# PID controller question

• posted

I have two questions regarding solving the problem

and comparing with the textbook's solution manual's approach

1. I have noticed that the PID controller shown in equation 6.6.16 is used to cancel out the plant's poles. Do you happen to know why the controller is used to cancel the plant's poles?

1. For second order systems, I recall the characteristic equation is
^2%20+%202\zeta\omega_{n}%20+%20\omega_{n}^2=%200). However, as the problem provides the specification that the bandwidth is 2 rad/s, I was under the impression of making the middle term in that equation to be 2. However, the solutions made it to be 2.08, and the third term to be

2.56. Do you think these two terms can be anything that would work in terms of making the controller?

• posted

It looks interesting but we can't see what you are talking about. Always post links on a separate line.

Canceling poles is done by placing zeros at the same point as the poles. Reducing a pole reduces the phase lag. If one can cancel all the poles but one then the closed loop transfer function, CLTF, will have a response like a single pole low pass filter and will not over shoot or oscillate. At least that is how it works in the text books. The internal model control, IMC, PID gain methods basically cancel poles. I am not a fan of canceling poles because you must know exactly where they are and sometime they move around as the system warms up or loads change. I prefer to place them in safe places like on the negative real axis. Then if they move a bit they cause no harm.

An extremely simple example of pole placement and how to post a link.

that the link is on a separate line.

Peter Nachtwey

• posted

It's easy to see that the link is broken. IOt's easy to fix the broken link. Why is anybody crabbing? Here:

Jerry

• posted

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Thanks for your explanation. I have no idea that including links in the same line as my description would cause such havoc. I will keep that in mind in the future.

• posted

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Thanks for your explanation. I have no idea that including links in the same line as my description would cause such havoc. I will keep that in mind in the future.

• posted

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Thanks for your explanation. I have no idea that including links in the same line as my description would cause such havoc. I will keep that in mind in the future.

• posted

It's also *usually* possible in most readers to force a single line by enclosing a long link in angle brackets, although even so,

still seems to fargle on the ^ character in Agent.

Interesting paper in your link. Which version of Mathcad is that? I stopped upgrading with 2001i, when they went with the crappy new licensing scheme -- although I had followed practically every upgrade since my original 5 1/4" Hercules graphics version.

• posted

That depends on the program you use to send email and use newsgroups. Thunderbird won't break any block of letters that doesn't include a space. for example:

Jerry

• posted

s

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=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 don't know how to use Thunderbird to post newsgroup messages to this group. I'm using Google Groups to get this done.

• posted

It's not so much that the link is on the same line as text, but that the link has been wrapped onto another line by your news client.

This is how I see part of your post when I look at it using slrn:

Notice how the first link has been wrapped between two lines.

Simon.

• posted

As Peter said. If you ever try this, direct pole cancellation, then also run simulations where the poles have moved because of age, manufacturing error, environmental changes, and with physical limits on the driver. Your trying to paper over a pothole instead of moving it; that might hurt. I've done it when I couldn't get the mechanical designers to modify the design and had to get the performance; I did complain and explain. It worked really great: for about six months, then they had to do the redesign. The point is, do the simulations with reasonable (or worst case) conditions early before effort is wasted on the wrong track. Also keep and structure the simulation process so you can redo it when the mechanical part of the design changes.

Ray

• posted

I kind of agree, but:

Pole placement can lead you to designing a system that's not very robust, unless you know from the get-go what sort of pole locations will work well with your system. Peter can do this because he's very familiar with the systems he designs for. The rest of us may want to do some checking of our design.

Pole-zero cancellation is _not_ a panacea -- in fact, it can easily be a garden path down which you can lead yourself. It _can_ be used to good effect _if_ the poles being canceled are within the loop closure frequency, and _if_ you take the expected real-world range of the plant's poles into account when you do your design. Even there, however, it's almost always better to feed the command forward, around the feedback controller, rather than use 'real' pole-zero cancellation. This complicates your design because you now have a two-input controller, but it allows the controller to 'goose' the plant on sudden command changes without putting too much gain in the feedback path.

Controlling a plant with a pole at +10 rad/sec with a controller that has a zero in the same place is right out, of course.

