- Email: [email protected]
Simplified IMC-PID tuning rules
ISA TRANSACTIONS* ELSEVIER
ISATransactions 33 (1994) 43-59
Simplified IMC-PID tuning rules Paul S. F r u e h a u f
*'a,
I-Lung Chien b, Mark D. Lauritsen c
" Applied Control Engineering, P.O. Box 520, Hockessin, DE 19707, USA Department of Chemical Engineering, Chang Gung College of Medicine and Technology, Kwetshan, Taoyuan, Taiwan, ROC c DuPont Engineering, 140 Cypress Station Drit:e, Houston, TX 77090, USA
Abstract We have significantly simplified the IMC-PID tuning rules. These new rules cover the vast majority of control loops encountered in the chemical industry. This work is the result of a great deal of experience in successfully applying IMC-PID tuning rules and an effort to prepare a training course on controller tuning. The simplified rules are very similar in form to the classic open loop Ziegler-Nichols rules and use the process reaction curve method for process testing. The two differences are that these rules are based on a less aggressive performance criteria and that we adapt the rules for some commonly encountered special cases. This paper presents the relationship between the simplified IMC-PID rules, the general IMC-PID rules, the Ziegler-Nichols rules and the Cohen-Coon rules. We show that the simplified rules are less sensitive to parameter mis-estimation than other more aggressive tuning rules. We also proposed rules for a fourth action; filtering. Filtering is available in digital controllers and smart field transmitters. We report that filtering and derivative action cancel each other and therefore should not be used together. We briefly outline the contents of the tuning course and finish the paper with an industrial example where the simplified rules have been successfully applied.
Key words: Controller tuning; Control algorithm
1. Introduction P I D c o n t r o l l e r s a r e still, by far, t h e most comm o n l y u s e d c o n t r o l l e r s in the c h e m i c a l industry. P r o b a b l y t h e single b i g g e s t r e a s o n for this is that a w e l l - d e s i g n e d a n d a d e q u a t e l y - t u n e d P I D cont r o l l e r m e e t s o r e x c e e d s most c o n t r o l objectives. W e d e s i g n c o n t r o l s t r a t e g i e s for n e w plants, m a k e c o n t r o l i m p r o v e m e n t s at existing p l a n t s a n d troubleshoot control problems. These experiences have shown us t h a t t h e r e is very l i m i t e d knowl-
* Corresponding author.
e d g e a b o u t c o n t r o l l e r t u n i n g at m a n y p l a n t sites. W e have i n v e s t i g a t e d c o n t r o l p r o b l e m s for p r o cesses t h a t r e c e n t l y u n d e r w e n t c o n v e r s i o n s to D i s t r i b u t e d C o n t r o l Systems ( D C S ) a n d f o u n d t h a t a m a j o r i t y o f t h e c o n t r o l Mops w e r e b e i n g run in m a n u a l simply b e c a u s e the loops w e r e poorly tuned. L a t e in 1992 we w e r e a s k e d to d e v e l o p a c o n t r o l l e r t u n i n g c o u r s e to h e l p a d d r e s s this p r o b l e m . T h e t a r g e t a u d i e n c e for t h e c o u r s e is i n s t r u m e n t technicians, o p e r a t o r s a n d e n g i n e e r s . O v e r the p a s t five y e a r s we have e m p l o y e d I n t e r nal M o d e l C o n t r o l - P r o p o r t i o n a l I n t e g r a l D e r i v a t i v e ( I M C - P I D ) c o n t r o l l e r t u n i n g rules with
0019-0578/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 0 1 9 - 0 5 7 8 ( 9 4 ) 0 0 0 0 4 - 6
44
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
great success to many control loops. Early in the course development we realized the general I M C - P I D method was complicated and would add significantly to the length of a training course. Our experience with applying these rules showed that they could be simplified and still be useful for the vast majority of control loops. The majority of the controllers we tune are PI only. However, on occasion, we find benefit in using derivative action or filter action. We therefore propose rules for all four actions. The main contributions of the simplified IMCPID tuning rules to the science of controller tuning are: the rules are based on rigorous IMCPID tuning and retain all the benefits of such, the equations for calculating the tuning constants are simple, the rules have been extensively tested on real processes, for most situations only two process dynamic parameters need to be estimated with only moderate accuracy and the rules work well for nonlinear processes because they emphasize the initial dynamic response. This p a p e r will show that the simplified IMCPID tuning rules are more robust than the rules of Ziegler and Nichols [1] and Cohen and Coon [2]. These older rules are less robust because they are based on a more aggressive tuning criteria (i.e., quarter wave decay). We, and others before us, have found this tuning criteria to be a bit too aggressive for most chemical industry applications. However, the contributions of these earlier works are major, please do not misinterpret the robustness analysis in this p a p e r to be a criticism of these works. On the contrary, the fact that the rigorous I M C - P I D tuning rules reduce to the same form as the original Ziegler-Nichols open loop rules further confirms just what a stroke of empirical genius this work was. Our interest in the work of Cohen and Coon came about because their tuning rules correct for one deficiency of the Ziegler-Nichols rules (i.e., sluggish closed loop response in the rare occasion when the process dead time is large relative to dominant open loop time constant). Our simplified IMCPID rules also correct for this deficiency. The first section of the paper reports the simplified tuning rules. The second section compares the robustness of the simplified tuning rules to
other rules. A brief description of the tuning method that we teach and how we apply the rules is included in the third section and we finish the p a p e r with an actual industrial example of the application of our rules. Appendix A is attached to document the derivation of the simplified rules.
