Stability of Stochastic Dynamical Systems: Proceedings of the International Symposium Organized by 'The Control Theory Centre', University of Warwick, ... 1972 (Lecture Notes in Mathematics, 294) 3540060502, 9783540060505

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich Series: Mathematics Institute, University of Warwick Adviser: D. B. A. Epstein

294

Stability of Stochastic Dynamical Systems Proceedings of the International Symposium Organized by "The Control Theory Centre", University of Warwick, July 10-14, 1972 Sponsored by the "International Union of Theoretical and Applied Mechanics"

Edited by Ruth F. Curtain University of Warwick, Coventry Warwickshire/England

Springer-Verlag Berlin.Heidelberg • New York 1 9 7 2

A M S Subject Classifications (1970): 34F05, 3 4 H 0 5 , 6 0 H 1 0 , 93-02, 9 3 D 9 9 , 93E05, 93E10, 93E15, 93E99

I S B N 3-540-06050-2 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-06050-2 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1972. Library of Congress Catalog Card Number 72-91895. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergsm

INTRODUCTION

The symposium on the "Stability of Stochastic Dynamical Systems'~ held at Warwick University, July 10 - ]4th, 1972, was sponsored by the International Union of Theoretical and Applied Mechanics, support we appreciate.

(IUTAN), whose

Following IUTAM policy, participation was by

invitation and was limited to around 60 so as to encourage lively discussion. A full list of participants is given~

The main theme of the symposium was the stability and other properties of differential equations with stochastic coefficients.

Both the general

mathematical aspects and applications were discussed.

The contents have been arranged according to the daily sessions, when a "key-note" speaker gave a one hour address to set the daily "theme". Other lectures were of twenty minutes duration.

As each contributor was

asked to submit a complete typed version of his address for the symposium~ these proceedings are in general more detailed than the actual talks in no need of further introduction.

Ruth F. Curtain,

1972

and are

V International Scientific Committee

Prof. J.A. SH~CLIFF (U.K.) Chairman Prof. P. BLAQUIERE (France) Dr. P. BRUNOVSKY (Czechoslovakia) Prof. K. ITO (Japan) Prof. R.E K A ~

(U.S.A.)

Prof. L. MARKUS (U.K. and U.S.A.) Prof. H. ROS~m3Roc~< (U.K.) Prof. v.v. SOLODOVNIKOV (U.S.S.R.) Prof. H.R. SCHWARZ (W. Germany) (IFAC Representative) List of Part icipan% s

Prof. S.T. ARIARATNAM

C aaada

Dr. L. ARNOLD

West Germany (FDR)

Prof. K.J. ~STROM

Sweden

Dr. M.C. AUMASSON

France

Dr. J.F. BARRETT

U.K.

Dr. A. BENSOUSSAN

France

Prof. P. BLAQUIERE

France

Prof. R. BROCKETT

U.S.A.

Dr. P. BRUNOVSKY

Czechoslovakia

Dr. HELGA BUNKE

DDR

Dr. P. CAINES

U.K.

Dr. J.M.C. CLARK

U.K.

Dr. RUTH CURTAIN

U.K.

Dr. M. DAVIS

U.K.

Prof. J.L. DOUCE

U.K.

Prof. T. DUNCAN

U.S.A.

Dr. R. ELLIOTT

U.K.

Prof. A. FRIED.MAN

U.S.A.

Dr. A.T. FULLER

U.K.

Dr. C.J. HARRIS

U.K.

Prof. P.J. HARRISON

U.K.

Prof. U. HAUBSMANN

Canada

Mr. D.B. HERNANDEZ-CASTANO

U.K.

Dr. M. HUGHES

U.K.

Prof. K. ITO

U.S.A.

VI Dr. J.G. JAMES

U.K°

Prof. F. KOZIN

U.S.A.

Prof. H.J. KUSHNER

U.S.A.

Prof. J.A. LEPORE

U.S.A.

Dr. P. MANDL

Czechoslovakia

Prof. L. MARKUS

U.S .A./U.K.

Prof. D. ~[AYNE

U.K.

Prof. R. MONOPOLI

U.S.A.

Dr. T. MOROZAN

Rumania

Dr. L.A. MYSAK

Canada

Dr. T. NAKAMIZO

Japan

Dr. P.C. PARKS

U.K.

Dr. A. PRITCHARD

U.K.

Prof. P. SAGIROW

West Germany (FDR)

Prof. P. SETHNA

U.S.A.

Prof. J.A. SHERCLIFF

U.K.

Dr. Ing. W. WEDIG

West Germany

Prof. P. WHITTLE

U.K.

Prof. J.C. WILLEMS

U.S.A.

Prof. J.L. WILL.IS

Belgium

Dr. D. WISHART

U.K.

Dr. J.A. Z ~ A N

Austria

Local Organizing Committee (University of Warwick)

Prof. J.A. SHERCLIFF Prof. L. MARKUS Dr. RUTH CURTAIN Dr. P.C. PARKS

(Chairman)

CONTENTS

I. M A T H E M A T I C A L K.

FOUNDATIONS

ITO:

Stochastic

Differentials

of Continuous

Local

Quasi-Martingales.

.

P. MANDL: An Application

of It$'s F o r m u l a

to Stochastic

Control

Systems

. .

8

A. P R I E D M A N : ~ S t a b i l i t y and A n g u l a r B e h a v i o r of Solutions of Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . .

14

T. NOROZAN: Boundedness

!I. K.J.

IDENTIFICATION

for Stochastic

Systems

. . . . . . . . . .

21

AND FILTERING

~STROM:~

System D.Q.

Properties

Identification

. . . . . . . . . . . . . . . . . . . . . .

35

NAYNE:

P a r a m e t r i z a t i o n snd I d e n t i f i c a t i o n of Linear M u l t i v a r i a b l e Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

A. ~ENSOUSSAN: O p t i m i z a t i o n of Sensor~' L o c a t i o n in a D i s t r i b u t e d F i l t e r i n g Prob]e~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T.E.

62

DUNCAN:

Some B a n a c h - V a l u e d

IIl.

STOCHASTIC

H.J.

KUSHNER:~

Stochastic

Processes

STABILITY ,,

with A p p l i c a t i o n s . . . . . . . . . .

I

Stability . . . . . . . . . . . . . . . . . . . . . . .

U.G. HAUSSNANN: S t a b J ] i z a t i o ~ of L i n e a r

85

Systems

with N u l t i p l i c a t i v e

Noise

97

. .125

VIII J.L.

WILLENS:

L y a p u n o v F u n c t i o n s and Global F r e q u e n c y D o m a i n S t a b i l i t y C r i t e r i a for a Clsss of S t o c h a s t i c F e e d b a c k Systems . . . . . . D.J.G.

139

JAMES:

Stability

of N o d e l - R e f e r e n c e

Systems

with Random

Inputs

....

147

W. WEDIG: R e g i o n s of I n s t a b i l i t y for a Linear S y s t e m w i t h R a n d o m Parametric Excitation . . . . . . . . . . . . . . . . . . . . . T. N A ~ M I Z O

and Y. SAWARAGI:

A n a l y t i c a l Study on n-th O r d e r L i n e a r S y s t e m w i t h S t o c h a s t i c Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . .

IV.

160

STOC}~STIC

STABILITY

173

!!

F. KOZIN: Stability Ch.J.

of the L i n e a r

Stochastic

System

. . . . . . . . . . .

P_&RRIS:

The F o k k e r - F l a n c k - K o l m o g o r o v E q u a t i o n in the A n a l y s i s of Nonl i n e s r F e e d b a c k S t o c h a s t i c Systems . . . . . . . . . . . . . . . J.A.

LEPORE

and R.A.

BROCKETT

Average P.R.

Value

230

STOLTZ:

S t a b i l i t y of L i n e a r C y l i n d r i c a l Shells S u b j e c t e d to S t o c h a s t i c Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . R.W.

186

and J.C.

WILLENS:

Criteria

for S t o c h a s t i c

Stability

. . . . . . . .

239

252

SETHNA:

U l t i m a t e B e h a v i o u r of a Class of S t o c h a s t i c D i f f e r e n t i a l Systems D e p e n d e n t on a P a r a m e t e r . . . . . . . . . . . . . . . . . . . .

273

H. BUNK-E: Stable P e r i o d i c S o l u t i o n s of W e a k l y N o n l i n e a r S t o c h a s t i c Differential Equations . . . . . . . . . . . . . . . . . . . . .

283

V. A P P L I C A T I O N S S.T. A R I A R A T N A M : S t a b i l i t y of M e c h a n i c a l Systems u n d e r S t o c h a s t i c P a r a m e t r i c Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . .

291

IX L.A. ~ S A K : Waves in a Rotating Stratified Fluid with LsteraJly Varying Random Inhomogeneities . . . . . . . . . . . . . . . . . . . . .

303

P.S. SAGIROW:

The Stability of a Satellite with Parametric Excitation by the Fluctuations of the Geomagnetic Field . . . . . . . . . . . . .

VI. OTHER CONTRIBUTIONS

(not presented

311

at the meeting)

V.G. KOLOMIETZ: Application of Averaging Principle in Nonlinear Oscillatory Stochastic Systems . . . . . . . . . . . . . . . . . . . . . . . .

317

V.V. SOLO DOVNIKOV and V.F. BIRIUKOV: Optimization of Multi-Dimensional Stochastic Systems and Stability of Solutions . . . . . . . . . . . . . . . . . . . . .

324

These were key-note

lectures

STOCHASTIC DIFFERENTIALS

OF CONTINUOUS LOCAL QUASI-MARTINGALES* A

KIYOSI ITO Cornell University The purpose of this paper is to unify the known results on stochastic differentials

in terms of differential quasi-martingales

so

that we can understand the intuitive meaning more easily. I. Let

[~t'

~-algebras. by

E t.

NOTATIONS AND DEFINITIONS

0 ~ t ~ i}

be a right continuous increasing family of

The conditional expectation relative to ~ t

is denoted

We consider the following classes of stochastic processes. is the class of all stochastic processes adapted to {~t}and

having continuous ~is

sample paths.

the class of all continuous

[ ~ t ] , namely all processes of martingales

X ~C~

local martingales

relative to

such that there exists a sequence

X n c g~ satisfying P(Xn(t ) : X(t) for every t) ~ i.

is the class of all

X ~

whose sample paths are of bounded

variation almost surely. is the class of all continuous all

X

local quasi-martingales,

such that X = M + L, M c ~ ,

L ~ g;

see D.L. Fisk [2], S. Orey [7] and K.M. dR

Rao [8].

is the class of all randominterval functions

(stochastic differentials)

induced by

dQ(s,t) Similarly for

d~,

d~

and

Q

= Q(t)-Q(s), d~.

*This work was supported by NSF GP-28109.

as follows: (s < t).

dQ

namely

2. Let us introduce DEFINITION

STOCHASTIC

DIFFERENTIALS

three basic operations

on stochastic

differentials.

dQ2 = edQ1 i f and only i f

i.

t

dQ2(s,t ) ~ Q2(t)

=f

-Q2(s)

dQl(e )

(s < t).

S

The integral is defined in the same way as stochastic Q1 = M1 + LI' such that both that

MI ~

and

LI c~

Ml(t )

and

EO dQl(O)=iAl~mo

~

S

i=l

A = {0 = e I < elO.


~ o, for ~ e R~, we have

~c(~(t,to,~))~ K(to,~)+

Zl b(r)P I where b(r)= inf c(x) J~l>.n.

Let Ar=Isup I~(t,to,~)l > r~ ,At,r=I Jx(t ~to,~)~ > r) = ~?~At, r Since F(Ar~:iim

P(Bm,r) , iim

b(r)P(Bm, r) ~K(to~)+ Theorem 2 is proved.

b(r)= ~,~ and , ~'or a l l

m >. to

and Bm,r=

25 Theorem 3o IZ If)

tim

in_f V(b,x) = be

(2) ~V(t+l,~[t+L,to,9))~_~EV(t,x(t,to,~))+F t oe where

z or all t e ~,

N, t ~ to, ~ e S o n ~

~ e(o,1), T¢[~ ~)

bhen the solubions of ( ~ ,N,~)

aze ablimatel,y bounded in probabill-

Proof . Let

t o ~ N, ~ > o,

~ C

S(~),qj~

From condition (2) it follows that

+ t-to

K(to,~)+

t-to-Z)

1-/~ for a l l t>~ t o , t 6. N.

ience

a(r)~ l

+ ....

~x(~,~o,~)l > r ]


o such that V(t,x) ~

for t 6 N

ixt >~ R o (2) EV(t+l,x(t+l,to,~))e~EV(b,~(t,to,~)) t >~ to, R r= S o D ~

then the solutions of ~

where , N,~)

for all teN,

toeN ,

~(o,i) aze equ.iul~.imatelF bounded in pro-

babilit~ Proof. Let to 6 N , c o, ~ 6 S(~.)G3~

26

We have $ ptl~(t,to,~)

I > ~ot~- E V ( t , ~ ( t , t o , ~ ) ) ~ " ~ t - t o

K(~o,~ ), for aZZ

teN, and thus, Theorem

#

t >. t o

is proved.

Oonsider the It6 system dx (t)=g( t ,x(t) )dt +B ( t ,~( t )dw( t )

(2)

where g: [o,~,)x R ~ --> R ~ is s continuous function, B(t,m) is a ~xpmatrix whose elements blm(t,x) are continuous functions and w(t) is the p dimensional process of Brownian Let W:[Os~)x

R~-~[o~)

motion,

be a function of class C 2, and

~heo~em 5. I_~ (~)

lira inf W(% ]~)= ~, &~

t >O

(2) ~W(t,x) ~ ~(t)

for all t ~ o, x e

!

where

~;[oj~)--~[oj~)

is a continuous function

with l ~ ( t ) d t ~

then the solutions of the sFstem (2) are strongly bounded in probaPro of.

I~et to>. o, ~ > o , ~ ~ S(~)EI~R ~ Let V(t,~)=W(t,~)+ •f~(s)ds,

a(r)=inf V(t,~), K(to,~)=su p V(~o,' t o ,

Ix(t,to,~)(~)l>

rJ and ~,~(t,~)=min{

By Ira's formula, we have

~,v(z~(t),~(z[~(t),to,~)}-V(to,~)~ J'~Av(s,~(S,to,~'))ds~_ o Hence

t,~.(~).}

27 :or all t >. t o But = lim P {Zo,~ ) ~ " 5 Thus

and Theorem 5 is proved. Gonsider the discrete system

~(t+l)=A(~)~(t)+~(t,~) ~(~,~,), t ~ N

(3)

where A(t) is a ~ - m a t r i x , G ( t , x )

is a @~-matrix,

whose elements

are continuous functions, and ~(t,~ ) is a p-dimensional random vector. Suppose that the trivial solution of the deterministic system :r(t+l)=A(t)y(t), t ~ :~ is exponentially stable, i.e. there exis~

ly(t,to,y)l~i~ W ~-t° Theorem 6. If

~o

, ~e(0,~)such tha~

~, for all t~N,to~N, t~t o, ~ R

sup l ~ ( t , x ) l ~ ; te N

~

sup E l ~ ( t ) } ~ ~eN

xe R ~ then the solutions of the s~ste m ~(~ are nltimat~ly

bounded in proba-

bility with ~espect to X. oo__._~f. Pr Let ~ e So, to~ N. Let x(t) be the solution of the system (3), with ~(to)(~ )= ~(~ and y(t,to,~) the solution of the system y(t+l)=~(t)y(t),

y(to,to,~): ~Let V(t,x)=sup(~y(t+s,t,x)|~ ~ )

t

e

It ms easy to see that ~x~V(t,~)_~

for t ~ N, x e

R~

N

:~£

with

28

v(u+i,y(u+~,to,~))-v(t,y(U,~o,~)) ~_ -(l-w)v(u,y(~,~o,yO), ~ ~o,teN,

Ye

Rf

; IV(~,XL)-V(~,~2) I ~_ ~ l ~ i - x 2 1 ,

~l,X2

for Rg

We may w£ite

v(t+i,~ (~+l))-v(t,~(~)) =v(u+l,y( u+i,y(u+~ ~-~,~(t)))-v(t,~(~)) + +v (t +i,~( t +l) )-W+l,y( ~+ L, u ,y(U))) Thus

v(~+].,y(u+].))-v(t,y(u))

__< - ( i - ~ ) v ( u , ~ ( u ) ) +

~I y ( u + l , ~ ( ~ ) ) -

-:~( t+ 1)l ~_ - ( 1-~r)v( u ,:~(~) ) +# I G@.,] (0)11 ,~( ~ )~ _z_ -(1.,¥) v ( t , X ( t ) ) + + ~ q I~(±)I where cl=su p iG(t,~)~ x~R ~ Hence

mV(U+i,](~+i)) z_~ L~(~,~(t))+ ~elc 2 c2

= sup ~ l?(t)i t~o

From Theorem 3 it 1"ollows that bhe solution of the system (5) are ultimately bounded in probability wibh respect

reX.

Theorem 7. If

(~)

sup IG(~,~)I~

M i~l'~

ten

(2) L - 7 - ~ z

M~>O

(3) Z_ ~. i ",/(L )1~ ~ ten ~he~ ~,hp ,enln~.innn nf b~e s,ystem (>) aft strongly bounded in probab ilib~ w i~h respce~

to X .

i)D 0 o f . Leb ~ o 6 N ,

~ £ S . Con~i(ier ~(~) and V(b,x) as in the proof of o

Theorem 6. We

have

v(t+l, :~(~+].))£ ~rv(u,:~(u))+#lm(~,x(~))~ I~(u)l_~v(u,:~(u)) +

29

Hence

_sv~.t+l,:,,(t+l)) --_ :~V(t,:~(t))-(L-~"

._ ~ ~)e~.x~,~.~l -, ~ ~rl. ~t~¢

Using Theorem 2, the theorem is proved. Consider the aiscrete system

:x('~+l)=A,'x(t)+a(t,',J)bJ;(~(t,~Z)~+Bg(|;,~ where A is a ~ - m a ~ r i x ,

B

),

6"(t)=c'x(t),

is a e,~-matrix,

teN,

b and c are

('q-) ~-dimens-

sional vectors, c' represents the transpose~ of the vector c, a(t,co), t e n

is a sequence of random variables,g(t,o~), t e N is a

seguence of p-dimensional random vectors and ~e ~& where ~ se~ of continuous functions f:RI--7 R1with

is the

o~-~'~)1

whereY(1)

~'=b-

+1

+

.4." O]+ o, a~ct (-.~.- -/~ g~)~-..~./-~>~ Let

%o>o such that

Let H=

A°

-~o(-L

,

/'

Z(cc,), ~ ( I ) ko

Obviously that M is a positive definite matrix . It is easy to see that

2h

- ~ b,Hb=Ao,A,HA_H=_qq, _ ~ ]

c where q = 4~

Let x ( t , ~ ) ,

as in the proof of the preceding theorem.

We have

4 x'(t,~) [A'HA-F~ ~(t,~ )+/~b'Hb fz(~t,~))+~'(t): = ~'(t,~) [A'~A-HJ~(t,~)+ i~(q(t,~)[/¢b'Hb-. + ]

where T(t)=IB'HBI

+

E Ig(t,~o)l ~

Hence

E

-(q,~(t,~))~ - ~o/~ ~(t,~ )1.2 ~o q'~(t,~)e(~(t~)) -~ol~(t,~)l~

where

~

= sup ~(t), tan

~i = ] ~~o

Using Theorem 3, we conclude that Theorem 9 is proved. We remark that it can be proved that the trivial solution of the

system (G), with g ( t , ~ ) ~ o a n d A

in mean square for all fE ~ Theorem 9 holds.

= o is e~ponentially stable

if and only if condition (2) of

33 Consider the following It6 system a~ : Ax + b ~ ( ~ ) d ~ ( t ) ,

where A is a ~-~-matrix,

(7)

~=c'~

w(t)

~d

b,c e R,

is t he pro-

tess of Brownian motion Theorem l o .

Th e triVl~laL ,So~,utilllO~

of t h e system ( 7 ) i s exlPonen-

tialLy stable in mean square fq~ all f e ~FA.

matrim

A is stable and

P~ OllO~i

~

if and only.if the

('/A-LA])'~I"~LA

~r

>0

(L=~-.~)

"

By ~ ( G )

-A'

will be denoted t~e matri~ which verifies

~(G)-

~(G)A=G.

wo

0

£#

J-~

Thus, by Theorem in [2] , it follows that if the trivial solution of (7) ~zor ~(~-)=k6- ,is exponentially stable in mean squarc~¢~ l l the matrix A is stable and (~c'(A-iZI) -I bi~dA >o j--Co

Suppose, now,that

~ 2~

h'A

Let

~ =b'~(cc')b,

and

(

the matrix A is stable and

.~,

i c'(~-i

~ I) -l

A---T- ÷ ~ ( i )

it is easy t o see t~at

= ,

>O

such that

- ~2

>o

' "h

Let R=

~~>~

~--b'~(T)b,

2 ..... ;o~o)L ~

bl ~ dA

"

-

and W(~)=x'~m. 2 - b ' HI3= ~ -'6---

c

A' l]+L~iii=-qq'- ~¢i,

~+b'~l~((c

) , ~=~'z

~lence

z - -F

f~ (~)]_~ _(q,

From this relation,

~- O}

Then it is not necessarily

Px{XT+t

x

be a Markov process whose

that w.p.l.

(3)

(Observe that

is slightly too broad a class of

by

x(~) = inf(t:

and suppose that

s ~ O, t ~ O.

Indeed,

(3) should

and we will restrict

to processes

where

our

(3) is true

T.

random variable

T

(defined on a set

~ C T

100

i.e.,

w

process

is a Markov time if we can tell whether or not x

s

up to time

Definition.

t

only, for each

T < t

by watching the

t > O.

If (3) holds for all Markov

times

T, then

xt

is said

to be a stron5 Markov process. Definition.

We will consider only strong Markov processes.

a Markov process for which the function of for each

t > O

x

given by

Exg(Xt)

and real valued continuous and bounded

If

xt

is

is continuous

g('), then

xt

is a

Feller process. A Feller process whose paths are continuous from the right is a strong Markov process.

