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Bode Diagram of Conditionally Stable System (Read 40703 times)
pancho_hideboo
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Bode Diagram of Conditionally Stable System
Sep 20th, 2007, 12:20am
 
Hi.

See attached figures. These are Bode Diagram and Nyquist Diagram of conditionally stable system.

In Bode  Diagram, Phase Margin at gain crossing point(green point) is positive.
So as far as seeing phase margin, this system is stable.
On the other hand, phase is across -180deg at red, yellow and blue points.
Although gain is smaller than 0dB at blue point, but  gain is larger than 0dB between red and yellow points.
I understand this system is stable at blue point.
But why is this system stable between  red and yellow point ?

While in Nyquist Diagram, locus doesn't enclose a point of -1+j0, so we could conclude this system is stable.
Also a root locus shows this system is stable.

In general the system which gain is larger than 0dB at phase crossing point(-180deg) could be unstable.

How should I interpret this bode diagram ?  :-/

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Condinally_Stable_System.jpg
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sheldon
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Re: Bode Diagram of Conditionally Stable System
Reply #1 - Sep 21st, 2007, 6:44am
 
Pancho,

  It has been a while but I think that the actual condition for stability is something
like a slope of 20dB/dec at unity gain. If the phase crosses through -180 degrees
at gains higher than 1, the system can't oscillate. In addition, I think that is also
a requirement that the slope requirement also applies at each multiple of -180
degrees of phase shift. Let me dig around and see if I can find a good reference.

                                                                         Best Regards,

                                                                            Sheldon
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monte78
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Re: Bode Diagram of Conditionally Stable System
Reply #2 - Sep 24th, 2007, 7:35am
 
Hi,

I think you can take a look to this discussion:

http://www.designers-guide.org/Forum/YaBB.pl?num=1182388268

Nyquist criteria is the correct way to determine if the system is stable.

Best Regards,

Monte

pancho_hideboo wrote on Sep 20th, 2007, 12:20am:
Hi.

See attached figures. These are Bode Diagram and Nyquist Diagram of conditionally stable system.

In Bode  Diagram, Phase Margin at gain crossing point(green point) is positive.
So as far as seeing phase margin, this system is stable.
On the other hand, phase is across -180deg at red, yellow and blue points.
Although gain is smaller than 0dB at blue point, but  gain is larger than 0dB between red and yellow points.
I understand this system is stable at blue point.
But why is this system stable between  red and yellow point ?

While in Nyquist Diagram, locus doesn't enclose a point of -1+j0, so we could conclude this system is stable.
Also a root locus shows this system is stable.

In general the system which gain is larger than 0dB at phase crossing point(-180deg) could be unstable.

How should I interpret this bode diagram ?  :-/


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pancho_hideboo
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Re: Bode Diagram of Conditionally Stable System
Reply #3 - Sep 24th, 2007, 8:42am
 
Hi, Monte.

Thanks for append.

But my question is not resolved.

Conditionally stable system(also called as Nyquist stable sytem) is stable as far as system is not nonlinear.
This can be also showed by root locus method.
But a positive feedback is formed at red and yellow points with larger than 0dB gain.
Why is a system stable ?
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« Last Edit: Sep 24th, 2007, 10:26am by pancho_hideboo »  
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HdrChopper
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Re: Bode Diagram of Conditionally Stable System
Reply #4 - Oct 3rd, 2007, 2:12pm
 
Hi,

Bode stability criterion cannot be applied in two particular cases (the hypotheses for stating such criteria are violated in cases 1 and 2):

1) If the open loop gain is unstable: that means it has at least one right-hand plane pole
2) If the open loop frequency response has more than one critical frequency, where the critical frequency is the one at which the phase is -180 degrees.

Both must be analyzed by means of the Nyquist criteria, as it was suggested before

The case you are showing corresponds to #2 where Bode stability criterion cannot be applied.

Regards
tosei
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Keep it simple
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pancho_hideboo
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Re: Bode Diagram of Conditionally Stable System
Reply #5 - Oct 5th, 2007, 7:19am
 
Hi, tosei.

Result of nyquist criteria is no more than result which shows a system is conditionally stable.

My question is why a system can be stable even though a positive feedback is formed at red and yellow points with larger than 0dB gain.

