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Does someone know how to analysis stability of closed loop s

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haulm

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Does someone know how to analysis stability of closed loop system.
I mean the closed loop system is analyzed directly and is not tranfered to the corresponding open loop which can be analyzed by Bode chart.
Is there any paper?
Thanks.
 

Re: Does someone know how to analysis stability of closed lo

I am not sure what you are starting with. Do you have a block diagram of the system with an open loop transfer function and a controller with feedback? All you have to do then is get the closed loop transfer function and check the poles. If all poles are in the left half plane, system is stable....

If the representation is in state space form, you close the loop and get the closed loop matrix A and check the eigenvalues; if all in the LHP, system is stable....

This the ABCs of a first course in control theory.

Dr. Salah Zenieh
 

Re: Does someone know how to analysis stability of closed lo

The Nichols chart is a tool for direct analysis of a linear closed loop system. It provides both gain and phase margin. Any good text on linear control systems will do. See, for example, D'azou and Houpis, or search for "Nichols Chart" on the web.
~
You can get a rough idea of the stability by looking at the "Q" (ratio of imaginary to real componets) of the closed loop poles. The higher the Q, the more oscillatory will be the response. Of course, the real components must be negative, otherwise you have an unstable system.
Regards,
Kral
 

Re: Does someone know how to analysis stability of closed lo

Thank Dr. Salah Zenieh and Kral.
Yes, there are many judging methods for closed loop system's stability. And Bode diagram is a simple way to decide closed system's stability by its open loop transfer function. I want to know how I can decide closed system's stability directly by its closed loop transfer function.
Because it is difficult to deduce the open loop transfer function when the system is controlled by multi-loop controllers. But the closed loop transfer function is easy to deduce by mathmatics way.
 

Re: Does someone know how to analysis stability of closed lo

I am still not sure why this is a problem in your case. I am not trying to make this a trivial matter. It is trivial when the problem is simply an example in a text book or a homework problem; most of the time that is. If you are looking at a complex real life problem, some problems may arrise.

It may help if you describe the problem in more detail... meanwhile, here is another advice.

If you are dealing with a large order system, transfer function representation is a bad idea to use for checking stability. For instance, say you have a 6th order transfer function with controllers in the feedforward and feedback paths, trying to close the loop numerically, say using Matlab, will most probably give you wrong asnswer. The numerical errors introduced in multiplying polynomials will get you.

Take for instance the following example...

p=[1 1];
for i = 1:49,
p=conv(p,[1 1]);
end;
roots(p)

save this code into an m-file in matlab. Let me explain. The polynomial is 50th order. This is simply obtained by asking Matlab to multiply (conv) 50 times (s+1)
factors. The produced polynomial is of order fifty. Clearly, the roots of this polynomial should be s = -1 repeated 50 times.... However, Matlab reports fifty *different/distinct* roots, none of them is -1!!!!

This is the numerical problem that Matlab has when dealing with polynomial multiplication. Hence, you should never attempt to close loops numerically using polynomial tools in Matlab. Instead, you must represent all the transfer functions involved in the loop description into state-space and do the stability analysis there.

Hope this helps, I can be of more help if you describe the problem in a bit more details.

Dr. Salah Zenieh
Dynamics and Control

Added after 1 minutes:

I am still not sure why this is a problem in your case. I am not trying to make this a trivial matter. It is trivial when the problem is simply an example in a text book or a homework problem; most of the time that is. If you are looking at a complex real life problem, some problems may arrise.

It may help if you describe the problem in more detail... meanwhile, here is another advice.

If you are dealing with a large order system, transfer function representation is a bad idea to use for checking stability. For instance, say you have a 6th order transfer function with controllers in the feedforward and feedback paths, trying to close the loop numerically, say using Matlab, will most probably give you wrong asnswer. The numerical errors introduced in multiplying polynomials will get you.

Take for instance the following example...

p=[1 1];
for i = 1:49,
p=conv(p,[1 1]);
end;
roots(p)

save this code into an m-file in matlab. Let me explain. The polynomial is 50th order. This is simply obtained by asking Matlab to multiply (conv) 50 times (s+1)
factors. The produced polynomial is of order fifty. Clearly, the roots of this polynomial should be s = -1 repeated 50 times.... However, Matlab reports fifty *different/distinct* roots, none of them is -1!!!!

This is the numerical problem that Matlab has when dealing with polynomial multiplication. Hence, you should never attempt to close loops numerically using polynomial tools in Matlab. Instead, you must represent all the transfer functions involved in the loop description into state-space and do the stability analysis there.

Hope this helps, I can be of more help if you describe the problem in a bit more details.

Dr. Salah Zenieh
Dynamics and Control
 

Re: Does someone know how to analysis stability of closed lo

Thank Salahz's help.
I have a controller with dual-loop for Inverter plant. I do not know how to analyze its stability. Can the feedforward influence the analyzing result?

See the attachment please.
 

Re: Does someone know how to analysis stability of closed lo

well i think you can find usefull information in Richard dorf'd Control systems engineering book
 

Re: Does someone know how to analysis stability of closed lo

below homepage is very good for analog design. Have a look.

**broken link removed**
 

Re: Does someone know how to analysis stability of closed lo

there is a paper discussing using laplas transform to analysis close loop directly, the paper's name is:

"Low-voltage low-power fast-settling CMOS operational transconductance amplifiers for switched-capacitor applications.pdf"

on IEE proceedings 20040414, by Yavari and shoaei, you can try it out.
 

Re: Does someone know how to analysis stability of closed lo

Thank very much.
Does someone have IEEE papers which analysis the close loop stability directly by using laplas transform ?
 

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