([5], Theorem 3.10).

Definition.

Let

measurable functions on

B

denote the Banach space of real valued bounded

X, and

B0

the subset of

Exf(X t) + f(x),

weakly as

t + O.

exists (i.e., there is pointwise convergence, h + O) and is in

infinitesimal operator Suppose w.p.1., and

for which

f c B0

If the weak limit

Exf(X t) - f(x)

as

B

T

f E ~(A),

BO, we say that A, and write

g(-)

g(x), and the left hand side is bounded is in the domain

~(~)

of the weak

Af = g.

is a Markov time and

ExT < ~, and

xt

is right continuous

(the continuity conditions can be weakened;

see [5], P.

133) then we have the important relation (4), known as Dynkins formula ([5], P. 133), T (4)

Exf(X ~) - f(x) = E x I ~f(Xs)dS" 0

101

The operator

A

plays a role for Markov processes,

similar to the role

the differentiation operator plays for differentiable non-random real valued functions.

Equation (4) is an analog of the deterministic

integral - differential

relationship and, as such, will play an important role in the sequel.

The non-homo6eneous case. to apply (4) to functions follows.

Define

Redefine

B, B 0

t

f(x,t)

If

xt

is non-homogeneous,

of both state and time, then we can proceed as

to be a state of the process (replace

appropriately,

let

f a~(~)

Ex~tf(Xs,t+s)

and

s ÷ O, and

- f(x,t)

conditions on

T

Ex,tg(Xs,t+s) + g(x,t)

X

g = Af

by

if

X × [0,~)).

f(.,.) e B

and

g(x,t)

S

weakly as

or if we wish

weakly as

s + O.

Then under the

in (4), T

(4')

Ex,tf(xT,t+T)

- f(x,t) = Ex, t

Af(xs,t+s)ds" 0

In (4') we understand that time is measured from the origin of time

2.

x

in (4') is the value of the state

S

s

t; i.e., the value

units of time after the initial

t.

A Few Sources of Stochastic Stability Problems Stochastic stability problems occur in almost all phases of physics,

control theory, numerical analysis and economics where dynamical models subject to random disturbances appear, and the process is of interest over a long period of time.

0nly a few simple problem types will be mentioned here. Suppose that

tion

~t

Yt'

is a Markov process which drives the differential equa-

= f(y~,yt,a), where

~

is a parameter,

driving term, or random variations interested in the range of

a

!

Yt

may represent an external

in some parameter of the equation.

for which

mains bounded in some statistical sense.

Yt ~ 0

w.p.1., or for which

For

fixed at

~

We may be lytl

re-

a0, we may be inter-

102

ested in the range of initial conditions

Py,y,{

E.g., Yt

~>t

sup

y,y'

for which (for some

lYtl i X )

>0

i > ~ > 0)

! 6.

may represent a stress in a mechanical structure, and it may be of in-

terest to keep the stress less than ~ >

0.

Also, it may be desirable to know

whether

Py,y,{

sup >

as

y

or

y'

t

lytl ~ X} + 0 >

0

or both tend to zero (a type of stability of the origin w.p.1.).

The above stability properties are all properties of the paths of the processes.

There are many problems of interest concerning the asymptotic behavior

of the moments and of the measures of the process - and even in cases where the process is of interest for only a finite time [i], [3]. A large class of stability problems arise in tracking situations. example,

For

suppose that we are driving on a road and sample our instruments and

errors (e.g., distance from the center of the lane and from other cars) somewhat irregularly (as is usually done), then can we track the center of the lane within a certain error, etc.

Tracking problems arise in radar and machine tool systems.

Many types of stochastic convergence can be studied. w.p.l, convergence mainly.

Here we deal with

Other stability problems deal with (a) convergence

w.p.l, to a set, (b) recurrence - the process always returns to a bounded set w.p.l.,

(c) no finite escape time w.p.l.,

tain moments of the process, to an invariant measure.

(d) convergence or boundedness of cer-

(e) convergence of the distributions of the process

Types (d-e) are usually more difficult to treat than

w.p.l, convergence, but also are of considerable practical importance.

3.

A Brief Review of Deterministic

Stability

Some results in deterministic

stability are briefly reviewed because,

in a certain abstract sense, the stochastic results are analogies of the deter-

103

ministic results. Let Rr

to

Rr

Rr

denote Euclidean

r-space, f(.)

a continuous function from

and suppose that there is a continuous solution to the homogeneous

differential equation

~ = f(x).

Let

V(-)

denote a continuous, non-negative,

real valued, continuously differentiable function on noted by

Vx('))

satisfying

fined by

Qk £ {x: V(x) < ~}

V(O) = O, V(x) > O, for

(whose gradient is de-

IxI # 0.

Let the set de-

be bou4qded with the derivative of

positive along trajectories in

(5)

for

Rr

V(x t)

non-

QI; namely

V(x t) = Vi(xt)f(x t) ~ -k(x t) ~ 0

x t g QI.

Let

V(xt)

x0 = x

be in

QI.

is non-increasing.

The following statements can be made:

Then

x t g Qk

for all

t ~ O.

From

t

(6)

V(x) - V(xt) = I k(Xs)dS > 0 0 I

we have that

Ik(Xs)dS < ~.

This, and the uniform continuity of

k(x s)

[o,~)

on

0

imply that

k(x s ) + 0

as

Furthermore as

s ~ ~ , and

x s + {x: k(x) : O}NQ~

x + O, the maximum excursions of

~ K~. IxtI decrease to

zero. Define an invariant set of points Then there is a function with

x t, t e (-~,~)

x 0 = x, and furthermore

xt £ G

G

in

Rr

as follows.

which satisfies the equation

for all

t g (-~,~).

entire trajectories over the doubly infinite time interval Let the trajectory and assume (5).

xt

Let

be bounded.

Thus

G

x e G. ~ = f(x)

contains

(-~,~).

In particular, let

x 0 = x e Qk,

Then the invariance theorem [7] states that the path tends to the

largest invariant set contained in

K k.

The theorem is important since it is often used to show that the tend to a much smaller set than sets to which

xt

can tend.

K k.

xt

It gives a very nice characterization of the

104

Example.

Define the differential equation on

R 2,

iI = x2 x2 = -g(xl) - ax2 where t

I g(s)ds

as

÷

t ~ ~,

sg(s)

> 0

for

s # 0

0 g(0) = 0,

a > 0.

Define the Liapunov function

v(x) =

+ 2 i1 g(s)ds. 0

Then

V'(x)f(x) = -k(x) = -2axe. x We can conclude that

x2t ÷ O.

But what about

It is natural to expect that

Xlt?

Xlt ÷ 0

also, and indeed (although the

Liapunov function argument does not directly yield it) it can be proved using a limiting argument, using the facts that

V(x t)

is non-increasing and

Yet it would be much simpler to merely substitute

x2t ~ 0

x2t + 0.

in the differential

equation, and see what trajectories are possible; namely, put the limit of into the equation, directly. conclude that

Xlt + 0

x2t

The invariance theorem allows us to do this, and to

also.

In examples involving functional differential or

more complicated systems, the invariance theorems can save an enormous amount of work. In the sequel, we will develop stochastic counterparts of all the concepts which we just used.

~ile

homogeneity is required for the invariance

theorem, there are straightforward non-homogeneous function theorems.

extensions of the Liapunov

i05

4.

Stopped Processes The weak infinitesimal operator

~

and Dynkins formula (4) will be used

to replace (5)~ (6) for the stochastic problem.

The domain

to be a subset of a set of bounded functions on

X.

tions

V(.)

~(~)

was defined

However, the Liapunov func-

which are most likely to be used, and to which

~

is to be applied,

are usually unbounded (as is usual in the deterministic case).

Even if

V(')

were

bounded, the process may have a stability property only in a bounded or compact set

Q

in

X.

I.e., ~f(x)

may be non-positive only in some neighborhood

Q

of the origin. There is no loss of generality in studying the process only while the paths are in such a set

Q.

For we can often (always, if

find a sequence of sets

Qn t X, and,

X

is

G-compact)

if desired, study the behavior of

X

by

studying the "limits" of the behavior of the process up to, say, T , where n inf(t: x t ~ Qn }.

Thus, we can bound

V(x)

for

x

or we can define a new process by merely stopping

T

"s u f ficiently far" from xt

on first exit from

=

n Q,

Q.

The latter approach is much more convenient. Let

Q

be a set in

conditions under which following.

Let

xt

X.

Dynkin ([5], Chapter 4) gives various general

T = inf{x t ~ Q}

is a Markov time.

be right continuous w.p.l.

(a)

Q

is open and has compact closure.

(b)

Q

is open, X

compact. (c)

Q

We mention only the

(Lemma 4.1)

is a metric space (or metrizable)

and

xt

is continuous.

Define the stopped process either (a)-(c) above hold, and let

xt = X t A T ' xt

(p. iii) where

t ~ T = min(t,T).

be right continuous w.p.l.

a strong Markov process ([5], Theorem 10.2).

Then

Let xt

is

Unless otherwise mentioned, ~Q

be used to denote the weak infinitesimal operator of the process

x t.

Let

will xt

Then to apply Dynkins formula to an unbounded function

V('), we only need check that the restriction of xt

is

(Lemma 4.1)

is open, and

be continuous w.p.l.

X - Q

V(')

to

Q

is in ~(~Q).

is right continuous w.p.l., we need to check whether the restriction of

If V(-)

106 to the union over ~(~Q).

5.

x = x0

in

Such verification

Stochastic

Q

of the almost sure range

usually seems to be straightforward

Stability and Asymptotic

Unless otherwise mentioned, this section.

Xs, s ! T,

After the theorems

is in

in examples.

Stability

we will use the following assumptions

are proved,

in

extensions to more general cases

will be discussed. (AI)

X

is Euclidean

(A2)

V(')

(A3)

Define Let

r-space.

is a non-negative

real valued and continuous

Q~ = {x: V(x) < X}

xt

of exit from

(A4)

QX.

A~

sup xaQ k

(where the definition

for some

strong Markov pro-

x0 = x

of

in

xt = x t A T x "

V(.) Q

opera-

is assumed res-

of the almost sure

xt).

Px( sup

t > s > 0

Observe that,

is not empty.

for the weak infinitesimal

TX = inf(t: x t ~ Q~}~ and

tricted to the union over range of

R r.

X, defined until at least the first time

Write

xt, where

V(') g ~ ( A x )

(AS)

Q~

denote a right continuous homogeneous

cess on the state space

tor of

and assume that

function on

[Ixs-xlI >

a} ÷ 0

as

t + 0

for any a > 0.

i_f_f Y ~ Qk' but is in the almost sure ran6e of

x 0 = x g Qk' then

~kV(y)

~ -k(y) = 0.

Xs, s ~ T k

We will use this fact implicitly

in.the followin6 theorems. Theorem i. tion as x

AkV(x ) t + ~.

e

Assume

(A1)-(A4).

Let

AkV(x) ~ 0

is defined for the stopped process).

Hence

V(xt)

Px {

sup

conver~es

Then

(recall that the operaV(~t)

conver6es w.p.l.,

for almost all paths remaining

in

Qk"

For

Qk'

(7)

l_~f V(O) = 0

o~ > t > 0

and

V(xt) >_ ~}

V(x) # 0

for

=

Px {

sup V(xt) >_ ~}

oo > t > 0

x # O, then as

i V(x)/~.

Ixl + O, the probability

in (7)

107

goes to zero (a type of stability of the origin). Proof.

Applying (4)

gives t

(8)

t~]T~

~V(Xs)dS < O.

ExV(X t) - V(x) = E X I ~A~V(xs)ds = Ex I ~ 0

0

Thus, w.p.l., :u

E~X V(X t) < V(~ s) S

or, equivalently, since measures

xs

is Markov, (-@s

is the smallest

G-algebra which

Xr, r _< s) %

E[V(xt+s) I ~ s ] < V(X~s), w.p.l.

Thus of

(V(xt),_~t} V(xt).

is a non-negative super-martingale.

(7) i s t h e s u p e r - m a r t i n g a l e p r o b a b i l i t y i n e q u a l i t y .

statements are obvious.

x

and

t

The r e s t o f t h e

Q.E.D.

N on-homo6eneouscase. pends on

This gives the convergence

or that

xt

Suppose that the Liapunov function is non-homogeneous.

V(',')

de-

We state the following

Theorem 2, without proof. Theorem 2.

Let the real valued continuous functions (o_~n Rr, R r x [0,~),

R r, resp.) VI('), V(',.), V2(. )

satisfy, for some real

t O _> 0, and

% > 0,

Vl(X) i V(x,s) i V2(x)

for

s > tO

and

x c Q~ = {x: Vl(X) < ~}. Le t

xt

be a right continuous stron6

Markov process defined until at least the first exit time ~enote the weak infinitesimal operator of the process stopped on first exit from t > t O • Then, for

t >to,

Q~.

Suppose

TI

from

Q~.

Let

(xt, tNT%), which is

V(x,t) g°-~(A%) and

AiV(x,t) SUPs > 0VI(Xs) >- ~} < Px't0{~ 0> ssu p>

V(Xs,S+t 0) >_ I)

i V(x,to)/l"

Also

V(xt,t+t0)

conver~es for almost all paths for which

t ~ 0, where we use

x0 = x

V2(x) ÷ 0

a_~s Ixl ÷ 0.

Let also

i - V(x,t0)/~.

and

Vl(X) > 0

for

Ixl # 0; then for any E > 0

and any nei6hborhood of the ori6in

AI, there is a neighborhood

i__ff x ~ A2, the probability of

ever leavin~

-k(x) ~ 0 Thus

i_nn QI.

k(x t) ~ 0

leave leavin~

QI.

< ~

(Asymptotic Stability). Then

k(~t) ÷ 0

(and also

V(xt)

so that,

is no ~reater than

Assume (AI)-(AS).

in probability and

V(~ t)

Let

s.

AI V(x)

conver~es

w.p.l.

conver~es) for almost all paths which never l-V(x)/X

be uniformly continuous in

Q~.

for almost all paths which never leave

QI"

and

k(-)

V(x) ÷ ~

a_~s Ixl + ~, then

to the probability of never

s

are unbounded.

n {x: k(x) < ~} = K E>0

w.p.l.

The

If they are bounded, replace

E > O.

K

c

= {x: k(x) > s > O} N Q1 -

is finite

This follows from the inequality t

(lO)

% V(x) _> -ExV(~t) + V(x) = E x I k(Xs)dS _> ~Ex T' (t,g),

0 where

Tx(t,E)

Rr ,

The key to the proof is the fact that the total time which the

can spend outside of the set

w.p.l, for any

x t ÷ [ n {x:k(x) < g } ] N ~>0 If the hypotheses hold for all

{x: k(x) : 0}.

by

Proof. % x

xt ÷

Then

two s e n t e n c e s i s i n t h e t o p o l o g y f o r t h e c o m p a c t i f i e d

QI n {x: k(x) < ~}

N{k(x) < ~}

process

AI

A2 C AI

QI.)

convergence in the last

if the

xt

(E~uation (7) 6ives a lower bound

Let Q1 - P1

to; thus

a_~s Ixl ÷ 0; then the right hand side of (9) ~oes to zero VI(0) = 0

Theorem 3.

for all

for the initial condition at the initial time

there is convergence with at least probability Let

V(xt,t+t 0) ~ ~

is the total time that

follows from (i0).

k(x s) >__ s

in

[0,t].

That

P k(x~t) ÷ 0

109 We next prove the first statement of the second paragraph of the theorem. Let

T(t,a)

w.p.l,

denote the total time that

for any

E > 0

and

x a Q~.

(A5), t h e u n i f o r m s t o c h a s t i c Let

C

k(~ ) > g s --

in

[t,~).

Then

T(t,E) + 0

The rest of the proof combines this fact with

stability

assumption, to yield the w.p.1,

convergence,

denote the set

C E = {x: k(x) < a} ~ Qk.

Assume that

k(x) < 0

for some

by uniform continuity of between 6(a))

Qk N cCa for

and

k(-)

C /2

0 4 e < eO' and

x E Qk, for otherwise the theorem is trivial. in

(C~

is the complement

Qk O C cc

Define the Markov times lows.

(If

dn

or

O'n

Qk, there is some of

so that the distance

Ca) is positive

(say

_>

is not empty.

~n' q'n

is not defined at

q0' = inf{t: ~x t a Cc/2} ,

a0

Then,

(finite on sets

On' ~'n' resp.) as fol-

~, set it equal to

~

there.)

d O = 0,

(~l = inf(t: x~ t ~ Qk n C c~, t _> (~} ' ~'n = inf{t: x t a Ca/2,

t >__ (~n-i } ' gn = inf{t: ~x t E Q1 n cC~, t > ~ n }, etc.

(ii)

There is some p >

sup Px { sup JXs-Xl x~Qk p > s > 0

0

so that

! 6(a)/2} L 1/2.

Define

A = {~: xg c Cc n Q~, o < s < p, o < ~}. n +s a/2 n n If

~ a An

infinitely often, then the total time out of

for the c o r r e s p o n d i n g p a t h E IA

n

~ ~

w.p.l,

i = O,...,n-1,

xt(m).

if and only if

are in

~

Then (~

n

~ a A n measures

c Ca/2 U Qk

only finitely

is infinite

often w.p.1.

% Xs, s ~ ~n' thus all

But

Ai,

) n

Z Px{Anl~q

([5], P. 398-399),

n

} ÷ ~

w.p.l.

and, by the strong Markov property

(Px+{A} = 0

for

t = ~ )

110

(12)

Z Px{AnI~(~

n

} > Z Px(~ {

sup p>s>O

n

1 > _ ~ Z I{a

Thus

Gn < ~

iXs_Xl

The remaining

follow easily from what we have already proved.

(1) k(x) = 0

See the hypothesis

S1

is deleted,

and

ing, so there is a discontinuity

k(.)

(2)

If the hypotheses

Ixl ~ ~, then

x t ÷ {x: k(x) = 0}

(3)

If

be continuous

for

k(')

X = Rr x ~ S 1.

Q~

Q~

may be unbounded.

subset

S

of

X.

t ~ Ws, ~.

Thus

topology of

in

Q~, then

(A5) may not hold.

Suppose that (A5) holds if

TS, C < ~

Q~

x t + {x: k(x) = 0} U {~}

w.p.1., w.p.1,

S2

is absorb-

X. and

V(x) ÷ ~

as

xt

for any state process.

Furthermore

{x: k(x) < ~}g

is replaced by any compact For any compact so that

S

and

xt ~ S - C

for

in the one-point-compactification

R r.

subsidiary

conditions

can be used to eliminate the point

Refer to the next section for the definition

k(x t) ~ 0.

~

k(xh) ~ 0

Then, we can obtain the following.

Sometimes

"invariant

The set

w.p.1.

is unbounded,

> O, there is a random variable

Q~, nor even that

has a hole in it, i.e.,

of Theorem 3 hold for all

AxV(x) ~ -k(x) ~ 0

If

in

on the boundary of

space, provided that the Dynkins formula is valid for the (4)

of the theorem

and proof of [1], Theorem 2, Chapter 2.

k(x) ~ 1 of

statements

Q.E.D.

There are examples arising in control theory where a target set

< ~o}

and Extensions

It is not necessary that

anywhere.

n

< oo} •

n

only finitely often w.p.l.

Discussion

> 6(~)_} I{q

set".

Let the measures

Then

xt

set whose s u p p o r t

is

of the

tends in probability contained

probability to the union of

in c QX

[{x:

of the terms "weakly bounded"

process

xt

be weakly bounded,

(~}.

and and let

to the support of the largest invariant

k(x)

c = O} n Qx ] u QX.

and a subset of

Thus,

xt

{x: k(x) = 0} N QX"

tends

in

111

The remarks and results for unbounded problem for a process of the type are Markov processes.

The process

not converge in any sense. ponent

Yt

If

(A5), if we replace

# = f(u,y), where ut

(u t,yt ) = x t

We may be concerned with the convergence of the com-

X

is a metric space, the proof still goes through under Rr

in (A2)

will not usually be bounded.

then the path Ql

u t, and the pair

may serve as a time varying parameter, and

by a metric space

verify (A5) and the uniform continuity of

Then

are motivated by the stability

only, but the Liapunov function may depend on both components. (5)

Ql

Ql

k(')

X.

It may be difficult to

in this case, and the closure of

But it sometimes happens that if

Xs, Tl > s ~ O, is contained in a bounded subset of

is "effectively"

contained in a bounded subset, and if

x = x 0 g Ql, Ql k(.)

continuous and (A5) holds on this subset, then the proof goes through. for a specific example. guarantee that

xt

(A2)-

w.p.l. is uniformly See [6]

(A5) plays a crucial role in the proof (since we need to c CE/2

does not jump (w.p.l.) from

to

C c£

and back to

C ce/2

infinitely often in a total integrated time which is finite), and some form of uniform stochastic continuity condition is probably essential.

II.

INVARIANT SET THEOREMS AND APPLICATIONS TO STOCHASTIC DYNAMICAL SYSTEMS In this Section we will develop a stochastic theory of invariance analog-

ous to the deterministic theory in [7], [8]. given conditions, the measures of the process measures,

and that

xt

The main conclusion is that~ under xt

tend to an invariant set of

tends to the closure of the support set of this set of

measures in probability as

t ÷ ~.

Note that we are using the terms "invariance" and "invariant" according to their usage in the general theory of dynamical systems.

The term has

nothing to do with the stochastic notion of invariant measure.

In this Section

x

t

will be a homogeneous

strong Markov process.

opment in [2], with some changes and corrections.

We essentially follow the devel-

112 i.

Definitions Let

Let

@

xt, be a separable metric space.

denote the initial measure of the process;

m(t,¢,.) time

X, the state space of the process

i.e., P{x 0 e A} = ¢(A).

denote the measure induced on the Borel sets of

t, with initial measure

¢.

X

Let

by the process at

The semigroup property +

m(t+s,~,.)

= m(t,m(s,~),.)

holds. Let in

~

f(.)

~

denote the space of probability measures on

is said to converge weakly to in

bounded if, for each e.

~ W(~)

if

f[~n ] ÷ f[~].

