Stochastic Systems: Theory and Applications 9810247427, 9789810247423

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STOCHASTIC SYSTEMS "Theory and Applications

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STOCHASTIC SYSTEMS V. S. Pugachev I. N. Sinitsyn Russian Academy of Sciencies

Translated by

|. V. Sinitsyna

Ve World Scientific NewJersey

«Londons Singapore e Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661

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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

STOCHASTIC

SYSTEMS

Theory and Applications

Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN

981-02-4742-7

Printed in Singapore by World Scientific Printers (S) Pte Ltd

PREFACE

The monograph is dedicated to the systematic presentation of the applied advanced theory and analytical methods for the stochastic systems, i.e. the dynamical systems described by the finite- and infinitedimensional stochastic differential, difference, integral, integrodifferential etc. equations. It is based on the results of the fundamental research performed in the Institute for Informatic Problems of the Russian Academy of Sciences in the context of the scientific program “Stochastic Systems” and the lecture courses-delivered by the authors in the domestic and the foreign Technical Universities. The book may be used as the textbook for the applied mathematics faculties of the universities. The unified procedure, thorough selection of the examples and the problems (over 500) and the large applications make this book useful for the graduates, the post-graduates and the lecturers. The book is of considerable interest also for the mathematicians who deal with the stochastic equations and their applications. The construction of general stochastic systems theory is based on the equations for the multi-dimensional characteristic functions and functionals. The stochastic differential equations with the arbitrary processes with the independent increments are studied. The equations with the Wiener and the Poisson processes are considered as a special case. The methods of the parametrization of the multi-dimensional distributions in the nonlinear stochastic systems based on the moments, the semiinvariants, the quasimoments and on the consistent orthogonal expansions are stated systematically. Special attention is paid to linear and reducible to linear stochastic systems theory based on the canonical representations (canonical expansions and integral canonical representations). Most attention has been concentrated on the structural theory of the composed stochastic systems on the grounds of the conditional probability measures. Chapter 1 is devoted to the mathematical models of the dynamic systems under conditions of the random disturbances and their characteristics. The linear and the nonlinear continuous, discrete and continuousdiscrete systems described by the stochastic equations in the finite- and the infinite-dimensional spaces are considered. Special attention is paid to the composed stochastic systems. The main notions of probability distributions theory of the random variables, the random processes and the random functions are stated in Chapter 2. After determining the probability spaces, the conditional

vl

Preface

probabilities, probabilities in the finite and the infinite products of the spaces are considered. The conditions of the existence of the regular probabilities are established. Further the different probability measures of the random functions and the probabilities of events connected with the random functions are studied in detail. The last sections are devoted to the distributions of the separable random functions, to the derivation of the criteria of the continuity and the differentiability of the random functions. Chapter 3 contains the account of the basic notions connected with the expectation of the random variables in Banach and topological linear spaces, the properties of the operators of the moments of the second and higher orders, the properties of the characteristic functionals and the correspondence between the probability measures and the characteristic functionals are established. The conditional operators of the moments and their properties are studied in detail. Special section is devoted to the elements of the probabilities distribution in the infinite-dimensional spaces. The properties of the normal distribution on the product of two spaces, the conditional normal distributions and the normal distributions in Hilbert spaces are considered. Approximate representation of the oneand the multi-dimensional distributions based on the consistent orthogonal expansions is treated. Special attention is paid to the semiinvariant and the quasimoments. A systemical presentation of general theory of the canonical expansions is given. Practical ways of the canonical expansions constructing are described. The joint canonical expansions are considered. Chapter 4 is devoted to the selected topics of stochastic calculus. The theory of the convergence, the continuity and the differentiability is considered. The integrability of the random functions over nonrandom and random measures is discussed. The theory of the stochastic Ito integrals, the symmetrized (Stratonovich) integral and the 6-integral is outlined systematically. The stochastic integrals over the Wiener and the Poisson measures are studied. The connection between the processes with the independent increments and the It6 integral is established.

The elements of the theory of the generalized random functions are stated. A systematic account to the integral canonical representations is given. The connection of the integral canonical representations with the spectral one is established. Practical ways of constructing the integral canonical representations and, in particular, the methods of the shaping

filters are considered. studied.

The joint integral canonical representations are

~~

Preface

vil

The general theory of the finite- and the infinite-dimensional stochastic systems is given in Chapter 5. At the beginning the differentiation of the Ito process is set out in detail. The Ito formulae for the differentiation of the composite functions in the cases of the Wiener, the Poisson and the general random process with the independent increments have a significant place. The theory of the stochastic integral, differential and integrodifferential Ito equations including the methods of reducing the state equations to the stochastic differential equations, change of variables, sufficient conditions of the existence and the uniqueness of the mean square solutions are considered. The central problem of stochastic system theory —he problem of calculation of the multidimensional distributions is stated. The elements of numerical integration of the stochastic differential equations are given. The theory of the one- and the multi-dimensional distributions in the continuous, discrete and continuous-discrete systems in the finite-dimensional spaces are stated in detail. For such systems the known equations (Fokker—Plank— Kolmogorov, Feller-Kolmogorov, Pugachev and their generalizations) are derived. The problems of one-dimensional and multi-dimensional distributions theory on the basis of the equations for the correspondent characteristic functionals in composite stochastic systems including the stochastic systems in the Hilbert spaces are considered. The main statements of the structural theory of the composed stochastic systems on the basis of conditionally probability measures are outlined. The last section contains the applications to optimal online filtering problems. Simple derivation of the Kushner—Stratonovich and the Zakai equations for the normal white noises is presented. Corresponding equations for the composite stochastic systems are given in Problems. Chapter 6 contains the account of the main methods of linear stochastic systems theory in connection with continuous, discrete and continuous-discrete systems. The spectral and correlation methods, the methods of general theory of linear stochastic systems are outlined. The correlation methods based on the canonical representations are considered with great attention. The general theory of the canonical representations of the linear transformations is given. The separate subsections are devoted to the solution of the linear operator equations by the canonical representations methods. The applications to the offline spectral and correlation analysis and the shaping linear filters design are given. Software for the analytical modeling problems is described. The Kalman-Bucy and the Liptser—-Shiryaev online filters are considered.

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Preface

Chapter 7 is devoted to the approximate methods of nonlinear finite- and infinite-dimensional stochastic systems theory based on the parametrization of the distributions. The normal approximation, statistical and the equivalent linearization methods are considered. The Genenormalization of the composite stochastic systems is treated. ral problem of parametrization of the multi-dimensional distributions is stated. The methods of the moments, the semiinvariants, the quasimoments and the orthogonal expansions are given: The practical problems of reducing the number of equations for the automatic derivation of the equations and the corresponding software for distribution parameters are considered. The solution of the structural parametrization problem of the multi-dimensional distributions is outlined on the basis of the ellipsoidal approximation method. The canonical representations methods for the nonlinear stochastic systems are considered. The ba-

sic methods are given for stochastic systems with the normal white noises. Corresponding methods for the composite stochastic systems are given in Problems. The applications of the given methods to the nonlinear stochastic systems offline and online analysis, modeling and filtering problems in mechanics, physics, biology, chemistry, finance and control are stated. Specialized software is described. Chapters 1, 5-7 give efficient toolkit for the analytical modeling and the design of the stochastic systems. To facilitate the mastering of the stated methods about 500 thoroughly selected examples and problems are given in the correspondent subsections, sections and chapters of this book. Appendices 1-7 contain auxiliary materials which include the information about tables of stochastic It6 differentials for the typical composite functions, the Hermite polynomials of a vector argument, the polynomials orthogonal to gamma and y?-distributions, the tables of the coefficients of statistical linearization for the typical composite functions, integrands evaluations in the equations of the basic methods of distribution parametrization, the tables of a basic characteristics and transformation the structural rules of the linear systems and the tables of some definite integrals and special functions. For information about the notions and theorems from vatious parts

of mathematics used in the book we advise (Korn and Korn 1968).

Bibliographical notes and References are given at the end of the book. The authors do not pretend in any way to provide a complete bibliography. Only those sources are given in the list which are cited in the text. Basic notations are presentented before Index.

~

Preface

ix

The book is divided into chapters, sections and subsections. The formulae (examples) are numerated with indication of the number of the chapter and the section, for instance, (1.2.3) means that formula (3) belongs to section 2 of chapter 1. The problems are numerated only with indication of the number of chapter. At each Appendix the separate numeration is accepted. For the convenience of the readers the formulations of all basic results and statements are given in the separate theorems and the corollaries or in the text italics. The beginnings and the ends of the evaluations, the proofs and the discussions which lead to certain results are indicated by triangular indices p and Let u(t, a ) be the solution of the homogeneous equation

= au

(1.2.2)

satisfying the initial condition u(r ie ) = I, i.e. the fundamental matrix of solutions

of Eq. (1.2.2). By the change of variables

7 = u(t, to)v the first Eq. of (1.2.1) is

reduced to the form

uv + uv = auv

+.a,z2+ a9

or by virtue of Eq. (1.2.2)

uv =ayr+ao. Remark.

It is known from differential equations theory that the determinant

of the matrix u(t, to) is expressed by t

A = |u(t,to)| = exp

J vra(ryar

(1:2:3)

to

where tr a(T) is the trace of the matrix alr): To derive this formula it is sufficient to notice that according to the differentiation rule of a determinant and the properties

of the determinants

A =

condition A(to) =

|u(to, to)| = 1 is determined by formula (1.2.3).

from this formula that

Atr a(r i The solution of this equation with the initial

It follows

A # 0 (we take for granted that the integral in (122.3)/is

finite at all t, 9 owing to which the exponential function does not vanish). Hence, taking into account that the matrix u is always invertible we get

v=u(aye+ ao).

1.2. Linear Differential Systems

25

Integrating this equation with the initial condition v(to) = u(to, to)~+2z(to) = 74 gives t

v(t) = zo + it u(r, to) | [ai(r)a(r) + ao(r) Jdr,

(1.2.4)

and

z(t) = u(t, to) v(t) = u(t, to)zo 1.2.5

+ u(t,to) fu(r, to)* [ar(z)e(r) +ao(r)|dr.< to : Let us prove that at any to, 7, t, tp

(

)

Putting u = u(t, to),

v= u(t, to).= u(t,to)4 , we have at any tp andt

> tp uv!

=

I.

(1.2.11)

Differentiating this formula we get:

uyve + uve = 0. So yielding by Eqs. (1.2.2), (1.2.11) we have

uv; = —auv’ = —a ort? = —u“la.

(i. 2612)

Hence by virtue of Eq (1.2.11) we obtain the differential equation for UV: vr = —-y!a,

Now we notice that according to (1.2.6) and (1.2.11) u(t, 7) = u(t, to)v(7, to)’. Differentiating this formula with respect to 7 and taking into account that according

to (1.2.12) 0;(7, to)? = —v(r7, to)? a(r), we find i,

(t,T) = u(t, to)vr(7, to)? = —u(t, to)v(7, to)? a(r) = —u(t, r)a(r).

