Confused about Pole Zero Compensation of Double Integrator

Somewhere on the web there's a nice page that goes into the myth surrounding the Barkhausen criterion. The Barkhausen criterion itself states that if a system's gain is 1 and it's phase shift around the whole loop is 0 (or 180 degrees to us control types who just assume that summing junction) then the system will be _marginally_ stable with a double, usually resonant, pole pair.

What the Barkhausen criterion does _not_ state* is that a phase shift to the summing junction of 180 degrees and a gain greater than 1 will lead to instability -- there are plenty of systems contrary to that, not least of which is a double integrator with PID compensation which has

270 degrees of phase shift at low frequencies, diminishing to less than 180 at the gain crossing, then going back up at higher frequencies. Such a system is a very stable and useful thing but it has _two_ gain margins -- the gain can neither drop too low nor raise too high if the system is to remain stable.

A Bode plot will tell you if a system that is stable _now_ will become unstable with a change in gain or phase. In general it won't tell you if a system is stable, or what you need to do to make an unstable system stable. If you know the number of unstable zeros ahead of time you can tell if a system is stable by use of a Nyquist plot. If you're handed a system in a bag, however, you're left trying to figure out the number of unstable zeros, in which case you're back to Bode plots**.

• Well, maybe Barkhausen said it but if so it was in the 1920s so he's excused.

** Although I've recently taken to plotting both a Nyquist plot and a Bode plot, with a circle of radius 1/sqrt 2 on the Nyquist plot as a quick & dirty indicator of gain & phase margins.

On Mon, 26 Sep 2005 17:31:57 +0100, Andy wrote (in message ):

The compensator has some effect on gain and phase even at frequencies well outside its region of operation. The phase shift is always a bit less than

180 degrees when the compensator is present, even at very low frequencies where the gain is much greater than 1. Personally, I always find it easier to see what's happening from a root locus plot than from the Bode plot. The root locus of the uncompensated system consists of a pair of loci which start at the origin and go along the positive and negative imaginary axis respectively. If you add a compensator consisting of a zero and a pole on the negative real axis (zero closer to the origin than the pole) the effect is to bend the loci into the left half plane so that for all gains greater than 0 the closed loop poles are to the left of the imaginary axis i.e. the system is always stable.

I suggest to download a copy of my root locus plotter program (RootLocs) and see the above for yourself - a picture is worth a thousand 1000 words. You can find it at:

snipped-for-privacy@btinternet.com/

It also does closed loop frequency and transient plots.

AAR

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I can think of no better way to visualize or explain conditional stability than using a Nyquist plot. I have plotted them for systems with unknown transfer functions from response measurements at selected frequencies and interpolating. (The same data give a Bode plot too.)

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Jerry

Yes, I see that, but I can't see the point in congratulating yourself over getting a clear

45 degrees of phase margin at one target frequency, when you are getting a lot less than that at other frequencies ( l.f. anyway ) in this particular example. The book I am using is a 1986 edition of 'Modern Control Systems' by Richard C. Dorf. I imagine the theory hasn't changed much since then. I shall reread the worked example they give, I'm sure it will eventually click why they bother pole-zero compensating a fixed 180degree phase system anyway when it looks ready to oscillate at all frequencies!

I have gone over the S-plane and Root-locus method and will try what you suggest. All the same, Bode plots seem easier to produce in the first instance ( without the aid of a computer program, that is ).

Thanks for that, I visited tour site and blagged three of your programs, Rootloc, LTinvert and the Parfrac program. All look very useful, thanks very much, should ease my learning a little bit!

Regards,

Andy.

On Wed, 28 Sep 2005 1:12:33 +0100, Andy wrote (in message ):

According to my book, Bode's first theorem implies that the phase margin at the crossover frequency (where the attenuation/log frequency curve crosses the the 0 dB line) is weighted more heavily towards determining system stability than it is at any other frequency, the weighting factor becoming progressively smaller the further one departs from the crossover frequency

So strictly speaking, as far as it is being used as a measure of stability, I think the concept of phase margin is valid only at the crossover point and its a mistake to apply it with the same "weighting" at other frequencies. In other words, the ground is effectively shifting under your feet with the Bode plot - which is another reason why I prefer root locus.

Hope this helps

Alan

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When I went to school, phase margin was the difference between actual phase shift and 180 degrees at *the open-loop zero-dB-gain frequency*. A Nyquist plot shows why that should be.

Jerry

On Thu, 29 Sep 2005 14:14:42 +0100, Jerry Avins wrote (in message ):

Jerry Avins wrote

I agree, and didn't intend to say otherwise. I was simply trying to convey the idea that at frequencies each side of crossover frequency, the difference between the actual phase shift and 180 degrees has a progressively lesser effect on stability than it does at the crossover frequency. Hence the term "phase margin" should be reserved for the crossover frequency only, just as you point out.

