Exploring the FOPDT Model With a Parameter Sensitivity Study

Step 3 of the PID controller design and tuning recipe is to approximate process bump test data by fitting it with a first order plus dead time (FOPDT) dynamic model.

Data from a proper bump test is rich in dynamic information that is characteristic of the controller output (CO) to measured process variable (PV) relationship. A FOPDT model is a convenient way to quantify (assign numerical values to) key aspects of this CO to PV relationship for use in controller design and tuning.

In particular, when the CO changes: the process gain describes the direction and how far the PV moves, the processs time constant describes how fast the PV responds, and the process dead time describes how much delay occurs before the PV first begins to move.

In this investigation, we isolate and study each FOPDT parameter individually to establish its contribution to the model response curve.

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Doug Cooper

Reply to
Doug Cooper
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Unless you have a plant that doesn't fit that model very well. Of course, any one who's going to present just one method of model identification isn't going to fess up that it may not be the best method, or even the appropriate one, for some plants.

Unless your plant has significant resonances or other important high-frequency behavior that will be masked by the inherent low-pass nature of the bump. Additionally, a plant with significant nonliniarities will not be fully characterized by a "bump" around it's nominal operating point -- trusting the information from a bump test with such a plant will leave you hanging out to dry for issues of system startup, fast settling, or system behavior in markedly different environmental conditions (e.g. in an outdoor plant, taking a bump test in the deep of winter may not help much for plant behavior in high summer).

Convenient. I'll give you that.

All true, but meaningless if the model isn't a good fit to the actual plant.

And when was the last time that Doug Cooper contributed something to this group other than advertisements for his intellectually restricted "training"?

I will grant this: The FOPDT model of plant behavior seems to be a very widespread method of modeling industrial plants. Presumably this means that it works well enough for most of the applications in the domain where it is popular.

But it's certainly not _the_ way.

I generally end up designing controllers for products, where you have to design one controller that will, without special tuning, work on a variety of plants. The controller needs to work in spite of variations is per-piece characteristics, variations due to temperature, supply voltage, light level, aging, what have you.

I don't use the FOPDT model. Ever. Instead I measure characteristics with frequency sweeps, and sometimes with open- or closed-loop step ("bump") tests. In the case of frequency sweeps I don't pretend that there's a closed-form transfer function that I can fit the data to; rather I just do my design in the frequency domain. In the case of step test data I will fit the data using an ARMA process, then I'll wish that I'd taken frequency sweeps.

To back up my measurements, I analyze the system mathematically, looking for important nonlinearities and opportunities for plant variation. Then I design around those.

Were I tuning controllers for process control loops I may well start by doing bump tests, and if it worked well enough I'd use the FOPDT characterization to justify my design choices, if not to arrive at them. But I'd sure as hell check to make sure that the model was a good fit to the data, I'd do analysis to check* that the plant nonlinearities wouldn't render the system unstable at some inconvenient time.

  • I originally said "make sure", but then I thought about all the times that I, and other smart people, have been wrong on this sort of thing...
Reply to
Tim Wescott

I don't think these posts or blog are intended for experts like you. I think it is meant for the casual PID tuner. After all, how can one identify a second order plus dead time system by just looking at it? I can't but my Mathcad or Scilab programs can. You and I know that there are very few first order plus dead time systems out there yet this is a good place for the casual PID tuner to start.

The control guru site only deals with low speed process control. A step in the control signal will have high frequency components.

That is true but if you look at the site you can see it address DLO or design level operation. The gains work only at that level.


I think he provides a lot of information and formulas that I haven't seen on this newsgroup. I refer PLC programmers to the control guru site often because it has the basic information need to get a PLC PID working with minimum effort.

No it isn't, but how many people can write their own system identification program where they can identify a non-linear SOPDT system? Not many. Look at the number of clueless students that will so be in the work force! I was clueless too way back.

