_0^{\ast }{B}_2^{\ast T}+\sum _{r=1}^{n^X}\left(B_{3r}^{\ast }{v}_0^{\ast }{B}_2^{\ast T}+B_2^{\ast }{v}_0^{\ast }{B}_{3r}^{\ast T}\right)E_{\ast }^X\\ & +\sum \sum _{r,s=1}^{n^X}{B}_{3r}^{\ast }{v}_0^{\ast }{B}_{3s}^{\ast T}\left(E_{r\ast }^X{E}_{s\ast }^X+K_{\mathit{rs}\ast }^X\right)=0\end{array}$$E57

and ordinary differential equation for $k^X\left(\tau \right)$, $\left(\tau =t_2-t_1\right)$:

$$\frac{{\mathit{dk}}^X\left(\tau \right)}{\mathit{d\tau }}=B_1^{\ast }{k}^X\left(\tau \right),k^X\left(0\right)=K_{\ast }.$$E58

Applying linear theory Pugachev (conditionally m.s. optimal) filtering [9, 10, 11, 12] to equations

$${\dot{X}}_t=A_0+A_1{X}_t+\left(A_2+\sum _{r=1}^{n^X}{A}_{3r}{X}_{\mathit{rt}}\right)V_t,$$E59

$$Z_t={\dot{Y}}_t=B_0+B_1{X}_t+B_2{V}_t,$$E60

We get the following normal filtering equations:

$${\dot{\widehat{X}}}_t=A_0+A_1{\widehat{X}}_t+{\beta }_t\left[Z_t-\left(B_{0t}+B_1{\widehat{X}}_t\right)\right],$$E61

$${\beta }_t=\left[R_t{B}_1+\left(A_2+\sum _{r=1}^{n^X+n^Y}{A}_{3r}{E}_r^X\right)v_0{B}_2\right]{\kappa }_{11}^{-1},$$E62

$$\begin{array}{l}{\dot{R}}_t=A_1{R}_t+R_t{A}_1^T-\left[R_t{B}_1^T+\left(A_2+\sum _{r=1}^{n^X+n^Y}{A}_{3r}{E}_r^X\right)v_0{B}_2\right]{\kappa }_{11}^{-1}\times \\ \times \left[B_1+B_2{v}_0\left(A_2^T+\sum _{r=1}^{n^X+n^Y}{A}_{3r}^T{E}_r^X\right)\right]+\left(A_2+\sum _{r=1}^{n^X+n^Y}{A}_{3r}{E}_r^X\right)v_0\left(A_2^T+\sum _{r=1}^{n^X+n^Y}{A}_{3r}^T{E}_r^X\right)\\ +\underset{r,s=1}{\overset{n^X+n^Y}{\sum \sum }}{A}_{3r}{v}_0{A}_{3s}^T{K}_{\mathit{rs}}.\end{array}$$E63

Here

$${\kappa }_{11}=B_2{v}_0{B}_0^T,{\kappa }_{22}=B_2{v}_0{B}_2^T.$$E64

For calculating (62) we need to find mathematical expectation $E_t^Q,$ covariance matrix $K_t^Q$ of combined vector $Q_t={\left[X_1,\dots ,X_{n^X},Y_1,\dots ,Y_{n^Y}\right]}^T$ and error ${\tilde{X}}_t,{\tilde{X}}_t={\tilde{X}}_t-X_t$ covariance matrix $R_t$ using equations

$${\dot{E}}_t^Q=a^Q{E}_t^Q+a_0^Q,$$E65

$$\begin{array}{l}{\dot{K}}_t^Q=a^Q{K}_t^Q+K_t^Q{\left(a_0^Q\right)}^T+c^Q{v}_0{\left(c^Q\right)}^T+\sum _{r=1}^{n^X+n^Y}\left[c^Q{v}_0{\left(c_r^Q\right)}^T+c_r^Q{v}_0{\left(c_0^Q\right)}^T\right]E_r^Q+\\ +\sum \sum _{r,s=1}^{n^X+n^Y}{c}_r^Q{v}_0{\left(c_s^Q\right)}^T\left(E_r^Q{E}_s^Q+K_{\mathit{rs}}^Q\right),\end{array}$$E66

where

$$a^Q=\left[\begin{array}{cc}0& B_1\\ 0& A_1\end{array}\right],a_0^Q=\left[\begin{array}{c}{B}_0\\ A_0\end{array}\right],{с}_r^Q=\left[\begin{array}{c}{B}_2\\ A_{1r}\end{array}\right]\left(r=0,1,\dots ,n^X+n^Y\right).$$E67