2. Simplified IMC-PID tuning rules The PID tuning rules based on the Internal Model Control (IMC) design method were first developed in 1986 by Rivera and co-workers [3]. Later extension to cover a wider range of process models and an extension suitable for time constant dominant processes were presented by Chien [4] and Chien and Fruehauf [5]. The IMCPID tuning method is very attractive to industrial users because it has only one tuning parameter: the closed loop speed of response. This parameter relates directly to the closed loop time constant for the response to setpoint changes and to the robustness of the control loop. A smaller closed loop speed of response decreases the robustness. Moreover the closed loop step load response exhibits no oscillation or overshoot. Experience indicates that this minimizes controller interactions and enhances overall process disturbance rejection. When asked to prepare the controller tuning course we began to investigate ways to simplify the I M C - P I D tuning rules. Our objective was to help the student tune the majority of control loops that they would encounter. To simplify the I M C - P I D tuning rules we used the following facts, assumptions and conclusions based on our experience: - The majority of processes are well approximated by dead time first order or dead time integrator models. - Tuning parameters do not need to be set to a high degree of accuracy to achieve satisfactory closed loop response. Accuracy to within 20% is adequate. - We have developed rules for setting the closed loop speed of response (i.e, Tcl) so it can be fixed. This eliminates the only tuning parameter that is required with the I M C - P I D rules.
P.S. Fruehauf et al./ ISA Transactions 33 (1994) 43-59
- ~
SloDe, R - -
/ f/
Definition
/
PI tuning rules The first and most commonly used rules are
""
/ Controlled Verlable
/f
45
1
May or May Not Level Out
Kc
(1)
2RL '
(2)
~'i = 5L,
I I:
I : ]
L, Apparent Dead Time
I 2" •,
,,,
,
Time Fig. 1. The open loop step response of a typical process.
- The most popular type of PID controller is the interacting form and the tuning settings are gain (not proportional band) and reset in minu t e s / repeat (not repeats/minutes). If one is working with tuning settings in alternative units, conversion is simple. - Open loop testing is used to determine the dynamic parameters for a process. The derivation of the simplified rules from the rigorous rules using these principles is included in Appendix A. Interestingly, the resulting rules reduce to a form that is very similar to the original Ziegler-Nichols [1] open loop tuning rules. Our rules are less aggressive and we modify the rules for some commonly encountered special cases. Before we present the tuning rules, we will briefly present the open loop procedure for process testing. With the controller in manual and the process as steady as possible, a step change in controller output is made. The process response will look something like Fig. 1. If the process levels out, or would if enough time were allowed, it is called dead time first order. If it would not, it is called dead time integrating. The tuning rules generally apply to both types of processes, though modifications to the rules are needed if the process is dead time first order and the time constant is short relative to the apparent dead time or when the apparent dead time is small (i.e., L < 0.5 minutes). As a result, we have developed three sets of rules.
where Kc is the controller proportional gain (dimensionless), ~i is the controller reset time (minutes/repeat), R is the change per minute in the process variable, (expressed as a % of the transmitter span), divided by the step change magnitude (expressed as a % of the controller output span) (1/minutes), L is the apparent dead time (minutes) These rules are derived from the IMC-PID rules for a dead time integrating process. However, as shown in [5], these rules apply to both dead time integrating and dead time first order processes. There are two types of less commonly observed responses which require modifications to the above rules. The first is when the process is dead time first order and the time constant is short relative to the apparent dead time. Fig. 2 illustrates this type of response. We have found, by performing many simulations, that the following rules give better closedloop response when the time constant divided by
Slope, R -,~,/ Final Response
63% of Final Response
_Z_~
It: :I
L, Apparent DeadTime
I Time Fig. 2. Open loop step response of a self regulating process.
46
P.S. Fruehauf et a l . / ISA Transactions 33 (1994) 43-59
the apparent dead time is 3 or less. The modified tuning rules are 1 Kc = 2 R L '
(3)
Ti =
(4)
become more conservative, thus enhancing the stability of the closed loop system. Please note that this last rule does not apply to flow loops. In summary, the simplified PI tuning rules are "t
'T,
where r is the process time constant (minutes). Notice that the only difference between this modification and the Eqs. (1) and (2) rules is that the reset time is, at times, significantly shorter which speeds up the closed-loop response. This rule is derived from the rigorous IMC-PID rules for dead time first order processes. Often when doing open loop testing it is difficult to observe the open loop time constant because of noise and other process variables that are changing that effect the dynamic response (i.e., the process never steadies out). If you cannot observe the time constant just use the first of set rules. On the occasion when you can observe the first order time constant and the ratio is small you can take advantage of the performance improvement that these modifications to the rules provides. The other special situation which requires tuning rule modification is when the apparent dead time is very small. In this case, the tuning rules in Eqs. (1) and (2) may result in excessive control action thus causing closed loop stability problems when even modest model mismatch is present. In addition, Eq. (1) can result in a very high controller gain that will produce large changes in controller output for small changes in setpoint. If the controller output is tied to a valve that adjusts the feed rate to a downstream unit operation this can cause large flow disturbances to the downstream unit. Also, when the apparent dead time is small the process is easy to control, so very tight tuning is not generally necessary. In this small dead time case, typically when L < 0.5 minutes, the tuning rules are modified to 1
Kc = ~,
(5)
r i = 4.
