Tuning Method -- Damped Oscillation

I am quoting the following passage out of a workbook from an Instrument Class taken from NTT. I would like to fully understand this method, but the passage leaves me with a few questions . Can anyone shed more light on this subject?

Tuning Method - Damped Oscillation

This Method is very similar to the Zieglar/Nichols method. It is used when sustained oscillations are not allowed and the Ultimate method cannot be used.

The integral and derivative are removed by setting the integral time to infinity and the derivative time to zero. The gain (or Proportional Band) is adjusted from a very low value until a quarter wave damped oscillation is observed when a Setpoint bump is applied.

Only the period, P, is measured with a stopwatch. The gain is then lowered to a very small value and then Integral and Derivative (if required) are set according to the below equations:
Ti = Pu /1.5 and Td = Pu /6.0

The gain is then increased until quarter wave damping is again observed in response to a setpoint bump.

Click to expand...

It says to adjust the gain (or proportional band) from a very low value until a quarter wave damped oscillation is observed when a Setpoint bump is applied.
- How do I do this? How do I know when I have a quarter wave damped oscillation? What values do I put on my trend? Do I scale the values 0-100%? What sort of period do I need on my trend to observe quarter wave damped oscillation?

Only the period, P, is measured with a stopwatch.
- How do I do this?

The gain is then lowered to a very small value? How low? 0 gain or 500% Pb?

Symbols:
Ti(integral ?) ?
Pu (gain ?) ?
Td (derrivative ?) ?

I am using a modicon PID2 controller.

my proportional band is entered in 5-500%
my reset is entered in 0.00 - 99.99 repeats per minute.

ATTENTION EVERYBODY - This is a ROUGH draft but I’m on a SHORT schedule - there are exceptions to all of these rules -

basic idea behind quarter wave dampening: if you hook up an automatic controller (PID) to a process - and enter a setpoint - and crank the controller gain up high enough - then the process will oscillate (you might have to kick start the oscillations to get them started - see “bumping” the setpoint below) - if the gain is too high, then the oscillations will get larger and larger and “run away” - if the gain is too low, then the oscillations will die out. suppose that the oscillations HAVE died out and the process is coasting along smoothly - now you “bump” the setpoint - example: raise it from 20% to 40% for a few seconds - then drop it right back down to 20% again - the controller will respond to this bump by increasing the output - the process will respond to the increased output by going up - then the output will drop off - and then the process will drop off - this will set up a series of oscillations - question: how many “ripples” did you get before the process settled down again? - if the answer is FOUR then you’ve just witnessed quarter-wave dampening - in other words, the first wave had a particular amplitude - the amplitude of the second wave was one quarter of the amplitude of the first wave - the amplitude of the third wave was one quarter of the amplitude of the second wave - the amplitude of the fourth wave was one quarter of the amplitude of the third wave - and so after four waves, the oscillations are approximately zero - of course this is all subject to nagging questions (is it really settled down? - did it take four waves, or four and a half?) but that’s the general idea - it’s sort of like adding salt to a recipe (“salt to taste” - what the heck is that all about? - whose taste?, etc.)

earlier we said that if the gain is too high, then the oscillations will get larger and larger and “run away” - if the gain is too low, then the oscillations will die out. suppose that the gain is JUST ENOUGH to cause the oscillations to continue in a nice sustained manner - this is what Ziegler-Nichols described as the Ultimate gain - that’s what the “u” stands for in the formula terms of your post -

you asked about what scaling to use on your trend - it doesn’t matter - the magic happens when the gain is the right value to cause the waves (whatever their initial height) to die out after four ripples -

you asked what sort of period you need on your trend in order to observe quarter wave dampening - the answer: (no sarcasm intended) “enough” - enough so that you can see the waves start up and then die out whenever you bump the setpoint - different processes (flow, for example) might only need a minute or so - a temperature loop might take an hour - but (and this is a MAJOR idea) each process has a characteristic frequency at which it will oscillate when under automatic control - it’s EXACTLY like a pendulum - as long as the length of the pendulum stays the same, then the frequency of oscillation will stay the same - which is why grandfather clocks were the most accurate timepieces available for many years -

you asked how to measure the period with a stopwatch - just get the process oscillating - then as soon as the trend hits the top of its swing, start the watch - stop the watch as soon as the trend hits the top of its swing again - that’s the period of oscillation - it’s just like trying to time the swing of the grandfather clock with a stop watch - but in my humble opinion, a better way is to print out the trend and measure the length of the x-axis (time) and then the length between the tops of successive waves - then set up a ratio and solve for the time of one period - now you’ve got something on paper to annotate - anyway, the time (you probably need to keep this in units of minutes) that you measure for the period is Pu in the formulas -

