Problems of Analysis and On-Line Conditionally Optimal Filtering of Processes in Nonlinear Stochastic Systems

where Y1 is a white noise usually independent of the process W, fJJ( (1, t) " 1f1 Cl, t) functions mapping RPx IL . t 0 Rnt ~n and R m6 respectively, d being the dimension of the white noise VI' The problem is to design an estimator for Z t or Z t+'i using at any

t

t> to

the observations X to = "") ,.. [t 0' t)} realizable in X( ={ ~: ... E real basis of time. The natural generalization of this estimation problem is stated in the following way: the (p+m) -dimensional random process [Z(t) 'I' y(t)'r]'r is defined by stochastic differential equations of the form

dy=
+

\'. S. Pug-ache \ , I. :-.I . Sinits\n and \ ' . I. Shin

10

ciz= CP(Y, Z,t)dt +

'feY,

z, t )dw .

in (2) and (14) by

VIC)WIC

+

( 11 )

The process Y being continuously observed at t > to an estimator for Zt is required using at any t > to the observations =[Y('f): 'fE[to,t)}.

Y/

o

V. I = [ VT" T.T

tc

IC

W tc

T

]

•

Therefore we shall write

(14) in sequel as

(15) is the same sequence of where {VIC } independent random variables as in (2).

The respective generalization of the extrapolation problem is possible only under the additional assumption that the functions cP and "If in the second equation of (11) are independent of Y and the process ~(t) consists of two independent blocks one of which enters only in the first equation of (11) and the other only into the second one (pugachevand Sinitsyn, 1985).

If [W/C} is a Markov sequence then it may be defined by a difference equation of the form of (2) in terms of independent random variables. The problem may be reduced to the case of state and observation equations of the form of (2), (15) by means of the extension of the system state vector Z IC including into it the noise W/C.

If the noise in the observations in different from a white noise then the observation equation (10) may be taken in a more general form, namely

The problem of simultaneous estimation of the state and unknown parameters of a system (2), (15) may be reduced to the problem (2), (5) in the usual way by extending the system state vector.

x

=

CfI

(

Z, N,

t)

(12)

where N= N(t) is the noise. Ne shall assume in this case that the random process N(t) is defined by an Ito stochastic differential equation of the form

dN= c.Po(N,t)dt +ljJo(N, t)dW1

(13)

~ et) being a '.'{iener process independent of the process Wet) in (9). In this case the problem may be reduced to the previous problem by differentiations of the observation equation (12) and by the respective extention of the observable vector X (Siluyanova, 1985; see also pugachev and Sinitsyn, 1985).

This problem of state estimation covers also the problem of simultaneous estimation of the system state and unknown parameters upon which tp , "If ,~ , 1ff may depend, Z being the extended state vector in this case containing all the unknown parameters as components.

3.2. Discrete Observations in a Discrete Stochastic system Let us consider a discrete stochastic system (2). suppose that a m -dimensional vector function is measured at the time moments t (IC) = /C T (IC = 1,2, ... ) with independent random errors. Then the observable process represents a sequence of random variables [XI<.] determined by

X IC

Z le , WIC ) (14) w- ) are some func tions

= WIC (

where W IC (.?:, mapping RP x R'1: into Rnt , [WIC} a sequence of independent random variables with known distributions. The problem is to design an estimator for Z 1C+1 or using at each time moment t ~0 the results of observations XI'''') XIC •

Z ~+j

The sequence of random variables {WIC ] in (14) is usually independent of the sequence{VIC } in (2). But this is not essential from the mathematical viewpoint since one may always replace

3.3. Discrete Observations in a Stochastic Differential System At last let us consider a stochastic differential system of the form of (9) in the case where the results of observations are described by equation (15), ZIC being the value of the process Z(t) at the time moment t(le) • The problem is to find an estimator for Z 1C+1 using the 0 bserva tions XI"", X le which is realizable in real basis of time. A more general problem is the estimation problem in a discrete-continuous system with the state vector Z(t):[Z'(t)7' ZN(t)""]'"

d Zl = CJl( z, t)dt +

Z

1/

(t)

00

=

1/

1f( z, t)clW ,

A

L.. Z 1C1A/C et) , /C=O

This problem consists in finding an estimator for the values Z /(+1 of the state vector Z(t) of the system at the moment t (/C+I) or for its future values Z/(,+J+1 using observations X I " " . ) X /C. defined by (15). To this problem is also reduced the estimation problem for system (9) when the results of observations are given by (14) with the sequence [W/C} determined by a difference equation of the form of (2) with independent random variables [V/C} , the sequence {VIC} being independent of the process W(t).

4. SOLUTION OF THE ANALYSIS PROBLEMS 4.1. Stochastic Differential Systems

11

Prublems of ."\'nah·sis

The finite-dimensional characteristic functions

9n- P1"'"

An. ; t 1 , ... , tfL ) r

n-

=.Mexp[iL.AICZt} 1C;1 IC

=

(n;1,2, ... )

1

of the random process Z(t) defined by equation (1), Zt=Z(t) , are successively determined at by the equations

t1 <. '"

<:

tn.-1

<:

tn.

a9",(?'1,···,An,; t 1 ,···,t n )18t n = =

M{L:~;a(Ztn-,t",)+ X(£(ZtfL,tn,r~lL;

t"J]exp[ii= fC;

1

A: ZtIC ]

(n;I,2, ... )

(17) p

- z:::

with the initial conditions

~=1

=

911.-/ A1'···' An.-f+ An,; (n= 2,3, '"

)

t 1 ,· .. ,

tn.-1 )

(18 )

where X (jL; t) is the logarithmic time derivative of the one-dimensional characteristic function of the process Hf(t) and go().) is the characteristic function of the initial value Z o = Z t 0 of the process Z(t) (Pugachev, 1944, 1965, 1980, 1981; Pugachevand Sinitsyn, 1985). Equations (17), (1 8 ) can be exactly solved only in some s pecial cases (Pugachev, 1944, 1980, 198 1 - ca s e of linear systems; Siluyanova, 1982 - case of some class of nonlinear systems). In the general case one should use approximate methods for solving equations (17), (1 8 ). The simplest approximate method for solving (17), (18) is the normal approximation method first proposed by pugachev (1944). This method is based on deriving from (17), (1 8 ) the equations for the first and the second moments and evaluating the expectations in these e~uat~ons by replacing the unknovm true d~str~but­ ion by the normal one (Pugachev and Sini t syn , 1985 ) • Generilizing the normal approximation method the method of moments and other approximate methods based on the parametrization of distributions are used (Demukh, 1965). Equation (17) at n= 1 yields the following e.quations for the initial and central moments

