Grasman, J. (Johan)

English

CWI's Institutional Repository

Estimates of large failure times from the theory of stochasticly nerturbed 
dynamical systems with and without feedback 
J. Grasman, Amsterdam 
Summary: In this naper the asymptotic theory of stochastic exit for dynamical 
systems with random perturbations is related to the aPalysis of life time of 
a physical system. In case of feedback the control is such that the exnected 
exit time is maximized. 
I. Introduction 
The life time analysis of nhysical systems is mostly seen as a ~ure 
statistical nroblem. In this paper we take a different approach and analyse 
the dynamics of a stochasticly perturbed system. 
In section 2 we characterize the type of noise that may act upon a 
system and show, by the example of a nonlinear spring, the bounds in state 
space that have to be satisfied in order to have a proper functioning physi-
cal system. 
In section 3 an asymptotic solution of the Fokker-Planck equation is 
presented, while in section 4 it is shown in which way this solution indi-
cates the most likely type of failure of the system. Moreover, it is demon-
strated how observed lifetimes for systems under strain (exneriments) can be 
used to predict life times under regular conditions. 
In section 5 we consider a linear control system and construct the feed-
back that maximizes the exit time. 
Finally, in section 6 stochastic difference equations are formulated, 
which are used in the Monte Carlo simulation of stochastic dynamical systems. 
2. Dynamical systems perturbed by noise 
We consider a system given by n state variables depending continuously 
235 
upon time and satisfying a system of counled nonlinear differential equations 
of the type 
(2. I) dx dt f(x), x(t) = (x 1 (t),. • .,xn (t)). 
Let for this system the origin x = 0 be a stable equilibrium point. Then we 
consider a bounded subdomain n of state space. This domain n contains the 
origin and no trajectories leave n for increasing t. 
Example I.I. A nonlinear snring satisfying 
(2.2) dz g(z, dt), p,(0,0) 0 
can be transformed into (2.1): 
(2. 3) 
(2.3b) 
Next we analyse the effect of small additive noise terms to the system 
(2. I). Thus, we will investigate a system of counled stochastic differential 
equations of the tyne 
(2. 4) dX. 
l 
m 
f. (X) dt + E l 
l j=l 
a .. (X) R • ( t) d t , 
lJ .l 
i J , ••• ,n 
with 0 < E << I. The noise terms R.(t), satisfying E{R.(t)} 
J J 
terized by the autocorrelation function 
T 
. I f 
= llm 2T 
T-><><> -T 
(2. 5) G(T) R(t)R(t+1)d1 
or the spectral function 
(2.6) S(w) f e-iwT G(T)d1, 
see Gardiner [I]. 
Q, are charac-
For G(T) = o(T) with o(T) the Dirac delta function, we have a so-called 
236 
white noise process: S(w) = I and all frequencies have equal intensity. In 
that case (2.4) is written as 
( 2. 7) dX. 
l. 
m 
f. (X) d t + E L G •• (X) d\J. ' i 
i j= I l.J J 
where dW. is the Wiener increment. 
J 
I,.~., n, 
Another possibility is to consider the noise as the output of a damned 
linear system forced by white noise. For one component this is a so-called 
Ornstein-Uhlbeck process: 
(2.8) dR -ciRdt + 13dW. 
It is easily verified that 
(2.9) G(T) 62 22· 
a +S 
Since in the spectral function the higher frequencies have lower intensities 
the process is called "red noise". It is noted that for a = R the red noise 
forces the dynamical system (2.4) with the same intensity as in the white 
noise case. Moreover, for a = S ~ 00 R tends to the white noise process. 
Example 2.2. We consider the nonlinear spring as part of the suspension of a 
car which is in a constant forward motion over a somewhat bumpy road, see 
fig. la. Let the spectral function of the random component of the force 
acting upon the sprinR be as depicted in fig. lb. Then we may take the random 
force as a red noise process. By using R x3/s we write (2.3) as 
(2. I Oa) dx 1 x2dt, 
(2.IOb) dx 2 -I {H g(x1,x2)+x3!dt, 
(2. IOc) dx3 -cix3dt + EcidH, 
which is a system of the type (2. 7). 
