janusz walczak, seweryn mazurkiewicz determination of moments of

JANUSZ WALCZAK, SEWERYN MAZURKIEWICZ
SILESIAN UNIVERSITY OF TECHNOLOGY, POLAND
DETERMINATION OF MOMENTS OF STOCHASTIC PROCESSES IN LINEAR STOCHASTIC
SYSTEMS OF FIRST ORDER
INTRODUCTION
Analysis of random dynamic systems concerns mainly in mechanics and control theory [2], [7]. So far, there have not been
much work dedicated to the analysis of random phenomena
in electronic [4], [5], energoelectronics [6] and electric circuits
[3]. These works usualy relate to the determination of the probabilistic characteristics of random processes occurring in the
presented systems. There are three groups of these works.
The first one consist in the determination of primary probabilistic characteristics, so that their multidimensional probabilistic
density functions. Determination efficient solutions in analytical form are difficult in this case [7]. The second one concerns
of the determination of moments of stochastic processes [7]
and it is conceptually simpler. It is sometimes possible to obtain effective results in the closed form. The last one relates
to determining the probabilistic characteristic in the frequency
domain [9] and is usually limited to the class of stationary processes.
Determination of moments of the processes can be carried
out using three methods [7]. The first one can be used when
analytical formulas of stochastic processes in the system are
already known. It is based on calculation of expected value
from the product of powers of processes. The next method is
based on the knowledge of the probability density function. It
is rarely used because it lead to great difficulties in determining the probability density function. The last of the methods
involves applying the Ito formula [7] to stochastic differential
equations. This method is used effectively to analyze the nonlinear problems [7], especially in mechanics.
The main issue of this paper is the using of first of mentioned method to determinate the first and second order moments of first order linear random dynamic system. It is a continuation of the previous works [10], applying to the determination of moments in nonlinear, random, noninertial systems.
1.
FORMALIZATION AND SOLUTION OF THE PROBLEM
The considered system, that is shown on the fig. 1, is described by stochastic differential equation, with random initial condition:
dY (t)
− aY (t) = X(t), Y (0) = Y0 , a ∈ R,
dt
It can be proved [1], [8], that the solution of random Cauchy
problem (in the mean square sense) for equation (1) is given
by the formula:
Zt
Y (t) = Y0 exp(at) +
exp(a(t − t0 ))X(t0 ) dt0 .
0
Fig. 1. SISO random system.
Hence moments of the first and second order are given by
eqations:
mY (t) = E[Y (t)] =
Zt
= E[Y0 ] exp(at) +
E[X(t0 )] exp(a(t − t0 )) dt0 ,
(3)
0
Cov(Y (t1 ), Y (t2 )) = E[Y (t1 )Y (t2 )] − E[Y (t1 )]E[Y (t2 )]
= E[Y02 ] − (E[Y0 ])2 +
Zt2 Zt1
(E[X(t01 )X(t02 )] − E[X(t01 )]E[X(t02 )])·
+
0
0
· exp(−a(t01 + t02 )dt01 dt02 exp(a(t1 + t2 )
(4)
where:
E[ ◦ ] – expected value operator.
Determination of a specific form of these moments is possible, when explicit description of process X(t) is known. This
method is illustrated by a following example.
2.
EXAMPLE
Considered system is shown on the fig. 2.
(1)
where:
a – deterministic parameter,
Y0 – random variable with known moments of the first and
second order (E[Y0 ] and E[Y02 ]),
X(t) – input stochastic process of the second order,
Y (t) – output stochastic process.
(2)
Fig. 2. RC Filter Circuit.
This system is supplied by the voltage source UX (t):
UX (t) = A sin(ωt) + W (t),
(5)
where:
ω, R, C – deterministic parameters of signal and system,
W (t) – white noise which is a derivative of the Wiener process B(t) (dB(t) = W (t)dt) with parameters:
E[B(t)] = 0, E[B(t1 )B(t2 )] = 2D min(t1 , t2 ), D ∈ R,
A, U0 – amplitude of sinusoidal signal and voltage on the
capacitor at the moment t = 0, being the random variables of second order.