So, Eqs. (61)–(67) define linear Pugachev filter for SDS USD with multiplicative noises reduced to SDS (59) and (60) (Proposal 8).

At last following [9, 10, 11, 12] let us consider linear Pugachev extrapolator for reduced SDS USD. Taking into account equations

$${\dot{X}}_t=A_0+A_1{X}_t+\left(A_2+\sum _{r=1}^{n^X}{A}_{3,n^Y+r}{X}_{\mathit{rt}}\right)V_1,$$E68

$$Z_t={\dot{Y}}_t=B_0+B_1{X}_t+B_2{V}_2$$E69

($V_{1,2}$ being independent normal white noises with $v_{1,2}$ intensities) and the corresponding result (Section 5) we come to the following equation:

$${\dot{\widehat{X}}}_t=A_0\left(t+\mathrm{\Delta }\right)+A_1\left(t+\mathrm{\Delta }\right){\widehat{X}}_t+{\beta }_t\left[Z_t-\left(B_0+B_1{\epsilon }_t^{-1}{\widehat{X}}_t-B_1{\epsilon }_t^{-1}{h}_t\right)\right].$$E70

Here ${\epsilon }_t=u\left(t+\mathrm{\Delta },t\right)$, $u\left(s,t\right)$ being fundamental solution of equation $\mathit{du}/\mathit{ds}=A_1\left(s\right)$,

$$h_t=h\left(t\right)=\underset{t}{\overset{t+\mathrm{\Delta }}{\int }}u\left(t+\mathrm{\Delta },\tau \right)A_0\left(\tau \right)\mathit{d\tau }.$$E71

Accuracy of linear Pugachev extrapolator (70) is performed by integration of the following equation:

$$\begin{array}{l}{\dot{R}}_t=A_1\left(t+\mathrm{\Delta }\right)R_t+R_t{A}_1{\left(t+\mathrm{\Delta }\right)}^T-{\beta }_t\left(B_2{v}_1{B}_2^T\right){\beta }_t^T+[\begin{array}{c}\\ \end{array}{A}_2\left(t+\mathrm{\Delta }\right)+\\ +\sum _{r=n^Y+1}^{n^X+n^Y}{A}_{3r}\left(t+\mathrm{\Delta }\right)E_r\left(t+\mathrm{\Delta }\right)\left]\begin{array}{c}\\ \end{array}{v}_2\left(t+\mathrm{\Delta }\right)\right[\begin{array}{c}\\ \end{array}{A}_2{\left(t+\mathrm{\Delta }\right)}^T+\sum _{r=n^Y+1}^{n^X+n^Y}{A}_{3r}{\left(t+\mathrm{\Delta }\right)}^T{E}_r\left(t+\mathrm{\Delta }\right)]\begin{array}{c}\\ \end{array}+\\ +\sum \sum _{r,s=n^Y+1}^{n^X+n^Y}{A}_{3r}\left(t+\mathrm{\Delta }\right)v_2\left(t+\mathrm{\Delta }\right)A_{3r}^T{\left(t+\mathrm{\Delta }\right)}^T{K}_{\mathit{rs}}.\end{array}$$E72

Equations (70)–(72) define normal linear Pugachev extrapolator for SDS USD reduced to SDS (Proposal 9).

