Model-Based Tuning Methods for PID Controllers
The manner in which a measured process variable responds over time to changes in the controller output signal is fundamental to the design and tuning of a PID controller. The best way to learn about the dynamic behavior of a process is to perform experiments, commonly referred to as “bump tests.” Critical to success is that the process data generated by the bump test be descriptive of actual process behavior. Discussed are the qualities required for “good” dynamic data and methods for modeling the dynamic data for controller design. Parameters from the dynamic model are not only used in correlations to compute tuning values, but also provide insight into controller design parameters such as loop sample time and whether dead time presents a performance challenge. It is becoming increasingly common for dynamic studies to be performed with the controller in automatic (closed loop). For closed loop studies, the dynamic data is generated by bumping the set point. The method for using closed loop data is illustrated. Concepts in this work are illustrated using a level control simulation.
FORM OF THE CONTROLLER
The methods discussed here apply to the complete family of PID algorithms. Examples presented will explore the most popular controller of the PID family, the Proportional-Integral (PI) controller:
In this controller, u(t) is the controller output and
is the controller bias. The tuning parameters are controller gain, ,
and reset time, .
Because is in
the denominator, smaller values of reset time provide a larger weight to
(increase the influence of) the integral term.
CONTROLLER DESIGN PROCEDURE
Designing any controller from the family of PID algorithms entails the following steps:
The form of the FOPDT dynamic model is:
where y(t) is the measured process variable and u(t) is the controller output signal. When Eq. 2 is fit to the test data, the all-important parameters that describe the dynamic behavior of the process result:
These three model parameters are important because they are used in correlations to compute initial tuning values for a variety of controllers . The model parameters are also important because:
DEFINING GOOD PROCESS TEST DATA
As discussed above, the collection and analysis of dynamic
process data are critical steps in controller
Suppose a controller output change forces a dynamic response in
a process, but the data file only shows
When generating dynamic process data, it is important that the
change in controller output cause a
It is essential that the test data contain process variable
dynamics that have been clearly (and in the ideal
It is important that the modeling tool display a plot that shows
the model fit on top of the data. If the two
NOISE BAND AND SIGNAL TO NOISE RATIO
When generating dynamic process data, it is important that the
change in the controller output signal
When generating dynamic process data, the change in controller output should cause the measured process variable to move at least ten times the size of the noise band. Expressed concisely, the signal to noise ratio should be greater than ten. In Fig. 1, the noise band is 0.25°C. Hence, the controller output should be moved far and fast enough during a test to cause the measured exit temperature to move at least 2.5°C. This is a minimum specification. In practice it is conservative to exceed this value.
Noise Band of Heat Exchanger PV
Process: Heat Exchanger Controller: Manual Mode
Figure 1 - Noise Band Encompasses ± 3 Standard Deviations Of The Measurement Noise
CONTROLLER TUNING FROM CORRELATIONS
The recommended tuning correlations for controllers from the PID
family are the Internal Model Control
The first step in using the IMC (lambda) tuning correlations is
to compute, , the
closed loop time
C τ is the larger of 0.1 P τ or 0.8 P θ
Moderate Tuning:C τ is the larger of 1.0 P τ or 8.0 P θ
Conservative Tuning:C τ is the larger of 10 P τ or 80.0 P θ
With ôC computed, the PI correlations for IMC tuning are:
Final tuning is verified on-line and may require tweaking. If the process is responding sluggishly to disturbances and changes in the set point, the controller gain is too small and/or the reset time is too large. Conversely, if the process is responding quickly and is oscillating to a degree that makes you uncomfortable, the controller gain is too large and/or the reset time is too small.
EXAMPLE: SET POINT TRACKING IN GRAVITY DRAINED TANKS
The gravity drained tanks process, shown in Fig. 2, is two
non-interacting tanks stacked one above the
Figure 2 - Gravity Drained Tanks Process
The design level of operation for this study is a measured level in the lower tank of 2.4 m while the pumped flow disturbance is at its expected value of 2.0 L/min. The control objective is to track set point steps in the range of 2.0 to 2.8 m. The process is currently under P-Only control and operations personnel will not open the loop for controller design experiments. Hence, closed loop set point steps are used to generate dynamic process data.
As shown in Fig. 3, the P-Only controller being used has a = 40 %/m and a bias value of 55.2% (determined as the value of the controller output that, in open loop, causes the measured level in the lower tank to steady at the design value of 2.4 m when the pumped flow disturbance is at its expected value of 2.0 L/min). With data being saved to file, the dynamic testing experiment begins. Specifically, the set point is stepped up to 2.8 m, then down to 2.0 m, and finally back to the design level of 2.4 m (set point sequences of other sizes and durations would be equally reasonable).
P-Only Set Point Step Test
Process: Gravity Drained Tank Controller: PID ( P= RA, I= off, D= off )
Figure 3 – Set point step tests on gravity drained tanks under P-Only control
Visual inspection of Fig. 3 confirms that the closed loop dynamic event is set point driven (as opposed to disturbance driven). Also, control action appears energetic enough such that the response of the measured process variable clearly dominates the noise.
FOPDT of Closed Loop Data
Model: First Order Plus Dead Time (FOPDT) File Name: closed.
Figure 4 – FOPDT fit of closed loop dynamic data generated in Fig.8.5
The dynamic data of Fig. 3 is fit with a FOPDT model using Loop-Pro software by Control Station. A plot of the model and closed loop process data is shown in Fig. 4. The model appears to be reasonable and appropriate based on visual inspection, thus providing the design parameters:
KP = 0.094 m/%
Time Constant,τP = 1.6 min
Dead Time,θP = 0.56 min
We first compute the closed loop time constant. Here we choose aggressive tuning, which is computed as:
τC = larger of 0.1ôP or 0.8èP = larger of 0.1(1.6) or 0.8(0.56) = 0.45 min.
Substituting this closed loop time constant and the above FOPDT model parameters into the IMC tuning correlations of Eq. 3 yields the following tuning values:
A reverse acting controller is required because KC is positive. Because the PI controller has integral action, the bias value is not entered but is automatically initialized by our instrumentation to the current value of the controller output at the moment the loop is closed.
Controller Design From P-Only Set Point Data
Figure 5 – Performance of PI controller in tracking set point steps
The performance of this controller in tracking set point changes is pictured in Fig. 5. Although good or best performance is decided based on the capabilities of the process, the goals of production, the impact on downstream units and the desires of management, Fig. 5 exhibits generally desirable performance. That is, the process responds quickly, shows modest overshoot, settles quickly, and displays no offset. Compare this to Fig. 3, that shows P-Only performance for the same control challenge.
INTERACTION OF PI TUNING PARAMETERS
One challenge of the PI controller is that there are two tuning
parameters to adjust and difficulties can arise
The center of Fig. 6 shows a set point step response that is
labeled as the base case performance. It is
PI Controller Tuning Map
Figure 6 - How PI controller tuning parameters impact set point tracking performance
The plot in the upper left of the grid shows that when gain is doubled and reset time is halved, the controller produces large, slowly damping oscillations. Conversely, the plot in the lower right of the grid shows that when controller gain is halved and reset time is doubled, the response becomes sluggish. This chart is called a tuning map because, in general, if a controller is behaving poorly, you can match the performance you observe with the closest picture in Fig. 6 and obtain guidance as to the appropriate tuning adjustments required to move toward your desired performance.
Understanding the dynamic behavior of a process is essential to
the proper design and tuning of a PID
1. Cooper, Douglas, “Practical Process Control Using Control
Station,” Published by Control Station,
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