When exploring the capabilities of the P-Only controller in rejecting disturbances for the gravity drained tanks process, we confirmed the observations we had made during the the P-Only set point tracking study for the heat exchanger.
In particular, the P-Only algorithm is easy to tune and maintain, but whenever the set point or a major disturbance moves the process from the design level of operation, a sustained error between the process variable (PV) and set point (SP), called offset, results.
Further, we saw in both case studies that as controller gain, Kc, increases (or as proportional band, PB, decreases):
▪ the activity of the controller output, CO, increases
▪ the oscillatory nature of the response increases
▪ the offset (sustained error) decreases
In this article, we explore the benefits of integral action and the capabilities of the PI controller for rejecting disturbances in the gravity drained tanks process. We have previously presented the fundamentals behind PI control and its application to set point tracking in the heat exchanger.
As with all controller implementations, best practice is to follow our proven four-step design and tuning recipe. One benefit of the recipe is that steps 1-3, summarized below from our P-Only study, remain the same regardless of the control algorithm being employed. After summarizing steps 1-3, we complete the PI controller design and tuning in step 4.
Step 1: Determine the Design Level of Operation (DLO)
The control objective is to reject disturbances as we control liquid level in the lower tank. Our design level of operation (DLO), detailed here for this study is:
▪ design PV and SP = 2.2 m with range of 2.0 to 2.4 m
▪ design D = 2 L/min with occasional spikes up to 5 L/min
Step 2: Collect Process Data around the DLO
When CO, PV and D are steady near the design level of operation, we bump the CO as detailed here and force a clear response in the PV that dominates the noise.
Step 3: Fit a FOPDT Model to the Dynamic Process Data
We then describe the process behavior by fitting an approximating first order plus dead time (FOPDT) dynamic model to the test data from step 2. We define the model parameters and present details of the model fit of step test data here. A model fit of doublet test data using commercial software confirms these values:
▪ process gain (how far), Kp = 0.09 m/%
▪ time constant (how fast), Tp = 1.4 min
▪ dead time (how much delay), Өp = 0.5 min
Step 4: Use the FOPDT Parameters to Complete the Design
Following the heat exchanger PI control study, we explore what is often called the dependent, ideal form of the PI control algorithm:
Where:
CO = controller output signal (the wire out)
CObias = controller bias or null value; set by bumpless transfer
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
Aside: our observations using the dependent ideal PI algorithm directly apply to the other popular PI controller forms. For example, the integral gain, Ki, in the independent algorithm form:
can be computed directly from controller gain and reset time as: Ki = Kc/Ti.
In the P-Only study, we established that for the gravity drained tanks process:
▪ sample time, T = 1 sec
▪ the controller is reverse acting
▪ dead time is small compared to Tp and thus not a concern in the design
• Controller Gain, Kc, and Reset Time, Ti
We use our FOPDT model parameters in the industry-proven Internal Model Control (IMC) tuning correlations to compute PI tuning values.
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. Tc 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 Response: Tc is the larger of 0.1·Tp or 0.8·Өp
If we seek a sluggish controller that will move things in the proper direction, but quite slowly, we choose conservative tuning (a big or long Tc).
▪ Conservative Response: Tc is the larger of 10·Tp or 80·Өp
Moderate tuning is for a controller that will move the PV reasonably fast while producing little to no overshoot.
▪ Moderate Response: Tc is the larger of 1·Tp or 8·Өp
With Tc computed, the PI controller gain, Kc, and reset time, Ti, are computed as:
Notice that reset time, Ti, is always equal to the process time constant, Tp, 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(1.4 min) or 8(0.5 min)
= 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(1.4 min) or 0.8(0.5 min)
= 0.4 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.
To address this, 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.
Since we already have Kc in engineering units, we employ the third option. 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 gravity drained tanks, PVmax = 10 m and PVmin = 0 m. The dimensionless Kc values are thus computed:
▪ moderate Kc = (3.5 %/m)∙[(10 - 0 m) ÷ (100 - 0%)] = 0.35 %/%
▪ aggressive Kc = (17 %/m)∙[(10 - 0 m) ÷ (100 - 0%)] = 1.7 %/%
We use the Kc with engineering units in the remainder of this article and are careful that our PI controller is formulated to accept such values. If we were using these results in a commercial control system, we would be careful to ensure our tuning parameters are cast in the form appropriate for our equipment.
• Controller Action
The process gain, Kp, is positive for the gravity drained tanks, indicating that when CO increases, the PV increases in response. This behavior is characteristic of a direct 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 decrease CO to correct the error. Such negative feedback is an essential component of stable controller design.
A process that is naturally direct acting requires a controller that is reverse acting to remain stable. In spite of the opposite labels (direct acting process and reverse acting controller), the details presented above show that both Kp and Kc are positive 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
The ability of the PI controller to reject changes in the pumped flow disturbance, D, is pictured below for the moderate and aggressive tuning values computed above. Note that the set point remains constant at 2.2 m throughout the study.
The aggressive controller shows a more energetic CO action, and thus, a more active PV response. As shown above, however, the penalty for this increased activity is some overshoot and oscillation in the process response.
Please be aware that the terms "moderate" and "aggressive" hold no magic. If we desire a control performance between the two, we need only average the Kc values from the tuning rules above. Note, however, that these rules provide a constant reset time, Ti, regardless of our desired performance. So if we believe we have collected a good process data set, and the FOPDT model fit looks like a reasonable approximation of this data, then Ti = Tp always.
While not our design objective, presented below is the set point tracking ability of the PI controller when the disturbance flow is held constant:
Again, the aggressive tuning values provide for a more active response.
Aside: it may appear that the random noise in the PV measurement signal is different in the two plots above, but it is indeed the same. Note that the span of the PV axis in each plot differs by a factor of four. The narrow span of the set point tracking plot greatly magnifies the signal traces, making the noise more visible.
Comparison With P-Only Control
The performance of a P-Only controller in addressing the same disturbance rejection and set point tracking challenge is shown here. A comparison of that study with the results presented here reveals that PI controllers:
▪ can eliminate the offset associated with P-Only control,
▪ have integral action that increases the tendency for the PV to roll (or oscillate),
▪ have two tuning parameters that interact, increasing the challenge to correct tuning when performance is not acceptable.
Derivative Action
The addition of the derivative term to complete the PID algorithm provides modest benefit yet significant challenges.
Again, the aggressive tuning values provide for a more active response.