Thus u(t, 7) as a function of T at fixed t is determined at T < t by the equation

u,(t,7) = —u(t, T)a(r)

leah)

with the initial condition u(t,t) = J att =t. 4 Transposing Eq. (1.2.13) we obtain

UAT)

=

alr) ulter)? «

Cle2ei4}

This equation is the adjoint equation to Eq. (1.2.2). Therefore u(t, 7)? as a function of T at fixed t is determined at 7 < t¢ by Eq. (1.2.14) adjoint to Eq. (1.2.2) with the condition u(t,t)” = I at the end. So u(t, 7) may be found by the integration of the adjoint Eq. (1.2.14) in reversed time

with the initial condition u(t,t) = I. 1.2.3.

Transformation of Linear System Equations into Cauchy Form

If the input-output relation of a system is represented in the form of a differential equation of higher than the first order then it usually contains not only the derivatives of the output but the derivatives of the input as well. Therefore, the problem arises to transform the equation

28

Chapter

1 Dynamical Systems and Their Characteristics

of a linear system into a set of equations of the first order which do not contain the derivatives of the input. Now we shall show how to do this. Let us consider a nonanticipative system whose behaviour is described by the linear differential equation of the form

See

Ybe,

K=O

k=0

(1.2.15)

where the coefficients ao, a1, ..., Qn, bo, b1, ... , bm depend in the general case on time t, and m < n.. It is evidently sufficient to consider the case of m = n. The case of m < n is the special case where Onis

Ort

=

());

> Let us introduce new variables

Z=Y—

QoL,

241

where qo, 41, ---, Yn—1

= 2% — get (K=1,...,n-—1),

(1.2.16)

are some functions which we shall determine from the

condition that the resulting equations of the first order contain no derivatives of the

input Z.

From Eqs.

(1.2.16) using the known formula for the derivatives of the

product of two functions

7

Uv Che SS en PuP)y Ae CrCee =

me

»

|

ee

1.2.17 ve

p\(r — p)!

(

we find

Y¥=%1+ 402, 8

Soon

yo) = Zs41 + Ge)

= 2540

em

r=0

Pg

Pal)

(1.2.19)

r=0,p=0

(SP

oe

eae

After changing the order of summation we get

NT NG +

Lorde? ((Sad

teen 1). » «(1-3.20)

Differentiating formula (1.2.19) corresponding to s = n — 1 gives

mee y\" ” ain

#9CES—r— 12)yet) =z, +

(of ("),t)

(1.2.21) Ahh ait 5mqd, + al Sor? p=1

r=p

1.2. Linear Differential Systems

29

Substituting the expressions (1.2.18), (1.2.20) and (1.2.21) into Eq. (1.2.15) we obtain

nin

+

> a-izi+e

AP

ip!

Ueeoee Da

ail

s=0

=

a >x) eos CPa,q’,”) >, = al?) , Sy

(1.2.22)

i)

Equating the coefficients at the corresponding derivatives of the input & in the leftand right-hand sides of Eq. (1.2.22) we obtain the equations for the functions gy:

Mee ce “r)=b, = (p=1,...,n).

Gres)

s=pfr=p

The last of these equations corresponding to p = 7 gives

Gh= an'b,.

(1.2.24)

For solving the other Eqs. (1.2.23) notice that

YS s=p

Lead? reppe Se r=p

s=p

i009

pI

1=0

n—pn—p—l

!)

.ye yg Pi teds--p-1 =) = 1=0 s=p+l

(=O

n—-pn—p—h

Fa

(1)

Dang pti aptlthTh =O

n—p—In—p—-—h

C.Pap erEn gy? = Gn n—p T yO h=0)

Y=0

h=0

>s Coeitptlthdy [=0

Substituting this expression into (1.2.23) and putting n — p= k-1k-h

=:

Onqk + Dey ny) Chaepian—ethtiq,. h=0

()

@) ip

(1.2.25) k we get

=

= on-k (K=1,-..,2—1).

7=0

(122.20) From

(1.2.26) we obtain the recursive formula for determining the functions q;,

+) 9n-1: k-1k—-h

h=0 [=0

(1.2.27)

30

Chapter 1 Dynamical

Systems and Their Characteristics

After determining in such a way the functions qo, 41,---,Qn—1

we transform

Eq. (1.2.22) into the form n

n

n-1

8

pee

Onin + 0 az = { ty Sod ree t=

l=1

sa)

Solving this equation for Zp, gives together with Eqs.

k =1,...,m-—1

hae)

1=0

(1.2.16) corresponding

to

the equation forming the set of the first order equations Ze = Ze41

+ Mee

Zn =

(kerk,

Ss Wy od

=

n

-Soa;

(1.2.29) Qj=121

+ nk ,

where

heathy

a) Ome ya, (1.2.30)

s—0

l=i

n-1

n

s

= an1 |bo— ) fangs — Soas Dain =

I

h=0

$=1

Si

Noticing that n

s

Sal

l=1

)

non

(a) I

n

n—l

n-1n-h

dias Doar = DOD oases=DOD ansiah? = DO DTansiah? (i)

gel!

i=h=)

(=O

a

(1.2.31)

we transform the formula (1.2.30) for gn into the form n—-1ln-h

Grit= Gawlibax My: Ds ansiqe?

:

(162.32))

h=0M=0

It is evident that this formula coincides with (1.2.27) at k = n. Thus all the functions

qi, --+ 5 Un are determined by (1.2.27). The

=---=

case

of ™


» Aay=

Bere

5

Qqn-1

ace eGOne

In

(1.2.35)

In the case of m = n the output y depends explicitly on the input zx. But this is a rare case since m < n practically always yields qg = 0. The

expression (1.2.34) of the output y does not contain z in this case. +,

In the special case of a stationary system all the coefficients aj, 2n; 00 01, +5 Unare Constant. ald thus Gp, d1,---,dn Gevermined

successively by formulae (1.2.24) and (1.2.27) are constant. qi? Oe Ut ree Ly ee considerably simpler in this case:

Therefore

an loniitiac (1.2.21) become

k-1

Go

Ge Dene Te te

(50-1-

So tm-tena) i ae reared Oe

Formulae (1.2.33) in this case take the form w=

7a

di =

(5.—

>a h =n—m

=n

me

oat

— 04

Gree

a

On

(k =n—m+1,...,

mn). (1.2.38)

32

Chapter 1 Dynamical

Systems and Their Characteristics

Notice that all previous calculations and consequently formulae (1.2.27), (1.2.33), (1.2.37) and (1.2.38) for go, 41, ---, Qn are valid both for the scalar and the vector input x and the output y. In the latter case ao, Gy js » Ope. B05 O15 1284 Ony Weds a> > Gea te plenent, matrices of the corresponding dimensions. Example

1.2.2. In order to transform the equation

a3 ¥ +aoy + ayy + aoy = bet + bit + bo

(1)

into a set of the first order equations we put according to (1.2.16) and (1 .2.33) OS

Ai,

22 = 21 — q12, 23 = 2 — Gov. Then by (1.2.33) we find

qo=0,

q1 = a3'b2, go = a3 '(b1 — aoqi — 20191),

q3 = 43 ‘(bo — 4191 — 4241 — 4391 — 4292 — 4342) As a result we obtain Eqs. (1.2.29) which has in this case the form 3 Ai—Zo7+Ne

,

‘ 232—23 4-090

,

: SI 23=—azg (aoz1

+4122

+4223)+

932 ,

Y

en =21.

(11) 1.2.4. Inverse Systems The system inverse to a system A is such a system whose output is equal to the input of the system A when its input is equal to the output of A (under the corresponding initial conditions). Thus the inverse system performs the transformation of signals inverse to that performed by a given system A. It is evident that if the system B is inverse to the system A then the system A is inverse to the system B. In other words, A and B represent two reciprocal or mutually inverse systems. The amplifier with the gain k and the amplifier with the gain 1/k, the differentiator and the integrator may serve as examples of pairs of mutually inverse systems.

If a system with the input «# and the output y is described by Eq.

(1.2.15) then the inverse system is described by the same differential equation in which y serves as its input and z serves as its output. If m = n, then the reciprocal systems described by differential equa-

tion (1.2.15) are of the same type (i.e. both differential systems). After transforming the equation into a set of the first order equations, as we have already seen in Subsection 1.2.3, the output of each of these two

1.2. Linear Differential Systems

33

systems will contain its input with some coefficient and the remaining part of the output will be determined by the corresponding set. of the first order differential equations. If m = 0 then the output of the inverse system is given by

2 = b5* > apy).

(1.2.39)

It goes without saying that for the existence of the inverse system in this case it is necessary that the inequality bo # 0 be valid at all t in the case of an one-dimensional system and that the matrix bo be invertible at all ¢ in the case of a finite-dimensional system. Under this condition the inverse system performs the linear differential operation

Lig) aD

ya Fa dte

(1.2.40)

k=0

> If 0 < m

< n then the operations which are performed by the inverse system

include the (n = m)-tiple differentiation of the input. As a result the output of the inverse system contains a linear combination of the input and its derivatives up to

the (n = m)*h order. In order to determine the inverse system it is necessary in this case to find the remaining part of its output. For this purpose we put nm—m

p=zt+nazt Do ay

(1.2.41)

1=0

and determine the coefficients Cg, Cj , ...

, Cn—m

In such a way that the variable Z

contains no linear combination of the input Yy and its derivatives.

Using the known formula (1.2.17) of the derivatives of the product of two func-

tions we find from (1.2.41)

Bie aom

ie Se ssCree y(t 1=0

r=0

=a? Ne Se Cre!(s—r) yltr) a= r=0)1=0

4

Cre os yl)

(1.2.42)

After this evident transformation of the double sum it is expedient to change again

the order of summation in order to obtain the coefficient at y6*) in the explicit form.

34

Chapter 1 Dynamical

Systems and Their Characteristics

For determining the limits of the inner sum we notice that0 k —n +m and

min(k,s)

ve

k=0

the order of summation

are possible we come to

Crea y(®),

Pe

s=0

0

min(k, s). As a result we obtain

or eltSyl) ,

r=max(0,k—n+m)

and

(1.2.43)

taking into account

that

k
0 weget

5Hb,x) = snbp z6*) s=0

+

3

k=0

min(k,s)

ha

DyPnnidesu CBee

k=0 s=max(0,k—n+m)

Eyes

masmenaeaay

r=max(0,k—n+m)

Let us substitute this expression into Eq.

(1.2.15) and equate the coefficients at

y™) | wens 5 y”) in the left- and the right-hand sides of the equality obtained.

taking into account that min(k, s) = s at s m

s

3

)

SF

Gthicaaaz

(b= sit pederp ny)

s=max(0,k—n+m) r=max(0,k—n+m)

In order to solve these equations for Cg, Cj , ... , Cn—m

order of summation.

m

Se

Then

m we get

(1.2.45) we change once more the

Remembering that r < 5 we get

m

Y Clb clbansea, (b= 9) oo)

(1.2.46)

r=max(0,k—n+m) s=r

or putting

s

=r+h

m

3. r=max(0,k—n+m)

Mm—Tf,

h=0

Clinbrgnce?,=a, (k= m,...,n).

(1.2.47)

1.2. Linear Differential Systems

At k =

n the sum

35

contains only one item corresponding

Therefore, we have at

k = n the relation bmCn—m

to

r =

m,

Gee eeb aa Separating in the item corresponding to

only one value

beCh om

h = 0 when r =

(1.2.48)

T = Mm and taking into account that h has

m—r

ys

De Cotte h=0

ol flWem

(1.2.49)

7 < ™), these equations can be solved for Cx_m. m—1

Cela = 07,

toages

>k—m

for all

As a result we obtain

m—r

ds

ec

r=max(0,k—n+m)

After determining Cyn—m

areal Cae be

.

