Showing posts with label Heat Exchanger. Show all posts

Integral (Reset) Windup, Jacketing Logic and the Velocity PI Form

A valve cannot open more than all the way. A pump cannot go slower than stopped. Yet an improperly programmed control algorithm can issue such commands.

Herein lies the problem of integral windup (also referred to as reset windup or integral saturation). It is a problem that has been around for decades and was solved long ago. We discuss why it occurs and how to prevent it to help those who choose to write their own control algorithm.

The PI Algorithm
To increase our comfort level with the idea that different vendors cast the same PI algorithm in different forms, we choose the independent, continuous, position PI form for this discussion:

Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value
e(t) = current controller error, defined as SP - PV
SP = set point
PV = measured process variable (the wire in)
Kc = proportional gain, a tuning parameter
Ki = integral gain, a tuning parameter

Note that Kc is the same parameter in both the dependent and independent forms, though it is more typically called controller gain in the dependent form.

Every procedure and observation we have previously discussed about PI controllers applies to both forms. Both even use the same tuning correlations. To tune Ki, we compute Kc and Ti for the dependent form and then divide (Ki = Kc/Ti).

Integral (Reset) Windup
Our previous discussion of integral action noted that integration is a continual summing. As shown below, integration of error means that we continually sum controller error up to the present time.

The integral sum starts accumulating when the controller is first put in automatic and continues to change as long as controller error exists.
If an error is large enough and/or persists long enough, it is mathematically possible for the integral term to grow very large (either positive or negative):

This large integral, when combined with the other terms in the equation, can produce a CO value that causes the final control element (FCE) to saturate. That is, the CO drives the FCE (e.g. valve, pump, compressor) to its physical limit of fully open/on/maximum or fully closed/off/minimum.

And if this extreme value is still not sufficient to eliminate the error, the simple mathematics of the controller algorithm, if not jacketed with protective logic, permits the integral term to continue growing.

If the integral term grows unchecked, the equation above can command the valve, pump or compressor to move to 110%, then 120% and more. Clearly, however, when an an FCE reaches its full 100% value, these last commands have no physical meaning and consequently, no impact on the process.

Control is Lost
Once we cross over to a "no physical meaning" computation, the controller has lost the ability to regulate the process.

When the computed CO exceeds the physical capabilities of the FCE because the integral term has reached a large positive or negative value, the controller is suffering from windup. Because windup is associated with the integral term, it is often referred to as integral windup or reset windup.

To prevent windup from occurring, modern controllers are protected by either:
▪ Employing extra "jacketing logic" in the software to halt integration when the CO reaches a maximum or minimum value.
▪ Recasting the controller into a discrete velocity form that, by its very formulation, naturally avoids windup.
Both alternatives offer benefits but possess some fairly subtle drawbacks that we discuss below.

Visualizing Windup
To better visualize the problem of windup and the benefit of anti-windup protection, consider the plot from our heat exchanger process below (click for a large view).
To the left is the performance of a PI controller with no windup protection. To the right is the performance of the same controller protected by an anti-windup strategy.
For both controllers, the set point is stepped from 200 °C up to 215 °C and back again. As shown in the lower trace on the plot, the controller moves the CO to 0%, closing the valve completely, yet this is not sufficient to move the PV up to the new set point.

To the left in the plot, the impact of windup is a degraded controller performance. When the set point is stepped back to its original value of 200 °C , the windup condition causes a delay in the CO action. This in turn causes a delay in the PV response.

To the right in the plot, anti-windup protection permits the CO, and thus PV, to respond promptly to the command to return to the original SP value of 200 °C.

More Details on Windup
The plot below (click for large view) offers more detail. As labeled on the plot:

1) To the left for the Controller with Wind-up case, the SP is stepped up to 215 °C. The valve closes completely but is not able to move the PV all the way to the high set point value. Integration is a summing of controller error, and since error persists, the integration term grows very large.
The sustained error permits the controller to windup (saturate). While it is not obvious from the plot, the PI algorithm is computing values for CO that ask the valve to be open -5%, -8% and more. The control algorithm is just simple math with no ability to recognize that a valve cannot be open to a negative value.
Note that the chart shows the CO signal bottoming out at 0% while the controller algorithm is computing negative CO values. This misleading information is one reason why windup can be difficult to diagnose as the root cause of a problem from visual inspection of process data trend plots.