Aside: it may appear that the random noise in the PV measurement signal is different in the two plots above, but it is indeed the same. Note that the span of the PV axis in each plot differs by a factor of four. The narrow span of the set point tracking plot greatly magnifies the signal traces, making the noise more visible.
Comparison With P-Only Control
The performance of a P-Only controller in addressing the same disturbance rejection and set point tracking challenge is shown here. A comparison of that study with the results presented here reveals that PI controllers:
▪ can eliminate the offset associated with P-Only control,
▪ have integral action that increases the tendency for the PV to roll (or oscillate),
▪ have two tuning parameters that interact, increasing the challenge to correct tuning when performance is not acceptable.
Derivative Action
The addition of the derivative term to complete the PID algorithm provides modest benefit yet significant challenges.
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.
Like the P-Only controller, the Proportional-Integral (PI) algorithm computes and transmits a controller output (CO) signal every sample time, T, to the final control element (e.g., valve, variable speed pump). The computed CO from the PI algorithm is influenced by the controller tuning parameters and the controller error, e(t).
PI controllers have two tuning parameters to adjust. While this makes them more challenging to tune than a P-Only controller, they are not as complex as the three parameter PID controller.
Integral action enables PI controllers to eliminate offset, a major weakness of a P-only controller. Thus, PI controllers provide a balance of complexity and capability that makes them by far the most widely used algorithm in process control applications.
The PI Algorithm
While different vendors cast what is essentially the same algorithm in different forms, here we explore what is variously described as the dependent, ideal, continuous, position form:
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
The first two terms to the right of the equal sign are identical to the P-Only controller referenced at the top of this article.
The integral mode of the controller is the last term of the equation. Its function is to integrate or continually sum the controller error, e(t), over time.
Some things we should know about the reset time tuning parameter, Ti:
▪ It provides a separate weight to the integral term so the influence of integral action can be independently adjusted.
▪ It is in the denominator so smaller values provide a larger weight to (i.e. increase the influence of) the integral term.
▪ It has units of time so it is always positive.
Function of the Proportional Term
As with the P-Only controller, the proportional term of the PI controller, Kc·e(t), adds or subtracts from CObias based on the size of controller error e(t) at each time t.
As e(t) grows or shrinks, the amount added to CObias grows or shrinks immediately and proportionately. The past history and current trajectory of the controller error have no influence on the proportional term computation.
The plot below illustrates this idea for a set point response. The error used in the proportional calculation is shown on the plot:
▪ At time t = 25 min, e(25) = 60-56 = 4
▪ At time t = 40 min, e(40) = 60-62 = -2
Recalling that controller error e(t) = SP - PV, rather than viewing PV and SP as separate traces as we do above, we can compute and plot e(t) at each point in time t.
Below is the identical data to that above only it is recast as a plot of e(t) itself. Notice that in the plot above, PV = SP = 50 for the first 10 min, while in the error plot below, e(t) = 0 for the same time period.
This plot is useful as it helps us visualize how controller error continually changes size and sign as time passes.
Function of the Integral Term
While the proportional term considers the current size of e(t) only at the time of the controller calculation, the integral term considers the history of the error, or how long and how far the measured process variable has been from the set point over time.
Integration is a continual summing. Integration of error over time means that we sum up the complete controller error history up to the present time, starting from when the controller was first switched to automatic.
Controller error is e(t) = SP - PV. In the plot below, the integral sum of error is computed as the shaded areas between the SP and PV traces.
Each box in the plot has an integral sum of 20 (2 high by 10 wide). If we count the number of boxes (including fractions of boxes) contained in the shaded areas, we can compute the integral sum of error.
So when the PV first crosses the set point at around t = 32, the integral sum has grown to about 135. We write the integral term of the PI controller as:
Since it is controller error that drives the calculation, we get a direct view the situation from a controller error plot as shown below:
Note that the integral of each shaded portion has the same sign as the error. Since the integral sum starts accumulating when the controller is first put in automatic, the total integral sum grows as long as e(t) is positive and shrinks when it is negative.
At time t = 60 min on the plots, the integral sum is 135 - 34 = 101. The response is largely settled out at t = 90 min, and the integral sum is then 135 - 34 7 = 108.
Integral Action Eliminates Offset
The previous sentence makes a subtle yet very important observation. The response is largely complete at time t = 90 min, yet the integral sum of all error is not zero.
In this example, the integral sum has a final or residual value of 108. It is this residual value that enables integral action of the PI controller to eliminate offset.
As discussed in a previous article, most processes under P-only control experience offset during normal operation. Offset is a sustained value for controller error (i.e., PV does not equal SP at steady state).
We recognize from the P-Only controller:
that CO will always equal CObias unless we add or subtract something from it.
The only way we have something to add or subtract from CObias in the P-Only equation above is if e(t) is not zero. It e(t) is not steady at zero, then PV does not equal SP and we have offset.
However, with the PI controller:
we now know that the integral sum of error can have a final or residual value after a response is complete. This is important because it means that e(t) can be zero, yet we can still have something to add or subtract from CObias to form the final controller output, CO.
So as long as there is any error (as long as e(t) is not zero), the integral term will grow or shrink in size to impact CO. The changes in CO will only cease when PV equals SP (when e(t) = 0) for a sustained period of time.
At that point, the integral term can have a residual value as just discussed. This residual value from integration, when added to CObias, essentially creates a new overall bias value that corresponds to the new level of operation.
In effect, integral action continually resets the bias value to eliminate offset as operating level changes.
Challenges of PI Control
There are challenges in employing the PI algorithm:
▪ The two tuning parameters interact with each other and their influence must be balanced by the designer.
▪ The integral term tends to increase the oscillatory or rolling behavior of the process response.
Because the two tuning parameters interact with each other, it can be challenging to arrive at "best" tuning values. The value and importance of our design and tuning recipe increases as the controller becomes more complex.
Initializing the Controller for Bumpless Transfer
When we switch any controller from manual mode to automatic (from open loop to closed loop), we want the result to be uneventful. That is, we do not want the switchover to cause abrupt control actions that impact or disrupt our process
We achieve this desired outcome at switchover by initializing the controller integral sum of error to zero. Also, the set point and controller bias value are initialized by setting:
▪ SP equal to the current PV
▪ CObias equal to the current CO
With the integral sum of error set to zero, there is nothing to add or subtract from CObias that would cause a sudden change in the current controller output. With the set point equal to the measured process variable, there is no error to drive a change in our CO. And with the controller bias set to our current CO value, we are prepared by default to maintain current operation.