We have been through this before. How can you do frequency sweeps on a temperature system or some other process control plant? How can you do frequency sweeps on a hydraulic system moving tons?

That is good and I am sure it works in your controlled applications. My realities is that the manufacturers do not have a clue as to how their system will work and it is up to the poor guy with a PLC or motion controller to make it work. Motion controllers usually have some sort of auto tuning to make life easier for the users. PLCs do not and all they have is the built in PID which I know is often not enough.

I think control guru's site is meant for the poor process control guy who has crude tools ( PLC ) and anything that can done in 15 minutes will help immensely.

Experts don't need this help. We can roll our own and can choose exactly the right control for the application.

I have been monitoring this newsgroup for years and years. There have been times when I thought there was no intelligent life here. I think the situation has got much better. Partly because of you and others like now use applied control theory in their applications and advice. Previously there were a lot of old timers with experience and their rules of thumb and whining about this or that can't be modeled because the system is non-linear or has a dead time.

I can remember when this Sinclair guy would post links to his book or site every week. I can remember being new to the newsgroup and actually making comments on his statements instead of ignoring them. I didn't think he anything to say except for buy my book. The control guru has information that the casual PID tuner can use.

Finally, a newsgroup is not a place for in depth tutorials or thesis. Everything get lost in the archives and one must search to find this info. I would much rather have the information I post on my website were I can just post links to it. I can update the information on my FTP site but the effort put into the replying on a newsgroup is gone for all practical purposes because people will not search. There were questions last year about minimizing integrators overshooting and a Girl in Ireland trying to tune her system. I would have to search for those threads but I have the Mathcad worksheets handy. So what do you want the control guru to do? He can't post the information here because the newsgroup will only display text. Tim, post links to where we can buy his book. I don't expect Tim to put the same effort into answer every newsgroup question like he did in his book.

Have any of you seen the formulas for calculating the PID gains for FOPDT and SOPDT systems anywhere else on this news group? I think the control guru has shared a lot of information on his blog.

Peter Nachtwey

Reply to
Peter Nachtwey

The 50+ articles in the controlguru.com table of posts

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represent perhaps 40% of a complete textbook.

It is all on-line. There are no pop-ups or in-your-face advertising. And it is all free. I post a new article twice a month and expect the project to continue for a few more years.

I currently have over 4000 unique visitors every month at

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I know of faculty from more than a dozen institutions who are migrating bit-by-bit from my textbook (currently used in 125 schools) to this online resource because of its practical orientation. I conclude from this that there is benefit from my efforts.

With that said, I acknowledge that electrical engineers, mechanical engineers and chemical engineers all approach control from a different perspective for valid reasons.


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site has a chemical/process orientation. The primary focus is on PID control and related architectures. Applications of interest include processes with streams comprised of gases, liquids, powders, slurries and melts. Final control elements for these applications tend to be valves, variable speed pumps and compressors, and cooling and heating elements. Industries that operate such processes include chemical, bio-pharma, oil and gas, paints and coatings, food and beverages, cement and coal, polymers and plastics, metals and materials, pulp and paper, and consumer and personal care products.

The control system at one of these plants may have from 20 to over 250 PID loops and from 50 to over 1000 measurements. Often, the controls people have a different supervision structure from the production staff trying to meet schedule. A control person seeking to tune a loop 'might' receive permission to make a couple of bump tests. On the other hand, they may be told to take a hike and be forced to dig out what they can from the data historian.

They are almost never given permission to conduct extensive testing. It is simply too expensive and sometimes even dangerous. A couple of bump tests can take much of a shift in many plants, and they are using expensive feedstock and consuming utilities the whole time. For one loop.

The PID controllers in these production facilities tend to be a horrifying mix from several of the roughly 20 major manufacturers. Each manufacturer uses different terminology and formulates the PID algorithm in its own fashion. Tuning these controllers is a challenge too often addressed with trial-and-error fiddling by the operators, technicians and engineers at these facilities.

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works to address the needs of these individuals.