2) When the SP is stepped back to 200 °C, it seems as if the CO does not move at first. In reality, the control algorithm started moving the CO when the SP changed, but the values remain in the physically meaningless range of negative numbers.
So while the valve remains fully closed at 0%, the integral sum is accumulating controller errors of opposite sign. As time passes, the integral term shrinks or "unwinds" as the running sum of errors balance out.
3) When the integral sum of errors shrinks enough, it no longer dominates the CO computation. The CO signal returns from the physically meaningless world of negative values. The valve can finally move in response.
4) To the right in the plot above, the controller is protected from windup. As a result, when the set point is stepped back to 200 °C, the CO immediately reacts with a change that is proportional to the size of the SP change. The PV moves quickly in response to the CO actions as it tracks the SP back to 200 °C.

◊ Solution 1: Jacketing Logic on the Position Algorithm
The PI controller at the top of this article is called the position form because the computed CO is a specific intermediate value between full on/open/maximum and closed/off/minimum. The continuous PI algorithm is specifying the actual position (e.g., 27% open, 64% of maximum) that the final control element (FCE) should assume.

● Simple Logic Creates Additional Problems
It is not enough to have logic that simply limits or clips the CO if it reaches a maximum (COmax) or minimum (COmin) value because this does nothing to check the growth of the integral sum of errors term.
I
n fact, such simple logic was used in the "control with windup" plots just discussed. The CO seems stuck at 0% and we are unaware that the algorithm is actually computing negative valve positions as described in item 1 above.

● Anti-Windup Logic Outline
When we switch from manual mode to automatic, we assume that we have initialized the controller using a bumpless transfer procedure. That is, at switchover, the integral sum of error is set to zero, the SP is set equal to the current PV, and the controller bias is set equal to the current CO (implying that COmin < CObias < COmax).

Thus, there is nothing to cause CO to immediately change and "bump" our process at switchover.

One approach to creating anti-windup jacketing logic is to artificially manipulate the integral sum of error itself. With our controller properly initialized, the approach is to flip the algorithm around and back-calculate a value for the integral sum of error that will provide a desired controller output value (COdesired), or:

Note that COdesired can be different in different situations. For example,
▪ We do not want tuning parameter adjustments to cause sudden CO movements that bump our process. So if tuning values have changed, COdesired is the value of CO from the previous loop calculation cycle.
▪ If the PI controller computes CO values that are above COmax or below COmin, then we must be concerned about windup and COdesired is set equal to the limiting COmax or COmin value.

The anti-windup logic followed at every loop sample time, T, is thus:

1) If tuning parameters have changed since the last loop calculation cycle, then COdesired = current CO. Back calculate the integral sum of error so CO remains unchanged from the previous sample time. This prevents sudden CO bumps due to tuning changes.

2) Update SP and PV for this loop calculation cycle.
3) compute:

4) If CO > COmax or if CO < COmin, then the anti-windup (integral desaturation) logic of step 5 is required. Otherwise, proceed to step 6.

5) If CO > COmax, then CO = COdesired = COmax. if CO < COmin, then CO = COdesired = COmin. Back calculate the integral sum of error using our selected COdesired and save it for use in the next control loop calculation cycle.

6) Implement CO

◊ Solution 2 - Use the Velocity (Discrete) Controller Form
Rather than computing a CO signal indicating a specific position for our final control element, an alternative is to compute a signal that specifies a change, ∆CO, from current position for the FCE. As explained below, this is called the velocity or discrete controller form.

We employ the dependent algorithm for this presentation, but the derivation that follows can be applied in an analogous fashion to the independent PI form. To derive the discrete velocity form, we must first write the continuous, position form of the PI controller to include the independent variable on the controller output, showing it properly as CO(t) to reflect that it changes with time:

Please note that this controller is identical to all dependent PI forms as presented in other articles. The only difference is we are being more mathematically precise in our expression of CO(t).

