Open access peer-reviewed chapter

Development of Ellipsoidal Analysis and Filtering Methods for Nonlinear Control Stochastic Systems

Written By

Igor N. Sinitsyn, Vladimir I. Sinitsyn and Edward R. Korepanov

Submitted: October 23rd, 2019 Reviewed: December 2nd, 2019 Published: February 13th, 2020

DOI: 10.5772/intechopen.90732

From the Edited Volume

Automation and Control

Edited by Constantin Voloşencu, Serdar Küçük, José Guerrero and Oscar Valero

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Abstract

The methods of the control stochastic systems (CStS) research based on the parametrization of the distributions permit to design practically simple software tools. These methods give the rapid increase of the number of equations for the moments, the semiinvariants, coefficients of the truncated orthogonal expansions of the state vector Y, and the maximal order of the moments involved. For structural parametrization of the probability (normalized and nonnormalized) densities, we shall apply the ellipsoidal densities. A normal distribution has an ellipsoidal structure. The distinctive characteristics of such distributions consist in the fact that their densities are the functions of positively determined quadratic form of the centered state vector. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters. For ellipsoidal linearization method (ELM), the number of equations coincides with normal approximation method (NAM). The development of EAM (ELM) for CStS analysis and CStS filtering are considered. Based on nonnormalized densities, new types of filters are designed. The theory of ellipsoidal Pugachev conditionally optimal control is presented. Basic applications are considered.

Keywords

  • conditionally optimal filtering and control
  • control stochastic system
  • ellipsoidal approximation method (EAM)
  • ellipsoidal linearization method (ELM)

1. Introduction

The methods for the control stochastic systems (CStS) research based on the parametrization of the distributions permit to design practically simple software tools [1, 2, 3, 4, 5, 6]. These methods give the rapid increase of the number of equations for the moments, the semiinvariants, and coefficients of the truncated orthogonal expansions of the state vector Y for the maximal order of the moments involved. For structural parametrization of the probability (normalized and nonnormalized) densities, we shall apply the ellipsoidal densities. A normal distribution has an ellipsoidal structure. The distinctive characteristics of such distributions consist in the fact that their densities are the functions of positively determined quadratic form u=uy=yTmTCym where m is an expectation of Y,C is some positively determined matrix. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters till QEAM=QNAM+nm1 and QNAM=rr+3/2 where 2nm being the number of probabilistic moments. For ellipsoidal linearization method (ELM), we get QELM=QNAM.

The theory of conditionally optimal filters (COF) is described in [7, 8] on the basis of methods of normal approximation (NAM), methods of statistical linearization (SLM), and methods of orthogonal expansions (OEM) for the differential stochastic systems on smooth manifolds with Wiener noise in the equations of observation and Wiener and Poisson noises in the state equations. The COF theory relies on the exact nonlinear equations for the normalized one-dimensional a posteriori distribution. The paper [9] considers extension of [7, 8] to the case where the a posteriori one-dimensional distribution of the filtration error admits the ellipsoidal approximation [4]. The exact filtration equations are obtained, as well as the OEM-based equation of accuracy and sensitivity, the elements of ellipsoidal analysis of distributions are given, and the equations of ellipsoidal COF (ECOF) using EAM and ELM are derived. The theory of analytical design of the modified ellipsoidal suboptimal filters was developed in [10, 11] on the basis of the approximate solution by EAM (ELM) of the filtration equation for the nonnormalized a posteriori characteristic function. The modified ellipsoidal conditionally optimal filters (MECOF) were constructed in [12] on the basis of the equations for nonnormalized distributions. It is assumed that there exist the Wiener and Poisson noises in the state equations and only Wiener noise being in the observation equations. At that, the observation noise can be non-Gaussian.

Special attention is paid to the conditional generalization of Pugachev optimal control [13] based on EAM (ELM).

Let us consider the development of EAM (ELM) for solving problems of ellipsoidal analysis and optimal, suboptimal, and conditionally optimal filtering and control in continuous CStS with non-Gaussian noises and stochastic factors.

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2. Ellipsoidal approximation method

This method was worked out in [1, 2, 3, 4] for analytical modeling of stochastic process (StP) in multidimensional nonlinear continuous, discrete and continuous-discrete (CStS). Let us consider elements of EAM.

Following [1, 2, 3, 4] let us find ellipsoidal approximation (EA) for the density of r-dimensional random vector by means of the truncated expansion based on biorthogonal polynomials pr,νuyqr,νuy, depending only on the quadratic form u=uyu=uy for which some probability density of the ellipsoidal structure wuy serves as the weight:

wuypr,νuyqr,μuydy=δνμ.E1

The indexes ν and μ at the polynomials mean their degrees relative to the variable u. The concrete form and the properties of the polynomials are determined further. But without the loss of generality, we may assume that qr,0u=pr,0u=1. Then the probability density of the vector Y may be approximately presented by the expression of the form:

fyfu=wu1+ν=2Ncr,νpr,νu.E2

Here the coefficients cr,ν are determined by the formula:

cr,ν=fyqr,νudy=Eqr,νU,ν=1N.E3

As pr,0u and qr,0u are reciprocal constants (the polynomials of zero degree), then always cr,0pr,0=1 and we come to the following results.

Statement 1. Formulae (2) and (3) express the essence of the EA of the probability density of the random vectorY.

For the control problems, the case when the normal distribution is chosen as the distribution wu is of great importance

wu=wxTCx=12πrKexpxTK1x/2;E4

accounting that C=K1, we reduce the condition of the biorthonormality (1) to the form

12r/2Γr/20pr,νuqr,μuur/21eu/2du=δνμ,E5

where Γ is gamma function [5].

Statement 2. The problem of the choosing of the polynomial system pr,νuqr,μu which is used at the EA of the densities (4) and (5) is reduced to finding a biorthonormal system of the polynomials for which the χ2-distribution with r degrees of the freedom serves as the weigh.

A system of the polynomials which are relatively orthogonal to χ2-distribution with r degrees of the freedom is described by series:

Sr,νu=μ=0ν1ν+μCνμr+2ν2!!r+2μ2!!uμ.E6

The main properties of polynomials Sr,ν are given in [2, 3, 4]. Between the polynomials Sr,νu and the system of the polynomials pr,νuqr,μu, the following relations exist:

pr,νu=Sr,νu,qr,νu=r2!!r+2ν2!!2ν!!Sr,νu,r2.E7

Example 1. Formulae for polynomials pr,νu and qr,νu and its derivatives for some r and ν are as follows [4]:

  • At r=2, ν2,

p2,νu=uν,q2,νu0,q2,ν'u0,q2,ν''u0;

  • At r2, ν=2

pr,2u=u2,qr,2u=18u2,qr,2'u=14u,qr,2''u=14.

