Stochastic analysis of systems described by the Roesser model

Brief Communication Stochastic Analysis of Systems Described by the Roesser Model by

N.

E.

MASTORAKIS

~2nd N.

J.

THEODOROU

National Technical University of Athens, Electrical Engineering 42 Patission Str., GR-10682 Athens, Greece

Department,

ABSTRACT : The Roesser state-space model for multidimensional systems, when Gaussian noise is added in their input and in their output, is considered in this work. The mean value and the covariance of the state-space vector as well as of the scalar output are evaluated.

I. Introduction

Multidimensional systems (m-D) theory has now arrived at a matured state, and many important results regarding the problems of m-D digital filter design, image processing, m-D system minimal realization, m-D polynomials factorization, m-D system stability and control, etc., can be found in the relevant technical literature (l-10). The Roesser state-space model (1) describes a multidimensional system in the case where we have a deterministic input. The case where a Gaussian noise is added to the deterministic input may be interesting for several systems (l-lo). The statespace variables are considered as stochastic variables, and they include a stochastic vector. For this vector, as well as for the scalar output, the evaluation of the mean and of the covariance is examined using the formulae of the m-dimensional system response.

ZZ.Stochastic

Analysis of MultidimensionaC Systems

Linear shift invariant state-space model (1) :

m-D discrete

$1The Franklin Institute00164032/93 %b.OO+O.W

systems

can be presented

by the Roesser

N. E. Mastorakis

and N. J. Theodorou

rxdn,, . . ..n.)) y(n,,.

.,n,)

= [C,

...

I

C,,]-

+D*u(n,,

. . .,n,)

(lb)

x,(nb...,n,) where x,(n,, . . . , n,), i = 1,. , , , m is the ith dimension’s state, u(n ,, . . . , nz) is the is the scalar output and A,, i,j = 1,. . . ,m, Bi, scalar input, y(n,, . . . ,n,) i= l,..., m,Ci,i=l ,..., m, D are real matrices with appropriate dimensions. Suppose that Gaussian noise is added in the input and in the output of this system : xl(n,+l,...,n,,) = I x,,,(n,,...,n,+l)

x~(n~,. . . ,n,) An,, . .,%A = [C,

!

Gl[

x,(n,,...,n,) +D*u(n,,

I

. . .,n,)+w,(n,,

where w,(n , , . . . , n,,,) is the Gaussian noise (i = 1,2). The following definitions are given for the Gaussian denotes the mean value of its argument) E[w(nl,...,n,)] 0

is a square

(2b)

noise (ll), (where p = E[ ]

=0

E[w(n,,...,n,)wT(n;,...,n~)l = ,+, 1.--, n,) i where R(E) = R(n ,, . . . , n,) duced :

. . .,n,)

matrix.

n,) Z (n;,...,&J

when

(n,,...,

when

(n,, . . . , n,,,) = (n’,, . . . , n;)

The following

notation

is intro-

xl(fll,...,n,> x(n)=x(n,,...,n,)= [ x,,(n I, .! . . , n,) 1.

x(n+ __ 1) = [:;;;;[:-y;;;] w,(n) = w,(nl,...,hJ,

w2(n)= w2(ni,. . . ,nm)

A$ = Eb#)l =

768

Journalof

the Franklin Institute Pcrgamon Press Ltd

Stochastic

Analysis of Systems

,a(n+ 1) = E[x(n+ ~ l)] = [ :;,+J;.l’:l;j, P&) = E[y@)l = p&,, . . . ,n,) = E[y(nl,. . . ,n,)l. From Eq. (2a) ; considering

mean values, we find that

.

‘..

.

A

Im

.

A mm I[

C1,tn,,...,nrJ h(nl,...,hJ 1

1 h(nl,..-,nm Cm]! 1 +

B, i

-u(n,,...,n,).

(3)

[ B,

From Eq. (2b), we have

PJ$r,,...,%J

= [C,

+D*u(n,,

..

[ Am

. . . ,n,).

(4)

,, . . . , n,)

Therefore, the mean values satisfy the state-space equations (1 a) and (1 b) without the noises w,($, w&). Therefore, the evaluation of &) = p(n,, . . . ,n,) and of X(H) = x(n,, . . . ,n,), y(11) = P#,, . . .s n,) is the same as the evaluation y(n) = y(n Ir . . ., n,) from Eqs (1 a) and (1 b), i.e. from the original equations without the noises w ,(& w&j. Now, in order to evaluate p(n ,, . . . , n,) and pY(n ,, . . . , n,), x(n,, . . . , n,) and y(n ,, . . . , n,) should be evaluated. For this reason, the following initial conditions are considered :

xlt0,n2,...,nm)

1 1 0

$(O,q,...,n,)=

9.V.) P(n,, . ..,n,-,,O)

!

