# What is the effect of the resonant frequency of the system function Porter diagram on the stability of the system and how to analyze it?

I am a novice in automatic control, the theoretical basis is not very good. We have a large electric clamping jaw, single degree of freedom, the motor is controlled by the torque output, the motor has a maximum stable clamping stiffness (This parameter is related to the gain margin, the greater the gain margin the greater the clamping stiffness can be). There is a rigid coupling between the motor and the gearbox.

We simulated the Porter diagram of the motor with matlab and got the curve shown in the first diagram, Gm is 78.9dB.

Then we changed the coupling to a flexible (elastic) coupling and got the second diagram below, Gm became larger, 87.7dB. My question is, when the coupling is changed to flexible, intuitively I feel it will cause resonance. But why is the gain margin bigger instead. Also, the part of the Gm increase appears a little to the right of the peak resonant frequency. Does this resonant frequency have any effect on stability?

In addition, there is another question. For the above figure, the gain margin of 87.7, when I set the maximum gain to 87.7 near the stability will become worse, or that the frequency of my control signal reaches the resonant frequency when the stability will be affected

• Would it be possible to draw a Nyquist plot for the same transfer function? Just because the phase barely escapes 180 deg near 60 rad/s doesn't mean that the gain margin occurs at 600 rad/s. It may be misleading. What would have been the gain margin of you had measured it in the second diagram at the same frequency as in the first diagram?
– AJN
Feb 18 at 9:30
• What you say does make sense. But what I mean is, if the assumptions of the above two graphs are met, that is, 87.7dB gain margin occurs at 600rad/s, then does this resonance summit have an impact on stability? This question is mainly because I do not quite understand the impact of resonance on system stability Feb 18 at 9:36
• For the second graph, it is not very obvious with the eye. After zooming in with matlab, we can find that the -180 degree crossover point appears near 600 rad. Feb 18 at 9:38
• Why do you call it a porter diagram? A quick web search doesn't tell me that it is related to bode diagrams.
– AJN
Feb 18 at 12:34

## "Actual" Gain Margin

Let the phase at 10 rad/s to 60 rad/s be -165 deg. i.e, 20 deg away from 180 deg. At those frequencies, this 20 deg distance from -180 deg will vanish if the system (for some reason) has an additional time delay of 35 ms and 6 ms respectively. Is the timing uncertainty in the various sub systems in your control system smaller than these numbers ? (I don't know how well your system is built and modeled).

If not, your actual phase cross over frequency could occur between 10 and 60 rad/s and hence your actual gain margin may be as small as 50 dB.

If the plot is from theoretical numbers, actual hardware could have phase cross over at 10 rad/s and increasing gain by 78 dB to get additional stiffness would lead to instability (possibly at oscillating at 10 to 60 rad/s rather than the resonant frequency).

Marked as A, D below. ## Behavior after increasing gain

For the above figure, the gain margin of 87.7, when I set the maximum gain to 87.7 near the stability will become worse, or that the frequency of my control signal reaches the resonant frequency when the stability will be affected.

When you increase the gain by 87.7 dB, the theoretical phase cross over frequency occurs at about 600 rad/s (marked as C in the figure above). At that frequency, 600 microseconds of timing uncertainty can eat away what ever gain (or phase) margin is left. Similar comment about the point marked A.

Also, note the points A and C. The new gain cross over frequency is at those points. i.e. you are operating your system at near zero phase margin. you can expect the system to have oscillations which are lightly damped and take several seconds to settle.

## Effect of increased gains on Phase margin

Text books tell you that you need to maintain 60 deg phase margin where possible. It is preferred not to go below 60 deg phase margin unless you can tolerate responses with overshoot and slowly converging oscillations.

From your figure, it appears that increasing gain by about 15 dB will reduce your phase margin to 60 deg. So that is possibly the limiting factor for increasing gains.

## Effect of flexible attachments

My question is, when the coupling is changed to flexible, intuitively I feel it will cause resonance. It appears that the peak response has increased in magnitude when flexible coupling is used. So, I think your intuition is correct to some extent.

• Your answer was great and answered a lot of my questions. I have another question about the flexible coupling, as marked in your diagram, the peak response has increased in magnitude when flexible coupling is used. But why does the gain margin increase when using a flexible coupling? How can I understand this result physically, I mean, why the use of a flexible coupling does not lead to a worse gain margin (if I disregard the delay and consider only the theoretical analysis of this diagram). Feb 18 at 13:41
• I am not familiar with the system being controlled here. My guess : With resonant systems superimposed/mounted on a rigid body, you could, in theory, get the opposite result also depending on the relative location of the control input and the sensor used for feedback. The terminology usually seen in literature is non co located actuator and sensor. Depending on the relative location, I would assume that one could get the opposite result also.
– AJN
Feb 18 at 15:20
• Analyzing this same system using Nyquist diagram/plot/contour may throw some light. You could also repeat this analysis with a different location of the sensor (i.e. torque measurement done on motor side of the coupling versus the clamp side of the coupling)
– AJN
Feb 18 at 15:20
• I have another question related to bode diagram, could you please help me answer it? Thank you very much. engineering.stackexchange.com/questions/54304/… Feb 19 at 6:26