On the boundeness in the mean square of the forced oscillations of linear systems with stochastic coefficients

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ARCHIEF

i

Appi. Sci. Res. 20

Technische Hogeschool

DeIft

March 1969

ON THE BOUNDEDNESS IN THE MEAN SQUARE

OF THE FORCED OSCILLATIONS

OF LINEAR SYSTEMS

WITH STOCHASTIC COEFFICIENTS

A. A. F. VAN DE VEN Dept. of Mathematics Technological University, Eindhoven, THE NETHERLANDS

Abstract

Systems of linear differential equations with parameters that are nonwhite,

Gaussian random functions are studied. Conditions sufficient for the

bounded-ness in the mean and in the mean square of the solutions are derived. A

second order system is given as an example to illustrate the methods used. § 1. Introduction

In recent years a substantial amount of study has been devoted to systems of linear differential equations with stochastic coefficients. In most of these systems restrictions are made by considering coef-ficients describing Gaussian white noise processes only, so that the

systems are Markovian and the Fokker-Planck equation may be

used (cf. [1][8J). By integrating this equation one finds systems of

first order linear differential equations for the mean and for the

mean square of the state variables. From these equations one can derive conditions for the stability in the mean square of the system, by using the Routh-Hurwitz conditions.

By means of a Liapunov operator for Markov processes Nevelson and Khasminskii [9] derived necessary and sufficient conditions for the stability in the mean square which are much simpler than those obtained by the Routh-Hurwitz conditions.

Investigations concerning systems with nonwhite coefficients are smaller in number. In 1963, Kozin [101 derived sufficient conditions for the almost sure stability of homogeneous stochastic systems.

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-Caughey and Gray [1 1] have extended the results of [10] and, in addition, they have given a generalization concerning

inhomogene-ous systems. But in the latter case they confined themselves to coefficients that are almost surely bounded, in other words,

Gaus-sian coefficients were excluded. Mehr and Wang [12] have

con-sidered the conditions for the almost sure boundedness of the so-lutions for the case that only the inhomogeneous part of the system is almost surely bounded.

In this paper we shall investigate systems with parameters which are nonwhite, Gaussian random functions. By writing the differ-ential equations as integral equations, and comparing the N-th

order system with a first order system we derive conditions that

are sufficient for the boundedness in the mean and in the mean square.

Differential equations with random coefficients arise in many fields, for example in mechanical, electrical, and chemical engi-neering. In mechanics a large variety of studies has been made, including random disturbances in rate gyros, randomly pulsating

loads on structures, control theory, rockets, and aircraft behaviour. We consider the following system of equations

+ ijxj = vj(t) - jj(t) x5,

(i,j = 1, ..., N)

(1)

where c are constants and the functions vi(i) and t) describe random processes with the following properties:

they are stationary and Gaussian,

the ensemble averages E[vj(t)] and E[5(t)] are zero. Writing (1) as an integral equation, we obtain

xj(t) = Fj(t) -f- G(t) - $ K(t, r) xj(r) dr, (3) o with F1(t) = N5(t) xj(0), (4) G(t) = $ N15(t - r) 'vj(T) dr, (5) o K5(t, r) = N11(t - r) ?7k;(T), (r t), (6) (2) while J with ô1 Usin expres: where Assum functic for the mi(i) In t for t bound proce will yi § 2. Fi Let where rando

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while N5(i) must satisfy

N1(t) + 1N1(1) = O, (7)

N15(0) = (8)

with ò = the Kronecker delta.

Using the method of successive iterations, we derive from (3) the

expression (cf. [131, 1.3) 00 t

= Fj(t) + «i(i) +

(- 1)n $ K.(t, r) .Fj(r) dr + n=1 O 00 i

+

(-1)' $ K(t, r) Gj(r) dr,

(9)

n1

O

where K°(t, r) is determined by the recurrence relations

K»(t, r) =

/KII,

ri) Kr1>(ri, r) dr1, (10)

K(t, r) = K15(i, T).