For r=2 at ν=3 we have

p2,3u=u3,q2,3u0,q2,3'u0,q2,3''u0;

at r=3

p3,3u=S3,3u,q3,3u=15040S3,3u,q3,3'u=15040S3,3'u,q3,3''u=15040S3,3''u,S3,3u=105+105u21u2+u3,S3,3'u=10542u+3u2,S3,3''u=42+6u;

at r=4:

p4,3u=S4,3u,q4,3u=19216S4,3u,q4,3'u=19216S4,3'u,q4,3''u=19216S4,3'u,S4,3u=197+144u24u2+u3,S4,3'u=14448u+3u2,S4,3''u=48+6u.

Following [5] we consider the H-space L2Rr and the orthogonal system of the functions in them where the polynomials Sr,νu are given by Formula (6), and wu is a normal distribution of the r-dimensional random vector (4). This system is not complete in L2Rr. But the expansion of the probability density fu=fyTmTCym of the random vector Y which has an ellipsoidal structure over the polynomials pr,νu=Sr,νu, m.s. converges to the function fu itself. The coefficients of the expansion in this case are determined by relation:

cr,ν=fupr,νudy/2ν!!r+2ν2!!r2!!.E8

Statement 3. The system of the functions wuSr,νu forms the basis in the subspace of the space L2Rr generated by the functions fu of the quadratic form u=ymTCym.

At the probability density expansion over the polynomial Sr,νu, the probability densities of the random vector Y and all its possible projections are consistent. In other words, at integrating the expansions over the polynomials Sh+l,νu and h+l=r, of the probability densities of the r-dimensional vector Y,

fy=12πh+lKeu/21+ν=2Nch+l,νSh+l,νu,u=ymTK1ym,y=yTyTT,E9

on all the components of the l-dimensional vector y, we obtain the expansion over the polynomials Sh,νu1 of the probability density of the h-dimensional vector Y with the same coefficients

fy=12πhK11eu1/21+ν=2Nch,νSh,νu1,u1=ymTK111ym,ch,ν=ch+l,ν,E10

where K11 is a covariance matrix of the vector Y'.

But in approximation (10) the probability density of h-dimensional random vector Y' obtained by the integration of expansion (9) the density of h+l-dimensional vector is not optimal EA of the density.

For the random r-dimensional vector with an arbitrary distribution, the EA (2) of its distribution determines exactly the moments till the Nth order inclusively of the quadratic form U=YmTK1Ym, i.e.,

EUμ=EEAUμ,μN.E11

(EEA stands for expectation relative to EA distribution).

In this case the initial moments of the order s and s=s1++sr of the random vector Y at the approximation (4) are determined by the formula:

αs1,,sr=αs=EY1s1Yrsry1s1yrsrwudy+v=2Ncr,vy1s1yrsrpr,vuwudyE12

Statement 4. At the EA of the distribution of the random vector, its moments are combined as the sums of the correspondent moments of the normal distribution and the expectations of the products of the polynomials pr,νu by the degrees of the components of the vector Y at the normal density wu.

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3. EAM accuracy

For control problems the weak convergence of the probability measures generated by the segments of the density expansion to the probability measure generated by the density itself is more important than m.s. convergence of the segments of the density expansion over the polynomials Sr,νu to the density, namely,

Awu1+ν=2Ncr,νpr,νuAfudy

uniformly relative to A at N on the σ-algebra of Borel sets of the space Rr. Thus the partial sums of series (2) give the approximation of the distribution, i.e., the probability of any event A determined by the density fu with any degree of the accuracy. The finite segment of this expansion may be practically used for an approximate presentation of fu with any degree of the accuracy even in those cases when fu/wu does not belong to L2Rr. In this case it is sufficient to substitute fu by the truncated density. Expansion (2) is valid only for the densities which have the ellipsoidal structure. It is impossible in principal to approximate with any degree of the accuracy by means of the EA (2) the densities which arbitrarily depend on the vector y.

One is the way of the estimate of the accuracy of the distribution approximation in the comparison of the probability characteristics calculated by means of the known density and its approximate expression. The most complete estimate of the accuracy of the approximation may be obtained by the comparison of the probability occurrence on the sets of some given class. Besides that taking into consideration that the probability density is usually approximated by a finite segment of its orthogonal expansion for instance, over Hermite polynomials or by a finite segment of the Edgeworth series [1, 2, 3, 4, 5] which contain the moments till the fourth order, the accuracy may be characterized by the accuracy of the definition of the moments of the random vector or its separate components, in particular, of the fourth order moments.

Corresponding estimates for these two ways of approximation are given in [2, 3].

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4. Ellipsoidal linearization method

Now we consider ellipsoidal linearization of nonlinear transforms of random vectors Y using mean square error (m.s.e.) criterion optimal m.s.e. regression of vector Z=φY on vector Y is determined by the formula [4, 6]:

mzY=h2Y,h2=ΓzyΓy1E13

or

mzY=h1Y+a,h1=KzyKy1,a=mzh1my.E14

where h1 and h2 are equivalent linearization matrices and my and Ky are mathematical expectation and covariance matrix detKy0. In case (14) coefficient h1 is equal to

h1=KzyKy1=zmzymyTKy1fzydzdy=mzymzymyTKy1f1ydyE15

where f1y is the density of random vector Y.

For ellipsoidal density f1y in (15) is defined by

f1y=f1ELy=f˜1ELuymyKyc.E16

In case (14) we get Statement 5 for ELM:

mzYm1zEL+h1ELmyKycY0,E17

where

h1EL=h1ELmyKyc=mzymzymyTKy1f1ELydy=mzymzymyTKy1f˜1ELuymyKycdy.E18

In case (13) we have Statement 6 for ELM:

mzYh2ELΓycY,E19
h2ELΓyc=mzyyTΓy1f1ELydy=mzyyTΓy1f˜1uymyΓycdy.E20

For control problems the following ELM new generalizations are useful:

  1. Let us consider for fixed dimension p=dimy and N in (2) with normal wu distinguish modifications of various orders ELM wp,2, ELM wp,3, …, ELM wp,N. In this case c=cp,v characterizes partial deviations from normal distributions of various orders v jointly for all p components of vector Y (be part of quadratic form UY.