Thus, when we switch from manual mode to automatic, we have "bumpless transfer" with no surprises. This is a result everyone appreciates.
Reset Time Versus Reset Rate
Different vendors cast their control algorithms in slightly different forms. Some use proportional band rather than controller gain. Also, some use reset rate, Tr, instead of reset time. These are simply the inverse of each other:
Tr = 1/Ti
No matter how the tuning parameters are expressed, the PI algorithms are all equally capable.
But it is critical to know your manufacturer before you start tuning your controller because parameter values must be matched to your particular algorithm form. Commercial software for controller design and tuning will automatically address this problem for you.
Implementing a PI controller
We explore PI controller design, tuning and implementation on the heat exchanger in this article and the gravity drained tanks in this article.
Today I am going to dicuss a very importnat issue.
In fact two of my colleagues have experienced the thisissue & almost got embarrased in front of the client.
Whle implementing PID youn should be very careful to implement SCALLING & UNITS to tyhe controller. My colleague was there to implement as PH control loop. After setting all the parameters they were not getting the correct reading out of PH transmitter & in turn they can't use these values to implement the control loop for the industry. After a long struggle at site they have called e. My first question was about scaling & guess What! They forgot to implement scaling. This simple mistake cost them & cluient more than two hours.
So remember , Controller Gain Is Dimensionless in Commercial Systems.
IIn modern plants, process variable (PV) measurement signals are typically scaled to engineering units before they are displayed on the control room HMI computer screen or archived for storage by a process data historian. This is done for good reasons.
When operations staff walk through the plant, the assorted field gauges display the local measurements in engineering units to show that a vessel is operating, for example, at a pressure of 25 psig (1.7 barg) and a temperature of 140 oC (284 oF).
It makes sense, then, that the computer screens in the control room display the set point (SP) and PV values in these same familiar engineering units because:
• It helps the operations staff translate their knowledge and intuition from their field experience over to the abstract world of crowded HMI computer displays.
• Familiar units will facilitate the instinctive reactions and rapid decision making that prevents an unusual occurrence from escalating into a crisis situation.
• The process was originally designed in engineering units, so this is how the plant documentation will list the operating specifications.
Knowledge Base Articles Compute Kc With Units
Like a control room display, the Control Station Knowledge Base presents PV values in engineering units. In most articles, these PVs are used directly in tuning correlations to compute controller gains, Kc. As a result, the Kc values also carry engineering units.
The benefit of this approach is that controller gain maintains the intuitive familiarity that engineering units provide. The difficulty is that commercial controllers are normally configured to use a dimensionless Kc (or dimensionless proportional band, PB).
To address this issue, we explore below how to convert a Kc with engineering units into the standard dimensionless (%/%) form.
The conversion formula presented at the end of this article is reasonably straightforward to use. But it is derived from several subtle concepts that might benefit from explanation. Thus, we begin with a background discussion on units and scaling, and work our way toward our Kc conversion formula goal.
From Analog Sensor to Digital Signal
There are many ways to measure a process variable and move the signal into the digital world for use in a computer based control system. Below is a simplified sketch of one approach.
Other operations in the pathway from sensor to control system not shown in the simplified sketch might include a transducer, an amplifier, a transmitter, a scaling element, a linearizing element, a signal filter, a multiplexer, and more.
The central issue for this discussion is that the PV signal arrives at the computers and controllers in a raw digital form. The continuous analog PV measurement has been quantized (broken into) a range of discrete increments or digital integer "counts" by an A/D (analog to digital) converter.
More counts dividing the span of a measurement signal increases the resolution of the measurement when expressed as a digital value. The ranges offered by most vendors result from the computer binary 2n form where n is the number of bits of resolution used by the A/D converter.
Example: a 12 bit A/D converter digitizes an analog signal into 212 = 4096 discrete increments normally expressed to range from 0 to 4095 counts.
A 13 bit A/D converter digitizes an analog signal into 213 = 8192 discrete increments normally expressed to range from 0 to 8191 counts.
A 14 bit A/D converter digitizes an analog signal into 214 = 16384 discrete increments normally expressed to range from 0 to 16383 counts.
Example: if a 4 to 20 mA (milliamp) analog signal range is digitized by a 12 bit A/D converter into 0 to 4095 counts, then the resolution is:
(20 - 4 mA) ÷ 4095 counts = 0.00391 mA/count
A signal of 7 mA from an analog range of 4 to 20 mA changes to digital counts from the 12 bit A/D converter as:
(7 - 4 mA) ÷ 0.00391 mA/count = 767 counts
A signal of 1250 counts from a 12 bit A/D converter corresponds to an input signal of 8.89 mA from an analog range of 4 to 20 mA as:
4 mA (1250 counts)∙(0.00391 mA/count) = 8.89 mA
Scaling the Digital PV Signal to Engineering Units for Display
During the configuration phase of a control project, the minimum and maximum (or zero and span) of the PV measurement must be entered. These values are used to scale the digital PV signal to engineering units for display and storage.
Example: if a temperature range of 100 oC to 500 oC is digitized into 0 to 8191 counts by a 13 bit A/D converter, the signal is scaled for display and storage by setting the minimum digital value of 0 counts = 100 oC, and maximum digital value of 8191 counts = 500 oC
Each digital count from the 13 bit A/D converter gives a resolution of:
(500 - 100 oC) ÷ 8191 counts = 0.0488 oC/count
A signal of 175 oC from an analog range of 100 oC to 500 oC changes to digital counts from the 13 bit A/D converter as:
(175 - 100 oC) ÷ 0.0488 oC/count = 1537 counts
A signal of 1250 counts from the 13 bit A/D converter corresponds to an input signal of 161 oC from an analog range of 100 oC to 500 oC as:
100 oC (1250 counts)∙(0.0488 oC/count) = 161 oC
As discussed at the top of this article, the intuition and field knowledge of the operations staff is maintained by using engineering units in control room displays and when storing data to a historian.
For this same reason, modern control software uses engineering units when passing variables between the function blocks used for calculations and decision-making. Calculation and decision functions are easier to understand, document and debug when the logic is written using floating point values in common engineering units.
Scaling the Digital PV Signal for Use by the PID Controller
Most commercial PID controllers use a controller gain, Kc (or proportional band, PB) that is expressed as a standard dimensionless %/%.
Note: Controller gain in commercial controllers is often said to be unitless or dimensionless, but Kc actually has units of (% of CO signal)/(% of PV signal). In a precise mathematical world, these units do not cancel, though there is little harm in speaking as though they do.