6. Normal nonlinear filtering and extrapolation

Let us consider SDS (2) reducible to SDS and fully observable measuring system described by the following equations:

$${\dot{X}}_t=a\left(X_t,Y_t,\alpha ,t\right)+b\left(X_t,Y_t,\alpha ,t\right)V_0,$$E73

$$Z_t={\dot{Y}}_t=a_1\left(X_t,Y_t,t\right)+b_1\left(X_t,Y_t,t\right)V_0.$$E74

Here, $a,a_1,b,b_1$ being known functions of mentioned variable; $\alpha$ being vector of parameters in Eq. (73); $V_0$ being normal white noise with intensity matrix $v_0=v_0\left(t\right)$.

Using the theory of normal nonlinear suboptimal filtering [10, 11, 12], we get the following equations for ${\widehat{X}}_t$ and $R_t$:

$${\dot{\widehat{X}}}_t=f\left({\widehat{X}}_t,Y_t,R_t,t\right)\mathit{dt}+h\left({\widehat{X}}_t,Y_t,R_t,t\right)\mathit{dt}\left[{\mathit{dY}}_t-f^{\left(1\right)}\left({\widehat{X}}_t,Y_t,R_t,t\right)\mathit{dt}\right],$$E75

$$\begin{array}{l}{\dot{R}}_t=\left\{f^{\left(2\right)}\left({\widehat{X}}_t,Y_t,R_t,t\right)-h\left({\widehat{X}}_t,Y_t,R_t,t\right)b_1{\nu }_0{b}_1^T\left(Y_t,t\right)h{\left({\widehat{X}}_t,Y_t,R_t,t\right)}^T\right\}\mathit{dt}+\\ +\sum _{r=1}^{n_y}{\rho }_r\left({\widehat{X}}_t,Y_t,R_t,t\right)\left[{\mathit{dY}}_r-f_r^{\left(1\right)}\left({\widehat{X}}_t,Y_t,R_t,t\right)\mathit{dt}\right].\end{array}$$E76

Here

$$f\left({\widehat{X}}_t,Y_t,R_t,t\right)={\left[{\left(2\pi \right)}^n|R_t|\right]}^{-1/2}\underset{-\infty }{\overset{\infty }{\int }}a\left(Y_t,x,t\right)exp\left\{-\left(x^T-{\widehat{X}}_t^T\right)R_t^{-1}\left(x-{\widehat{X}}_t\right)/2\right\}\mathit{dx},$$E77

$$\begin{array}{l}{f}^{\left(1\right)}\left({\widehat{X}}_t,Y_t,R_t,t\right)=\left\{f_r^{\left(1\right)}\left({\widehat{X}}_t,Y_t,R_t,t\right)\right\}\\ =[{\left(2\pi \right)}^{n_x}\mid R_t\mid \}{}^{-1/2}\underset{-\infty }{\overset{\infty }{\int }}{a}_1\left(Y_t,x,t\right)exp\left\{-\left(x^T-{\widehat{X}}_t^T\right)R_t^{-1}\left(x-{\widehat{X}}_t\right)/2\right\}\mathit{dx},\end{array}$$E78

$$\begin{array}{l}h\left({\widehat{X}}_t,Y_t,R_t,t\right)=\{{\left[{\left(2\pi \right)}^{n_x}|R_t|\right]}^{-1/2}\underset{-\infty }{\overset{\infty }{\int }}\left[{\mathit{xa}}_1{\left(Y_t,x,t\right)}^T+b{\nu }_0{b}_1^T\left(Y_t,x,t\right)\right]\times \\ \times exp\left\{-\left(x^T-{\widehat{X}}_t^T\right)R_t^{-1}\left(x-{\widehat{X}}_t\right)/2\right\}\mathit{dx}-{\widehat{X}}_t{f}^{\left(1\right)}{\left({\widehat{X}}_t,Y_t,R_t,t\right)}^T\}{\left(b_1{\nu }_0{b}_1^T\right)}^{-1}\left(Y_t,t\right),\end{array}$$E79