(6)
Notice that K c does not have L in the denominator and thus is smaller and '/'i is larger compared to Eqs. (1) and (2). The controller settings
"g
-->3 L 1
Kc ri
2RL 5L
--<3 L 1
2RL r
L<0.5 1
R 4
The above tuning rules work very well for the vast majority of the control loops encountered in the chemical industry. For specific control loops like flow and level we provide other recommendations later in this paper. For rare occasions when the open loop step response exhibits overshoot, oscillatory, or inverse behavior, we recommend modeling the process dynamics as a more complicated second order response and refer to Table 1 in [5] for controller settings. 2.1. F i l t e r a n d d e r i v a t i v e t u n i n g rules
Before we discuss the filter and derivative tuning rules it will be helpful to discuss filter action, derivative action and the relationship between the two. Filter action
We use signal filtering in some loops to improve performance. One common example is in level loops where the measurement is often noisy. Without filter action the noise is passed by the proportional action of the controller directly to the control valve causing it to move unnecessarily. Reducing this movement by decreasing the controller gain is undesirable because of the destabilizing effect this has on the level control loop. Filter action is a superior solution. We are not aware of any rules on how to set filter time constant. We believe that filtering is now used more often because it is available in many digital controllers and smart field instruments. It has become a fourth controller action. The task of setting the filter time is one of using as much as you dare without degrading the performance of the loop. The goal is to specify a
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
filter time constant that provides the benefits of filtering, without adversely impacting closed loop response. We used time domain and frequency analysis to develop a general rule. A dead time first order process and a dead time integrating process were simulated, with the process feedback signal being filtered in each case. The tuning rules were used to provide the PI controller settings. Fig. 3 illustrates one of the plots used for this analysis. It shows the time domain responses for a dead time
47
first order process subjected to a step change in setpoint. A filter time constant less than or equal to 50% of the apparent dead time was selected as the general rule because it provides the desired filtering action without severely impacting loop performance. Derivative action
Derivative action can improve the performance of a control loop by shortening the closed loop natural period. The performance improve-
Response to Step Change in Setpoint Deadtime First-Order Lag Process
1.2
I
/
Tf
.T£
:
2*L
-
M
e a s u r e d
4*L
ffi
L
0.8
0.6
f
-
0.5*L
f
= 0.2*L
0.4
0.2
t
I
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
90
100
~me Fig. 3. S e t p o i n t r e s p o n s e s ~ r d i f f e r e n t v a l u e s o f f i l t e r t i m e c o n s t a n t .
48
P.S. Fruehauf et al./1SA Transactions 33 (1994) 43-59
ment is not dramatic; however, in some situations it can help enough to warrant using it. We have used the rigorous I M C - P I D tuning rules to derive rules for setting derivative time which say to make it equal to one half the apparent dead time. In order to limit the number of rules, we elected not to incorporate the slight gain increase and integral decrease given by the rigorous rules. These changes are small relative to the required accuracy for setting these parameters.
Relationship between filter and derivative action Filter and derivative action are essentially the mathematical inverse of each other. They are in series in the feedback circuit so they cancel each other. Therefore, we propose rules that specify one or the other but not both. Derivative action causes the controller output to bounce more than when it is not used, but improves control performance. Filter action dampens this bouncing but degrades performance. Often the output of one loop is a disturbance to another loop. Derivative action could improve one loop's performance but could hurt another. Filter action does the opposite. The decision to use filter or derivative action becomes a compromise between performance of an individual loop versus the system as a whole. Often the best compromise is when neither is used. This is one reason why we feel the PI only controller is so popular. In summary, the rules for setting filter or derivative time are Tf : 0.5L or "/'d= 0.5L, where % is the filter time constant (minutes), ~'d is the derivative time (minutes).
robustness of the simplified I M C - P I D rules to other tuning rules in two different ways. The first section uses time response plots to show the effect of model mismatch for PI tuning rules. There is a model mismatch when the process model parameters change from what they originally were. This mimics what can happen when the operating conditions of a chemical process are changed, because almost all processes are non-linear to some degree. When we tune we approximate the dynamics at a particular operating condition with a linear model. The second section compares PID tuning rules using stability plots. This comparison is also different from the first, in that we are looking at mismatch between what the actual process dynamics are and what we estimate them to be. This measure of robustness tells us how accurately we must estimate the process dynamics when doing an open loop step test to prevent unstable closed loop response.
PI tuning rule sensitivity to model mismatch In this section, the simplified I M C - P I D tuning rules are demonstrated using three numerical examples with different process dynamic characteristics. These tuning rules will be compared to the popular tuning methods of Ziegler and Nichols [1] and Cohen and Coon [2]. Fig. 4 shows the feedback control structure for all the simulations in this section. y is the controlled variable, u is the controller output, Ys is the setpoint, d is the disturbance, G c is the feedback controller (PI), Gp is the process transfer function, G L is the disturbance transfer function.
