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A proportional, integral, and derivative (PID) controller can feature three terms.)

We sometimes see only two of the terms in use. For example, the derivative is disregarded for a PI controller. While I know what each of the three terms do, I am not sure about the situations that warrant their removal and the control implications from their removal.

Please let me know if this question requires more clarity or elaboration.

chicks
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HFOrangefish
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    Usually this question is asked - what are the effects when these terms are *added*... – Eugene Sh. May 16 '22 at 15:48
  • Your question reminded me of [this video](https://youtu.be/fusr9eTceEo) which I found at one point when working on a linear actuator controller. You may find it helpful, too. – JYelton May 16 '22 at 16:15
  • You forgot to include the conditions for each K factor like SS, Ramp, Step , impulse, but that's OK since I mentioned this so you should have waited – Tony Stewart EE75 May 16 '22 at 17:39
  • I remember a story I read on a hand-out at my high-school (vocational) about this... An old man was manning (regulating) the water in a dam - and did it perfectly. He got ill, and his three sons had to take over. The first one would slowly fill the water at a constant rate. The second would open fully and overfill. The third I don't remember. But when all three worked together, they would work as a PID regulator and fill it perfectly. BTW, has anybody seen a story similar to this? (been trying to find it) – Baard Kopperud May 16 '22 at 23:09
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    Best explanation of a PID controller on the internet : [PID Without a PHD](https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pdf) – Tejas Kale May 17 '22 at 09:52

5 Answers5

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If you start with a PID and:

  • remove the D action: the controller won't respond quickly to quick changes in the input. This is sometimes desired if the input is noisy and you don't wont the output to be that jerky. Sometimes D action is needed for a quick reaction.
  • remove the I action: the controller may not cancel any error between input and output beyond what the P action can. D action doesn't fix errors in the long term.
  • remove the P action: Not usual in my opinion. sometimes only I controllers are used and are a bit slower in reaction than PI because of the lack of P action to an error. The I action will always tend to correct the error, although it might also create instability. I don't know of any practical case of DI controllers.
Joan
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  • An interesting post: https://robotics.stackexchange.com/a/5232/31961 – devnull May 16 '22 at 12:55
  • A DI controller can be used to create a "limiter" that only subtracts from the setpoint of another controller. – Jeroen3 May 16 '22 at 12:55
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    "the I action will always correct any error". Except when it overcorrects, and far too late. I've built a couple of those... –  May 16 '22 at 15:20
  • @user_1818839, completely right – Joan May 16 '22 at 16:06
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    Joan is correct about DI being not practical, but you should have said "**Not Possible**" because unless you know the limits or the open-loop characteristics for the transfer function due to disturbances, **your setpoint has no proportional feedback for that dimension.** It would be like accelerating and braking blindfolded with all other sensors blocked and unable to control the speed. – Tony Stewart EE75 May 16 '22 at 17:24
  • Isn't derivative control also used to prevent overshooting? – BillThePlatypus May 16 '22 at 22:53
  • I don't know about practical *linear* DI controllers, but it may be useful to have a non-linear control system which primarily responds to changes in the input *stimulus*, but also attempts to zero the long-term difference between the input *stimulus* and *feedback*. This may be useful if the system's response to the control output is very predictable, but the usable frequency response of the feedback is limited. – supercat May 17 '22 at 15:13
  • @BillThePlatypus: Indeed, the D term does double duty, though I think it's important to note that in practice the D term almost never behaves as an "actual" derivative (a pure 6dB/octave high-pass filter with a cut-off frequency of zero) but will almost always behave as a combination of high-pass and low-pass filters. If one passed the stimulus and feedback through identical linear filters, the effect of filtering them separately and adding the results would be the same as adding them and then filtering the difference, but if one uses different filtering networks, then one can... – supercat May 17 '22 at 15:22
  • ...tailor them to best fit the system's behavior. I don't know why I don't usually see that discussed, since the filter characteristics necessary to yield best input response are often different from those required to yield the best damping behavior, and since most control systems have a long enough lag between input and response that the actual behavior of the "d" term doesn't really match the mathematical predicitons. – supercat May 17 '22 at 15:26
  • @supercat If you can estimate Proportional output by any other means linear or non-linear, then it becomes proportional feedback gain , Kp. Neglecting Kp will not work unless the setpoints you know you can get close to like zero and max. then it is open loop. That's my logical analayis for analog control. Perhaps there is a digital analog. ;) – Tony Stewart EE75 May 17 '22 at 17:53
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The P term is the term which provides the main drive to drive the process in a direction which reduces the error signal. Integral response is inherently slow and so the proportional term is needed to speed the response of the system to a change in the set point.

The integral term responds to what has happened in the past and drives the steady state error to zero and so without an I term there will be a steady state error which will allow the P term to generate some output to hold the system in balance when in its steady state.

With a 3 term controller (PID), when it's in its steady state, there will be no output from the P or D terms and all the drive to the process will come from the I term which will hold the system in a state with no steady state error and therefore no error signal into the inputs of either P or I or D.

The derivative term can be viewed as a look ahead term, it responds to what will happen in the future. Increasing proportional gain will reduce rise time but will increase overshoot but adding the D term will reduce overshoot without affecting the rise time too much. So if you leave out the D term you must have a smaller proportional gain and slower rise time in order to keep the overshoot low if low overshoot is important to you.

