Lecture 3 Identification
-
Upload
latisha-carter -
Category
Documents
-
view
214 -
download
0
Transcript of Lecture 3 Identification
-
8/9/2019 Lecture 3 Identification
1/6
Lecture #3 Identification
ECE 842
Ali Keyhani
Identification is the determination on the basis of input and output, of a process (model), within a specified
class of processes (models), to which the process under test is equivalent.
y(.)
u(.)
w(.)
y(.)
u(.)
w(.)
Process
Equivalent
Model
The equivalent model is a model which has captured the underlying mechanism which has generated
the original process.
Three Step Modeling Procedure for Process {y(.)} excited by white noise
1. Model IdentificationMeans the determination of n and m for process.2. Parameter Estimation; The determination of { ^ )(ka } from { Y(j), J < k }3. Diagnostic Checking
Checking the {w(.)} sequence to establish that the noise indeed is white.
-
8/9/2019 Lecture 3 Identification
2/6
Classical Approach to Identification and Modeling
Given
Build The Model OverThis Portion of PastHistory
Test The Model OverThis Portion of PastHistory
A 3 Step Procedure for Modeling IS:
Identify A CandidateModel
Estimate The Parameter ofThe Identified Model
Diagnostic CheckingIs The Noise
White?
Go To IdentifyAnother Model
Done
No
Yes
k
Y(k)
Covariance: A partial indication of the degree to which one variable is related to another is given by the
covariance, which is the expectation of the product of the deviation of two random variables from their
means.
{Y(.)} random process Y(.)
{x(.)} random process x(.)
Cov {x(k) Y(k)} = E [(x(k) ux) (Y(k) uy)]
= E [x(k) Y(k)] ux uy
Where
ux = E [x(k)]
uy = E [Y(k)]
-
8/9/2019 Lecture 3 Identification
3/6
Cross Correlation Coefficient
yx
yxuukYkxE
=
)]()([
Variance of x ]))([( 22 xx ukxE =
Variance of y ]))([( 22yy
ukYE =
Notes:
1) The correlation coefficient is a measure of the degree of linear dependence between x and y.2) If x and y are independent, = 0 (the inverse is not true).3) If Y is a linear function of x, = 1
Zero Mean Process: Given process {Y(.)}, if the mean of the process is estimated and then subtracted
from each Y(k), we obtain a zero mean process.
Ex.
N
kkY 11 )}({ =
= )(1
11
^
kYN
u y
1
^
1 )1()1( yuYY =
1
^
1
1
1
^
1
)()(
.
)}({.
.
)2()2(
y
N
k
y
ukYkY
processmeanzeroaiskY
uYY
=
=
=
For zero mean processes {x(.)} and {Y(.)}
Cov { x(k) Y(k) } = E [ x(k) Y(k) ] = xy
xy is called cross covariance or correlation between x and Y variables
The basic cross correlation or covariance equation
E [ x(k) Y(k+j) ] = xy (j)
E [ Y(k) x(k+j) ] = yx (-j); i.e. xy (j) =yx (-j)
-
8/9/2019 Lecture 3 Identification
4/6
For j = 0
E [ x(k) Y(k) ] = xy =xy (0)
The sample estimate of cross correlation for covariance,
mjjkYkxjN
jjN
k
xy .,.........1,0)()(1
)(1
^
=+
=
=
mjjkxkYjN
jjN
k
yx .,.........1,0)()(1
)(1
^
=+
=
=
Sample cross correlation coefficient of zero mean {Y(.)} and {x(.)} processes.
yx
xy
kYkxE^^
^^ )]()([)0(
=
=
==N
k
xykYkx
NkYkxE
1
^^
)()(1
)]()([)0(
yx
xy
xy ^^
^
^ )0()0(
=
=
==N
k
xxkx
N 1
22^
0
^
))((1
1
=
==N
k
yykY
N 1
22^
0
^
))((1
1
J Period apart cross correlation for process {x(.)} and {Y(.)} with zero mean.
xy (j) = E [ x(k) Y(k+j) ] = xy (j)
Cross correlation coeff.
yx
xy
xy
jj
)()( =
Same cross correlation coeff.
yx
xyxy ^^
^
^
)0()0(
=
-
8/9/2019 Lecture 3 Identification
5/6
Historically, the correlation function is defined without restriction on mean and without normalization:
Correlation function mjjkYkxjN
jRN
k
xy .,.........1,0)()(1
)(1
1
^
=+
=
=
)]()([)(^^
jkYkxEjRxy +=
The covariance function has the mean value removed
mjujkYukxjN
jCjjN
k
yxxyxy .,.........1,0])(][)([1
)()(1
^
=+
==
=
In practice, the mean value usually has been subtracted from the data.
The correlation coefficient function
yx
xy
xy
jj
)()( = , )0(xx = , )0(yy =
j Period Apart, auto-covariance for process {Y(.)} with zero mean E [Y(k)] = 0
(j) = e [ (Y(k) Y(k+j) ]
Autocorrelation coefficient
0
)()(
jj =
Sample correlation coefficient
0
^
^
0
^
^^ )()]()([
)(
jjkYkYE
j =+
=
Where
..........2,1,0)()(1
)]()([)(1
^^
=+
=+=
=
jjkYkYjN
jkYjYEjjN
k
yy
=
=N
k
kYN 1
2
0
^
))((1
1
Note that j period apart autocorrelation is a partial indication of the degree to which present is dependent to
the past. i.e.
1) The autocorrelation is a measure of the degree of linear dependence between the present and thepast.
-
8/9/2019 Lecture 3 Identification
6/6
2) If Y(k) and Y(k+j) are independent, then 0)(^ =yyy .3) 20)0( =YYR i.e the mean-square value of the random process can always be obtained by setting the
j = 0.
4) )()( jRjR YYYY = 5) )0()( YYYY RjR the largest value of the autocorrelation always occurs at j = 0.