3. Robustness
Robustness is the term used to describe the sensitivity of closed loop response to changes. A set of tuning rules are more robust if they can tolerate larger changes in the dynamics parameters (e.g., L and R) and still maintain adequate response. The next two sections compare the
YS
~
~
~_*
Y
Fig. 4. Feedback control structure used for simulations illustrated in Figs. 5-10.
P.S. Fruehauf et al./ ISA Transactions 33 (1994) 43-59
Example 1. Time-constant-dominant process:
sponse for both controlled variable and controller output. On the contrary, both Ziegler-Nichols and C o h e n - C o o n methods result in overshoot and an oscillatory closed-loop response. In order to test the robustness of the proposed tuning rules, + 50% model mismatches in process gain (related to R) and dead time were simulated. Fig. 6 illustrates the effect of the process gain mismatch. The figure clearly shows the superior closedloop behavior of the simplified IMC-PID settings under severe model mismatch conditions.
10e-2S
Gp=G/= 5 0 0 s + l " From the step response, R, L and T are calculated to be R = 0 . 0 2 , L = 2 and T = 5 0 0 . The resulting PI controller settings are
Kc ri
Simplified IMC-PID
Ziegler-Nichols
Cohen-Coon
12.5 10
22.5 6.7
22.5 6.6
Fig. 5 shows the disturbance closed-loop response for the simplified IMC-PID, ZieglerNichols and C o h e n - C o o n rules. In the simulation, d is changed from 0 to 1 at time equal to zero. Notice that the proposed tuning rules give very smooth closed loop re-
0.08
49
Example 2. Process with short time constant and significant dead time: e-4S Gp=Gt-
2s+l"
IMC-PID
0.06 0.04,
/ /
0.02
Both Z-N and C-C
\\\ , ~"
0 -0.02
.
.
.
.
!
.
.
.
.
10
!
•
•
•
•
I
.
30
20 Time
.
.
.
I
.
.
4,0
.
.
I
50
(minutes)
0
•I .I ""0":]
~
_.. IMC-PID
\ 'X
",
-is
/f
"
0
f
// /
X
-20
"-
Bolh Z-N
"
"
I
i0
"
I
"
"
"
•
20 Time
I
30 (minutes)
Fig. 5. Load response for ExampLe 1.
"
|
40
•
I
5O
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
50
IMC-PID
o--1 /
Both Z-N and C-C
',,
/ ', ",,l
-0.05
-0.I
•
|
ii
i
|
•
•
•
'
I
"
'
"
IMC-PID,..,
/"
/
I
/*%, / /
',\
\
"
"
\£ "
\ \ \ \
I
'
/ 1 I II
\
'
"
"
I
4,0
50
(minutes)
Both Z-N and C-C.
f
// %
/,~, I
0 =
-10
/
,
"
30
Time
i0--
'
20
10
0
/
/ / /7
/
\
1
~'
=
-20"
-30
-
'\\/
\\ /I' I
.
.
.
~
\\
.
I
10
.
.
.
.
I
20
T1hae
I
\\
~
I
"
t
\ "
30
"
I
I
.
.
.
.
40
I 50
(minutes)
Fig. 6. Load response for Example 1 for a + 50% gain mismatch.
For this example, the controller settings are
K, ~'i
Simplified IMC-PID
Ziegler-Nichols
Cohen-Coon
0.25 2
0.45 13.3
0.45 2.9
Fig. 7 shows the disturbance rejection closedloop responses using the three tuning methods. Notice again that the proposed tuning rules give very smooth closed loop response for both controlled variable and controller output, while Ziegler-Nichols settings give very sluggish response. C o h e n - C o o n settings provides improvement over Ziegler-Nichols settings in this situation but are still a little oscillatory. Fig. 8 shows the closed loop responses for a + 5 0 % model mismatch in dead time. The simplified tuning rules still provide smooth closed-loop response while Ziegler-Nichols settings are still very slug-
gish and C o h e n - C o o n settings are more oscillatory. Example 3. Process with small dead time: 10e-0.16s Gp = G /
15s + 1
In this example, the process dead time is very small (i.e., about 10 seconds). The controller settings for the three tuning methods are
K,, ri
Simplified IMC-PID
Ziegler-Nichols
Cohen-Coon
1.5 4
8.43 0.53
8.44 0.52
Fig. 9 shows the closed-loop responses for the three tuning methods. Notice that Ziegler-Nichols and C o h e n - C o o n settings result in an almost identical response.
P.S. Fruehauf et a l . / ISA Transactions 33 (1994) 43-59
the closed loop response becomes unstable. We estimate these parameters as part of the open loop tuning procedure we use. A set of tuning rules are more robust if larger errors in the estimates can be tolerated. These plots were developed because we wanted a general way to assess the robustness of the simplified IMC-PID rules before we included them in our tuning course. The time response plots in the previous section only test the rules for specific values of mismatch. The plots in this section illustrate the robustness for all values of mismatch of interest. Tuning rule robustness plots have been prepared for the three example processes presented in the previous section. Fig. 11 is the plot for the example one process. The X-axis is the ratio of the estimated dead time (Lo) to the actual dead time (La). The Y-axis is the ratio of the estimated
Although their responses are much faster than the proposed tuning method, the controller output is rather oscillatory. Fig. 10 shows the response for a + 5 0 % model mismatch in dead time. We should mention that this kind of dead time mismatch (i.e., process having dead time of 15 seconds versus model having dead time of 10 seconds), is reasonable and can be expected to happen from time to time. With this small model mismatch the Ziegler-Nichols and C o h e n - C o o n settings become unstable, while the simplified IMC-PID rules still provides smooth and stable closed-loop response.