From point of view of life time analysis of the spring we 
237 
wish to have an estimate for the frequency by which the spring leaves the 
normal operation range. This range is determined by the requirements of 
bounded displacement and bounded acceleration: 
( 2. I I a) 
(2. I I b) 
r 
z + z 
rest 
s 
1 
w-
(a) The nhysical system (b) Snectral function of random 
force upon spring 
Fig. I. Influence of road upon spring 
3. The Fokker-Planck equation 
For the •tochastic state variables Xi(c) satisfvinR 
m 
(3. I a) dX. f.(X)dt + € l a .. clW., l l j=I lJ J 
(3. I b) x. (0) 
l 
= 0, I, a •• ,n 
the probability density p(x,t) of being at x at time t satisfies the ~okker-
Planck equation 
n 
:J 2n n ( 3. 2) rip 2 l I f.(x)n, p(x,O) 6(x) :it e: a .. ax. ax. ax:-lJ l i, j=I l J i=I l 
with m 
2 T a .. 0 ik 0 kj' lJ k=l 
238 
It is noted that we took oij independent of X, which is not an essential re-
striction for the method of analysis. It simplifies some of the comnutations. 
For t ~ ~ a stationary distribution p(x) exists with 
,., r~ ') n 
(3. 3) M sp ! € - l a. 3-p - - . f. (x)n o. ij (lx. ax. '· '.:ix. 1 i, j =I 1 
.1 i=I 1 
'faking a WKB-Ansatz, we assume that asymntotically 
0 
(3.4) -O(x)/c~ p(x) "'w(x)e · , ()(0) 0. 
Substitution in (3.3) yields 
n 3Q ClQ n aQ (3. 5) (" + \ f. (x) 0, ) a .. £ £ I £ i, J =I lJ l l J i=I l 
which is the eikonal equation known from geometrical optics. Its solution is 
positive definite for !xl > 0. For small values of Ix! we take 
(3.6) 
with p 
( 3. 7) 
where 
(3. 8) 
n 
n ( x) 
"' 
! ) 
i, ]=I 
H-1 satisfying 
FH + HFT + A 
A=ia .. ), lJ nxn 
P .. x. x. 
lJ l J 
o, 
F { "f. I , ;Jx~ 0 Jnxn" J x= 
To compute () for larger values of x we have to integrate along rays in state 
snace starting at a small sohere in the origin where (3.6)-(3.8) hold. The 
ray method is based upon the observation that we may write (3.5) as 
n n 
(3. 9) H(x,p) I a .. Pi pJ. + L fi(x)pi 
i,j=l lJ i=I 
and that along a ray 
o, P· l 
3Q 
~ 
l 
239 
dx. 
3H (3. I Oa) l cl;- dp:- , 
l 
dp. 
aH (3. I Ob) 1 cl;- -~ , 
l 
dO n (3. !Oc) d~ l a .. p. p.' i=I lJ 1 J 
see Ludwig f4] and r.rasman and Lankelma f2]. 
4. Expected exit time and life time 
In this section we construct an asymptotic estimate for the exnected 
time of residence in the domain n. 
( 4. I) 
Let ()(x) take its minimal value in an at x and let 
K min ()(x) 
:rn 
Q(~), 
then x r an is the most likely point oF exit and for E + O the distribution 
of exit points tends to li(x-~), see Annendix A. Thus, in case of failure of 
a system, we may conclude in this way about the most likely type of failure. 
The expected time, needed to reach the boundary, is asymptotically 
( 4. 2) T O<r:<<l, 
where a is determined by w(x), which follows from hiRher order l~B-approxima-
tion, see Appendix B. The exit time is asymptotically the same for all points 
of Q bounded away from an. This is understood from the fact that the drift 
towards the equilibrium is of a larger order of magnitude than the diffusion. 