It is easily found that the solution

of Eq. (2a) is :

(it,. . . , id # (nl,...,nm) “m “I

a,, ...,n,) .B,s%--so+. Vol. 330, No. 4. pp. 767-773, Printed in Great Britain

=

j~o-20 m

. .+A n, -i,.n,-i,

,..., n,-i,-

I.n,-i2.....nm-im

(,~l-il-

1 BO,....O.

‘)

1993

769

N. E. Mustorakis und N. J. Theodorou

n,,,

; . . . , Co -w,(i,,. . . , i,,,)+ ;,=”

A”I.n2-i2.....nrn-L

??I

*+i(O, iZ, . ..,i,)+...+

2

““, 1

..'

i,=o

L>

I

Atl-i,.l12

-1*. ..I& ,-I,,. ,.n,,

I =0

- S(i, ,...,L1,O) where the following

(5) definitions

A’I.--.*~T~ = 0

are used :

if and only if

A’,.. ,’ = I

i, < 0

or..

. or

i, < 0,

(I is the identity matrix)

A’,,.... I,,,= Al.%..-.OA’,-&..

.‘,,r+...+A

0

,.. .)A’,...., $,>.,.L--

and

ALO

,..., 0 =

I’ .:“j,..., LA,,,! .;.Am] A”vo.’

Bl.0,

Considering

. ..o

!IJ )...,

BO. ..*l).l =

the mean value for both members (i,,...,

p(~,,...,%) .Bl.o..

=

=

.O+...+A

=

of Eq. (5), we find that

L) Z (n,, . . . , f~)

i;.

..’

co

(A”]

i,-l.n2-r2

. ...n,,,-i,,,

‘8, =

n,-i,.n,-l, ,.... Q-i,,,-

IBO...

,“.I)

(6) where 770

Journal

of the Franklin lnstttute Pcrgmon Press Ltd

Stochastic Analysis of Systems

F(il,..., i,_l,O) =

0 :

dil,

. ..,L,,O). i

~,(O,i,,...,Q 0 @(O,i2,

.

..,i,)

= 0t

-

1. form : i.e.

Now, we rewrite Eqs (2a), (2b) in a more compact

x(n+ 1) = A*x(n)+B*u(c)+G.w,(n)

(7a)

_

y(n) = C*x@)+D-u@)+w,@>.

G’b)

Now, Eq. (3) may be rewritten

p(n+ 1) = A*y(c)+B.u(p). The covariance

matrix of a multidimensional

(8)

stochastic

vector is defined as follows

:

PxCn> = Wx(p) -~@)>(x(p)-i4nNTl. From Eqs (7a) and (S), if we define R(n) = E[w,(@+~(nJ], P,(n+)

=

E[(A[x(rt)--(p)l+Gw~(n))(A[x(l?)-P@)I+GwI(~))~I

= ELWx($-/@NW> -A#AT

=

we have

AP,(n)AT _

+ .W@) -P@))~:($G’

+ GR(n)GT. _

Note that some terms are equal to zero since w,($ P,(n+

is independent

of x@). So

1) = AP,(c)AT+GR(n)GT.

(9)

Observe that for the evaluation of P,@) the input z&z) does not play any role. A simple explanation is that the term B * U(H) in the right-hand member of Eq. (7a) does not introduce any uncertainty to the vector xw). It is desirable now to find the expression of P,(n). For this purpose the following difference is considered : “2 x(p)-p(c)

= x(n,,...,n,)-p(n,,

. . ..n.)

=

C i,=O

.-.

$j

An1-"2-i2~..-"~npi"~

l"‘= 0

"8% I .(5(0, iz, . . . , i,) -@(O, iz, . , im)) +. . . + fiJ . . . in,Tco AnI ~il.n2~i2.....nm-l -in,- ),nrn i,=o

Vol. 330, No. 4, pp. 767-773, 1993 Pnnted in Great Britain

771

N. E. Mastorakis

and N. J. Theodorou

tnt,...,n,)

tit,. . .,i,J f .(%(i ,,...,

i,-,,O)-l.i(i,

.(A”,~‘,--l.n2~i

7v

,...,

i,_,,O))+...+

2*-..‘n,,,-r,.G,,0

. . . . . o+

,gy ,~,-‘~,“~~i~,

. ..+A

,$, ,?I

. ..“.,,mi,,-

IGO,.

.o. ,)

,(i,, . , , im>.

Therefore P,(n) = E[(x(+~(~))(x(~)-&J))~]

= 2

...

,,=o

*E[(rZ(i ,,.,.,

i,,_,,O)-@(i

,,...,

"i'

A"1-',."2-~2...,.",,,-~,- I.",

i",_, = 0

i,,-,,O))(i(i,

,...,

i,n_,,O)-p(i,

,...,

i,,,_,,O))T]

H”, .(A”,-il.n2-i2

,.... n,,,-i,,,

. .+

,.n,,)T+.