AAR

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OK: we're tuned to the same channel. That still leaves Andy wondering how there can be in-phase feedback without oscillation at some frequency with gain. I haven't given it enough thought to write a short explanation and don't have time for a long one. It's 1:40 AM after a long day.

Jerry

On Fri, 30 Sep 2005 6:39:50 +0100, Jerry Avins wrote (in message ):

Good.

That still leaves Andy wondering

It shouldn't. At frequencies above the crossover the gain is less than 1. At frequencies below the crossover the phase is always less then 180 degrees due to the compensator. At very low frequencies the phase is only very slightly less than 180, but because we are a long way from the crossover frequency the effect of this small "margin" (dangerous word!) on stability is less than it would be at the crossove frequency (according to Bode's first theorem). That is the point I'm trying to make.

AAR

Andy, Could I suggest looking at Nichol's charts? They don't seem popular but I have found them to be the best in evaluating/picturing margins, sensitivity, etc..... I haven't found a web site extolling the virtues and uses yet; perhaps I should write a wikipedia article. This contains an introduction:

are to construct and program; and in fact several popular (some free) provide them for you.

Ray

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Succinctly, why is phase shift remote from the unity-gain point unimportant? How do I put into words the information from a Bode plot that a Nyquist plot extracts and makes clear?

(For the curious, an Nyquist plot is a polar graph. The radius is gain and the angle is phase. Frequency, if shown, is given by tick marks along the curve. The same curve makes sense in Cartesian coordinates as the usual s plane. It's just another way to look at it. A system is stable if the point s = [-1, 0] is not enclosed by the curve. See

Jerry

A Nichols chart is a Cartesian plot of gain vs. phase. A Nyquist plot is either polar gain vs. phase or Cartesian real vs. imaginary, which are the same things. The information Nichols and Nyquist convey is plainly the same, and (without frequency ticks) less than a pair of Bode plots.

I'm sure it's partly a matter of what one is used to that determines one's preference. I can say for myself that the easy transition from Re/Im to magnitude/phase attracts me to Nyquist plots.

Jerry

On Fri, 30 Sep 2005 15:46:43 +0100, Jerry Avins wrote (in message ):

I didn't say it was unimportant, only that its effect on system stability is less. I can't give a physical explanation of why that is - you would need to stare a long time at the complex mathematical expression that is Bodes first theorem to work that one out.

I don't know. Personally I'm no great fan of either of them so I don't want to argue their relative merits.

AAR

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I thought you did, and I agreed. As far as stable or not, it doesn't matter at all. As far as stability goes either [-1. 0] ([1, pi]) is enclosed, or is isn't. The decay time of transients is another matter.

Jerry

On Fri, 30 Sep 2005 21:22:47 +0100, Jerry Avins wrote (in message ):

Jerry, I'm afraid I don't follow your point. Are you questioning my answer to the original question and if so could you be more explicit.

AAR

Haven't looked at Nichol's charts yet but they're on the list. Jerry has summed up my dilemma: A system with two integrators has 180 degrees of phase shift at all frequencies. Even if compensated out at the unity gain frequency, with a phase boost circuit, that leaves a phase shift of 179.9 degrees or whatever at some low frequency where the compensator has run out of puff, and where a double integrator has plenty of gain. A step input applied to this system ( in my imperfect understanding ) ought at the least to make the output ring with a very underdamped oscillation.

I have looked at it on an S-plane root-lous diagram and the same problem occurs ( as it should ). A compensated double integrator pulls the root locus away from the jw axis and into the left hand plane, but it still snakes back to graze the jw axis at low frequency ( near the origin ) as it connects up with the poles at the origin, so it still looks marginally stable at low frequencies. I can accept that it will not oscillate, but the settling times might be large.

The question occurs, since a normal system with say, three poles will easily manage 180 degrees of phase shift at one particular frequency, and if there is gain at that frequency it will oscillate at that frequency.

A double integrator in theory, uncompensated, could oscillate at any and all frequencies up the the unity gain frequency. What am I to make of that? A strange beast indeed. I haven't revisited the example given in Dorf yet as I want to take a fresh look at it. In the mean time I'm just working through some more straightforward examples of lead, lag, and PID compensation.

Thanks for all the comments,

Andy.

One point is that the open-loop phase at DC or any frequency other than the zero-dB point(s) unrelated to closed-loop stability. It's not less important; it's not important at all. The other point is that this is not an intuitive result.

Jerry

On Sat, 1 Oct 2005 0:03:41 +0100, Jerry Avins wrote (in message ):

That may be so but I don't think it answers the original poster's question, which is essentially WHY is it that only the phase margin at the crossover matters. I think you have to look to Bode's theorem for that and then you get into the weighting factor thing.

There's a nice explanation of Bode's theorem (and the weighting factor inherent in it) in the book "Control System Design" by Stanley M. Shinners, which is probably out of print now.

AAR

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It was a very good question and I wish I had a better answer. It's true that the answer is buried there in the math, but I don't really feel that I understand something if math is the only explanation I know.

Jerry