More details about the nroblem of stochastic exit are found in Schuss rsJ. 
Thus, we have found that 
(4.3) lnT K/E 2 + 0(1). 
This formula can be used to find expected exit times for very small s values 
from experiments with the n. hysical system for lar",er s sav E without 
" ' · exp.' 
240 
any analytical knowledge of its dynamics. Finding an average exit time Texp. 
we obtain 
2 
E 
(4.4) lnT exp. " = - 2- -<-nT £ exp. 
Carrying out experiments at two £-values one may also eliminate a in in (4.2) 
and obtain a more accurate value of T(c). The average frequency of leaving 
-I the domain of regular operation, T(E) , is a measure for the life time of 
the physical system. The present approach suggests that for life times a 
formula similar to (4.4) holds. 
Example 4,1. For a bi-stable system the expected time of residence Ti in the 
domain of attraction of the two stable equilibria can be computed. Let us 
consider the motion of a point with unit mass at a surface V(x), see fip,.2, 
(4.Sa) 
(4.Sb) 
(i) Let Qi be the domain of attraction of x , i = I,2 (for the deterministic 
system). Necessarily the minimum value of Q at 3Q(i) is attained at the un-
stable equilibrium x(O). He have now 
(4.6a) lnT. K. /£2 + 0 (I), 
l 1 
(4.6b) K. = min Q(x) Q (x (O». 
l 
an(i) 
From the theory it is deduced that the senaratrix is most likely crossed at 
the saddle point x(O). It is also most likely that the system is then almost 
at rest. It will take a time 
(4. 7) 
to leave a neighborhood of x(O) Starting at x = x(O) the probability of 
arriving near x(l) or x(Z) is fifty-fifty. Consequently, the bi-stable 
241 
system may as well be modeled by a three states Markov chain with transition 
matrix 
( 4. 8) M 
and with residence times in those states as given above. 
v 
x-
X(O) 
Fig.2. A bi-stable system 
5. Linear control systems 
Krtolica f3] investigates the linear control system 
(5. I a) dX FXdt + u(Y)dt + rn.,dv, 
('>. 1 b) dY O<c<<I, 
where 
The vector type white noise processes are 
dV dW = (d\.J , ••• ,dH ) . 1 q 
Moreover, the pair (F,C) is assumed to be observable and (F,oV) is control-
lab le. 
242 
We take the linear control law 
(5.2) udt -RdY 
and construct the matrix R that maximizes the expected residence time in 
1l c fin with the origin contained in n. 
For (5.1)-(5.2) we write 
( 5. 3) dX 
or, replacing dVi and ~/j by n Wiener increments dUk, 
(5.4) 
with 
(5.5) 
dX 
2 
a 
(F-RC)Xdt + sadU 
For the system (5.4) the stationary Fokker-Planck equation is solved, as in 
section 3. Since the system is linear the function Q(x) of the WKB-apnroxima-
tion is of the form 
(5.6) Q(x) T -I x H x 
with H satisfying 
(5. 7a) T T (F-RC)H+H(F-RC) +RG1l +GV o, 
(5. 7b) 
We have to choose R such that the minimum value of Q at Cl>l is maximized: 
(5. 8) K max min Q(x). 
R 3'2 
We may interchange the minimization and maximization, as the boundary an is 
independent of R. From results by lfonham r6J on Riccati equations, it is 
243 
deduced that Q(x) is at its maximum at an for 
(5. 9) R 
with P satisfying the Riccati equation 
(5. I 0) 
6. Stochastic difference equations 
Dynamical systems with small random nerturbation can be simulated with 
the Monte Carlo method. To perform the simulation, the Wiener increment dW 
has to be replaced by a pseudo random generator G(t). Euler's method can then 
be applied giving the following system of stochastic difference equations 
(6. I) xi (t+h) X.(t)+hf.(X)+c/h G.(t), i 
l. 1 l. 
l ' ... ,n. 
The time step h gives an error in X of order O(h). He define the stochastic 
variable 
(6.2) 6X. = X.(t+h)-X.(t). 
l. l. l. 