Tuning rule stability plots In this section, tuning rule stability plots are used to compare how much we can misestimate the process dead time and process slope before
f
51
IMC-PID ~ Z-N
0.5"
0
°::1 .
.
.
.
I
0
"
"
"
10
'
I
.
.
.
.
I
'
•
30
20
Time
I
"
•
I
50
40
(minutes)
O r
-0.5
-1
-1.5
-2
•
0
I
10
.
.
.
.
I
"
"
"
20
Time
I
30
(minutes)
Fig. 7. L o a d r e s p o n s e for E x a m p l e 2.
"
•
I
4o
•
I
50
52
P.S. Fruehauf et aL / ISA Transactions 33 (1994) 43-59
process slope (Ro) to the actual process slope (Ra). The curves are boundaries between the regions of stable and unstable estimates. The stable regions are above and to the right of the curves and the unstable regions are below and to the left of the curves. The curves are defined by the equations for the Bode stability criteria (i.e., magnitude ratio of unity at the resonant frequency) using the different tuning rules. These plots can be used to compare the robustness of tuning rules in the following way. Assume that we have just done an open loop step test on a process and the response looks something like Fig. 1. The process dead time is misestimated, due to signal noise, to be 1.5 minutes. Assume that the actual dead time is 2 minutes. This defines a line perpendicular to the X-axis at 0.75. This line intersects the I M C - P I D rule curve
at 0.4 and both the Ziegler-Nichols and C o h e n Coon rule curves at 1.25. If the true process slope is 1, the graph tells us that if we mis-estimate the process slope to be less than 0.4 the I M C - P I D rules will result in unstable tuning, whereas the other rules will only allow a mis-estimation of less than 1.25 before unstable tuning results. For the Ziegler-Nichols and C o h e n - C o o n rules if we estimate the process slope to be exactly what it is we will calculate tuning settings that produce unstable response! We conclude that the I M C - P I D rules are more robust because they allow for a larger mis-estimation of the slope than the other rules. More generally, the farther the stability boundary line is from the point (1, 1) (i.e., the point which corresponds to exact estimates of dead time and slope) the more robust the tuning rules are. Fig. 12 is the robustness plot for the Example
-_
.
.
f
.
IM@PID '
'
I
0
"
"
"
10
'
I
.
.
.
.
I
20
Time
0 ~
-0.5
r.
/
"
. "
"
"
30 { m i n u t e
I
"
'
'
40
"
.
50
B }
Z-N
=
.1
-1.5
=
•
•
"
I
10
.
.
.
.
I
"
"
20 T i m e
"
I
.
.
.
.
30 ( m i n u t
e B )
Fig. 8. Load response for Example 2 with a + 50% dead time mismatch.
I
40
"
"
5O
P.S. Fruehauf et a l . / ISA Transactions 33 (1994) 43-59
2 process where the process time constant is short relative to the dead time. The Ziegler-Nichols and C o h e n - C o o n rules are more robust for this case but are still significantly less robust than the simplified IMC-PID rules. For brevity the Example 3 process plot is not shown. The simplified IMC-PID rules are intentionally conservative for this type of process and, therefore, are much more robust than the other rules.
For general loops, we advocate that open loop testing be used. This is essentially the reaction curve testing method proposed by Ziegler and Nichols. We teach general loop tuning as a three step process. The steps are (1) to learn to run the process with controller in manual and estimate L and R, (2) to perform open loop step tests to accurately determine L, R and r (if it is possible to observe r), and to calculate tuning and (3) to test new tuning with the loop in automatic by making setpoint changes. The first step is critical and far from trivial; however, most tuning discussions we have seen do not even address this step. This method is used for general control loops, e.g. temperature, pressure, composition loops, etc. For special classes of control loops like flow and level, we provide the following controller tuning recommendations.
4. Tuning method taught in training course The tuning method we teach breaks loops up into three general categories: general loops, flow loops and level loops where averaging level control is desired.
f 0.05
m
0.03
"
53
IMC-PID
0.01
-0.01
I
.
.
.
.
I
10
.
.
.
.
20
Time
I
30
(minutes)
O~
-0.5"
j[~V
I
B o t h Z-N and C-C
-1-
-1.5
-,2
! IMC_PID? I
"
"
"
10
I
20
Time
(minut
era)
Fig. 9. Load response for Example 3.
"
"
I
30
54
P.S. Fruehauf et aL / ISA Transactions 33 (1994) 43-59
For flow loops we teach that the dynamics are generally the same for all loops and therefore so is the tuning. We recommend that the gain be set between 0.5 to 0.7 and that the reset be set between 0.2 and 0.3 minutes. Fine tuning can then be done by a closed loop trial and error method. Level loops are generally tuned so the tank will absorb flow disturbances so that they are not passed on to downstream unit operations. When this is the case, we recommend that the following equations be used: 100% × 2 Kc ALR ' 4×V zi
Kc Q '
where A L R is the allowable level range (%), V is
f
IMC-PID
the measured tank volume (gallons) and Q is the maximum flow through level control valve (gallons/minute). The allowable level range is the range of level that can be allowed to vary without any adverse effects. For a surge tank, the range is often selected to be 80%. This type of level controller tuning is referred to as averaging level control. In the rare case when tight level control is required the general loop method should be used.
