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Rational Polynomial LSE Approximation of Complex Functions 25
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Frequency w (in radians/secondl
O'--='~-,-'-_--L_--L_--L__~_:'.,-_-:':--_----!
0.1 0.2 0.4
2 4 10 20 40 100
Frequency response characteristics of a dynamic system with
a transfer function given as
l+jw
FU"')~
1+2(0.5)U",/ 10) + U"'/ 10)'
Magni- Phase
k "'k tude Angle Rk lk
0 0.0 1.00 0 1.00 0.000
I 0.1 1.00 5 1.00 0.090
2 0.2 1.02 10 1.00 0.177
3 0.5 1.12 24 1.02 0.450
4 0.7 1.24 31 1.05 0.630
5 1.0 1.44 39 1.10 0.900
6 2.0 2.27 51.5 1.4\ 1.78
7 4.0 4.44 50.5 2.82 3.42
8 7.0 8.17 28 7.23 3.82
9 10.0 10.05 - 6 10.00 -1.00
10 20.0 5.56 -59 2.85 -4.77
11 40.0 2.55 -76 0.602 -2.51
12 70.0 lAS -82 0.188 -1.43
13 100.0 1.00 -84 0.09\ -1.01
Rk=(Magnitude at wk) X cos(phase angle at "'k)
I k == (Magnitude at Wk) x sin(phase angle at Wk).
Figure 2.9. Frequency response and discrete data for a second-degree system. [From Levy, etc.,
IRE Trans. Auto. Control, Vol. AC~4, No. I, p. 41, May ]959. © J959 IRE (now IEEE).]
The table of values is the given data. Although measured data are often
inaccurate (noisy), this particular data set was computed from the F(w) values
shown in Figure 2.9 for purposes of illustration. The graph shows the magnitude
and angle components of the function. The technique will be to approximate
only the magnitude function by finding the unknown coefficients of
(2.47).