Prior to executing the PID controller calculation, the PV signal must be scaled to a standard 0% to 100% to match the "dimensionless" Kc. This happens every loop sample time, T, regardless of whether we are measuring temperature, pressure, flow, or any other process variable.
To perform this scaling, the minimum and maximum PV values in engineering units corresponding to the 0% to 100% standard PV range must be entered during setup and loop configuration.
Example: if a temperature range of 100 oC to 500 oC is digitized into 0 to 8191 counts by a 13 bit A/D converter, the signal is scaled for the PID control calculation by setting the minimum digital value of 0 counts = 0%, and the maximum digital value of 8191 counts = 100%.
Each digital count from the 13 bit A/D converter gives a resolution of:
(100 - 0%) ÷ 8191 counts = 0.0122%/count
A signal of 1537 counts (175 oC) from a 13 bit A/D converter would translate to a signal of 18.75% as:
0% (1537)∙(0.0122%/value) = 18.75%
A signal of 1250 counts (161 oC) from a 13 bit A/D converter would translate to a signal of 15.25% as:
0% (1250)∙(0.0122%/value) = 15.25%
Control Output is 0% to 100%
The controller output (CO) from commercial controllers normally default to a 0% to 100% digital signal as well. Digital to analog (D/A) converters begin the transition of moving the digital CO values into the appropriate electrical current and voltage required by the valve, pump or other final control element (FCE) in the loop.
Note: While CO commonly defaults to a 0% to 100% signal, this may not be appropriate when implementing the outer primary controller in a cascade. The outer primary CO1 becomes the set point of the inner secondary controller, and signal scaling must match. For example, if SP2 is in engineering units, the CO1 signal must be scaled accordingly.
Care Required When Using Engineering Units For Controller Tuning
It is quite common to analyze and design controllers using data retrieved from our process historian or captured from our computer display. Just as with the articles in this e-book, this means the computed Kc values will likely be scaled in engineering units.
The sketch below highlights (click for a large view) that scaling from engineering units to a standard 0% to 100% range used in commercial controllers requires careful attention to detail.
The conversion of PV in engineering units to a standard 0% to 100% range requires knowledge of the maximum and minimum PV values in engineering units. These are the same values that are entered into our PID controller software function block during setup and loop configuration. The general conversion formula is:
where:
PVmax = maximum PV value in engineering units
PVmin = minimum PV value in engineering units
PV = current PV value in engineering units
Example: a temperature signal ranges from 100 oC to 500 oC and we seek to scale it to a range of 0% to 100% for use in a PID controller. We set:
PVmin = 100 oC and PVmax = 500 oC
A temperature of 175 oC converts to a standard 0% to 100% range as:
[(175 - 100 oC) ÷ (500 - 100 oC)]∙(100 - 0%) = 18.75%
A temperature of 161 oC converts to a standard 0% to 100% range as:
[(161 - 100 oC) ÷ (500 - 100 oC)]∙(100 - 0%) = 15.25%
Applying Conversion to Controller Gain, Kc
The discussion to this point provides the basis for the formula used to convert Kc from engineering units into dimensionless (%/%):
Example: the moderate Kc value in our P-Only control of the heat exchanger study is Kc = - 0.7 %/ oC. For this process, PVmax = 250 oC and PVmin = 0 oC
Kc = (- 0.7 %/ oC)∙[(250 - 0 oC) ÷ (100 - 0%)]
= - 1.75 %/%
Example: the moderate value for Kc in our P-Only control of the gravity drained tanks study is Kc = 8 %/ oC For this process, PVmax = 10 m and PVmin = 0 m
Kc = (8 %/ m)∙[(10 - 0 m) ÷ (100 - 0%)]
= 0.8 %/%
Final Thoughts
Textbooks are full of rule-of-thumb guidelines for estimating initial Kc values for a controller depending on whether, for example, it is a flow loop, a temperature loop or a liquid level loop. While we have great reservations with such a "guess and test" approach to tuning, it is important to recognize that such rules are based on a Kc that is expressed in a dimensionless (%/%) form.
In a previous article, we looked at the structure of the P-Only algorithm and we considered some design issues associated with implementation. We also studied the set point tracking (or servo) performance of this simple controller for the heat exchanger process.
Here we investigate the capabilities of the P-Only controller for liquid level control of the gravity drained tanks process. Our objective in this study is disturbance rejection (or regulatory control) performance.
Gravity Drained Tanks Process
A graphic of the gravity drained tanks process is shown below :
The measured process variable (PV) is liquid level in the lower tank. The controller output (CO) adjusts the flow into the upper tank to maintain the PV at set point (SP).
The disturbance (D) is a pumped flow out of the lower tank. It's draw rate is adjusted by a different process and is thus beyond our control. Because it runs through a pump, D is not affected by liquid level, though the pumped flow rate drops to zero if the tank empties.
We begin by summarizing the previously discussed results of steps 1 through 3 of our design and tuning recipe as we proceed with our P-Only control investigation:
Step 1: Determine the Design Level of Operation (DLO)
Our primary objective is to reject disturbances as we control liquid level in the lower tank. As discussed here, our design level of operation (DLO) for this study is:
- design PV and SP = 2.2 m with range of 2.0 to 2.4 m
- design D = 2 L/min with occasional spikes up to 5 L/min
When CO, PV and D are steady near the design level of operation, we bump the CO far enough and fast enough to force a clear dynamic response in the PV that dominates the signal and process noise. As detailed here, we performed two different open loop (manual mode) dynamic tests, a step test and a doublet test.
Step 3: Fit a FOPDT Model to the Dynamic Process Data
The third step of the recipe is to describe the overall dynamic behavior of the process with an approximating first order plus dead time (FOPDT) dynamic model. We define the model parameters and present details of the model fit of step test data here. A model fit of doublet test data using commercial software confirms these values:
- process gain (how far), Kp = 0.09 m/%
- time constant (how fast), Tp = 1.4 min
- dead time (how much delay), Өp = 0.5 min
Following the heat exchanger P-Only study, the P-Only control algorithm computes a CO action every loop sample time T as:
CO = CObias Kc∙e(t)
Where:
CObias = controller bias or null value
Kc = controller gain, a tuning parameter
e(t) = controller error, defined as SP - PV
· Sample Time, T
Best practice is to set the loop sample time, T, at one-tenth the time constant or faster (i.e., T ≤ 0.1Tp). Faster sampling may provide modestly improved performance. Slower sampling can lead to significantly degraded performance.
