Control Theory - Nyquist Plot - Speed of Response from Low Frequency Asymptote

[Master1022]

[Moderator: moved from a homework forum. This does not sound like homework.]Homework Statement:: Why is it the case that when the low-frequency response is to the right of the M = 1 line that the 'speed of response is slow'?
Relevant Equations:: M-cirlces

Hi,

Hope you are doing well and staying safe.

I just wanted to ask a quick conceptual question within the topic of M-circles in Control Theory. We have been told that one can get qualitative (as well as quantitative) information about the closed loop response of a transfer function from its open loop nyquist plot. Moreover, we have been told that when the low frequency asymptote of our open loop response lies to the right of the M = 1 lines ($x = -0.5$) that this will lead to a slower closed loop response. Subsequently, we can speed up the response by shifting this low frequency asymptote closer to the M = 1 line. I wanted to know why that is the case?

I have tried to reason about what the location of the low frequency asymptote means beyond the DC gain. Does it have to do with the fact that it might take longer for a Nyquist plot to decrease to 0 if it is approaching from far away? I don't know if that is a rigorous reason as I am sure we can construct a function that has a high DC gain whilst still approaching the origin (on the Nyquist plot) quickly.

I believe this should be a simple question, but I haven't been able to figure it out conceptually.

Any help is greatly appreciated.

 

[Adrmay]

Thanks!The reason for the slow speed of response when the low-frequency response is to the right of the M = 1 line is that the phase margin of the system is reduced. The phase margin is a measure of the stability of the system and is defined as the difference between the phase of the open loop transfer function at the frequency where the magnitude response is equal to 1 (M=1). If the phase is greater than -180° at this frequency, then the system is considered stable. When the low-frequency response is to the right of the M = 1 line, the phase margin is reduced and thus the system is less stable and has a slower speed of response.