Confused about Pole Zero Compensation of Double Integrator

Andy · 2005-09-26T16:31:57+00:00

Hello All, I have been delving into control theory for the first time since university and am slowly making progress. Until last night, I thought I had pole-zero compensation cracked: if your system has 180 degrees of phase shift with a loop gain of more than one, due to poles causing phase lag as frequency increases, then add some lead phase shift to the system around that point with a pole-zero compensator, right? Gives you a bit of phase margin.So, I now come across a system with a G(s) of k/s^2, and a feedback H(s) = 1. We now have a system that suffers from 180 degrees at ALL frequencies. Yet my book shows it being compensated around the frequency where the gain falls to 1 with a standard pole zero compensator.My question is ( and I'm sure the answer is staring me in the face ) why won't such a compensated system oscillate at frequencies far above or below the compensator's region of operation? OK, say sytsem gain falls below 1 at HF, ruling that out, why won't it oscillate at low frequency, where there is plenty of gain and 180 degrees of phase shift from the double pole at the origin?Andy.

Sep 26, 2005 23 Replies

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AAR 20 years ago

On Fri, 30 Sep 2005 23:27:51 +0100, Andy wrote (in message ):

Yes, it does behave in a very underdamped way, but only if you make the gain too high or too low. Suppose the compensator consists of a zero at -1 and a pole at -10, then with RootLocs you can easily see that for a minimum damping ratio of 0.7 say, the gain must be not less than 20 and not more than 50 (rounded values). Note: these are the values of K in the open loop transfer function when it is expressed with all coefficients of S as 1, i.e. K(S+A)/S^2(S+B). You can get the same results from the Bode plot by shifting it up/down until the phase margin (at the crossover frequency) drops below a certain value. But its not immediately evident what that value is for a closed loop damping ratio of 0.7

AAR

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Andy 20 years ago

Aah, good point. I remembered I have a graph of damping ratio versus phase margin for a second order system, 0.7 is achieved with a phase margin of 65degrees. I had forgotten that a root locus diagram shows all values of gain, so by picking that carefully one can keep to a well-damped part of the curve.

cheers,

Andy.

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RRogers 20 years ago

Neither plot is usefull for design unless the frequency tics are included. I guess the reason that I like Nichol's charts is that the closed loop gain and phase are explicitly shown, gain changes are just vertical displacements, it shows exactly what modifications are needed, and gives me an intuitive/visual way to evaluate margin/robustness. In the end though, since I could achieve that same results with any of the techniques, I just like the picture of a open/closed loop map with the storm center; aesthetics.

Enough said by me. Ray

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Jerry Avins 20 years ago

I think that the more ways one can see a thing, the better one's grasp of it is likely to be. Each representation highlights certain aspects; together they evoke a more complete picture, even if any one of them has all the information.

And by me.

Jerry

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