/C

where m = M Z is the expectation of the state vector Z of the system, ~ the vector subscript, <:: = [Z.1 '" Z-r J'I' .> <5(z,)= <:1+ .. . + 'l: the vector , e oc r whose all the components are equal to zero except the ~th component which is equal to unity. The method of moments consists in evaluating the expectations in (19), (20), (21) by replacing the known true distribution by some appro~­ mating function which is completely determined by the moments of orders 1, ••• ••. , N . Then equations (19) corresponding to '(,1"'" 'tp ~o,1, ... , "'; fi('t)= 1, .... , N or equations (20), (21) correspond~ng to

w:-

"(,1, . . . ,

Z- P

= 0, 1, ... ,

N ;

($( 't ) = 'l, .. . , N

become a closed set of ordinary differential equations approximately determining with the respective initial conditions all the moments of orders 1 J • • • , N and thus determining the function approximating the one-dimensional distribution of the system state. In the special case of "';2 equations (19) and (20), (21) represent the equations of the normal approximation method. Similarly the approximate equations for the moments of other finite-dimensional distributions are derived (Pugachev and Sinitsyn, 1985). As a function approximating the unknown distribution a truncated orthogonal expansion of a density, in particular the truncated Hermite polynomial expansion or Edgworth series may be used. Another efficient approximate method of stochastic differential system analysis is the method of orthogonal expansions (method of quasimoments) (Boguslavsky, 1969· Pugachev and Sinitsyn, 1985). This meth~d is based upon the approximate representation of the unknown density .£(:l;0 or fn.(Zf,..·,:tIl.;tf, ... ,tn.) ly a partial sum of some its orthogonal expansion, ~

By M.r we denote the expectation of the random variable )( •

(21 )

M(1/C(Z,t)jU1:_e

f1(I;t)~ff (:l;8)=

\', S.

12

Pl:~ache\,

I. :\, Sinitsm and \', I. Shin

(21) at

=UT(:l)[1+f L, CyPy(:l)} 10=3

(22)

G'(~)=/C

e

where is the vector parameter consisting of the components of the expectation m of the vector Z t ,of the elements of its covariance matrix}{ and of all the coefficients C~ involved in (22), WO) a density with the same expectation ~ and covariance matrix K as f l (.l;t), [py(Z)} a sequence of polynomials satisfying \'Iith another sequence of polynomials {q,-\) (l)} the biortho~onality condition 00

Jw-eX)?:

.y

- 00

(:l)Cf

fl

(z)dZ =

D-)l

(23)

'P'

'5~fl being the Kroneker symbol, 0-v-v = 1 ~ 'by = 0 at.JIL *' -» , 'Y,,.J-l the vector ~bscripts .:;l = [-\)1'" :;)p J'1', ~=LfL1

f'

.. , JA-p and G'(-v)= The coefficients C""

~1 +

.. , +,:;)r

•

in (22) are de-

4.2. Discrete Stochastic Systems The ftnite-dim~nsional characteristic functlons of tne sequence of random variables [ZIC} defined by difference equation (2) a~e determined at le, < ", < /Cl!, by the equatlons (PUgachev, 1986) Tt-I

alC

(J 0 ' ' ' ' 1C",+1

+i.A:W/C

I!,

0()

(24)

-00

~p =0,1, ... , IV; 6'CV)=3, .. " I\'

become a closed set of approximate differential equations determlning with the respective initial conditions all the parameters m, K, C y (~(Y)=3"." N). nimilarly the approximate equations for ~he coefTicients of orthogonal expansions of the finite-dimensional densities are derived (pugachev and Sinitsyn, 1985). Equations (19), (20), (21), (25) may be approximately solved by any method of numerical integration of ordinary differential equations, for instance by the Runge-Kutta method.

termined by

c-v= 5Sll;t)q,:yCZ)c.lz ~ M~}Z)

and equations (25) at

0('0= 2.

~1"'"

0

1 , ... ,

AfI,

(Z.cfI, ,V,o; )1 ~

)

=

T

Mexp { i f=1 L. Ae Z ICe

(26)

(1t=1,2, .. , )

f!,

wi th given di tions

91 (/I)

+

and the initial con-

In particular, in the case of the normal density 1JJ(1) , N(m, K) , (22) represents the partial sum of the Hermite polynomial expansion of II (l; t ) . From (17) and (24) follow the ordinary differential equations for the coefficients C-y

+

+

(25) where in addition to previous notations

et.;,/It. CZ)

K

and
(It = 2, :3, '"

(27)

).

Equations (26), (27) can be exactly solved for linear systems (2). In the general case of a non-linear equation (2) approximate methods should be used. The same approximate methods as those used for stochastic differential systems may be applied to solve appoximately equations (26), (27). In particular, the simplest method, i.e. the normal approximation method consists in deriving from (26) the difference e quations for the expectation nL and the covariance matrix K of the state vector Z of a system and evaluating the expectations involved by replacing the unknown true distribution by the normal one yielding the closed set of difference equations for m, K • This method is the special case of the method of moments which will now be briefly outlined. From (2) or (26) follow the equations for the initial and central moments of the random vectors Z /C (Le. of the onedimensional distribution of the sequence

{ZIt} ) 't

't 1

'tf

d.,/G+1 :!r1WC1 (Z ... ,Y/G)'" W/Gp (Z/G'Y,,),

(28)

m/G+1 =!rJW/G(Z/C, V/C), ~

(29)

r

't 1

fi.cd =!r1lwC1 (Z ... , ~)-MWC1(ZA:'~)] )( 'Gp

... x[w.:p(Zc, Ye)- MUJ ep (Ze, Y::)]

(30)

13

Problems of .\Ilah sis

The method of ~om~nts is bas~d ~pon closlng equations ,28) or l~9), ,30) by replacing the unknown true distribution in the expectations by some approximating function completely determined by the moments of orders 1, ••. , N • Then equations (28) at (.,1"'" 't p = 0,1, ... , N ; 6'(,,)= 1, . .. , N or equations (29) and (30) at "1" " ) 'l-p = 0,1, ... , N; C)(~)= 2, •.. , N wi th appropriate ini tial condi tions determine approximately all the moments involved and thus determine the approximate expression of the sought onedimensional distribution of the random sequence [Z le} • In the special case of N= il. equations (28)-(30) represent equations of th e normal approximation method. Similarly the equations for the moments of other finite-dimensional distributions of [Z/C} may be derived.