$$\begin{array}{ll}{f}^{\left(2\right)}\left({\hat{X}}_t,Y_t,R_t,t\right)=& \left[\right(2p{)}^{n_x}\mid R_t\mid {\}}^{-1/2}\underset{-\infty }{\overset{\infty }{\int }}\left(x-{\hat{X}}_t\right)a{\left(Y_t,x,t\right)}^T\\ & +a\left(Y_t,x,t\right)\left(x^T-{\hat{X}}_t^T\right)+b{\nu }_0{b}_1^T\left(Y_t,x,t\right)\times \\ & \times exp\left\{-\left(x^T-{\hat{X}}_t^T\right)R_t^{-1}\left(x-{\hat{X}}_t\right)/2\right\}\mathit{dx},\end{array}$$E80

$$\begin{array}{l}{\rho }_r\left({\widehat{X}}_t,Y_t,R_t,t\right)=\left[\right(2p{)}^{n_x}\underset{-\infty }{\overset{\infty }{\int }}\{\left(x-{\widehat{X}}_t\right)\left(x^T-{\widehat{X}}_t^T\right)a_r\left(Y_t,x,t\right)+\\ +\left(x-{\widehat{X}}_t\right)b_r{\left(Y_t,x,t\right)}^T\left(x^T-{\widehat{X}}_t^T\right)+b_r\left(Y_t,x,t\right)\left(x^T-{\widehat{X}}_t^T\right)\}\\ \times exp\left\{-\left(x^T-{\widehat{X}}_t^T\right)R_t^{-1}\left(x-{\widehat{X}}_t\right)/2\right\}\mathit{dx}\left(r=\overline{1,n_y}\right),\end{array}$$E81

$${\widehat{X}}_0={\mathrm{E}}_N\left[X_0\mid Y_0\right],R_0={\mathrm{E}}_N\left[\left(X_0-{\widehat{X}}_0\right)\left(X_0^T-{\widehat{X}}_0^T\right)\mid Y_0\right],$$E82

where $a_r$ being $r$ th element of line-matrix $\left(a_1^T-{\widehat{a}}_1^T\right){\left(b_1{\nu }_0{b}_1^T\right)}^{-1}$; $b_{\mathit{kr}}$ being element of $k$ th line and $r$ th column of the matrix $b_1{\nu }_0{b}_1^T$; $b_r$ being the $r$th column of the matrix $b_1{\nu }_0{b}_1^T{\left(b_1{\nu }_0{b}_1^T\right)}^{-1}$, $b_r=\left[b_{1r}\dots b_{\mathit{pr}}\right]$ $\left(r=\overline{1,n_1}\right)$.

Proposal 10. If vector SDS USD (2) is reducible to Eqs. (73) and (74) then Eqs. (75)–(81) at conditions (82) define normal filtering algorithm. The number of equations is equal to

$$Q_{\mathit{NAM}}=n_x+\frac{n_x\left(n_x+1\right)}{2}=\frac{n_x\left(n_x+3\right)}{2}.$$E83

Hence, if the function $a_1$ is linear in $X_t$ and function $b$ does not depend on $X_t$ all matrices ${\rho }_r=0$ and Eq. (76) does not contain ${\dot{Y}}_t$ (Section 3).

Analogously Section 6 we get from [12] corresponding equations of normal conditionally optimal (Pugachev) extrapolator for reduced equations

$${\dot{X}}_t=a\left(X_t,Y_t,t\right)+b\left(X_t,t\right)V_1,$$E84

$$Z_t={\dot{Y}}_t=a_1\left(X_t,Y_t,t\right)+b_1\left(X_t,Y_t,t\right)V_2,$$E85

where $V_1$ and $V_2$ are normal independent white noises.