● Deriving the Discrete Velocity Form
The first step in deriving the discrete form is to take the time derivative of the continuous form. In physics, the time derivative (rate of change) of a position is a velocity. This is why the final form of the PI controller we derive is often called the velocity form.

Taking the derivative of the continuous PI controller above with respect to time yields:

Since CObias is a constant and the derivative of a constant is zero, then:

Removing this term from the equation results in:

If we assume discrete or finite difference approximations for the continuous derivatives, then the controller becomes:

where ei is the current controller error, ei-1 is the controller error at the previous sample time, T, and ∆ei = ei - ei-1.
Recognizing that loop sample time is T = ∆t, then the PI controller becomes:

Rearranging, we arrive at the discrete velocity form of the PI controller:

● Reason for Anti-Windup Protection
Discrete velocity algorithms compute a ∆CO that signals the FCE to move a specific distance and direction from its current position. As we can see from the PI controller form above, the computation does not keep track of the current FCE position, nor does it mathematically accumulate any integral sums.

In a sense, the accumulation of integration is stored in the final control element itself. If a long series of ∆CO moves are all positive, for example, the valve, pump or compressor will move toward its maximum value. And once the FCE reaches its maximum limit, any ∆CO commands to move further will have no impact because, as stated in the first sentences of this article, a valve cannot open more than all the way and a pump cannot go slower than stopped. It is the physical nature of the FCE itself that provides protection from over-accumulation (i.e., windup).

As long as the CO never reaches COmax or COmin, the continuous position and discrete velocity forms of the PI controller provide identical performance. A properly jacketed continuous position PI controller will also provide windup protection equal to the discrete velocity form. Implicit in these statements is that sample time,T, is reasonably fast and that T and the tuning values (Kc and Ti) are the same when comparing implementations.

● Concerns with Discrete Velocity PID
Unfortunately, the usefulness of the discrete velocity form is limited because the method suffers problems when derivative action is included. We find that we must take the derivative of a derivative, yielding a second derivative. A second derivative applied to data that contains even modest noise can produce nonsense results.

Some vendors implement this form anyway and include a signal filter and additional logic sequences to address the problem. Thus, even with the anti-windup benefits of a discrete velocity algorithm, we find the need to jacket the algorithm with protective logic.

Tuesday, 22 March 2011 // Labels: action, basics, behavious, Heat Exchanger, integral, pid, tuning // 1 comments //

PI Control of the Heat Exchanger

PI Control of the Heat Exchanger

We investigated P-Only control of the heat exchanger process and learned that while P-Only is an algorithm that is easy to tune and maintain, it has a severe limitation. Specifically, its simple form permits steady state error, called offset, in most processes during normal operation.

Then we moved on to integral action and PI control. We focused in that article on the structure of the algorithm and explored the mathematics of how the proportional and integral terms worked together to eliminate offset.

Here we test the capabilities of the PI controller on the heat exchanger process. Our focus is on design, implementation and basic performance issues. Along the way we will highlight some strengths and weaknesses of this popular algorithm.

As with all controller implementations, best practice is to follow our proven four-step design and tuning recipe as we proceed with this case study.

Step 1: Design Level of Operation (DLO)

Real processes display a nonlinear behavior. That is, their process gain, time constant and/or dead time changes as operating level changes and as major disturbances change. Since controller design and tuning is based on these process Kp, Tp and Өp values, controllers should be designed and tuned for a specific level of operation.
Thus, the first step in our controller design recipe is to specify our design level of operation (DLO). This includes stating:
▪ Where we expect the set point, SP, and measured process variable, PV, to be during normal operation.
▪ The range of values the SP and PV might assume so we can explore the nature of the process dynamics across that range.
We will track along with the same design conditions used in the P-Only control study to permit a direct comparison of performance and capability. As in that study, we specify:
▪ Design PV and SP = 138 °C with range of 138 to 140 °C
We also should know normal or typical values for our major disturbances and be reasonably confident that they are quiet so we may proceed with a bump test. The heat exchanger process has only one major disturbance variable, and consistent with the previous study:
▪ Expected warm liquid flow disturbance = 10 L/min

Step 2: Collect Data at the DLO
The next step in the design recipe is to collect dynamic process data as near as practical to our design level of operation. We have previously collected and documented heat exchanger step test data that matches our design conditions.