A",."L~i2.....fl,n-ln,

$ i,= 0 $20

*E[(i(O, i2,. . . , i,“) -@(O, i2,. . . , i,))(S(O,

i2,. . . , im) -fi(0,

i2,. . . , i,,,>>T]

(it,. ..,i,,> f tn,,...,n,) . (An1.n2-iz

. . . i,,to

j.

.G

1.0,....0

+.

-,(.nl-r2,....“nl~i,,l-1

. . +A”,

. ..O)T(An.-!,-

. ((G’,O.

. (~n,-r,.rr-i~

I.n2~i*....,n,-r,

(A”,-iI-

. . . . n,,,pin,,)T+

‘.tI-1

,.... n,“-I,.-

2

,....

.,,,-i,,)T

-G”.....o,’ +

_. . +

-E[w,(i,,

. . . , i,,,)w,(i,,

..,

im)T]

(GO..-..O,‘)T

I)‘)

(10)

since different initial conditions are independent. Using Eqs (2b) and (6), the output y(n,, . . . , n,) is found : my = [C, , . . ., Cnl *

+...+

5 i,=o

tit,. ..,L)

+

...

,5....imfo A”I.“2-iZ...,.%‘-i)“~(0, _4n-i

, .,i,_,,O)

I.“ml*p(i,,.

i,_,=O

# tn,,...,n,)

;,:. . . .

$.

+. . .+~“1-il.“~~i2.....“nl-‘rnAlso, after some manipulation can be evaluated. Thus

772

,,.... IT,_,-i,

‘“c’

i2, . . . , i,,,)

(A”]

-iI -

‘Jr-i,

,....

.,-‘,gl.o

‘B”~~~.-o~‘)u(i,, . . . , i,,,)

the covariance

,.... 0

+D*

u(n,,

. . , ,n,).

P,@), that is, E([( y - L)(Y -

/+ITI),

Journal ofthe FranklinInstitute PressLtd

~ergamon

Stochastic

Analysis

of Systems

(11) where P,($

is given by Eq. (10).

III. Conclusion The paper refers to the m-D discrete systems when an m-D Gaussian noise is present in their input and in their output. We find the mean and the covariance of the state-space vector x(c) and the output y(n). More specifically, one observes that in the case where a Gaussian noise is added in the input and in the output of an m-D system, ,u($ [the vector mean of x@)] and /.+,@z)[the mean of ye)] satisfy the original state-space equations (without noise). Considering the x@)-, y(A)response, the mean &I) and &II) as well as the covariance of x@) and y@ can be evaluated. References (1) R. P. Roesser, “A discrete state-space model for linear image processing”, IEEE Trans. Autom. Control, Vol. AC-20, No. 1, pp. I-10, 1975. (2) K. Hirano, M. Sakane and M. Z. Mulk, “Design of three dimensional recursive digital filters”, IEEE Trans. Circuits Syst., Vol. CAS-31, pp. 55&561, 1984. (3) H. Mutluay and M. M. Fahmy, “Frequency domain design of N-D digital filters”, IEEE Trans. Circuits Syst., Vol. CAS-32, pp. 12261233, 1985. (4) M. E. Zervakis and A. N. Venetsanopoulos, “Design of 3-D IIR filters via transformations of 2-D circularly symmetric rotated filters”, Proc. 1986 IEEE Int. Conf. Acoustics, Speech, Signal Processing, Tokyo, Japan, pp. 545-548, 1986. (5) D. S. K. Chan, “The structure of recursible multidimensional discrete systems”, IEEE Trans. Autom. Control, Vol. AC-25, pp. 663-673, 1980. (6) S. G. Tzafestas, “State-feedback control of three dimensional systems”, in “Multidimensional Systems, Techniques and Applications”, (Edited by S. G. Tzafestas), pp. 161-232. Marcel Denner, New York and Basel, 1986. (7) Q. Liu and L. T. Bruton, “Design of 3-D planar and bean recursive digital filters using spectral transformations”, IEEE Trans. Circuits Syst., Vol. CAS-36, pp. 365-374, 1989. (8) C. M. Trikha, Y. C. Chopra and J. S. Bajwa, “Three-dimensional unilateral linear iterative circuits”, Int. J. Syst., Sci., Vol. 14, No. 2, pp. 117-139, 1983. (9) S. G. Tzafestas and N. J. Theodorou, “An inductive approach to the state space representation of multidimensional systems”, Control Theory Adv. Technol., Vol. 3, No. 4, pp. 293-322, 1987. (10) Karl J. Astrom, “Introduction to Stochastic Control Theory”, Academic Press, New York, 1970. (11) Peter S. Maybeck, “Stochastic Models, Estimation and Control”, Academic Press, New York, 1979. (12) T. Kaczorek, “Two-dimensional linear systems”, in “Lecture Notes in Control and Information Sciences”, Vol. 68, Springer, Berlin, 1985. Received : 1 July 1992 Accepted : 4 January 1993

Vol. 330. No. 4, PP. 767-773. Printed in Great Britain

1993

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