7. Examples

Let us consider scalar system

$$\phi \left({\dot{X}}_t,X_t\right)\equiv {\phi }_1\left({\dot{X}}_t\right)+{\phi }_2\left(X_t\right)+U_{1t}=0$$E86

$${\dot{U}}_{1t}={\alpha }_{10}+{\alpha }_{11}{U}_{1t}+{\beta }_1{V}_{1t}.$$E87

Here, $X_t,{\dot{X}}_t$ being state variable and its time derivative; $U_{1t}$ being scalar stochastic disturbance; $V_{1t}$ being scalar normal white noise with intensity $v_{1t}$; ${\phi }_1$ and ${\phi }_2$ being nonlinear functions; ${\alpha }_{10},{\alpha }_{11},{\beta }_1$ being constant parameters. After regression linearization of nonlinear functions, we have

$${\phi }_1\approx {\phi }_{10}+k_{\dot{X}}^{{\phi }_1}{\dot{X}}_t^0,{\phi }_2\approx {\phi }_{20}+k_X^{{\phi }_2}{X}_t^0.$$E88

At condition $k_{\dot{X}}^{\phi }\ne 0$ we get from (86) and (88) equations for mathematical expectation $m_t^X={\mathit{EX}}_t$ and centered $X_t^0=X_t-m_t^X$:

$${\phi }_{10}+{\phi }_{20}+m_{1t}^U=0,$$E89

$${\dot{X}}_t^0=a_t{X}_t^0+b_t{U}_{1t}^0,$$E90

where

$${\phi }_{10}={\phi }_{10}\left(m_t^{\dot{X}},D_t^{\dot{X}}\right),{\phi }_{20}={\phi }_{20}\left(m_t^X,D_t^X\right),$$E91

$$a_t=a_t\left(m_t^X,m_t^{\dot{X}},D_t^X,D_t^{\dot{X}},D_t^{U_1},D_t^{{\mathit{XU}}_1}\right)=-k_X^{{\phi }_2}{\left(k_{\dot{X}}^{{\phi }_1}\right)}^{-1},b_t=-{\left(k_{\dot{X}}^{{\phi }_1}\right)}^{-1}.$$E92

Equations (87) and (90), for $U_{1t}^0=U_{1t}=m_t^{U_1}$ $\left(m_t^{U_1}={\mathit{EU}}_{1t}\right)$ and ${\overline{X}}_t={\left[X_t{U}_{1t}\right]}^T$ may be presented in vector form

$${\dot{m}}_t^{\overline{X}}=A_{0t}+A_t{m}_t^{\overline{X}},$$E93

$${\dot{\overline{X}}}_t^0=A_t{\overline{X}}_t^0+B_t{V}_{1t},$$E94

$$A_t=\left[\begin{array}{cc}{a}_t& b_t\\ 0& {\alpha }_1\end{array}\right],B_t=\left[\begin{array}{c}0\\ {\beta }_1\end{array}\right].$$E95

Covariance matrix

$$K_t^{\overline{X}}=\left[\begin{array}{cc}{D}_t^X& K^{{\mathit{XU}}_1}\\ K_t^{\dot{X}{U}_1}& D_t^{U_1}\end{array}\right]$$E96

and matrix of covariance functions

$$K^{\overline{X}}\left(t_1,t_2\right)=\left[\begin{array}{cc}{K}_{11}^{\overline{X}}\left(t_1,t_2\right)& K_{12}^{\overline{X}}\left(t_1,t_2\right)\\ K_{21}^{\overline{X}}\left(t_1,t_2\right)& K_{22}^{\overline{X}}\left(t_1,t_2\right)\end{array}\right]$$E97

satisfy to linear equations for correlation theory (Section 3)

$${\dot{K}}_t^{\overline{X}}=A_t{K}_t^{\overline{X}}+K_t^{\overline{X}}{A}_t^T+B_t{\nu }_{1t}{B}_t^T,K_{t_0}^{\overline{X}}=K_0^{\overline{X}},$$E98

$$\frac{\partial K^{\overline{X}}\left(t_1,t_2\right)}{\partial t_2}=K^{\overline{X}}\left(t_1,t_2\right)A_{t_2}^T,K^{\overline{X}}\left(t_1,t_1\right)=K_{t_1}^{\overline{X}}.$$E99

