Determining K from Breakaway Point in Second-Order Systems

To determine K using the breakaway point, follow these steps for a second-order system with a dominant pole:

1. Find the maximum value of the steady-state gain (Kss) by analyzing the output response to a constant input.
2. Locate the breakaway point, which is the gain margin value at the frequency where the phase margin is zero.
3. Determine the frequency (ωba) corresponding to the breakaway point, where the gain margin is zero.
4. Use the equation K = Kss/√(1-ωba^2/ωn^2) to solve for K, where ωn is the natural frequency of the system.

Note: This method assumes a dominant pole that significantly outweighs other poles or zeros in magnitude.