Isn't a Closed Loop System with Two Poles at Origin Unstable (even with Zero?)

In a simple negative feedback system, you have transfer function:
G(s)/(1+G(s))

Gain Margin is the gain below unity when phase of G(s) is 180deg.
Phase Margin is amount of phase needed to reach a total of 180 deg at unity gain.

But if you have two poles at the origin, isn't it stable from the start?
This is because two poles at origin means you are at 180deg at DC where your G(S) gain is above unity. So you have negative Gain Margin from the start!

Now if you add a Zero to G(s) to compensate, it is obviously possible to add to the phase so that it goes back to 90deg. And thus by the time you get to unit gain, you will have enough difference from 180 to get you good Phase Margin.

But still, how does the Zero help with the fact that at DC, you had negative Gain Margin? It seems that this fact alone should make any feedback system with two poles at origin unstable! What am I missing???

A system can have any number of poles and zeros below the unity gain crossover. For stability, we need only a good first order roll off around unity gain crossover. Your example of (1+s/wz)/s^2/wu^2 is perfectly stable. This is what happens in a pll or in a multipath opamp/feedforward opamp. Another example (1+s)^2/s^3.

One cannot determine stability by looking at a single frequency. Only Nyquist criterion can give the complete picture of stability.

deba_fire said:

A system can have any number of poles and zeros below the unity gain crossover. For stability, we need only a good first order roll off around unity gain crossover. Your example of (1+s/wz)/s^2/wu^2 is perfectly stable. This is what happens in a pll or in a multipath opamp/feedforward opamp. Another example (1+s)^2/s^3.

One cannot determine stability by looking at a single frequency. Only Nyquist criterion can give the complete picture of stability.

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Thanks for your answer.

My simplistic understanding was that in a closed-loop feedback system, you have stability if you have both positive gain margin AND positive phase margin.

In the case of CL(s) = G(s)/(1+G(s)H(s))

It means that when phase(G*H) = -180, you must have db(GH) < 0 (below unity gain) so that you have positive gain margin. But from what you are saying, that isn't actually required. You CAN have a stable system even with negative gain margin. Clearly a system with multiple poles at the origin starts off at or below -180 degrees while gain is above unity so the gain margin rule seems to have exceptions?

The other requirement is that at db(GH)=0, phase(GH) > -180 (so if phase(GH) = -135, then you have PM of 45deg). Is this also a hard requirement or are there exceptions? What I mean is that suppose at db(GH)=0, your phase was actually -225deg for a PM of -45. Is that always unstable?

As for Nyquist, is that simply testing for the possibility of 1+G(s)H(s) = 0? Is that all?
Or is it testing for 1+G(s)H(s) has RHPs?

Thanks for any further help in advance!

For a minimum phase system, the definition of GM and PM, and their implications are perfectly correct.

But for non-minimum phase system, even if the PM is negative we cannot say the system is unstable. I don't have an example for such a system. But for analyzing such systems, Nyquist plots are used.

Having any poles at exactly zero Hz is physically / electrically
unrealizable.

Not really! A VCO has an ideal integrator behaviour from input control voltage to output phase.

I'd personally like a better understanding of this issue as well.

I can say from experience that the answer is no, it's not necessarily unstable because I inherited what was essentially a current source supply feeding a cap (which represents a pole at the origin) which had a control compensation with it's own pole at the origin. This really confused me until I decided to accept it.

I rationalized it by deciding that phase margin is only measured at the crossover, and that theoretically while the two poles at the origin should approach 180 they will never hit it exactly. So the 'phase margin' at very low frequencies approaches zero, but that's not where phase margin matters (right??).


Though why phase margin matters only at the crossover is another thing that I don't find very intuitive. One implication of that is that a system can become unstable (or have greatly reduced phase margin) if gain is reduced, because it may push the crossover to lower frequencies where the two origin poles may dominate (right??).



