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In some questions (in linear control theory) it is desired to design a controller to change the system overshoot (Mp), my question is:

Is it possible to find Mp or damping factor directly from Bode diagram without estimating the transfer function of the system?

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Kevin Reid
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SMA.D
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1 Answers1

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The question assumes there is one damping factor i.e. the transfer function is dominated by a 2nd order system. In practice there could be several interacting 2nd order systems so care has to be taken here.

Assuming it is dominated by a single 2nd order system, this diagram may be useful: -

enter image description here

This picture allows you to calculate \$\zeta\$ by looking at the peak amplitude value in the bode plot. From this you can calculate \$\omega_n\$ using the 2nd formula. Once \$\omega_n\$ is known you can double check the value of Q (1/2\$\zeta\$) because, for a 2nd order filter like this Q IS the magnitude of the amplitude response at \$\omega_n\$.

Here's the bigger picture showing how the bode plot and pole zero diagram are related: -

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Andy aka
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  • Thank you for answering. What about higher values of \$\zeta\$ where diagram has no peak value? – SMA.D Jul 14 '16 at 08:48
  • I don't know what you mean. – Andy aka Jul 14 '16 at 09:55
  • The 5th diagram that you showed has no peak. (You have marked it "amplitude Q at \$\omega_n\$") – SMA.D Jul 14 '16 at 11:00
  • Oh yes it does have a peak - look carefully along the 0dB line (Q=1) at about 0.7\$\omega_n\$ – Andy aka Jul 14 '16 at 11:03
  • Just another question, what is the \$\zeta\$ values in the 5 plotted diagrams (just to compare the order of peaks for different cases). – SMA.D Jul 14 '16 at 11:13
  • Convert the dB scale to a real ratio at \$\omega_n\$ to give Q. So the next one up occurs about 3 dB hence Q = 1.4142 and zeta is 1/2Q or 0.354. The top trace peaks about 24 dB up hence Q is about 16 and zeta is 1/32 or 0.03. Also take note that there has to be a recognizable peak or the formulas become "out of bounds". – Andy aka Jul 14 '16 at 11:36