Assuming that the initial values xj(0)

are independent of the

functions Thi(t) and vj(t), and taking the average of (9), we obtain for the mean mi(i) of xi(t)

00 t

mi(i) E[xi(t)] = E[Fj(i)] + (- 1)n $ E[K>(t, r)l E[F(r)] dr +

ni

O

00 _t

I

V (-1)$E[K°(t,r)Gj(r)]dr.

(11)

n=1 O

In this paper we investigate the behaviour of the expression (11)

for t -- 00, and from this we derive certain conditions for the

boundedness in the mean of the solutions of the system (1). Before proceeding we will first pay attention to a first order system, which will yield some results useful for our further considerations.

§ 2. First order system

Let

+ o4 = V(t) - Y(t)

, (12)

where c is a constant and the functions V(t) and Y(t) describe

random processes which satisfy the conditions (2). in e-to 'S-m h r-th at n e, g r.

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The solution of this equation is

E(t) = E(0) e_0'0 + $ V(r) e_t_T)_Z(TM dr, (13)

o

where

Z(r, t) _{= $} Y(r') dr'. (14)

Averaging (13) gives

m(t) E[E(t)] = ni(0)

e_tE[e_O.t)] + $ etE[V(i-) e't] dr.

o

(15) Here it is assumed that the initial value E(0) is independent of V(t) and Z(t).

For the calculation of the expectations in (15) we need the

concept of the characteristic function. The characteristic function of a random variable

y

(yi, y ...yx)

is defined by

qy(uj, U2, ..., UK) =E[eIuhu5],

(i = J_1 j = 1, ..., K).

₍₁₆₎ If y is Gaussian with zero mean, then , has the form (cf. [14] p. 75)

Uj) = exp[Ajiujui],

(17)

with

= E[yjy1].

Taking K = I and y = Z(O, t) in (16) and (17), we obtain

j - q,(ui = i) = eu",

where

I,

= E[Z2(O, t)] = $ $ E[Y(ri) Y(r2fl dr1 dT2 =

2$(t - r) Ryy(r) dr,

with

Ryy(r) = E[Y('ri) Y(r2)],

r =

-Similarly, for

K = 2,

y = Z(r, t),

Y2 = we find

E[V(T) e_t] =

1 ((u u2) '\

2eu" i

_\

uU _Ju2=O w ai w F e' 1,

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p.75)

where

o

Writing (12) in the form of an integral equation, using the method of successive iterations and averaging, we obtain

g ,

m(t) =rn(0)e{1 ±

_{$ 5... $}

E[Y(ri)...Y(rn)]drn...dri}-n=2,4... 0 0 0 00

trj

t,,

-

_{S $} - Se

ti)

x n'=1,3...O O O X E[Y(ri) ... Y(rn) V(r+1)] drn+1 ... dr1. (24) In the first (second) series the summation must be carried out over

even (odd) n only, because of the fact that the expectation of an

odd number of Gaussian processes is zero (cf. [14], p. 82).

Comparing the expression (23) with (24) and observing that the value of in(0) is arbitrary, we arrive at the following equations

00 t ti tn-i

I +

$ $ ... $ E[Y(ri) ... Y(rn)] dr ... dr1

=

e,

(25) n='2,40 O O and 00 1 ti t $ $ ... $ E[Y(r1) ...

Y(r) V(r+1)] dr+1

... dr1

=

n».1,3...O O O = $ p(r) eT et< di-. (26) o (13) (14) rt)j dT. t g-,

=

E[V(r) Z(r, t)] _{= $} E[V(r) Y(r')] dr'

_{= $}

Ry(r') dr',

T o

and, in (20),

g g

=

E[Z2(r, t)] _{= $} _SE[Y(ri) Y(r2)] dr2 dr1

=

T 1.