}

(n.=1,2, ... )

(31)

Eva luating the expectations by replacing the unknol"ffi densi ty 0 f Z K. by (22) equation s (29), (30) at d(,,)= ~ and (31) become a closed set of difference equations approximatly determining with the respective initial conditions all the parameters in (22). In the same wayother finite-dimensional distributions of the sequence {Z Ie.} may be approximately determined by the me thod of orthogonal expansions (PUgachev , 1986) .

(fL=2,3,

Jio

A:T

=[A~ ... J5j;

Systems rhe methods of statistical a nalysis of stochastic differential systems may be extended to discrete-continuous stochastic systems described by mixed sets of differential and difference equations of tne form of (4) (pU[;UClleV, Shin and Si nitsyn, 1986). For this purpose we introduce the step random process

ct

IC=O

.

=

rt

C'A 1 ,.·.,A n ;

(33)-(35) determine successively the evolution and the jump-wise increments at the points t(f)= et= 1,2., ... ) of all the finite-dimensional characteristic functions of t he process Z(t).

eT

8quations (33) at 11.= 1 yield the equations for the initial and central moments of the extended state vector Z of the system (4) (i.e. of its one-dimensional distribution) (pugachev, Shin and Sinitsyn, 1986)

Mf ),~7'a(Ztn, tn)

+

,· .. )

f)

~1+

.. .

+ (,~

+

J,

1'1"

•

o '1"1 ] L::l / 'l"Z/} xCv(Z,t) A ; t) e ;{=o

)=

MZ

C+ 1, 1

'"

+

x

(z~+ 'tP+:!t"

( 't 1 + 'lP+1

(C+o

Z e+ 1, '.t"

~

"("p

Z";+1

IC

tf

/\P+1 ... Ap+1L ] 'I'

(32 )

Then in the same ·;lay as for differential and discrete systems the finite-dimensional characteristic functions of the random process Z(t) = [Z/(t)'l" ZIl(t)'T' Z///(t)'" ] '1" are determined at t1 <: . .• ' " < t n, by the equations

8S

Ap

Eq~ations

cl. ~ Ct

(

L, Z/C 1 ACt)

A"+1 ...

cl(,=M[O(i~f)'(,I ... a(iAJit;r [LA ~(Z,t)

4.3. Discret e - cont inuous 3tochastic

Zffl )=

(35 )

)

where (J> ) is the characteristic function of the initial value Zo of the process Z(t)= [Z/Ct)'T' Z\t)'T" ] T at t=O and Ak:,=[:A~'1" A~Tr

•

00

(33 )

with the initial conditions

As a function approximating unknown distribution a truncated orthogonal expansion, in particular Hermite polynomial expansion or 3dgwor th series is e;enerally used. Similarly (2) or (26) yield the e quations for coefficients C~ of the approximate expression (22) of the density f 1 (:l;t CIC »)J t CC )= IC T)

,

tn )/at n

X(£(Ztn,tn)TA~; tn)] x

• We1

(Ze,Ve)···Wt,p_x(Ze,Ve ) (36 )

(t=O,1, .. . )? m'=MQ,(Z,t) , nte+1

. -M{

jL'(,-

=

/J'l,1+ ... +'t.,.

MW~ (Ze, Ye)

O(iAf)Z1 ... q(i~~/x

[i

J

IT t A a,(z, )

+X(Hz,tf:/;t)]eLA''rcz/-ml)} ::\/=0

(37)

+

x

\" , S, Pu gacl le\, I. :\ , Sini lS \'ll a nd \ ', I. Shi n

nal d i s t r i bution s of the process Z (t) may be approxima t e l y determined by t he method of moment s or by the me t ho d of ort hogona l ezpansions. 4 . 4 . Hereditary St ochastic Differentia l Sy s t ems

ce

=

where

0 , f , ' ..

Z1 et)" ... ,

(38 )

)

z p. . ~ (t)

are the com-

ponent s of the vector process

Z (t),

'f

z'ct) '" [Zj Ct) .. . Z:n; (t) ] .,. ZpCt)

'1' ]

>

Z "(t) = [Z'HCt)

Ifl

) Z (t) : [ZP+1(t) ,.,

Zp+,,(t)

]'1' J

It i s i mpossi bl e t o dedu ce equat i on s similar t o (17) for the fi ni te- dimen s i onal d i s t ribut i ons of the sta t e ve c tor of the heredi t a~y system de8c r ibed by a eeneral s t ochastic di fferential equa t ion of the form of (5) . Nevertheless equa tions (5) - (7) may be reduced to a stochastic diffe r ential equations of t h e form of (1) i n a wide class of problems (Si ni tsyn , 1986) . Let us consider at first the hered i tary stochast i c system (5) - (7) for the case I"/here t he func ti ons A(t,'!:",:t, U) and B(t, "t,:C, 'U) have the form

A(t,'f,:c,'U)= aa,'!:") ':P(:l"

U ,'f)"

B(t,'f,:i,u)=r(t, r)'lfU, u,'C) W e1 , ... , Wt, p-5i: are the compo !lents of the vector function We.

and rCt,'!:") ar e matrix functions, often ca l led hereditary kerne l s , Cji (:t, u,'f) is a Z- - di mensiona l function , V( 1., u, 'l") is ~ "9- matrix [u.l1ction . In prac ti ce hereditary kernels usua lly satisfy the fol l owinG conditions : Z-.

