Hardware Reference
In-Depth Information
One can rewrite equation 2.9 as,
b
0
(jω
k
)
m
+b
1
(jω
k
)
m−1
+···+b
m
= G
fr
(ω
k
)
¡
(jω
k
)
n
+ a
1
(jω
k
)
n−1
+ ···+ a
n
¢
,
(2.11)
or, equivalently
b
0
(jω
k
)
m
+ ···+ b
m
−G
fr
(ω
k
)
¡
a
1
(jω
k
)
n−1
+ ···+ a
n
¢
=(jω
k
)
n
G
fr
(ω
k
).
(2.12)
Using vector notation,
φ
T
(jω
k
)[b
0
a
n
]
T
= x(ω
k
)+jy(ω
k
),
b
1
···
b
m
a
1
a
2
···
(2.13)
where
x(ω
k
)+jy(ω
k
)=(jω
k
)
n
G
fr
(ω
k
)and
φ(jω
k
)=
[(jω
k
)
m
(jω
k
)
m−1
··· 1 −G
fr
(ω
k
)d
1
(jω
k
)
n−1
··· −G
fr
(ω
k
)d
n
]
T
.For
each frequency, φ is a vector of complex numbers. Equating real part of φ with
x and imaginary part of φ with y,wecanrewriteequation2.13as
∙
¸
∙
¸
φ
R
(ω
k
)
φ
I
(ω
k
)
x(ω
k
)
y(ω
k
)
· θ =
,
(2.14)
a
n
]
T
where, θ =[b
0
b
1
···
b
m
a
1
a
2
···
represents the parameter
vector.
The frequency response is measured for N different frequencies, and we get
N sets of the above equation. That is,
Φ
2N×np
Θ
np×1
= Y
2N×1
, (2.15)
where np is the number of parameters to be identified. This is a linear in the
parameters (LIP) model and can be solved using linear least-squares method,
i.e., to find the estimate Θ of the parameter vector by minimizing the cost
function,
J
LS
=(Y −ΦΘ)
T
(Y −ΦΘ).
(2.16)
Solution of this least squares problem is,
Θ=(Φ
T
Φ)
−1
Φ
T
Y.
(2.17)
It should be noted that the measurement vector Y
2N×1
in this LIP model
is obtained by multiplying the experimental frequency response by (jω
k
)
n
.In
other words, true measurement of the frequency response is weighted by a
frequency dependent factor which increases as ω
k
increases. The estimation
algorithm puts higher weights on the measurements in the higher frequency.
This becomes problematic particularly in cases where measurement data span
several decades of frequencies. Because of the multiplication by (jω
k
)
n
,both
Φ and Y differ widely in magnitude. This may lead to failure to achieve
good fit between the measured response and model's response. A remedy is
to filter Y and Φ by some approximation to the filter
1
A(jω)
and iterate this
procedure [103].