  2. At decomposition of vector Y on l1,l2,,lr random subvectors, Y=Yl1TYl2TYlrTT, we distinguish ELMwl1,,lr,N. Coefficients cl1,v,,cl2,v characterize partial deviations of subvectors from normal distribution.

  3. For matrix transforms Z=φY=φ1YφqYT,φiY=φi1YφipYT(i=1,q)¯,dimφ=p×q, we have the following formulae for ELM:

mzYm1zEL+H1ELmyKycYm;E21
mzYH2ELY.E22

where

H1ELmyKyc=h11ELmyKych1qELmyKycE23
H2ELΓyc=h21ELΓych2qELΓyc,E24

(h1iEL and h2iEL(i=1,q)¯ are determined by formulae (18) and (19)).

  1. For transforms depending on time process t, it is useful to work with overage ELM coefficients miz and hiEL for time intervals.

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5. EAM and ELM for nonlinear CStS analysis

Let us consider nonlinear CStS defined by the following Ito vector stochastic differential equation:

dYt=aYttdt+bYttdW0+R0qcYttυP0dt,Yt0=Y0.E25

Here YtΔy is (Δy is a smooth state manifold) W0=W0t is an r – dimensional Wiener StP of intensity v0=v0t, P(Δ, A) is simple Poisson StP for any set A, Δ=t1t2, P0ΔA=P0ΔAμPΔA, μPΔA=EP0ΔA=ΔvPτA. Integration by υ extends to the entire space Rq with deleted origin, a and b are certain functions mapping Rp×R, respectively, into Rp,Rpr, and c is for Rp×Rq into Rp.

Following [4] we use for finding the one-dimensional probability density f1yt of the r-dimensional Yt which is determined by Eq. (25). Suppose that we know a distribution of the initial value Y0=Yt0 of the StP Yt. Following the idea of EAM, we present the one-dimensional density in the form of a segment of the orthogonal expansion in terms of the polynomials dependent on the quadratic form u=yTmTCym where m and K=C1 are the expectation and the covariance matrix of the StPYt:

f1ytf1EAMu=w1u1+ν=2Ncp,vpp,νu.E26

Here w1u is the normal density of the p-dimensional random vector which is chosen in correspondence with the requirement cp,1=0. The optimal coefficients of the expansion cp,v are determined by the relation

cp,v=f1ytqp,vudy=Eqp,vU,ν=1N.E27

The set of the polynomials pp,vuqp,vu is constructed on the base of the orthogonal set of the polynomials Sp,vu according to the following rule which provides the biorthonormality of system at p2 given by (5). Thus the solution of the problem of finding the one-dimensional probability density by EAM is reduced to finding the expectation m, the covariance matrix K of the state vector, and the coefficients of the correspondent expansion cp,v also.

So we get the equations

ṁ=φ10mKt+ν=2Ncp,vφ1νmKt,E28
K̇=φ20mKt+ν=2Ncp,vφ2νmKt,E29
ċp,κ=cp,κ12pκcp,κp×trK1φ20mKt+K1ν=2Ncp,vφ2ν(mKt)+ψκ0mKt+ν=2Ncp,vψκνmKt,κ=2,,N,E30

where the following indications are introduced:

φ10mKt=aytw1udy,φ1νmKt=aytpp,vuw1udy,

φ20mKt=aytyTmT+ymaytT+σ¯ytw1udy,
φ2vmKt=aytyTmT+ymaytT+σ¯ytpp,vuw1udy,
σ¯yt=σ¯yt+R0qcytυcytυTvPt,σyt=bytν0tbytT,E31
ψκ0mKt=q'p,κu2ymTK1ayt+trK1σyt+2q''κuymTK1σytymw1udyψκvmKt=q'p,κu2ymTK1ayt+trK1σyt+2q''κuymTK1σytym}pp,vuw1udy.E32

Eqs. (28)(30) at the initial conditions

mt0=m0,Kt0=K0,cp,κt0=cp,κ0κ=2NE33

determine m,K,cp,2,,cp,N as time functions. For finding the variables cp,κ0, the density of the initial value Y0 of the state vector should be approximated by Formula (26).

So we get the following result.

Statement 7. At sufficient conditions of existence and uniqueness of StP in Eq. (25), Eqs. (28)(33) define EAM.

For stationary CStS we get the corresponding EAM equations putting in Eqs. (28)(30) right-hand equal to zero.

Example 2. Following [4, 14, 15] in case of vibroprotection Duffing StS:

X¨+δẊ+ω2X+μX3=U+V,Xt0=X0,Ẋt0=Ẋ0,

(δ,ω2,μ,U are constants, V is the white noise with intensity v) with accuracy till 4th probabilistic moments, ellipsoidal approximation of one-dimensional density is described by the set of parameters:

m1=EX,m2=Ẋ,K11=EX02,K12=EX0Ẋ0,K22=EẊ02andc2,2.

These parameters satisfy the following ordinary differential equations:

ṁ1=m2,m1t0=m10,ṁ2=Uω2m1+μm13+3m1K11δm2,m2t0=m20;
K̇11=2K12,K̇12=K22ω2K11+3μK11K11+m12δK12+24μc2,2K112,K̇22=ν2ω2K123μK12K11+m12+δK22+48μc2,2K11K12,K11t0=K110,K12t0=K120,K22t0=K220;
ċ2,2=c2,2K11νK5δK=K11K22K122,c2,2t0=c2,20.

At U=0 stationary values are as follows:

m¯1=0,m¯2=0,K¯11=ω2ω46μν/δ6μ,K¯12=0,K¯22=ν2δ,c¯2,2=0.

At conditions

1. U=0;μ=0.1;ω=3;δ=1;ν=0.5;2. U=0;μ=0.5;ω=3;δ=1;ν=0.5;3. U=0;μ=1;ω=3;δ=1;ν=0.5

And at zero initial conditions, the results of analytical modeling for K11, K12, K22 are given in Figures 13. Mathematical expectations m1 and mn are equal to zero.

Graphs (1) are the results of integration of NAM equations at initial stage. Then for nongenerated covariance matrix K integration of EAM equations (2). Graphs are the results of EAM equation integration at the whole stage.

Figure 1.

K11 graphs for at 0,1 (a); 0,5 (b); 1,0 (c).

Figure 2.

K12 graphs for at 0,1 (a); 0,5 (b); 1,0 (c).

Figure 3.

