Question 10.18: Find the transfer function of the subsystem whose Bode plots...
Find the transfer function of the subsystem whose Bode plots are shown in Figure 10.57.
Let us first extract the underdamped poles that we suspect, based on the peaking in the magnitude curve. We estimate the natural frequency to be near the peak frequency, or approximately 5 rad/s. From Figure 10.57, we see a peak of about 6.5 dB, which translates into a damping ratio of about ζ = 0.24 using Eq. (10.52). The unity gain second-order function is thus G_{1} (s) = ω^{2}_{n} / (s² + 2ζω_{n} s + ω^{2}_{n}) = 25/ (s² + 2.4s + 25). The frequency response plot of this function is made and subtracted from the previous Bode plots to yield the response in Figure 10.58.
M_{p} = \frac{1}{2ζ \sqrt{1 – ζ²} } (10.52)
Overlaying a −20 -dB/decade line on the magnitude response and a −45°/decade line on the phase response, we detect a final pole. From the phase response, we estimate the break frequency at 90 rad/s. Subtracting the response of G_{2} (s) = 90/(s + 90) from the previous response yields the response in Figure 10.59.
Figure 10.59 has a magnitude and phase curve similar to that generated by a lag function. We draw a −20-dB/decade line and fit it to the curves. The break frequencies are read from the figure as 9 and 30 rad/s. A unity gain transfer function containing a pole at −9 and a zero at −30 is G_{3} (s) = 0.3(s + 30)/(s + 9). Upon subtraction of G_{1} (s) G_{2} (s) G_{3} (s) , we find the magnitude frequency response flat ±1 dB and the phase response flat at −3°±5°. We thus conclude that we are finished extracting dynamic transfer functions. The low-frequency, or dc, value of the original curve is −19 dB, or 0.11. Our estimate of the subsystem’s transfer function is G(s) = 0.11G_{1} (s) G_{2} (s) G_{3} (s) , or
G (s) = 0.11 (\frac{25}{s² + 2.4s + 25 } ) (90 \frac{1}{s + 90} ) (0.3\frac{s + 30}{s + 9} ) (10.89)
= 74.25 \frac{s + 30}{(s + 9) (s + 90) (s² + 2.4s + 25)}
It is interesting to note that the original curve was obtained from the function
G (s) = 70 \frac{s + 20}{(s + 7) (s + 70) (s² + 2s + 25)} (10.90)
Students who are using MATLAB should now run ch10apB8 in Appendix B. You will learn how to use MATLAB to subtract Bode plots for the purpose of estimating transfer functions through sinusoidal testing. This exercise solves a portion of Example 10.18 using MATLAB.
