Really Simple Control Theory Question?

On Thu, 7 Apr 2005 19:35:48 +0100, Fred Stevens wrote


Perhaps I can chip in here.
For simplicity, consider just a single complex pole. I know they always exist in pairs but visualisation is easier with just one. Now if you plot |F(s)| at every point of the S plane on an axis out of the plane, you get a surface which looks like a stretched rubber membrane being pushed up from below to infinity at the pole. On the jw axis (sigma=0) this ||F(s)| surface is |F(jw)|, which by definition is the magnitude part of the frequency response.
Alan
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We seem to be at cross purposes here somehow, I'm not sure how to resolve this. Firstly, my mention of the system Bode plot being apparent from slicing along the jw axis is not something I have deluded myself into believing, I have a book in front f me 'Analog Electronics, Ian Hickman, Newnes' that shows the pole zero plot in the s-domain of a first order low pass filter, and they point out that in the special case where sigma =0 i.e. we can only traverse the jw axis, the transfer function directly above that line is a Magnitude, or Bode Plot. In fact a good quote is "The vertical slice through the F(s) surface, along the jw axis is the same as the amplitude response".
Why this should be difficult to see I don't quite understand: the height of the pole zero plot is the transfer function, or gain, or Vout/Vin of the filter, and the input sinusoid frequency varies along the jw axis, so we see the filter's gain change with input frequency.
As for your example, I'm not an expert on this subject as you may have already guessed, but you seem to be suggesting that your example with a complex pole pair has two different gain factors at the same frequency. How can a real system have two different gains simultaneously at the same frequency???? If you took a second order system, measured the gain at a particular w, then changed the damping factor and measured the gain again, yes, the gain would have changed, but that would be a whole new system, a whole new pole zero plot. Surely the pole-zero plot for a system is single-valued at any point? Apologies if I've misunderstood your point,
Andy
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Okay, I see what you are getting at - then we are saying the same thing, I think, in different ways.
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Fred Stevens wrote:

Does http://cnyack.homestead.com/files/alaplace/laptr1.htm help?
You can play around with the plot a bit at http://cnyack.homestead.com/files/alaplace/lt1rpm.htm
http://dspcan.homestead.com/files/Ztran/zlap.htm might also interest.
Jerry
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Aaah, thanks Jerry, a picture is worth a thousand words. I shall have a better look at these sites tonight and see if they're a good resource.
Fred, the above link takes you to the exact pole zero plot I am investigating. If you look at the 'Surface Plot of Magnitude", you will see a red line, which is the surface contour directly above the jw axis. This is what I am calling a Bode Plot, frequency versus response. HTH,
Andy

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