Vector Eq. (98) is equal to the following scalar equations:

$${\dot{D}}_t^X=2\left(a_t{D}_t^X+b_t{K}_t^{{\mathit{XU}}_1}\right),{\dot{D}}_t^{U_1}=2{\alpha }_1{D}_t^{U_1}+{\beta }_1^2{\nu }_{1t},{\dot{K}}_t^{{\mathit{XU}}_1}=a_t{K}_t^{{\mathit{XU}}_1}+b_t{D}_t^{U_1}+{\alpha }_1{K}_t^{{\mathit{XU}}_1}.$$E100

From Eq. (90) we calculate variance

$$D_t^{\dot{X}}=a_t^2{D}_t^X+b_t^2{D}_t^{U_1}+2a_t{b}_t{K}_t^{{\mathit{XU}}_1}.$$E101

Thus, for MAM algorithm we use Eqs. (89), (91), (98)–(101).

Let system (86) and (87) is observable so that

$$Z_t={\dot{Y}}_t=X_t+V_{2t}.$$E102

Then for Kalman-Bucy filter equations (Proposal 4), we have

$${\dot{\widehat{X}}}_t=A_t{\widehat{X}}_t+{\beta }_t\left(Z_t-{\widehat{X}}_t\right),{\beta }_t=R_t{\nu }_{2t}^{-1}\left(det{\nu }_{2t}\ne 0\right),{\dot{R}}_t=2A_t{R}_t+{\nu }_1-{\nu }_2{\beta }_t^2.$$E103

For Kalman-Bucy extrapolator equations are defined by Proposal 5 at $u\left(t,\tau \right)=e^{-a\left(t-\tau \right)}$.

In Table 1, the coefficients of statistical linearization of for typical nonlinear function are given.

