PID spotlight, part 11: How a PID controller works with an integrating process

 

PID control of an integrating process insights

• Understand that controller gain alone will allow a proportional-integral-derivative (PID) controller to follow setpoint changes, but will not eliminate error when a process disturbance occurs. Raising controller gain does lower the error, but raising controller gain too high will cause instability (swinging). When swinging the process variable (PV) and output (OP) peaks line up.
• Learn how integral alone is always unstable (swinging). When swinging the PV and OP peaks do not line up.
• Explore how derivative alone will slow the rate of change of the PV. Too much derivative will cause instability.

---

In PID spotlight part 3 on process types I stated that “how we tune a PID controller depends on the type of process response.” In other words, once you mate a PID controller to a process the mated pair behave in a way that is unique to that (type of) process. Furthermore, a PID/integrating process behaves, and is therefore tuned, differently than a PID/self-regulating process. There are different things to look for and a different set of rules.

As we did for self-regulating processes let’s explore how controller gain, integral, and derivative individually affect the process, beginning with controller gain. As with self-limiting processes we will use this information to figure out how to blend them together to get a functioning controller.

Gain (proportional) interaction with an integrating process

There are several things to discuss here beginning with the definition of process gain (K_p) for an integrating process (see Figure 1). Self-limiting process gain (K_p) is defined as the change in process variable (PV) relative to the change in controller output (OP) expressed as ΔPV/ΔOP (%/%). If you change the controller output connected to an integrating process the PV will eventually go to infinity (or negative infinity) if left long enough. The self-limiting process definition of process gain is useless for an integrating process because the answer will always be “infinity.”

What we really need to know is when we change the controller output how much will the direction of the PV change? This is defined as K_p = (ΔPV/minute)/ΔOP {(%/min)/%}. For simplicity we usually express process gain for integrators as %/minute with the understanding that this is per a one percent change in the controller output (note that disturbances are also expressed in percent change in controller output equivalent).

When we look at the change in controller setpoint (SP) at the five-minute mark the one thing that should really jump out at us is the PV gets to the SP without the need for integral action. This is radically different from a self-limiting process where integral is required to drive the PV to SP. What’s going on here?

A material (or energy) balance process like a level holds an inventory that does not change as long as the input and output flow rates match. A SP change does not change the uncontrolled flow (input or output), it is merely a request to change the inventory. The controller does this by (in this case) reducing the output flow until the inventory (PV) matches the SP, at which point when the error drops to zero the gain contribution drops to zero and the OP is back where it started. The material balance has been restored.

A nontrivial surprise? How we think about tuning an integrating process

I am emphasizing this seemingly trivial point because it significantly affects how we think about tuning an integrating process (more importantly you must forget how controller gain works with a self-limiting process). One thing to realize is that when you make a SP change to an integrating process integral action is actually counterproductive (are you surprised?)

At the 30-minute mark the inlet flow increases 10% (OP equivalent). To match the change in inlet flow the OP must increase 10%. This can only happen when the error between the PV and the SP multiplied by the controller gain equals the change in inlet flow {ΔOP = ΔInlet Flow = (PV – SP) * K}. Just like a self-limiting process a disturbance to an integrating process will result in a permanent offset from SP unless integral action is used.

Finally at the 60 minute mark the SP is changed again. The PV parallels the change in SP, maintaining the constant offset that was caused by the change in inlet flow at the 30-minute mark.

Two things to know about gain-only control of an integrating process

What we have learned about gain only control of an integrating process is:

• The PV will follow SP changes without the need for integral action. In fact integral action will likely be counterproductive.
• A disturbance will result in a persistent offset between the PV and SP. Integral action will be required to eliminate the offset.

This presents us with a couple of problems that we didn’t have with self-limiting processes. One is that if we have to balance tuning for SP changes versus disturbance rejection we have a balancing problem; how much integral can we include to remove disturbances without impacting the response to SP changes? Fortunately we almost never change SP’s on integrating processes (at least in refineries), so this is generally not an issue. The second problem is we cannot use a SP change to test controller tuning for disturbance rejection (see Figure 2).

As in Figure 1 the input flow changes 10% at the 30-minute mark. The controller output (OP) changes 10% after some initial oscillation to match the change in input. The higher controller gain reduces the PV offset from SP, however it does not and cannot eliminate it. This behavior is identical to how a self-limiting process behaves; raising controller gain reduces PV offset from SP, but at the risk of increased controller oscillation (see Figure 3).

Figure 3 shows that, similar to self-limiting processes if the controller gain is raised too high the controller goes unstable. Furthermore, the PV and OP peaks line up, which is a key visual cue to use during heuristic controller tuning efforts.

Integral (Reset) and integrating processes

Figure 4 shows us that integral only control of an integrating process doesn’t work. One quirk of integrating processes is integral only control is always unstable no matter how slow the integral tuning constant (T_i) is set. One common tuning problem of integrating processes is inadequate controller gain relative to the integral tuning constant, which will result in a continuous slow roll of the process. (Figure 4 was made using a parallel PID to show how integral action doesn’t work without controller gain.)

Similar to a self-limiting process when there is too much integral action (relative to controller gain) the PV and OP peaks do not line up.

Figure 5 shows what happens when a little controller gain is added to complement integral control of an integrating process (PI control).