In this study, T ≤ (0.1)(1.4 min), so T should be 8 seconds or less. We meet this specification with the common vendor sample time option:
▪ sample time, T = 1 sec
· Control Action (Direct/Reverse)
The gravity drained tanks has a positive Kp. That is, when CO increases, PV increases in response. When in automatic mode (closed loop), if the PV is too high, the controller must decrease the CO to correct the error (read more here). Since the controller must move in the direction opposite of the problem, we specify:
▪ controller is reverse acting
· Dead Time Issues
If dead time is greater than the process time constant (Өp > Tp), control becomes increasingly problematic and a Smith predictor can offer benefit. For this process, the dead time is smaller than the time constant, so:
▪ dead time is small and thus not a concern
· Computing Controller Error, e(t)
Set point, SP, is manually entered into a controller. The measured PV comes from the sensor (our wire in). Since SP and PV are known values, then at every loop sample time, T, controller error can be directly computed as:
▪ error, e(t) = SP - PV
· Determining Bias Value, CObias
CObias is the value of CO that, in manual mode, causes the PV to remain steady at the DLO when the major disturbances are quiet and at their normal or expected values. Our doublet plots establish that when CO is at 53%, the PV is steady at the design value of 2.2 m, thus:
▪ controller bias, CObias = 53%
· Controller Gain, Kc
For the simple P-Only controller form, we use the integral of time-weighted absolute error (ITAE) tuning correlation:
Moderate P-Only:
Using our FOPDT model values from step 3, we compute:
And our moderate P-Only controller becomes:
▪ P-Only controller: CO = 53% 8∙e(t)
Implement and Test
To explore how controller gain impacts P-Only performance, we test the controller with the above Kc = 8 %/m. Since the correlation tends to produce moderate performance values, we also explore increasingly aggressive or active P-Only tuning by doubling Kc (2Kc = 16 %/m) and then doubling it again (4Kc = 32 %/m).
The ability of the P-Only controller to reject step changes in the pumped flow disturbance, D, is pictured below (click for a large view) for the ITAE value of Kc and its multiples. Note that the set point remains constant at 2.2 m throughout the study.
As shown in the figure above, whenever the pumped flow disturbance, D, is at the design level of 2 L/min (e.g., when time is less than 30 min) then PV equals SP.
The three times that D is stepped away from the DLO, however, the PV shifts away from the set point. The simple P-Only controller is not able to eliminate this "offset," or sustained error between the PV and SP. This behavior reinforces that both set point and disturbances contribute to defining the design level of operation for a process.
The figure shows that as Kc increases across the plot:
- the activity of the controller output, CO, increases,
- the offset (difference between SP and final PV) decreases, and
- the oscillatory nature of the response increases.
Offset, or the sustained error between SP and PV when the process moves away from the DLO, is a big disadvantage of P-Only control. Yet there are appropriate uses for this simple controller (more discussion here).
While not our design objective, presented below is the set point tracking ability of the controller (click for a large view) when the disturbance flow is held constant:
The figure shows that as Kc increases across the plot, the same performance observations made above apply here: the activity of CO increases, the offset decreases, and the oscillatory nature of the response increases.
Different manufacturers use different forms for the same tuning parameter. The popular alternative to controller gain found in the marketplace is proportional band, PB.
If the CO and PV have units of percent and both can range from 0 to 100%, then the conversion between controller gain and proportional band is:
PB = 100/Kc
Thus, as Kc increases, PB decreases. This reverse thinking can challenge our intuition when switching among manufacturers.
Many examples on this site assign engineering units to the measured PV because plant software has made the task of unit conversions straightforward. If this is true in your plant, take care when using this formula.
Integral Action
Integral action has the benefit of eliminating offset but presents greater design challenges.
The P-Only Control Algorithm
The simplest algorithm in the PID family is a proportional or P-Only controller. Like all automatic controllers, it repeats a measurement-computation-action procedure at every loop sample time, T, following the logic flow shown in the block diagram below (click for large view):
Starting at the far right of the control loop block diagram above:
- A sensor measures and transmits the current value of the process variable, PV, back to the controller (the 'controller wire in')
- Controller error at current time t is computed as set point minus measured process variable, or e(t) = SP - PV
- The controller uses this e(t) in a control algorithm to compute a new controller output signal, CO
- The CO signal is sent to the final control element (e.g. valve, pump, heater, fan) causing it to change (the 'controller wire out')
- The change in the final control element (FCE) causes a change in a manipulated variable
- The change in the manipulated variable (e.g. flow rate of liquid or gas) causes a change in the PV
The goal of the controller is to make e(t) = 0 in spite of unplanned and unmeasured disturbances. Since e(t) = SP - PV, this is the same as saying a controller seeks to make PV = SP.
The P-Only Algorithm
The P-Only controller computes a CO action every loop sample time T as:
CO = CObias Kc∙e(t)
Where:
CObias = controller bias or null value
Kc = controller gain, a tuning parameter
e(t) = controller error = SP - PV
SP = set point
PV = measured process variable
Design Level of Operation
Real processes display a nonlinear behavior, which means their apparent 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 Kp, Tp and Өp values, controllers should be designed and tuned for a pre-defined level of operation.
When designing a cruise control system for a car, for example, would it make sense for us to perform bump tests to generate dynamic data when the car is traveling twice the normal speed limit while going down hill on a windy day? Of course not.
Bump test data should be collected as close as practical to the design PV when the disturbances are quiet and near their typical values. Thus, the design level of operation for a cruise control system is when the car is traveling at highway speed on flat ground on a calm day.
Definition: the design level of operation (DLO) is where we expect the SP and PV will be during normal operation while the important disturbances are quiet and at their expected or typical values.
Understanding Controller Bias
Let's suppose the P-Only control algorithm shown above is used for cruise control in an automobile and CO is the throttle signal adjusting the flow of fuel to the engine.
Let's also suppose that the speed SP is 70 and the measured PV is also 70 (units can be mph or kph depending on where you live in the world). Since PV = SP, then e(t) = 0 and the algorithm reduces to:
CO = CObias Kc∙(0) = CObias
If CObias is zero, then when set point equals measurement, the above equation says that the throttle signal, CO, is also zero. This makes no sense. Clearly if the car is traveling 70 kph, then some baseline flow of fuel is going to the engine.
This baseline value of the CO is called the bias or null value. In this example, CObias is the flow of fuel that, in manual mode, causes the car to travel the design speed of 70 kph when on flat ground on a calm day.
Definition: 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.