A set

Define an

u-limit set

if there is a sequence

as a set

tn ÷ ~

K

C X

W(~)

so that

X.

M = {~a}

e > 0, there is a compact set ++

in A

ures

ME

~

m'(t,'),

m(t,m'(s,~),.)

is an invariant set if for each for

t ~ (-~,~)

= m'(t+s,.)

where

for any

xt

x e X

is in the support set

S(~)

N

x.

of

set

S(~)

Similarly

in ~

+

for which

~a(X-K ) < e

with the property:

f[m(tn,~)] + f[~]

(~

for all

is a f(.) e C X.

m'(0,') = ~(.), the initial measure, and t ~ 0

and

s ~ (-~,~). t e (-~,=)

and initial condition of

~

if

~(N) > 0

~.

Thus for each

Exf(Xt)

is continuous in ~

x

is written for

for

~(')

~ £ M,

and satisfying the Let

~

be in

-~.

for each neighborhood of

U S(~) is the support set of a set Q in ~ . ~gQ S(Q) is not necessarily closed. The process x t

Occasionally for simplicity or

is weakly

S(Q) =

is closed, but

Feller process if

for every

~ e M, there is a sequence of meas-

there is a trajectory of measures defined for all law of motion of the process

{~n )

We may abbreviate

weak limit of a sequence of measures taken along the trajectory) A set

A sequence

ff(X)~n(dX ) ÷ ff(x)~(dx)

CX, the space of continuous bounded functions on

the convergence relation as

for all

~

X.

t > 0 and

and

The is a

f(.) e Cx.

m(s,$)

for

m(s,~,.)

m(s,~(.),').

++It is important to keep in mind that the trajectory of measures.

~-limit set is an

~-limit set of a

113

Next, the main theorem and a useful corollary will be given.

Then the

conditions of the theorem will be replaced by more easily verifiable conditions.

2.

The Invariance Theorem Theorem 4.

Assume (BI)-(B3).

(BI)

The trajectory

{m(t,¢), t ~ 0}

(B2)

For each

i_n_n C X, f[m(t,¢)]

finite

f(') t

is continuous in

(t ~ O) interval, unif0rm!y in

weakly bounded (B3)

is weakly bounded.

f[m(t,@)]

@, for

¢

for each fixed

W

[l.e., a_~s @n ÷ @' f[m(t'@n)] + f[m(t,@)]

Then

W(@)

~(t)

f(') s CX, a_~s t Proof.

for each

t ~ 0. f(') s C X

weakly compact invariant set

i_n_n W(~), t ~ O, so that

f[m(t,#)]

for all

in an~

t > 0.]

is a non-empty, weakl~bounded~

and there is a sequence

¢

0n any

set.

is weakly continuous ~n

and each

t

- f[~(t)]

÷ 0

~.

According to Theorem !, Section I, Chapter 9 of [i0], a suffici-

ent condition for a sequence in -~ to have a weakly convergent subsequence is that it be weakly bounded. Let

{E.}

each

G.

I ~

W(@)

is not empty.

denote a real sequence which tends to zero.

i

are compact sets

Thus

G.l

so that

Gi+ I ~ Gi

and

there is a countable family ~ !

and dense in

i

CG .

Each element of ~ m

By (AI), there

m(t,@,X-G i) --< si' all

t _> 0.

For

of continuous functions, defined on

C i,

can be extended to a continuouS function

i

X

without increasing its norm (using the normality of the metric space and [9],

Theorem 1.5.3). U ~i" i

Write

Let

.°~ i

G = U G i. i

denote the countable family of such extensions and Observe that, for any

f(.) g C X,

-~=

oi

114

(*) also holds for

m(t,@,-)

replaced by an element in the weak closure of

(m(t~¢,')). Let

m(t ,¢,') n

Define the function

converge weakly to

Fn(.,.)

4(')

in the

u-limit set

W(¢).

by

Fn(t'f) = I f(x)m(tn+t'@'dx)"

If

tn-T _~ O, then

Fn(t,f) = J f(x)m(t+T,m(tn-T,@),dx).

is weakly bounded, (B2) uniformly in

implies that

n, for each

f(').

Fn(t,f)

is continuous in

[-T,T].

verges to a continuous function of

an

fg(')

in ~

t, F(t,f)

interval.

for which

and

for all

n

con-

and uniformly g ~ O, there is and

t h - t n,

f e CX. 4(t,.)

~(t,A) =

IA

[-T,T]

Fn(t,f)

f(') E ~ ,

f(') g C X

IFn(t,f s) - Fn(t,f) I < e

Define the set function

f(') g CX, and

on

t n) for which

for each

Since, for any

the asserted convergence is for all

where

t

By successive applications of the

diagonal procedure, we can extract a subsequence (of

[-T,T]

(m(tn-T,@))

Thus Ascoli's Theorem implies that there is a

uniformly convergent subsequence on

on any compact

Since

by

inf F(t,f) f h IA

is the indicator function of the Borel set

A

in

X.

The argument in [lO], pp. 441-444, can be used to prove that, for each t ~ (-~,~), 4(t,')

is a unique probability measure, 4(t,G) = 1

and

F(t,f) = I f(x)4(t,dx)

for each

f(') c Cx.

4(0,') = 4('). ported in in

Cx,

G.

Thus

W

m(t +t,¢ ") + ¢(t,') n

The weak closure of

~

{m(t,~,')}

for each

t g (-~,~)

where

is also weakly bounded and is sup-

Thus, by (B3), we can write, for any

t c (.~o ~), s ~ O, and

f(')

115

f[m(s,m(tn+t,¢))] ÷ f[m(s,~(t))] f[m(s,m(tn+t,¢)) ] = f[m(O,m(tn+t+s,¢)) ] f[m(O,@(t+s))] = f[~(t+s)],

which implies that

~(t+s,') = m(s,~(t),')

mine the measures uniquely. ~(t)

Thus

{~(t)}

since the continuous functions deterobeys the law of the process and each

is in an invariant set. Let

(thus

f[~n(')]

{~n }

f[~(')]

converge for each

are weakly bounded).

on

CX, and

~(G) = 1.

W

m(ti(n),¢,') ~ ~n (')

as

f(.) ~ CX, as

There is a measure

n ÷ =, where

~(')

We need to show that

for which

~(') E W(¢).

i ÷ =, for some real sequence

t.(n) ~ ~. i

~n (') e W(~) f[~n(')] +

For each

n,

Since

lim lim If(x)m(ti(n),¢,dx) = lim If(X)~n(dX) n i n :

for each for which

f g ~ , we can extract a subsequence

If(x)~(ax), (t }

of the double sequence

m(t ,~,.) ~ $(.). Only the last assertion of the theorem remains to be proved.

there is a sequence or

{ti(n)}

{t n}

so that for any subsequence

{t~}, and some

Suppose that f(')

in

CX,

(*)

By weak boundedness of to some

~(') E ~g .

lim sup

inf

n

~Ew(¢)

Jf[m(t~,¢)] - f[~(')]l >

o.

{m(t',¢,.)}, there is a subsequence which converges weakly n This

~(')

must also be in

W(¢), a contradiction to (*).

Q.E.D. Theorem 5. (i)

P

Assume (B1)-(B3) of Theorem 4. --

Then

x t ÷ S(W($)) ~ C, the closure of the support set of the invariant

116

set W(@)

i.e.,

P.{inf

'

@

(ii) X

and let

Let

Gn .

of an invariant

as

t ÷ ~, for an~

be a real valued, non-nesative Let

Then

G

xt

conver6es

in ~robability

X

that, for each

Let

NE(C)

denote an

m(t,¢,X-Gn)_


0,

(*)

lim P@{x t e X-N£(C)} t ÷ ~ ~

since (*) implies so that

for which

function on

to the lar6est support

n

(i)

and continuous

denote compact sets in

n

set whose s u p p o r t i s c o n t a i n e d i n

Proof.

e > 0.

--

k(')

P k(x t)_ + O.

en ÷ O, Gn+l D

Ixt-Y I > g} ÷ 0

yeC

P@{x t

(i).

Suppose

= 0,

(*) is violated.

e X-~e(C)} ~ e 0 > O.

Then there are

There is a function

t

n

f(') e C X

+ ~

and

£0 > 0

satisfying

n 0 ! f(x) ~ i, f(x) = 0 {t~}

of

on

{tn}, m(t~,@,')

f[~(')] ~ e 0 > O.

Thus

in the support set of

Ne/2(C),

f(x) = i

on

X-Ne(C).

converges weakly to a

~(')

X-Ne(C) , which is disjoint @('), a contradiction

in

from

For some subsequence W(@)

and

~, contains

to the definition

of

Under the conditions

of Theorem i, if

for the stopped process are weakly bounded, to the stopped process. each

~

is compact,

k(')

in Theorem

is not bounded, from

Ql

If the conditions

then

{m(t,~,.)}

(5) is the

k(')

Theorem 6.

Proof.

Let

Q

some point

Q,E.D.

of Theorem 4. is compact,

then the measures

and we can apply the invariance theorem of Theorem i hold for all

is weakly bounded.

of Theorem 3.

it may be that the measures

are weakly bounded.

(BI)-(B3)

+

C.

(ii) follows easily from (i), and the proof is omitted.

Discussion of the Conditions

f[m(t~,¢)]

Usually,

Furthermore,

for the process

I < ~, and the function

even if each

Ql

stopped on exit

See Example 2 in [6].

(B3) holds for a Feller process on any topological W

Cn (') + ¢(').

We must show that

state space.

117

I

(*)

f(x)m(t,¢n,dX) - Jf(x)m(t,¢,dx) ÷ 0

for all

f(') e Cx.

Write (*) as

I

[f(y)m(t,x,dy)](¢n(dX)

- @(dx))

= lht(x)[@n(dX) - ¢(dx)] m(t,x,.) is in

CX

denotes the measure with initial condition

x

O. and

ht(x) = Exf(Xt)

which

W

by the Feller property, and the convergence follows since

Cn(" ) + ¢(.).

Q.E.D. Remark. Theorem 6 implies that condition (B3) is not very restrictive. Theorem 7.

Let

(*) as

Px{[Xt-X] > g) ~ 0 t ~ 0, uniformly for

K c X, let the family Proof.

Let

x

in any compact set.

{m(t,x,'), x g K, t ~ T} {@a}

For each real

T > 0

be weakly bounded.

Then (B2) holds.

denote a weakly bounded set of measures.

second hypothesis implies that the family

{m(t,¢a,'), t ~ T, all

a}

bounded (we omit the proof, which is not hard). Write, for

and compactum

t > O, s > O, s+t < T,

IIf(x)[m(t+s,~G,dx ) - m(t,~,dx)] I IIExf(Xs ) - f(x) Im(t,~G,dx) = I IExf(Xs) - f(x) Im(t'¢a'dx) G' + I IExf(Xs) - f(x)Im(t'~a'dx)" X-G'

Then the is weakly

118

Choose compact

G'

to make the second term less than

Then, using the first hypothesis, choose for

s < sO

and

x E G'

sO > 0

~, for all

so that

IExf(Xs) - f(x) I ~

thus proving the right continuity of

To prove left continuity, write, for

a, t ~ T. E

Exf(Xt)

T > t-s > 0, s > O, - -

m

- -

IIf(x)[m(t,(~c~,dx) - m(t-s,@a,dx)]l IExf(Xs)-f(x)Im(t-s,¢a,dx).

~ I IExf(Xs)-f(x)Im(t-s,@a,dx) + I G' Choose compact

G'

and then choose

X-G' E ~ ~

so that the second term is

sO

so that

E IExf(X s) - f(x) I < ~

for

0 ~ t-s ~ T, and all s ~ sO

for

and all

x g G'.

Q.E.D.

III. Example i.

EXAMPLES

A relatively simple example is the diffusion process given

by the It$ equation

dx I = x2dt dx 2 = -g(Xl)dt - ax2dt - x2cdz where t I g(s)ds ÷ ~

as

t ÷ ~, sg(s) > O, s # 0, g(0) = 0,

0 and in

g(') R 2.

satisfies a local Lipschitz condition. Then

xt

Let

Q

be a bounded open set

can be defined up until the first exit time from

stopped process is a continuous Feller process and (BI)-(B3) hold.

V(x) = x~ + 2 il g(s)ds 0 is in ~(AQ)

and, for

x E Q,

~Qv(~I = x2(c 2 2 - 2a).

a,

Q, and the The function

119

Let

c

2

< 2a.

Then

Px { sup V(xt) _> l} _< V(x)/l ÷ 0 ~>t >0 and

xt

can be uniquely defined on

[0,~)

satisfy a global Lipschitz condition.

as ~ ÷

w.p.1., even though

g(.)

does not

It is a continuous Feller process and (B1)-

(BB) hold. Let

c 2 < 2a.

Then

x2t -~ 0

w.p.l, and by Theorem i, x t

tends in pro-

bability to the smallest invariant set whose support satisfies x2t = O, for all P Thus x t -~ O. This and the w.p.l, convergence of V(xt) implies t h a t x t + 0

t.

w.p.l.

Example 2.

For the second example, we take a problem arising in the

identification of the coefficients of a linear differential equation. The system to be identified is the scalar input, scalar output asymptotically stable, reduced form, system

n-1

(i)

di

di

m

(dtnd + i=0~ai dt i)y = (~0 bi 1---r)u' dt n > m,

where

u(t)

c.u.(t) i

i

'

is the input. where

~(t)

We wish to know the

ai,b i.

is a stationary Markov process.

The input

u(t)

is

The "equation error" method

of P. M. Lion ("Rapid Identification of Linear and Nonlinear Systems", Proc. 1966 Joint Automatic Control Systems Conference, University of Washington, Seattle) will be used.

For this method, Some estimate of the derivatives of a smoothed input and

output are needed. Let

H(s)

denote a transfer function the degree of whose denomenator ex-

ceeds the degree of the numerator by at least the "derivatives of the smoothed

u,y

n.

For any real number

as

Yk(S)

= H(s)(s+c)ky(s),

k = 0 .....

n

Uk(S)

= H(s)(s+c)ku(s),

k = 0 .....

m

c, define

120

and the equation error

c(t)

(2)

as

e(t) = Yn(t) +

where {~i,Bi }

are to be prescribed.

n-i m Z ~iYi(t) + Z Biui(t) 0 0

Let the system

(y,y(1),...,y(n-l))

state variablized by the minimal order, (with asymptotically stable

be

A y) i y =

AYx y + BYu, y = HYx y, and write

y(s)

N(s)

:DqiTu(s)

n-1Qi(s)

+

Z

D 7xi(O)

0

where the last term goes to zero exponentially. Let us impose the following conditions: (CI)

](t)

is a right continuous stationary Feller strong Markov pro-

cess with

Elu(t)l 2 = M O < ~.

formable paths w.p.l. all (C2)

~

w.p.l, for

0

sup ~>h>O

lu(h)-ul > e} + O u = u(O)

as

6 ÷ 0

uniformly for the initial

in any compact region.

E[7(t+T)7'(t)lT(s) , s ! 0] ÷ ~(T), the covariance of the cesses.

(C4)

In particular, f e-ktlu(t)Idt < ~

k > O.

condition (C3)

Thus, the paths are Laplace trans-

Let

E~(t) = 0.

pro-

(This condition is not essential.)

Su(~), the spectral density of

There are real numbers

7(t)

{a2,B~}

u(t), is nonzero over some interval.

so that

g(t) ~ 0

if all

x.(0)l = 0.

To see this write the Laplace transform of (2) where we have

e(s) = Yn(S) + = H(s)

if

n-I m o o ( s) + ~ ~iui(s) Z ~iYi 0 0 s+c)n N(s)

D--UV[+

n-i

o

Z ~i(s+c)

0

i N(s)

m

o

+

D-77Y + ~ Si(s c) 0

u(s) = 0

121 m

N(s) _

0

o i 8i(s+c) ~

~

n

o i Z a.(s+c) 1 0 For {e°,6°} 1

1

used in (2), g(t) ÷ 0

systems generating

Yi(t), ui(t)

exponentially.

o n

n

=i.

In fact, we suppose that the

are connected to their inputs at

that their initial conditions do not depend on the process non-random.

=

u(t).

The condition can be relaxed to allow for random

t = 0, and Then

e(t)

is

Yi(O), ui(O) , at

some extra complication in the analysis. The parameter adjustment procedure is

k 32 dj = - ~ 3---~= -k g yj

(B) k 892

Define the column vectors

o o . ,Bm_8~ ) z = (aO-&O,...,&n_ I - an_l,.. w = (yo,...,Yn_l, u 0 ..... Um). Then (4)

= -kwe = -kw{[Yoa O +...+ Yn_l~n_l +...+ Um~m]

÷ Yn ÷ lY0 0 ÷÷ (5)

where

Um °l

= -kww'z + ~t o 6t = -kW[Yn + YO~O +'''+ Um~ ]" We can assume that the

stable systems of the form Yk' Uk' y' u, z namely

- Ey0 o ÷÷

Yk(t), uk(t)

are the outputs for asymptotically

.Yk AYkxYk Yk Yk Yk x = + B y, Yk(t) = H x , etc.

Thus all

are state variabilized, and the composite state variabilization,

x(t), is a right continuous strong Markov process and Feller.

122

Furthermore, it is uniformly stochastically continuous in the sense of (C2). + Let of

E ~(0) I2 < ~.

£ = -kww'z.

Then

Let

¢(t,s)

l¢(t,s)l ! 1

denote the fundamental matrix solution

and co

Iz(t)I

u,B,Qp-IQBu

where etA'pe tA dt,

--co

then there is no stabilizin$

control.

Consider now the case

R

m

unstable modes of the corresponding Rm,

implies

that

C(u)

matrix A,

is sufficiently

Assume that

imaginary eisenvalues. stabilizin$

Restricting i.e.

(ii) of theorem 4.2 is satisfied.

satisfied provided

THEOREM 4.3.

# R n.

control.

If

DD'

large.

is positive

R m # R n,

the system (i.i) to the

to a certain complement With

P = I

(4.1) can be

The final theorem results.

definite and that

then for

of

C

sufficiently

A

has no purely larse there is no

130 References

[i]

U.G. Haussmann, Noise",

[2]

"Optimal Stationary Control with State and Control Dependent

SIAM. J. Control, 9(1971), pp. 184-198.

.....,

"Stability of Linear Systems with Control Dependent Noise",

Ibid., forthcoming.

[3]

W.M. Wonham,

"Optimal Stationary Control of a Linear System with State-

Dependent Noise", SIAM. J. Control, 5(1967), pp. 486-500.

[4]

,

Random Differential Equations in Control Theory, Probabilistic

Methods in Applied Mathematics, vol. II, A.T. Bharucha-Reid, ed., Academic Press, New York, 1969.

[5]

M. Zakai,

"A Lyapunov Criterion for the Existence of Sta%ionary Probability

Distributions for Systems Perturbed by Noise", SIAM. J. Control, 7(1969), pp. 390-397.

LYAPUNOV

FUN,CTIONS A N D

GLOBAL

A CLASS

FREQUENG, y , D O ~ A I N

OF S T O C H A S T I C

FEEDBACK

STABILITY

CRITERIA

FOR

SYSTENS

J A c Q ~ % L. WIL~MS University Gent,

of Gent

Belgium

ABSTRACT This paper deals with the stability of a particular class of stochastic systems;

feedback systems are considered which have a feedback gain with a

deterministic

gain which may be nonlinear and/or time-varying and a stochas-

tic component which is white noise. Lyapunov functions are constructed and criteria for global stability are derived similar to the results available for related deterministic

feedback systems,

erion, the Popov criterion,

such as the Routh-Hurwitz

crit-

and the circle criteria. I. INTRODUCTION

The difficult step in the application of Lyapunov theory to analyse the stability of deterministic systems

systems as well as the stability of stochastic

(Kushner 1967), is the construction of a suitable Lyapunov function.

There is indeed no general systematic procedure functions.

For deterministic

for generating Lyapunov

systems very interesting results,

Popov criterion and the circle conditions,

such as the

have been obtained for a partic-

ular class of feedback systems containing a linear time-invariant path element and either a nonlinear or a time-varying

forward

feedback element

(Zames 1966, J.C. Willems 1971a, J.L. Willems 1970). This is the motivation for considering a similar class of stochastic

feedback systems,

where the

gain of the feedback element has a stochastic white noise component. the purpose of this paper to investigate constructing Lyapunov functions

whether or not the procedures

for the deterministic

for the stability analysis in the stochastic the path integral method

(Brockett

for

case remain useful

case as well. In particular

1970) and the Kalman-Yacubovitch-Meyer

lemma (Kalman 1963) are used to generate useful Lyapunov functions stochastic

It is

for the

feedback system.

Section 2 deals with the case where the deterministic

part of the feed-

back element is a constant gain; a Lyapunov function is obtained which proves a necessary and sufficient

condition

for mean-square

stability;

the white

132

noise component of the feedback gain has a destabilizing effect on the meansquare stability properties. The criterion is also sufficient for Lyapunov stability with probability one, but not necessary. In Section 3 a nonlinear time-varying feedback element is considered;

stability criteria are obtained

which are closely related to the circle criterion for deterministic systems (Zames 1966), but which also show a destabilizing effect of the white noise gain component. A particularity of this analysis,

which does not appear in

deterministic stability theory, is that different solutions of the spectral factorization problem in the path integral method or of the Kalman-Yacubovitch-Meyer lemma yield non-equivalent stability criteria. Section 4 deals with the stability analysis when the deterministic

feedback gain component

is either nonlinear and time-invariant or linear and time-varying.

Criteria

similar to the well known Popov criterion are derived. Some possible extensions and generalizations are discussed in Section 5. 2.