=

2 ₅

(t - r -

T')

Ry(r') dr',

(21)

with

I (15) Rvy(r)

=

E[V(t) Y(t + r)].

of V(t) Hence, if we define

d the

(p(t) _{= $}Rvy(r) dr, o (22) inction

=

2_$(t - r) Ryy(T) dr, o equation (15) becomes

m(t) =

m(0)

-

$

q(r) eT

dr. (23)

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These two relations enable us to get an estimate of mj(t) from (II), as shown in the next section.

§ 3. N-th order system

Substitution of (10), (7), (5), and (4) in (11) yields the following ex-pression for mi(i)

00 t ,.j

mi(t) = N15(t) mj(0) + _{$ $} Ti) J\1J1k (TI - T9)

n=2,4... O O N1k(Tfl_1 Tj) E[7k, _?7knjn(fl)J N11(T) mj(0) dT ... dri 00 fT1 Yn

-

- r) NJ,k,(T1 - T2) n1,3.. O O O Njkn+i(T1Z - T+1) E[rik ?llCnjn(Tfl) Vk,(Tfl+1)] dri,(i, j, k1,

kn1, ji...fn = 1, 2, ..., N), (27)

with mg(0) = E[xg(0)].

In the sequel we impose the following restrictions on the system (1)

mj(0) m0, (i = 1, ..., N), (28)

IN11(t)

-L

et,

(i, f = 1, ..., N). (29) In (28) and (29), m0, w, and ß are real positive constants.

There exist stationary Gaussian random functions (t) and (t) with zero mean, so that

E[jg(Tl) kz(T2)]I E[rj(r1) 'ìj(T2)]

R(Tl - T2)

0,

E[ijg(ri) i'k(T2)]I s E[r1(i-i) v(r2)]

R(Ti - T2)

0,

and (30)

E[vj(Tl) vg(T2)]j E[v(ri) v(T2)] -_ R(Tl - T?) 0,

(i, j, k, ¿ = 1 ...2V).

The condition (29) means that the deterministic part of the

system (1), i.e. (1) with v1(i) = i(t) = 0, is stable.

If the conditions (30) are met, then

?71(TflI

E[ri)

... 7(T 0,

7g(Tn) Vk,(Tn+i)]l E[7J(Tl) ... ?7(Ti) V(Tn±1)] 0.

To zero nuni corr witl V mec Th k1, coi po en an w and (31)

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To prove (31) we use the fact that ij() and vj(t) are Gaussian with

zero mean. Then it is possible to write the average of an even

number (> 2) of functions as combinations of the two-dimensional correlation functions (cf. [14], p. 83). For example (Xj(t) Gaussian with zero mean)

E[Xi(ti) X2(t2) X3(t3) X4(t4)]

=

= E[X1(t1) X2(t2)] E[X3(3) X4(4)] + + E[X1(t1) X3(t3)] E[X2(2) X4(t4)] +

+ E[X1(t) X4(t4)] E[X2(t2) X3(t3)]. (32) With the conditions (28), (29), and (30) satisfied, the following inequality can be derived from (27)

Nm0 mj(tfl

et+

et x

C,) w t rl n-1 00 _N2n+1 n=2,4,., fl I Tn

JJ

Je+1)

OU O

...J E[(ri) ...

(T) ...dr1. (33)

The term N2n+1 _{in (33) stems from the summat.ions in (27) over}

k1...k, j,

...,

j, j (or k±1) from i to

N. In general, some of the

coefficients 1(t) or j(t) in the system (1) are zero. In this case it is possible to replace N2n+1_{by a smaller term (cf. the example at the}

end of this paper). Let

N2(T) Nv(r)

Y(i-) = , and V(i-) = , (34)

w O)

and define

t t

= $

R(r) dT,

and

F(t) = 2 $ (t - T) R(r) dT.

(35)

O O

\Vith (22) this leads to

N3 N4 9(t)

₌

_, (t), and /(t) = F(t). (36) w-00 _N2n+1

+

COn+l

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Hence, (33), together with (25) and (26), is readily found to imply

Nm0 N

rní(t)l e

e42 +J(r) eT e4'(T) dT.