A P-Only controller bias (sometimes called null value) is assigned a value as part of the controller design and remains fixed once the controller is put in automatic.
Controller Gain, Kc
The P-Only controller has the advantage of having only one adjustable or tuning parameter, Kc, that defines how active or aggressive the CO will move in response to changes in controller error, e(t).
For a given value of e(t) in the P-Only algorithm above, if Kc is small, then the amount added to CObias is small and the controller response will be slow or sluggish. If Kc is large, then the amount added to CObias is large and the controller response will be fast or aggressive.
Thus, Kc can be adjusted or tuned for each process to make the controller more or less active in its actions when measurement does not equal set point.
P-Only Controller Design
All controllers from the family of PID algorithms (P-Only, PI, PID) should be designed and tuned using our proven recipe:
- Establish the design level of operation (the normal or expected values for set point and major disturbances).
- Bump the process and collect controller output (CO) to process variable (PV) dynamic process data around this design level.
- Approximate the process data behavior with a first order plus dead time (FOPDT) dynamic model.
- Use the model parameters from step 3 in rules and correlations to complete the controller design and tuning.
Moderate P-Only:
This correlation is useful in that it reliably yields a moderate Kc value. In fact, some practitioners find that the ITAE Kc value provides a response performance so predictably modest that they automatically start with an aggressive P-Only tuning, defined here as two and a half times the ITAE value:
Aggressive P-Only: Kc = 2.5 (Moderate Kc)
Reverse Acting, Direct Acting and Control Action
Time constant, Tp, and dead time, Өp, cannot affect the sign of Kc because they mark the passage of time and must always be positive. The above tuning correlation thus implies that Kc must always have the same sign as the process gain, Kp.
When CO increases on a process that has a positive Kp, the PV will increase in response. The process is direct acting. Given this CO to PV relationship, when in automatic mode (closed loop), if the PV starts drifting too high above set point, the controller must decrease CO to correct the error.
This "opposite to the problem" reaction is called negative feedback and forms the basis of stable control.
A process with a positive Kp is direct acting. With negative feedback, the controller must be reverse acting for stable control. Conversely, when Kp is negative (a reverse acting process), the controller must be direct acting for stable control.
Since Kp and Kc always have the same sign for a particular process and stable control requires negative feedback, then:
- direct acting process (Kp and Kc positive) −› use a reverse acting controller
- reverse acting process (Kp and Kc negative) −› use a direct acting controller
If the wrong control action is entered, the controller will quickly drive the final control element (e.g., valve, pump, compressor) to full on/open or full off/closed and remain there until the proper control action entry is made.
Proportional Band
Some manufacturers use different forms for the same tuning parameter. The popular alternative to Kc found in the marketplace is proportional band, PB.
In many industry applications, both the CO and PV are expressed in units of percent. Given that a controller output signal ranges from a minimum (COmin) to maximum (COmax) value, then:
PB = (COmax - COmin)/Kc
When CO and PV have units of percent and both range from 0% to 100%, the much published conversion between controller gain and proportional band results:
PB = 100/Kc
Many case studies on this site assign engineering units to the measured PV because plant software has made the task of unit conversions straightforward. If this is true in your plant, take care when using these conversion formula.
Implementation Issues
Implementation of a P-Only controller is reasonably straightforward, but this simple algorithm exhibits a phenomenon called "offset." In most industrial applications, offset is considered an unacceptable weakness. We explore P-Only control, offset and other issues for the heat exchanger and the gravity drained tanks processes.
he P-Only Control Algorithm
The simplest algorithm in the PID family is a proportional or P-Only controller. Like all automatic controllers, it repeats a measurement-computation-action procedure at every loop sample time, T, following the logic flow shown in the block diagram below (click for large view):
Starting at the far right of the control loop block diagram above:
- A sensor measures and transmits the current value of the process variable, PV, back to the controller (the 'controller wire in')
- Controller error at current time t is computed as set point minus measured process variable, or e(t) = SP - PV
- The controller uses this e(t) in a control algorithm to compute a new controller output signal, CO
- The CO signal is sent to the final control element (e.g. valve, pump, heater, fan) causing it to change (the 'controller wire out')
- The change in the final control element (FCE) causes a change in a manipulated variable
- The change in the manipulated variable (e.g. flow rate of liquid or gas) causes a change in the PV
The goal of the controller is to make e(t) = 0 in spite of unplanned and unmeasured disturbances. Since e(t) = SP - PV, this is the same as saying a controller seeks to make PV = SP.
The P-Only Algorithm
The P-Only controller computes a CO action every loop sample time T as:
CO = CObias Kc∙e(t)
Where:
CObias = controller bias or null value
Kc = controller gain, a tuning parameter
e(t) = controller error = SP - PV
SP = set point
PV = measured process variable
Design Level of Operation
Real processes display a nonlinear behavior, which means their apparent 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 Kp, Tp and Өp values, controllers should be designed and tuned for a pre-defined level of operation.
When designing a cruise control system for a car, for example, would it make sense for us to perform bump tests to generate dynamic data when the car is traveling twice the normal speed limit while going down hill on a windy day? Of course not.
Bump test data should be collected as close as practical to the design PV when the disturbances are quiet and near their typical values. Thus, the design level of operation for a cruise control system is when the car is traveling at highway speed on flat ground on a calm day.
Definition: the design level of operation (DLO) is where we expect the SP and PV will be during normal operation while the important disturbances are quiet and at their expected or typical values.
Understanding Controller Bias
Let's suppose the P-Only control algorithm shown above is used for cruise control in an automobile and CO is the throttle signal adjusting the flow of fuel to the engine.
Let's also suppose that the speed SP is 70 and the measured PV is also 70 (units can be mph or kph depending on where you live in the world). Since PV = SP, then e(t) = 0 and the algorithm reduces to:
CO = CObias Kc∙(0) = CObias
If CObias is zero, then when set point equals measurement, the above equation says that the throttle signal, CO, is also zero. This makes no sense. Clearly if the car is traveling 70 kph, then some baseline flow of fuel is going to the engine.
This baseline value of the CO is called the bias or null value. In this example, CObias is the flow of fuel that, in manual mode, causes the car to travel the design speed of 70 kph when on flat ground on a calm day.
Definition: 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.
A P-Only controller bias (sometimes called null value) is assigned a value as part of the controller design and remains fixed once the controller is put in automatic.
Controller Gain, Kc
The P-Only controller has the advantage of having only one adjustable or tuning parameter, Kc, that defines how active or aggressive the CO will move in response to changes in controller error, e(t).