you asked how low a value to use for the gain - don’t go all the way to zero - you won’t get ANY control at all that way - but you want something reasonably low so that you can increase it a little bit at a time - testing the process as you go by bumping the setpoint after each increase - again, it’s like adding salt to a recipe - YOU’RE going to have to judge when it’s JUST RIGHT -

and incidentally - you should be aware that the gain is the reciprocal of the proportional band - so when I say use a small gain - (let’s say 1.50 for an example) - then the proportional band would be 1 / 1.5 or 0.666 - different controllers use different units but I think that’s right based on what you posted - that means for MORE action - you need a LARGER value for GAIN - but a SMALLER value for PROPORTIONAL BAND -

you mentioned a few terms such as Ti and ----------

oops - out of time

hope this helps - I did the best I could with what I had to go on - I think I could do better over the phone using a fax or two to illustrate -

Ron has a real good description going. I'm just going to toss in a couple more thoughts.

In your example you are dealing with proportional band. As Ron said, with values of proportional band the correction decreases as the proportional band increases. So when your instructions talk about a low value, they really mean a high number. Proportional band is defined as the change in the process variable (measured input) required to produce a 100% change in the output. So the larger the proportional band gets, the larger the change in the input needs to be to get a given output change.

The classic Ziegler-Nichols closed-loop tuning method would have you change your proportional band until the process goes into a sustained oscillation, as Ron said. The tuning method you list assumes that the process will not tolerate a sustained oscillation, so it says change the proportional band until you get a quarter wave oscillation. This oscillation is important because it is the only way you have to measure the Ultimate Period (Pu), which is the reciprocal of the charateristic frequency Ron mentions. The Ultimate Period is the basis for both the Reset Time (Ti) and the Derivative Rate (Td). Rons suggestion of printing out the trend is a very good one and I highly recommend you do that. It is much easier to see the point that the oscillation turns around looking at a trend.

Once you have Ti and Td set you are told to start bringing the proportional band back into play. Given that you now have a derivative term you would expect that you will be able to tune a smaller value of proportional band than you came up with early in the process. I'm kind of pulling a number out of the air on this but I would start with a value about 1.25 to 1.5 times the proportional band value that set up the quarter wave oscillation when you started. This should be low enough response that you don't go into oscillation right away but high enough that you won't be playing with proportional band numbers all day.

Ti and Td are also reciprocol quantities (now that's a confusing phrase). As you decrease these values the correction gets more aggressive. This is just a note for future tuning.

Also note that Ziegler-Nichols style methods are not intended to give you the perfect gain settings. The method by design produces a stable response that is not going to oscillate. It basically gets the process under control for you so you can start fine tuning. Some processes will be tuned closely enough using this tuning method that further adjustment is not necessary. However, don't be disappointed or surprise if fine tuning is necessary.

Good luck,
Keith

I think I might have got it.

Thanks for you help. Still have a couple questions left.

It was explained to me once, that the period was just 1/2 of the sign wave (is my memory serving me correct). And that 1/4 decay ratio was only that the second hump (lets say the negative arc) is 1/4 the amplitude of the first positive arc. I could measure this by measuring the time from the top of the first arc to the bottom of the second arc, and this should give me the ultimate period (Pu)?

Now, the controller I am using is the modicon. So in finding a Ti (integral) value, I just need to take this period (in minutes?) and divide it by 1.5 (my integral setting is in repeats per minute). Or do I need to take the reciprocal of 1.5, or better yet, just multiply by 1.5 instead of divide.

Side question: Pb in the modicon is from 5-500%. Converting this number to Kc yields me a number from .2 - 20% Is this correct?

Now for the bumpless transfer questions......

The period you are looking for is for a full sine wave, from crest to crest or trough to trough.

A quarter-wave decay simply states that a wave crest will be 1/4 the height of the previous crest. So if the first crest of your oscillation is 1 unit high, the 2nd crest will be 1/4 unit high, the 3rd crest will be 1/16 unit high, the 4th crest will be 1/64 unit high, ... This will continue until the system losses are greater than the oscillation energy.

However, for the purposes of measuring Pu any oscillation will do. The important thing is that you get enough oscillation that you can accurately determine the crests of the waves so you get a reasonably accurate determination of period.

Your integral and derivative values are in terms of time. So follow the equations you listed originally, that is divide you Pu by the numbers listed.

As for your side question, as you said a Pb of 5-500% will give you an equivalent Kp (proportional gain) of 20-0.2. There is no percent; it is a real number.

I hope this helps.
Keith