Hardware Reference
In-Depth Information
2.2.2
Identi fi cation of Transfer Function Model
Frequency response in the low frequency range (Figure 2.10) show approxi-
mately -40 dB/decade slope in magnitude and −180 phase suggesting a rigid
body, double integer model
k
s 2 . The zero cross-over of the magnitude response
occurs at about 400 Hz, i.e.,
¯
¯
=1forω c =2π× 400 or k =6.3 × 10 6 .
The measurement was carried out with the LDV resolution set to 0.1V/µm.
So if we define the transfer function in units of µm/V then k =6.3 × 10 7 .
This double integrator model or, as used in chapter 3, a second order model
with poles on the left hand side of the complex plane is often used as the
nominal model for the sake of controller design. However, knowledge of the
flexible mode dynamics is also crucial. There are well established methods for
identification of a transfer function from frequency response data [163], [172].
These methods finds the coeﬃcients of the transfer function G(s)= B(s)
k
(jω c ) 2
A(s) such
that the frequency response of the identified transfer function matches as close
as possible to the frequency response obtained experimentally. The frequency
response data include two vectors:
1. the vector [ω k ]fork =1, ···,N contains all frequencies for which mag-
nitudes and phases are measured and
2. the vector [G fr k )] contains the frequency response measured at each
of the N frequencies.
The transfer function model G(s) is the ratio of two polynomials of Laplace
Transform parameter s,
A(s) = b 0 s m + b 1 s m−1 + ··· + n m
G(s)= B(s)
,
(2.8)
s n + a 1 s n−1 + ···+ a n
with m ≤ n for proper transfer function. The frequency response of a system
is equal to its transfer function evaluated at the points along the positive
imaginary axis of the complex plane, i.e., the response at any frequency ω k is
b 0 (jω k ) m + b 1 (jω k ) m−1 + ··· + b m
(jω k ) n + a 1 (jω k ) n−1 + ···+ a n
= G fr k ).
(2.9)
The parameters of the transfer function G can be obtained by solving the least
squares estimation problem that minimizes the error criterion
¯
¯
N
X
2
B(jω k )
A(jω k )
−G f r(ω k )
.
(2.10)
k=1
This is a nonlinear least squares problem and can be solved iteratively. Com-
mercial softwares are available for solving such problems, e.g., Frequency Do-
main Identification Toolbox from MATLAB TM [115].
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