= O,1, ... ,N ; 1.1 + •• • +l. p"''' =1 • • • • ,111 or equa~ions (37) and (38) at z,1""''(,f+r.= '" 0) 1, . .• , N; ~i" .. . + 'Gp .. r. =~, ,.,' III become closed set of equations determi ninG al l t he ::loments involved . In the specia l case of N=2. equati ons (36) - (38) represent the equations of the normal approxi mation method . In the same v/ay the equations for the coefficients C~ in the approxi mate e~pression of the form of (22) of the de n sity f 1( i. ; t) al"e derived from (33) (l'ugachev , 311in and 3init'3yn , 1986)

Cv =M{ 9-~ ([a/iaA' Zll(t{ Zllf(t{

G(t:r)

where

f~eplac i ng the unknown one - dimenaional distribu ti on i n the expressi on s of expectat i ons i n (36) - (38) by some approximat i ng func t ion which is c omp let e l y d e termi n ed by the moments of orders 1, ••• ••• N equations (36) at '(.1r'" ?op ... " =

f)

le

(40)

~

~ (t, et) =0 ,

r (t, '[ ) = 0

(4 1 )

(non- anticipat i veness)3 00

-L I Qij (t,

'C)/ der <

OQ

.,

<>0

J I fi.j(t,'L)Jdc;;

<

00

(stability)4(42)

-00

I"/here Gii (t, er) and fii ( t) 'f') are the elements of the matrices Get, 'i) and t, er) .

rc

For sta ti onary kerne l s, c;Ct,'C)=QC.EL r(t, 'f') = f(.~ ) ~ : t - 'l:', whose Lap l ace transfo~ms are rational funct i ons of a complex variable ~ , i.e . admit the representation o£ the form

)([~A''fQ,(z,t)+~(g(Z,tr:A/; t)]e~~/'I'z'} ,+ +

Mq,~(Z)m+ h[Mq,~(Z)K})

C.y ( t

Z /1' t+1


]'1')

([

=

A=O

w~(Zt ,Vt)

0, 1, ... ) •

'T'

(33 )

Replacing the unknown one - dimensional distribution in the expressi ons of expec tations by i ts approximate expression of the f'Jr~; of (22) e1uations (37) , (38) a t O('G): ~ a nd (39) become a closed set of equ:c!. t ions determininc all the unknorm parameters in the approximate expression of the den sity Si Cl ; t) . Simi l ar l y a l l the other f i nit e - dimens i o-

J This condition i s somel"l13.t sUl?erfluous

since the inte2ration in (7) lS extend ed only over the interval [ to, t ]. ,le ment i on this condition to emphasize that a ll the systems considered are non- anticipative .

4 1'he stabili t~· in t:1is sense adopted in the contro l theory is equive.lent to the asymptotic stability in Lyapunov sense .

15

Problems of Anahsis

ff(~)e-ad.s=Q(;l)-1p(.,)

(43)

o

the hereditary stochastic differential system (5) - (7), (40) may be reduced to the follow~ng stochastic differential system (Sinitsyn, 1986b):

dz= a(Z,u,t)cU U=UI+U':

Q U 11 =

P

B(t,'t:, I, u)=

2:

Jr.=" GIt (t.. er)

t ) W.

(44)

Let a non-stationary hereditary kernels (H t '() and t, "f: ) a t fixed ~ b; the solutions of the linear difderential equations

r(

(45) These kernels at fixed ed by the formulae

t

are determin-

it

F(t,'t)= p«

r

,

(48)

c

where and rh t, 't:) are hereditary kernels of one of two types considerea.. Due to the lack of prior information the functions A ( t, '!, l, U) and B (t, t",.l, u) are usually Imown approximately and may often be approximated by formula (40)or(48) with G(t,'t),ra,'t) of one of two types considered. Then (5)-(7) may be reduced to a stochastic differential equations of the form of (44). Another special case in which a hereditary stochastic differential system may be reduced to an ordinary stochastic differential system of the form of (1) by means of extending the state vector is the case where ACt, 'r, 2, U), B Ct, '1:, .l,u) admit the representation

ACt, «~~, u) =A+(t) A-('t:,:t, u) ~ B(t,'t,:f:, u )=8 +(t)B -('f,~, u) at t> r:

.

(49)

t

yl=f A-(
Ct,
(46)

where the functions r;tt, '!) and rl(t er) at fixed t are the solutions of the linear differential equations

r'f;* G'(t,'f;) = I ~ SCt-er),

yll= Jt 8-('t,Z, U)d W('l').. (50) to we reduce the hereditary stochastic differential system to the following stochastic differential system (Sinitsyn, 1986):

dz

Q~ r'Ct,'C)= I"l. bet-et).

(47)

Here ft=rtCt,:D), Ht=HtCt,:D) and Qt= Qt(t '])) Pt=Pt(t 1» are some '"(, ~ "l. ma'trix linear' differential opera tors of orders n., m (n. > m. ), le ~ (~> e. ) respectively the sub;cipt t at the operator indicating that the operator acts on the function considered as a function of t at fixed er ,by asterisl~ the adjoint opera tor is marked I ~ is the unit matrix of order ~ • In this case (7) may be replaced by the equations (pugachev, 1965)

f t U I = Ht

rlt.(t,'r)tf4 Cl,U,!')..

In this case putting

GCt,'!)= H.,;"" C;; / (t,'!),

t ) ..

/ Q t U " = Pt 'If (Z, u, t ) W . As a result the hereditary stochastic differential system will be reduced to the stochastic differential system (44) vd th non-stationary opera tors ft, Ht,

Pt·

t; c,h.(t,1:)Pk(l,U,'f:), N

+ g(z,u,t)dW~

Here ,... H , et. , P are 't ~ z, matrix differential operators of orders n.. m (n:> m), /C, t (/C:> e) respectively. Applying the known methods the latter two equations may be transformed into the Chauchy form for the extended state vector and after that written in the usual form of Ito stochastic differential equation (pugachev and Sinitsyn, 1985).

Qb

N

A(t,'i,I,U) =

F'U'=H'P(Z,U,t),

yJ ( Z, U,

U = U I + U "/,

For more complicated functions ACt, 't, ~ u) and 8 (t,'f:", 2, U) in (7) it'is often useful an approximation of the form

'=

a(z,~

t)dt-t- B(z,U, t)dW-,

U=At(t)yl + 8+(t)Y//,

(51 ) Since A (t.. 'f:, :l, U) and 8 (t .. 't, :l! 'U) in (7) practically always may be ~pproximated by the expression Of. the form (49) or by more general expresslon '" Aft+(t)A - (Z(!"), U(f).. 'f:), A(t, '!, Z('t), Y(!'») =L. k 4-1

N

E

+

_

B(t, er, Z(~), U(f:» = 8J" WB", (Z(r), V('!).. 'r) (52) equations (5)-(7) practically always may be reduced to a stochastic differential equation of the form of (1) by extending the state vector of the system.