5. Industrial
example
We have successfully applied these simplified I M C - P I D tuning rules to many different processes. As one example, we have applied these rules to a new 'paper' drying oven. The 'paper' is
Bolh Z-N and C-C
0.07
i I I
0.05
It
0.03
t t
I
0.01
-0.01
i
|
I 10
20 Time
1--
IMC-PID I1 It t I lt Jl F /[
t| 30
(minutes)
,,f
I
Bo'lh Z-N
m'KI C4
II I I I I I I I
-1-
-2
I I
[{ II
I ] I I I I
L
t I
f I
I lO
20
Time (m Inut ee ) Fig. 10. L o a d response for E x a m p | e 3 for + 5 0 % dead time mismatch.
30
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
made from a polymer. Multiple or single rolls of paper are heated using radiant heaters and then pressed to produce the product which is used in high performance composite materials. Many different gauge products are made with throughput rates varying substantially. This created some concern that the controller tuning might need to be changed for different products. The oven is divided into multiple heating zones. Infrared temperature sensors are used to measure the
55
paper temperature at various locations in the oven. PID controllers are used to control these temperatures by manipulating the upstream heater intensity. A complex control strategy, that will not be discussed, sets the setpoints of these controllers. The process is controlled with a Honeywell TDC 3000 DCS. Shortly after startup, one of the authors was asked to provide assistance in tuning these controllers. The process was not making quality
Example #1 Tp = 500 min, La = 2 min, Kp = 1
1.8
1.6
1.4
1,2 \\\x \ •
0.8
0.6
0.4
0.2
L
I
I
I
1.4
1.6
1.8
2
I 0.2
0.4
0.6
0.8
1
1.2 Lo/La
IMC Rules
-- Z-N Rules
Cohen-Coon I
Fig. 11. Stability plot for Example 1.
56
P.S. Fruehauf et al. /1SA Transactions 33 (1994) 43-59
product because there was too much variability in the p a p e r temperatures. One problem with the tuning was that both filter and derivative action were being used. In addition, they had the same tuning settings causing them to completely cancel each other. In this particular case we judged that filtering was more helpful because of the large amount of measurement noise. The derivative action was removed by setting the tuning value to zero. We applied open loop tests to each and every
loop following the procedure already outlined. From these tests we estimated the dead time and process slope and calculated tuning parameters. Fig. 13 illustrates the result of one such test. Originally all the loops were not tested because it was thought that the untested loops would have similar responses. It turned out that they did not. This is a common pitfall in tuning. Some of the loops in the middle of the oven had much larger apparent dead times than the other loops, which resulted in significantly slower tunExample #2
T p = 2 m i n , La = 4 m i n , Kp = 1
1.8
1.6
1.4
1.2
\
0.8
\ 0.6 N
\~
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Lo/La - - IMC Rules
- - Z-N Rules
Cohen-Coon I
Fig. 12. Stability plot for Example 2.
1.8
57
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
LOOP T U N I N G
ADO PU, SP fiND OP TRgCES OELETE RLL TRACES
Ip
ROD
1
fiRggETER.P~
I SELECT
TINEDASE] 1247TC
DELETE | GREEN TRgCE/
I SELECT ' ADD [DATA SOURCE] I C X A N G E ZONE] RT RT RT
5
MIN
lOg.OR
-
PU 175.n t OP l g g . $ $ PU 68.5Z
J ggagigl
171.34
OEG C
PU 145.$g
OP
$.gg$
SP 158.77 OP 4g.ggg
-
i
~
I
1
L247TC g.g TO 255.g OEG C CZ ZONE 2 SECT C TERP
t
I
I
I
i
i
g .gga
I
INTEGRRL iIOERIUgTIOE
lilllillil
T3 ' g i l l '
]1
KPP l i i l i l
TS e g g l g l
KNL l i i g i l
TO Igggll
6.$$ PU OP
Q.85
liillill
171.3 ; m i n i m 4O.O , .~=i i
~ g , ,
FIT RESPONSE WITH TWO STRAIGHT LINE Fig. 13. Open loop step response test for the industrial application example.
ing. The original tuning for these loops, which was extrapolated from tests on other similar loops, produced very oscillatory response. The important lesson to remember is to test each loop. We have seen many cases where 'identical' loops had very different dynamic responses. Once the testing was complete the machine started making quality product. The tuning required about 12 hours. They had been making bad product for two weeks costing approximately $10 K / d a y . Interestingly, the tuning was found be good for all types of products. We speculate that this is partially due to the added robustness of these rules.
especially for his historical perspective on this subject. The Straight-Line Control Co., Inc. has recently published a booklet on controller tuning and control loop performance [6] that was influenced by this work. The booklet is highly recommended.