LINEAR TIME-INVARIANT FEEDBACK

In this section the stability is considered of a system consisting of a

linear time-invariant forward path element with rational transfer function

H(s) and a multiplicative feedback whose gain is the sum of a deterministic constant k and a stochastic white noise component. This system is described by the stochastic It8 differential equation dx = (A x - kbcx)dt - b c x d 6

(I)

where matrices and vectors are denoted by underlined symbols. The triple

Ao,~,~

is a realization of the transfer function H(s) = c(Is-Ao)-1~, and

denotes a scalar Wiener process with independent increments and E((~(t)-~(~)) 2) = ~ I t - ~ l ,

E(d~(t)) = 0

(3)

Let G(s) be the transfer function of the deterministic closed loop system G(s) = p ~ )

= £(!S-Ao-kbc)-1~ = H(s)/(kH(s)+1)

Suppose that p(s) is strictly Hurwitz, and denute the polynomial p(s) by p(s) = s n + Pn_1 sn-1 + ... + Po ' q(s) = qn_1 sn-1 + ... + qo where, without loss of generality, p(s) is assumed a monic polynomial. Then (I) is equivalent to

133

p(D)y dt + q(D)y d ~ =

O

(4)

where D = d/dt. In the sequel either equation (I) or equation (4) is used, whichever is more convenient. Both the state vectors ~ and ~ = [ y

Dy °..

Dn-ly IT are used; they are related by a linear transformation, and they are identical, if A has the companion form (Brockett 1970), ~ = [qo qn_1 IT, and ~ = [O

0 ... 0

ql "'"

I] . For the stability analysis of a stochastic

equation, such as (I), instead of the derivative of the Lyapunov function in deterministic stability theory, the sign definiteness of LV(~) should be considered (Kushner 1967), where LV(x) is the sum of the derivative for zero noise and

?Yn

e

where Yi denotes the i

th

component of the vector ~. Hence, if quadratic

Lyapunov functions are used, it is the aim to find a Lyapu~ov function whose deterministic derivative contains a term proportional to (q(D)y) 2. This can be achieved by means of the path integral technique (Brockett 1970). Let the polynomial h(s) be the solution of the set of linear equations ½[h(s)p(-s) + h(-s)p(s)]~ q(s)q(-s)

for all s

(5)

A unique solution exists if p(s) has no zeros on the imaginary axis. Using the Lyapunov function Ft(~) V(~) = J t(O)[ p(D)z h(D)z -(q(D)z) 2] dt

(6)

m

(where the notation is as introduced by Brockett), we obtain LV(~) = -(q(O)y) 2 + ½ hn_ I ~2(q(D)y)2

(7)

Considering the path independence of the integral in (6) and evaluating this integral along solutions of p(D)z = O, the positive definiteness of V(~) is easily proved. By means of the Lyapunov theorems for stochastic systems (Kushner 1967) the following criterion is readily established : Criterion I

The null solution of system (I) is mean-square stable

in the

large and stable in the large with probability one, if p(s) is strictly Hurwitz, and ~'2hn_i/2 ~ I

(8)

134

To prove asymptotic stability pole-shifting techniques can be applied, or the following Lyapunov function is used

v(~) :

s

t(~)

[p(O)z(h(O)z+am(O)z) - (q(O)z)2 -a (z2~Dz)2+...

t (~)

+ (Dn'Iz)2)] dt

where m(D) is the solution of

½[p(D)m(-D)÷p(-O}m(D)~ = ~

(-~)i D2i

i=O Suppose ~ h n _ I / 2

~2qn_l/a = ~w(O)/2

(22)

E,t

where w(t) denotes the inverse Laplace

transform of G(s).

This criterion is proved by means of the following quadratic

Lyapunov

function v(~)

: ftt(~)[q(D)z (~)

where r(s) is the negative of p(s) and q(~)

p(D)z - (r(D)z) 2 ] dt

spectral

(23)

factor of the even part of the product

:

r(s) : ~ [p(s)q(-sl+p(-s)q(s)]T h i s Lyapunov f u n c t i o n y i e l d s

the desired

result

(24) since

LV(~) = -(r(D)y) 2 - (k(y,Dy .... ,t)-~qn_i/2)(q(D)y)2

140

The asymptotic stability is proved as in Section 2. Remark 6

The criterion reduces to a well known stability criterion for det-

erministic feedback systems if the stochastic component of the feedback gain is absent. Criterion 3 is stricter than its deterministic counterpart; indeed w(O) or qn-1 is positive if G(s) is positive real. The condition on the deterministic part of the feedback gain gets more restrictive as ~

increases.

The conditions of Criterion 3 require the passivity of the linear part of the open loop and a sufficient degree of passivity for the deterministic or average feedback element; the degree of passivity required depends on the statistics of the noise component of the feedback gain but also on the forward path element,

since w(O) appears in (22) and (23). In deterministic

theory the passivity conditions of the forward path and the feedback path are uncoupled. Remark 7

The criterion is equivalent to a corollary obtained by Willems and

Blankenship (1971, Criterion 4) for systems where the deterministic component of the feedback gain is constant.

I

(Note that a factor ~ was omitted in that

reference in the statements of corollaries 3 and 4). Here it is shown that the criterion is valid for a much larger class of stochastic systems. Remar~ 8

The Lyapunov function can also be used for the system dE = (A_~x - k(~,t)bcx) dt - f(~,t)bcx d ~

(25)

The stability conditions are : (i) G(s) positive real. (ii) For all x and t : ~2 k(x,t)~ ~--qn-1 (f(x,t))

(26)

For asymptotic stability it suffices that either of the conditions holds in the strict sense, as explained above. Remark ~

Consider the case where the mean is non-zero. For system

with E(d~) = m, the function

G(s)/(mG(s)+1)

real. For system (25), inequality k(x,t)~ ~2w(O)

(19)

is required to be positive

(26) should be replaced by

(f(x,t)) 2 - mf(x,t)

For k(~,t) : O the system described by (16) is obtained. Using the same technique for Lyapunov function construction as in the proof of Criterion 3 the following related results are readily derived :

141

Criterion 4

The stochastic

system (19) has a null solution which is mean

square stable in the large and stable in the large with probability one if (i) G(s) -I - a

is a positive real function,

and

(ii) for all x and t :

k(~,t)~ a + ~2w(0)/2 for some (positive or negative)

(27)

constant a. The stability properties are

asymptotic if either of both conditions are true in the strict sense. Criterion ~

The null solution of the stochastic system (19) is mean square

stable in the large and stable in the large with probability one if there exist constants k I and k 2 such that (i) The function kiG(s)+1 k2G(s)+1 is positive real. (ii) For all x and t k1+k 2 k1+k 2 ---~--. - b ~ k(~,t) ~ --~--- + b

with

[~i_k2 b =

(28)

J

]~

[---~-- - ~- (qn_1(k1+k2)+2Pn_1-2rn_1)

where rn_ 1 is the coefficient

(29)

of s n-1 in the polynomial r(s)

:

r(s) :~[(klq(s)÷p(s))(kaq(-sl+p(-s)l+(klq(-s)÷p(-s))(kaq(s) +p(s) ] + that is the positive spectral

(klq(s)+p(s))(k2q~s)+p(~)).

Remark

10

factor of the even part of the polynomial

The frequency domain conditions of Criteria 4 and 5 have an

interesting interpretation in terms of circle conditions. criterion 4, requires the frequency response G(jW) completely within the circle shown in Fig.

Condition

(i) of

for positive a to lie

I. For negative a, the frequency

response G(j~) should lie completely outside the circle shown in Fig. 2, and should encircle the disk as many times as G(s) has poles with positive real parts (unstable open loop poles). This corresponds to a criterion obtained by Blankenship and Willems

for a much smaller class of stochastic

tems. The geometric interpretation

of condition

sys-

(i) of Criterion 5 is shown

142

Fig.

Fi~. 2

I

Im

Re

J

Frequency Fig.

domain

condition

of Criterion

4.

I : a positive

Fig. 2 : a negative

Fig. 3 Frequency

domain

condition

of Criterion

~ ~ Q(jw / I

-~2

I

kl,k 2 positive

~e

5

G(jw) -1

kl

k I positive, k 2 negative

kl,k 2 negative

-I ~Re

k2

143

in Fig. 3. Remark 11

The condition on the feedback element is different depending on

whether the positive or negative spectral factor is used for generating the Lyapunov function. This is a particularity which does not appear for deterministic systems. This condition is least restrictive if the positive spectral factor is used. This same conclusion is obtained if the Kalman-Yacubovich-Meyer lemma is used for constructing the Lyapunov function as is shown below. The choice of the spectral factor has no effect on the stability condition if k I or k 2 are infinite,

as in the cases considered in Criteria 3

and 4. Remark 12

The same stability criteria and Lyapunov functions can also be

obtained by means of the Kalman-Yacubovich-Meyer lemma and its generalizations instead of the path integral technique. Only Criterion 5 is discussed here. The positive real character of (kiG+1)/(k2G+1) is sufficient

for the

existence of a negative definite symmetric matrix P satisfying the matrix inequality (Willems J.C°

1971b)

P_~A + ATp - (Pb+mcT)(~T~+m~) + klk2cTc ~ O

(3o)

where m = (k1+k2)/2 . The stability criteria are then obtained from the Lyapunov function V(~) = -xTpx_ __ . Here also the stability result depends on the choice of the solution ~ of the matrix inequality (30). Willems (1971b) has shown that the smallest(-~) is obtained if the inequality

(30) holds with

the equality sign and if that particular solution is chosen such that the eigenvalues of A-b(bTp+mc) have negative real parts; this corresponds to the choice of the positive spectral factor as the solution of the factorization problem in the path integral method. 4. THE STOCHASTIC POPOV CRITERION It is wellknown in deterministic stability theory that less restrictive stability results than the circle criteria can be obtained if the deterministic feedback gain is assumed to be either nonlinear and time-invariant or time-varying and linear. In this section it is shown that similar conclusions are obtained for the stochastic feedback system considered in Section 3. The case of a nonlinear time-invariant deterministic

feedback is considered first,

and a result similar to the Popov criterion is derived. Consider the system considered in Section 3, but suppose that the deterministic feedback component is nonlinear, memoryless and stationary. The system is described by the It8 equation

144

d~ : (Ax-f(cx)bcx)

dt - bcx d ~

(31)

or p(D)y dt + f(q(D)y)q(D)y Criterion

6

dt + q(D)y d ~

The null solution of the stochastic

(32)

= 0

system described

by (31) or

(32) is mean square stable in the large or stable in the large with probability one, if there exist a positive

constant

a such that

(i) the function (1+as)q(s)/p(s) is positive

real,

and

(ii) for all u :

f u)

2 (aqn-2Pn-1+qn-l-2rnrn-1 +a(f(u")+u dr(u), 2 ) du )qn-1

where r n and rn_ I are the coefficients

of D n and D n-1 in the positive

ral factor of the even part of the polynomial The stability is asymptotic

(33) spect-

(1+as)q(s)p(-s).

if either of the conditions

holds in the strict

sense. The criterion is proved by means of the Lyapunov

function

(t(y) V(~) = J

dt [(1+aD)q(D)y

p(D)y - (r(D)y) 2 + aq(D)yf(q(D)y)q(D)y]

Co) where r(s) = ~ 2 [(1+as)q(s)p(-s)+(1-as)q(-s)p(s)] Remark

I~

Condition

The criterion

reduces to the Popov criterion if ~2 vanishes.

(33) is much simpler if qn-1 or w(O) vanishes,

is of degree

(n-2). Then f(u) ~

Criterion

+

that is, if q(s)

(33) requires

aqn_2Pn_1~2/2

6 reduces to the conditions

of Criterion 3 if a = O.

Next we consider the case of a linear deterministic

feedback gain compon-

ent. Then the system equation is d~ = (A_~x-k(t)bcx) dt - bc_~x d~

(34)

145

and the following stability criterion is obtained : Criterion 7

The null solution of the stochastic system (34) is mean square

stable in the large and stable in the large with probability one, if there exists a positive constant a such that the function

(1+as)q(s)/p(s)

is

positive real, and the gain k(t) is nonnegative and satisfies (with the same notations as above) k(t) ~ 2 ~ ( t )

~2

+ ~- (aq

p +q -r r ) n-2 n-1 n-1 n n-1

(35)

for all t. It is asymptotically stable if either of both conditions is true in the strict sense. R e m a r k 14

Criterion 7 reduces to C r i t e r i o n 3 i r a

= O. Otherwise the cond-

ition on G(s) is less restrictive, but the condition (35) is stricter. means of the techniques used in the stability analysis of deterministic

feed-

back systems the criteria obtained in this section can be transformed and extended in various ways. Here Criteria 6 and 7 are given to illustrate the possible criteria one can obtain. 5. DISCUSSION In this paper some stability criteria are derived for systems having a stochastic element in a feedback structure. The analysis has shown that the techniques for generating Lyapunov functions developed for the stability analysis of deterministic feedback systems also yield interesting stability results in the stochastic case. The analysis of this paper could be extended and generalized in various ways. Without conceptual problems, multivariable feedback systems can be dealt with; the path integral method is not well suited for this case, but the procedures indicated in Remark 6 and Remark 12 apply to multivariable

feedback systems as well. Discrete systems could also

be considered (Willems and Blankenship 197~); the analysis is even more straightforward,

since the subtleties of ItScalculus disappear. It would be

interesting to consider other types of noise (Blankenship 1972), and to see how the frequency domain conditions are affected. The weak point of the analysis of this paper is that only quadratic Lyapunov functions are generated, and all conditions are sufficient

for mean square stability. It would

be interesting to consider other types of Lyapunov functions in order to obtain less conservative conditions for stability with probability one, which would not necessarily imply mean square stability (Kushner 1967).

146

REFERENCES Brockett, R.W., Finite Dimensional Linear Systems , New York : Wiley, 1970. Blankenship, G.L., Stability of Uncertain Systems, Ph. D.thesis, M.I.T., Report ESL-R-448, June 1971. Blankenship, G.L., Asymptotic properties of stochastic systems : a nonlinear integral equation, Technical Memo 24, Systems Research Centre, Case Western Reserve University, Cleveland, Ohio, 1972. Kalman, R.E., Proceedings of the Nat. Academy of Science of the U.S.A., 49, pp. 201-205, 1963. Kleinman, D.L., IEEE Trans. on Automatic Control, AC-14, 429-430, 1969. Kozin, F., Automatica , 5, 95-112, 1969. Ku~hner, H.J., Stochastic Stability and Control, New York : Academic Press, 1967. Rabotnikov, Iu. L., Prikl. Mat. Mekh., 28, 935-940, 1964. Willems, J.C., The Analysis of Feedback Systems, Cambridge, Mass. : M.I.T. Press, 1971(a). Willems, J.C., IEEE Trans. on Automatic Control, AC-16, 621-634, 1971(b). Willems, J.C., and Blankenship, G.L., IEEE Trans. on Automatic Control, AC-16, 292-299, 1971. Willems, J.L., Stability Theory of Dynamical Systems, London : Nelson and New York : Wiley Interscience,

1970.

Zames, G., IEEE Trans. on Automatic Control, AC-11, 228-238 and 465-477, 1966.

STABILITY OF MODEL-REFERENCE S Y S ~ M S WITH RANDOM INPUTS

D,J.G.

JAMES

Lanchester Polytechnic, Coventry, England

I.

Introduction In recent years model-reference adaptive control systems have proven to be one

of the most popular methods in the growing fie ld of adaptive control.

The input to

the system is also fed to a reference model, the output of which is proportional to the desired response; the outputs of the model and system are then differenced to form an error signal.

Since this error signal is to be zero when the system is in

its optimum state it is used as a demand signal for the adaptive loops which adjusts the variable parameters in the system to their desired values. Various methods of synthesizing the adaptive loops have been proposed but the one that has proven most popular is that developed by Whitaker et.al. (1961) and referred to as the sensitivity or MIT rule.

Here the performance criterion is taken

as the integral of error squared and this leads to a rule that a particular parameter be adjusted according to the rule Rate of change of parameter

=

- Gain x (error) x ~ (error) ....

(~rameter)

Although the MIT rule results in practically realizable,systems mathematical analysis of the adaptive loops, even for simple inputs, prove to be very difficult. A stability analysis for sinusoidal input has been previously considered by the author (James 1969, 1971); however, in practice, a more realistic input is a random one and the purpose of this paper is to investigate stability for such an input. Since the object is to pose the problem and illustrate the difficulties involved we shall limit our discussiGn to a first order MIT system with controllable gain.

2.

L i n e a r stochastic,,,,,,,,,s,ystems All the stability problems c o n s i d e r e d i n t h i s

paper reduce to one of

investigating the stability of a system of linear differential equations with random coefficients.

A vast amount of literature dealing with such systems has been

148 published in recent years and various types of stability have been proposed (Kozin 1966, 1969).

In this work we shall confine ourselves to three types of stability,

namely, stability in mean, stability in mean square and almost sure asymptotic stability (a.s.a.s.) and will investigate such stability u~der two types of random coefficient variations, viz: (i) Gaussian white noise processes and (ii) Gaussian no n - wh i t e p r o c e s s e s . 2.| ~auseian white noise processes We shall be concerned with a system of equations of the form

w~.r.A.,

is an ~ . n

constant . ~ , ~

an lqXtl

m a t r i x and /~ = ~/~:0{I:)~

= ~ m } ,'

=

, ~ ,j = I, 2, ....-

1, 2 .... ,rl

an ~

column vector of

Gaussian white noise processes having properties

e[

o,

--

Such a system has been studied extensively in the literature (Ariaratnam and Graeffe 1965, Caughey and Dienes 1962, Bogdanoff and Kozin 1962) and we shall confine ourselves here to a brief outline of the method of stability analysis. The response of system (I) is a continuous n-dimensional Markov process and such processes are completely described by the Fokker-Planck equation ,1

rl

appropriate

81:-.o In stability

~i:--..o

6t;

i n v e s t i g a t i o n a knowledge of t h e moments o f t h e system r e s p o n s e i s

usually sufficient

and a system o f f i r s t

the moments o f o r d e r

K , X~a.. . . . . "~nKM entire

JC

state space.

N

order differential

equations determining

are r e a d i l y o b t a i n e d from (2) by m u l t i p l y i n g throughout by

i h(,+~a÷

"-

-+Kn=~

, and i n t e g r a t i n g by p a r t s over the

Necessary and sufficient conditions for stability in the mean

and mean square are then readily obtained by applying the Routh-Hurwitz criteria to

149 the system of differential equations obtained in the cases ~ = I, 2 respectively. 2.2 Gaussian non-white processes If the coefficient variations are Gaussian but non-white then the response of the system no longer forms a Markov process.

However it is possible to construct

linear time invariant filters which, with Gauesian white noise as input, will have the required non-white Gaussian coefficient variations as output.

The response of

the total system, which includes these linear filters, will then form a Markov process.

Unfortunately, the additional state variables introduced by the filters

render the system equations non-linear with the result that the moment equations, obtained from the appropriate Fokker-Planck, can no longer be solved recursively and one cannot obtain criteria for stability in the mean square. We shall be concerned with linear systems of the form

] where

A~. is an

processes and

~

nxrl

constant stahility matrix,

"

V~ ~ L~o,-,)

(3)

~A(~) are stationary ergodic

constant matrices.

Sufficient conditions guaranteeing a.s.a.s, of (3) have been obtained by many authors and the most recent improvements in the stability criteria obtained appear to be those due to Infante (1968) and Man (1970).

By applying the results of the

extremal properties of the eigenvalues of pencils of quadratic form Infants showed that a sufficient condition for a.s.a.s, in the large of system (3) is that

i.~-#

where .~..]~)

denotes the far, st ei~nv~ue of the ~tri~

symmetric positive definite matrix.

~'} and ~

is a

By simultaneously reducing two quadratic forms

to diagonal from Man extended the development ~

Infante to obtain the sufficient

condition far a.s.a.s, in the form II

i

(5)

~,---o

where

and G

are positive definite constant matrices satisfying

150

3.

,GaSn adaption model reference s~stem Consider a model and system to be governed r e s p e c t i v e l y by the equations

(6) where the time constant "I" and model gain K

are constant and known, but the

process gain Ke is unknown and possibly time varying.

The Iroblem here is to

determine a suitable adaptive loop to control K c so that

kv ~c eventually equals

the model gain ~ . The MIT rule gives

where 4.

= @Kv/~

, and this leads to the scheme of f i g . 1 .

Random Input

4.1 Stability analysis If to the system of fig.1 a general random signal g,.(l:J,@~l~ are zero and

when but ~

where

KV~a~ K

~(~

is applied at time

t=O,

and if subsequently K¥ remains constant

is adjusted according to (7) then the system equations (6) become

~£B-

K - K vK~(t~.

Despite the recent progress in stochastic stability theory, methods of investigating the stability of (8), where the system is not asymptotically stable when the noise terms are equated to zero, are not forthcoming.

However, digital simulation

of the system, which will be discussed in section 4.2, suggests that stability boundaries exist fcr such an input. we shall assume that

.he..

~ >>T

In order to have a first look at the probl~,

N

a~ A . ( . = o, I, 2, .... N)

are r~dem v a r ~ b l s s drawn from an

amplitude probability distribution l~(~). Substituting (9) in (8) and solving within the time interval ~ ~ ~ ~ • ( ~ . ~ furnishes the following recurrence relationship for

3c (~" "~..'~

151 Fig.1

First order system - MIT ~ain adaption

(gin(t)

K I+Ts

>

>

E)i(t)

gs(t)l Kv I +Ts

,ZT which on successive application leads to &,o

~T

Thus in order to investigate the stability of the system we must examine, by letting T-~ ~

the convergence of the infinite product

~T

+'=o

By considerin~ a large number of terms and their distribution and by considering the logarithm of the product of these terms we are led to consider the integral

If

~ >0

then ~

to zero,

~(~-bO

diverges and the system is unstable; if ~ < O ~

~1 ~oozand the system is stable.

If

~

then -~ diverges has a Gaussian

distribution with zero mean and variance o-z then (10) leads to the stabili~ criterion

~e-~ 7"Vs

1.2

157

Fig.3

Stab ilit~ boundaries for random input

5-

b-

T

Runge-Kutto

7%7 7Y 6 =175

CrGnk-Nicolson

\

.

0

I

28

!

!

56

84

!

1

112

140 >

-7-~-6

168

CD 0 b.) 0

o

c) c~

CD CD-

0

0

cD I

o

r

I

c--

c~ c-3 I

I

o

"lJ

"1

-1"1 o x-

o

o

I

I

Z_ °

o

?