As the sum here contains only Cc; with | greater than k — m (k —r items since

we

emp.

h=0

eal

(1.2.50)

by (1.2.48) we may find successively Cp—m—1,---,

Co using formula (1.2.50).

0.

™ we transform Eqs. (1.2.45), k > be z'*) =

»» a,y”) ;

k=0

k=0

(1.2.51)

where

m—1

ide

min(k,s)

3

Sy

‘ea ane.

(1.2.52)

s=max(0,k—n+m) r=max(0,k—n+m)

it 2.0piiynenotan— 1): Changing the order of summation here and taking into account that max min(k, s) s

=katk Giese”

r=max(0,k—n+m) s=r

(1.2.53)

Systems and Their Characteristics

Chapter 1 Dynamical

36

or

k

m—r

r=max(0,k—n-+-m)

h=0

Ge in), —

Sy

SiC hidrthle alk SU gleg cnc Mm),

(1.2.54)

Formula (1.2.41) shows that the system inverse to the system described by Eq. (e215)

at 0
|o)

lie ee

2 lrg56 ala let Hy |

y; (S )

7:

6)

—>

hs

yHy,ty,-);

Ys;

y)

— > y=y,ty,ty, y;

Fig. 1.3.3 A block diagram is called a graphical image of the system structure or its blocks. Here the structure is regarded the aggregate of the blocks and the connections representing the loops. The elements of a block

diagram are (Fig.1.3.3): e Linear dynamic links presented by the blocks with the operators describing the functions filled in the blocks or in the description applied (Fig.1.3.3a). e Linear and nonlinear functional generators (Fig.1.3.3b). e The summators presented by the circles divided into the sectors. Generally the summators realize the algebraic summation. Some times we use + signs in circles (Fig.1.3.3c). e Couplings which transmit the directed action without changing (Fig.1.3.3d). e Branching points on the coupling lines (Fig.1.3.3e). Equally with the block diagrams the oriented graphs are often used (Fig.1.3.4a,b,c). The main elements of an oriented graph are edge, node and arrow. The oriented graphs do not need special notations for summators and couplings as in the block diagrams.

1.3. Conneétions of Systems

45

qg)

Fig. 1.3.4 Transformation rules for block diagrams and graphs are given in Subsection 1.3.4 and Appendix 6. 1.3.2.

Weighting Functions of Linear Systems Typical Connections

Consider at first a series connection of two linear systems whose weighting functions gi(t,7) and go(t,7) are given (Fig.1.3.5). Find the weighting function g(t, 7) of a composite system.

> For this purpose we find its response on the unit impulse. As a result of the action of the unit impulse on the input of the first system its output will represent its weighting function 91 (t, af). Consequently, the input of the second system is equal

a(t

gityn).

(17371)

Here the input of the second system according to general formula (1.1.8) is expressed by formula [oe)

foe}

yo(t) = J s2lt,o)22(0) a0 = J 92lt,0)ni(0,7) do. —00

—oo

(1:3:2)

46

Chapter 1 Dynamical

Systems and Their Characteristics

But the output of the second system represents also in a given case the output

of the whole series connection of the considered systems, i.e. the weighting function obtained as a result of the connection of a system

(1.3.3)

yo(t) = g(t,7).

Comparing this formula with (1.3.2) we obtain the following formula for the weighting function of a series connection of two linear systems

gts

J salt-o)au(er) do. 4

(1.3.4)

Thus the weighting function of two connected in series linear systems is determined by formula (1.3.4). In the special case of nonanticipative systems we have g;(c,7) = 0 at o < 7 and go(t,0) = 0 at o > t. Consequently, for nonanticipative systems an integrand in (1.3.4) differs from zero only in the limits +

For determining the weighting function g(t, ai) of a composite system we shall find its response on the unit impulse 6(t = a) Accounting that while delivering the unit impulse 6(¢ = T) on the inputs of connected systems the outputs are equal to

the corresponding weighting functions g1 (t, T) and ga(t, T) we obtain

g(t, 7) = gi(t, 7) + ga(t, 7). 4

(1.3.7)

Thus at the parallel connection of the linear systems thetr weighting functions are added. For studying the systems with a feedback we use the notion of the

reciprocal (mutually inverse) systems (Subsection 1.2.4). Consider a series connection of two reciprocal linear systems. The weighting function of the linear system inverse relatively to the system with the weigh-

ting function g(t,7) we shall denote by the symbol g~ (t, rT). The weighting function of a series connection of two reciprocal systems is equal to 6(t—7). On the other hand the weighting function of a series connection may be calculated using formula (1.3.4). As a result we obtain [oe)

ifg(t, 0)g(0,7) do = 6(t — 7). —oo

(1.3.8)

48

Chapter 1 Dynamical

Systems and Their Characteristics

This relation represents the main property of the weighting functions of the reciprocal linear systems. Changing the order of the connection of the systems we have another relation co

/ g(t,0)9 (0,7) do = 6(t — 7).

(1.3.9)

—0o

Thus the wetghting functions of two reciprocaldinear systems satisfy two

integral relations (1.3.8) and (1.3.9) which in general do not coincide. The weighting function g(t, 7) of the system with a feedback mapped in Fig.1.3.7 may be defined as in the previous cases after observing the passage the unit impulse. But this way is rather difficult and leads to

an integral equation relative to the sought weighting function g(t, 7). It is rather easy to find the weighting function of an inverse system (Subsection 1.2.4).

Bigs 13.7

Fig. 1.3.8

> It is evident that the system which is inverse relative to the system with a feedback is the parallel connection of the system relative to the system in the direct loop and the system in the feedback loop (Fig.1.3.8).

Really, denoting the input

of the considered system by Z, the output by y and the output of the system in a feedback loop by Z we see that © — Z will be the input of the system in the direct loop and y will be its output.

While supplying the output

y on the input of the

parallel connection of the systems with the weighting functions g; and g2 we shall obtain on the output of the first system the signal 2 — Z, and on the output of the second system the signal z. Adding these signals will give on the output of a parallel connection the signal X (Fig.1.3.8).

Thus a series connection of the systems shown

in Fig.1.3.7 and 1.3.8 give an ideal servomechanism.

are reciprocal.

Consequently,

these systems

Applying formula (1.3.7) for the weighting function of the parallel

connection of the linear systems we find

g(t,7)= 9; (t,7) + ga(t,7).4

(1.3.10)

1.3. Connections of Systems

49

This formula determines the weighting function of the linear system which is inverse relative to the system with a feedback shown on Fig.1.3.7.

1.3.3.

Connections of Stationary Linear Systems

At first let us consider a series connection of two stationary linear

systems with transfer functions ®;(s) and ®2(s) (Fig.1.3.9).

Suppose

that on the input of this series connection a signal representing an exponential function of time e*’ acts indefinitely for a long time. This signal while passing through the first system is multiplied by the transfer func-

tion ®;(s). Thus on the output of the first system and on the input of the second system will be the function ®;(s)e**. By virtue of the linearity of the second system its output will be equal to the product of its response on the exponential signal e** by the constant multiplier ®2(s),

i.e.

©1(s)®2(s)e*'.

Thus a series connection of the stationary linear

systems gives the stationary linear system whose transfer function ®(s) as equal to the product of the transfer functions of connected systems

&(s) = 1 (s)2(s).

Sao

(1.3.11)

oo

(s) = ®, (5) ®, (S) Fig. 1.3.9 This formula is easily spread to a series connection of any number of the stationary linear systems. Namely, if the transfer functions of

n connected in a series systems are equal to ®(s),...,®n(s) then the transfer function of the connection ®(s) is determined by formula

@(s) = ©1(s)b2(s)---@,(s) = ]] O:(s).

(1.3.12)

p=

Thus the result of a series connection of the stationary linear systems does not depend on the order of their connection.

50

Chapter 1 Dynamical

Systems and Their Characteristics

As the modulus of the product of the complex numbers is equal to the product of the moduluses of the factors, and the argument is equal

to the sum of the arguments of the factors then from (1.3.12) follow the formulae for the gain- and phase-frequency characteristics of a series connection nm

|&(iw)| = [] ee (iw)], k=

(1.3.13)

\

Ig [B(iw)| = Solg|®x(iw)|, arg @(iw) = So arg, (iw). Formulae (1.3.14) show that at a sertes connection

(1.3.14)

of the statio-

nary linear systems their dectbel-log and log phase-frequency characteristics are summarized.

®, (s) es

e

®,(s) oa.

D(s)e

:

®, (S) e.

(s) = ®, (5) + ®,(S) Fig. 1.3.10 For determining the transfer function of the parallel connection of the stationary linear systems with the weighting functions ®,(s) and 2(s) (Fig.1.3.10) we suppose that on the input of this connection the signal representing the exponential time function e*’ acts indefinitely for a long time. As a result the outputs of the connected systems will be equal to ®;(s)e%’ and ®(s)e*' correspondingly. The output of the connection will be equal to the sum of these expressions. Consequently, the transfer function of a parallel connection of the stationary linear system is equal to the sum of the transfer functions of connected systems:

B(s) = O;(s) + 9(s).

(1.3.15)

1.3. Conriéctions of Systems

51

Evidently, this formula is also spread to any number of parallelly connected systems:

(s) = 5 ;(s).

(1.3.16)

Let us pass to the determination of the transfer functions of the stationary linear systems closed by a feedback. At first we consider the system consisting of the stationary linear system with the transfer function ®;(s) closed by the nonnegative feedback containing a statio-

nary linear system with the transfer function ®2(s) (Fig.1.3.11).

For

determining the transfer function (s) of this system let us consider an

inverse system which has the transfer function 1/®(s) (Subsection 1.3.2). This inverse system represents a parallel connection of the system with

the transfer function 1/;(s) and the system with the transfer function ®5(s). Therefore using formula (1.3.16) we get

(5) °@ (s)'® +1 (s)'®

= (s)®

Fig. 1.3.11

mane = ee + @2(s) = i+ ®1(s)P2(s)

®(s)

®i(s)

#1 (s)

From here we find the transfer function of the system with a feed-

back:

Aiea

RAS,

1+ ©1(s)®o(s)

(1.3.17)

At a rigid negative feedback it is necessary to put ®(s) = 1 in formula (3.17): Thus at any connections of the stationary linear systems we always obtain the stationary linear systems, whose transfer functions and frequency response characteristics are determined by means of elementary

52

Chapter

1 Dynamical Systems and Their Characteristics

algebraic operations on given transfer functions (on frequency response characteristics respectively) of connected systems. At all possible connections of the stationary linear systems described by ordinary differential equations as a result we always obtain stationary linear systems whose behaviour is also described by ordinary differential equations. Really, in this case the transfer functions of all connecting systems represent the rational functions of s. Therefore all derived for-

mulae for the transfer functions of the connections determine the transfer function of the connection ®(s) in the form of the rational function of s. After presenting this function in the form of the ratio of two polynomials and replacing s by the operator of the differentiation with respect

to time D we shall find the operators F(D) and H(D) and Eq. (1.2.60) for the input — output relation. 1.3.4. Structural Transformation Rules

It is expedient to transform the block diagram into more convenient form by means of the partition of the composite systems into more simple ones (decomposition), the connection of the simple systems into one (aggregation) and another transformations for the sake of simplicity of the structure on retention of the number and the order of included systems. The main principle of structural transformations consists in the fact that all the transformations of the block diagram of the linear sys-

tem (Subsection 1.3.1) should be performed in such a way that all the inputs and the outputs of each transforming section of the block diagram will remain invariant. With regard to Subsections 1.3.2 and 1.3.3 from this principle follows the rules of the structural transformations of linear systems given in Appendix 6.