Yes and in the digital world it's quite easy to implement an integrator as a simple accumulator which would be ideal up until the point where the resulting output is saturated.

deba_fire said:

A system can have any number of poles and zeros below the unity gain crossover. For stability, we need only a good first order roll off around unity gain crossover.

Click to expand...

Can you expand on this point? I've never heard of it.

So you are saying that if you have -20dB/decade rolloff of the open loop gain of G(s)H(s) around the unity gain frequency that the closed loop system is always stable?

What is the mathematically basis for this statement? Never heard of this rule until now.

Around the unity gain cross over, the closed loop response is 1/(1+P), P being the open loop response. So phase margin(PM) just tells how much away you are from the critical -180 degree.

Now for minimum phase systems, (systems which have no pole/zero in right half plane), there is a unique relation between magnitude and phase of the system. That is, for minimum phase systems, only magnitude information is enough to know about the system. These systems are also unidirectional, meaning you increase the open loop gain and beyond some gain they become unstable. But non-minimum phase system, which are also called conditionally stable, the stability as the open loop gain varies is not unidirectional. That is, for certain ranges of gain the system is stable. This are easily understood from Nyquist diagrams, in terms of how far or close you come to the (-1,0) point. More information is in Automatic Control Systems 9e (WSE), Kuo.

So for good robust performance, against parameter variations, if one maintains a -20dB/dec around the gain cross over point, this buys enough margin for stability. Phase margin is still a good metric to check for any system.

deba_fire said:

Now for minimum phase systems, (systems which have no pole/zero in right half plane), there is a unique relation between magnitude and phase of the system. That is, for minimum phase systems, only magnitude information is enough to know about the system. These systems are also unidirectional, meaning you increase the open loop gain and beyond some gain they become unstable.

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IIUC, minimum phase system is basically a system where phase plot should be monotonically decreasing. And I can see how this happens with RHP pole/zero. But phase response with multiple poles at origin and LHP zero also won't be monotonically decreasing. So are systems like (1+s)^2/s^3 also non-minimum phase systems? (Or maybe you define RHP as including poles on the jw axis?)

But here is the thing that confuses me and others.

Let's say you have a non-minimum phase system so that there is negative gain margin when phase crosses 180 at w1. But at unity gain wc, phase is well away from 180 so you have good phase margin. And wc > w1.

But it would seem that if there is some noise at w1, in negative feedback you have a total of 360 (180 from loopgain and 180 from negative feedback) around the loop. And furthermore, you have positive gain. So shouldn't a signal at w1 keep reinforcing itself with positive feedback until it blows up and saturates.

Yes you don't get this at wc. But shouldn't w1 cause a blowup???

The reasoning intuitively is ok, but is wrong. For ex: Assume the open loop response P, at some frequency is -10 i.e. P(jw1) = -10. By your reasoning, it will be unstable. But lets say P(s) = -10 for all frequencies. The closed loop loop response is 10/9, it is stable.

Looking at a single frequency, one cannot say anything about stability of the system. The blowup phenomenon is only applicable when loop transfer function is exactly equal to -1. And that's why phase only at gain crossover is important. In other words, for oscillations, P=-1 is only a necessary condition.

deba_fire said:

The reasoning intuitively is ok, but is wrong. For ex: Assume the open loop response P, at some frequency is -10 i.e. P(jw1) = -10. By your reasoning, it will be unstable. But lets say P(s) = -10 for all frequencies. The closed loop loop response is 10/9, it is stable.

Looking at a single frequency, one cannot say anything about stability of the system. The blowup phenomenon is only applicable when loop transfer function is exactly equal to -1. And that's why phase only at gain crossover is important. In other words, for oscillations, P=-1 is only a necessary condition.

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Where do you get 10/9? Isn't it 1/(1+P) = 1/(1-10) = -9?

Also you say that for oscillations, only 1+P=0 is necessary. But what if 1+P has zeros on the RHP, wouldn't that also be unstable?