0

(.11 0 I

(aL.) C~ C) I

~7Y8

?

laJ

9

o o

r.

o

159

The digital simulation results for the case when the input is purely random suggests that an outstanding problem is that of obtaining stability criteria for the linear system

~(k~ ~ A~t~ ~ )

, where the time-varying elements of

~)

are

correlated Gaussian non-white processes and the system is not asymptotically stable when the time-varying elements of A~)

are made identically zero.

ACKNOWLEDGMENT The author would llke to express his gratitude to Dr P. C. Parks, University of Varwick, for his helpful discussions concerning the work described in this paper. REFERENCES ARIARATNAM, S.T., and GRAEFE, P.W.U., 1965, Int.J.Control, I, 239; 1965, Int.J.Control, 2, 161; 1965, Int.J.Control, 2, 205. BOGDANOFF, J.J., and KOZIN, F., 1962, J.Acoust.Soc.Am., 34, 1065. CAUGHEY, T.K., and DIENES, J.K., 1962, J.Math. and Phys., 41 , 300. INFANTE, E .F., 1 968, A.S.M.E.Jour.App.Mech.,

5, 7.

JAMES, D.J.G., 1969, Int.J.Control, 9, 311; 1971, Amer.Inst.Aero.Astro.,

9, 950;

1972, Int.J.Control, to be published. KOZIN, F., 1966, Paper 3A, ~rd IFAC Congr., London; 1969, Automatica, 5, 95. MAN, F.T., 1970, A.S.M.E.Jour.App.Mech.,

37, 541.

PARKS, P.C., 1 966, IEEE Trans .Aut .Cont., AC-I I , 362. WHITAKER, M.P., OSBURN, P.V., and KEZER, A., 1961, Inst.Aero.Sci., paper 61-39.

REGIONS

OF I N S T A B I L I T Y

WITH RANDOM

FOR A LINEAR

PARANETRIC

SYSTE~

EXCITATION

W. WEDIG

Universit~t

Karlsruhe

(TH),

Karlsruhe,

Germany

INTRODUCTION The

dynamic

loads has b e e n leads

stability

to a linear

ternal

so-called

efficient.

second

order

stability

for

subjected

properties

are w e l l - k n o w n

by the M a t h i e u

flJ we k n o w those

diagram

frequencies the

and amplitudes°

system

becomes

bility

regions

quency

of the

If the stability The

zero

square,

equation

all second

stability

siam white The

trated

noise

(see fig. with

old value. constant

!),

But,

in the

of the above,

remain

of Gaus-

excitations,

above

the

It is p h y s i c a l l y

frequencies

such s t a b i l i t y

of the s t a b i l i t y

specillus-

density

suitab-

Comparing

we find the

of w h i c h vary

coefficients

in m e a n

case

spectral system.

her.

noise

the

coefficient.

zero and a flat

mits

for n o n - w h i t e

fre-

manner,

may be g r a p h i c a l l y

and we find no i n s t a b i l i t y that

insta-

f2J.

limit

excitation

of

itself.

coordinates

stability

obvious,

excitation

the natural

a stochastic

the critical

to harmonic

corresponding

said to be stable

condition

map m e n t i o n e d

excitations

important,

twice

of its state

frequency

is a uni-

in a stochastic

the m e a n value

if we r e c o r d

in contrast

and most near

has been a p p l i e d

stability

co-

of e q u i l i b r i u m

frequency

contains

to infinity

w h i c h has

stability

two,

is then

order m o m e n t s

by the n a t u r a l

the

above

definition

f3J,

for the

fluctuates

system

tends

corresponding

ly m u l t i p l i e d figure

mentioned of the

to the

of values

situated

ex-

force.

According

the natural

excitation

if the time

This

are

the

for periodic

the p o s i t i o n

The first

diagram

and near

external

when

bounded,

trum.

system

position

for w h i c h

unstable.

of this

ranges

axial

study

dependent

problems

to a time v a r y i n g

equation.

this

in w h i c h

as a time

such stability

system

fluctuating

cases,

equation,

appears

of this

Strutt

under

In simple

differential

excitation

example

column,

structures

investigated.

parametric

A technical

form p i n - e n d e d The

of elastic

extensively

this

same thresh-

we have here regions,

in a similar limits

must

equation.

a

the liman-

exist

These

li-

161

mits

are to be d e r i v a t e d

If we select

in the following.

STATE

EQUATIONS

the i n i t i a l l y

mentioned

a simple

example

equation

m a y be taken

of stability

problems,

column under

the

end thrust

corresponding

as

stability

in the f o r m

y ÷ 2w, O 2 . w$[1+ cx(t)]y = O. The u n k n o w n transverse

time f u n c t i o n motion

of the

parameter

dimensionless

coefficient

For lightly

damping

nential

of the

damped

as well,

excited

(E, DLLT).

then

~t is its n a t u r a l

problems

and w e a k l y to one

equation

external

in the m a t e r i a l

such m e c h a n i c a l

in c o m p a r i s o n

of this

column,

dimensionless a viscous

y(t)

(1)

excitation

by means of the

describes

frequency,

the ¢ is a

Ex(t) and D is a

of which we are a s s u m i n g

column.

we can restrict systems,

our interest

to

so that $ and D are small

It is then possible

to make

the

expo-

substitution

y(t) = T(t) expGcu~DO for

y(t}

and %o convert

the original

(2) equation

i) into

the

shorter

form

T÷~211÷ ¢,x(t)]T = e,=¢/(1-09, containing of the state

the small

system.

This

=

parameter stability

¢7 and the damped equation

natural

can be w r i t t e n

frequency

as a set of

equations

T-z~÷z 2, t= iv,(z~-z2), (3a)

z; where

z 7 and

z2

[zr

x rz, . z/2].

are the g e n e r a l i z e d

(3b) coordinates

of the

system.

162

LOW-PASS

We

shall first

Such p r o c e s s e s through

having

of all discuss

can be o b t a i n e d

a low-pass

a low-pass

by p a s s i n g

x(t)

process

~(t). noise ~(t)

G a u s s i a n white

=

filter

an a r b i t r a r y

ter's r e s p o n s e

PROCESSES

limit

frequency

is s t a t i o n a r y

~g. A f t e r

and its spectral

a short

time,

density has

the fil-

the w e l l -

known form

Herein

the

coefficient

its a u t o - c o r r e l a t i o n Since the input

x(t)

response

form M a r k o v p of all

= ~(~

ker-Planck

function

~(t)

z 2 and

3

i=l OZi

the

equation

the e x p e c t a t i o n

the c o n d i t i o n a l

z,

and

z2

probability

z 3 is given by the

a i

there

its

of (3) density

corresponding

available. values

and

Fok-

For

E z,l,

b, , --limS E[~zj Azl, •J

bgj are the

indicated

4t~.0~¢

J

incremental

moments

and

in [4].

our purposes,

of all state

of s o l v i n g however,

coordinates

the F o k k e r a knowledge

is sufficient.

of

Accor-

of these m o m e n t s =

to m u l t i p l y

a,- lira "

is no g e n e r a l m e t h o d

M.,.~.k7 E[z~'zsk~z~"] state

2

by the m e t h o d

ding to the d e f i n i t i o n

of the

coordinates

z -(z,;z2,z3 )r,

coefficients

computed

At present,

we have

of

is G a u s s i a n white noise,

state

L ~j=J uLiua~ i

p..- p(z,t;'Zo,to),

Planck

as the

~(t), so that E[~(tl)~(t2) ] = So6(tT-t2).

spectrum

equation

C1~

in w h i c h

is g i v e n by

Therefore,

zl,

3

m a y be

constant

of the f i l t e r

as well

processes.

coordinates

So is the

both

coordinates

L#z~z22z;~ p d~dz2dzs, sides

of the F o k k e r - P l a n c k

and to integrate

e q u a t i o n by p o w e r s

it over the

entire

state

plane. As a result equations

of this procedure,

of the moments,

which

we then o b t a i n

are best w r i t t e n

the d i f f e r e n t i a l

in m a t r i x

form [5].

163

(~)

2z~ " = A kt~fk. Tlg=/ k(k_ l)M k_2* ¢~L~k., '

¥,-1Efz, z,#A, "

~,-

LE[z22z~]J

i, ,ool -kr~

They contain the d i m e n s i o n l e s s ance

~=

~ g ~ / 2 of the low-pass

are n o t e d above.

moments

of the system

limit frequency process.

0

.

-1

~g=~g/2~and

the vari-

The m a t r i c e s A~ and R of (4)

decision.

of the equation's

It becomes

of linear differential

evident,

is the exponential

If k is equal

vector are the second order

that these moments

general

are weakly

so that we actually have a

equations with constant

APPROXIMATE A sufficiently

process.

in-

(3), whose b e h a v i o u r we have to examine for the

coupled by E 7 with higher order moments, sequence

e-~

the power of the low-pass

the components

stability

,

The index k of the moment vector ~k is a positive

teger and indicates to zero,

0

0-i-k~J

coefficients.

SOLUTION

solution of the h o m o g e n e o u s

equations

(4)

fumction

l~k(t) = ~k exp(2~ 9~ with an amplitude multiplied equations

by

vector

2~.

(5)

~k and the u n k n o w n

W h e n we introduce

(4), we get an infinite

eigenvalue

9 adequately

this f u n c t i o n into the m o m e n t ~

system of h o m o g e n e o u s

algebraic

equa-

tions. ~C, '

2_~.E

For

~T equals zero,

~-~,.E

the latter can exactly be solved.

obtain three times the infinite magnitude

~- -k,7,,.~,,,

In this case we

of eigenvalues

164

and the

corresponding

c(O)

_2~ (k÷21)!

amplitude

vectors,

"n(n+l)/2]

as n o t e d

below.

~-',ik÷21.,=O, n-O,± 1, k~l--O~l,...

n(n-l)/2J The n u m b e r n has the roots tive

integer

plitude

proximate genvalue

The

£1 is not equal

solution

of its

can be solved It can easily

real

to zero,

part.

Thus,

the

the three for each posi-

comoonents

values

of

of the am-

k.

it is reasonable

to seek an ap-

method

the u n k n o w n

amplitude

expanding vector

of (5)

in power

eise-

El'

coefficients

results

(o(O)E.A JC(o + ~'.o-j)p(j)

¢[..

that

for i n c r e a s i n g

by a p e r t u r b a t i o n

small p a r a m e t e r

comparisom

which

are u n b o u n d e d

and determines

Ak, which we can obtain

9 and the c o r r e s p o n d i n g

of the

n =~I

values

matrices

k. It should be remarked,

vectors

In case

ries

three

of the diagonal

in the recurrence

2

~

formula

(,-~

step by step. be shown,

only

that

the first

the a p p r o x i m a t i o n s

eigenvalue

of this

has the greatest

eigenvalue

need to be

noted here.

9(')=[i In particular, vanishing These

the

solutions

and the others

results

contain

are c a l c u l a t e d

only positive

xp - l/(~÷i/p), are complex,

of an odd degree

conjugate

STABILITY As m e n t i o n e d

up to the fourth

real parts,

~p - I/?rt:i/p:

respectively

above,

because

are

approximation. the values

(p= 1,2)

complex.

IN M E A N

the first

of a p p r o x i m a t i o n

SQUARE

eigenvalue

has the greatest

real

of

165

part.

It therefore

order m o m e n t s with the

determines

of the

initially

state

the

increasing

equations

introduced

behaviour

of all secomd

(3) and it determines,

exponential

function

(2),

together

the b e h a v i o u r

of the moment

E[y2(t)]:E#z,+z2}~exp(-2u6DO of the

original

position

stability

of the system

equation

(i) will

(i) as well.

be stable

Consequently,

in m e a n

the

zero

square

fim=E[y2(t)]= k= const, if th@ value

of this root

is not

greater

than the

damping

coefficient

D.

2~9-2~D~0 In this

stability

solution

condition

of the first

we now have

eigenvalue.

to introduce

If we restrict

terms up to the order

~

tical v a r i a n c e

of the parametric

Upon density

(~I~} 2

computation

in a stability spectral

map

densities

point

system.

(fig.

This

excitation

we have

been

coefficients

The

value

to those

for the

cri-

e x(t) [6].

the critical

see that

however,

smaller

densities

of its limit the

near

twice

spectral

frequency of all

kmown

white

the

frequency

stability

of

con-

noise [3] and sto-

[7].

varying

spectral

demsities

do not

endanger

the stability

without

~o

critical

region,

the natural

is the already

the bandwiths,

may be,

envelope

of an instability

for G a u s s i a n

w i t h weakly

processes,

as much. spectral

derived

values

the limit

is s i t u a t e d

threshold

chastic

Low-pass

i). We

defines

of w h i c h

having

their

equation

2

it for the v a r i o u s

dition~

system

of this variance,

~ 2~(~,~

and we can plot

the

our solution

a quadratic

as well

~

lowest

, we obtaim

the approximate

the greater destabilizing

of the

the m a x i m u m

of

the

In

system.

166

I

0 1~

"

ol

4

Figure the case

of small

cess will

first

per p r o x i m i t y

limit

touch,

of twice

From

frequencies, and then

an i n c r e a s i n g

cross,

the natural

OF THE POWER

point

of view, which

For this

t e r m of the it since

ments's

the r e c u r r e n c e

to simplify

equations

= T and of the

purpose,

expansions.

formula

the real

dimensionless

add up two of these

Id

22

sum of the q u a d r a t i c

SERIES these

results

of course,

2

moments

are only en-

the power

series

(6)

to k n o w the

impossible

to cal-

(7) is too complicated. we must ~=

first

coordinates t/~

of the

so that we obtain

equation

of the proin the up-

of the system

conditions

state

velocity

equations,

limit

it would be n e c e s s a r y

It is,

this formula,

by t a k i n g

processes

spectrum

stability

frequency

under

In order

for the

the

the m a t h e m a t i c a l

are convergent. culate

m a p of low-pass

if we can determine

general

'

......

i: Stability

CONVERGENCE

sured,

~''---

transform

the mo-

of the p o s i t i o n system.

a single

We t h e n

differential

167

E[(~2 ÷J~2)x~ = Dk exp(2~ 9~ Ok and

the a m p l i t u d e

the

eigenvalue

9 of w h i c h we shall

investigate

in the following. The m i x e d moment, is smaller

than half

/E~

remaining

on the right

the absolute

side

of the

sum of the quadratic

If on the r i g h t - h a n d

side we n o w increase

by means

side decrease

eigenvalue,

simpler

(8),

moments.

~2)xk÷]//2 _~ E[~x~'~

and on the l e f t - h a n d the

equation

recurrence

by the

of this relation, we t h e n obtain

formula

=

for a m a j o r a n t

of the

t h a n zero.

coefficient

The

amplitudes,

the indices

Do and the

of which

eigenvalue

are greater

~ itself

are given

by

Applying

the p e r t u r b a t i o n

two terms

method,

we can exactly

~k " uk ;,(o)÷¢,u~ ;,(1)÷ E;T _2F,(2) u k +..., and must

then estimate

the h i g h e r

-(I) 2(1÷1), D;;÷,=(1,.2,1÷I)~ '/2~9 , -(0)

ed by c o m p a r i n g

and

the

we introduce

formula

terms

once

again. 1+

2'_1!,[i... j -I1, . _ .

1

21

/-I (2j-l)2j],

:n÷t) n

!

inequalities

of this

coefficients

of the same powers

this

and o b t a i n

~2,.J The most

second majorant

second majorant two inequations

important

part

n

2 _~ 2177 1..

H[1÷1+2i(2i.3)] - i=t F[[1÷2i(2i÷3)]-

i=1

D(°)..- 1, D.o(° 0 d-1,2...)

=(2n) n O~, =(1,2;1)1a; 21(a~/rle) 2nj;~[,

=(2n,I)

The r e s p e c t i v e purpose

the first

O;;.,=(l;2,1÷l)~(l"3)a,~(a'~"~, )2 j~.2[1÷2j(2j÷l")]" l..-O,l, J2~.., n..-1,2,3 ....

2/

O;,,(1,2,1)ax,

rence

calculate

of the' expansions

into the

I. For this

corresponding

for the amplitudes

of the first

12- .].

of

may be prov-

2t

2 .1

leads

inequality

.

.I .

l-

to

],

=" -=. 1"711 ÷ 2 i ( 2 i ÷ I ) ] , Z i=2

recur-

-'~-D~÷tJ

0,1,2..., n = 3/~,S...

(9)

168

and shall The result

be n r o v e d first

for

the

ln'7(n~l): In the (9)

of

are

to

p=n-land

easily

1"-2(n~-2):

p=n-3we

obtain

s u m on the

its two

able

{P can p=n-2in of

~-- O,

snd a double and

following.

coefficients

powers

1-I

case

glected we

two

in the

the

side.

on b o t h

Its

first

sum°

power

of

we

obtain

the

inequality

of

so t h a t

of the inequality.

1"-3(n~-3): E-'[1. 2i(2i.,. 3) ÷ 2j(2j .3)]- ~ (n-2) Z2i(2i+ n-, n-, 3) ~ ~.. 22i(2i+I) Similarly,

side

t e r m m a y be n e -

to a s i n g l e

sides

~j=l{i,q)

and

n-l~-n/2.

can be r e d u c e d

sums

calculated

sum on the r i g h t - h a n d

left-hand

last ~rms

compare

a single

be

i=l

Z 1=2

of the

coefficients

"

of a g e n e r a l

{.

n-I

n-1

In-p(n-~ p) : (n-p+ l) Z 2 i,(2 ~+3). . . Zip_fl2ip_2+3)_~ ~ 2 2i, . . . 2 ip.2(2ip_2+l).

,,~"J~".,.~p~--l ~-2J.,

It is o b v i o u s , smaller comes

that

n/2.

than

greater.

this

For

calculate

on the r i g h t - h a n d

(p-l)

side

finite

p-I

last

inequality

increasing

Consequently.

•

the

sums

side

is f u l f i l l e d ,

values

it m u s t

-

of p its

be p r o v e d

? 2ip_2(2ip .2+3) =- .

I ° 22i,(2~+3)... i7,...i~>.2=1 (J,,--J,-2) W h e n we

"-'~' f,i. .,p.2--2 .,,,.2:

on b o t h

a single

finite

left-hand

o n l y for p

is

side beequals

n.

:,;/).

~-;

sides

(n-p+1)

.....2,;

i,,. • JpL2=2

(g-

if

of t h i s

pro@uct

inequality , we o b t a i n

a n d on the

left-hand

products.

p-1

p-I

p-1

TT2i(2i.3)+ 17"2i(2i+3)+ ... + 172i:2i+3) ~ p TT2i(2i+l).

i=l(i*U

The

first

i=lfi.2)

two

terms

i=1 (i#p-1)

of t h i s

(2p÷1)/5 +(2p+1)/K The

other

inequality

By means is an u p p e r

may

of t h e s e

bound

+

sum

of the

suffice

...

we

= i--2

to v e r i f y ~-

be p r o v e d

results

Z

p/2,

the

inequality.

p..-3,,(,,...n.

in a s i m i l a r m a n n e r . arrive

amplitudes

at a c l o s e d

as w e l l

as of the

solution,

eigehvalue

self.

21÷I

oo I 2n n i (1,2;l+I)G, 0

the b o u n d n e s s

analytically.

different

-

w h e r e w(s)

n

XiqkLFk~(O)

t e r m of the c h a r a c t e r i s t i c

do is to c o m p u t e

Invoking

n

[

i n t e g r a l must v a n i s h

of the

system,

i.e.,

i ... + a2s+ a 1

for k+~=odd.

Frequency Domain Stability Criterion:

frozen

(25)

Here we a r r i v e

at:

Assume that the frozen system is stable.

Then the linear stochastic system of Eq. (1) is mean square stable if and only if n

i -

n

~ ~ (-I) m-k q k & K m > 0 k=l &=i

where Km =

(j~)m-l~(j~0

2d0a

-oo

ii.

There

is an a l t e r n a t i v e

to the N e v e l s o n minantal

calculations, A(0)

Aij(0)

After

criterion•

we can o b t a i n = 2n H

This will

w a y to c o m p u t e D(0).

& Khasminskii

performing

lead us

some d e t e r -

as*

(26)

n

= (_i) n-k 2 n-I Hij

where a n an_ 2 an_4

Hn

•..

i

an_ I an_ 3 ...

0

an

an_ 2 . • .

0

0

0

°.•

0

e n e n _ I en_ 2 ...

e1

0

I

an_ I an_ 3 ...

0

0

an

an_ 2 ...

0

0

0

0

aI

0

,

Hij

aI

=

...

in w h i c h *This is not difficult, but somewhat lengthy.

So it is omitted here (cf. [12,13])

179

em = It should for the

be noted frozen

its first

for

i + j : 2m

0

for

i + j # 2m

that

system,

H n is equal

fact

Algebraic Criterion:

order

H.. is r e s u l t e d ij (en,en_l,...,el).

Hurwitz determinant

from H

by r e p l a c i n g n Thus we can compute

Hij Hn

: (-I) n-k

gives

to the n-th

and that

row by a row vector ~ij(0)

The above

I

as

(27)

:

Assume that the frozen system is stable.

Then the linear

stochastic system of Eq. (I) is stable in the mean square sense, if and only if i n

(-1)n-k

n

(k+g=even) which is very relevant to Nevelson & Khasminskii criterion [14] Some other discussions on stability will be includes in appendices. 4. E Q U I V A L E N T 4.1

Derivation

of E q u i v a l e n t

the d i f f e r e n t i a l

equation

dnxe ~+ d tn where

b k ~ 0, and

In some Eq.(28)

instead

criterion

constant

of the

described

fl(t) is a Gaussian

white

it is of interest

form

covariance

selection

taken

here

e{X_e(t)}= {xe(t) , ~e(t)} that

noise

with E{B(t)fl(T)}= qS(t-T).

to be asked q should

here

of

is how the

be selected.

The

that

E{~(t)}

= coy

Eq.(29)

(28)

to use the r e p r e s e n t a t i o n

The q u e s t i o n

verified

by

dk-lxe

of Eq.(1).

coY

a system

coefficients

b k and the noise

for the

It is easily

Consider

k~I= b k dtk_-------i--: bo + B(t)

applications,

coefficients

System

with

n

SYSTEM

{[(t),

holds

(29)

~(t)}

for the

same

initial

condition

if we choose

bk = ak n q = qoo- 2 ~ qokmk(t)+ k=l It is also p r o v e d equivalent

that

in the sense

k = 0,i, n k=l

the constant that

...

