For a system with G(s) = (s^2 + 2*s + 2)^-1 and Gc(s) = K, for the damping ratio, I have calculated the one for the open-loop to be 1/ sqrt(2) and the closed-loop one being 1/sqrt(2+K). I'm wondering why is the damping ratio for the open-loop system < damping ratio for the closed-loop system (or is this even the case?). Much appreciated to hear your thoughts.
Draw the root locus and it will become obvious.
The plot starts at the poles and goes to infinity because there are no zeros. Thus the angle from the close loop poles, through origin to the x-axis is always decreasing. The damping ratio = cos(theta) where theta is this again.
See any book that discusses lines of constant damping ratio, ex: Ogata, Modern Control Engineering, 1970, p 487.
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