Fig. 1.3.12 Example

1.3.1.

For transforming

the flexible feedback

by the rigid

one (Fig.1.3.12a) we interchange the branching point and A» system in a feedback (Fig.1.3.12b).

~

1.3. Connections of Systems

53

Fig. 1.3.13 Example

1.3.2.

Consider the translation of the system Ag along the

connected in a series stationary linear systems Aj, ...

, An (Fig.1.3.13a).

As the

stationary linear systems connected in a series may be arbitrary interchanged then the mutually inverse (reciprocal) systems introduced by the translation of the branching point and the summator via the one and the same system are mutually compensated. So for the translation of the parallel loop from Ay to Aj or vica versa it is necessary to consider the translation of the branching point via 4; and summator through Ap

(Fig.1.3.13b).

b)

Fig. 1.3.14 Example

1.3.3.

For the system (Fig.1.3.14a) analogously we conclude

that for the translation of Ag in a feedback from A, to Ap and vica versa along the connected in a series stationary linear systems Az, ...

, Ap it is necessary to

54

Chapter 1 Dynamical

consider only the summator

Systems and Their Characteristics

translation via A, and the branching point via An

(Fig.1.3.14b). Example

1.3.4. Block diagram of the linear system with two inputs and

one output (Fig.1.3.15a) may be transformed into the block diagram on (Fig.1.3.15b).

Fig. 1.3.15 Example

input and two outputs

1.3.5.

For the block diagram of the linear system with one

(Fig.1.3.16a) the transformed

block diagram is given on

Fig.1.3.16b.

b)

Fig. 1.3.16 Example

1.3.6.

Fig.1.3.17a presents the block diagram of the linear

system with a parallel loop and the additional branching point in one of the loops. Fig.1.3.17b and Fig.1.3.17c show the sequential stages of the summator translation. Example

1.3.7.

Block diagram of the composed linear system with a

feedback loop having the internal summator is shown on Fig.1.3.18a. The sequential

stages of the summator translation are given on Fig.1.3.18b and 1.3.18c.

1.4. Stochastic Differential Systems

Fig. 1.3.17

55

Fig. 1.3.18

1.4. Stochastic Differential Systems 1.4.1. General Form of Equations of Stochastic Differential Systems Stochastic models of systems involve the action of various random factors. While using models described by differential equations the inclusion of random factors lead to the equations which contain random functions, i.e. such functions whose values at given values of the argu-

ments are random variables (Chapter 2). Eqs. (1.1.4) for a stochastic system must be replaced in the general case by the equations

GAP

( ZEAyeyves GZ),

(1.4.1)

where F'(z,a,t) and G(z,t) are random functions of the p-dimensional vector z, n-dimensional vector x and time t (as a rule G is independent

56

Chapter 1 Dynamical

Systems and Their Characteristics

of x). In consequence of the randomness of the right-hand sides of Eqs. (1.4.1) and also perhaps of the initial value of the state vector 7) = Z(to) the state vector of the system Z and the output Y represent the random variables at any fixed moment t. This is the reason to denote them by capital letters as well as the random functions in the right-hand sides of

Eqs. (1.4.1). The state vector of the system Z(t) and its output Y(t) considered as functions of time ¢ represent random functions of time t (in the general case vector random functions)» In every specific trial

the random functions F'(z,«,t) and G(z,t) are realized in the form of some deterministic functions f(z,z,t)°and g(z,t) and these realizations determine the corresponding realizations z(t), y(t) of the state vector

Z(t) and the output Y(t) satisfying the differential equations (which are the realizations of Eqs. (1.4.1.): z = f(z,a,t), y = g(z,t). Thus we come to the necessity to study the differential equations with random functions in the right-hand sides. In practical problems the randomness of the right-hand sides of the differential equations arises usually from the fact that they represent known functions some of whose arguments are considered as random variables or as random functions of time t and perhaps of the state and the output of the system. But in the latter case these functions are usually replaced by the random functions of time which are only obtained by assuming that their arguments Z and Y are known functions of time corresponding to the nominal regime of system functioning. In practical problems such an assumption usually provides sufficient accuracy. Thus we may restrict ourselves to the case where all uncertain variables in the right-hand sides of differential equations may be considered

as random functions of time. Then Eqs. (1.4.1) may be written in the form

Z = f(Z,x,Ni(t),t), Y = g(Z, No(t),t),

(1.4.2)

where f and g are known functions whose arguments include random functions of time N;(t) and N2(t). The initial state vector of the system Zo in practical problems is always a random variable independent of the

random functions N;(t) and N2(t) (independent of random disturbances acting on the system). Every realization [m1 (t)? no(t)? ie of the random function [Nj (t)? N2(t)"]” determines the corresponding realizations f(z, «, ni(t),t), g(z, no(t), t) of the functions f(z, x, Ni(t),t), g(z, No(t),t), and in accordance with this Eqs. (1.4.2) determine respective realizations z(t) and y(t) of the state vector of the system Z(t) and its output Y(t).

~~

1.4. Stochastic Differential Systems

57

Example 1.4.1. The equations of motion in Examples 1.1.4 and 1.1.5 describe the stochastic models of motion of various mechanical systems if the forces and the moments

in the right-hand sides of these equations represent random functions

of time.

1.4.2. Stochastic Linear Differential Systems The differential equations of a linear stochastic system differ from Eqs. (1.2.1) of a linear deterministic system by additional random items: Z=aZ+ayxr+ag+

ay Ni(t),

Y =bZ+b).+

bi No(t) ,

(1.4.3)

In these equations N;(t) and No(t) are random functions of time (in general they are vector random functions). Introducing the composite vector

random function W(t) = [ Ni(t)? No(t)? |” and block-matrices a4 = [a20], bj = [06,] where 0 is the matrix whose elements are all equal to zero we represent the

random items of Eqs.(1.4.3) in the form a2Ni(t) = a,N(t), 6: No(t) = b},.N(t). Therefore without loss of generality we may omit the indices at random functions and write Eqs. (1.4.3) in the form Z=aZ+ajxr+anp+a2,N(t),

Y=bZ+b0+b:N(t).

(1.4.4)

In practical problems the deviations of a nonlinear system from the required regime may sometimes be assumed as sufficiently small. In such a case the equations of the system may often be linearized relative to random deviations from the required regime and relative to random disturbances acting on the system. In such cases the nonlinear equations describing the behaviour of the system are replaced by approximate linear equations in the deviations. It gives the opportunity to study the nominal regime of system functioning using the deterministic nonlinear

model (1.1.4) and then to study random deviations from the nominal regime using the much simpler linear stochastic models (1.4.4). 1.4.3. Linear Systems with Parametric Notses In practical problems we have to deal with linear systems in which the noises depend linearly on the state vector of the system. In such cases we have to use for the description of the behaviour of a system linear differential equations with fluctuating coefficients. The fluctuations of the coefficients of the differential equations of a linear system are usually

58

Chapter 1 Dynamical

Systems and Their Characteristics

called parametric noises. For such systems the matrices az and 6, in Eqs. (1.4.4) depend not only on time, but are also linear functions of the state

vector of a system. Thus Eqs. (1.4.4) in the case of parametric noises are replaced by the equations P Z=aZ + ajz + ao + (000+ Sanz] N(t),

ia Y =bZ+bo+ (i+ SZ) N(t).

(1.4.5)

P

(p=)

Example

1.4.2.

The motion of a pendulum with vibrating suspension

point in viscous medium is described by the equation

Ag+ Bg + mgl(1 + No)sing + mglN, cosy = 0,

(1)

where ¢ is the deviation angle of a pendulum from the vertical, A is the centroidal

moment

of inertia, B is the coefficient of moment

is the static moment, gN; vertical components

of forces of viscous friction, mgl

= gN, (t) and gN2 = gN2(t) are the horizontal and

of the acceleration vector of the suspension point representing

random functions of time.

At small angles Y putting in (I) sin p & », cos y & 1 and assuming as state variables

p =

Z}, yg =

Zy we obtain a linear stochastic differential system with

additive and parametric noises whose equation has the form

lela alla; ofl S]4) Lm(1) 2, 2

|..0,

ot

Z

2a

ia

0

0

N,

where we = mgl/A, 26 = B/A.

1.4.4. Stochastic Bilinear Differential Systems

The systems in which the vectors Z, Y, Ni, N2 in Eqs. allowing the decomposition of the form ae

|zz

vu

a a

lev tee

aT?

aN

(1.4.2)

[Ni2"0| ,

are called so, with the components Z’, Y' and Z”, Y” satisfying the linear equation of the form Eq. (1.4.3) and the bilinear equations, respectively:

~

1.5. Stochastic Discrete Systems

Z =aZ' +ayx+ao+a.N{, :

Z" =|

P

Act) AZ|

59

Y'=b2Z' + bo +b,N5, P

Z", Y" =|

Bot >> BZ, |2”.

jvesl

(1.4.6)

ivemt!

Here a, aj, a2, Ap, Ax, bo, b1, Bo, By are the matrices, ao, bo are the vectors of the suitable dimension. The bilinear stochastic systems described by Eqs. (1.4.6) are an important special case of the stochastic systems with random parameters,

Z = f(Z,2, Ni(t),U,tY, Y = 9(Z, No(t), U,2),

(1.4.7)

U being a vector of random parameters. Example

1.4.3. The equations of a body motion around a fixed axis in

viscous media have the form

d*o _

Ady

(1)

di? bral ate Here I is an axial moment

of inertia, ) is an angular velocity, A /T is specific coeffi-

cient of viscous friction which is a random time function satisfying to linear differential equation

A=-a\+hN, where @ and hare some positive coefficients,

A=A/T,

(1)

NV= N (t) is a random time function.

Denoting Z; = ~, Z2 = A we reduce the Eqs.

(I) and (II) to the following two-

dimensional bilinear stochastic differential equations:

EVAL

IN

(II)

1.5. Stochastic Discrete Systems 1.8.1. Discrete Systems

It is known that for a discrete insertion of the inputs into a system it is necessary to modulate some parameters of the input impulses in accordance with their values. A device forming the sequence of the impulses which depends on an input is called an impulse element. Usually the impulses form remains invariable at the modulation. We may distinguish three main types of the impulses modulation:

60

Chapter 1 Dynamical Systems and Their Characteristics

e pulse-amplitude modulation (PAM) at which the amplitude of the impulses a; depends linearly (with the gain pam) on the input value at the moment of the impulse action t;, a, = az (tx) = Epama(te); e pulse-width modulation (PWM) at which the duration of the impulse 7; linearly depends (with the gain pwm) on the input value at the moment of the impulse action, T;(t,) = €pwm2(tx);

e time-impulse modulation (TIM) at which the time shift 7 of an impulse linearly depends (with the gain tm) on the input value at a definite time moment, Ts = €yimz(tx). At PAM and PWM the modulating signal changes the area (i.e. the intensity) of the impulses, and at TIM the area of the impulse remains constant. The dependence of modulating parameter of the impulses produced by an impulse element on the correspondent discrete input values is called a characteristic of an impulse element. The latter may be linear or nonlinear. The impulse element with linear characteristic is linear, and with a nonlinear characteristic is nonlinear. The impulse elements vary also in the form and the character of the impulses modulation, in impulses frequency and their relative duration. Usually the impulse elements are periodically generate one impulse at each period. The period of the impulses succession TJ; is called a period of

repetition (a tact) of a discrete system. resents a repetition rate. called a relative duration

The variable w, = 22/T, rep-

The parameters y, y = T;/T, and 1 — y are and impulses ratio.