$\phi$	${\phi }_0$
${\dot{Y}}^3$	$m\left(m^2+3D\right)$
$sin\omega \dot{Y}$	$exp\left(-\frac{{\omega }^{2}D}{2}\right)sin\mathit{\omega m}$
$cos\omega \dot{Y}$	$exp\left(-\frac{{\omega }^{2}D}{2}\right)cos\mathit{\omega m}$
$\dot{Y}exp\left(\alpha \dot{Y}\right)$	$\left(m+\mathit{\alpha D}\right)exp\left(\mathit{\alpha m}+\frac{{\alpha }^{2}D}{2}\right)$
$\dot{Y}sin\omega \dot{Y}$	$\left(msin\mathit{\omega m}-\mathit{\omega D}cos\mathit{\omega m}\right)exp\left(-\frac{{\omega }^{2}D}{2}\right)$
$\dot{Y}cos\omega \dot{Y}$	$\left(mcos\mathit{\omega m}-\mathit{\omega D}sin\mathit{\omega m}\right)exp\left(-\frac{{\omega }^{2}D}{2}\right)$
$sgn\dot{Y}$	$2\mathrm{\Phi }\left(\frac{m}{\sqrt{D}}\right)$
${\dot{Y}}^{2}sgn\dot{Y}$	$2D\left\{\left(\frac{m^2}{D}+1\right)\mathrm{\Phi }\left(\frac{m}{\sqrt{D}}\right)+\frac{1}{\sqrt{2\mathit{\pi D}}}exp\left(-\frac{m^2}{2D}\right)\right\}\left(m=m_{\dot{Y}},D=D_{\dot{Y}}\right)$
$\left\{\begin{array}{c}\frac{l}{D}\dot{Y},\\ l,\\ -l\end{array}\begin{array}{c}\mid \dot{Y}\mid \le d;\\ \dot{Y}<d;\\ \dot{Y}<-d\end{array}\right.$	$\begin{array}{l}l\left\{\left(1+m_1\right)\mathrm{\Phi }\left(\frac{1+m_1}{{\sigma }_1}\right)-\left(1-m_1\right)\mathrm{\Phi }\left(\frac{1-m_1}{{\sigma }_1}\right)\right.+\\ +\left.\frac{{\sigma }_1}{\sqrt{2\pi }}\left[exp\left\{-\frac{1}{2}{\left(\frac{1+m_1}{{\sigma }_1}\right)}^2\right\}-exp\left\{-\frac{1}{2}{\left(\frac{1-m_1}{{\sigma }_1}\right)}^2\right\}\right]\right\}\end{array}$
$\left\{\begin{array}{c}\gamma \left(\dot{Y}+d\right),\\ 0,\\ \gamma \left(\dot{Y}-d\right)\end{array}\begin{array}{c}\mid \dot{Y}\mid <-d;\\ \mid \dot{Y}\mid \le d;\\ \dot{Y}>d\end{array}\right.$	$\begin{array}{l}\gamma \left\{1-\frac{1}{m_1}\left[\left(1+m_1\right)\mathrm{\Phi }\left(\frac{1+m_1}{{\sigma }_1}\right)-\left(1-m_1\right)\mathrm{\Phi }\left(\frac{1-m_1}{{\sigma }_1}\right)\right]\right.+\\ +\left.\frac{{\sigma }_1}{m_1\sqrt{2\pi }}\left[exp\left\{-\frac{1}{2}{\left(\frac{1-m_1}{{\sigma }_1}\right)}^2\right\}-exp\left\{-\frac{1}{2}{\left(\frac{1+m_1}{{\sigma }_1}\right)}^2\right\}\right]\right\}\end{array}$
$\left\{\begin{array}{c}-l,\\ 0,\\ l\end{array}\begin{array}{c}\dot{Y}<-d;\\ \mid \dot{Y}\mid \le d;\\ \dot{Y}>d\end{array}\right.$	$l\left[\mathrm{\Phi }\left(\frac{1+m_1}{{\sigma }_1}\right)-\mathrm{\Phi }\left(\frac{1-m_1}{{\sigma }_1}\right)\right]$
${\dot{Y}}_1{\dot{Y}}_2$	$m_1{m}_2+K_{12}$
${\dot{Y}}_1^2{\dot{Y}}_2$	$\left(m_1^2+K_{11}\right)m_2+2m_1{K}_{12}$
$sin\left({\omega }_1{\dot{Y}}_1+{\omega }_2{\dot{Y}}_2\right)$	$exp\left[\frac{{\omega }_1^2{K}_{11}+2{\omega }_1{\omega }_2{K}_{12}+{\omega }_2^2{K}_{22}}{2}\right]sin\left({\omega }_1{m}_1+{\omega }_2{m}_2\right)$
$sgn\left({\dot{Y}}_1+{\dot{Y}}_2\right)$	$2\mathrm{\Phi }\left({\zeta }_{1,2}\right),{\zeta }_{1,2}=\frac{m_1+m_2}{\sqrt{D}},D=K_{11}+2K_{12}+K_{22}$

Table 1.

Coefficients of statistical linearization for typical nonlinear functions [12, 13, 14].

Let us consider normal scalar system

$$F\equiv \mathrm{\Phi }\left({\dot{X}}_t\right)+a_t{X}_t+u_t=0.$$E104

Нere random function admits Pugachev normalization

$$\mathrm{\Phi }\left({\dot{X}}_t\right)\approx {\mathrm{\Phi }}_0+k^{\mathrm{\Phi }}{\dot{X}}_t^0+{\mathrm{\Delta \Phi }}_t^0,$$E105

where ${\mathrm{\Delta \Phi }}_t^0$ being normal StP satisfying equation of forming filter

$$\dot{\mathrm{\Delta }}{\mathrm{\Phi }}_t^0=a_t^{\mathrm{\Delta \Phi }}{\mathrm{\Delta \Phi }}_t^0+b_t^{\mathrm{\Delta \Phi }}{V}_t.$$E106

Note that functions ${\mathrm{\Phi }}_t^0$ and $k^{\mathrm{\Phi }}$ depend on $E_t^{\dot{\mathrm{\Phi }}}$ and $D_t^{\dot{\mathrm{\Phi }}}$. Equations (104) and (105) are decomposing on two equations. First equation at condition $k^{\mathrm{\Phi }}\ne 0$ is as follows:

$${\mathrm{\Phi }}_0+a_t{E}_t^X+u_t=0,{\mathrm{\Phi }}_0=k^{{\mathrm{\Phi }}_0}{E}_t^{\dot{X}}.$$E107

