First I find directly 'all' derivatives using process identification techniques via least-square approximation methods. Basic for that is a measured process transfer function data set for e.g.
A6*v1^(6) + ... + A2*v1'' + A1*v1' + A0*v1 = v2
See Example yellow line:
formatting link
Any Ai is known and used for
- feedforward
- disturbance compensations
As far as controlling is concerned I use 'State Observation Control'.
That means using all coefficient Ai found per program that I have written for that purpose.
Yes, I call it a "tail". I got burnt on that early in my career and have avoided pole cancellation ever since; unless I had an outside loop to kill the tail. It's best not to take it too casually even then. Sometimes you have to make sure _that_ zero moves to _that_ pole when the loop gain is changed; and doesn't wander off.
No, I couldn't accurately identify that many poles. I would model the dominate poles and compute the gains for that. Then I would use a smooth target generator so I wouldn't excite the other poles.
You approximate that with a Taylor's approximation.
I don't see how you can tell were one pole begins simply by looking at the Bode plot. Finding real poles is hard enough. It is harder yet to pick imaginary poles. It is too hard to pick the pole locations accurately from a plot.
You ignore feedback resolution for starters. Try computing the 6th derivative in your example below when the feed back isn't a 17 digit floating point number.
You are estimating the coefficients for the plant but you are not calculating the derivatives of the process variable.
Just use 'appropriate' equipment. It depends on the task. (Note: Another one of my programs use 49 digits.)
Read again:
In other words: 100% mathematics! No further improvements are possible!
formatting link
The dotted black line is the solution of the measured 'real' points (blue dots). The differential equation is shown in the yellow bar. That is the mathematically found process transfer function.
The data used you find on the mentioned page below. (Note: The mentioned techniques can also be applied to non-linear differential equations changing them to linear differential equations.)
You make my point, then. With an approximate model you cannot do pole positioning with arbitrarily fast poles. From past discussions in this forum I have no doubt that you can figure out how far you can push things using your calibrated eyeball to tell you where the danger zone is -- I just think that there are better and more easily taught ways of getting to the same end.
When I do Bode plot design I discard any pretense of knowing where the poles are -- I just design the system in the frequency domain, then (if necessary) double-check the results in the time domain to find out if there are any interesting behaviors like the long tail previously mentioned.
One should know their system well enough to know what poles exist. The higher poles are not a problem if you don't excite them. This is why target generators are so important.
I don't see most systems. We sell a motion controller and the customer installs it. We provided tuning wizard that helps. Our customers are usually able to tune the systems by themselves if their hydraulic systems doesn't have too many flows. Since we control hydraulic systems that are moving TONS instead of pounds it is not wise to do a frequency sweeps. Also the gains change as a function of direction.
Motors are easy to tune. Here is an example of auto tuning a motor in torque mode.
formatting link
can see when I move the cursor across the plot the target and actual position are almost the same.
OK, so I ignored the RL time constant. The tuning is still almost perfect. This is because the RL time constant is much shorter than the time constant due to inertia.
I would like to see how you do this. I have motors and data that I can send or post. I would be interested in seeing if you can get the same results for tuning this motor.
I haven't seen the long tail behavior.
Tuning this motor is easy. Underdamped second order systems are much harder. I would like to see how you handle that. Like I said, I have real systems with real data. I have customer with data for all sorts of systems.
PolyTech Forum website is not affiliated with any of the manufacturers or service providers discussed here.
All logos and trade names are the property of their respective owners.