, n

(30)

n ~ qk~[mk~( t)+ m k ( t ) m ~ ( t ) ] ~=I

it will

coefficient have

the

s y s t e m with same

Eq.(30)

correlation

is

matrix

180

and hence the same cross spectral densities this sense, the system defined by Eq.(28)

(in the steady state).

is said to be equivalent

In to

the original system defined by Eq.(1). Application to State Estimation

Briefly the problem is to estimate the

n state vector ~(t) of the stochastic

linear system described by Eq.(1),

provided that the m vector valued observation: d~(t) = H~(t)dt + Rdl(t) is available.

(31)

Here H and R are mxn and mxs matrices respectively,

and

~(t) is the s vector independent Wiener process. The equivalent

system to Eq.(1)

d~e(t) where

~ = (0,0,

is given by Eq.(28) with Eq.(30):

= A~e(t)dt + ~[aodt+ dB(t)]

... ,i)'

32

Similarly the equivalent

system to Eq.

31

can be expressed by dYe(t) = H~e(t)dt + Rd~(t) Thus the equivalent Eqs.(32)

33

Kalman filter can be immediately obtained based on

and (33) as dL(t)

= AL(t)dt+

~a odt+ K[d~e(t)- HX_e(t)dt]

(34

where K = PH'R -I

(35

= AP + PA'- PH'R-IHp + uu'q Since the actual observation process is given by Eq.(31), ~e(t)

in Eq.(34)

izable filter.

the process

should be replaced by ~(t) to obtain the physical realThus

d~(t) = [AS(t)+ ~ao]dt + K[d[(t)- H~(t)dt]

(36)

It is also not difficult to prove that the estimate x(t) governed by Eq.(36)

is the optimal linear estimate

The use of the equivalent

in the minimal variance

sense.

system provides with a simple method for

determining the optimal linear estimate of the stochastic

linear system.

In appendix A.3 extension will be made to the more genral case. 5. CONCLUSION Some analytical results have been presented here for the single input-single

output linear stochastic

system.

with the derivation of moment differential moment of output variables

is presented

The description begins

equation from which the second

in somewhat

compact form.

the correlation matrix can be expressed in a similar way.

~en

It turns out

181

that the output frozen

system:

spectral however

An explicit mean square

density

the magnitude

stability.

The resultant

for estabilishing

the stability computation

it is very doubtful

for the general stochastic

is increased

form of the condition

Thus the required However

has the same shape

vector

of the sigle

whether

case.

the equivalent

stochastic

system,

some simple

order

is presented.

be straightforwardly

extended

a simple method output

criterion

system.

does exist

to the feedback

et al [15].

system

is derived

as ones

to the state

Some of results to the more

the

conslderablly.

attempt

statistics

and its application

noisy observation

gives

Input-single

An alternative deterministic

that it has the same second

for quaranteeing

labour can be lessened

system has been made by Sawaragi

Finally

by the COnstant factor.

is obtained criterion

as that for the

general

in the sense

for the original estimation

presented case,

from the

here may

as indicated

in

appendix. APPENDICES A.1 Stochastic Liapunov Method i

Consider

the free system as

dx i = xi+idt n

(37)

dx n = -k~l[akdt+ An equiblium

solution

in question. probability tisfies

is evidently

speaking

the equiblium

LV(x)

if there

= - k(~)

Let the Liapunov V(£) where

of Eq.(37)

Roughly one,

exists

,

P is an nxn symmetric --

the first frozen

=

a Liapunov

system.

whose

solution

function

(Ax)'V

--

be of the quadratic

stability

is stable

V(~)

Assume

and

sa-

form

positive

definite

matrix.

Then

x + Pnn ~ ' Q ~

(38)

is related

that the frozen

to the stability

system is stable,

of the

then the matrix

so that

the covariance

from Eqs.(37)

with

> 0 which

(A~)'V x = - x'Qx

because

is

P = (Pij)

term on the RHS of which

P can be chosen

~=0,

< 0 (cf. Kushner[10]).

function

= x'Px

LV(x)

dak(t)]x k

matrix

(39)

Q is non-negative

definite.

It follows

(38) that

LV(~)

= (Pnn- l)x'Qx

from which the stability

condition

(40) is obtained

as

Pnn > I.

Thus the

182

problem is only to evaluate

the element Pnn"

From Eq.(39)

aiPjn + ajPin- Pi-l,j- Pi,j-i = qij

(41)

(i,j = 1,2,...,n) which is a set of equation putation,

for n(n+l)/2

unknown Pij"

By a direct

the Pnn can be solved as (cf. Nakanizo[12]) n n l)n- k 1 X [ (Pnn = 2 ~ n k= 1 ~=i qk~Hk~

Thus we arrive at the same algebraic should be noted that the stability lity with probability

criterion

(42) as presented

of second moments

in 2.

It

implies the stabi-

one.

Stability of r-th moments*

A.2

com-

It follows t

from Eq.(5)

that

~{V(x)} - ~{v(~o)} =I E{LV(x_)}dt O

If E{V(~)}

~ n~ (_l)m_kqkiKm k=l 4=1 (k+~=2m)

A.3 Application of EquivalentSystem to State Estimation system may in general be described by n dxi(t) =j=i ~ [aijdt+ dwij(t)]xJ+ dWio(t)

A linear sZochastic (47)

(i = 1,2,...,n) where aij are non-random coefficients, and wij(t)[i=l,...,n:j=0,1,...,n] are the standard Brownian motion processes. It is supposed that there is a set of noisy observations n

dYi(t ) = ~l[hijdt+ dvij(t)]xj+ dVio(t ) (48) j= (i = 1,2,...,m) where hij are constant, and vij(t)[i=l,...,n:j=0,1,...,m] are Brownian motion processes. Equations (47) and (48) can be written in vectormatrix form

d~(t) = A~(t)dt+ dW(t)~+ dWo(t)

(49)

d~(t) = H~(t)dt+ dV(t)~+ d~o (t)

(5O)

The incremental covariances of Brownian motion processes are given by ~{dwij (t)dWk~(t)) = qijk~dt (i,k=l,...,n: J,~=0,1,...,n) ~{dvij(t)dVk~(t)} = rijk~dt (i,k=l,...,m: j,~=0,1,...,m) and

184

~{dwij(t)dWk~(t)}

= Sijkzdt (i=l,...,n; J=0,1,...,n; k=l,...,m; ~=0,1,...,m) The present problem is to find the minimal variance linear estimate of the state x(t) provided that the process {y(T) t /to llx(t; Xo 'to ) II > c

Definition I n.

}

< c

(z. 3)

L zapunov Stability in the M e a n

The equilibrium solution is stable in the m e a n if the expectation exists and given

e > 0 , there exists

6(e, to) such that

llxo II< 5 implies

z ~t t s~p ll~t; x o, %)II } < ~ >~t ° Definition I

a. S,

c2.4)

A l m o s t Sure L y a p u n o v Stability

The equilibrium solution is said to be almost surely stable if lira

sup

P{ ilXo II ~o t >~t o llxlt; =o, to)II=

0 }= 1

12.5)

A l m o s t sure L y a p u n o v stability states that the equilibrium solution is stable for almost all sample systems.

This is the s a m e as saying the Definition

I holds with probability one. Asymptotic stability can be extended to the stochastic case in a similar way.

Definition IIp

Asymptotic. .Stability . . . . . .in . .Probability .

T h e equilibrium solution is said to be asymptotically stable in probability if ID holds and if there exists

5 > 0 such that

llx° II < 5 implies

191 lira T'*~ for any

{sup t~T

P

11 x ( t ; x o , t o )

II > ~

}

(z.6)

: 0

C > 0 .

Definition ~rn"

Asymptotic Stability in the M e & n

T h e equilibrium solution is said to be asymptotically stable in the

= e a n if ~

holds and if there e=sts lira E ~ sup

T~

T

D e f i n i t i o n TI

a.s°

IIx(t; % , % )

~ > 0 such that

II

llXo II < ~

~plies

: 0.

(Z.7)

A l m o s t Sure A s y m p t o t i c S ~ b i l i t 7

The e q u i l i b r i u m s o l u t i o n is said to be a l m o s t s u r e l y a s 3 a ~ p t o t i c a l l y stable i f

I

a. s.

holds and

P~.T-.~o

t>~T

llx(t;Xo'to)]]:°

~l

.

(z. 8)

T h e definitions above, as w e have stated before, are direct transitions to the stochastic setting of L y a p u n o v stability. These stability concepts are concerned with sample behavior on the half line. In the early stages of the development of this subject, m o s t studies w e r e concerned with the stability of various statistics of the solution process at a given time, rather than of the samples.

This is probably due to the fact

that it is easier to study statistical behavior than to study sample behavior. T w o typical stability concepts related to the m o m e n t s

are as follows;

[ 53, [15~ . Definition Ill. L yapunov Stability of the M.gan T h e equilibrium solution is said to possess stability of the m e a n if the expectation exists and lira ,,"Xo II ~

0 E t ll~It; x o, to~ I~ : 0

for allt >ito .

Iz. 9~

192 D e f i n i t i o n IV.

Exponential

The equilibrium

Stability of the M e a n

solution is said to possess

mean if the expectation

exists and if there

exponential

exists constants

s t a b i l i t y of t h e

a,

8,

5,

all

llx° II < 6 implies

greater than zero such that

E{ IIx(t;x o, to~ II} < ~ llXoII exp :- ~lt-t o) S for all t > t

iz 10~

o

Although the stability definitions IH and IV above do not appear to be as strong a restriction on the solution process as given in Ip-I a. s. ' IIp-IIa. s. ' there are significant implications in IIl and IV for s a m p l e stability behavior [ 16~, [ 17~ . H o w e v e r ,

stability of the m o m e n t s

alone does not always provide

a satisfactory intuitive basis u p o n w h i c h to judge the stability characteristics of the s y s t e m .

This can easily be illustrated by the simple first order linear

Ito differential equation, dx = axdt + ~x

w h e r e a,

dB

,

(2.11)

q are constamts, and the

process with

E{ Bzlt/} : t,

The solution process o b t a i n e d v i a t h e Ito c a l c u l u s

B -process is the zero m e a n

Wiener

~14~ . t o (2. 11) w i t h i n i t i a l c o n d i t i o n x ( o ) = x °

, is

as

x(t) = x ° exp [(a-qZ/z)t + v B(t) ] , with probability

one.

Furthermore,

E From

{

(2. IZ)

xn(t

t h e nt h m o m e n t s

,}

= x n exp o

E, o

of (2. 12) a r e e a s i l y s h o w n t o b e Z 2

/2)nt + ~

t

L . A

(2. 13)

(2.13), w e find that there is exponential stability of the n

a n d only i f

th

moment

if

2

a < ~

(n-n 2)

(2. 14)

Thus, for a < 0 , the first m o m e n t are unstable.

is exponentially stable, but higher m o m e n t s

F o r a < _(yZ , the first and s e c o n d m o m e n t s

stable, a n d higher m o m e n t s

are unstable, etc., etc.

are exponentially

It s e e m s

difficult to

associate a physical m e a n i n g to the s y s t e m behavior, k n o w i n g only that the

193

first m o m e n t N

moments

even m o r e

is stable but the s e c o n d m o m e n t are stable a n d all higher m o m e n t s

is unstable, or that the first are unstable.

To make

matters

interesting, it is well k n o w n that the s a m p l e functions of the B r o w n i a n

m o t i o n g r o w no faster than ~ t log log t , with probability one.

Therefore, the

stability of the s a m p l e solutions (Z. iZ) are d e t e r m i n e d by the algebraic sign of 2 ~2 a - Z , b u t (3. 1 5 ) , p o s s e s s e s

case is also illustrated development,

in

F i g . II.

dramatic

w i t h (3.15) since the

a horizontal

For further

asymptote.

details concerning

s e e [19] .

What has been accomplished that the sufficiency

b y (3. I6) i s a l s o c o m p a r e d

in this section?

conditions that have previously

Basically, appeared

we have seen

in the literature

for linear systems with stationary ergodic coefficients apparently are quite conservative w h e n c o m p a r e d with the yet to be determined, true stability region. Simply by adding slightly m o r e of the coefficient process, previous results. [iZ],[ 6 ] and

detail concerning the statistical structure

w e have achieved a rather dra/-natic advance over

The succession of stronger sufficiency conditions beginningwith

[ ii ] and n o w the conditions (3. ii), (3. 16) bring with t h e m the

question "where does it finally end? ".

W h a t is the true stability boundary for

the second order system (3. 6 ) with a stationary ergodic Oaussian coefficient? T o this question, unfortuantely, w e have no a n s w e r at the present time. But, let us return to the equality (3. 2 ) . U p o n dividing by t , w e obtain

log/Ix(t)lip log I~(oIIIp _

t

i = T

,~

t

J O

x'[(A+F(s))'P

+

(3.17)

P(A +F(s))] ds

x 'PX

If it can be established that the quotient on the left hand side of (3.17) remains negative as t approaches infinity, with probability one, then it m u s t follow that

199

liml itll 0

t'~c°

,

1318

with probability one, yielding the almost sure asymptotic stability of the equilibriu~ solution of (3. i), since (3.18) implies Lyapinov stability for linear systems. If the quotient remains positive as t approaches infinity, with probability one, the equilibrium solution is unstable. Hence, the algebraic sign of the limit, as t approaches infinity, of the integral on the right hand side of (3.17) b e c o m e s the necessary and sufficient condition for almost asymptotic stability. That is, if the limit is negative, w e have stability and if the limit is positive, w e have instability. O n e must, therefore, establish the existence of the limit with probability one, and then evaluate this limit. At this time, there does not appear to be an easy w a y to accomplish this for the systems that w e have been investigating in this section. However,

quite recently Khaz'minskii has recognized that such a limit

can be studied for linear Ito stochastic differential equations [ 20]. The next section is devot ed to applications of 14~az'minskii's results o

200 IV.

Linear It~ Differential Equations In this section w e shall be c o n c e r n e d

with linear stochastic Ito ~ differential

equations, w h i c h can be written in differential f o r m for i = 1. . . . .

~ , as

L b! n L dx. = Z x: d r + E Z ~ x. dB , z j=l I ~ r=l j=l ir j r where

b!1 ' ~ r

are constants a n d the

eesses f o r w h i c h

0

It is well k n o w n

B

(4.1)

are mutually independent W i e n e r

E

:

lt-sl

pro-

•

~ I ] that the~unique, solution process to the stochastic

s y s t e m (4. i) is a M a r k o v

diffusion process, with an associated generator < ,

defined by t Z ai~(x) i, j=l

u = (Bx, g r a d u) + where

&

aij{x) = E k, s =i

n rE= l

.2 x

(4. z) j

k s g i r ° j r x k Xs

and B = (b~) i.

U p o n applying the It~ differential f o r m u l a [22]

log llxll, where

x is the solution p=ooess to the system (4. iI and llxll is

the Euclidean n o r m

(x, x) I/2 , one obtains the expression in differentials,

d log IIx ll~(log IIx/I dt whero,

to t h e function

k = x/llxl!,

n

+ z

r=l

(o(r)L ~)dBr(t)

i

~ t r ) - - ( q r ) , i, j ; 1

.....

(4.3) ~,

a n % i s givenby (4. Z).

In particular,

~Y~

log llx II :

Q(~)

=

1

~

(B~, ~) + ~ - Z:l aij(~) - i,Ej:l

a i j (k) kik j

(4.4)

If w e substitute (4.4) into (4.3), integrate the resulting equation, a n d divide by

t , one obtains

201

log Ilk(t)II - log Ilx(o)U

f

1

J L(BX(s), k(s))+~i~=laij(X(s)

)

O

~

a

i, j=]

t + O

lj

(X(s))Xi(s)~j(s) as (4.5)

n

r=l

( a ( r ) k(s), k(s) dBr(S ) ,

which is the analogue to the formula (3. 17 ) of Section Ill. Thus it follows that the stability properties that w e are attempting to determine will be implied by the limit of the integrals on the right hand side of (4.5) as

t approaches infinity.

A/though, w e w e r e stopped at this point for stochastic systems of the type studied in Section Ill, K_haz'minskii has s h o w n the w a y through the d i l e m m a for systems of the type studied in the present section, chapter 6~. He recognized that the vector

[ see [ 20], and [23],

k = x/ llxll , w h e r e

process to (4. i), itself satisfies an ItAo differential equation.

x is the solution That is,

k

satisfies an equation of the form, dl = Al(k)dt + AZ(X)dB Hence, the sphere

(4.6)

k process is a M a r k o v process defined on the surface of the

IIX II = i.

ditions.

,

Furthermore,

this process is ergodic under certain con-

W e note, however, that the ergodic properties of the

determined by its singularities. determined by the nature of the larities.

In particular,

k - process are

the ergodic properties are

X - process in the neighborhood of its singu-

A singularity of a M a r k o v process is defined as a point at which

the diffusion component vanishes. singularities are the solutions to

For the X-process,

given by (4.6), the

AZ(X ) = 0 .

Since the second integral on the right hand side of (4.5) approaches

202 zero with probability one as t approaches infinity [Z3] it follows f r o m (4.4), (4.5) that

lira t-*~

log llx(t) II

- log t "

l[ x ( o ) Ii

= E ~Q(k)i"

(4.7)

-

with probability one, in the ergodic case. This is exactly the formula w e want since it yields the necessary and sufficient condition for almost sure sample asymptotic stability in terms of

the e x p e c t a t i o n E t Q ( K ) ) . If t h e e x p e c t a t i o n is n e g a t i v e , the d e s i r e d s t a b i l i t y property follows. probability one.

If the expectation is positive, the samples are unstable with W e m u s t evaluate the arithmetic sign of the expectation.

As a sim~e

example,

c onsider the first order linear ItAo equation

(2. II). F o r this case the generator is simply

~=ax

d ~

o"2 +-2--x

Z dE d-~x "

One easily finds f r o m (4.4), Z

Q(X) = (a - S Z - ) kz Recalling that l Z = 1 , w e immediately obtain the well k n o w n conditions >0

E~Q(X)} = ( a - 4 )

,

~2/2

a becomes

9t k'i"

dr: ÷

~,

Ki -~

i

~. ~'~.~j EL 9i.~-' i.,]

I"

K'i"

j

Nj

l" s and E2(t,t)__ = O.

Hence

b

E, ~c) = ¢p. e,cp { - C~-s~I - "/CF,~'~./~pct*- t'~.a,., = , Lz (t

Ez &l

-

_I

exp.(-zc~-s 0).

There is however an alternative input/output des-

cription which, although it has roots going back at least as far in time as do the concepts of transfer function and impulse response, has become particularly prevelant in the last half decade.

This descrlntlon gives the so-called

f!ankel matrix

of ZI defined by: cANB

CAB CAB

H ~-

: cANB

CA2B

...

:

.o° .°.

cAN+IB

.o•

cAN+IB

~.,IIO = [W(i+J-2)(0)] .

cA2N-IB

It turns out that many qualitative input/output properties of Zlare most easily described in terms of H. It is well-known that there exist many minimal realizations ~A,B,C}

of a given

G(s), W(t), or H, but that they all may be recovered from one of them by the transformation group {A,B,C} R nxn. the

~ {SAS-I,sB,CS -I} with S an arbitrary invertible e l ~ e n t of

The dimension of a minimal realization of a given transfer function is called

M~4illan degree. We will consider the following class of systems El:

Definition 2:

Z I is said to be

completely symmetric if m=p and*

The infinite matrix H is said to be

nonnegative definite (denoted by ~ 0) if all its

finite truncations are nonnegative definite, i.e. if N and for all sequences {zl}~'.

H = H' ~ 0.

N ~ z~ CAi+JBzj ! 0 for all i,j=0

256

The following lemma gives a very useful alternative characterization of completely symmetric systems.

Its proof, which is not germane to our purposes, is

an immediate consequence of some known facts in realization theory and is left to the reader. Lemma I: C(Is-A)-IB

I is completely symmetric if and only if its transfer function G(s) admits a realizatio~ {AI,BI,C I} with

=

A I = A~ and B 1 = C~.

Thus Z 1 is completely symmetric if and only if there exists a nonslngular matrix S such that SAS -I = (SAS-I) ' and SB - (CS-I) '

(nxn)

Completely svmmetrlc systems

have the property that the el~envalues of A are all real.

Thls Is in fact also the

case after applying symmetric feedback and it may be shown that E 1 is completely symmetric if G(s) = C'(s) and If A-BKC has real ei~envalues for all K = K'.

Note

also that Z Is completely svmmetrlc if and only if its transfer function admits the k Ri partial fraction expansion G(s) = i=l~ ~ with R i = R i' _> O. If m=p=l then ~lis completely symmetric if and only if the poles and the zeros of the transfer function G(s) are real and interlace, i.e. If ~i,%2 .... 'In are the Doles and If Zl,Z 2 ..... z r are the zeros of G(s), then r = n-l, h i and z i are real, and ~i > Zl > 12 > "'" > Zn-i > ~n"

Thls pole-zero pattern is illustrated in Figure 2. Im

•

n

Figure 2:

Zn-I

1

Re

i

Typical pole~zero pattern of a completely symmetric system,

Completely symmetric systems are a natural generalization of relaxation systems (see Wlllems

[1972]) which are completely svmmetrlc systems which satisfy the

additional stability requirement Re %[A] ~ O.

Thus E 1 is a relaxation system if and !

only if its transfer function admits a realization {AI,BI,C I} wlth A 1 = A I ~ O and B 1 = C I.

There are various other ways of defining a relaxation system.

It may be

The backgrou~materlal of realization theory used here may be found in Brockett [1970], Chapter 2, or Kalman [1969], Section i0.Ii.

257

shown that Z I defines a relaxation system if and only if H = H' ~ 0 end oH = ~ ' ~ O, where @H denotes the shifted Hankel matrix of Z I, i.e., ~ with the first block row (or column) deleted.