16

\', S, Pugachn', 1. :\, Sillits\1l alld \', 1. Shill

Thus a hereditary stochastic differential system of the form of (5)-(7) practically always may be approximated by a suitable stochastic differential system of the form of (1), and then any method of section 4.1 for stochastic differential system analysis may be used.

4.5. Stochastic Differential Systems with

setting up the equations for the distribution parameters. But even with such programs the number of equations for multi-dimensional systems may be too large to be manageable. Table 1 shows the dependence of the number of equations on the dimension P of the system state vector and the maximal order N of the parameters involved 5 • Number of Equations as a FUnction of P and N

TABL3 1

Time Lags stochastic differential system with time lags (8) is a special case of system (5)-(7) where the function A (t,'t, ~,U) in (7) is a linear function of 2' whose coefficients represent f5 -functions of the form 5' ( t - 1:' K. - 't) and B ( t er z u.) = 0 • Hone of the approximations of previous Section is applicable in this case. So a special approach to such systems is necessary.

PJJ:;f

~

1

2 4 6 8 10

2 4 6 8

10

2

3

4

5

5 9 14 34 27 83 44 164 65 285

14 69 209 494 1000

20 125 461 1286 3002

6 27 209 923 3002 8007

A uGual approach to reducing a system

with time lags to a differential system in control theory is replacing of the transfer function e-5'i' of the delay e lement \Vi th the time lag et by a polynomial

e

~
=

1+

d'f + '" +

11. 11./ "'.I d 'f

(53 )

Using this approximation and introducing new variables Y1 , ••• , Y fTt, determined by differential equations

(C[: J)f!,jn,! + '" + 't", :D + 1)Y/G" Z

(54)

(1C=1,. .. ,m)

we replace (16) by the equations (54) and

cl. Z = a (Z, Y1, ... , Ym +

cc Z, Y

f , ... ,

Y

/Tt, ,

, t ) cl t

+

t )dW .

(55)

Equations (54), (55) with the initial conditions

(56) describe the stochastic differential system approximatinc the system with time lags (8). Replacing (16) by (54), (55) one may use then any of the methods of stochastic differential system analysis to find approximately the finite-dimensial distributions of the state vector of a stochastic differential syste!.1s with time lags .

4.6 • .softv/are for Solving Analysis problems The number of equations for distribution parameters in all tile approximate methods of analysis of stochastic systems considered in previous sections grows enormously with increasinG dimension of a system. This necessitates an automatic setting up the equations for the distribution parameters. So the soft'./are 101' c:olving stochastic system analysis problems should include programs for an automatic

~ 2 4 6 8

7 35 329 1715 6434

8 44 494 3002

9

10

54 714 5004

65 1000 8007

20 230 10625

For larger p the numbe r of equations amounts to many millions. 80, for instance,for p =100, N =6 the number of equations is of order 2· 109 . Juch tremendous numbers of equations render the above methods of stochastic system analysis practically inapplicable to systems of very large dimensions at the modern l evel of computer abilitie s . And even for computers of future Generations such numbers of equations to set up and solve numerically are too l arge, at least in the nearest future. 3 0 the reduction of the number of equations is of the first necessity. To reduce the number of parameters in the approximate expressions of the distributions of multi-dimensional random vectors ;'lal' chikov proposed (1973) to characterize the interdependence of different components of a random vector only by their covariances ne g lecting the fine structure of their interdependence. '1'11is may be achieved in several different \'lays. :\part fro;;] the ori ginal :,1al' chikov's approximation of a multi-diuensional dis tribution involvinG only the covariances of the components of the rando m vector and no other mixed moments, ,Lt is expedient to use a partial sum of some orthogonal expansion of the density of a random vector, generally of Hermite polynomial expansion or of ':;dS':iort!l series \'Ii th mixed moments of orders lligher than the second replaced by their expressions in terms of the covariances either valid for normal distributions or obtained by equalizing to zero all the 'nixed semi invariants of orders hicher than the

5 The number of equations for the para-

meters of the one-dimensional distribution is e~ual to 1 = Cp+N)!/(p!Nl)-

-1.

C;'N -

17

Problems of ,-\nah'sis

second . rhe first approach leads to the formula for the central moments of even orders 6'('t)-= '1: 1 + '" + 't p -= :2 ~ (57) where th~ surn is extended over a ll the permutations of ~ -:'. subscripts 1.,1' f l , ...

'" h d ,e;5,

of ';ihich are equal to 1, <>2, 1 are eqClal to 2 , ••• , "t p are equal to f .. !eke beine t he covariance of the components Z,,-, of the random vectol' Z . ,~ ll t:1e iTlo;nents of odd orders hi0her than the Recond are t hen equal to zero . The other (Kashkarova and shin) leads to the recursive formula [6("t)/.!l] It, I (-1) 't

Ze

.f'-z = Z1! ..•

Z,f '

~

k

1t=2

.Ji::Je ;Ve1 !".::Ve p !

h.

X~1+~="t,~1

(58)

the inner sum beinG extended over a l l P - dimensional vector subscripts =V t , ... , ~I.-' ~1C=[~C1"'~"'f']'r the sum of which is equal to Z, , the integers ~1C1, , ( IC - 1 It) satisfying the • • • -VIeP "", conditions v.c~ ~O, e>"(-))1C)==\)"1+ ... + + -:Yrp ~ 2.,

jUelL+ee

=

6 kJt.e, 0(1,)= 61+ ...+"t p ~ -4.