Conclusions We have greatly simplified the IMC-PID tuning rules. These rules cover most loops encountered in the chemical industry. The tuning rules are For general loops: "g
We would like to acknowledge the help of Dave W. St. Clair, formerly of DuPont, and now with Straight-Line Control Co., Inc., for his help with developing the simplified tuning rules and
T
-->3 L
Acknowledgments Kc ri
,rf
1
1
2RL
2RL
5L =
--<3 L
0.5L or ~'d = 0.5L.
r
L<0.5 1 4
P.S. F r u e h a u f et a l . / ISA Transactions 33 (1994) 4 3 - 5 9
58
For flow loops: K c
= 0.5 to 0.7,
From numerous simulations and industrial application, we find setting rcj = 2 L provides very good closed-loop performance, the resulting Kc and T i are
ri = 0.3 to 0.2.
For level loops: 100% × 2 K c -
ALR
4× V ,
Ti
5 9RL ' which we elected
K~
K c × Q "
We have shown that the simplified I M C - P I D tuning rules are more robust then the rules of Ziegler-Nichols and C o h e n - C o o n . We have shown that filter and derivative action cancel each other and should not be used together and we have proposed rules for setting these actions. We have developed rules for setting filter action so that maximum filtering can be obtained without adversely affecting closed loop performance. We have shown an example where the rules have been successfully applied to an industrial process.
1 to simplify to K~ - 2RL
(A.3) (A.4)
and r i = 5 L . Tuning rule when r / L < 3
When self regulating process behavior is observed and the dead time is relatively large compared to the process time constant, the above tuning rules will be too conservative. A better set of tuning rules is to tune the control loop based on first order plus dead time response. From [5], the PI tuning rules for process model of first order plus dead time are Kp
Appendix A. Derivation of the simplified tuning rules from the rigorous IMC-PID rules
for a process model of K c
e
(rs +
Ls
1) '
T
(A.5)
K c -
Kv(rcl + L)
'
Tuning rule when r / L > 3
(A.6)
Ti :T.
From our experience, a large portion of Chemical Plant control loops are time constant dominant processes, thus the initial portion of the dynamic step response can be approximated as a system with integrator. The importance of the initial response for controller tuning purposes has been pointed out by Chien and Fruehauf [5], thus controller tuning based on a system with integrator can provide excellent closed loop performance. The original tuning rules for a system with integrator and dead time are Re -
Ls
for process model of - S
Kc
2rcl + L R(.rcl + L)2 ,
r i = 2r d +L.
(A.1) (A.2).
It can easily be shown that R = K p / T choosing rot = L from experience, then
1 2RL
Kc r i = r.
and by
(A.7) (A.8)
Tuning rule when L < 0.5
For a process with negligible dead time, typically when L < 0.5 minute, by choosing rc~ = 2 and L = 0 in Eqs. (A.1) and (A.2), we enhance the closed-loop stability. The resulting K c and rg are
1 K c = ~, "J'i =
4.
(A.9) (A.10)
P.S. Fruehauf et aL /1SA Transactions 33 (1994) 43-59
References [1] J.G. Ziegler and N.B. Nichols, "Optimum settings for automatic controllers", ASME Trans. 64 (1942) 759. [2] G.H. Cohen and G.A. Coon, "Theoretical consideration of retarded control", ASME Trans. 75 (1953) 827. [3] D.E. Rivera, M. Morari and S. Skogestad, "Internal model control, 4. PID Controller Design", Ind. Eng. Chem. Proc. Des. Dev. 25 (1986) 252.
59
[4] I-L. Chien, "IMC-PID controller design - An extension", IFAC Proc. on Adaptive Control of Chemical Processes 147, Copenhagen, Denmark, August 1988. [5] I-L. Chien and P.S. Fruehauf, "Consider IMC tuning to improve controller performance", Chemical Engineering Progress 33 (October 1990). [6] D.W. St. Clair, Controller Tuning and Control Loop Performance, 2nd Edition, © (1993) by the Straight-Line Control Co., Inc., 3 Bridle Brook Lane, Newark, DE 19711-2003.
ISATransactions 33 (1994) 43-59
Simplified IMC-PID tuning rules Paul S. F r u e h a u f
*'a,
I-Lung Chien b, Mark D. Lauritsen c
" Applied Control Engineering, P.O. Box 520, Hockessin, DE 19707, USA Department of Chemical Engineering, Chang Gung College of Medicine and Technology, Kwetshan, Taoyuan, Taiwan, ROC c DuPont Engineering, 140 Cypress Station Drit:e, Houston, TX 77090, USA
Abstract We have significantly simplified the IMC-PID tuning rules. These new rules cover the vast majority of control loops encountered in the chemical industry. This work is the result of a great deal of experience in successfully applying IMC-PID tuning rules and an effort to prepare a training course on controller tuning. The simplified rules are very similar in form to the classic open loop Ziegler-Nichols rules and use the process reaction curve method for process testing. The two differences are that these rules are based on a less aggressive performance criteria and that we adapt the rules for some commonly encountered special cases. This paper presents the relationship between the simplified IMC-PID rules, the general IMC-PID rules, the Ziegler-Nichols rules and the Cohen-Coon rules. We show that the simplified rules are less sensitive to parameter mis-estimation than other more aggressive tuning rules. We also proposed rules for a fourth action; filtering. Filtering is available in digital controllers and smart field transmitters. We report that filtering and derivative action cancel each other and therefore should not be used together. We briefly outline the contents of the tuning course and finish the paper with an industrial example where the simplified rules have been successfully applied.