K22 graphs for at 0,1 (a); 0,5 (b); 1,0 (c).

The results of investigations for c2,2 are given in Figure 4 for the following sets of conditions:

Figure 4.

C22 graphs for at sets № 1 (a); 2 (b); 3 (c); 4 (d).

1. U=0;μ=1;ω=3;δ=0,5;ν=0,5;T=020zeroinitialconditions;2. U=0;μ=1;ω=3;δ=0,5;ν=1;T=020zeroinitialconditions;3. U=0;μ=1;ω=3;δ=0,5;ν=1;T=020zeroinitialconditionsexceptm10=0,2;4. U=0;μ=1;ω=3;δ=1;ν=1;T=020zeroinitialconditions.

For the stationary CStS regimes, EAM gives the same results as NAM (MSL). EAM describes non-Gaussian transient vibro StP at initial stage.

Methodological and software support for analysis and filtering problem CStS for shock and vibroprotection is given in [4, 14].

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6. Exact filtering equations for continuous a posteriori distribution

Following [7, 8, 9, 15], let the vector StP XtTYtTT be defined by a system on vector stochastic differential Ito equations:

dXt=φXtYtΘtdt+ψXtYtΘtdW0+R0qψXtYtΘtvP0dtdv,Xt0=X0,E34
dYt=φ1XtYtΘtdt+ψ1XtYtΘtdW0+R0qψ1XtYtΘtvP0dtdv,Yt0=Y0.E35

where Yt=Yt is an ny-dimensional observed StP YtΔy(Δy is a smooth manifold of observations); XtΔx(Δx is a smooth state manifold), W0=W0t is an nw-dimensional Wiener StP nwny of intensity ν0=ν0Θt;P0ΔA=PΔAμPΔA,PΔA for any set A represents a simple Poisson StP, and μPΔA is its expectation,

μPΔA=EPΔA=ΔνPτA;

vPΔA is the intensity of the corresponding Poisson flow of events, Δ=t1t2; integration by υ extends to the entire space Rq with deleted origin; Θ is the vector of random parameters of size nΘ;φ=φXtYtΘt,φ1=φ1XtYtΘt,ψ=ψXtYtΘt, and ψ1=ψ1XtYtΘt are certain functions mapping Rnx×Rny×R, respectively, into Rnx,Rny,Rnxnw, Rnynw; ψ=ψXtYtΘtv, and ψ1=ψ1XtYtΘtv are certain functions mapping Rnx×Rny×Rq into Rnx,Rny. Determine the estimate X̂t StP Xt at each time instant t from the results of observation of StP Yτ until the instant t,Yt0t=Yτ:t0τ<t.

Let us assume that the state equation has the form (34); the observation Eq. (35), first, contains no Poisson noise ψ"10; and, second, the coefficient at the Wiener noise ψ'1 in the observation equations is independent of the state ψ'1XtYtΘt=ψ'1YtΘt, and then the equations of the problem of nonlinear filtration are given by

dXt=φXtYtΘtdt+ψXtYtΘtdW0+R0qψXtYtΘtvP0dtdv,Xt0=X0,E36
dYt=φ1XtYtΘtdt+ψ1YtΘtdW0,Yt0=Y0.E37

The known sufficient conditions for the existence and uniqueness of StP defined by (36) and (37) under the corresponding initial conditions [1, 3, 16] are satisfied.

The optimal estimate X̂t minimizing the mean square of the error at each time instant t is known [10, 11, 12, 13, 14] to represent for any StP Xt and Yt.

An a posteriori expectation StP Xt: X̂t=EXtYt0t. To determine this conditional expectation, one needs to know pt=ptx and gt=gtλ, the a posteriori one-dimensional density, and the characteristic function of the distribution StP Xt.

Introduce the nonnormalized one-dimensional a posteriori density p˜txΘ and a characteristic function g˜tλΘ according to

p˜txΘ=μtptxΘ,g˜tλΘ=EΔxpteiλTXtμt=μtgtλΘ,E38

where μt is a normalizing function and EΔxpt is the symbol of expectation on the manifold Δx on the basis of density ptx. Then, by generalizing [11] to the case of Eqs. (36) and (37), we get the following exact equation of the rms optimal nonlinear filtration:

dg˜tλΘ=EΔxp˜tiλTφXYtΘt12ψν0ψT(XYtΘt)+R0qeiλTψXYtΘtv1iλTψXYtΘtvνPΘtdveiλTXdt+EΔxp˜tφ1XYtΘtT+iλTψν0ψT(XYtΘt)eiλTXψν0ψT1YtΘtdYt.E39

If by following [15, 17] the function ψ in (36) admits the representation

ψ=ψωΘv,E40

where P0ΔA=P0(0t]dv, then Eqs. (36) and (37) take the form

Ẋt=φXtYtΘt+ψXtYtΘtVΘt,Xt0=X0,E41
Ẏt=φXtYtΘt+ψ1YtΘtV0Θt,Yt0=Y0.E42

with V0Θt=Ẇ0Θt;VΘt=W¯̇Θt,

W¯Θt=W0Θt+R0qωΘvP0(0t]dv,E43

where νPΘtvdv=μΘtv/tdv is the intensity of the Poisson flow of discontinuities equal to ωΘt; the logarithmic derivatives of the one-dimensional characteristic functions obey certain formulas

χW¯ρΘt=χW0ρΘt+R0qeiρTωΘv1iρTωΘvνPΘtvdv,E44

where

χW0ρΘt=12ρTν0Θtρ.

In this case, the integral term in (39) admits the following notation:

γ=R0qeiλTψXtYtΘtωΘv1iλTψXtYtΘtω(Θv)νPΘtvdv.E45

For the Gaussian CStS, the condition γ0 is, obviously, true, and we come to the well-known statements [11, 15, 17].

Statement 8. Let the conditions for existence and uniqueness be satisfied for the non-Gaussian CStS (36) and (37). Then, the equation with a continuous rms of the optimal nonlinear filtration for the nonnormalized characteristic function (38) is given by (39).

Statement 9. Let the non-Gaussian CStS (41) and (42) the conditions for existence and uniqueness be satisfied. Then, the equation with continuous rms of optimal nonlinear filtration for the nonnormalized characteristic function is given by (39) provided that (45).