In Figure 5, we see that adding a little bit of controller gain to the controller stabilizes the controller while retaining an integral constant of 0.05 repeats/minute (parallel PID algorithm). The response to the SP change at the 5-minute mark still swings a little bit; raising controller gain further will stop this. Note that the integral contribution slows the return of the OP to 70% between 5 and 12 minutes. This results in the PV overshooting the SP (PV will always overshoot the SP after a SP change when integral is present); the PV doesn’t return to SP for another 20 minutes as the integral contribution returns to zero. In short, the presence of integral hinders the response of the controller to a SP change.

The response to the disturbance at the 60-minute mark is identical to the response of a self-limiting process. The controller gain responds first (as expected) followed by the integral action trying to eliminate the offset. After some swinging the controller gain contribution drops to zero as the PV settles on SP (the error has returned to zero) and the integral contribution (and change in OP) matches the size of the disturbance (10%).

This illustrates an issue with tuning an integrating process; integral is required for eliminating PV offset caused by a process disturbance, but it hinders response to a SP change. Fortunately, the SP of most integrating processes are rarely changed.

Derivative (re-act) interaction with an integrating process

Figure 6 shows how derivative alone interacts with an integrating process.

Figure 6 shows how derivative penalizes PV movement. The 4% disturbance to inlet flow that starts at the 30-minute mark causes the PV to ramp upward 2%/minute. A derivative constant of 4 minutes moves the OP enough to reduce the ramp rate to 0.67%/minute. (The 60 second derivative filter was used to slow the derivative response enough for you to see the impact of adding derivative. Note that very few control systems allow adding a filter to derivative.)

Derivative action on changes in PV (the recommended option for derivative control) does not respond to the SP change at the 5-minute mark.

Figure 7 shows how raising derivative affects controller performance.

Figure 7 shows that too much derivative will result in controller oscillation. Further increases to derivative will result in controller instability. Similar to self-limiting processes the visual cue for too much derivative is rapid swings of the OP caused by high derivative chasing small changes in the rate of change of the PV.

Note that increasing derivative did further reduce the rate of change of the process variable (PV).

As a general rule derivative is never used for level control. Derivative causes small changes in inlet flow to become large changes in outlet flow, which will upset downstream processes. Levels are typically tuned to avoid rapid changes to outlet flow regardless of what the inlet flow is doing, thus dampening any upsets. This surge flow management requires special attention to controller tuning to avoid overflowing or emptying the vessel while still minimizing changes in flow to the downstream process. There are, of course, exceptions to this rule, but these depend on the process. We will look at tuning to minimize the impact of disturbances and tuning to minimize controller output movement (surge management).

Summary: Blending gain, integral and derivative? 4 similarities, 3 differences

We have learned that integrating processes respond to PID control in some ways similar to self-limiting processes, and in other ways quite differently. The similarities include:

• Controller gain cannot eliminate the error between the process variable (PV) and setpoint (SP) that is caused by a process disturbance. Integral action is required to eliminate the error.
• Too much controller gain will cause swinging. The visual cue that the controller gain is too high is the process variable (PV) and controller output (OP) peaks will line up.
• Too much integral will cause swinging. The visual cue that integral is too high is the PV and OP peaks do not line up.
• Too much derivative will cause swinging.

The differences are:

• Controller gain alone is adequate for SP changes. The PV will settle back on SP without the use of integral. In fact, integral will result in the PV overshooting the SP and delay the eventual return of the PV to SP.
• Integral alone is always unstable on an integrating process. Controller gain is required to maintain stable control. What may appear to be too much integral may in fact be too little controller gain (heuristic tuning will not be as simple as just slowing down the integral action).
• Derivative, by penalizing PV movement, will slow down the ramp rate caused by a disturbance.

As with self-limiting processes, good control loop tuning means blending controller gain, integral, and (rarely) derivative together to get the ‘best’ controller response. One challenge will be determining what ‘best’ is as part of the goal will be to avoid unnecessarily disturbing downstream processes. We will see that our thought process for tuning an integrating process will in some ways be similar to, and in other very different from, how we tune a self-limiting process.

 

Ed Bullerdiek is a retired control engineer with 37 years of process control experience in petroleum refining and oil production. Edited by Mark T. Hoske, editor-in-chief, Control Engineering, WTWH Media, [email protected].

 

CONSIDER THIS

Integrating processes when controlled by a PID controller do not behave like self-limiting processes. What are the similarities, and what are the differences? How will this affect how we think about tuning PID controllers mated to integrating processes?

 

 

ONLINE

PID series from Ed Bullerdiek, retired control engineer

Part 1: Three reasons to tune control loops: Safety, profit, energy efficiency

PID spotlight, part 2: Know these 13 terms, interactions

PID spotlight, part 3: How to select one of four process responses

PID spotlight, part 4: How to balance PID control for a self-limiting process

PID spotlight, part 5: What does good and bad controller tuning look like?

PID spotlight, part 6: Deadtime? How to boost controller performance anyway

PID spotlight, part 7: Open-loop tuning of a self-limiting process

PID spotlight, part 8: Closed-loop tuning for self-limiting processes

PID spotlight, part 9: Heuristic tuning for a self-limiting process (part A on heuristic tuning)

PID spotlight, part 10: Heuristic tuning in a self-limiting process

 

Aug. 1 RCEP webcast available for one year: How to automate series: The mechanics of loop tuning

 

More on PID and advanced process control from Control Engineering.