₍₃₇₎

o

It is now possible to derive from (37) sufficient conditions for the

boundedness in the mean of system (1) for various forms of the

correlation functions. For example, choose

R(t)

= Po e'1t1,

R(t)

penhl, (38)

with Po' P and i real positive constants. Then (35) becomes

where C1 and C2 are constants and

We ca: square \Ve on TI fo f iì (i

(t) = -- (1

- e),

(ii ¡u In the

t - - (1 - et),

(for t 0). (39) to a se Substitution of (39) in (37) leads to § 4. E We ci

exp -ßt ±

- t -

(1 - e

Nm0

[

N4 (p

p _Pt)

₊

suspe w w2 u _¡u- _{of moi} t i N41p p

ßr +

r

-+

N3p0 1(1

- eM) exp [_

2

(I -

e_PT)}]dT( where

wu i

issta

fying

exp[(ß-2 t1+

exp - -- JTIdT=

Nm0 _{PN4) i} _N30

J

[

(

pN4\ i

Den (,J

j

_{jtw2J J} o we cai = C1 + C2 (40) pN4

ßo=ß

. ¡uw-Compc

In this way we have derived the following result:

The solution of system (1) is bounded in the mean for all t O

if the conditions (28), (29), (30), and (38) are fulfilled and if, in _{are no} (41)

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addition,

p ßo

/1 N

We can derive sufficient conditions for the boundedness inthe mean square in a way completely similar to the one demonstrated above. We omit the lengthy calculations and give the result only:

The solution of the system (1) is bounded in the mean square for all t O if the conditions (28), (29), (30), and (38) are

ful-filled and if, in addition,.

E[xi(0) xj(0)] z a, (i, = i, ..., 1V; a 0),

= IE[v(t) v(t + r)] P1,

ßw2

i 2N

In the next section an example is given to illustrate theapplication to a second order system.

§ 4. Example, second order system

We consider a mathematical pendulum with a randomly moving

suspension. This system can be described by the linearized equation of motion

+ 2b + [1 + (t)1 x = v(t),

(45)

where b is a positive constant, so that the deterministic part of (45) is stable, and the functions (t) and v(t) are random variables satis-fying (2) and (38).

Denoting

xl = X, we can rewrite (45) in the form

X1 = X2,

X2 = Xj - 2bx2 + v(t) - i(t) xi.

(46)

Comparing (46) with (1) we see that only the coefficients

i2i(t) = (t)

and v2(t) = v(t)

are not identically zero. Hence, the number N in (33) becomes 1.

X2 = X,

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For system (45) the expression (9) becomes = x() = N11(t) xi(0) + N12() X2(0) +

+ jN12(t -

T) v(r) dT + O 00 t ti Tn_i

+

(-1)nf$

...

S N12(t-1)

ii=1 (J G O

Ni2(T_i -

T) i(Ti) ... 7(Tn) X X {Nll(T) xi(0) + N12(T) X2(0)} dT ... (171 + co

in

Tn

+

(-1)n $ $ ... $ N12(t - Ti) ...

Ni2(rn -

T+1) X O O O

X (Ti) ... (Tn) V(Tn+1) dT+i ... di1. (47)

We assume that the initial values satisfy all conditions posed in

Sec. 3.

We investigate separately the cases

(1) 0 ( b i 1;

(ii) b = 1; (iii) b ' I

O < b < I.

In this region

N11(t) = et cos(Qt) +

ebt sin(Qt), Q = (1

-and (48)

N12(t) = - e-bt sin(Qt).

Q So,

N11(t) I = e_bticos(Qt) + sin(Qt) I

e,

IN12(t) = e !sin(Qt) e-u.

Hence, putting co = Q and ß = b in (44), we conclude that x(t)

is bounded in the mean square if

<b(I_b2),

(O<b<i).