Formula (57) and (58) are exped i ent t o use Nhi l e applying the method of moments. If the method of orthogona l expan s ion s i s used then it is preferab le to equalize to zero a ll the coefficients C-v whose vector subscript .:y has more than one component different from zero. The above assumpt i ons considerably reduce the n~~ber of equations for the- di st ribution parameters. Tab l e 2 shows the dependence of reduced number of equations upon the di mension p 0 f the state vector of a system and the highest order of moments N involved. TABL3 2

~ 2 4 6 8

10

Reduced Humber of ;;; quations as a Function of P and N 10

20

30

40

r~

230 270 3 10 350 390

495 555 615 675 735

a60 940 1020 1100 1180

0)

85 105 125 145

~

50

100

150

200

2 4 6 8 10

1325 1425 15 25 1625 1725

5150 5350 5550 5750 5950

11475 11775 12075 123 75 12675

20300 20700 21100 21500 21900

Table 2 shows that the :.lal 'ch ikov 's idea to s impli fy the approximate repreBelltations of multi - dimens i ona l distributions iR very effic i ent resu ltin ~ in very essentially diminished numbe r of equations for distribution parameters. 30 , for illstanc e , in the above example of p =100, N=6 the coefficient of r educti on of the TIUJlibar of equations is of order 2'10- 5 • This reduction of the n~~ber of equations allows to apply the Inethods of statistical analysis of stochastic sys tems to systems of very l arge dimensions, certainly if the software includs an automatic setting up of equations . .Jo the f irst feature ol til'3 soft\'l?re necessary for s tochastic system analysis is that it must include stl3.ndA.rd programs for !:!utoma t ic settine up of equations for dis-, trioui; i on parameters. ~he first Guch pro;;ram was produced by Dashevskii (1976).This pro gram is designed for st06hastic diffe rential systems (1) of rather low dimensions (not larger than 6) with polynomial nonlineari tie s \'! i th the account of moment s of orders up to 10. Shin compiled the program ANALYSIS rea li zing the automatic setting up and solving the equat i ons of the no:rmal approximation method for stochast ic differential systems ( 1) of any dimensions vii th polynomial nonlineari ties of a special kind ( Shin and Sinitsyn, . 1985) • In t he general case of any nonlinea~ities in ~ifferent i al, difference , mixed differential - difference, integro-differentia l equations the automatic setting up of equations for distribut i on parameters require.'] the computation of a numoer of inte gra ls of the form 0()

-

J CjJ(I)H:y(I) 1.iJ0 1)dZ 00

where cP ( Z ) is a bi ven nonlinear function of the state vector of a sys t em , UJ'N (.l) the multi-d i mensional normal density, ftI(m.,K), Hy(l) (cJ(-Y)=3, ... ,N) the corr8sponding He~mite polynomials. The number of such int egra l s is l arge in the case of a large dimension of the s tate vector of the system . This necessitates to include in softv,!:!re the possibility of a l arge number of parallel calculations ~~d requires an adequate multi-processor computer. Perhaps special micro-processors fo r these operations shou ld be created. 5. SOLUTIon OF FILT.2:RHm AND EXTRAPOLAT I ON PROBLEMS

e~

the p -dimensional vector is denoted whose all the com~?nents are equa l to 0 except the ~ n component which is equal to 1.

6 ae ca ll tha t by

We shall not touch up here the usual approaches to filterin g problems leading either to exact solutions which are absolutly usele ss for on-line estimation or to rather poorly justified suboptimal solutions like extended Kalman - BUCy

\'. S. Pugache\', J. :-.J. Sinitsm and \'. J. Shin

18

filters and others (see for example the book by Pugachev and sinitsyn, 1985). The requirement of computational simplicity of estimators naturally leads to the idea of conditionally optimal estimation (Pugachev, 1978a, 1978b, 1982a, 1982b).

In the case of a stochastic differential or differential-difference system with discrete observations we define the class of admissible filters by (59) and (61) as well as in the case of a discrete stochastic system.

The main idea of conditionally optimal filtering consists in restricting the class of admissible filters in such a way that any admissible filter could realize on-line estimation directly in experiments. So we refuse the absolute optimality of estimators for the sake of their computational simplicity. Filters determining estimates as the solutions of rather simple differential equations (case of continuous observations) or difference equations (case of discrete observations) may serve as examples of filters satisfying the requirement of computational simplicity.

Defining classes of admissible filters in such a way the problem of finding an optimal filter is reduced to the determination of optimal coefficients cL., J3, 1) in (60) or 0, l' in (61).

case of a stochastic differential with continuous observations we the class of admissible filters formula A

Z=AU

(59)

and the differential equation

cL u = ci. .§ (Y, U, t ) d t

+

+j3~(Y, U, t)clY + rclt where A matrix,

The last question to deside is the question of the criterion of optimality. It is quite natural to try to minimize the mean ! square of the error (MSE) M Zt - Zt

I"

5.1. Classes of Admissible Filters In the system define by the

As for the choice of functions ..§, It, .:TA: we recomend following Dashevsky (1983) to take them in such a way that equations (60) and (61) be as close as possible to the corresponding relations given by the optimal filtering theory.

(60)

is some constant f'l( N p ~ of the rank p , .:5Cg,u,t), 'l,(/;/,u,t) are some'tfunctN~

ions mappinc; R,P x R.N x ~ into R, and fl~m respectively, a. • .JJ, l' arbitrary matrices of dimensions N'Jr't .. "Jr~ and respectively depending on time t . }e shall suppose that the function Jh u~ t) has continuous derivatives of the first order with respect to t and of the second order with respect to the components of the vectors !I, U. Any choice of ~,J6, T as functions of time determines the corresponding admissible filter. Any choice of the matrix A and the functions .§(/I,u"t), 'l,(V,u,t) (and therefore of the integers JV, ~ ~ ) defines the corresponding class of ~dmiss­ ible filter.

"-1

.s<

In the case of a discrete stochastic system we define the class of admissible filters by formula (59) and the difference equation

where J~ ( ~, U) are some functions mapping 1//'L.It Il N into Il ~, b.e, 81C arbi trary ma trice s of dimensions 't and N1( 1 respectively. Any choice of the sequence s {Ore}, {1're} determines an admissible filter. Any choice of the matrix A in (59) and the functions .s1C(~,U) (and therefore of integers 'l#t''t )defines a class of admissible f 1. er.

"'.It

( M I Z IC+1- Z re+1

I!')

