Nyquist by hand

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A Nyquist plot is a polar plot of frequency and phase response. Magnitude -- linear, not dB --is the radius and phase is the angle.

For a chosen frequency, place a dot in the right place on polar paper. (I write the frequency as a label.) Plot enough dots to see the curvature -- more where the curve bends more sharply, then connect the dots.

Jerry

What I meant: A Nyquist plot is a polar plot of amplitude vs. phase response with frequency as a parameter. Magnitude -- linear, not dB -- is the radius and phase is the angle.

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Jerry, I believe that is so called Nichols plot. Nichols plot gives amplitude vs phase chart. That is different from Nyquist plot. For example if feedback transfer functions is T =3D G/(1+GH), usually in order to determine stability of a system, its Nyquist plot is drawn for GH and observed if it encircles point (-1,0) and how many times. Nyquist plot represents simple graphical form of GH(s) mapping. Since s =3D a+jw, GH(s) maps that complex domain into new complex variable domain u+jv. Nyquist plot represents Im{GH(s)} vs Re{GH(s)} plot or "v" vs "u" plot. Procedure to draw it by hand can be very tedious and with many software pacjkages available today, I'm surprised that someone ask this question. However there are some formal rules that can be used when drawing Nyquist plot for simple systems. Usually you devide GH(s) into portions for s =3D jw. Then you observed how portion from w =3D 0+ to w =3D +inf is mapped. Then you try and see how portion from w =3D +inf to w =3D -inf is mapped. You do this by findind limit values (lim). Nyquist plot is always symmetrical around x (Real) axis. Like I said, constructing Nyquist plot by hand is very complex and difficult for most but few simple examples. I usually use tables with graphical plots or some powerful software tools such Matlab or Scilab. I would recommend Scilab, because it is very powerful and entirely free alternative to expensive Matlab.

Best regards, Asim Vodencarevic

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Plotting Re vs. Im in rectangular coordinated gives the same curve.

Jerry

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A comment--I would think everyone here is taking | has taken one or more control system classes. Therefore I would assume everyone has at least a student version of Matlab available. Isn't this correct?

dave y.

I graduated in 1962. There was no Matlab then.

Jerry

=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF I graduated in 1975. There was no Matlab then either that I know of. Go ahead and post Matlab script. Scilab can translate most Matlab to Scilab.

Does anybody really use Nyquist or Nichols plots? I know poor students are forced to waste their time with them. I have never seen the need.

Peter Nachtwey

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Well it is very easy to do both in Matlab and Scilab. Function is called "nyquist". For example system is g =3D 1/s(s+1) Here's Matlab code: num =3D 1; den =3D conv([1,0],[1 1]); g =3D tf(num,den); nyquist(g);

Scilab code is: s =3D poly(0,'s'); g =3D syslin('c', 1/(s*(s+1))); nyquist(g)

Except to pass exams, I didn't see any real use of Nyquist plot. I know far better and easier ways to check stability. Gain and phase margins are very easy to obtain from Bode plot. Bode plot has much more information to an engineer than Nyquist (or Nichols) plot.

Regards

Even in 1988, when I got my degree, one was expected to construct Nyquist plots with a calculator and graph paper.

I use Nyquist plots all the time. If you're going to do frequency-domain tuning, then you have to use Nyquist and Bode plots extensively.

But you'll spend less time installing Scilab and learning how to make it cough up a Nyquist plot then you will doing three or four Nyquist plots by hand.

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I can see using Bode plots. It would be much easier to see the inflection points on a Bode plot.

Peter Nachtwey

You can glance at a Nyquist plot and see the maximum sensitivity -- just look at how close the trace gets to the -1 point. Bode is better for seeing the phase and gain crossings, for predicting how fast your system will operate, and for getting an idea of how you next need to adjust the controller (or that you're done tuning that particular controller topology).

I usually plot both when I'm tuning a system.