I guess maybe the best answer is that a closed loop TF is unstable if 1+P=0+0j
OR if 1+P has some root with real positive part.

And Nyquist is simply telling us if 1+P=0+0j or has a root with positive real part.

(Also can you tell me if you consider RHP to also include poles at the jw axis?)

Closed loop response is P/(1+P). P is the open loop gain. I wrote it wrongly the first time. My apologies.

And Nyquist is simply telling us if 1+P=0+0j or has a root with positive real part. This is what Nyquist plot does. It's a way of figuring out closed loop poles by looking at how the open loop Nyquist plot encircle the (-1,0) point. The criteria of 1+P=0 is only telling if oscillations are possible or not. If 1+P has zeros on the RHP, it will be unstable for sure.

The common intuitive arguments about stability which like
1) if you increase the gain, system becomes less stable.
2) Looking at a single frequency and if the gain is larger than 1 and phase -180, system is unstable
They are true for all pole systems, where the loop magnitude response and phase decreases monotonically. With zeros, these arguments doesn't hold water.

celebrevida said:

Thanks for your answer.

My simplistic understanding was that in a closed-loop feedback system, you have stability if you have both positive gain margin AND positive phase margin.

In the case of CL(s) = G(s)/(1+G(s)H(s))

I!

Click to expand...

The gain margin is evaluated where the gain is 1, not at s=0.

An intuitive way to think of it is this: the phase delay at the inverting input is about equal to the open loop phase divided by the excess gain. For an opamp, the OLG is usually much greater than the loop gain at low frequencies, so the phase shift at the inverting input is small, even if the OL phase shift is large! Thus, the inverting input is in phase with the non-inverting input, and no oscillation results.

Perhaps, if the loop gain is very large, the opamp would oscillate at a low frequency!

bnevins said:

The gain margin is evaluated where the gain is 1, not at s=0.

Click to expand...

The gain margin is evaluated at the frequency where the loop gain phase is -360deg (including the sign inversion at the inv. input).
The phase margin is evaluated at the frequency where the loop gain is unity.

Systems with more than one croosover frequencies or frequencies which give -180º should be analized with Nyquist stability criterion, NOT with Bode.

The foregoing examples indicate that it is not possible to
formulate a Bode stability criterion that is simple to use and
applicable to all possible cases at the same time

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By JUERGEN HAHN, THOMAS EDISON, THOMAS F. EDGAR 2001.

LvW said:

The gain margin is evaluated at the frequency where the loop gain phase is -360deg (including the sign inversion at the inv. input).
The phase margin is evaluated at the frequency where the loop gain is unity.

Click to expand...

Thanks for correcting that!

- - - Updated - - -

CataM said:

Systems with more than one croosover frequencies or frequencies which give -180º should be analized with Nyquist stability criterion, NOT with Bode.


By JUERGEN HAHN, THOMAS EDISON, THOMAS F. EDGAR 2001.

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could you give a link to a good explanation of Nyquest criterion?

I have studied Nyquist criterion with my class notes and books... I do not have link.
Books which has Nyquist Stability criterion explained:

Modern Control Engineering by Ogata
Modern Control Systems by Dorf
Control Systems Engineering by Nise

I haven't had time to fully read this but searching for app notes for related subjects turned up a couple Burr Brown (now TI) app notes that directly address this in the context of composite amplifiers which exhibit two poles at the origin.

https://www.ti.com/lit/an/sboa015/sboa015.pdf

Burr Brown said:

PHASE ONLY MATTERS AT THE INTERCEPT
Another concept fundamental to op amp feedback in composite-amplifier
circuits becomes apparent when you examine
phase shift and stability. Composite amplifiers such as
the one in Figure 10 produce a –40dB/decade slope over
wide ranges both before and after the 1/β intercept. Because
this slope corresponds to a 180° phase shift, frequent concern
over stability conditions arises at points other than that
of the critical intercept.

Click to expand...

And it continues by trying to explain why that's ok