α = −1/RC, β = ωRC .
For t1 > t2 , variable t1 must be exchange onto t2 and reverse. The graphs of the expected value function and the covariance function Cov(UY (t1 ), UY (t2 )) for E[A] = E[UY0 ] =
1, E[A2 ] = E[UY20 ] = 1/2, R = C = 1, ω = 3, D = 1/4 are
shown on fig. 3, 4.
3.
SUMMARY
The method of determination moments of stochastic processes, discussed in the article, can be used when the explicit
solution of stochastic differential equations are known. This
method also allows to determine the higher orders moments.
However, the results are quite complex.
4.
REFERENCES
1. Bobrowski D.: Wstep
˛ do losowych równań różniczkowych
zwyczajnych. PWN, Warszawa 1987.
2. Grigoriu M.: Applied Non-Gaussian Processes. Prentice–
Hall, New Jerscy 1995.
3. Kolarova E.: Modeling RL Electrical Circuits by Stochastic
Differential Equations. Int. Conf. EUROCON, November
22–24, Belgrade, Serbia 2005, pp. 1236–1238.
4. Maffezzoni P., D’Amore D.: Phase–Noise Reduction in
Oscillators via Small Signal Injection. IEEE Trans. on CAS
Part I, Vol. 58, No. 10, Oct. 2011, pp. 2498-2507.
Fig. 3. Expected value function.
5. Mehrotra A.: Noise Analysis of Phase–Locked Loop.
IEEE Trans. on CAS. Part I, Vol. 49, No. 9, Sept. 2002,
pp. 1309–1316.
6. Skowronek K.: Obwody elektryczne w ujeciu
˛
stochastycznym. Monografia. Wyd. Pol. Pozn., Poznań 2011.
7. Socha L.: Równania momentów w stochastycznych układach dynamicznych. PWN, Warszawa 1993.
8. Soong T. T.: Random Differential Equations in Science
and Engineering. Math. in Science and Eng. Vol. 103.
Academic Press. New York 1973.
9. Swiesznikow A. A.: Podstawowe metody funkcji losowych. WNT, Warszawa 1965.
10. Walczak J., Mazurkiewicz S.: Transformacje momentów
sygnałów stochastycznych w losowych ukladach
˛
nieliniowych. Kwartalnik Elektryka Pol. Śl., nr. 3, 2011 (w druku).
5.
Fig. 4. Covariance function.
The determined moments, based on the equations (3) and (4),
of process UY (t) are given by equations (for t1 < t2 ):
mUY (t) = E[UY0 ] exp(αt)+
+
E[A] (β exp (αt) + sin(ωt) − β cos(ωt))
,
1 + β2
Cov(UY (t1 ), UY (t2 )) =
(6)
SHORT ABSTRACT
In this paper a method of determining moments of stochastic processes in linear, random dynamic systems is described. The first order dynamic system with random initial condition and random excitation is considered. Random excitation contains white noise and sinusoidal signal
with random amplitude. The equations of the expected value and covariance function are also presented. The obtained results are shown by the example.
exp (α(t1 + t2 )) E[UY20 ] − (E[UY0 ])2 +
E[A2 ] − (E[A])2 2
β exp (α(t1 + t2 )) +
(1 + β 2 )2
prof. dr hab. inż. Janusz Walczak
mgr inż. Seweryn Mazurkiewicz
Politechnika Ślaska
˛
Wydział Elektryczny
Instytut Elektrotechniki i Informatyki
ul. Akademicka 10
44–100 Gliwice
+β exp (αt1 ) sin(ωt2 ) − β 2 exp (αt1 ) cos(ωt2 )+
+β sin(ωt1 ) exp (αt2 ) + sin(ωt1 ) sin(ωt2 )−
−β sin(ωt1 ) cos(ωt2 ) − β 2 cos(ωt1 ) exp (αt2 ) −
−β cos(ωt1 ) sin(ωt2 ) + β 2 cos(ωt1 ) cos(ωt2 ) +
+αD exp (α(t1 + t2 )) − αD exp (−α(t1 − t2 ))
where:
(7)
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