Most all processes comprised of gases, liquids and such are overdamped and self regulating. The time constants have units expressed not in milliseconds or rarely even seconds, but most often in minutes (and sometimes hours). Dead time, measurement noise, and nonlinear behavior are always complicating issues. If underdamped behavior shows up in a trend plot, it likely means that an upstream controller is cycling and needs to be checked out. It is almost never the characteristic behavior of the process itself.

No loop in these plants is truly first order plus dead time (FOPDT). Not a single one. In fact, if modeled from first-principles including the thermo, transport and kinetic equations, we would likely find such processes to be accurately described with tenth or even twentieth order differential equations.

But in my experience, 85%+ of PID loops when the application is a process with streams comprised of gases, liquids and such can be well-tuned using the FOPDT approximation. If we introduce non self regulating overdamped processes (also discussed on

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we are up to maybe 90%

  • of the processes.

If readers start at the top of the table of posts

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they will find, presented in a logical order and free of any charge, the details, methods and practices I believe provide the best general methods for the application domain I am focused upon.

I hope readers will visit and judge its value as a resource for themselves.

And I know my posts on this newsgroup announcing the publication of the next installment of the e-book are valued by some of the readership. So it is a practice I intend to continue.

Doug Cooper

Reply to
Doug Cooper

Boy Howdy. I've used your methods and practices successfully, Doug.

Permission to bump a loop in a huge food process plant is frowned upon where I am now. HAACP concerns are the most frequent reason I'm given when I ask to bump a loop so I can approximate a FOPDT fit. With so many other processes that are dependent on the loop you would like to test operations people are justified in there hesitancy to allow it. This holds true for new processes as well, although I do manage to get approval for it *sometimes*. Make the product with minimal waste and make it now.

Err, yep. I don't call it "fiddling" however. My preferred term is "knob dicking."

It most certainly does.

Taking personal time to help people in their endeavors is a wonderful thing, and definitely not something that should be frowned upon. And for that I thank you, sir. Keep up the good work.

Reply to
Ursa Major

-- good rebuttal, snipped --

As usual when I work myself up to a screed, I'll bet I forgot to include any of the positives of your site.

Just the fact that you have responded tells me that you don't consider this newsgroup to be a dumping ground for your advertisements, which was a big part of my concern about your posts.

Because I _don't_ come from the process control world, and because it is at least as economically important than individual widgets with built-in controllers, I read most of the process control posts with interest. I'll admit that when it comes to network management questions my eyes glaze over, but I do at least skim the first few posts in those threads, too.

My only real concern is that the web is so flooded with information about "process control" methods of tuning that when someone is ready to tune a loop on a design for a widget that's going to be built in quantity they'll often approach it with one of the "process control" methods -- and in so doing they'll end up with a widget design that either under performs or that can't even maintain stability over normal manufacturing variations.

If their business is still standing at this point, they then have to either fumble around with tuning, or try to get an expert to help them (often too late, because much of the virtue of control engineering comes from verifying and tweaking the design of the plant before any tuning is done).

Reply to
Tim Wescott

On Tue, 27 Mar 2007 12:19:02 -0700, Tim Wescott proclaimed to the world:

Good points and well said. One value of the time I spend in this group is the exposure to different logic processes an engineer picks up from working in his field. Our fields have evolved. Your specialty requires a mathematical precision that most offshoots of control have not required. While the math involved in PID is very important now, particularly in your area, it was not so important in mechanical control systems past the engineering phase. The guys making this stuff work thought about the math involved rarely.

Understanding why and how the differences in different aspects of controls have evolved makes it easier to adapt and make use of them.

Reply to
Paul M

I think this is almost a perfect newsgroup. Fairly small, mostly composed of reasonable experienced people. Sometimes I ask for help; and occasionally I comment when I think it will help. The traffic is reasonable and I appreciate different viewpoints on the idea of automatic control; food, chemical, motors, etc.....

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