I

in the case of

\1 CMI Z

-

filtering and M I Z t -Zt +~ 1 \C+ of - Z ' in the case of extrapolation. Ie+ J'" 1 But if we pose the problem of minimizing

I)

the 1,1SE at each time moment t (t(re+i») then we come to a multi-criterion problem which has no solution in the general case. So we shall pose the problem to find the values of cL, ft, '0 at any moment t minimizing the MSE at the infinitesimally close next moment ~ > t , ~ _ t in the case of continuous observations. In the case of discrete observations the corresponding problem is to find the values of OIC, 'tIC for each IG minimizing the MSE at the next moment t (IC ... -f) • The filters determined in such a way are called conditionally optimal filters. such filters are pareto-optimal filters for the problem of multicriterion optimization considered. Our approach to defining classes of admissible filters allows to design conditionally optimal filters of various orders. In particular, taking IV= f -' A .. I equations (60),~(61) determine directly the estimates Z t, Z'" /C+ 1 ·AIl the admissible filters are of dimension p in this case (recall that all the suboptimal nonlinear filters are at least of order p ( p ... .3 )/2 ). On the other hand taking the appropriate N and the functions ,!, 'l in (60) or TIC in (61) it is possible to construct the class of admissible filters containing any given suboptimal filter. The conditionally optimal filter in this class will certainly be better that the given suboptimal filter. Thus the theory of conditionally optimal filtering allows to improve the accuracy of any given suboptimal filter. FUrthermore, taking /'I=- f', A: I and the functions ,S, ~ or .:fIG depending only on a part of components of the vector Z'" U i t is possible to design conditionally optimal filters for estimating only the respective part of the components of the state vector Z . Such filters will certainly be of lower than

19

Problems of .-\ na h sis

f

dimensions . (62)

5.2. Conditional l y Optimal Filter for a Stochastic Differentia l system

where

From (59) and. (60) follows with the accuracy up to infinitesimals of higher order

E:c=M [J:e (~,UIC)-tlC ] Jc (YII:JU~) ~

Z

j

-

Zt

=

Ao(,t.s (Yt , Tlt,t)

mIC = MZIC+j+1 ) tIC =.M1:c (¥JCJ U/C),

(:)-t)-r

AjJt 'lC'ft,u", t)LlW+ At(j-t) where ~ W = W(j)- W(t). The prob+

lem of minimizing

Mlz;1- Z~+-rlt

Mlz;s-z~l!

represents the

pro blem of finding the re regression of A

Z

6+'; -

Z

t

or

lin~ ar

Z~-

mean squa -

Zt

of or on the random vector

T'!']" [£ (Yt> U b t)TAVJ' 'l,(Yt,Ut,t) •

The routine techniques yields the linear algebraic equations for 01.. t" fit, whose coefficients involve the expectations of some functions of the random variables Y t , Zt, Ut. To find the distribution of these random variable3, i. e . the one- dimensional d i stribution of the random process [y(t)'" Z(t)'r U(t)'" ] '" we substitute into (60) d:c from the first equation of (11) and use the equat ions (25), (26) for the one-dimensional characteristic function 91(~1'::\1'::\3; t) of the process (y(t)" Z(t)" U(t)'" rr determined by the obtained set of stochastic differential equations . The simultaneous solution of equations for cl,t, fit,

at

1'/:, 9f(~{,A%,.A3; t) gives the solution of the pro bl em of conditionally optimal filter design. In the problem of cond itionally optimal extrapolator design the coefficients of the equations for optimal oI"t .. .J3t, involve the expectat ions of some functions of ¥t,. Z t , Zt+"C, Ut. To find the distribution of these random variables we use equations (25), (27) for the two-dimensional characteristi c f unction 92, (AI' A.2J ;:/,3.? ftf J .JIZJ.JI3; t:)) of the process [y(t)'" Z(t)'" U(t)'I']~ The simultaneous so lution of equations for o/,t~ fit, ~t, 31(A"AflJAjjt) and 82(A1J~I.'.A.3'p{.? .JIlJJ'J;t,j) Cives the solution of the problem o~ conditionally optimal extrapo lator desi gn. Any approximate method of Section 4.1 for t.he solution of equations (25)-(27) may be used for this purpose (pugachev, 1982b, 1984; Pllgtl. chev and Sinitsyn, 1985) .

re

5.3. Conditionally Optimal Filter for a Discrete Stochastic System The problem of finding optimal SIC, DIC in (61) is in fact the problem of finding the linear mean square re gression of Z 1<.+1 or Z I<.+J+1 on the r a ndom vector le CY l e , U.. ) • The standard techniques yields the equations

.:r

LIC =M [ZIC+i+ l - mle ]

h (Y

U/e/'.

IC '

(63)

To find the expectations in these formulae in the case of filtering ( = 0 ) vie suhstitute into them Z 1e+1 fr om (::» and use equation (26) for the one-dimensional characteristic function of the ranT dom mq'..lcnce [[X; VIe ]'''} determined by differ~nce equations (14), (2), (61). The simu lte.neo'.ls solution of equations (2G), (62) , (63) yields the solution 0;: the problem of conditionally optimal filter desien. To find the expectations,in (63) in the ca3e of extrapolation (J;' 0) we add to previous equations the equations (26), (27) for the two-dimensional characteristic fWlction of the sequence

i

Z;

[ [X reT

z;

u-: ]

T } . The simul taneOU8 so lution of the obtained equations yield the solution of the problem of conditiona lly optimal extrapo lat or design. Any of the approximate methods of section 4 . 2for solving e~uations (26),(27) may be used for thi s purpose (pugachev , 1978b; 1979).

5 . 4 . Conditionally Op timal Discrete Filter for a Discrete-Continuous s tochast ic system

8'A;' 1'1e

rhe problem of finding optimal in (61) in the case of a discrete-continuous stochastic system described by equations of tne form of (16) differs from the same problem in the case of a discrete system only in the method of finding the expectations in (63). To determine the joint distribution of the random variables X /Cl Z IeJ Z 1I:+IJ U Ie in the filtering problem (i = 0) we use equations (33), (18) for the one - dimensiona l characteristic f~~ction of the random proce3s [X(t)T Z{t)" U(t)T Z"'(t)'''] 'I' where coo

X(t)=

L. XIC 1A JC (t), 11:=0 00

Z//,o'(t) = 00

L. z' i

.4C=o

V(t)- L: U/C 11:=0

IC

Ale

(t) .J

[(/C)

iA et), A~= t , t

(IHI)) .J

(64)

IC

defined. by equations (16), (15), (61), (64). Simultaneous solution of equations (62), (63) ana equations of the fo rm of (33), ( 18) for the one-dimensional cha~acteristic function of the process

(X(t)T Z(t)T Z'I/(t)'l' U(t)T JT solve the problem of conditionally optimal filter design .