Electrocardiograms usually involve several traces plotted against time. Some years ago, my father-in-law, a cardiologist asked what I thought of a new scheme some company was pushing called a vector cardiogram. He explained that it was made by plotting one "lead" against another. Some cardiologists were better able to make certain diagnoses from it. I showed him how to plot a vector cardiogram after the fact if it was wanted; no new machine was needed. He concluded that while old hands (like him) could see all the detail in a standard 'gram, he could see how the new plot might bring out detail that a greenhorn might miss. Nevertheless, by discarding time, there was actually less information, and so aspiring cardiologists (he was also a professor) would do better to learn to read the vs.-time tapes well enough not to need the "vector" plots. Nowadays, when the computer collecting the data to make the standard plots can just as easily print out a vector cardiogram as well, the technique is returning to favor. My own cardiologist thought it was the cat's meow until I explained to him what it is. Scary, no?

The relation between vector and standard cardiograms is the same as that between Nyquist or Nichols and Bode plots. Nevertheless, while the first two contain less information than than last, they can make certain features more evident.

Jerry

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So what do you do with it? How do you calculate the gains then? You have never said. I still prefer my pole and zero placement.

Peter Nachtwey

I use the Nyquist plot to see how close to the 3dB or 6dB sensitivity circle I am, and I use the Bode plot to guide me in tuning my gains.

I use a heuristically guided stochastic conjecture refinement process (I make a wild-ass guess, then I use my experience to guess again until I get it right).

Which requires that you make an educated guess at the plant model, as well as placing (guessed-at) constraints on your target pole locations. At that point, if someone say "how much can things vary before this system won't perform to spec" I can't see how you can do anything but shrug.

I prefer to put my guesses where they don't have as much effect on the stability analysis.

But to each their own -- if you believe it works best, it probably does for you.

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Of course there are those of us who think that Nichols is that best way to go, but have given up on convincing anybody with words. Looking at Wikipedia, maybe I will spruce that entry up and see if that helps to convince people.

RayR

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I'll ask you the same thing I asked Tim. Why bother? What do you do with it? How do you use it to calculate gains? Where do you get the information to plot it? What does an ideal Nichols plot look like. How does one achieve the ideal or desired Nichols plot?

I prefer Nichols plots over Nyquist too but they are both the showing the same information in different ways. Which way allows the student to calculate gains?

Peter Nachtwey

Let me refine/expand the question a little. In the controls text is have Nise (4th ed) there is reference to a d-plot that encircles the entire positive real axis. The magnitues of the vectors are then calculated (approximated) as you move around the d in a clockwise direction, this corresponds to increasing frequency. The examples I have seen and tried to follow seem inconsistent. I searched online for a tutorial or other explanation but didnt come up with anything very detailed. Astrom and Haglund (2 ed) reference using the plots for loop tuning, this is what renewed my curiosity. The largest assumption that I am making assumes that I can define a transfer function. Thanks to many for your prior and follow up discussions

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for material for the Wikipedia entry I came across the above. I think Fig 6 answers your questions better than I could in words. Scilab has the plot as a command so you just have to enter the open- loop xfer function. There are minor problems with black and chart; but I can give you hacked replacements. I should probably fix the hack for black though. Since you are interested I will expand. The amount of gain required for a 3 db peaking is obvious as is the positioning of breakpoints to extend the bandwidth or to cool off peaking. The phase error on closing the loop is immediately evident. The Nichols chart like the Bode plot can be done with experimental data and any uniform gain is simply movement in the vertical direction. Of more importance to me; the "topology" or nearness of a solution to the "black hole" gives a constant feedback about how stable the closed loop system will be when various breakpoints or gain are varied. Several alternative compensations can be overlayed for comparison. There is a paper on the internet exploiting this by drawing tubes around the curves to insure a certain levels of robustness.

I can go on (and on ...) :) RayR

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I think that the Nichols chart is one of the most useful and neglected tools in control system design!

fred.