Transfer function T1 has a zero in right half plane, hence it is a non-minimum phase system, and bode plot is not applicable for such systems. Note that, bode plot is only sufficient, but not necessary condition for system stability/instability. You can go for Nyquist criterion or eigenvalue analysis for stability assessment.

Refer to this document: https://picture.iczhiku.com/resource/eetop/ShKYEeUGEgWPTVnv.pdf

Dear Mr. Lokesh,

As an initial thought regarding the concept involved in this problem, I would like to suggest this article which I have read few days ago. This also contains one of the bode plots you have considered. It may be useful in getting the clarity. Please go through this link:

https://www.quora.com/Can-a-control-system-be-stable-with-negative-gain-margin-and-positive-phase-margin

Please let me know your thoughts after seeing this. I would always look forward for a healthy discussion for mutual benefit

Regards

Satya

Thanks for sharing the link Malyala Venkata Satya Sai Chandra .

The post in there also says that the margins are just how far from the unstable point i.e., GH = 1∠∓180°, irrespective of their (GM and PM) signs.

But, the stability inference from the bode plot of system function T1 in my post is totally contradictory with the above verdict, infact this is why I made this post.

Lets see for others response.

Dear Lokesh Nalla

The missed ingredient of your analysis and interpretation, is to ignore whether the transfer-function is for an open-loop system or a closed-loop system (?). The other missed component is to ignore controller gain in your Bode plots. What is the controller gain K associated to your generated plots (?)

I have attached an image with some screen-shots from Ogata control engineering text book 4th edition, where you may spot that, the gain K is important in the shape of the Bode plot of an open-loop or closed-loop system.

Bode plot is exploited for two main purpose:

1. Experimental determination of transfer functions, based on open-loop investigations.

2. Stability analysis and controller design, for various selections of the gain K.

So instead of concluding and interpret stability condition based on phase-margin and gain-margin, try to get deeper insight into your control system and its block-diagram as well as the controller gain K.

Some remarks:

University professors are accustomed to conduct lectures through a pre-defined machinery routine. Most of them do not care about their teaching outcome, but they are more concerned about to get elevated from an assistant professor to an associate professor and then the full professorship. Some specific time constraints and profit is also accompanied, which is a concern both for teachers and students.

Such as you, I have been taught and told many concepts in university, but when I try to get deeper and independently fathom those concepts, I find much of those college taught concepts devoid of a sturdy foundation. So we may learn too many fundamental concepts through self-study, endeavor and perseverance. As you are honorably doing by your question, and I am doing also by my questions through my ResearchGate platform.

Thanks.

First let me thank you Saeb AmirAhmadi Chomachar for sparing your time in sharing your views.

Let me answer the questions you raised on. The plots shown in the post are for open loop system, one is with controller and the other is without controller. Like everyone does, I also followed the steps of finding stability of open loop system first and then the controller is designed/chosen based on my desired specs.

Here in this post, my concern was and ofcourse is about to know the exact stability inference conditions from the bode plot as what I was taught is failing for some cases.

Dear Lokesh Nalla

I assumed T1 as the transfer function of the open-loop system, and calculated the closed-loop transfer function by a simple unity negative feedback.

T.F. (closed-loop)= GH/(1+GH),

where I assumed G=T1, and H=1.

For a simple T1=(s+a)/(s^2+bs+c).

And calculated the transfer function T.F. which is simply of the same format of T1, but with different coefficients. Hence, your controller block, might be a bit complicated, while you have not reported your controller transfer function. Moreover, the denominator of your closed-loop Transfer Function has no constant term:

2.483e-05 s^4 + 77.99 s^3 + 1364 s^2 + 3.142e06 s + (constant number?),

hence, a pole-zero cancellation might have occurred in your closed-loop T.F. resulting in an unstable sub-system (zero-dynamics).

Moreover, as Malyala Venkata Satya Sai Chandra referred you to:

https://www.quora.com/Can-a-control-system-be-stable-with-negative-gain-margin-and-positive-phase-margin

based on those discussions, it appears only a fixed judgement that for a stable system both the Gain Margin (GM) and Phase Margin (PM) should be positive, is not always satisfactory as in too many cases such an assertion might be on a loose base.

positive Gain Margin (GM) and positive Phase Margin (PM), only characterize and quantify the behavior of a system when encountered with disturbance and destabilizing inputs, or in other words degree of robustness and resistance against disturbance. Hence, I guess you should not be prepossessed with certain interpretations on these values, such as what you kindly posed in your question.

Feel free to share with me any further comment or query you have on.

Thanks.

Dear Lokesh Nalla

Please take a look at the image attached. It is a screen-shot from Ogata control engineering text-book, 4th edition. Thanks.