1.5.2. Characteristics of Discrete Linear Systems Let us denote g,(t) the response of a linear discrete system on a short-time input signal equal to 1 and acting only during the k'® impulse action. Then its response on a short-time signal equal to x(t;) and which acts only during the time of the k*® impulse action will be equal to g(t )a(t~) on the basis of the superposition principle. The response of a linear discrete system on the whole sequence of the impulses modulated by the input disturbance «(t;) on the basis of the superposition principle

(Subsection 1.1.2) will be determined by the formula co

y(t)= S > ge(t)e(te)-

(1.5.1)

k=-00

The functions g;,(t) characterize completely a linear discrete system as knowing these functions we may calculate the response of a discrete

1.5. Stochastic Discrete Systems

61

linear system on any input disturbance x(t). The functions g,(t) determine the part of the input values acting at different time moments t, in the formation of the output at any time moment t. That is why the functions g;(t) are called the wetghting coefficients of a linear discrete system.

Any nonanticipative (physically realizable) discrete system (1.5.1) may response at a given moment t only on the signal acting on it at the previous instant. Therefore for any nonanticipative discrete system

ge(t) =0 at t> ge(ti)a(te). k=—0o

(1.5.10)

64

Chapter 1 Dynamical

Systems and Their Characteristics

For brevity setting 2, = x(tz), yr = (ty), ge(t;) = gie we can rewrite formula (1.5.10) finally in the form

Teng

i OME

ERPS gS

(1.5.11)

k=-00

In the case when the instants of an output fixing coincide with the in-

stants of the impulses action, t; = t; (J = 0,+1,+2,...), formula (1.5.11) at t = t; will give the following expression for the values of an output of a nonanticipative linear discrete system: if

yt =p einGinte fle

a 2 ath he

(1.5.12)

Roa)

Formula (1.5.12) and the analogous formulae for the discrete systems with many inputs and outputs represent, in particular, a digital computer as a discrete linear ystem because the results of the calculations depend linearly on the initial data. In this case the initial data introduced at each step of the calculations are the input signals and the results of the calculations are the outputs signals. The weighting coefficients gj), determine the program of the calculations. At the sufficiently large impulses repetition rate any discrete system may be considered approximately as a continuous one. Really, the response of any system on a short-term unit signal acting during one period of the impulses repetition TJ, at infinitesimal 7}. is an infinitesimal of the order T;,. Therefore the function g,(t) at small T, = At may be expressed by formula

ge(t) © g(t, te)T> = g(t, ty) At.

(1.5.13)

Here g(t,t,) is some function independent of T,. So, for instance, in the case of a series connection of an impulse element and a continuous linear system with the weighting function g;(t,7) we can express the dependence of the impulses of a given form on the period of the repetition

of the impulses assuming the function n(t) in the form n(t) = A(t/T;), where h(€) is a function different from zero only at the interval 0 < €

[v1 yo)", =

[0, Q |

is the input,

0) B, C are constant coefficients deter-

mining inertial, dissipative and position forces respectively. Show that the matrix of

fundamental solutions u(t, wf) and the matrix of transfer functions ®(s) have the form

u(t,7) = e~*-7) cos W_(t — T) + ns sinw,(t — T) x

oo sinw,(t — 7)

2

4

se sinw,(t — T)

&(s) =

cosw,(t — T) — ee sinw,(t — T)

(II)

1 ee

As’ + Bs+C | (9¢ = B/A,w? = w? — €?,w? = C/A).

As°+Bs+C

(111) 1.6. Show that for a linear nonstationary system

i =y2,

Yo=—cet

"yi+2,

t>0,

the elements of the matrix of fundamental solutions u(t, T) ab 2S

(I) Ky |1 = Ac|,

Cal /4 are determined by the formulae:

uii(t,7) = z ds

ja+ 27) (=) — (1-27) (=) |;

watt = Spe [Clon | rt)i/2

Wai(b; Fl ale(< 2 () |,

usatts)= @ (5) a

)-a-2(9)'].

a

~

Problems

91

Find u(t, 7) at c > 1/4. 1.7. Prove that for a nonstationary linear system

Y1 =

UPS

Y=

—t-2y,

—ttyta,

t>

0,

(I)

the matrix of fundamental solutions u(t, T) is equal to

alter)

cos In=t

—711sin In =

: 7 sin Int=

(I)

¢os In at

-z

1.8. Prove that the weighting function g(t, T) of a nonstationary linear system

ytt'yt+(l—n7t-*)y=a2,

y=z,

(1)

is determined by:

g(t,7) = 5[In(7)Na(t) — Nn(t)In(t)]7

(11)

where Jn, (7) and N, (t) are the Bessel functions of the first and the second kind respectively. 1.9. For a stationary linear system ¢

Peaseor, y= wile = 242983

a

hl, B= |2296s

fu

|G

(I)

where a is a constant square matrix, b a scalar constant, y the output, show that the elements of the matrix of transfer fucntions (s) =

=

—(a => sI))~'b at A(s)

la = sl| are determined by the formulae

®11(s) = —b |s? — s(azq + a33) + 422433 — d23432| /A(s) ,

®12(s) = b(—say2 + a12433 — a32013)/A(s) , $13(s) = —b(sa13 + a12423 — 413422)/A(s), o1(s) = 6(—sag1 + 421433 — 431423)/A(s) , ®o0(s) = —b[ s* — s(a11 + a33) + 411433 — 413431|/A(s),

o3(s) = b(—saz3 + 411423 — 421413)/A(s), $31(s) = —b(sag1 + a21432 — 431422)/A(s) , 30(s) = b(—sag2 + 411432 — 412431)/A(s) ,

@33(s) = —b [s? — s(ay1 + 22) + a11422 — ai2a21 |/A(s).

(II)

92

Chapter 1 Dynamical Systems and Their Characteristics

1.10. Show that for the stationary system of Problem 1.9 at 4d}; = @31 = 413

="a39 = 0) 419 = Go3 = lye

—w?, a92 = —2€, 433 = —Q

the elements of

the matrix of transfer functions are determined by

$11(s) =—b[s* + (26 + a)s + 2ae] /A(s),

12(s) = —b(s + a)/A(s), P13(s) = —b/A(s), o1(s) = bwo(s + a)/A(s), O22(s) = —bs(s +a)/A(s), Do3(s )= —bs/A(s), 3) = O35 => 0, $33(s) ) = —b(s? + 2es +.w2)/A(s), A(s) = —(s +a) [we +

(2e + s)s] ,

(1)

Check that the elements of the matrix of fundamental solutions u(t, tp) are given by uia(t, T) =

(- 1/w? ) [uri(t, T) + a/wouar(t, T) a

31 = U32 = 0, u3a(t, T) =

uza(t, ®) iy

exp {—a(t a T)} )

ug3(t, 7) = (—1/w?) {uai(t,7)(1 —

ea /w?) — w[usi(t, 7) — usa(t, T)]} ;

(II) the functions U;1, U12, U21, U22 being determined in Problem 1.5. 1.11. Show that for the stationary linear mechanical system with n degrees of

freedom the differential equations of motion have the following form:

(a) In Lagrangian variables

Ag + (B+ B’)g-+ (C+ C)g=Q;, where

¢ =

og On

[q edn

e is the vector of generalized coordinates,

is the vector of generalized forces. In this case the vector

the output, and the vector

(I) and Q =

[Qi

y = q represents

£ = Q represents the input, A is the symmetric matrix of

the inertia coefficients, B is the symmetric matrix of the dissipative or acceleration forces, B’ is the antisymmetric matrix of gyro forces, C’ is the symmetric matrix of the conservative position forces, and C’ ' is the antisymmetric matrix of nonconservative position forces. (b) In canonical variables

sla feet beh eT ledealaane®

where g and p =

Ag are the vector of generalized coordinates and the vector of

generalized impulses respectively.

In this case [q? pt]? represents the state vector

of the system, the input and output being the same as in the case of the Lagrangian variables.

Problems

93

1.12. Show that for the stable stationary system of Problem 1.11 at n = 2

a Ai2

Ai2 es i A22 | | Qe a

Bi, Biz

Bi Boe

4

0 —Byan»

Ci1 Cie | OC 12 (|Cho, C22 dare i2 O

Bip 0

q1 q2

=

qi q2 Q1 Qe |

(1)

the elements of the matrix of transfer functions ®(s) are determined by the formulae

11(s) = {Aj ({C| + C42) — s[Cii(C22 +5)

— (Biz + By2)(C12 — Ch) /A(s), 12(s) = -{Aj,(|C] + C'ts) — s[C11(Biz — Bip) — (Bi + s)(Ci2 — Cia) }/A(s),

o1(s) =—{-Aj,(|C] + C42) + s[(Ci2 + C{.)(Baa + 8)

(II)

— C2o(b12 + Biy)|}/A(s), o0(s) ={AZ(|C] + Cq2) + s[(C12 — Clp)( Biz — Bip) — C2(Bi1 + s)]}/A(s) where A(s) = s*4 +1159 + 1728? + 173s +14,

ry = Byy Az, + BorAj, — 2B12 Aj, ,

ro = [BiAg — (Bia + Biz) Ate] |Bo2Ary — (B12 — Bip) AyD] + (C1149 + C22Ay, — 2C12Ajp) , r3 = [By Aj — (Bio + Biz) Aja] [(Bi2 — Biz) Ad, — B22 AQ | a [ BooAy

(Biz wy By») At | [Cir Ax x (Ci2 im Cia) Ain

“+ [(Biz - Bi) Are — B22 Ajp| [CiAi

a (Cie te Cya)An

+ {[ Bi Az, + (Bio + Bia) Ary] + [(Bia + Bi) AT — Br Ax] } x [C2245 — (C12 — Ci2) AQ] , ra = (IC] + Ale Aj; =

Agen

|A|

(4,9 =

|We 3

|A|

=

Aj1A22—A?,

IC | =

(111)

CiiCo9—C-,

1.13. Show that if the matrices A~! B and A7~ 1C in Problem 1.11 are commute and B’ = 0, C’ = 0, then there exists always a linear transformation of the output

y = L¢ satisfying the conditions

TWA fe dele Blas diag (266s 5... 2En),

TE OP

diag {uen rasu2 Js,

I

94

Chapter 1 Dynamical

Systems and Their Characteristics

reducing the set of equations of motion to n independent linear differential secondorder equations

Cy + 2enCn

where

Up

=

up

,

(h7= 1)

ane

un(t) is the A*® element of the column-matrix

corresponding equations at 1.14.