Alternatively,

Z1 defines a relaxation system if and only if

its impulse response W(t) = ceAtB is a completely monotonic ~unction on [0,~), i.e. dk W(t) ~ W'(t) and ( - l ) k - - W(t) > 0 for all t > 0 and k = 0,1,2, Relaxation dt k -_ ..- . systems play an important role in physics. They describe the response of various classes of systems

such as R-C and R-L electrical networks, viscoelastic materials

thermal systems, and chemical reactions. We now state the main result of this section. Theorem i:

Assume that Z I i8 completely 8y~ynetric and that K - K' almost surely.

Let ~max A

~{Xmax[A_BKC]}"

max

Then Z is almost surel~. asFmptoticall~ . . . stable if

m a x-- %1 with eaualitv holding if and only if K = 0 almost surely.

Note also that Theorem 2 is easily extended to the

case where K does not possess a density function. 4.

n-i qn-i s +'''+qo Let g(s) = n. n-i s *Pn_l s +"'+Po

and the zeros of g(s).

and let %1,...,%n and Zl,...,Zn_ I denote the poles

Thus %1 > Zl > %2 > "''>

% (K) denote the zeros of p(s)+Kq(s). n

Zn-i >

%n"

Let %max(K) - hi(K) > ...>

From root-locus considerations it is easily seen

260 that z i < li(K) < li(0) < li(-K) < zi_ 1 for K > 0 and i=1,2 .... ,n (where we have put n

ZO

~ and zn ~ -~).

n

Since -

ing uDDer bound for I

max

~ I i + Kgn_ I : - ~ li(K) we thus obtain the followi:l i-I (see Figure 3):


0

qn-i

which requires in particular that l I < 0. Examples:

i.

If K is uniformly distributed between the limits K_ and K+ then Z is

almost surely asymptotically stable if: Z+

Z_

+

g(z+) g(z_)

[Z_ /z+

The limiting behavior of I as K-~ +max

+ !

1 max where a =

0

This ine0uality is easily verified directly

where z+ ~ Xmax(K+) and z_ ~ Xmax(K_). 1 from the graph of f ( o ) g(~) . 2.

d~ -g(~) ->

is given by (see Figure 3):

for K + ~

Zl

for K + -~

i -Kan- i+~

n n-i qn-2 [ li - ~ zi = - - Pn-l" i=l i=l qn-I

Thus as K becomes more and more distributed

at large absolute values we see that almost sure asymptotic stability results if: n ziP+ + (i!iki

n-i -

[0

i~l= zi)P- - qn-I j_~

where P+ __A P(K > 0) and P_ A= P(K < 0).

K p (K)dE

For the uniformly distributed case studied in

Example i with K+ > 0 and K_ < 0 this condition requires n •

3.

n-I

K2

+

qn-1

Consider the equation studied by Infante [1968], p. ii:

. f (t)-8

n + Xc

< 0

;

B 6 = ~n-

},c

261 where 8, £, % > 0. problem.

This equation describes the kinetics of a simple nuclear reactor

It is easily seen that Theorem 2 applied to this case with 1

s+l ~

g(s) " £

s(s+ ~ +

~)

and k(t) - -f(t).

Thus almost sure asymptotic stability results if:

where p(.) denotes the density function of f. 1.2

A Frequency-D~ain

Stabillt X Crlterion

In this section we will derive another criterion for almost sure asymptotic stability of the system

Z.

We first recall the definition of a positive real

function: Deflni,,tion 3: s.

Let H(s) be a matrix of real rational functions of the complex variable

It is said to be positive real if H(s) + H'(~) ~ 0

for all Re s ~ 0, s ~ Doles

of N(s). There exist various equivalent conditions for positive realness. may be found in most books on electrical network synthesis [1957], Chapter i, or Newcomb

[1966]).

Such conditions

(se~ for example, ~uillemin

Positive real functions play a fundamental

role in the theory of passive systems, particularly in the analysis and synthesis of electrical networks.

They have recently also shown to be an essential tool for

obtaining frequency-domain stability criteria for feedback systems.

A time-domaln

condition for positive realness is given in the following lemma, the celebrated

Kalman-Yacubovich-Popov L e ~ a 2:

lemma:

Consider the minimal system: = Fz + Gv

and let ~ be a real number. there exists a solution

; w = Hz,

Then H(l(s-O)iF)-Ic is positive real if and only if

0 = O' > 0

to the relations:

F'0 + QF ~ -200 OC = H' For a proof of L e n a

2 we refer the reader to Willems

[19721.

262 The value of the above lemma in stability analvsls lles in the fact that the quadratic form induced by the matrix 0 yields a very suitable candidate for a LvaDunov function.

It Dlays a crucial role in the following theorem which is the

main result of this section: Theorem 3:

Let m = p.

Then E i8 almost surely a8ymptoticall~ stable i~ there exists

a constant (m~m) matrix A and a real n~mber ~ such that: (i)

(il) and

(iii)

Proof:

A + A'

> 0

;

F(s-O) ~ G(s-o)(l-A(s-o)G(s-~)) -I is positive real; ~{mln[~,~min[(K+K')(A+A')-l]]}

> O

o

We will ass~ne that (I-ACB) is invertlble and that the McMillan degree of

F(s) is n.

The general case may be resolved by a subsequent limiting argument which

is left to the reader• It is easily seen that F(s) is the transfer function of the system: = Az + B(v+AO)

w = Cz ,

;

or

= (A+B(I-ACB)-IACA)z+B(I-ACB)-Iv

; w = Cz .

This system is minimal since the McMillan degree of F(s) is assumed to be n. by condition

Thus

(ll) and Lemma 2 there exists a matrix 0 = O' > 0 such that [A+B(I-ACB)-IAcA]'Q+Q[A+B(I-ACB)-IAcA]

< -2~0

and OB(I-ACB) -I Let S be an invertible

=

C'

(nxn) matrix such that S'S = O and let x I = Sx.

The

equation for x I is given by: ~i

TM

(AI-BIK(t)CI)Xl

where A I = SAS -I, B I * SB, and C I = CS -1. (AI+C~ACIAI)' + (AI+C{ACIA1) ~ -2ol.

Moreover, B I = C~(I-ACIB I) and

Consider now the derivative of V(x I) = X~Xl+

y~Ayl, where Yl = ClXl' along solutions of the above differential equation. calculation using the above relations shows that: V(x I) ~ -2OXlXl-2YlK(t)Y 1

-2~V(Xl)+2y~(oA-K(t))y I .

A simple

263 Let %(t) = Imln[(K(t)+K'(t))(A+A')-i ] and let P be a nonsingular matrix such that P'P =A +A'

Since %(t) = %min[P-l(K(t)+K'(t))(P') -I] it thus follows that YiK(t)Yl

%(t)Yily I for all YI"

Hence

V(xI) ~-2OV(x I) + 2(o-l(t))y~Ay I • We now distinguish two cases:

(1) (li)

and

l(t) > C which imnlies V(x I) _< -2OV(x I) ; %(t) ~ 0 which, since V(x I) ~ Yi A YI' implies:

V(x 1) < -2SV(x 1) + 2 ( O - t ( t ) ) V ( x 1) = - 2 t ( t ) V ( x 1) • Hence

V(xI) ! - 2 m i n [ C , ~ ( t ) ] V ( x 1) and

V ( X l ( t ) ) _< V ( X l ( t o ) ) e x p ( -

2

f

t

t0

min[l,c(t)]dt)

.

By the ergodic hypothesis and condition (lii) this indeed imnlies that lim V(Xl(t))=O almost surely.

Thus lira Xl(t) = S l i m x(t) = 0 almost surely, which proves the t-~m t-~o

theorem. • Notes:

5.

If K + K' > £I > 0 almost surely and if G(s) is positive real then

Theorem 2 predicts almost sure asymptotic stability by considerin~ the limit o ÷ 0 and A ÷ 0.

In this sense Theorem 2 is thus a generalization of the circle criterion.

The advantage of the theorem is that it allows the gain K(t) to become negative provided however this is compensated by K(t) being sufficiently positive at some other time. One of the disadvantages of Theorem 3 is the inherent difficulty in verifying the average value condition from the distribution of K since %mln{[(K+K')(A+A')-I]} is a very nonlinear function of K.

In the scalar case however one may resolve the

various conditions in Theorem 3 much further•

Thus we arrive at the following

more explicit criterion for systems with a single stochastic parameter: qn_iSn-l+.•.+~ Theorem 4: Assu~e that m = ~ ~ 1 and let g(s) = C(Is-A)-IB = o sn+nn_Isn-l+...+po

denote the transfer function of ~I • Then Z is almost surely asymptotically stable if there exists a real constant 6 such that

(i) 8{min[B,K] } > C, ;

264

(ii) and

the pole8 of g(s)

(iii)

lie

i n Re s < - q n _ l 8 ;

the locus of G(J~-qn_lS) ,

-~ < ~ < m

does not encircle

or

intersect the closed disc centered on the negative real axis of i the complex plane and passing through the origin and the point - ~ . Proof:

By Theorem 3 it suffices to show that there exists a constant ~ > 0 such

that F(s-~) = g(s-~)(l-l(s-~)g(s-s)) -I is positive real and ~{min[u%,k]} > ~. that this implies u > 0. 1

g(s-O)

Now F(s-~) is Dositive real if and only if F-l(s-@)

Note -

r(s-~)

~(s-~) is ~ositive real.

Since F-l(s-O) = (___~i _ ~)(s-O) + q(s-~) with qn-i r(s) a polynomial of degree at most (n-l) it follows that X < ~ and that F-l(s-o) --

On_ I

i will be positive real for some % if and onlv if it is nositive real for ~ = ~ , an- I which is thus the optimal value of X to consider. The condition ~n-i > 0 follows from the frequency domain condition (ill) as a result of the behavior of g(Jm-o) for + ~.

Pick now s = 8qn_l.

In order to complete the proof of the theorem it suffices to show that F-I(s-G) = 1

1

g(S-qn-IB)

qn-I

hess

s + 8

is ~ositive real.

By one of the test of positive real-

this can be achieved by proving that Re F-l(s-O)]s=j~

~ 0 and (since F-l(s-o)

has no more zeros than Dole~ that the roots of q(s-o) lie in Re s < ~.

The real part

condition comes down to asking g(s-O)]s=j~ to have the non-intersection property stated in condition

(iii).

kq(s-O) lie in Re s < ~

By the non-encirclement

By letting k + m this implies that the roots of

for k > B.

q(s-O) lle indeed i n Re s < O.

condition the roots of p(s-o)+

By the non-lntersectlon property g(J0a-~) ~ 0 for

-co < ~ < ~ and we conclude that the roots of q(s-o) indeed lie in Re s < @ as desired. • Notes: 6.

It may be shown that conditions

(if) and (iii) of Theorem 4 will be veri-

fled for 8 ~ 81 if they are verified for 81 .

Thus the optimal 8 to consider is the

smallest number which satisfies condition (i) of the theorem. 7.

If K has density function p(K) then condition

h(S) A S

p(K)dK +

(i) of Theorem 4 requires that:

Kp(K)dK > 0 --co

dh(B)

N o w - - ~ - - - - ~ 0 , h ( 0 ) ~ 0 and h ( ~ ) = ~{K}. if

and o n l y i f

B > B*.

~{K} > 0, and i f

Thus Theorem 4 w i l l

Thus t h e r e e x i s t s

so, then there exists

predict

a B

a fl s u c h t h a t

such t h a t

almost sure asymptotic stability

h(B) > 0

h(8) > 0 f o r of I

265

if ~[K] > 0, if the poles of g(s) lie in Re s < qn_l ~* and if g(J~-qn_16*) satisfies the frequency domain condition of Theorem 4.

This procedure lends itself very nicely

to the graphical analysis illustrated in Figure 4.

h(fl)

Re

--- Im

/

-I/

Figure 4: E_xamples:

4.

qn-i s*)

I~lustrating the application of Theorem 4.

Assume that K is uniformly distributed between K

and K+ with K_ ~

and K+ + K_ ~ O.

Then B* = K+ -~IK~-K~ . Expressed in terms of the spread A K = K + - K K+-K_ (~and the mean M = 2 ....... this yields ~* = AK _ ¢~ )2 which in the range of interest AK * > M > 0 shows that ~ increases with AK for fixed M. 2 --

-

-

This again indicates the

destabilizing effect due to the uncertainty in K. 5.

Let Z 1 be a completely symmetric system as defined in Section i.i.

Then con-

ditions (ii) and (iii) of Theorem 4 will be satisfied as long as qn_l ~ < -%1 with %1 11 the largest pole of g(s). The stability condition then becomes ~{min[- - - ,K]} > 0 qn-I which is similar t~ but more conservative tha~ the condition obtained in Note 4. Thus Theorem 2 which only applies to completely symmetric systems gives a sharper stability estimate than Theorem 4 which applies to general systems. 2.

ANALYSIS OF THE MEAN AND THE COVARIANCE EOUATIONS

This last section of the paner is concerned with the stability analysis of the mean and the covariance of the state of ~ where K(t) is assumed to be a white stochastic process.

For simpliclt~ we will consider only the case in which the

266 process K(t) is scalar valued, but we will treat the non-stationary case.

If we

denote the mean of K(t) by k(t) and the variance by ~2(t) then Z is described by the stochastic differential equation: Z' : dx = (A-k(t)bc)x dt + q(t)bcx dR , where A ¢ R nxn, b ¢ R nxl , c ~ R Ixn, and ~ denotes a Wiener process with zero mean and unit covariance.

This stochastic differential equation is to be interpreted in

the sense of Ito and we will take it as the startin~ point of our analysis. It is well-known that if k(t) and q(t) are sufficiently smooth (e.g., locally integrable) then for all given X(to) there exists a unioue solution to Z' for t > t . --

o

Let ~(t) ~ ~{x(t)},

r(t) ~ ~{x(t)x'(t)}, and R(t) ~ ~{(x(t)-~(t))

(x(t)-~(t))'} denote respectively the mean, the second moment matrix, and the

covariance matrix of x(t).

These are governed by the equations:

= (A-~(t)bc)~

;

= (A-~(t)bc) r+r (A-~(t)bc) '+q 2(t)bcrc 'b ' and

;

R(t) = r(t)-~(t)~' (t),

with initial conditions ~(t o) = x(t o) and r(t o) = X(to)X'(tO)We will be concerned with the asymptotic properties of these variables.

The

relevant stochastic stability concepts are now defined: Definition 4:

square,

E' is said to be asymptotically stable in the mean, in the mean

or in the covariance if, respectively, lim ~(t) = 0,

lim r(t) = 0, or

llm R(t) = 0 for all given initial conditions X(to). t-~o It is easily seen from the relations r(t) ~ R(t)+~(t)~'(t) and R(t) = R'(t) ~ 0 that mean square asymptotic stability implies stability in the mean and in the covariance.

The stability of the mean is a standard deterministic stability

problem for which many criteria have been derived.

These criteria involve the trans-

fer function g(s) = c(Is-A)'~ and properties of k(t) as, for example, its bounds (e.g. in the circle criterion: see Brockett [1970], Section 35), bounds on its derlvative~ or its periodicity.

The stability of the differential equation which

expresses the evolution of the second moment matrix P(t) is much more intricate

267 to analyze and we will show how criteria like the multlvariable circle criterion may be used.

If q2(t) = 0 then its stability is equivalent to the stability of

the mean equation, whereas if q2(t) # 0 then more stringent conditions will have to be imposed. 2.1

Multilinear_System Theory It is easy to see that if x I and x 2 are vectors which satisfy the linear

equations: ;

Xl = Al(t)Xl and

;

x2 = A2(t)x2

XleR x2eR

nI n2

, ,

then the product XlX 2' satisfies also a linear equation, namely:

_dddtXlX2 " Al(t)xlx2 + XlX2A2(t)

"

By taking x I = x 2 we see that if x satisfies a linear equation, then so does XX

I .

This idea generalizes from auadratic forms to homogeneous p-th degree forms. These facts have been known at least since Lyapunov's thesis, but they have to the present time been used very little in system theory.

They may for example be

exploited in the minimization of homogeneous performance measures of degree p > 2 for linear dynamical systems. The above ideas may be used in setting up transfer functions for a class of bilinear systems. ~. M

We will make some use of the Kronecker product denoted here by

Thus the Kronecker product of M e R nxm and R £ R Dx~ is the element ~

R E R np:~nq defined by: mllR

M~ R~ ~2-~f-_ m22R mplR

mlqR

ml2R °.°

mp2R

m2qR

m

Pq

R

The main use of this notation is that if an (nxn) matrix 0 is written in lexo2 graphic notation as the n -vector

268

0v = c°l(qll' q12 . . . . . then (MQ) v ~ (I ~

qln . . . . .

qnl' qn2 . . . . ' qnn )

M)O v.

Consider now the following lemma: Lemma 3:

Let {A,b,c} be a minimal realization of the transfer funotion g(s) =

c(Is-A)-ib.

Then the differential equation: = AQ + 0A + bv' + vb'

defines a minimal realization on the

;

w = cO ,

dimensional space of symmetric (nxn)

n(n+l)

2

matrices of the transfer function: g[2l(s) = ( c ~ I + I ~ c ) ( I s - I ~ A - A ~

l)-l(b~

We will not give a detailed proof of this lemma.

l+l~b)

.

The proof exploits the fact

that the above matrix eauation describes the 5ilinear system d - - xx' = Axx' + xx'A' + bux' + xu'b' dt

;

yx t = CXX' where ~ = Ax + bu; y = cx. The dynamical system identified in the statement of Lemma 3 plays an important role in the analysis of the covarlance e~uation under cons~deratlon. from this lemma that controllability

and observabilltv will be preserved.

poles of g[2](s) are given by {%I(A)+lj(A)},

i,J=l ..... n.

convenient general formula for deriving g[2](s) from g(s). however, Example:

The

There appears to be no In a specific case

it is a relatively straightforward matter to calculate g[2](s). 6.

Let [A,b,c] be the standard controllable representation

[1970], n. 106) of Q(s) = 1 s 2+as+b

~[2] (s) = -

2.2

We know

Then "

i s3+3as2+(2a2+4b)s+4ab

s(s+2a)

The Circle Criterion for the Covariance Equation We now return to the covariance eouation: = (A-k(t)bc)F + F(A-k(t)bc)' + q2(t)bcrc'b'

?s

(see Brockett

269 which we model as the feedback system: E{ : Q = AO + QA'+bv'+vb'+bwb'

z~ : v=-~(t)y,

;

y - cO, z = c O c ' ,

w=~2(t)~

It follows from Lemma 3 that l I' is comnletelv controllable and completely observable. Let ~(s) =A ICll (s)

Gl2(S)]

LG21(s)

~22(s)j

,

where

y(s) = Gll(S)V(S ) + Cl2(S)W(S)

and

z(s) = G21(s)v(s) + G22(s)w(s) !

denote the transfer function of Z I.

G(s) = |

c~

c

,

It is easily calculated that G(s) is given by:

(Is-A@

I-I@A)

-1

[b ~ I + I ~ b

I b $ b]

L

Thus t h e s t a b i l i t y deterministic

of t h e c o v a r i a n c e

feedback system with

equation

is equivalent

to the stability

(n+l) feedback loons, with transfer

nj

of a

function

G(s) i n t h e f o r w a r d l o o o and R a i n m a t r i x k(t)I F(t) = 0

_~2 (t

in the feedback loop. The multivariable

circle criterion and its various generalizations

immediately applicable to this situation.

is thus

We will illustrate this only in the

simplest case.

Let If" If denote some norm on R n+l and let matrix norms be

induced norms.

The small loop gain theorem due to Zames [1966] thus leads to:

Theorem 5:

Assume that Re l[A] < 0.

Then ~' is asymptotically stable in the mean

square if:

( sup tl~(J~)ll)( -eo 0.

The qualitative similarity between

these conditions and those for harmonic excitation is worth noting.

Stability with probability one may be investigated by the Liapunov method as discussed in Kushner (1967).

For the system defined by Eq.(7), the

scalar function

VCa

n

o : 1

satisfies the requirements of a Liapunov function. stochastic Liapunov functions, i f ~ { V ( a ) } where~is

According to the theorem on

~ 0 then a ÷ [a:~{V(a)} = 0] w.p.l.,

the differential generator of the process a(t).

the necessary differentiation, we obtain

After performing

298 n n 2 [V(a)] = -¢ ~ ~ Aija. i=l j=l ) where the coefficients A.. are as defined by (8). 19 for asymptotic stability w.p.l are ~ Aij > O,

Hence, sufficient conditions

j = 1,2,...,n,

that is, k~. n k.. k.. k.. dil'- -!~! S ( 2 w i ) - 2 3"=i ~ 4wi 13( wI)i S~lj + ~ JISij) > O, i = 1 . .2. . 1 j~i

.,n.

(13)

For a single degree of freedom system, the result reduces to the condition for stability in the mean.

For w

O

= Wr ± Us, (13) leads to the pair of conditions

k k k rs ( rs + Sr)s(~r±~s) drr - 4~ " ~ - w

> 0

k k k sr sr rS)s(~ri~s) dss " 4-~- (~--- ± w

> 0

r

S

r

S

(i0)

s

(14)

r

Gzroscopic Szstems The class of systems considered has equations of motion of the form 2 2 ql + gSql - 29~2 + (~I -~ )ql + f(t)ql = 0

(1S) q2 + ~Bct2 + 2 ~ 1

2 2 + (w2-fi)q2

+ f(t)q2

= 0

where f(t) is a stationary stochastic process with zero mean.

These equations

represent the transverse flexural motion of a light elastic rotating shaft of unequal flexural rigidities carrying a single rigid mass at its mid-length and subjected to a randomly varying axial thrust, see for e.g. Dimentberg

(1961).

The coordinates ql' q2 denote the transverse displacement of the mass measured with respect to rotating principal axes. ~ is the angular velocity of the shaft and ~i,~2,

(~I < ~2 ) are the natural frequencies of transverse vibration;

represents the coefficient of internal damping, Fig.4.

8

299

{2

q2 ~

q

l

~l .... S'rJJi

Fig. 4

The eigenfrequencies ml, ~2 of the system (15) for ¢ = O, f(t) = 0 are given by the roots of the equation

. 2 2

2-2

(Wl+~2+2fl)~

+

(@a2)(@~2)

= O.