Second equation at condition $k^{\mathrm{\Phi }}\ne 0$ is as follows: $k^{\mathrm{\Phi }}{\dot{X}}_t^0+{\mathrm{\Delta \Phi }}_t^0+a_t{X}_t^0=0$ may be presented as

$${\dot{X}}_t^0=a_t{\left(k^{\mathrm{\Phi }}\right)}^{-1}{X}_t^0-{\left(k^{\mathrm{\Phi }}\right)}^{-1}{\mathrm{\Delta \Phi }}_t^0.$$E108

Equations (106) and (108) for $Z_t^0={\left[X_t^0{\mathrm{\Delta \Phi }}_t^0\right]}^T$ leads to the following vector equation for covariance matrix

$${\dot{K}}_t^Z={\mathit{AK}}_t^Z+K_t^Z{A}^T+B{\nu }^V{B}^T,$$E109

where $A=\left[\begin{array}{cc}-a_t{\left(k^{\mathrm{\Phi }}\right)}^{-1}& -{\left(k^{\mathrm{\Phi }}\right)}^{-1}\\ 0& a^{\mathrm{\Delta \Phi }}\end{array}\right],B=\left[\begin{array}{c}0\\ b^{\mathrm{\Delta \Phi }}\end{array}\right]$. Eqs. (107) and (109) give the following final relations:

$$E_t^{\dot{X}}=\left(a_t{E}_t^X+u_t\right){\left(k^{{\mathrm{\Phi }}_0}\right)}^{-1},$$E110

$$\begin{array}{l}{D}_t^{\dot{X}}=a_t^2{\left(k^{\mathrm{\Phi }}\right)}^{-2}{D}_t^X+{\left(k^{\mathrm{\Phi }}\right)}^{-2}{D}_t^{\mathrm{\Delta \Phi }}-2a_t{\left(k^{\mathrm{\Phi }}\right)}^{-2}{K}_t^{X\mathrm{\Delta \Phi }},\\ {\dot{D}}_t^X=-2\left(a_t{D}_t^X+K_t^{X\mathrm{\Delta \Phi }}\right),\\ {\dot{D}}_t^{\mathrm{\Delta \Phi }}=2a_t^{\mathrm{\Delta \Phi }}{D}_t^{\mathrm{\Delta \Phi }}+{\left(b_t^{\mathrm{\Delta \Phi }}\right)}^2{\nu }^V,\\ {\dot{K}}_t^{X\mathrm{\Delta \Phi }}=a_t^{\mathrm{\Delta \Phi }}{K}_t^{\mathrm{\Delta \Phi }}-\left(a_t{K}_t^{X\mathrm{\Delta \Phi }}+D_t^{\mathrm{\Delta \Phi }}\right){\left(k^{\mathrm{\Phi }}\right)}^{-1}.\end{array}$$E111

8. Conclusion

Models of various types of SDS USD arise in problems of analytical modeling and estimation (filtering, extrapolation, etc.) for control stochastic systems, when it is possible to neglect higher-order time derivatives. Linear and nonlinear methodological and algorithmic support of analytical modeling, filtering, and extrapolation for SDS USD is developed. The methodology is based on the reduction of SDS USD to SDS by means of linear and nonlinear regression models. Special attention is paid to SDS USD with multiplicative (parametric) noises. Examples illustrating methodology are presented. The described results may be generalized for systems with stochastically unsolved derivatives and stochastic integrodifferential systems reducible to the differential.

Acknowledgments

The author is grateful to experts for their appropriate and constructive suggestions to improve this template. Research is supported by the Russian Academy of Sciences (Project-AAAA-A19-119001990037-5). Also, the author is much obliged to Mrs. Irina Sinitsyna and Mrs. Helen Fedotova for translation and manuscript preparation.