Open access peer-reviewed chapter

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

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

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

DOI: 10.5772/intechopen.90732

## 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=yTmTCymwhere mis an expectation of Y,Cis some positively determined matrix. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters till QEAM=QNAM+nm1and QNAM=rr+3/2where 2nmbeing 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.

## 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=uyfor which some probability density of the ellipsoidal structure wuyserves as the weight:

wuypr,νuyqr,μuydy=δνμ.E1

The indexes νand μat 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 Ymay be approximately presented by the expression of the form:

fyfu=wu1+ν=2Ncr,νpr,νu.E2

Here the coefficients cr,νare determined by the formula:

cr,ν=fyqr,νudy=Eqr,νU,ν=1N.E3

As pr,0uand qr,0uare reciprocal constants (the polynomials of zero degree), then always cr,0pr,0=1and 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 wuis 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 Γis gamma function [5].

Statement 2. The problem of the choosing of the polynomial system pr,νuqr,μuwhich 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 rdegrees of the freedom serves as the weigh.

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

Sr,νu=μ=0ν1ν+μCνμr+2ν2!!r+2μ2!!uμ.E6

The main properties of polynomials Sr,νare given in [2, 3, 4]. Between the polynomials Sr,νuand 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,νuand qr,νuand its derivatives for some rand νare 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=2at ν=3we 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 L2Rrand the orthogonal system of the functions in them where the polynomials Sr,νuare given by Formula (6), and wuis 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=fyTmTCymof the random vector Ywhich has an ellipsoidal structure over the polynomials pr,νu=Sr,νu, m.s. converges to the function fuitself. 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,νuforms the basis in the subspace of the space L2Rrgenerated by the functions fuof the quadratic form u=ymTCym.

At the probability density expansion over the polynomial Sr,νu, the probability densities of the random vector Yand all its possible projections are consistent. In other words, at integrating the expansions over the polynomials Sh+l,νuand 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,νu1of the probability density of the h-dimensional vector Ywith the same coefficients

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

where K11is 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 Nthorder inclusively of the quadratic form U=YmTK1Ym, i.e.,

EUμ=EEAUμ,μN.E11

(EEAstands for expectation relative to EA distribution).

In this case the initial moments of the order sand s=s1++srof the random vector Yat 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,νuby the degrees of the components of the vector Yat the normal density wu.

## 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,νuto the density, namely,

Awu1+ν=2Ncr,νpr,νuAfudy

uniformly relative to Aat Non 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 Adetermined by the density fuwith any degree of the accuracy. The finite segment of this expansion may be practically used for an approximate presentation of fuwith any degree of the accuracy even in those cases when fu/wudoes not belong to L2Rr. In this case it is sufficient to substitute fuby 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].

## 4. Ellipsoidal linearization method

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

mzY=h2Y,h2=ΓzyΓy1E13

or

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

where h1and h2are equivalent linearization matrices and myand Kyare mathematical expectation and covariance matrix detKy0. In case (14) coefficient h1is equal to

h1=KzyKy1=zmzymyTKy1fzydzdy=mzymzymyTKy1f1ydyE15

where f1yis the density of random vector Y.

For ellipsoidal density f1yin (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=dimyand Nin (2) with normal wudistinguish modifications of various orders ELM wp,2, ELM wp,3, …, ELM wp,N. In this case c=cp,vcharacterizes partial deviations from normal distributions of various orders vjointly for all pcomponents of vector Y(be part of quadratic form UY.

2. At decomposition of vector Yon l1,l2,,lrrandom subvectors, Y=Yl1TYl2TYlrTT, we distinguish ELMwl1,,lr,N. Coefficients cl1,v,,cl2,vcharacterize 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

(h1iELand 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 mizand hiELfor time intervals.

## 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 YtΔyis (Δyis a smooth state manifold) W0=W0tis 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 υextends to the entire space Rqwith deleted origin, aand bare certain functions mapping Rp×R, respectively, into Rp,Rpr, and cis for Rp×Rqinto Rp.