The D term will respond to a step input and resulting error signal by giving the system a kick in the right direction. After the initial kick the error signal into the 3 terms will be reducing and so the rate of change of error (which the D term responds to) is negative and so the D term is acting in the opposite direction to the P and I terms. The D term is putting the brakes on to reduce overshoot.

Rapid changes in the set point will initially result in the D term giving the system a kick in the right direction but this can create quite a problem in a noisy environment.

5
  • Ki nulls output steady-state(SS) error but causes a ramp for step disturbances.
  • Kp controls the loop gain for error correction with SS error reduced with a higher gain (e.g. Op Amp with Aol=1e6 so inputs are a virtual null but error is actually reduced by 1e6 for unity gain.)
    • This is the best feedback because it is a zero-order response and thus error correction is in phase with output. Removing it is a problem because this has your target reference to perform error correction.
  • Kd predicts error by phase lead or differentiation of error but also amplifies HF noise disturbances often used with partial gain bandwidth constraints for lead-lag compensation near unity gain for phase margin improvements. so the slope is __/--- in a limited range.

I can think of one DCDC regulator I simulated recently that does not use Ki, Kp, or Kd. That is a hysteretic regulator where the setpoint is the average between the upper and lower thresholds of hysteresis.

Invert the result in this table (1) for decreasing K gain ratios.

enter image description here

Bonus

P Cheung, London Imperial College
p5/6 c/o G. Ziegler & N. B. Nichols(1942)

enter image description here

Extra Bonus

Using your Automobile experience to understand Control Theory

https://www.youtube.com/watch?v=XfAt6hNV8XM

Tony Stewart EE75
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P is the base term and obvious way to control a device. As in, if the motor is going too slowly, it must be sped up. It is nearly impossible to control devices without P (that is, using only I or D). So, your question should be, what if you just use PI or PD control.

The I adds more power based on the time you are off of the set point. This is important for example if you have a motor that is closely matched to the application and outside forces (wind drag) may be slowing the system's effort to reach the set point.

The D term prevents overshoot. It lowers the gain of P if you are approaching the set point too quickly. This dampens the oscillations and gives you a "soft landing" as you reach the set point. A D term is important if your motor is much more powerful than your application needs (more power is good if you have a good control system but oscillations can occur of you don't have a D term).

GT Electronics
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    It is absolutely possible to control something without P. Using only I works perfectly fine, it's just a little slower. – Hearth May 16 '22 at 14:57
  • It really only works with "I only" if the motor is really designed (or set) to hold the load near the set point without a lot of variation to load. Otherwise, you are right that it responds a little slower (at the beginning) but as the Wikipedia author in PID writes, as it gets to the set point, the result can be "brutal". – GT Electronics May 16 '22 at 15:26
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    There are situations where an I controller is what you need. I'm not sure why you're talking about a motor, though. Regardless, your claim that it's "impossible to control anything without P" is plainly false. – Hearth May 16 '22 at 15:31
  • I stand by my comment since your exception is so narrow. Assume Motor as any load or device that needs to be controlled. – GT Electronics May 16 '22 at 17:08
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    The only narrow exception is the list of things where I-only control would be a good idea. The list of things where it would in fact control the plant to zero steady-state error, on the other hand, is quite large. So it's very much false to claim that "It is impossible to control anything without P", as you do above. Almost never the best option, sure, but impossible, definitely not. – Hearth May 16 '22 at 19:12
  • I'll edit my answer. – GT Electronics May 16 '22 at 22:55
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    @GTElectronics I've been working with closed-loop control of piezo actuators for the last 9 years. We've historically used I control almost exclusively. Whilst you're correct that P makes things faster, this isn't always a good thing, because it gives you higher jerk which excites any undamped mechanical resonances in the system (and *every* real system has some of those). Our more recent control system has S-curve move shaping to deal with that, in common with other motion controllers, but we still often find that I-only is the best overall solution when we calibrate. – Graham May 16 '22 at 23:03
  • As I said, it is a very unusual case to the majority of the engineering world but it seems to be very important to your very specific case and to you. You and all of the other people responding to my answer to justify I-only control systems can... what, oh, it's only you. Umm. Ok. Specific. – GT Electronics May 17 '22 at 03:16
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    @GTElectronics I'm giving one personal example, to add to all the others. At some point, multiple examples become "generic" and not "specific". :) In fact mathematically a P term on its own can't hit a setpoint unless the system under control has some very specific characteristics. Barring those unique cases, the maths proves that a P term on its own will always give you a residual error. Of the three terms, the I term is the only part of a PID which (in a typical system) guarantees settling to the setpoint. Not that P isn't useful, but I is non-optional if you need to hit a target. – Graham May 17 '22 at 11:13
  • If you get two friends to agree with you here, I'll let you have the last word. – GT Electronics May 17 '22 at 17:17
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The second and third answers are based on my past experience in process control (many years ago now!).

Removing the Proportional term: system will never reach a steady-state at the set point; it may "hunt" around it, but never settle. I can't remember an actual control system without the P term; but my understanding of closed-loop systems says this term is essential.

Removing the Integral term: Depending on the process being controlled, removing the I-term can lead to a decaying oscillatory response to set-point change, or even continuous oscillation.

Removing the Derivative term: this can lead to a sluggish response, particularly in a high-inertia system. (This would include a high thermal inertia system, e.g., a furnace.)

(Some simple systems, e.g. voltage regulators, can operate satisfactorily with just the proportional term, but most industrial processes will require at least the P and I term.)

andychannelify
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