Markov Vector Approach-4
Docsity.com
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community
Ask the community for help and clear up your study doubts
University Rankings
Discover the best universities in your country according to Docsity users
Free resources
Our save-the-student-ebooks!
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Approx. Method for Analyzing Lightly Damped Nonlinear Systems with Broad Band Excitations, Slides of Stochastic Processes
Equations and discussions related to the analysis of lightly damped nonlinear systems under broad band excitations using an approximate method. Topics such as fpk equation, backward kolmogorov equation, first passage times, and safe regions.
Typology: Slides
2012/2013
1 / 56
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Approx. Method for Analyzing Lightly Damped Nonlinear Systems with Broad Band Excitations and more Slides Stochastic Processes in PDF only on Docsity!
Markov Vector Approach-
Docsity.com
2
0
0
;^
0;
0
;
0
~
1;
~
;
~
1
0;
2
t
ij
m m
MX
CX
KX
W
t
t^
X
X
X
X
X
t
N
n
m
W
t
m
W
t
W
t W
t
D
0
1
2
1
2
;
0 &
0
0;
2
x
x
w t
t
x
x
w t
w t
w t
D
t
t
0
0
;
0;
0
;
0
.
0;
2
x
xf
H
g
x
w t
t
x
x
x
x
w t
w t w t
D
0
0
;^
0;
0
&
0
1
;
0;
2
;
2
1, 2,
,
2
j^
j^
j^
j^
j^
j^
j^
j^
j^
j
j
t
j^
j^
k^
jk
jk
j^
j jj
U
m X
m
X
W
t
t
X
x
X
x
X
U
X KX
W
t^
W
t W
t^
D
m
j
n
D
Solution ofFPK equationsTransient & Steadystate solutions Steady statesolutions
Docsity.com
4
0
2
2
3
1
1
3
2
2
3
2
3
1
3
3
3
3
3
3
1
4
4
5
( );
0;
0 ( )
,
2
2
2
2
2
3
6
3
3
6
j
X
X
w t
t
X
X
dX
t
Xdt
dB t
f
X
d
h
h
h
h
X
t
t
X
D
dt
t
X
X
m
X
X
m
m
X
X
X
D
m
D
m
X
X
X
Dm
X
X dt
X
X
m
m
m
m
Dm
m
Example
3
3
2
4
6
2
4
12
4
4
12
X
X
X
Dm
m
Dm
m
Docsity.com
5
Moment equations form an infinite hierarchy of equations which at no stage provide sufficient
CL
number of equations
to solve for the moments:
This is the characteristic feature of
OSURE PROBLEM
Remarks
moment equations
of non linear systems driven by random excitations
Closure approximations
Assume that the higher order moments, beyond a given order,are related to the lower order on
es as though they
obey an adhocly specified pdf (Ex: Gaussian closure)Neglect cumulants beyond a specified order
Docsity.com
7
2
1
1
2
1
2
2
2
2
1
1
1
1
2
1
2
1
2
2
2
1
1
1
2
1
Example
specified.
i^
j^
ij
ij
t
X
a X
a
w
t
X
w
t
t
X
X
dX
X dt
dX
a X dt
a X dt
X dB
t
dB
t
dX
X
dB
t
dt
dX
a X
a X
X
dB
t
dB
t dB
t
D
i j
GDG
X
11
2
22
1
11
1
22
t
D
D
X
D X
D
Docsity.com
8
1
2
1
2
2
2
1
1
1
2
10
2
01
01
2
2
1
1
2
01
1
10
20
2
1
11
11
2
2
2
2
1
1
1
02
2
11
1
20
2
20
2
2
1
1
2
1
11
22
2
02
1
11
dX
X
dB
t
dt
dX
a X
a X
X
dB
t
m
X
m
m
a X
a X
a m
a m
m
X
X
m
m
X X
a X
a X
X
m
a m
a m
m
a X
a X
X
X D
D
a m
a m
D
11
20
22
m
D
Docsity.com
10
10
01
01
2
01
1
10
01
10
20
11
11
11
02
2
11
1
20
02
1
20
20
2
02
1
11
11
20
22
2
02
11
20
22
0
0
0
0
0 &
0
0
2
0
0
0
0
0
0
2
2
0
2
0
m Question: Are the steady state mom
m
m
a m
a m
m
m
m
m
m
m
m
a m
a m
m
a m
m
a m
a m
D m
D
a m
D m
D
Steady state
ents realizable?
That is. are these moments stable?
perturb the solutions and see if
Strategy : perturbations die out.