(50) Ii b = 1. In thE and Usin when from and The N11 and N12 and In y ditio

Intl

e =

(11)

In this case

N11(t) = et ± t e,

(51)

and

N12(t) = t e.

Using the inequality

t e-t <

et,

for t 0, (52)

e(1 e)

where e is an arbitrary positive number smaller than 1, we infer

from (51)

Nn(t) <

e(1

e)

and (53) 1N12(t)!

e.

e(1 -

e)

In view of (44), with w = e(1 - e) _{and ß} e, we obtain the

con-dition

<e2e(1 e)2,

0<e< 1).

(54)

I-L

In the region O <e < I the function e(1 - e)2 has a maximum for

e = 1/3. Consequently, x(t) is bounded in the mean square if

2e-= 0.548, (ß = 1). (55)

¡i 27

(iii) b> 1.

Then

Nii(t)

_{= W'}

e__')t [b ± Q' - (b - Q')

e2Q'tJ, (Q' = (b2 - I))

and (56)

N(t)

_{= W'}

e

(I - e2').

(b+Q')

JV11(t)

e')t,

4 _2Q' and (57) N12(t) 2Q'

(12)

Hence, in this case (44) with w = 2Q' and ß = b - Q', leads to

<2(h2_ 1)(b(b2 i)).

(58)

Recapitulating, the following sufficient conditions have been found for the boundedness in the mean square of the solution of

system (45)

for 0<b<1,

IL

-<0.548,

for b = 1, (59) Ja <2(b2

1)(b - (b2 - l)),

for b> 1.

Comparing (44) with (42) we see that we can find the conditions

for boundedness in the mean by multiplying the right hand sides

of (59) by two.

These results are shown in Fig. 1.

Fig. 1. Diagram showing the sufficient conditions for the bou ndedness in the

mean and the mean square of system (45).

§ 5. Coni We hay mean sc lations that the are Gau in this ç the con ample. Received Bo cAu LEI GR ARI (19. Ws GR Gs al I NE Ko CA Min Mi LA. Hil

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I,

§ 5. Conclusions

We have derived sufficient conditions for the boundedness in the mean square, and, consequently, in the mean, of the forced

oscil-lations of linear systems with stochastic coefficients for the case

that the deterministic part ofthe system is stable and the coefficients are Gaussian. It should be borne in mind that the conditions given in this paper are sufficient onlyand not necessary. This follows from

the conditions in the neighbourhood of b = I in the foregoing

ex-ample.

Received 9 February 1968

REFERENCES

B000ANOFF, J. L. and F. KOzIN, J. Acoust. Soc. Am. 34 (1962) 1063. CAUGHEY, T. K. and J. K. DIENES, J. Math, and Phys. 41 (1962) 300. LEIBOWITZ, M. A., J. Math. Phys. 4 (1963)852.

GRAY, A. H. and T. K. CAUGHEY, J. Math, and Phys. 44 (1965) 288.

ARIARATNAM, S. T. and P. W. U. GRAEFE, mt. J. Control 1 (1965) 239, ibid. 2 (1965) 161, ibid. 2 (1965) 205.

WEIDENHAMMER, F., Ing. Arch. 35 (1966) 1. GRARFE, P. W. U., Ing. Arch. 35 (1966) 276.

GRAEFE, P. w. u., Linear Stochastic Systems,Mech. Eng, report M.K,-20, Nation-al Research Council of Canada,Ottawa 1966.

NEVELSON, M. B. and R. T. KHASMXNSKII, P.M.M. 30 (1966) 487. KoziN, F., J. Math, and Phys. 42 (1963) 59.

CAUGHEY, T. K. and A. H. GRAY, J.A.M. 32 (1965) 365. MEHR, C. B. and P. K. C. WANG, J.A.M. 33 (1966) 234. MIKHLJN, S. G., Integral Equations, Pergarnon, London 1957.

LANI o, J. H. and R. H. BATTIN, Random Processes in Automatic Control, McGraw-Hill, New York 1956.