Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
N(0,1)
Normally distributed
random generator
zn Xn
Xn = µXn−1 + σ � 1 − µ 2 zn
C.5 Ornstein-Ulhenbeck Process
Colored
noise
Figure C.1: Simulation of colored noise via block wise programing.
The law of the random variable at time 0 is supposed the invariant probability associated
to (C.1) such that Xt is a stationary process.
Equation (C.1) can be rewritten:
and then
d � e ρt � ρt
Xt = αe dBt,
Xt = e −ρt
� � t
X0 + αe ρs �
dBs .
The stochastic process Xt is such that [28]:
1. If X0 is a gaussian variable with zero mean and variance equals to α2
2ρ , then X(t) is
gaussian and stationary of covariance:
2. X(t) is a markovian process.
0
E[X(t)X(s + t)] = α2
2ρ e−rρ|s| .
3. When X(0) = c, the stochastic law of X(t) is a normal law with mean e−ρtc, and
variance α2
�
2ρ 1 − e2ρt � .
As explained in [59], Part 3, equation (3.15), for a sample time∆ t, the exact updating
formulas of the Ornstein-Uhlenbeck process is given by:
where:
− µ = e −ρ∆t ,
− γ 2 = � 1 − e −2ρ∆t� � α 2
2µ
�
,
X(t + ∆t) =X(t)µ + γzn
− zn is the realization of a standard gaussian distribution (i.e. N(0, 1)).
163
(C.2)