I have a basic question about stability of a system whose phase approaching DC is more negative than -180 degrees. For example, consider a the transfer function (s^2 + 2s +25)/(s^3). It is obvious looking at a root locus for this system that it is conditionally stable (i.e. stable for sufficiently large gain). The Bode plot of the open loop system has phase that starts at -270 and eventually approaches -90 degrees. Picking a large feedback gain, the system is stable and notice that the open loop gain can be greater than 1 when the phase passes through -180 degrees.
Clearly this system can be made stable for sufficient gain, so my question is: what is wrong with the following thought experiment? Let say the open loop gain at the -180 phase crossing is 1.5 (which is clearly possible). Then I inject a sinusoid into the closed loop system at the frequency which corresponds to the -180 phase crossing. This signal will be amplified and will then be a larger input at the same frequency and phase at the input to the system. This will continue and lead to unbounded oscillation. Obviously this does not happen. Why? That is, why is the 180 phase crossing (with larger than unity gain) stable for an increasing crossing, but not for a decreasing phase crossing.
As you see, a Bode plot can't tell you if a system is stable or not (unless you see a gain of exactly -1). It will only tell you how far you need to go to make a known stable system unstable or visa-versa.
I can't make the math fit to your intuition, but I can recommend that you check out the Nyquist plot. It lets you determine the stability of the system where the Bode plot cannot.
By the way, both the Bode and Nyquist plots can be used on sampled-time systems as easily (easier in the case of the Nyquist) as with continuous-time systems: instead of sweeping the frequency along the imaginary axis of the s-plane, sweep it along the unit circle on the z-plane. In either case you're interested in seeing if all the poles are in one region (to the left or inside, respectively), and that's really what both techniques are doing.
---------------------------------------- Tim Wescott Wescott Design Services
In your continuous time system I don't think you can consider the signal passing round the loop in the manner you describe. The frequency response you is a steady state response (particular integral in differential equation terminology), there is an initial transient response (complimentary function) which, if stable, leaves the closed loop in its final state. If not stable then it really doesn't matter what the input is!
I agree with Tim, the Nyquist is the way to test/check if a transfer funstion of a system is stable, a quicker way is also to use the root locus!! But Nyquist is more powerful, but also more complicated.
Having earned your admiration I hesitate to say this, but I don't usually use the Nyquist plot. This is partially because it's complexity can be confusing (particularly if you have or suspect unstable zeros), but also because it doesn't tell you about what frequencies your margins are happening. Once you've done it a few times you can glance at a Bode plot and have a good idea of the general character of the time-domain response; you can't do this with Nyquist.
What I usually do is verify that the system is stable at a point, then use that as a reference when shoving Bode plots around -- and I usually try to allow for greater gain margins at higher frequencies, because that's generally where a system will show the most variation over temperature and manufacturing variations.
The conceptual problem you are having is that you are trapped into thinking that a system is stable at a certain frequency and unstable at another. A linear system is either stable or it is not. If it is unstable, then when perturbed, its response will grow and become unbounded and its behaviour will be governed by its closed loop pole positions.
A Bode plot gives an indication of stability for minimum phase systems where the vertical part of the Nyquist D-contour is evaluated, so here one is only using part of the contour. However, it suffices quite often unless there are "winding number" and NMP zero issues in which case the final test is the full Nyquist criterion complete with number of encirclements etc.
Oh, I fully understand that a linear system is either stable or it isn't. But when you're designing to a real-world plant you can expect that the plant's gain and phase will be more controlled and predictable at some frequencies than others, and that in almost all cases where it does show the most uncertainty is at the higher frequencies. So you really want to design for more gain margin at those uncertain frequencies.
Also, while from a theoretical point of view it's a black-and-white stable or not situation, from a practical point of view the frequency, amplitude and character of the oscillation will yield incredibly valuable information about the root cause of the problem (assuming the oscillation isn't the desired effect). The note of the singing, the size of the swing, whether it's raspy or clacking or smooth, these will all help pinpoint the place where your real system diverges from theory, which in turn will tell you if parts are broken or binding, electronics burned, bolts untightened, etc.
You're also correct in stating that using only a Bode plot for analysis will only give you a yes/no indication about system stability if the system in question is minimum phase. A Bode plot will, however, _always_ give you gain and phase margins for the points where a system will transition from whatever state its in (stable or unstable) to the other. So if you already have some side information for the system then the Bode plot provides all the rest.
So if you have a brand-new system with presumably unknown stability characteristics you can use a Nyquist plot and see, and you can also use it to design for absolute gain and phase margins. The problem (if you're me) is that you have to remember how many encirclements to count. Since I'm of the "count every sign change and add one to correct for the one you missed" camp, I prefer a method that's harder to muck up.
What you _cannot_ do with a Nyquist plot without modification is to design for one gain margin at low frequencies and a greater one at high frequencies. You _can_ do this with a Bode plot. Also, if you have a "black box" plant with _really_ unknown characteristics then you don't necessarily know if or how many unstable zeros there are, so measuring it with a frequency sweep and creating a Nyquist plot does you little good. It's either suitable (and stable) as it is, or you need to wrap it with _some_ controller to tame it before measuring it's characteristics. Either one of these cases immediately gives you the one and only piece of information that's missing from a Bode plot analysis.
So I'll continue to use my current method: first get the thing stable, either by verifying one stable transfer function on paper or by looking at a non-singing system in front of me, then use Bode plots to give me appropriate gain and phase margins at all frequencies, taking expected plant variations into account.
Don't forget that in the real world things are multivariable, bode and Nyquist work well for single input - single output, but looking at each input and output pair in isolation can be proven to be arbitrarily poor for multivariable-types of disturbances and uncertainly.
Fortunately the bode type plots can be replaced with singular value plots for multivariable systems, one of which the 'sensitivity' function' is a good measure of the distance of the Nyquist locus from the critical point.
So far I've only had to deal with SISO systems, or fairly simple MIMO systems. As such I've been able to stick to "modern" control (1950's "modern". Presumably you're talking about "post modern" control). Can you recommend a good text for the practicing engineer on this subject?
Declaring bias (at least one page of my work in there!) - I can recommend
MULTIVARIABLE FEEDBACK CONTROL Analysis and Design
by
Sigurd Skogestad and Ian Postlethwaite
Published by Wiley 1996
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not sure how relevant the more detail chapters but the first 3 chapters are very relevant to the topic of this usenet thread - and they are available for free!
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