For a given value of e(t) in the P-Only algorithm above, if Kc is small, then the amount added to CObias is small and the controller response will be slow or sluggish. If Kc is large, then the amount added to CObias is large and the controller response will be fast or aggressive.
Thus, Kc can be adjusted or tuned for each process to make the controller more or less active in its actions when measurement does not equal set point.
P-Only Controller Design
All controllers from the family of PID algorithms (P-Only, PI, PID) should be designed and tuned using our proven recipe:
- Establish the design level of operation (the normal or expected values for set point and major disturbances).
- Bump the process and collect controller output (CO) to process variable (PV) dynamic process data around this design level.
- Approximate the process data behavior with a first order plus dead time (FOPDT) dynamic model.
- Use the model parameters from step 3 in rules and correlations to complete the controller design and tuning.
Moderate P-Only:
This correlation is useful in that it reliably yields a moderate Kc value. In fact, some practitioners find that the ITAE Kc value provides a response performance so predictably modest that they automatically start with an aggressive P-Only tuning, defined here as two and a half times the ITAE value:
Aggressive P-Only: Kc = 2.5 (Moderate Kc)
Reverse Acting, Direct Acting and Control Action
Time constant, Tp, and dead time, Өp, cannot affect the sign of Kc because they mark the passage of time and must always be positive. The above tuning correlation thus implies that Kc must always have the same sign as the process gain, Kp.
When CO increases on a process that has a positive Kp, the PV will increase in response. The process is direct acting. Given this CO to PV relationship, when in automatic mode (closed loop), if the PV starts drifting too high above set point, the controller must decrease CO to correct the error.
This "opposite to the problem" reaction is called negative feedback and forms the basis of stable control.
A process with a positive Kp is direct acting. With negative feedback, the controller must be reverse acting for stable control. Conversely, when Kp is negative (a reverse acting process), the controller must be direct acting for stable control.
Since Kp and Kc always have the same sign for a particular process and stable control requires negative feedback, then:
- direct acting process (Kp and Kc positive) −› use a reverse acting controller
- reverse acting process (Kp and Kc negative) −› use a direct acting controller
If the wrong control action is entered, the controller will quickly drive the final control element (e.g., valve, pump, compressor) to full on/open or full off/closed and remain there until the proper control action entry is made.
Proportional Band
Some manufacturers use different forms for the same tuning parameter. The popular alternative to Kc found in the marketplace is proportional band, PB.
In many industry applications, both the CO and PV are expressed in units of percent. Given that a controller output signal ranges from a minimum (COmin) to maximum (COmax) value, then:
PB = (COmax - COmin)/Kc
When CO and PV have units of percent and both range from 0% to 100%, the much published conversion between controller gain and proportional band results:
PB = 100/Kc
Many case studies on this site assign engineering units to the measured PV because plant software has made the task of unit conversions straightforward. If this is true in your plant, take care when using these conversion formula.
Implementation Issues
Implementation of a P-Only controller is reasonably straightforward, but this simple algorithm exhibits a phenomenon called "offset." In most industrial applications, offset is considered an unacceptable weakness. We explore P-Only control, offset and other issues for the heat exchanger and the gravity drained tanks processes.
A controller seeks to maintain the measured process variable (PV) at set point (SP) in spite of unplanned and unmeasured disturbances. Since e(t) = SP - PV, this is equivalent to saying that a controller seeks to maintain controller error, e(t), equal to zero.
A controller repeats a measurement-computation-action procedure at every loop sample time, T. Starting at the far right of the control loop block diagram above:
- A sensor measures a temperature, pressure, concentration or other property of interest from our process.
- The sensor signal is transmitted to the controller. The pathway from sensor to controller might include: a transducer, an amplifier, a scaling element, quantization, a signal filter, a multiplexer, and other operations that can add delay and change the size, sign, and/or units of the measurement.
- After all electronic and digital operations, the result terminates at our controller as the "wire in" measured process variable (PV) signal.
- This "wire in" process variable is subtracted from set point in the controller to compute error, e(t) = SP - PV, which is then used in an algorithm (examples here and here) to compute a controller output (CO) signal.
- The computed CO signal is transmitted on the "wire out" from the controller on a path to the final control element (FCE).
- Similar to the measurement path, the signal from the controller to FCE might include filtering, scaling, linearization, amplification, multiplexing, transducing and other operations that can add delay and change the size, sign, and/or units of our original CO signal.
- After any electronic and digital operations, the signal reaches the valve, pump, compressor or other FCE, causing a change in the manipulated variable (a liquid or gas stream flow rate, for example).
- The change in the manipulated variable causes a change in our temperature, pressure, concentration or other process property of interest, all with the goal of making e(t) = 0.
The steps of the controller design and tuning recipe include: bumping the CO signal to generate CO to PV dynamic process data, approximating this test data with a first order plus dead time (FOPDT) model, and then using the model parameters in rules and correlations to complete the controller design and tuning.
The recipe provides a proven basis for controller design and tuning that avoids wasteful and expensive trial-and-error experiments. But for success, controller design and tuning must be based on process data as the controller sees it.
The controller only knows about the state of the process from the PV signal arriving on the "wire in" after all operations in the signal path from the sensor. It can only impact the state of the process with the CO signal it sends on the "wire out" before any such operations are made in the path to the final control element.
As indicated in the diagram at the top of this article, the proper signals that describe our complete "process" from the controller's view is the "wire out" CO and the "wire in" PV.
Complete the Circuit
Sometimes we find ourselves unable to proceed with an orderly controller design and tuning. Perhaps our controller interface does not make it convenient to directly record process data. Maybe we find a vendor's documentation to be so poorly written as to be all but worthless. There are a host of complications that can hinder progress.
Being resourceful, we may be tempted to move the project forward by using portable instrumentation. It seems reasonable to collect, say, temperature in a vessel during a bump test by inserting a spare thermocouple into the liquid. Or maybe we feel we can be more precise by standing right at the valve and using a portable signal generator to bump the process rather than doing so from a remote control panel.
As shown below, such an approach cuts out or short circuits the complete control loop pathway. External or portable instrumentation will not be recording the actual CO or PV as the controller sees it, and the data will not be appropriate for controller design or tuning.
Every Item Counts
The illustration above is extreme in that it shows many items that are not included in the control loop. But please recognize that it can be problematic to leave out even a single step in the complete signal pathway.