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7. EAM (ELM) for nonlinear CStS filtering

EAM (ELM) for approximate conditionally optimal and suboptimal filtering (COF and SOF) in continuous CStS for normalized one-dimensional density is given in [11]. Let us consider the case of nonnormalized densities:

p˜txΘptuΘ=wuΘμt+ν=1Ncνpνu.E46

Here, w=wuΘ is the reference density and pνuqνu is the biorthonormal system of polynomials, Ct=Kt1; Kt is the covariance matrix and cν is the coefficient of ellipsoidal expansion

cν=μtEEAqνUt=qνUλg˜tEA(λΘ)λ=0,E47

with the notation

u=xTX̂tTCtxX̂t;Ut=XtTX̂tTCtXtX̂t;Uλ=T/iλX̂tCt/iλX̂t;E48

EEA is the expectation for the ellipsoidal distribution (46).

According to [11], in order to compile the stochastic differential equations for the coefficients cν, one has to find the stochastic Ito differential of the product qχug˜tλ bearing in mind that u depends on the estimate X̂t=mt/μt and the expectation mt and the normalizing function μt obey the stochastic differential equations. Therefore, one has to replace the variables x and u and the operators /iλ and Uλ, carry out differentiation, and then assume that λ=0.

So by repeating [11], we get that the equations for mt and μt with the function φ̂1 obey the formula

φ̂1=EΔxptφ1,E49

with regard to the notation

σ0=ψν0ψT,σ1=ψν0ψ1T,σ2=ψ1ν0ψ1TE50

and the equation for g˜tλΘ is representable as

dmt=fdt+hdYt,dμt=bdYt,E51
dg˜t=Adt+BdYt.E52

It is denoted here that

f=μtf0+ν=1Ncνfν,h=μth0+ν=1Ncνhν,b=μtb0+ν=1Ncνbν,
f0=f0YtX̂tΘt=EΔxwφ,fν=fνYtX̂tΘt=EΔxwpνφ,
h0=h0YtX̂tΘt=EΔUwσ1YtΘt+Xφ1(XYtΘt)Tσ2YtΘt1,hν=hνYtX̂tΘt=EΔUwpνσ1XYtΘt+Xφ1(XYtΘt)Tσ2YtΘt1,
b0=b0YtX̂tΘt=EΔUwφ1XYtΘtTσ2YtΘt1,bν=bνYtX̂tΘt=EΔUwpνφ1XYtΘtσ2YtΘt1,A=EΔxp˜tiλTφXYtΘt12λTψν0ψTXYtΘtλR0qeiλTψXYtΘtv1iλTψXYtΘtvνPtdveiλTX,B=EΔxp˜tφ1XYtΘtT+iλTψν0ψ1T(XYtΘt)eiλTXψ1ν0ψ1T1XYtΘt.E53

The equations for coefficient of MOE in (46) and (47) in virtue of [11] have the form

dcχ=EΔxp{qχu2φTCtXX̂t+trCtσ0+2qχuXTX̂tTCtσ0CtXX̂t+R0qqχu¯qχu2qχuXTX̂tTCtψνPtdvqχuXTX̂tTCth+X̂tbφ1/μt+qχutrh+X̂tbσ1TCt/μt+2qχutrh+X̂tbσ1TCtXX̂tXTX̂tTCt/μt}dt+12ncχ1+2χcχtrĊtKt+cχ12ntrCthσ2hT2X̂tTCthσ2bT+X̂tTCtX̂tbσ2bT/μt2dt+EΔxpqχuφ1T+qχuXTX̂tTCtσ1σ21dYt.E54

In addition to the notation (54), we assume that

γχ0=γχ0YtX̂tΘt=EΔxw{qχu2φXYtΘtTCtXX̂t+trCtσ0XYtΘt+2qχuXTX̂tTCtσ0XYtΘtCtXX̂t+R0qqχu¯qχu2qχuXTX̂tTCtψ(XYtΘtv)νPtdv},γχν=γχνYtX̂tΘt=EΔxwpν{qχu2φXYtΘtTCtXX̂t+trCtσ0XYtΘt+2qχuXTX̂tTCtσ0XYtΘtCtXX̂t+R0qqχu¯qχu2qχuXTX̂tTCtψ(XYtΘtv)νPtdv},εχ0=εχ0YtX̂tΘt=EΔxw{qχuσ1XYtΘtTφ1(XYtΘt)XTX̂tT+2qχuσ1XYtΘtTCtXX̂tXTX̂tT},εχν=εχνYtX̂tΘt=EΔxwpν{qχuσ1XYtΘtTφ1(XYtΘt)XTX̂tT+2qχuσ1XYtΘtTCtXX̂tXTX̂tT},ηχ0=ηχ0YtX̂tΘt=EΔxwqχuφ1XYtΘtT+qχuXTX̂tTCtσ1XYtΘtσ2YtΘt1,ηχν=ηχνYtX̂tΘt=EΔxwpνqχuφ1XYtΘtT+qχuXTX̂tTCtσ1XYtΘtσ2YtΘt1.E55

and then we can rearrange Eq. (54) in

dcχ=μtγχ0YtX̂tΘt+ν=1NcνγχνYtX̂tΘt+tr[μt(h0YtX̂tΘt+X̂tb0(YtX̂tΘt)+ν=1NcνhνYtX̂tΘt+X̂tbνYtX̂tΘtεχ0YtX̂tΘt+ν=1Ncνεχν(YtX̂tΘt)/μtCt]+12ncχ1+2χcχtrĊtKt+cχ12ntrCth0YtX̂tΘt+ν=1Ncνhν(YtX̂tt)/μtσ2YtΘt×h0YtX̂tΘtT+ν=1Ncνhν(YtX̂tΘt)T/μt2X̂tTCth0YtX̂tΘt+ν=1Ncνhν(YtX̂tΘt)/μt×σ2YtΘtb0YtX̂tΘtT+ν=1Ncνbν(YtX̂tΘt)T/μt+X̂tTCtX̂tb0YtX̂tΘt+ν=1Ncνbν(YtX̂tΘt)/μtσ2YtΘt×b0YtX̂tΘt+ν=1Ncνbν(YtX̂tΘt)T/μtdt+μtηχ0YtX̂tΘt+ν=1Ncνηχν(YtX̂tΘt)dYtχ=1N.E56

The modified ellipsoidal suboptimal filter (MESOF) is defined by Eqs. (51), (52), and (56) and the relation X̂t=mt/μt under the initial conditions

mt0=EX0Y0,μt0=1,cχt0=cχ0χ=1N,E57

(cχ0χ=1N are the coefficients of the expansion (46) of the probability density p˜t0x=p0xY0 of the vector X0 relative to Y0).