Following [4] we use for finding the one-dimensional probability density f1ytof the r-dimensional Ytwhich is determined by Eq. (25). Suppose that we know a distribution of the initial value Y0=Yt0of 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=yTmTCymwhere mand K=C1are the expectation and the covariance matrix of the StPYt:

f1ytf1EAMu=w1u1+ν=2Ncp,vpp,νu.E26

Here w1uis 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,vare determined by the relation

cp,v=f1ytqp,vudy=Eqp,vU,ν=1N.E27

The set of the polynomials pp,vuqp,vuis constructed on the base of the orthogonal set of the polynomials Sp,vuaccording to the following rule which provides the biorthonormality of system at p2given 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 Kof the state vector, and the coefficients of the correspondent expansion cp,valso.

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,Nas time functions. For finding the variables cp,κ0, the density of the initial value Y0of 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,Uare constants, Vis 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=0stationary 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, K22are given in Figures 13. Mathematical expectations m1and mnare equal to zero.

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

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

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].

## 6. Exact filtering equations for continuous a posteriori distribution

Following [7, 8, 9, 15], let the vector StP XtTYtTTbe 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=Ytis an ny-dimensional observed StP YtΔy(Δyis a smooth manifold of observations); XtΔx(Δxis a smooth state manifold), W0=W0tis an nw-dimensional Wiener StP nwnyof intensity ν0=ν0Θt;P0ΔA=PΔAμPΔA,PΔAfor any set Arepresents a simple Poisson StP, and μPΔAis its expectation,

μPΔA=EPΔA=ΔνPτA;

vPΔAis the intensity of the corresponding Poisson flow of events, Δ=t1t2; integration by υextends to the entire space Rqwith deleted origin; Θis the vector of random parameters of size nΘ;φ=φXtYtΘt,φ1=φ1XtYtΘt,ψ=ψXtYtΘt,and ψ1=ψ1XtYtΘtare certain functions mapping Rnx×Rny×R, respectively, into Rnx,Rny,Rnxnw, Rnynw; ψ=ψXtYtΘtv,and ψ1=ψ1XtYtΘtvare certain functions mapping Rnx×Rny×Rqinto Rnx,Rny. Determine the estimate X̂tStP Xtat each time instant tfrom the results of observation of StP Yτuntil 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'1in 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 X̂tminimizing the mean square of the error at each time instant tis known [10, 11, 12, 13, 14] to represent for any StP Xtand Yt.

An a posteriori expectation StP Xt: X̂t=EXtYt0t. To determine this conditional expectation, one needs to know pt=ptxand 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˜txΘand a characteristic function g˜tλΘaccording to

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

where μtis a normalizing function and EΔxptis the symbol of expectation on the manifold Δxon 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 ψin (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/tdvis 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 γ0is, 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).

## 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=wuΘis the reference density and pνuqνuis the biorthonormal system of polynomials, Ct=Kt1; Ktis the covariance matrix and cνis 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

EEAis 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˜tλbearing in mind that udepends on the estimate X̂t=mt/μtand the expectation mtand the normalizing function μtobey the stochastic differential equations. Therefore, one has to replace the variables xand uand the operators /iλand Uλ, carry out differentiation, and then assume that λ=0.

So by repeating [11], we get that the equations for mtand μtwith the function φ̂1obey the formula

φ̂1=EΔxptφ1,E49

with regard to the notation

σ0=ψν0ψT,σ1=ψν0ψ1T,σ2=ψ1ν0ψ1TE50

and the equation for g˜tλΘis representable as

dmt=fdt+hdYt,dμt=bdYt,E51

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/μtunder the initial conditions

mt0=EX0Y0,μt0=1,cχt0=cχ0χ=1N,E57

(cχ0χ=1Nare the coefficients of the expansion (46) of the probability density p˜t0x=p0xY0of the vector X0relative 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 nof 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>3and N=5, we have n+N+1nn+3/2. We conclude that for n>3and 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 Θwith the expectation mΘand covariance matrix KΘthe 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 ρand 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).