A simple scaling element that multiplies the signal by a constant value, for example, may seem reasonably unimportant to the overall loop dynamics. But this alone can change the size and even the sign of Kp, thus having dramatic impact on best tuning and final controller performance.
From a controller's view, the complete loop goes from "wire out" to "wire in" as shown below.
Every item in the loop counts. Always use the complete CO to PV data for process control analysis, design and tuning.
Pay Attention to Units
As detailed in this related article, signals can appear in a control loop in electronic units (e.g., volts, mA), in engineering units (e.g. oC, Lb/hr), as percent of scale (e.g., 0% to 100%), or as discrete or digital counts (e.g. 0 to 4095 counts).
It is critical that we remain aware of the units of a signal when working with a particular instrument or device. All values entered and computations performed must be consistent with the form of the data at that point in the loop.
Beyond the theory and methods discussed in this e-book, such "accounting confusion" can be one of the biggest challenges for the process control practitioner.
Processes with streams comprised of gases, liquids, powders, slurries and melts tend to exhibit variations in behavior as operating level changes. This, in fact, is the very nature of a nonlinear process. For this reason, our recipe for controller design and tuning begins by specifying our design level of operation.
Controller Design and Tuning Recipe:
- Establish the design level of operation (DLO), which is the normal or expected values for set point and major disturbances.
- Bump the process and collect controller output (CO) to process variable (PV) dynamic process data around this design level.
- Approximate the process data behavior with a first order plus dead time (FOPDT) dynamic model.
- Use the model parameters from step 3 in rules and correlations to complete the controller design and tuning.
The dynamic behavior of the gravity drained tanks process is reasonably intuitive. Increase or decrease the inlet flow rate into the upper tank and the liquid level in the lower tank rises or falls in response.
One challenge this process presents is that its dynamic behavior is nonlinear. That is, the process gain, Kp; time constant, Tp; and/or dead time, Өp; changes as operating level changes. This is evident in the open loop response plot below.
As shown above, the CO is stepped in equal increments, yet the response behavior of the PV changes as the level in the tank rises. The consequence of nonlinear behavior is that a controller designed to give desirable performance at one operating level may not give desirable performance at another level.
Nonlinear Behavior of the Heat Exchanger
Nonlinear process behavior has important implications for controller design and tuning. Consider, for example, our heat exchanger process under PI control.
When tuned for a moderate response as shown in the first set point step from 140 °C to 155 °C in the plot below, the process variable (PV) responds in a manner consistent with our design goals. That is, the PV moves to the new set point (SP) reasonably quickly but does not overshoot the set point.
The consequence of a nonlinear process character is apparent as the set point steps continue to higher temperatures. In the third set point step from 170 °C to 185 °C, the same controller that had given a desired moderate performance now produces a PV response with a clear overshoot and some oscillation.
Such a change in performance with operating level may be tolerable in some applications and unacceptable in others. As we discuss in this article, "best" performance is something we judge for ourselves based on the goals of production, capabilities of the process, impact on down stream units and the desires of management
Nonlinear behavior should not catch us by surprise. It is something we can know about our process in advance. And this is why we should choose a design level of operation as a first step in our controller design and tuning procedure.
Step 1: Establish the Design Level of Operation (DLO)
Because, as shown in the examples above, processes have process gain, Kp; time constant, Tp; and/or dead time, Өp values that change as operating level changes, and these FOPDT model parameter values are used to complete the controller design and tuning procedure, it is important that dynamic process test data be collected at a pre-determined level of operation.
Defining this design level of operation (DLO) includes specifying where we expect the set point (SP) and measured process variable (PV) to be during normal operation, and the range of values the SP and PV might typically assume. This way we know where to explore the dynamic process behavior during controller design and tuning.
The DLO also considers our major disturbances (D). We should know the normal or typical values for our major disturbances. And we should be reasonably confident that the disturbances are quiet so we may proceed with a bump test to generate and record dynamic process data.
Step 2. Collect Dynamic Process Data Around the DLO
The next step in our recipe is to collect dynamic process data as near as practical to our design level of operation. We do this with a bump test, where we step or pulse the CO and collect data as the PV responds.
It is important to wait until the CO, PV and D have settled out and are as near to constant values as is possible for our particular operation before we start a bump test. The point of bumping a process is to learn about the cause and effect relationship between the CO and PV.
With the process at steady state, we are starting with a clean slate. As the PV responds to the CO bumps, the dynamic cause and effect behavior is isolated and evident in the data. On a practical note, be sure the data capture routine is enabled before the initial bump is implemented so all relevant data is collected.
Two popular open loop (manual mode) methods are the step test and the doublet test.
For either method, the CO must be moved far enough and fast enough to force a response in the PV that dominates the measurement noise.
Also, our bump should move the PV both above and below the DLO during testing. With data from each side of the DLO, the model (step 3) will be able to average out the nonlinear effects as discussed above.
- Step Test
- Doublet Test
Step 3: Fit a FOPDT dynamic model to Process Data
In fitting a first order plus dead time (FOPDT) model, we approximate those essential features of the dynamic process behavior that are fundamental to control. We need not understand differential equations to appreciate the articles on on this site, but for completeness, the first order plus dead time (FOPDT) dynamic model has the form:
Where:
PV(t) = measured process variable as a function of time
CO(t - Өp) = controller output signal as a function of time and shifted by Өp
Өp = process dead time
t = time
When the FOPDT dynamic model is fit to process data, the results describe how PV will respond to a change in CO via the model parameters. In particular:
- Process gain, Kp, describes the direction and how far PV will travel,
- Time constant, Tp, states how fast PV moves after it begins its response,
- Dead time, Өp, is the delay from when CO changes until when PV begins to respond.
Step 4: Use the model parameters to complete the design and tuning
In step 4, the three FOPDT model parameters are used in correlations to compute controller tuning values. For example, the chart below lists internal model control (IMC) tuning correlations for the PI controller and dependent ideal PID controller, and dependent ideal PID with CO filter forms:
The closed loop time constant, Tc, in the IMC correlations is used to specify the desired speed or quickness of our controller in responding to a set point change or rejecting a disturbance. The closed loop time constant is computed:
- aggressive performance: Tc is the larger of 0.1·Tp or 0.8·Өp
- moderate performance: Tc is the larger of 1·Tp or 8·Өp
- conservative performance: Tc is the larger of 10·Tp or 80·Өp
The FOPDT dynamic model of step 3 also provides us the information we need to decide other controller design issues, including:
- Controller Action
- Loop Sample Time, T
- Dead Time Problems
- Model Based Control