+2 Cn =

B = Meys\. ei

Consider a mechanical

(II)

EO}

Write the

nC :

system with n degrees of freedom

described by

Lagrange equations

d

(OL

OL

a (i) — 3 = 2 where q =

-

[q1 soli ‘ieis the vector of generalized coordinates,

L = T'— II is the

Lagrange function, 7’ = T(q, q, t) is the kinetic energy, I] = II(q) is the potential

energy; Q =

[Qi MGS

servative forces.

ie Q.=

Show that Eq.

Q(4; q; t) is the vector of generalized noncon-

(I) may be transformed into the set of Hamilton

equations in canonical variables gq and

G==-, Here

p = OT /Oq

p=->-+Q.

H = H(q, p, t) being the Hamilton function,

(II)

H = pq -T.

1.15. The vertical oscillations of the body of a car while moving on the rough road are described by the equations

My + 20(y — ¢) + 2v(y- 9) =0, where M/ is the car mass,

w(y = q) and ply = q) are the nonlinear functions

determining damping and restoring forces respectively,

g =

q(c) is the function

characterizing the microprofile of the road in the vertical plane,

7 =

vt, VU being

the car velocity. We assume that both pairs of wheels of the car undergo the same vertical displacement at every instant due to the roughness of the road (i.e. the road

profile in the vertical plane has a very long correlation interval).

Find the transfer

function of the car at equalities w(y = q) = b(y = q) and ply = q) = c(y = q); b and c being constant coefficients, assuming q as an input and Y as an output. 1.16.The equations of motion of the body with one degree of freedom equipped

with the nonlinear dynamic damper have the form

pa + ba + cx + pa(@ — Za) + vale — vq) = F(t),

Hata — pa(@ — £a) — Ya(x@ — 2a) = 0,

“wn

Problems

95

where & and £q are the displacements of the body and of the damper, pi and [lq their

and C the coefficients of viscous friction and of restoring forces respectively,

masses,

wa(x = La) and a(x = La) the functions characterizing the forces of viscous friction and the restoring forces, and F(t) the disturbing force. —%q=

a(x

ba(« _

La) and pa(r = La) =

In the special case where

cae. — La), with bg, cq being

constant coefficients find the transfer function of the system considering the force F(t) as an input, and the displacements © and &q of the body and damper as the components of the output.

1.17. Show that the nonlinear equation n—-1

Oe

Nea

See yy

fee, et)

k=n-—m

m

=D da(ysy’, 0. yO", 1), (m 0 that P(Q) = 1.

If a nonnegative finite or o-finite measure v is determined in the measurable space (Q,S) then the probability P = P(A) may be presented in the form of the sum of the v-continuous and v-singular measures, and the v-continuous part represents an integral over the measure v with

respect to some function f(w)

P(A) = | tev@s) + P(A).

(2.1.1)

A When the probability P(A) is v-continuous, P;(A) = 0 and

P(A) = :;Jobin

(2.1.2)

A

As it is known from the functional analysis in this case the function f(w) represents Radon-Nikodym derivative of the probability P with respect to the measure vy and is called a probability density with respect

to measure v in the space (2S, P). Any subset N of the set No € S for which P(No) = 0 is called a zero

set.

If zero set N belongs to the o-algebra S of the probability space

(Q,S, P) then, evidently P(N) = 0. But zero set does not need belonging to S (if No C S then the subsets of the set No do not necessarily belong

to S). On such zero sets the probability P(N) is not determined. The probability space (Q,S, P) is called a complete one if the o-algebra S contains all zero sets of this space. If the probability space (Q,S, P) is not complete then it may be always supplemented after adding in S all zero sets and all the unions of zero sets with the sets from S and determining the probability P on the supplemented sets.

102

Chapter 2 Random

2.2. Random

Variables

Variables, Processes and Functions

2.2.1. Definitions

In elementary probability theory a random variable is a variable which assumes as a result of a trial one of possible values and it is im-

possible (before a trial) to know what value it will assume namely. Here it is supposed that in the space of possible values of each random variable there exists a probability distribution. In order to come to an exact definition of a random variable notion we shall refer to the examples from elementary probability theory and random functions theory. Example

2.2.1. The number of the occurrences of the event A (in one

trial) in Examples 2.1.1 and 2.1.8 represents a random variable which assumes the value 1 when A appears and the value () when A does not appear. It is easy to notice that this random variable represents a function of a point in the probability space

(Q, S:, iP) of Example 2.1.8 equal to 1 at w = W, and 0 at WwW = Wo. Example

2.2.2.

nm triales in Bernoulli 0,1,...,m.

scheme

This random

The

number

represents

of the occurrences

a random

variable

of the event

A at

with possible values

variable is a point function in the probability

space

(Q, Ss) P) of Example 2.1.9 equal to.m at all points W to which correspond the se-

quences

A,,...,

=O

lees8):

Example sidered

An

containing

letters

A

and

n

—

m

letters

A

(m

2.2.3. The finite-dimensional random vector X may be con-

as the point function

Example 2.1.10.

‘m

r(w)

=

wW in the probability

space

(Q, On P) of

Each component X, of the random vector X represents also the

point function r,(w) in the probability space (Q, S, Vane Example

2.2.4.

Infinite-dimensional random

vector X

(the sequence

{xX x} of the scalar random variables) may be also considered as the point function z(w) = w in the space QQ of all numerical sequences

{xz}. It is evident that any

component of an infinite-dimensional random vector is the function of a point in the space 2 (Lm at any fixed M

is the function of the sequence

{7x }): The same is

true, in particular, for the sequence of the events in Example 2.1.6.

The considered examples turn us to the following definition of a random variable. A random variable with values in a measurable space

(X,A) is called (S,A)-measurable function of a point in the probability space (Q,S, P) mapping Q into X and determined everywhere in Q besides may be the set of zero probability P. The value of a random variable at any given point w of the space (2 (i.e. the value which it assumes when as a result of a trial the elementary event w appears) is called a realization of this random variable.

2.2. Random Variables

103

It is evident that any random variable in (X, A) is at the same time a random variable in (X,.A’) where A’ is any o-algebra in X containing APA’ eA: We shall often denote a random variable by the same letter as the space of its values. While considering the set of the random variables with the values in one and the same space the denotations of these variables will be supplemented with the indexes. So, for instance, X, Xq represent the random variables with the values in the space X. If X, Xq are the random variables in (X,.A) then according to the definition X

= z(w), Xq = Lq(w) where x(w), w(w) are (S,A)-measurable functions mapping 2 into X. a 2.2.2. Probability Measure

By virtue of (A, S)-measurability of a random variable

X = 2(w)

in (X, A) the probability of the fact that it will take the value from the set A is determined by the following formula:

P(X 6A),= Plar(A))sucAce A, where as usual z~1(A) is an inverse set A in the space 2. Formula

f,(A) = P(X € A) = P(27"(A)),

AEA,

(2.2.1)

determines the probability uz, induced by the random variable X = z(w) in the space of its values. The triple (X, A, zz) represent the probability

space induced into (X, A) by the random variable X = a(w). The space X of the values of the random variable X = a(w) is called its phase space or the space of the realizations. The probability pz in the phase space is called the probability measure or the distrubution of the random variable X. The probability space (X,A, uz) is called the phase probability space of the random variable X = x(w). It is evident that any probability space may be considered as a phase space of some

random variable. If the random variable

X = «(w) is considered in its

phase probability set then the argument w in its denotion naturally is not indicated. In such cases we shall denote it simply by X and its realizations by z. If the nonnegative finite or o-finite measure v is determined in the

space (X,.A) then the probability measure pz of the random variable X may be presented by formula

entAye= /f(a)v(dz)+ps(A), A

AEA,

(2.2.2)

104

Chapter

2 Random

Variables, Processes and Functions

where (A) is the v-singular measure. In the special case when the measure jl, is v-continuous, 5(A) = 0, we have

Ay= [semwees

“Aten

(2.2.3)

Here f(x) is Radon—Nikodym derivative of the probability measure pz with respect to the measure vy which is called a probability density of the random variable X with respect. to the measure v. Remark.

Inelementary probability theory the probability Aensity of

variable is always considered in a finite-dimensional phase space (Example

arandom 2.1.10).

In problems which are connected with one random variable it is always convenient to assume its phase probability space as the probability space.

But in the problems

connected with the set of the random variables (possibly with different phase spaces) it is expedient to study all of them in one probability space. Example

2.2.5.

The two-points set {0, Hi serves as the phase space

of the random variable in Example 2.2.1, and the probability measure is completely

determined by its value in one of these two points: a0

Hae({1}) = 0), Hr ({0}) =),

lle

Example

2.2.6.

The sett {0,are

as n} serves as the phase space of

the number of the occurrences of an event in 7 trials in Example 2.2.2, and the probability measure [/, is completely determined by its values in these points: Ha ({m})

SOR

pt ote

T(t = nl) ml i — a)

Example

2.2.7.

In Examples

sO:

cehatt

2.2.3 and 2.2.4 the space of elementary

events () itself serves as the phase space of the complex (correspondingly real) random

vector X, and the complex plane (correspondingly real axis) serves as the common phase space of its components.

2.2.3. Equivalence of Random

Variables

Let X = z(w) be a random variable with its phase space X. The inverse mapping z~'(A) of the o-algebra A represents the o-algebra which is a part of the o-algebra S in the space of elementary events

, «1(A) Cc S. In the general case 2~!(A) does not coincide with S. Thus the variable X = x(w) induces in the space 2 some o-algebra

S; = a '(A), as a rule more poor than the o-algebra S. Therefore the random variable mined

on the same

Y = y(w) in the measurable space (Y, B) deterprobability space

(Q,S, P)will be not obligatory

(Sz, B)-measurable (from the fact that y~!(B) € S does not follow that y~'(B) € Sz). So if the random variable Y in (Y, B) whose phase space

a“

2.2. Random

Variables

105

Y being a separable B-space is (S,,B)-measurable then it is (A, B)measurable function of the random variable X. The random variables which are equal almost surely, i.e. with the probability 1, are called equivalent. Thus two statements are valid.

Theorem 2.2.1. If the random variable Y = y(w) in (Y,B) whose phase space Y being a separable B-space is (S,,B)-measurable then it ts equivalent to some (A, B)-measurable function of the random variable

aX eel Ga) Theorem 2.2.2. Jf (A, B)-measurable function valent to such a function) there are none restrictions

the random variable Y = y(w) in (Y,B) ts of the random variable X = x(w) (or ts equithen it is (S,,B)-measurable. In this case on the phase space of the random variable Y.

It is evident that all equivalent random variables have one and the same probability measure.

If the random variable Y = y(w) in (Y,B) is (A, B)-measurable function of the random variable X = x(w), Y = y(X) then it is arandom variable in the probability space (X,A,p,).

Therefore the probability

measure fly of this random variable may be expressed in terms of the probability measure pz of the random variable X by formula (2.2.1)

Hy(B) =ps(p*(B)),

BeB.

