This paper considers systems whose input signals are fuzzy stochastic processes of second order. The analysis is entirely restricted to discrete time linear time-invariant systems. Convergence conditions of the output are given. The equations on the mean value functions and the covariance functions are derived. The representation of fuzzy stochastic processes is also discussed.
Analysis and design of complex systems often in-
volve two kinds of uncertainty: randomness and fuzzi-
ness. The randomness models stochastic variability
and fuzziness models measurement imprecision due to
linguistic structure or incomplete information. In some
8elds of application, suchas reliability modeling, de-
cision making, data analysis, software reliability and
earthquake prediction [2,3,7,10,13,18,19], the uncer-
tainty arises from bothrandomness and fuzziness si-
multaneously, and exceeds the realm of the classical
probability theory and fuzzy set theory. The concept of fuzzy random variables were introduced by Kwark-
ernaak [11], Puri and Ralescu [15] to describe fuzzy
random quantities. The studies of fuzzy random vari-
ables ranged from expectation, limit theorem [8] to
martingale [4,16]. For analytical treatment of systems
subjected to fuzzy random excitations, Wang and
Zhang [17,20] studied the general theory of fuzzy
stochastic processes and fuzzy stochastic dynamical
systems. However, neither the covariance de8ned
in Ref. [20] is a fuzzy number, nor does it ful8ll
some main properties of covariance. Recently, Korner
[9], Feng et al. [6] de8ned a real-valued covariance
of fuzzy random variables that makes well-de8ned
sense.
In this paper, we will consider a linear time-
invariant dynamical system withone input
u
and one
output
y
. Assuming that the system is characterized
by its weighting function
h
, the input–output relation f the input signal
u
is a fuzzy stochastic process, the
response
y
is also a fuzzy stochastic process (Fig. 1).
The problem is then to 8nd the relationships between
the fuzzy random properties of the output
y
and those
of the input
u
. Because of the intrinsical nonlinear-
ity of fuzzy linear algorithms, some conclusions of
stochastic systems [1] cannot be directly extended to
their fuzzy counterparts.
The present paper is organized as follows. In
Section 2, we state some results of fuzzy stochastic
processes of second order. In Section 3, conditions
for the output to be a fuzzy stochastic process of
second order are speci8ed. Then the characteristic
equations are derived for fuzzy stochastic systems of
nonnegative weighting function, symmetric member-
ship or general case. Illustrative examples are given.
Section 4 is dedicated to the representation of fuzzy
stochastic processes.