For real ~, fl should be outside the range ~i ~ ~

~ ~2

Considering only these values of ~ (for which the unloaded shaft is stable), Eqs. (IS) may be converted to standard form by the transformation

ql = al sin 91 + a 2 sin 9 2 q2 = ~lal cos ~I + ~2a2 cos 9 2 ql = alml cos 91 + a2~ 2 cos 9 2

q2

=

-elalJ1 sin ~1 - ~2a2~2 sin ~2'

where

91 = ~it+~1,

%

= ~2t+%,

--2 . 2 ~2, ~o1- (o~1-~)

--2

o~2 =

. 2 ~2. o~2- (COl-~ J 2~2

2~ 1

2f,~2 =

--2 . 2 2. a)2- (~2-~)

300 Proceeding now exactly as for non-gyroscopic systems and assuming the spectral density of f(t) to be eS(co), averaged equations may be set up for the squares of the amplitudes al, a 2 from which the following conditions for stability in the mean may be obtained: For co = O

2~.1

(i = 1,2):

2 2.2 col-co2J B > 16f~2~2 S(2~ i)

if ~ < col

i

2

2 2

(col-co2) B >

--2 --2 2 --2 S(2coi) 2 (col-co2+4~ )col

if ~ > co2"

For coo = col+co2: --2 2 2 > (colco2+Wl-~ ) 2 2 -4 (col-~) colco2 . • --

>

S(COI+CO2) if ~ < col

2 ~2.2~4

8 [colco2+col-~ ) ~ (~2_w~) [16~4_ .--2 --2.2~-(col-co2) Icolco2

S(COI+CO2) if ~ > co2

For coo = Wl-co2: . • --

>

2 ~2.2~4

8 Lcolco2-col+~ ) ~ (co~_~2) [ 1 6 ~ 4 _

--2 2 2 > (colco2-col+~) 2 2 -4 (~ -col)colco2

.--2 --2.2 ~-- -Lcol-co2J Jcolco2

S(COl-CO2)

if ~ < col

S(col-co2) if ~ > co2"

Sufficient conditions for asymptotic stability w.p.l may also be obtained as 22 before using the Liapunov function V(a) = al+a 2 but are not presented here. Again, the qualitative similarity between the stochastic and the deterministic results may be noted.

However, while in the case of sinusoidal excitation,

instability near coo = col+co2 is present only when ~ < col and that near coo=~i-$2 only when ~>co2' see Mettler

(1967), in the stochastic case there is

no such distinction; both forms of instability can occur in both ranges of values of ~.

301

CONCLUSIONS This survey has dealt with the stability of coupled linear oscillatory systems under stochastic excitation of small intensity.

The systems considered

are typically encountered in the study of the dynamic stability of elastic structural and mechanical systems subjected to randomly fluctuating loads. Certain similarities between the stability conditions for the case of stochastic excitation and those for deterministic sinusoidal excitation have been emphasized. The results presented here are obtained using the method of averaging which is valid to the first approximation only and hence correspond only to instabilities of the first order.

To obtain conditions corresponding to higher

order instabilities, higher approximations must be made in the solution of the governing equations in standard form.

It may then be possible to investigate

the influence, if any, of the values of the excitation spectrum at higher multiples and fractions of the natural frequencies and combination frequencies. Evidence that the spectrum at these frequencies may influence the stability is provided by some recent results of Bolotin (1971) for a single degree of freedom system excited parametrically by filtered Gaussian white noise.

For such a

system, differential equations governing the response moments were derived by Bogdanoff and Kozin (1962). Using a truncated linearised form of these equations, Bolotin has obtained approximate conditions for second moment stability and shown them graphically on a plot similar to the Strutt diagram for the Mathieu equation for various values of the damping parameter.

The stability boundaries reveal a

dip when the centre frequency of the excitation is close to the system natural frequency,

in addition to the one near twice the natural frequency.

While

these results do indicate a qualitative trend, the validity of the approximations made in obtaining them seem somewhat drastic and questionable.

It appears that

Wedig (1972) has also obtained similar results in his paper at this Symposium. Non-linear oscillatory systems under harmonic parametric excitation possess several complicated and interesting instability phenomena, Mettler (1965, 1967).

A corresponding general investigation of such systems under stochastic

excitation appears to be lacking. ACKNOWLEDGMENTS The research for this paper was supported (in part) by the National Research Council of Canada under Grant No. A-1815.

The assistance of Mr. D.S.F. Tam

and Mr. S.C. Oen in the preparation of the paper is gratefully acknowledged.

302

REFERENCES Ariaratnam, S.T. and Graefe, P.W.U., 1965, Int. J. Control, ~, 161. Ariaratnam, S.T., 1965, Proc. Int. Conf. on Dynamic Stability of Structures, Ed: G. Herrmann, Pergamon, 1966, 255. Ariaratnam, S.T., 1967, Proc. Can. Cong. Appl. Mech., Vol. III, 163. Ariaratnam, S.T., 1969, Proc. IUTAM Symp. on Instability of Continuous Systems, Ed: H. Leipholz, Springer-Verlag, 78. Bogdanoff, J.L. and Kozin, F., 1962, J. Acous. Soc. Am., 34, I063. Bolotin, V.V., 1964, The Dynamic Stability of Elastic Systems , Holden-Day, Inc. Bolotin, V.V., 1971, Study No. 6, Stability, Ed: H. Leipholz, Solid Mechanics Division, University of Waterloo, 385. Caughey, T.K. and Dienes, J.K., 1962, J. Math. Phys., 40-41, 300. Caughey, T.K. and Gray, A.H. Jr.,1965, J. Appl. Mech., 32, 365. Chelpanov, Dimentberg,

I.B., 1962, P ~ , 26, 762, English Translation,

1145.

F., 1961, Flexural Vibration of Rotating Shafts, Butterworths.

Graefe, P.W.U., 1966, Ing. Arch., 3_SS, 202. Gray, A.H. Jr.~ 1967, J. Appl. Mech., 34, 1017. Infante, E.T., 1968, J. ?ppl. Mech., 35. Kozin, F., 1963, J. Math. Phys., 42, 59. Kushner, H.J., 1967, Stochastic Stability and Control, Academic Press. Man, F.T., 1970, J. Appl. Mech., 37, 541. Mettler, E., 1965, Proc. Int. Conf. on Dynamic Stability of Structures, Ed: G. Herrmann, Pergamon, 1966, 169. Mettler, E., 1967, Proc. Fourth Conf. on Nonlinear Oscillations, Academia, 1968; 51.

Prague,

Samuels, J.C., 1960, J. Acoust. Soc. Am., 32, 594. Samuels, J.C., 1961, J. Acoust. Soc. Am., 33, 1782. Stratonovich, R.L., 1963, Topics in the Theory of Random Noise, Vol. I, Gordon and Breach. Stratonovich, R.L., and Romanovskii, Yu.M., 1958, Nauchnye doklady vysshei shkoly, Fiziko-mat. nauk., 3, 221, Reprinted in Non-Linear Transformations of Random Processes, Ed: Kuznetsov, P.I., R.L. Stratonovich and V.I. Tikhonov, Pergamon, 1965, 332. Wedig, W., 1972, Proc. IUTAM Symposium on Stability of Stochastic Dynamical Systems, Springer-Verlag. Weidenhammer,

F., 1964, Ing. Arch., 33, 404.

WAVES IN A ROTATING STRATIFIED FLUID WITH LATERALLY VARYING RANDOM INHOMOG~EITIES By Lawrence A. Mysak Department

of Mathematics and Institute of Oceanography University of British Columbia Vancouver, B.C.,Canada ABSTRACT

We discuss the propagation and stability of internal waves in a rotating stratified unbounded fluid with randomly varying buoyancy frequency, N • The first order smoothing approximation is used to derive the mean wave dispersion relation when ~ is of the f o r m N Z = No2(f÷~j~{) , where N o = constant~ O ~ £ z ~ and~ is a centered stationary random function of the horizontal direction X This form f o r . ~ represents a stochastic model of the lateral variations in the temperature and salinity microstructure in the ocean. From the complex dispersion relation, expressions are obtained for the phase speed change and spatial growth rate ( ~ 2); in particular, attention is focused on the asymptotic behaviour of these expressions for short and long correlation lengths ( ~ 3). i. INTRODUCTION It is now well established from observations that on the gross temperature and salinity depth profiles in the ocean, there are superimposed small step-like variations

(e.g., see Gregg and Cox, 1972, and their references).

That is, below

the surface mixed layer it is apparent that in many regions of the ocean, the density stratification consists of thin, sharply-defined homogeneous thicknesses varying from a few centimeters to several meters.

layers with

Consequently,

observed depth profiles of the Brunt-V~is~l~ or buoyancy frequency, which involves the vertical density gradient and which is of fundamental

importance in the theory

of internal waves, exhibit highly irregular fluctuations about their mean values (e.g., see Gregg and Cox, 1972; McGorman and Mysak,

1972,hereafter referrred to

as I). Although it is not yet fully understood how this so-called or 'microstructure'

arises in practice,

'fine-structure'

it is believed that both convective and

diffusive processes are involved in its formation (Gregg and Cox, 1972; J.S. Turner and C.F. Chert, private communication).

O~very

plausible mechanism is the so called

salt finger instability which is caused by the different rates of diffusivity of heat and salt (Turner and Stommel~1964).

If a warm salty body of water overlies

a cold, fresh and initially heavier body of water, then, because of the rapid diffusion of heat downward, the system becomes gravitationally unstable and long narrow convection cells (salt fingers) are formed. Under certain conditions the salt fingers become self limiting and a fairly regular step-like structure in the temperature and s~linity profiles is formed.

Fine-structure

suggestive of salt

fingering has been reported, for example, west of the Strait of Gibraltar in the depth range 1 2 0 0 - 1 5 0 0 ~ ~ which is just below the warm salty core of the Mediterranean

304

water (Tait and Howe, 1968).

The inverse situation, in which warm salty water

overlies lighter, cold fresh water, also appears to lead to layering.

In this case

sharp interfaces separating layers in turbulent convective motion are formed.

Such

layering has been observed recently in the Arctic ocean in the depth range 200500 ~

, which is just above the warm saline core of the Atlantic Intermediate

Water (Neshyba et aI)1971).

In an attempt to understand this inverse situation,

Drs. Chen and Turner have recently shown in the laboratory that horizontal gradients in the diffusing components can lead to a fairly well defined (though changeable) layer structure. A particularly interesting feature of the Chen~Turner experiments was the existence of horizontal variations in the microstructure.

This is to say, the

thickness of a particular layer varied considerably in the horizontal direction and often two layers merged into one.

Similar remarks also apply to the horizontal

behaviour of microstructure observations recently made in the Pacific off Vancouver Island (Nasmyth, 1973).

Thus, in view of the considerable observational,

experimental and theoretical interest in internal waves (see Turner, 1973), it appears worthwhile at this time to determine the effect of the horizontal variability in the microstructure on the propagation and stability of internal waves.

Below,

we shall discuss this problem via a study of the internal wave dispersion relation that is implied by a stochastic wave equation.

The starting point of our analysis

is the assumption that in the internal wave equation, the buoyancy frequency, ~ has the form

N where

% =

-

constant, O < f Z ~


----

d Z/ 4

(2.3)

0 and~Z={l~Iz-~z-{Z))O. The

the frequency passbands

~z~

O ' ~ v ~ / ~ (PBI) and

PBI is the usual oceanic situation.

Since#o~

kinetic energy is proportional to~o~Z

latter inequality holds for

~/o~

~Z

e - d ~ for N = ~

(PBII);

and the wave the factor e #~/z

( ~ = wave amplitude),

in (2.2) ensures that the kinetic energy is conserved. we do not set ~ = ~

~Z L

In the subsequent analysis

(the Boussinesq approlimation), so that (2.2), which we shall

call the deterministic solution, implies amplification for ~ > O propagating wave phase) and attenuation for ~ phase). In terms of the polar coordinates

(upward

(downward propagating wave

Ko)~ where(~=~o(dos~e))eq.(2.3)

takes the form

KO =

d / ~ 7 ( ~,.)Z g,o_gq,.-~ -- .f;~ 7--.,O)I/Z

The deterministic phase velocity thus has magnitude

C =ff/~ o

which is along the direction of energy propagation,

is given by

=

2-

(2.4)

and the group velocity,

q"

(2.5)

It is easy to show that the group velocity is always orthogonal to the local tangent of the O-= constant curves as given by (2.3); for relatively short waves (K~J/2~),

~C-o and~o D are essentially orthogonal.

306 For ~ 2

of the form (I.i), eq. (2.1) becomes a stochastic differential

equation for ~ ) ~ J

, where

¢ ~X) ~ j ~ ) = ~ X ~ ) e -,/- /A'~) ~.J = 0

:~

:

where

-i~-~ ~ - ~ O ) :

al

-

-

(2.6)

a e,

l

~ p C < ~ ( e ~- :2")-' "-

M

.[

(2.7)

are r e s p e c t i v e l y d e t e r m i n i s t i c and s t o c h a s t i c d i f f e r e n t i a l o p e r a t o r s , a n d < j ~ } = O l where < . >

denotes ensemble average.

According to Keller and Veronis (1969), (2.6) admits plane wave solutions of the form ( ~ > - - ~ ÷ ~ is correct to O ( f 2 )

provided that the following dispersion relation, which , is satisfied:

(2.8) where

Do

-~ e - ~

e ~

q, = : . : k x r . e ~ ) and ~ - 1

is the i n t e g r a l operator oO

de:

(z.~)

--o0 in which ~

is the following Green's function:

~ :×,~)

- g 1

310

From the above tables we thus draw the following geophysically relevant conclusion:

For PBI internal waves and for short correlation lengths (which is

the case in practice f o r ~ = / ~ )

(see I) and probably also the case forl/~

=~{X)j

although this point needs to be confirmed), vertical and lateral random fluctuations in

~

have the same average effect on the propagation and stability of the

waves.

The random fluctuations make the waves travel slower and enhance the

deterministic growth/decay behaviour. Acknowledsements This paper was partially supported by the National Research Council of Canada and was written while the author was a Senior Visitor during 1971-1972 in the Department

of Applied Mathematics and Theoretical Physics, University of

Cambridge.

REFERENCES Frisch, U. Probabilistic Methods in Applied Mathematics,

Vol i. (ed. A.T. Bharucha-

Reid), Academic Press, New York (1968) Gregg, M. C.jand Cox, C. S. Deep-Sea Research, 19, 355-376 (1972) Keller, J. B.~and Veronis, G. J" Geophys.

Research, 74, 1941-1951 (1969)

McGorman, R. E.~and Mysak, L. A. Geophys. Fluid Dynamics, Nasmyth, P. In proceedings University,

in press (1972)

of 'Fourth Colloquium on Ocean Hydrodynamics',

Liege

in press (1973)

Neshyba, S., Neal, V. T.~and Denner, W. J. Geophys. Research, 76, 8107-8120 (1971) Tait, R. I.~ and Howe, M. R. Deep-Sea Research, 15, 275-280 (1968) Turner, J. S. Buoyancy Effects in Fluids, Cambridge University Press, in press (1973) Turner, J. S.jand Stommel, H. Proc. Nat. Acad. Sci. USA, 52, 49-53 (1964)

THE STABILITY

OF A SATELLITE

BY THE FLUCTUATIONS

WITH PARAMETRIC

OF THE GEOMAGNETIC

EXCITATION FIELD

Peter S. Sagirow University

The motion

of a satellite

field by a satellite domains ferent

1. Satellite Satellites

stabilized

fixed magnetic

corresponding definitions

of Stuttgart,

to different

with magnetic

field in two different

ways

interaction duced

H

a damping

to

moment

with respect

depending

above are described

(1) ~

properties

k

and

As these

velocity

by the

to

in-

currents H~ .

are

It is

of the satellite

in a circular

equatorial

caused by the moments

orbit

mentioned

[6]

~

is the moment I

is caused

field.

is a positive

of the shell,

in the pitch

B

rod inter-

proportional

and the eddy currents

+ kH2Y ' + HI~inY

is the yaw angle,

magnetic moment

is proportional

of the satellite

B~"

and to dif-

by the geomagnetic

by the field.

on the angular

by equation

models

A second moment

of the earth

case of a satellite

the yaw oscillations

to the yaw axis,

a restoring

the second moment

In the most simple

Here,

causing

to the geomagnetic

The stability

are compared.

are influenced

of the satellite

H

integral

[1]. The stabilizing

the field

in the shell

proportional

linearization

of the field.

between

to the geomagnetic

Field

elements

acts with the field directly to the intensity

with respect

rod is considered.

of the stochastic

in the Geomasnetic

Germany

constant

of inertia with respect

depending

is the moment

axis of the satellite.

on the magnetic

of the magnetic

The intensity

H

rod fixed

can be described

by

(2)

H = Ho ( t * ~ )

where

H o = const

field.

Transforming

( 3)

where time.

~

denotes

the fluctuations

the time and introducing

of the geomagnetic

the constant

c = k ( FIo)'/~(B I ) v z

we obtain

(4)

and

the normed equation

Y +cC1+~)~ 9 +(I+~) si~ = 0 the dot denotes

the derivative

with respect

to the transformed

312 2. Linearization

of the Nonlinear Noise Term

Suppose the fluctuations The nonlinear term

ferent ways. Assuming v

by

~

q(t)

to be Gaussian with

v : (I+~)2

= 1+2~ .

~

: I+2q +~2

E~ : O, E ~ ~ : ~ .

can be linearized in dif-

to be small we can neglect

~z

and replace

A more sophisticated method is the statistical

earization proposed by Kazakov and determine the constants

Ev=Ev

[3]. We replace

~i • ~i

v

by

lin-

v i : ~i+ ~i~

either by the requirements

E(v - Evil= E(v-Ev)

,

or by the requirement I

E ( v - v a )~

In the first case we obtain case ~

~3 : I + ~ Z , ~3

= ( ~ z + ~3)/2

The coefficients

•

2 Thus,

"-

rain.

~, : I+~'

~z: K 4 + 2 ~ i

A further possibility

and in the second

[3] is

~

1+~ z

four linearized models are obtained:

~i , ~i

are given below:

mode I

'~i

~i

1

t

2

2

4+~ z

3

I + 6 '2

2

4

I ÷g2

4 • ¢FVFiT

3. Introduction of the White Noise The fluctuations ing frequencies

q (t)

of the geomagnetic

and amplitudes

field have rapidly

[2] and can be approximated

chang-

by white

noise

where

w(t)

is a normed Wiener process with

Then• equation

(6)

Ew(t)

(5) reads as

~ + c(~i+~i6w)6~ +(1+6~vlsinW " 0

= O, Ew2(t)

= t

3t3 or written correctly as a stochastic

(7 ) where

=

dx

( "

- sin x~ - c=i x=

x : (x~,x 2)'r

)

: (q~, ~)T

For small oscillations

the eqs.

system

df

+

and

T

(

°

)

-~'sinx~-~ex

z dw(~')

denotes transposition.

(6) and (7) can be linearized:

(8) or

(9)

=

4. Stability

(° -1

-c=i

° 1

×dw~

-~1~i¢

in the Mean Square

The mean square stability Khasminski

-0"

can be checked by the criterion of Nevel'son-

[5]. Interpreting

obtain the stability (10)

eq.

(8) in the sense of It8 we directly

condition 2c~i

>

c 2 ~ i = ~ z + 6,z

To the linearized models derived above correspond

the three explicit

stability conditions

~




0

I shows the stability domains in

the (c, ~z)_plane. Interpreting eq.

(8) in the sense of Stratonovich

to derive the equivalent

(11)

It8 equation.

[7], first we have

This equation is

~*('r..,(xi-~6'~'CZpi~'*gc~iw ) ~ + ('J-~i'62"Cl~i+~v)~

and leads to the Nevel'son-Khasminski

The stability domains

stability

condition

corresponding to the different

models are shown in Fig. 2.

= 0

linearization

314

3 0.5

2 1

0

m

0

1

2

3

Fig. I: Domains of stability in mean square for eq. interpreted in the sense of It8

4-

c

(8)

0 l,

0,5 3 4 2

.....

I

1

....

I

i

2

3

Fig. 2: Domains of stability in mean square for eq. interpreted in the sense of Stratonovich

,,,,

,i

4

(8)

c

315

5. Stabi!it[.in

Probability

The necessary and sufficient

condition for stability in probability

can be obtained introducing the polar coordinates = tan-4(x4/xz) and

~

and considering the processes

= inr(t)

r = (x~ +x ~ )4/2 , ~

= ~(t)

on

r : 1

This method is proposed in [4] and leads to a sta-

bility condition in integral ficulties evaluating

form. However,

the occuring integrals

due to the numerical difthis condition turns out

to be non effective. Sufficient

stability conditions

techniques.

can be obtained by stochastic Liapunov

As Liapunov function for the linear system (9) can serve

the quadratic

form

v(x) = Ax~ + Bx4x , + Cx~

(13) The stability

condition depends on the constants

A,B,C

and is de-

rived applying the L-operator to (15) and requiring the positive definiteness

of the functions

-Lv(x)

(~4)

v(x)

and

= ( 5 - C G i ) x~ + (Bc~ i - 2A+2C-2C~2c~i ) x , x z + * (2c~;

- cc~i~

However. the best possible choice of

~ - 5 ) x~

A.B.C

leads to the old con-

d i t i o n (I0). Considering the stability of the nonlinear system (7) and using the Liapunov function

(15) with

A-c~i/2 the stability

B=t

in probability

once more condition stability

,

,

C=O+cZ~)/2c=i,

D=~C(1-~2c~i)

can be shown to hold for

~

satisfying

(I0). This result follows also directly from the

of the linearized system.

6. Conclusions All results obtained agree qualitatively with the general insight in the considered motion. c ~&

and

c~

,

All stability bounds

e.g. for

k40

~2(c)

tend to zero for

(vanishing damping)

and for

I ~O

(vanishing magnetic moment of the stabilizing rod). The maximum of 62(c)

which must be expected between the mentioned minima occurs for

all models in the relatively small intervall and

k

the value

c = 0,4

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