:2()

\', S, 1'1Igachl'\. L :\. Sinitsyn and \', L Shin

To determine the expectations in (63) the joint distribution of the random variables XI(., ZIC~ Zf(;+i+l, U/C is necessary in the general case. In order to find this distribution we use equations (J3), (18) for the two-dimensional characteristic function of the process (X(t{ Z(t)'r Z///(t)T U(t)" ] 'r defined by equations (16" (15), (61" (64). Simultaneous solutlon of equatlons (62), (63) and equations of the form (33~ (18) for the one- and two-dimensional characteristic functions of the process [X(t)'I' Z(t)'I' Z"/(t)'I' U(t)"'] 'r solve the problem of conditionally optimal extrapolator design. 5.5. Prior ::;stimation of Accuracy of Filtering The approximate methods of Section 4 for stochastic system analysis permit to determine the accuracy of filtering and extrapolation of processes in stochastic systems considered by any filters described by differential or difference equations. In fact the knowledge of the joint distribution of the state vector of a system (or its future state vector) and its estimate Z is sufficient to calculate the mean square of the error of filtering and to find confidence regions for the stata vector. These calculations are based only on prior data and do not involve observations, and therefore may be fulfilled while designing a filter. This gives the opportunity to compare different filters from the viewpoint of accuracy. In particular the methods of Section 4 allow to estimate the accuracy of the conditionally optimal filters and extrapolators while designing them. 5.6. Software for Conditionally Optimal Filter Design The calculations necessary to find conditionally optimal filters are rather cumbersome and time consuming. But these calculations are based only on prior data and do not involve observations. consequently all these calculations may be fulfilled beforehand while designing conditionally optimal filters. Nevertheless these calculations cannot be realized without the corresponding software especially in problems of large dimensions. First of all it is easily seen from the discussion in sections 5.2, 5.3, 5.4 that th~ software for solving analysis problems should be included into the software for designing conditionally optimal filters. This software should provide an automatic setting up the equations for respective parameters of distributions and solving them together with the equations for optimal coefficients of filter equations. secondly, the software for conditionally optimal filter design should include the software for an automatic evaluating the joint distributions of the state vector of a system and its estimate and for finding confidence regions (usually rectangular) and the corres-

ponding confidence probabilities. It is also necessary to investigate the question of possibility and necessity of developing special problem oriented microprocessors for various stages of conditionally optimal filter designing . 6. conCLUSIons The survey presented here of problems of analysis and on-line filtering and extrapolation of processes in stochastic systems of the types considered are of prime importance for informatics and control theory. The wide practical use of the methods considered is impossible without special software desiGned for solving such problems and perhaps would require special problem oriented computers. The necessary features of this software were discussed. RZF3,{SHCGS Bo guslavskii, I. A. (1 969 ) • s tatistical analysis of a multivariate dynamic system u s ing Hermite polynomials of many variables. Autom. and Remote Control, 11, 1751-1/64. DasneVskil, i'fl . L. (1 976). Technical realization of the moment-semiinvariant method for analysing random processes. Autom. and Remote Control, 37, 10, 1490-1492. Dashev skii, 1,1.L. (198 3). Desi gn of conditionally optimal fi·lters implementing optimal nonlinear filtering equations. Autom. and llemote Control, 44, 10,

1330-133/.

-

Demukh, V.I. (1965). An approximate method for analysing accuracy of nonlinear systems. Autom. and Remote Control, 26, 6,.1012-1016. Maltchikov"J. I . (1 973). Det ermina tion of distribution of output variables of multidimensional nonlinear system. Autom.and Remote Control, 34, 11, 1724 1729. pugachev, V. S. (1944). Random func nonS determined by ordinary differential equations. Trudy Academii N. ye. Zhukovskogo, 118, 3-36 (in Russian). pugachev, V. S. (1965a). s tatistical methods in automatic control. Froc. 2 Internat. Coneress of IFAC, 27 aug .4 sept., 1963, Eas e l, s witzerland. Non-Linear s ystem Theory, Butterworths, London, pp. 1-13. pugachev, V.S. (1965b). Theory of Random Functions and its A Ilcatlon to contro Fro ems. pergamon press, Lon on. pugachev V. S. (1978a). 2 sti~ation of . variables and parameters In stOChastlC systems described by differential equations. soviet iliath. Dokl., ..!2, 4, 967-971. pugachev, V.S. (1978b). Recursive estimation of variables and parameters in stochastic systems described by difference equations. soviet I.la th. Dokl., 19, 6, 1495-1497. pugachev, V.S. (1979). J<; stimation of variables and parameters in discretetime nonlinear systems. Autom. and Remote Control, 40, 4, 512-521. pugachev, V.S. (1980). Finite-dim~nsional distrlbutions of processes deflned by stochastic differential equations, and the extrapolation of such processe~. soviet. !.lath. Dokl., ~, 2, 382-386.

Problems of :\nah'sis

pugachev , v . s . (1 981) . The finite -dimensional distributions of a random process determined by a stochastic diffe rential equation and their application to contro l problem s . probl. Control and Informat. Theory , la, 2 , 95 -11 4. puGacnev, v . s . C198 2a ). A-generalizat ion of the theory of conditlonally optimal estimation and extrapolation. s oviet rJa th. Dokl ., 25 , 1, 79- 82 . pugacnev , v . s . (1982b) . c onditionall~ optimal estimation in stochastic aifferential sys t ems . Automatica, 18, 6 , 685 - 696 . -pugachev , V. S. ( 1984) . Conditionally optimal filtering and extrapolation of continuous processes . Autom . and Remote Con trol, 45 , 2, 212 -21 8 . puga chev, V. S. and I . n . 3initsyn ( 1985) . Stochasti c Differential stems . Nau a , i.losco\'l in nussian ; :;;ng lish translation to be published by J • .iiley & sons) . Shin, V. I . and I . N. Sinitsyn (1985) . The experience of application of normal approximation me t hod in pr oblems of analysis of processes in multi - d i mensional non-linear stochastic systems . 2-nd Al l-Union Conference "Perspective methods of planning and analysis expe riment s at investiga tion of random fie lds and ~rocesses. Abs t acts . Moscow Energe tlca Ins tltut e , part 1, pp.106-1 07 . Si luyanova, I . D. (1982). 'r he fini t e - dimens ional distributions of the outputs of one class of non-linear sys t ems . Prob l. Contro l. and Informat . Theory , 1~ 407- 418 . Sini t syn I I. ~J . ( 1986 ). Fini te - dimensional distrlbutlons of the proces ses in sto chastic integral and inteero-differential systems (this issue).

21