Upon solution of Eqs. (51), (52), (56), and (57), the rms optimal estimate of the state vector and the covariance matrix of filtration error in MESOF obey the following approximate formulae:

X̂t=mt/μt;Rt=EΔxwXmtμtXTmtTμt+ν=1NcνμtEΔxwpνXmtμtXTmtTμt.E58

Note that the order of the obtained MESOF, especially under high dimension n of the system state vector, is much lower than the order of other conditionally optimal filters. It is the case at allowing for the moments of up to the 10th order. Then, already for n>3 and N=5, we have n+N+1nn+3/2. We conclude that for n>3 and N=5, MECOF has a lower order than the filters of the method of normal approximation, generalized second-order Kalman-Bucy filters, and Gaussian filter. Thus, the following theorems underlie the algorithm of modified ellipsoidal conditionally optimal filtration.

Statement 10. Under the conditions of Statement 8, if there is MECOF, then it is defined by Eqs. (51), (52), and (56) under the conditions (57) and (58).

Statement 11. Under the conditions of Statement 9, if there is MESOF, then it is defined by the equations of Statement 10 under the conditions (45).

The aforementioned methods of MESOF construction offer a basic possibility of getting a filter close to the optimal-in-estimate one with any degree of accuracy. The higher the EA coefficient, the maximal order of the allowed for moments, the higher accuracy of approximation of the optimal estimate. However, the number of equations defining the parameters of the a posteriori one-dimensional ellipsoidal distribution grows rapidly with the number of allowed for parameters. At that, the information about the analytical nature of the problem becomes pivotal.

For approximate analysis of the filtration equations by following [11] and allowing for random nature of the parameters Θ, we come to the following equations for the first-order sensitivity functions [11]:

dΘX̂s=ΘAX̂sdt+ΘBX̂sdYt,ΘBX̂st0=0,dΘRsq=ΘARsqdt+ΘBRsqdYt,ΘRsqt0=0,dΘcκ=ΘAcκdt+ΘBcκdYt,Θcκt0=0.E59

Here the procedure of taking the derivatives is carried out over all input variables, and the coefficients of sensitivity are calculated for Θ=mΘ. It is assumed at that the variance is small as compared with their expectations. Obviously, at differentiation with respect to ΘΘ=/∂Θ, the order of the equations grows in proportion to the number of derivatives. The equations for the elements of the matrices of the second sensitivity functions are made up in a similar manner.

To estimate the MESOF (MECOF) performance, we follow [5, 8] and introduce for the Gaussian Θ with the expectation mΘ and covariance matrix KΘ the conditional loss function admitting quadratic approximation, the factor ε=ε21/4, as well as

ρX̂s=ρX̂sΘ=ρmΘ+ii=1nΘρi'mΘΘs0+i,j=1ρij''mΘΘi0Θj0.E60

It is denoted here

ε2=EEAρΘ2ρmΘ2,EEAρΘ2=ρmΘ2+ρmΘTKΘρmΘ+2ρmΘtrρmΘKΘ+trρmΘKΘ2+2trρmΘKΘ2.E61

At that, in (61) the functions ρ and ρ" are determined through certain formulas on the basis of the first and second sensitivity functions. Therefore, we come to the following result.

Statement 12. Estimation of MESOF (MECOF) performance under the conditions of Statements 10 and 11 relies on Eqs. (59)(61) under the corresponding derivatives in the right sides of Eq. (59).

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8. New types of continuous MECOF

Based on Statements 10 and 11 in [18], continuous MECOF were described. We consider the problem of continuous conditionally optimal filtration for the general case of Eqs. (34) and (35) where it is desired to determine the optimal estimate X̂t of process Xt at the instant t>t0 from the results of observation of this process until the instant t, that is, over the interval t0t, in the class of permissible X̂t=AZt estimates and with a stochastic differential equation given by

dZt=αtξYtZtΘt+γtdt+βtηYtZtΘtdYtE62

under the given vector and matrix structural functions ξ and η and every possible time functions αt,βt,γt (αt and βt are matrices and γt is a vector). The criterion for minimal rms error of the estimate Zt is used as the optimality criterion. It is common knowledge that selection of the class of permissible filters defined by the structural functions ξ and η in Eq. (62) is the greatest challenge in practice of using the COF theory [1, 3, 11]. In principle they can be defined arbitrarily. One can select ξ and η at will so that the class of permissible filters contained an arbitrarily defined COF. In this case, COF is in practice more precise than the given COF. At the same time, by selecting a finite segment of some basis in the corresponding Hilbertian space L2 as components of the vector function ξ and elements of the matrix function η, one can obtain an approximation with any degree of precision to the unknown optimal functions ξ and η. This technique of selecting the functions ξ and η on the basis of the equations of the theory of suboptimal filtration seems to be the most rational one. At that, the COF equations obtained from the equation for the nonnormalized a posteriori characteristic function open up new possibilities.

To use the equations obtained from nonnormalized equations for the a posteriori distribution, one needs to change the formulation of the COF problems [3, 11] so as to use the equation for the factor μt. For that, we take advantage of the following equations to determine the class of permissible continuous MECOF (62):

dμt=ρtχYtZtΘtdYt,E63
X̂t=AZt/μt,E64

where χYtZtΘt is a certain given structural matrix function and ρt is the row matrix of coefficients depending on t and subject to rms optimization along with the coefficients αt,βt, and γt in the filter Eq. (62).

Relying on the results of the last section and generalizing [7], one can specify the following types of the permissible MECOF:

  1. This type of permissible MECOF can be obtained by assuming Zt=mt,A=In and determining the functions ξ,η,χ in Eqs. (62) and (63) and obeying Eq. (51) which gives rise to the following expressions for the structural functions:

ξ=ξYtZtΘt=μtf0YtZt/μtΘtTf1(YtZt/μtΘt)TfN(YtZt/μtΘt)TT;E65
η=ηYtZtΘt=μth0YtZt/μtΘtTh1(YtZt/μtΘt)ThN(YtZt/μtΘt)TT;E66
χ=χYtZtΘt=μtb0YtZt/μtΘtTb1(YtZt/μtΘt)TbN(YtZt/μtΘt)TT.E67

At that, the order of MECOF defined by Eqs. (62) and (63) is equal to n+1. This type of MECOF may be designed for Zt being constant Z0 and A=I<n.