Step 3: Fit an FOPDT Model to the Design Data
Here we document a first order plus dead time (FOPDT) model approximation of the step test data from step 2:
▪ Process gain (how far), Kp = -0.53 °C/%
▪ Time constant (how fast), Tp = 1.3 min
▪ Dead time (how much delay), Өp = 0.8 min

Step 4: Use the Parameters to Complete the Design
One common form of the PI controller computes a controller output (CO) action every loop sample time T as:

Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value; set by bumpless transfer as explained below
e(t) = current controller error, defined as SP - PV
SP = set point
PV = measured process variable (the wire in)
Kc = controller gain, a tuning parameter
Ti = reset time, a tuning parameter

• Loop Sample Time, T
Best practice is to specify loop sample time, T, at 10 times per time constant or faster (T ≤ 0.1Tp). For this study, T ≤ 0.13 min = 8 sec. Faster sampling may provide modestly improved performance, while slower sampling can lead to significantly degraded performance. Most commercial controllers offer an option of T = 1.0 sec, and since this meets our design rule, we use that here.

• Computing controller error, e(t)
Set point, SP, is something we enter into the controller. The PV measurement comes from our sensor (our wire in). With SP and PV known, controller error, e(t) = SP - PV, can be directly computed at every loop sample time T.

• Determining Bias Value
Strictly speaking, CObias is the value of the CO that, in manual mode, causes the PV to steady at the DLO while the major disturbances are quiet and at their normal or expected values.

• Bumpless Transfer
A desirable feature of the PI algorithm is that it is able to eliminate the offset that can occur under P-Only control. The integral term of the PI controller provides this capability by providing updated information that, when combined with the controller bias, keeps the process centered as conditions change.

Since integral action acts to update (or reset) our bias value over time, CObias can be initialized in a straightforward fashion to a value that produces no abrupt control actions when we switch to automatic. Most commercial controllers do this with a simple "bumpless transfer" feature. When switching to automatic, they initialize:
▪ SP equal to the current PV
▪ CObias equal to the current CO

With the set point equal to the measured process variable, there is no error to drive a change in our controller output. And with the controller bias set to our current controller output, we are prepared by default to maintain current operation.

We will use a controller that employs these bumpless transfer rules when we switch to automatic. Hence, we need not specify any value for CObias as part of our design.

• Computing Controller Gain and Reset Time
Here we use the industry-proven Internal Model Control (IMC) tuning correlations. The first step in using the IMC correlations is to compute Tc, the closed loop time constant. All time constants describe the speed or quickness of a response. The closed loop time constant describes the desired speed or quickness of a controller in responding to a set point change or rejecting a disturbance.

If we want an active or quickly responding controller and can tolerate some overshoot and oscillation as the PV settles out, we want a small Tc (a short response time) and should choose aggressive tuning:
▪ aggressive: Tc is the larger of 0.1·Tp or 0.8·Өp
Moderate tuning is for a controller that will move the PV reasonably fast while producing little to no overshoot.
▪ moderate: Tc is the larger of 1·Tp or 8·Өp
If we seek a more sluggish controller that will move things in the proper direction, but quite slowly, we choose conservative tuning (a big or long Tc).
▪ conservative: Tc is the larger of 10·Tp or 80·Өp

Once we have decided on our desired performance and computed the closed loop time constant, Tc, with the above rules, then the PI correlations for controller gain, Kc, and reset time, Ti, are:

Notice that reset time, Ti, is always set equal to the time constant of the process, regardless of desired controller activity.

a) Moderate Response Tuning:
For a controller that will move the PV reasonably fast while producing little to no overshoot, choose:
Moderate Tc = the larger of 1·Tp or 8·Өp
= larger of 1(1.3 min) or 8(0.8 min)
= 6.4 min
Using this Tc and our model parameters in the tuning correlations above, we arrive at the moderate tuning values:

b) Aggressive Response Tuning:

For an active or quickly responding controller where we can tolerate some overshoot and oscillation as the PV settles out, specify
Aggressive Tc = the larger of 0.1·Tp or 0.8·Өp
= larger of 0.1(1.3 min) or 0.8(0.8 min)
= 0.64 min
and the aggressive tuning values are:

Practitioner's Note: The FOPDT model parameters used in the tuning correlations above have engineering units, so the Kc values we compute also have engineering units. In commercial control systems, controller gain (or proportional band) is normally entered as a dimensionless (%/%) value.

For commercial implementations, we could:
▪ Scale the process data before fitting our FOPDT dynamic model so we directly compute a dimensionless Kc.
▪ Convert the model Kp to dimensionless %/% after fitting the model but before using the FOPDT parameters in the tuning correlations.
▪ Convert Kc from engineering units into dimensionless %/% after using the tuning correlations.

CO is already scaled from 0 - 100% in the above example. Thus, we convert Kc from engineering units into dimensionless %/% using the formula:

For the heat exchanger, PVmax = 250 oC and PVmin = 0 oC. The dimensionless Kc values are thus computed:

▪ moderate Kc = (- 0.34 %/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)] = - 0.85 %/%
▪ aggressive Kc = (- 1.7%/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)] = - 4.2 %/%

We use Kc with engineering units in the remainder of this article and are careful that our PI controller is formulated to accept such values. We would be mindful if we were using a commercial control system, however, to ensure our tuning parameters are cast in the form appropriate for our equipment.

• Controller Action
The process gain, Kp, is negative for the heat exchanger, indicating that when CO increases, the PV decreases in response. This behavior is characteristic of a reverse acting process. Given this CO to PV relationship, when in automatic mode (closed loop), if the PV starts drifting above set point, the controller must increase CO to correct the error. Such negative feedback is an essential component of stable controller design.
A process that is naturally reverse acting requires a controller that is direct acting to remain stable. In spite of the opposite labels (reverse acting process and direct acting controller), the details presented above show that both Kp and Kc are negative values.

In most commercial controllers, only positive Kc values can be entered. The sign (or action) of the controller is then assigned by specifying that the controller is either reverse acting or direct acting to indicate a positive or negative Kc, respectively.

If the wrong control action is entered, the controller will quickly drive the final control element (FCE) to full on/open or full off/closed and remain there until a proper control action entry is made.

Implement and Test

Below we test our two PI controllers on the heat exchanger process simulation. Shown are two set points step pairs from 138 °C up to 140 °C and back again.
The first set point steps to the left show the PI controller performance using the moderate tuning values computed above. The second set point steps to the right show the controller performance using the aggressive tuning values. Note that the warm liquid disturbance flow, though not shown, remains constant at 10 L/min throughout the study.
(For comparison, the performance of the P-Only controller in tracking these set point changes is pictured here).

The asymmetrical behavior of the PV for the set point steps up compared to the steps down is due to the very nonlinear character of the heat exchanger.

If we seek tuning between moderate and aggressive performance, we would average the Kc values from the tuning rules above.

But if we believe we had collected good bump test data (we saw a clear response in the PV when we stepped the CO and the major disturbances were quiet during the test), and the FOPDT model fit appears to be visually descriptive of the data, then we have a good value for Tp and that means a good value for Ti.

If we are going to fiddle with the tuning, we can tweak Kc and we should leave the reset time alone.

Tuning Recipe Saves Time and Money
The exciting result is that we achieved our desired controller performance based on one bump test and following a controller design recipe. No trial and error was involved. Little off-spec product was produced. No time was wasted.

Soon we will see how software tools help us achieve such results with even less disruption to the process.

The method of approximating complex behavior with a FOPDT model and then following a recipe for controller design and tuning has been used successfully on a broad spectrum of processes with streams composed of gases, liquids, powders, slurries and melts. It is a reliable approach that has been proven time and again at diverse plants from a wide range of companies.