I know the the structure of a 3rd order system is:

\$Q_s=(s+a)(s^2+2\xi\omega ns+\omega n^2)\$

but what do I do if I have something like this?


How do I measure its natural frequency?

  • 1
    \$\begingroup\$ Natural resonant frequency only really applies, as a concept to 2nd order filters. \$\endgroup\$ – Andy aka May 7 '16 at 9:04
  • 3
    \$\begingroup\$ Natural frequency is the frequency that a 2nd order term would oscillate at continuously if the damping were zero. So, theoretically, you could combine any two 1st order terms from three, and define a natural frequency. But the \$\omega_n\$ derived from any one of these combinations would not have any practical implication. \$\endgroup\$ – Chu May 7 '16 at 9:53

If a system only has real roots the response is comprised of exponential terms and not oscillatory terms. In this case

$$y(t) = C_1e^{-4t} + C_2e^{-5t} + C_3e^{-3t},$$

thus there is no such thing as a natural frequency of this system because there are no oscillating terms in the solution.


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