  1. 2. To obtain a wider class of permissible MECOF, rearrange Eq. (56) in

dcχ=Fχ0YtZtΘt+ν=1NcνFχνYtZtΘt+λ,ν=1NcλcνFχλνYtZtΘt+cχ1λ,ν=1NcλcνFχλνYtZtΘtdt+μtηχ0YtZtΘt+ν=1Ncνηχν(YtZtΘt)dYtE68

with the following notations:

Fχ0YtZtΘt=μtγχ0YtZtΘt+μttrh0YtZtΘtZtb0YtZtΘtεχ0YtZtΘt;FχνYtZtΘt=γχ0YtZtΘt+tr[hνYtZtΘt+ZtbνYtZtΘtεχ0YtZtΘt+(h0YtZtΘt+Ztb0YtZtΘt)εχνYtZtΘt]+12nδχ1,ν{trbνKt+Cth0YtZtΘtσ2(YtΘt)h0(YtZtΘt)T2ZtTCth0YtZtΘtσ2YtΘtb0YtZtΘtT+ZtTCtZtb0YtZtΘtσ2YtΘtb0YtZtΘtT}+1nχδχνtrĊtKt;Fχλν=trCt(hλYtZtΘt+Ztbλ(YtZtΘt))εχν(YtZtΘt)/μt+1nδχ1,λ{trCth0YtZtΘtσ2(YtΘt)hν(YtZtΘt)TZtTCt(h0YtZttσ2YttbνYtZtΘtT+hνYtZtΘtσ2YtΘtb0YtZtΘtT+ZtTCtZtb0YtZtΘtσ2YtΘtbνYtZtΘtT}/μt;
Fχλν=12n{trCt(hλYtZtΘt+σ2(YtΘt)hν(YtZtΘt)T2ZtTCthλYtZtΘtσ2YtΘtbνYtZtΘtT+ZtTCtZtbλYtZtΘtσ2YttbνYtZtΘtT}/μt2.E69

By taking as the basis for the type of permissible MECOF Eqs. (51), (52), and (68), one has to regard all components of the vector Zt as all components of the vector mt and coefficients c1,,cN so that Zt=mtTc1cNT. At that, the order of all permissible filters is equal to n+N+1. Putting Zt=Z0 and A=Ill<n, one gets the corresponding MECOF.

  1. 3. The widest class of permissible filters providing MECOF of the maximal reachable accuracy can be obtained if one takes the function ξ in (62) as the vector with all components of the vector functions μfχ0,cνfχνχν=1N in Eqs. (65) and (66) all addends involved in the scalar functions Fχ0,cνFχν,cλcνFχλνχλν=1Ncχ1cλcνFχλν, χ1λν=1N in (68) and as the function η in (62), the matrix whose rows are the row matrices μthχ0,cνhχνχν=1N in (68) and all row matrices μηχ0,cνηχνχν=1N in (69). As for the function χ in Eq. (63), it is determined through (67) as in the case of the simplest types of permissible filters. The so-determined class of permissible filters has ECOF defined by Eqs. (51), (52), and (69), at that ECOF is more precise than ESOF. We notice that this class of permissible filters can give rise to an overcomplicated ECOF because of high dimension of the structural vector function ξ. So we distinguish the following new type of permissible filters.

  2. 4. Components of the vector function ξ are all components of the vector functions μtfχ0,cνfχνχν=1N and all scalar functions Fχ0,cνFχν,cλcνFχλνχλν=1N, cχ1cλcνFχλνχ1λν=1N without decomposing them into individual addends. This class of permissible filters also includes ECOF (51), (52), and (69).

To determine the coefficients αt,βt, and γt of the equation MECOF (62), one needs to know the joint one-dimensional distribution of the random processes Xt and X̂t. It is determined by solving the problem of analysis of the system obeying the stochastic differential Eqs. (62) and (63). As always in the theory of conditionally optimal filtration, all complex calculations required to determine the optimal coefficients of the MECOF Eq. (62) or (63) are based only on the a priori data and therefore can be carried out in advance at designing MECOF. At that, the accuracy of filtration can be established for each permissible MECOF. The process of filtration itself comes to solving the differential equation, which enables one to carry out real-time filtration.

Consequently, we arrive to the following results.

Statement 13. Under the conditions of Statement 8, the MECOF equations like (62) and (63) coincide with the equations of continuous MECOF where the structural functions belong to the four aforementioned types.

Statement 14. The rms MECOF of the order nx+1 coinciding with MECOF is defined for CStS (34) and (35), Eqs. (62)(64), and the structural functions of the first class.

Statement 15. The rms of MECOF of the order nx+N+1 coinciding with MECOF obeys for the CStS (34) and (35), Eqs. (62)(64), and the structural functions of Statement 14.

Statement 16. If accuracy of MECOF determined according to Statement 14 is insufficient, then the functions of the Statement 15 can be used as structural ones.

Statement 17. The relations of Statement 12 underlie the estimate of quality of MECOF under the conditions of Statements 13–15, provided that there are corresponding derivatives in the right sides of the equations.

Example 3. The presented MECOF for linear CStS coincide with Kalman-Bucy filter [2, 3, 4, 11].

Example 4. MECOF for linear CStS with parametric noises coincide with linear Pugachev conditionally optimal filter.

Finally let us consider quasilinear CStS (36) and (37), reducible to the following differential one:

Ẋt=φXtΘt+ψtΘV1EL,E70
Ẏt=φ1XtΘt+V2ELE71

where V1 and V2 are non-Gaussian white noises. In this case using ELM and Kalman-Bucy filters with parameters depending on mtx,Ktx and c1tx, we get the following interconnected set of equations:

ṁ1x=φ00mt0x=m0x,ṁ1y=φ10mt0y=m0y,E72
Ẋt0=φ01Xt0+ψtΘV1,Ẏt0=φ11Xt0+V2EL,Yt00=Y00,E73
K̇tx=φ11Ktx+Ktxφ11T+ψtΘG1ELtΘψtΘT,Kt0x=K0x,E74
X̂̇t=φ00φ01mtx+φ01X̂t+RtG2ELtΘ1Ẏtφ11X̂tφ10+φ11mtx,X̂0=MELX̂t0,E75
Ṙt=φ01RtRtφ01Rtφ11G2ELtΘ1φ11Rt+ψtG1ELtΘψtT,R0=MELX0X̂0X0X̂0T.E76

Here the following notations are used:

mtx=MELXt,mty=MELYtandKtx=MELXt0Xt0T,Rt=MELXtX̂tXtX̂tTE77

being the mathematical expectations, state, and error covariance matrices

φ00=φ00mtxKtxc1txtΘ,φ10=φ10mtxKtxc1txtΘ,φ01=φ01mtxKtxc1txtΘ=φ01mtx,φ11=φ11mtxKtxc1txtΘ=φ10mtxE78

being ELM ecoefficiencies, GiEL (i=1,2) are intensities of normal EL equivalent white noises. So Eqs. (72)(78) define the corresponding Statement 18.