Docsity.com
11
0
0
0
The moment equations can also be used to study stabilityof the system in terms of response moments.
small perturbation
exp
Y Perturbations do not g
AY
B
Y
Y
t
Y
A Y
B
A
t
st
A
sI
A
sI
row in time if the real part of the
eigenvalues of
are all
A
Docsity.com
13
1
1
0
1
1
1
1
0
1
1
1
1
0
1
2
1
2
0 0
2
2
1
2
0
1
2
0
1
2
0
2
2
1
1
0
1
1
1
2
0
1
2
0
1
2
0
1
1
|^
m
n
m
n
m
n m
n
X
t
X
t
x
p x
t t
dx d
p x
t t
p x
x
t t t t
dx d
p x
x
t t
t t
p
t t
dx d
X
t
X
t
x
p x
x
t t
t t
p
t t
dx d
d
2
2
1
2
0
1
1
2
0
1
2
0
1
2
0
1
2
|^
n
x d
v x
x
t t p x
x
t t
t t
p
t t
d
d
Docsity.com
14
1
1
0
1
1
2
0
1
2
1
2
0
1
1
2
0
1
2
0
1
2
0
1
2
2
1
1
1
,
; ,
,
; ,
,
; ,
|^
,
; ,
,
; ,
Consider the FPK equation
,
It follows that
m
m
m
n
n
n
n
t
j^
ij
j^
i^
j
j^
i^
j
X
t
X
t
x v x
x
t t
dx dx
v x
x
t t
p x
x
t t
t t
p
t t
d
d
p
f
x t
p
GDG
p
t
x
x
x
v
0
2
1
1
1
1
2
0
1
1
1
2
2
1
2
0
0
1
2
1
1
2
0
0
,
lim
,
; ,
,
;
,
,
;
,
n
n
n
t
j^
ij
j^
i^
j
j^
i^
j
n
t^
t
n
f
x t v
GDG
v
t
x
x
x
v x
x
t t
x
x
p
t
t
d
d
x p
y
y
t
t
Docsity.com
16
1
2
3
3
3
1
1
3
3
2
2
2
2
1
1
2
3
3
2
2
3
3
1
2
1
2
3
3
1
2
2
Let
be a scalar Markov process. For
we have
|^
|^
|^
|^
|^
|^
X
t
t
t
t
p x
t
x
t
p x
t
x
t
p x
t
x
t
dx
p x
t
x
t
p x
t
x
x
x
t
p x
t
x
t
x
Backward Kolmogorov equation and reliabilityfunction
2
1
2
1
1
2
2
2
2
2
1
2
1
(^22)
x
x
x
x
p
x
x
p
x
x
o
x
x
x
Docsity.com
17
(^2)
1
2
1 2
1
3
3
1
1
3
3
2
2
2
2
1
1
2
3
3
1
2
2
1
2
2
1
1
2
2
2
2
2
2
1
2
2
1
1
2
2
1
(^22)
3
3
1
2
3
3
1
1
2
1
2
2
;
|^
;
;
|^
;
;
|^
;
;
|^
;
;
|^
;
1
;
|^
;
2
1
;
|^
;
;
|^
;
1
;
x
x
x
x x
x
p x
t
x
t
p x
t
x
t
p x
t
x
t
dx
p
p x
t
x
t
x
x
p x
t
x
t
dx
x
p
x
x
p x
t
x
t
dx
o
x
x
x
p x
t
x
t
p x
t
x
t
t
p
x
x
p x
t
x
2
1
2
1
1
2
2
2
2
1
2
2
1
2
2
1
1
2
(^22)
|^
;
1
;
|^
;
0
x
x
t
x
t
dx
o
x
x
p
x
x
p x
t
x
t
dx
t
x
t
Docsity.com
19
0
2
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
1
,
,
;
0;
0
,
,
; |
;
ICS :
;
|
;
BCS :
1
n
n
n
t
j^
ij
j^
i^
j
j^
i^
j
n
i^
i
i
dX
t
f
t X
t
dt
G t X
t
dB t
t
X
X
p
p
p
f
t
x
GDG
t
x
t
x
x
x
p
p x t
x
t
p x t
x
t
x
x
j
Vector version of Backward Kolmogorov equation
0
0
, 2,
,
, lim
; |
;
0
j x
n
p x t
x
t
Docsity.com
20
If we are interested in pdf of
for a given initial condition
we use forward equation.
If we are interested in the pdf of time for first passage across a threshold, we use backward equation.
X
t
Remarks
Docsity.com