I would like to get some clear reason or explanation why a system can be stable.
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« Last Edit: Oct 5th, 2007, 10:04pm by pancho_hideboo »  
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aaron_do
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Re: Bode Diagram of Conditionally Stable System
Reply #6 - Apr 29th, 2009, 8:30am
 
Hi pancho_hideboo,


did you ever figure out the answer to this question? I happened across this post and it peaked my curiosity. The furthest I can get is that the closed-loop response has no right-hand-plane poles, but I realize that this is not really the kind of intuitive answer that you are looking for. I would really like to know the answer too.


thanks,
Aaron

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Re: Bode Diagram of Conditionally Stable System
Reply #7 - Apr 29th, 2009, 7:24pm
 
Let me ask questions to approach the answer.

Is that "gain" your:
  open loop gain
  'loop gain', or
  closed loop gain ?

---What is your feedback factor?    (see Return Ratio discussions)
If FBF takes you to the phase inflection you are definitely in trouble.

If it is a unity gain config, you may be OK.
Interesting to ponder.   Huh
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Re: Bode Diagram of Conditionally Stable System
Reply #8 - Apr 29th, 2009, 7:44pm
 
Its not my diagram, but that should be the loop gain, so feedback factor is already taken into account. The system is stable, the question is simply, "why is it stable if the overall feedback is positive and the gain is more than unity at certain frequencies?"...


Aaron
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pancho_hideboo
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Re: Bode Diagram of Conditionally Stable System
Reply #9 - Apr 29th, 2009, 8:25pm
 
wave wrote on Apr 29th, 2009, 7:24pm:
Is that "gain" your:
  open loop gain
  'loop gain', or
  closed loop gain ?
Of course, This is Open Loop Gain.

wave wrote on Apr 29th, 2009, 7:24pm:
If FBF takes you to the phase inflection you are definitely in trouble.
I can't understand what you mean.

This Bode Diagaram and Nyquist Diagram are very typical example of conditionally stable system which we often see in common control therory text book.
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« Last Edit: Apr 30th, 2009, 6:50am by pancho_hideboo »  
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pancho_hideboo
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Re: Bode Diagram of Conditionally Stable System
Reply #10 - Apr 30th, 2009, 6:49am
 
aaron_do wrote on Apr 29th, 2009, 8:30am:
Hi pancho_hideboo, did you ever figure out the answer to this question?
I still don't have any explanation which I can be enough satisfied with. Undecided

However the following is a little related to this conditionally stable system.
http://www.designers-guide.org/Forum/YaBB.pl?num=1234428781/7#7
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« Last Edit: Apr 30th, 2009, 7:59am by pancho_hideboo »  
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wave
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Re: Bode Diagram of Conditionally Stable System
Reply #11 - Apr 30th, 2009, 4:01pm
 
pancho_hideboo wrote on Sep 20th, 2007, 12:20am:
Hi.

But why is this system stable between  red and yellow point ?

How should I interpret this bode diagram ?  :-/



Pancho - for your specific Q, the operating point are the conditions of In-stability.  Depending on your gain configuration (FeedBack Factor), will determine where you are operating.  G-BW is constant, but you have to pick which gain to avoid that BW region.
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pancho_hideboo
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Re: Bode Diagram of Conditionally Stable System
Reply #12 - Apr 30th, 2009, 7:20pm
 
wave wrote on Apr 30th, 2009, 4:01pm:
Pancho - for your specific Q, the operating point are the conditions of In-stability.
This sytem is stable. And we can't define Q.

wave wrote on Apr 30th, 2009, 4:01pm:
Depending on your gain configuration (FeedBack Factor), will determine where you are operating.
G-BW is constant, but you have to pick which gain to avoid that BW region.
???
FeedBack Factor is 1.
Do you understand a concept of "conditionally stable system" ?
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Re: Bode Diagram of Conditionally Stable System
Reply #13 - May 9th, 2009, 10:18pm
 
here's my thought,

If you have an input signal at the red dot frequency, it will get amplified and then add to the input after going through the loop. In an ideal system, this will cause the input and output to rise indefinitely. However, in a real system, the system will enter its non-linear region where the gain at the red dot frequency will start to decrease to the point where it becomes zero (i.e., it latches). If you exclude latching up from being unstable then the system is stable but latch-able!

I'm not an expert in control theory so I'd really like if someone can simulate this kind of system in cadence or matlab and see what actually happens.
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pancho_hideboo
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Re: Bode Diagram of Conditionally Stable System
Reply #14 - May 9th, 2009, 11:10pm
 
ndnger wrote on May 9th, 2009, 10:18pm:
If you have an input signal at the red dot frequency, it will get amplified and then add to the input after going through the loop.
In an ideal system, this will cause the input and output to rise indefinitely.
Not correct.
From nyquist criteria, there is no pole in RHP(Right Half Plane) for closed loop transfer function.
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