## 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 X̂tof process Xtat the instant t>t0from the results of observation of this process until the instant t, that is, over the interval t0t, in the class of permissible X̂t=AZtestimates and with a stochastic differential equation given by

dZt=αtξYtZtΘt+γtdt+βtηYtZtΘtdYtE62

under the given vector and matrix structural functions ξand ηand every possible time functions αt,βt,γt(αtand βtare matrices and γtis a vector). The criterion for minimal rms error of the estimate Ztis used as the optimality criterion. It is common knowledge that selection of the class of permissible filters defined by the structural functions ξand ηin 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 ξand ηat 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 L2as components of the vector function ξand elements of the matrix function η, one can obtain an approximation with any degree of precision to the unknown optimal functions ξand η. This technique of selecting the functions ξand ηon 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 χYtZtΘtis a certain given structural matrix function and ρtis the row matrix of coefficients depending on tand subject to rms optimization along with the coefficients αt,βt, and γtin 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=Inand determining the functions ξ,η,χin 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 Ztbeing constant Z0and 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 Ztas all components of the vector mtand coefficients c1,,cNso that Zt=mtTc1cNT. At that, the order of all permissible filters is equal to n+N+1. Putting Zt=Z0and 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 ξin (62) as the vector with all components of the vector functions μfχ0,cνfχνχν=1Nin Eqs. (65) and (66) all addends involved in the scalar functions Fχ0,cνFχν,cλcνFχλνχλν=1Ncχ1cλcνFχλν, χ1λν=1Nin (68) and as the function ηin (62), the matrix whose rows are the row matrices μthχ0,cνhχνχν=1Nin (68) and all row matrices μηχ0,cνηχνχν=1Nin (69). As for the function χin 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 ξare all components of the vector functions μtfχ0,cνfχνχν=1Nand all scalar functions Fχ0,cνFχν,cλcνFχλνχλν=1N, cχ1cλcνFχλνχ1λν=1Nwithout decomposing them into individual addends. This class of permissible filters also includes ECOF (51), (52), and (69).

To determine the coefficients αt,βt, and γtof the equation MECOF (62), one needs to know the joint one-dimensional distribution of the random processes Xtand 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+1coinciding 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+1coinciding 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 V1and V2are non-Gaussian white noises. In this case using ELM and Kalman-Bucy filters with parameters depending on mtx,Ktxand 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.

## 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 Xis the nonobservable state vector; Yis the observable vector; UDis the control vector; W0being the Wiener StP, P0ABbeing the independent of W0centered Poisson measure; φ,ψ1,ψ2and φ',ψ'1,ψ'2being the known functions. Integration is realized in Rqspace with the deleted origin. Initial conditions X0and Y0do not depend on Xand Y. Functions φ,ψ1,ψ2in (79) as a rule do not depend on Y, but depend on Ucomponents that are governed by Eq. (79). Functions φ',ψ'1,ψ'2in Eq. (80) depend on Ucomponents 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 nUis the unit vector of external normal for boundary Din point U; 1DUis the set indicator.

Conditionally optimal criteria is taken in the form of mathematical expectation of some functional depending on Xt0t=Xτ:τt0tand Ut0t=Uτ:τt0t:

ρ=EℓXt0tUt0tt,E83

where Eis the mathematical expectation and is the loss function at the given realizations xt0t,ut0tof Xt0t,Ut0t.

So according to Pugachev we define COC as the control realized by minimization (83) by choosing α,β,γand by satisfying (82) at every time moment and at a given α,β,γfor all preceding time moments. For the loss function (83) depending on Xand Uat the same time, moment tis necessary to compute ellipsoidal one-dimensional distribution of Xand Yin 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.

## 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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Igor N. Sinitsyn, Vladimir I. Sinitsyn and Edward R. Korepanov (February 13th 2020). Development of Ellipsoidal Analysis and Filtering Methods for Nonlinear Control Stochastic Systems, Automation and Control, Constantin Voloşencu, Serdar Küçük, José Guerrero and Oscar Valero, IntechOpen, DOI: 10.5772/intechopen.90732. Available from:

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