One problem in relationship between step transient response and frequency response

[threekingtiger]

In control theory, the relationship between step transient response and frequency response has been elaborated specifically, especially for 1st order and 2nd order systems.

However, for linear, time-invariant, high-order systems having a dominant pair of complex congjugate closed loop poles, some relationship is too vague. For instance, some author claims that in terms of the open-loop frequency response, the damped natural freqency ωd in the transient response is somewhere between the gain crossover frequency(ω@unity gain) and phase crossover frequency(ω@-180°). Could anybody explain the reason for the location of ωd ?

 

[leo_o2]

I think ωd is a frequency that the loop can sustain oscillating. At the freq, the positive-feedback gain should be bigger than 1. However the accurate theory is unclear. The guess between ω@unity gain and ω@-180° seems reasonable. However the actual oscillating freq depends large-signal characteristic.