This variable has first and second moments 
(6.3a) E{6X.(t)} hf.(x)+E/h E{G.(t)} = hf.(x), 
l. l. l. 1 
(6.3b) 
Consequently, in unit time the expectation of 6Xi equals the local vector 
field f(x) while its variance equals £ 2. The average exit time over a large 
number of runs is approximated for £ small by the asymptotic expected exit 
time, which we computed in section 4. 
244 
We consider the singularly perturbed Dirichlet problem 
n n 
;ju I I f. (x) 0 in (,\I a) L u ' a. + I. I. lj l 3x. i,j=I i=I l 
(Al b) u = h (x) at 
~·ith all trajcct1Jries of the deterministic system (:?. 1) entering ~--~ for in-
creasing t on their way to the stable equilibrium x = 0. 
Let for the system (3. la) q(~,x) be the probability density of leaving 
. at x '- ;,;-,, if started at x ' r:. Then we have that 
(A2) q(x,x)h(x)dS u(x). 
From singular perturbation theory it is known that the asymptotic solution 
of (Al) has the form 
•) 
(A3) u"" \h(x)-C}e-p/E- + c, 
where is the distance of x to ri and C an unknown constant. This constant 
is determined by the divergence theorem using the asymptotic solution p(x) 
of the stationary Fokker-Planck equation, see (3. 3)-(3. 10): 
(A4) r 2 du a pL,u-uM,pdV = i jc (p -- u .....E.) + puf(x) 0 \ 1dS, 
, , J 3n 3 n 
311 
where ·. is the outward normal at an and 3·/3n the co-normal derivative 
(A5) 
From (A4) we derive, using (A3), 
(A6) c 
fagpf(x)·~h(x)dS 
f ac;Pf (x) •\'dS 
which, because of (3.4), is asymptotically equivalent with 
(A7) c 
245 
where x is the point at an with minimal Q. 
Appendix B 
The expected time T(x;s) needed to reach an, if starting at x E n, 
satisfies 
(B 1) L T 
E 
-1 in n, T 0 at "'' 
with the elliptic operator Ls given in (Ala). Applying the divergence 
theorem, as we did in (A4), we obtain 
(B2) I I 2 1T pL.T-TM pdV = jE (p .:___T C. E an apl cin + pTf•vdS. 
II 311 
The formal asyn1pt.otic solution of (BJ) has the form 
(B3) -p/E2 T "" -Ce + C, 
where p is the distance of x to 3(i. and C an unknown constant. Substitution 
of (3.4) and (B.3) in (B2) yields 
(B4) c 
The volume integral has its largest contribution from an <:-neighborhood of 
the origin, while the integral over (JQ has its minimal value at~ E an, so that 
(B5) c "" 
References 
[I] Gardiner, C.W.: Handbook of stochastic methods for physics chemistry and 
the natural sciences. Berlin: Springer-Verlag 1983. =Springer Series 
in Synergetics vol. 13 
[2] Grasman, J.; Lankelma, J.V.: The exit problem for a stochastic dynamical 
system with almost everywhere characteristic boundaries. Amsterdam: 
Centre for Math. and Comp.Sc. 1984. = Report AM-8403 
[3] Krtolica, R.: A singular perturbation model of reliability in systems 
control. Autornatica 20 (1984) 51-58 
246 
[4] Ludwig, D.: Persistence of dynamical systems under random perturbations. 
SIAM Rev. 11. (1975) 605-640 
[5] Schuss, Z.: Theory and applications of stochastic differential equations. 
New York: Wiley 1980. 
[6] Wonham, W.M.: On a matrix Riccati equation of stochastic control. SIAM 
J. Control 6 (1968) 681-697 
Department of Applied Mathematics 
Centre for Mathematics and Computer Science 
Kruislaan 413, 1098 SJ Amsterdam 
The Netherlands 


Estimates of large failure times from the theory of stochastically perturbed dynamical systems with and without feedback

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