(2.2.4)

The same formula determines a probability measure of any random va-

riable Y equivalent to (A, B)-measurable function p(X) of the random variable X. 2.2.4. Two Types of Convergence of Random Sequences The sequence of the random variables {X;}, X; = %,(w) with the values in the topological space is called a.s. convergent (almost surely, with probability 1) to the random variable X = z(w) if the sequence of

the functions {xz,(w)} converges to «(w) almost everywhere on {2 relatively to the measure P. The sequence of the random variables {X;}, X, = xx(w) with the values in the separable B-space X is called convergent in probability to the random variable X = x(w) if at any € > 0

lim P(\| Xe — X [I> €) = Jim P({w : |]24(w) — (w) ||> €}) = 0. The following statements are valid.

106

Chapter 2 Random

Variables, Processes and Functions

Theorem 2.2.3. For the a.s. (respectively in probability) convergence of the sequence of the random variables {X;,} with the values in the separable B-space X it is necessary and sufficient that this sequence will be fundamental a.s. (respectively in probability). Theorem 2.2.4. Any a.s. convergent sequence of the random variables {X;,} with the values in the separable B-space ts also convergent in probability. The inverse is in general case not true. But from any sequence of random variables convergent in probability we may separate such sequence which a.s. converges. Example

2.2.8.

Let Xn be the random variable with two possible va-

lues 0 and n whose probabilities are equal to 1 — n-? and n~? respectively.

The

sequence of the random variables {Xn} corresponding ton = 1,2, ... , converges in probability to 0 as at any € > 0

P(|Xnl

1

26) = P(Xn =n) ==Z70

at

noo.

n

This sequence also a.s. converges as at any €,0 < € €

ifevenatone

n>p)
p)

PUXal

Let (Xe A) be an arbitrary measurable space, r(w) bea

(ST, A) measu-

rable function in infinite product of the spaces Q? with the values in X. Suppose that

there exists a countable class of the sets { Fi,} generating the o-algebra A. io

PR

Then

will be a countable class of the sets in Q7 generating the o-algebra

te (A) & S™ induced in QT by the function z(w). There exists such sequence

of the countable subsets {Ly} of the set 7’, that zo We put L = ULx. Then we shall have a—1(F,)

such countable set

L C T' that the o-algebra S

(k == pe Pe a As Zor foe (A) e So

(F;.) e SU

(k = lade. ne

E S” at all k. Thus there exists

contains all the sets a= 1( Pi)

A) is the minimal 0-algebra containing all ae

Oy) then

Consequently, the function r(w) is (S™ A)-measurable. Thus any

measurable function z(w) in Q7 with the values in the measurable space (X, A) in which the o-algebra A is generated by the countable class of the sets is (S¥ . A). measurable at some countable

D C ‘Ts i.e. depends on no more than a countable set

2.5. Probabilities in Infinite Product of Spaces

TE

of the coordinates of the point W in OF eat particular, any measurable functional in ee depends on no more than a countable set of the coordinates of the point W in

Ors Now

we pass to the construction of the probability in OD Suppose that at (Q¢, Sz) the probability P,( Az) and the regular conditional probabilities Paster els (At lw, fonds wi, ) correspondent to all hy, meal ad. (n each space

Sly dives .) are given. ‘Then applying formula (2.4.18) sequentially we find the probabilities in all possible finite products of the spaces {);:

Patt, Peso

As) = ijPtg,t, (At, lw) Pr, (der) ,

th lad XtvenXaAen)

=

Nae

At, XX At, x Ps...

oe

ane

(2.5.1) eer

Ee

he

ER

| Wr—1)

1

tn-1

(dw, ».G VOR

iegee ent

DS dwn-1)

a Space fice oeeng

After determining the probability 1 ,tz on the measurable rectangles Os, x oF by

the first formula (2.5.1) at first we shall extend it on the o-algebra St, Xx Sia Then using the second formula on the measurable

(2.5.1) at

m =

3 we detemine the probability Psy ,ta,t3

rectangles Qs, x Q:, x Qt, and extend it on the 0-algebra

Si, x St, x S; ,- Continuing this process we determine consequently the probabilities Pay peeytn atn=

4,5,....

It is easy to understand that the conditional prob-

abilities F eeks ae, td (Azur, tnd ies) (t, Ls

Gh

ote

i Seatise :) should be

consistent in such a way that they will give one and the same probability tae eeeat ss in the product of the spaces OQ, DK ores XK ey

(n Be 2, ose: Ps in any ways.

Then

the obtained set of the probabilities in all finite products of the spaces (Q4, S:) will satisfy the conditions of the consistence and the symmetry.

Let us assign now the function of the set P on the set of all measurable rectangles of the infinite product of the spaces Qr by formula Pe

where

Ae

x

Pax

Ay),

(2.5.2)

Ft is the rectangle with the sides At, seuss. 5 At, which le in the spaces

MtHe cme

Wem correspondingly.

set of the rectangles may

This function is additive as the bases of any finite

be considered

as belonging to one and the same

product of the spaces, for instance Sire X:++:& is additive in (Q4, XX

finite

G2 me and the probability jb sete

Ue ; St, X-22X St, MesFurthermore if we restrict ourselves

128

Chapter 2 Random

Variables, Processes and Functions

to measurable rectangles whose all bases belong to one and the same finite product of the spaces Q:, KISLEX

Q:, then the function P(R) is O-additive by virtue of

the @-additivity of the probability lady ,.:,tn+

Consequently, the function P(R) is

the function on the set of all measurable rectangles of the space (a>, Sf) which have the basis in one and the same finite product of the spaces Q:, Xoo

X GPAs

According to the theorem from functional analysis about the measure extension it may

be uniquely extended over the d-algebra St, x

cylinders of the space Q? with the bases in OF > ea

aX

St, of all measurable

Chey.4 A

It is evident that the

extension is determined by formula

PCR

re

Sa

eee

Bye)

(2.5.3)

where Cr, ,.1, tn is the cylinder in Q7 with the measurable base Bi, Oo. eke (Q4,

KPO

Seen SE):

Let us consider now the problem of the possibility of the function P (R) exten-

sion determined on the class of the measurable rectangles of the space (QT ; Sy by formula (2.5.2) on the whole o-algebra SE. The class C of all measurable rectangles is a semi-algebra and satisfies the conditions of the known from functional analysis theorem about the measure extension. In order that it may be possible to use the theorem about the measure extension for determining the probability on the whole o-algebra ST it remains only to prove

the -additivity of the function of the set P on the class of all measurable rectangles

which is determined by formula (2.5.2).

For this purpose it is sufficient to show

that P is continuous in zero, i.e. that for any decreasing sequence of the rectangles eae gsRp a) Ry at p < q with the empty intersection lim PCr

= 0. Hence it

is sufficient to consider the case when the set of the spaces {Qa} in which lie the sides of the rectangles Ry, is infinite (the case of a finite set of the spaces {Qz, } we have already considered).

We denote by AM*) the side of the rectangle Ry, in the

space Oe and put BS) = AbE+) KATE

Alt) where 7, is the number of the

sides of the rectangle R,,. Applying formula (2.4.18) we may present formula (2.5.2) at

R=

R, in the form

P(Rn) = / P(BO\or)Pa (der) = f1400 es)P(Blen) Pa, (der), AY) where

1

P (Bo Nw)

is

Saye o Se Dy au St, y I
0 that Pia)

> € atalln.

As

Pe thts) = | then it follows that there exists such point @] in Q;, that f(@1) > €. It is evident that ©, E€ AW at all n as f(#1) = 0 if wy does not belong to AW

beginning with some

{P(BS

mn.

But the sequence of the conditional probabilities

jz) is nonincreasing and lim P(BY

—

a, =

f(@1).

Therefore from

ds) ==

f(@1) > € follows P(Bs No) > € at all n. Further on the basis of (2.4.20) we have

P(BY ay) = i P(BY) |, w2) Pes ,19(dw2|@1) ,

(2.5.4)

A?) where

She

P(BY

ay, we) is a conditional probability in (Q45 X-++X

St, ). From P(BY?

Here Wy E Awe

St, :

a) > € follows the existence of such point W2 in Qy,

that P(By? lo, We) > €atalln andW2 we shall get the sequence

Qe,

€ Al?) at all n. Continuing this process

of the points {@;, } in the space Q4, (k aan

Np

he 4 3)

Consequently, all the points W of the infinite product of the space

QT? which have the coordinates Carn

ays Wp in the spaces ay Pee

to all rectangles Rt, whose sides lie in the spaces Oe Rota

Qe, belong

Q4,. Hence it follows

that the points of the space Q? which have the coordinates W. k in all the spaces

8

(k ct

.) belong to all the rectangles R,,, i.e. the intersection of all the

rectangles R,, is not empty. So at lim PCRs) of 0 the intersection of all the rectangles R,, cannot be emply. Consequently,

lim P (Rn)

=

0 if NR,

=

@

what proves the continuity of the

function P in zero and at the same time its 0-additivity on the set of all measurable rectangles of the space Q?.

Thus formula (2.5.2) determines the probability in the

space QT on the set of all measurable rectangles C. Hence there exists an unique extension of the probability P determined by formula (2.5.2) on the 0-algebra st

generated by the set C of all measurable rectangles. It follows from formula (2.5.2) that the probability es ,.:,tn projection of the probability P on the subspace (Q:, X-°:X

represents the

eae Phas ae St, )

of the space (OF. Siu. Hence it follows that the probability Py, in the subspace (Stee ars) of the space vine So}, LC of the probability P on (QO% Soh

T determined in such a way is the projection

130

Chapter

2 Random

Variables, Processes and Functions

The requirement of the regularity of all conditional probabilities in each space

is essential as formula (2.5.4) and all analogous formulae which are necessary for unbounded extension of the process in given proof of the continuity of P in zero have the sense only if all conditional probabilities (Qe, S:) are regular. Now let us consider the random vector X = x(w) with the countable set of the components X;

=

Lp (w) (k i

Me

a oD Suppose that on the 0-algebra of

Borel sets A of the phase space of this random variable

X its conditional distribution

pF(Alw)

relative to some O-algebra F C S of the probability space (Q, Se P) is

assigned.

The space (X ; A) may be considered as the countable product of one-

dimensional spaces Ce 5 Be):

x=R?=][

ke,

A=B”

where

Ry

= R

axis (k sl

is the real axis; By

=J] Br, k=1

pel

=

Wap ae Ns In this case we may

B is the o-algebra of Borel sets of this ensure that any countable-dimensional

random variables have conditional distributions. In other words there exist the regular conditional probabilities determined on the correspondent ¢-algebras of Borel sets in any countable-dimensional probability spaces.
Finally let us consider the random surable space

variable X

(Xx; A) and its conditional distribution

=

z(w) in arbitrary meaF(Alw)

relative to some

g-algebra F C S of the sets of the probability space (Q, S, Py; Suppose that there

2.7. Probability Measure of Random

Function

133

exists such finite- or countable-dimensional random variable (Q, A, ft) in the probability space and that the o-algebra A,, = ua? (BN) induced by it coincides with

A. According to (2.2.4) formula

v¢(Blw) determines

=pr(u-'(B)lw),

Be B",

the conditional distribution of the variable

UV =

u(w).

(2.6.4) In conformity

with the proved theorems there exists the regular conditional distribution v'-( Blw) of the variable UV. Naturally the question arises: whether is it possible to determine

the regular conditional distribution Hr (Alw) of the variable

X by formula (2.6.4).