• posted

I found myself doing this a lot at one point in my career. I took to getting my work done, then summing up my final status report with a statement of the real problem, and a prediction of how long it would be before I was tinkering with the design again.

After a while, they started letting me in to the mechanical design reviews.

• posted

I notice. Now see how Thunderbird treats the same original text:

I have two questions regarding solving the problem

and comparing with the textbook's solution manual's approach
Jerry

• posted

I learned a bit about pole-zero cancellation pushing op-amp circuits to peak performance. It can do great things for phase margin, but it can have a strange effect even with no component aging. The phase margin isn't much affected by a slight failure of exact cancellation, but the last bit of settling to final value is greatly extended. I won't go into the math, but the system "sees" another time constant about equal to the

*difference" between the exact zero and the actual one. That's long!

Jerry

• posted

I don't understand this statement. I would like to see an example. If you place the closed loop poles on the negative real axis away from

0 the system will be safe. The response will be like a multi-pole low pass filter. Yes, a lot depends on where you place the poles.

Peter Nachtwey

• posted

What you basically need is a 'real' process identification. Let's think you have found a nasty one of

1,319279E-04 v1'' + 0 + 0 = v2 (non-self regulating)

Then you must compensate the disturbances z appropriately.

See this example:

The optimization routine would have tuned the PID controller more sensitive. Therefore I stopped optimization to have robust controlling.
• posted

In general terms, however, the faster the poles that you're placing -- no matter how well damped they are -- the more gain you need and the more predictability that you're demanding from the plant at high frequencies. Since most plants tend to get unpredictable at high frequencies, this isn't necessarily a safe thing to do blind.

In your case, IIRC, you usually close hydraulic systems at around 10Hz. As an extreme example, what happens when you place your poles at 100Hz instead? 1000?

Without getting too close to reality, consider a system with a real transfer function

1 1 H(s) = - --------------- s (s/1000 + 1)^4

Say that your best model gives you a transfer function H(s) = 1/s.

Now do a simple proportional-only controller with a pole placed at 10Hz

-- all is well. But try to push that pole placement out to 50Hz and you're down to less than a 30 degree phase margin, with over 6dB of peaking, bad behavior, and who knows what will happen on a really cold day. Push the placed pole out to 80Hz and you're at almost 20dB of peaking, on the ragged edge of stability.

You could use the 'five pole' model to push the pole placement out much further -- but if you aimed for that 80Hz loop closure you'd have a system that's exquisitely sensitive to relatively small changes in the pole location.

(note that to do pole placement with five poles you need at least a four- pole controller; if you've got that many poles in your controller then you're probably leading yourself down the garden path unless you have a _really_ good idea of what your plant is going to do over environmental, aging, and possibly per-part variation).

• posted

Don't confuse the technique with the execution. Obviously the poles can't be placed too far to the left. There are practical limitations such as sample rate, sample resolution, output limitations and other un-modeled features.

A system like your H(s) requires 5 gains if the integrator is not used and the sample rate needs to be 10K. Your example has another practical limitation that JCH often ignores. How does one measure second, third, forth and fifth derivative?

Peter Nachtwey

• posted

Well, that's my point exactly -- as a technique, pole placement works great _if_ you already have a good idea of where the poles can be placed. I contend, however, that it gives you no means of determining where those poles _should_ be placed without using other techniques (such as robust design techniques or Monte-Carlo design tries).

Obvious from the _outside_ of the pole placement algorithm, yes -- that's what I was trying to point out. From _inside_ the algorithm there's no guidance whatsoever, as contrasted to robust design techniques which demand that you start with an awareness of the parameter variation of your plant and go from there.

One could write a book on them.

A 5th-order system would need five gains _for pole placement_, yes. But many robust control design techniques (not excluding Bode plot design, although it is arguably not 100% robust) work very well to find reduced- order compensators. Given a 5th-order system definition would you insist on making a 4th-order controller each and every time? Does this mean that if you had a 10th-order system definition you'd insist on a 9th- order controller? What about H(s) = (e^(0.001s))/s? Where do you find the infinity of poles for your controller, to place the infinity of poles in your plant? Bode plot design (and other robust techniques) take care of this with little problem.

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