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].