As by supposition any set A € A is thé inverse of some set B € i

then formula

p'p(Alw) =ve(Blw), A=u-1(B),

determines p'-(Alw) for all sets

A € S .

determines Ue (Alw) uniquely not for all

u—!(B)

(2.6.5)

But in the general case this formula

A € S. For the existence of the regular

conditional distributions of the random variable X it is sufficient that such a finite-

or countable-dimensional function u(x) in (X Ai He) be measurable relative to A and the corresponding ¢-algebra of Borel sets B” will exist such that its range A, will be Borel set and the o-algebra A,

=

uw? (BE) induced by it will coincide

with A. As any probability space may be considered as the phase probability space of

some random yariable then for the existence of the regular conditional probabilities

in the probability space (Q, S, i it is sufficient the existence of finite- or countabledimensional random variable having Borel range and inducing the o-algebra S.
p-1 dkyk, Where the coefficients equations

d,...,d,

Ms Kepnasedp ah (eygt)

satisfy the linear

(ease... 1),

k=1

It is easy to check that formulae (2.7.1) give the same normal multidimensional distributions of the random process X;. Consequently, the random process X; is normally distributed and its probability measure is determined by formula (2.7.12). Remark.

These results are easily generalized on the random

processes

with any consistent set of the multi-dimensional probability densities f, (a1 sitar

t;,...,tn)

(or

with

any

consistent

set

of

the

conditional

frl@i thoi, so, GagSiss+ 4, Sn). end my thas care ai(ayes

=

fr(eisti) fr @o5t2|21;th) oA Gran (nite les, cap tn_15

gs

distributions

re thao

158.

a2 1))-

The probability measure of a random process of the set of all measurable cylinders in the functional space (Xxf : Al) is determined in the general case by formula

(2.7.16) By the formula of such a type we may also approximate with any degree of the accu-

racy the value of the probability measure of the random process on any measurable

set of the space (Xe ; ALY.

2.7.7. Streams of Events Let us consider some notions of the theory of the stochastic streams of the events which are important for the stochastic systems of queueing.

A stream

of the homogeneous events may be characterized by the distributions

of the random instants of the events occurrences.

Let us assume JQ as the initial in-

stant and suppose that the probability measure 1’; (A) of the moment 7} of the occurrence of the first event and the conditional probability measures l,, (Alt, sess ini)

of the instants of the occurrences T;, (n = 2, 3, . . .) of all following events are known. All these conditional distributions are regular as they are determined on the d-algebra B of Borel sets of the real axis R (Section 2.4). Considering T;, as a random function

of an integer argument NM we may determine its multi-dimensional distributions by formula

~

2.7. Probability Measure of Random

Pe ates,

ON

“ Un

d(T,

Function

2111, --

143

5 ine)

An-2

x ‘h Bie

fiat

net nies I) [Ti so*

the o-algebra B™

D4

Bx.) we may extend uniquely this measure on

of the space R™.

Let us consider a special case of a homogeneous

stream in conformity to the

recurrent stream of the events. Recall that a stream of the events is called recurrent if all the intervals between adjacent events are the independent random variables. Below (A = T) means that as usual a set of the numbers obtained by the subtraction of T from all the numbers

contained in A.

Then for such a stream of the events

with one and the same distribution lV of all the intervals between adjacent events we

have v1(A) = v(A), Vn (Alty, i. ti 2 40) = Uy(A — boa) (n eee

.) and

formula (2.7.17) for 1/1...» will take the form

Vi nar = [van Ai

CAR|

/ v(dt2)... A2-T1

/

Vi An — TmO1)v(dtme1)-

An-1-Tn-2

(2.7.20)

If there exists the probability density f(t) of the time interval between adjacent events

the

(A= ffet, A

then formula (2.7.20) may be rewritten in the form

".n(Arx:-*xXAn)=

eer

[-f(n)dn | f(t2—m1)dr::.

eee

| f(tm—tm-1)dt.

Cmpay

144

Chapter

2 Random

Variables, Processes and Functions

It is convenient to connect the stream of the events with the random function

X (t) which represents the number of the events occuring at the time interval [to stl. Evidently, all the realizations of this random function are the step functions which increase by unit jump at each instant of the occurrence of the next event (by 7 if the

moments of the occurrence of n events coincide). Let us find the multi-dimensional distributions [l4,,...,t

of the random function X (t). As at any moment

n

ft the ran-

dom function X (t) may have only integer non-negative values then for determining Lt,...,tn it is sufficient to find the probabilities

‘

n

Pty,..utn(™M1,---,™Mn) = P( () {X (te) = me})

(2.7.22)

k=1

for all m;, = 0 Aleem)

(= 9ny ans

gS

But these probabilities are easily determined by means

ae |, of the known

ee distributions

Vky,...,bn Of the instants of the events occurrences found before. It is easy to see that j EL ermal

eC

CDSs)

Mn

)—

i

x(ti5 te |(t2, ta|) 6

ey

RO

els MD

(tant

ag!

5 ¥ Gesip ty ]

ene CO)

271-20)

For the recurrent stream of the events formula (2.7.23) takes the form ty

.

ti-T1

ti-Tm,-1

Playtest) = f(dn) ff v(dn)... farms) to

0

t2-Tm,

x

t2-Tm,+41

iA (dtm, +1)

‘) (dT m,+2)

ti-Tmy

0

tn-Tmy—y

Dee

eS

0

tn-Tmy—y

ii

VAG fe

mare

ae

/

tn-1-Tm,_1

(oe)

VA diay)

0

In the special case of the stationary Poisson stream

/ tn-Tmn

of events

Claes

e

(2.7.24) f(t) =

\ew*#

where A is the intensity of a stream (the mean number of the events in an unit of

time) formula (2.7.24) gives Pty,..,tn(™1,

AM (tr —

ito

2 eens Mn)

I

(tare ty) eee alt A4 eee =! “Reh sts) m,!(mz —m})!...(™Mn — Mn-1)!

oe

OC

OO

n

:

(2.7.25)

~

2.7. Probability Measure of Random

Function

145

(0< mi) e(h)}) < 6(h)

(2.9.13)

that at any h € (0,7)

at allt,t+7 €T,|| 7 ||=A (The Kolmogorov—Loéve criterion). > For proving that at the fulfillment of conditions (2.9.12) and (2.9.13) almost all realizations of the random function w(t w) are continuous it is sufficient to show that almost at all w

sup

|| z(t’,w)—a(t,w) || 70 at h-0.

t,t’Er

|| t/—-4 ||e})

= 0at7 50.

Therefore any countable set L dense in T is a separant for a(t, w). In particular, the set L of all the points of the region T' with the dyadic-rational coordinates may serve as a separant, i.e. with the coordinates of the type k2~”, where k is any integer,

and p is any natural number.

Thus we may restrict oneselves by the values “ACs w)

on the set I, on the points 7 with dyatic-rational coordinates. Let us introduce the variables

eee lO

spe

SES

Pn

el et

gee Kec

Fy ae

BO)

hn wt 8 eee i)

|

(ea

where the upper bound is taken over all the points | =

(E12

—:

Ke2e* ) ET

and over all q from | till n. It is evident that Zp 2 e(27P) if and only if when even one of the variables

SOE Oo mE ie is larger or equal e(27?).

kg el) eee, KDA)

ie

WEAR

pes

ics instar)

a AE on

Nall

For calculating the number of the variables ae

we denote by 7}, Lebesgue measure of the projection of the set 7’ on the axis m

(m FS

ek n). Then the number of different values of the m*? coordinate of

the type k,,27? in the region 7’ does not exceed 2?T,,.

(ki2—*, aloe E22)

So the number of points

in the region T' does not exceed 2"?7; ...T7;,.

such point corresponds 7 variables ed the random variables Yes ;,,

As to each

oe (q — 1) ree n) then the number of

does not exceed n2"PT, ...T;,. So on the basis of

(2.9.13) we find

Pepe eOPt)) Shy PIV as

beatal2 4) < per PGT, 622). nr

Let us introduce the events

Ap = {Zp = €(2°?)} = {w : Hw) 2 (2°? )},

164

Chapter

2 Random

Variables, Processes and Functions

B, = U 4p, N=limB,= () By. p—=4q

quik

According to the proved above PAG) co

PLB, ys SD EPCAD

1 then by virtue of (2.9.12) P(N) = (). Thus the set NV co

is zero. Ifw € N thenas N = UB, then W € By at any g. But B, =

() Ap.

p=4q Therefore W belongs to all Ap at p > q. It means that Zp = Zp (w) < efr?) at any point WEN

at all sufficiently large p.

Let us take now ie ial odbr

3 | v-l

two arbitrary points I, l! with dyadic-rational

coordinates,

||< 271 then there exists such point lp) with the coordinates

k127-4,..., kp, 274% that all the components of the vectors of the differences | — Ip and I’ — Ig is smaller than 2~%.

Here the difference of the mth coordinates of the

points | and Ig will be expressed by the dyadic number of the type Im

a ys eyaeeS el Rea) B p=qtl where each of the numbers Tp

is equal to 0 and 1. If we perform the transition from

the point /p into | by the sequential steps of the length Tmp Dak (p = qt+l1, q4+2,.. .)

at first on one axis further on the other and so on, on the n*® axis and use repeatedly the inequality of the triangular then we get

ICD En

ouk eae m=1

seme p=q+1

ae nO: p=qtl

The similar inequality we obtain for | z(l’,w) = t(Ip,w)

|. Consequently,

|| (0) —aw) || e}) = 0.

(2.10.2)

The random function X(t) = «z(t,w) is called differentiable in probability on

T if it is differentiable in probability at all t € 7’.

The random function X(t) = a(t,w) is called a.s.

differentiable

(differentiable with probability 1) at the point t € T if there exists such

random function Xj,(t) = x/,(t,w) that

P({w: ah (t,w)

a! (t,w)

at

h—-O})=1.

The random function X(t) = a(t,w) is called a.s.

(2.10.3) differentiable

(differentiable with probability 1) on T if it is a.s. differentiable at all teT.

The random function X(t) = x(t, w) is called differentiable if almost all its realizations are differentiable: P({w:2(t,w) Remark.

differentiable on

T}) =1.

(2.10.4)

It follows from the general properties of the convergence

in

probability and a.s. (Section 2.2) that the random function a.s. differentiable is also differentiable in probability.

The inverse in the general case is not valid.

It is also

evident that the differentiable random function is differentiable a.s., and consequently, in probability.

But it does not follow from the a.s.

differentiability of a random

function that it is differentiable as the uncountable union of zero sets Ny =

{w ;

oP (, w) ee oh (ie w)} at h — 0 even at one m}, t € T may be not zero set.

168

Chapter 2 Random

Example rentiable.

2.10.1.

Variables, Processes and Functions

The random process X (t) of Example 2.7.2 is diffe-

Consequently, it is differentiable a.s.

and in probability.

The same refers

to the random process X(t) of Example 2.7.4. Example

2.10.2.

The random process Y(t) of Example

2.7.3 is a.s.

differentiable, and consequently, in probability at any ft “2 Q. Therefore it is a.s. differentiable and in probability at any interval 7' which does not contain the origin.

But it is not differentiable on any interval (to, t1), to > 0, t) > 2to as none of its realizations corresponding to |w|