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9. Ellipsoidal Pugachev conditionally optimal continuous control

The idea of conditionally optimal control (COC) was suggested by Pugachev (IFAC Workshop on Differential Games, Russia, Sochi, 1980) and developed [13]. The COC essence is in the search of optimal control among all permissible controls (as in classical control theory) but in the restricted class of permissible controls. These controls are computed in online regime. At practice the permissible continuous class of controls may be defined by the set of ordinary differential equations of the given structure.

So let us consider the following Ito equations:

dX=φXYUtdt+ψ1XYUtdW0+R0qψ2XYUtvP0dtdv,E79
dY=φ'XYUtdt+ψ'1XYUtdW0+R0qψ'2XYUtvP0dtdv.E80

Here X is the nonobservable state vector; Y is the observable vector; UD is the control vector; W0 being the Wiener StP, P0AB being the independent of W0 centered Poisson measure; φ,ψ1,ψ2 and φ',ψ'1,ψ'2 being the known functions. Integration is realized in Rq space with the deleted origin. Initial conditions X0 and Y0 do not depend on X and Y. Functions φ,ψ1,ψ2 in (79) as a rule do not depend on Y, but depend on U components that are governed by Eq. (79). Functions φ',ψ'1,ψ'2 in Eq. (80) depend on U components that govern observation.

The class of the admissible controls is defined by the equations

dU=αξYUt+γdt+βηYUtdYE81

without restrictions and with restrictions

dU=αξYUt+γdt+βηYUtdYmax0nUTαξYUt+γdt+nUTβηYUtdYnU1DU.E82

Here nU is the unit vector of external normal for boundary D in point U; 1DU is the set indicator.

Conditionally optimal criteria is taken in the form of mathematical expectation of some functional depending on Xt0t=Xτ:τt0t and Ut0t=Uτ:τt0t:

ρ=EℓXt0tUt0tt,E83

where E is the mathematical expectation and is the loss function at the given realizations xt0t,ut0t of Xt0t,Ut0t.

So according to Pugachev we define COC as the control realized by minimization (83) by choosing α,β,γ and by satisfying (82) at every time moment and at a given α,β,γ for all preceding time moments. For the loss function (83) depending on X and U at the same time, moment t is necessary to compute ellipsoidal one-dimensional distribution of X and Y in Eqs. (79), (80), and (82) using EAM (ELM). This problem is analogous to COF and MCOF design (Section 8).

For high accuracy and high availability CStS especially functioning in real-time regime, software tools “StS-Analysis,” “StS-Filtering,” and “StS-Control” based on NAM, EAM, and ELM were developed for scientists, engineers, and students of Russian Technical Universities.

These tools were implemented for solving safety problems for system engineering [19].

In [18, 20] theoretical propositions of new probabilistic methodology of analysis, modeling, estimation, and control in stochastic organizational-technical-economical systems (OTES) based on stochastic CALS informational technologies (IT) are considered. Stochastic integrated logistic support (ILS) of OTES modeling life cycle (LC), stochastic optimal of current state estimation in stochastic media defined by internal and external noises including specially organized OTES-NS (noise support), and stochastic OTES optimal control according to social-technical-economical-support criteria in real time by informational-analytical tools (IAT) of global type are presented. Possibilities spectrum may be broaden by solving problems of OTES-CALS integration into existing markets of finances, goods, and services. Analytical modeling, parametric optimization and optimal stochastic processes regulation illustrate some technologies and IAT given plans. Methodological support based on EAM gives the opportunity to study infrequent probabilistic events necessary for deep CStS safety analysis.

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10. Conclusion

Modern continuous high accuracy and availability control stochastic systems (CStS) are described by multidimensional differential linear, linear with parametric noises, and nonlinear stochastic equations. Right-hand parts of these equations also depend on stochastic factors being random variables defining the dispersion in engineering systems parameters. Analysis and synthesis CStS needs computation of non-Gaussian probability distributions of multidimensional stochastic processes. The known analytical parametrization modeling methods demand the automatic composing and the integration of big amount interconnected equations.

Two methods of analysis and analytical modeling of multidimensional non-Gaussian CStS were worked out: ellipsoidal approximation method (EAM) and ellipsoidal linearization method (ELM). In this case one achieves cardinal reduction the amount of distribution parameters.

Necessary information about ellipsoidal approximation methods is given and illustrated. Some important remarks for engineers concerning EAM accuracy are given. It is important to note that all complex calculations are performed on design stage. Algorithms for composition of EAM (ELM) equation are presented. Application to problems of shock and vibroprotection are considered.

For statistical CStS offline and online analysis approximate methods based on EAM (ELM) for a posteriori distributions are developed. In this case one has twice reduction of equation amount. Special bank of approximate suboptimal and Pugachev conditionally optimal filters for typical identification and calibration problems based on the normalized and nonnormalized was designed and implemented.

In theoretical propositions of new probabilistic methodology of analysis, modeling, estimation, and control in stochastic OTES based on stochastic CALS information technologies (IT) are considered. Stochastic integrated logistic support (ILS) of OTES modeling life cycle, stochastic optimal of current state estimation in stochastic media defined by internal and external noises including specially organized OTES-NS (noise support), and stochastic OTES optimal control according to social-technical-economical-support criteria in real time by informational-analytical tools (IAT) of global type are presented. Possibility spectrum may be broaden by solving problems of OTES-CALS integration into existing markets of finances, goods, and services. Methodological support based on EAM (ELM) gives the opportunity to study infrequent probabilistic events necessary for deep CStS safety analysis.

Acknowledgments

The authors would like to thank Russian Academy of Science for supporting the work presented in this chapter. Authors are much obliged to Mrs. Irina Sinitsyna and Mrs. Helen Fedotova for translation and manuscript preparation.

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Written By

Igor N. Sinitsyn, Vladimir I. Sinitsyn and Edward R. Korepanov

Submitted: October 23rd, 2019 Reviewed: December 2nd, 2019 Published: February 13th, 2020