Key words: Controller tuning; Control algorithm
1. Introduction P I D c o n t r o l l e r s a r e still, by far, t h e most comm o n l y u s e d c o n t r o l l e r s in the c h e m i c a l industry. P r o b a b l y t h e single b i g g e s t r e a s o n for this is that a w e l l - d e s i g n e d a n d a d e q u a t e l y - t u n e d P I D cont r o l l e r m e e t s o r e x c e e d s most c o n t r o l objectives. W e d e s i g n c o n t r o l s t r a t e g i e s for n e w plants, m a k e c o n t r o l i m p r o v e m e n t s at existing p l a n t s a n d troubleshoot control problems. These experiences have shown us t h a t t h e r e is very l i m i t e d knowl-
* Corresponding author.
e d g e a b o u t c o n t r o l l e r t u n i n g at m a n y p l a n t sites. W e have i n v e s t i g a t e d c o n t r o l p r o b l e m s for p r o cesses t h a t r e c e n t l y u n d e r w e n t c o n v e r s i o n s to D i s t r i b u t e d C o n t r o l Systems ( D C S ) a n d f o u n d t h a t a m a j o r i t y o f t h e c o n t r o l Mops w e r e b e i n g run in m a n u a l simply b e c a u s e the loops w e r e poorly tuned. L a t e in 1992 we w e r e a s k e d to d e v e l o p a c o n t r o l l e r t u n i n g c o u r s e to h e l p a d d r e s s this p r o b l e m . T h e t a r g e t a u d i e n c e for t h e c o u r s e is i n s t r u m e n t technicians, o p e r a t o r s a n d e n g i n e e r s . O v e r the p a s t five y e a r s we have e m p l o y e d I n t e r nal M o d e l C o n t r o l - P r o p o r t i o n a l I n t e g r a l D e r i v a t i v e ( I M C - P I D ) c o n t r o l l e r t u n i n g rules with
0019-0578/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 0 1 9 - 0 5 7 8 ( 9 4 ) 0 0 0 0 4 - 6
44
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
great success to many control loops. Early in the course development we realized the general I M C - P I D method was complicated and would add significantly to the length of a training course. Our experience with applying these rules showed that they could be simplified and still be useful for the vast majority of control loops. The majority of the controllers we tune are PI only. However, on occasion, we find benefit in using derivative action or filter action. We therefore propose rules for all four actions. The main contributions of the simplified IMCPID tuning rules to the science of controller tuning are: the rules are based on rigorous IMCPID tuning and retain all the benefits of such, the equations for calculating the tuning constants are simple, the rules have been extensively tested on real processes, for most situations only two process dynamic parameters need to be estimated with only moderate accuracy and the rules work well for nonlinear processes because they emphasize the initial dynamic response. This p a p e r will show that the simplified IMCPID tuning rules are more robust than the rules of Ziegler and Nichols [1] and Cohen and Coon [2]. These older rules are less robust because they are based on a more aggressive tuning criteria (i.e., quarter wave decay). We, and others before us, have found this tuning criteria to be a bit too aggressive for most chemical industry applications. However, the contributions of these earlier works are major, please do not misinterpret the robustness analysis in this p a p e r to be a criticism of these works. On the contrary, the fact that the rigorous I M C - P I D tuning rules reduce to the same form as the original Ziegler-Nichols open loop rules further confirms just what a stroke of empirical genius this work was. Our interest in the work of Cohen and Coon came about because their tuning rules correct for one deficiency of the Ziegler-Nichols rules (i.e., sluggish closed loop response in the rare occasion when the process dead time is large relative to dominant open loop time constant). Our simplified IMCPID rules also correct for this deficiency. The first section of the paper reports the simplified tuning rules. The second section compares the robustness of the simplified tuning rules to
other rules. A brief description of the tuning method that we teach and how we apply the rules is included in the third section and we finish the p a p e r with an actual industrial example of the application of our rules. Appendix A is attached to document the derivation of the simplified rules.
2. Simplified IMC-PID tuning rules The PID tuning rules based on the Internal Model Control (IMC) design method were first developed in 1986 by Rivera and co-workers [3]. Later extension to cover a wider range of process models and an extension suitable for time constant dominant processes were presented by Chien [4] and Chien and Fruehauf [5]. The IMCPID tuning method is very attractive to industrial users because it has only one tuning parameter: the closed loop speed of response. This parameter relates directly to the closed loop time constant for the response to setpoint changes and to the robustness of the control loop. A smaller closed loop speed of response decreases the robustness. Moreover the closed loop step load response exhibits no oscillation or overshoot. Experience indicates that this minimizes controller interactions and enhances overall process disturbance rejection. When asked to prepare the controller tuning course we began to investigate ways to simplify the I M C - P I D tuning rules. Our objective was to help the student tune the majority of control loops that they would encounter. To simplify the I M C - P I D tuning rules we used the following facts, assumptions and conclusions based on our experience: - The majority of processes are well approximated by dead time first order or dead time integrator models. - Tuning parameters do not need to be set to a high degree of accuracy to achieve satisfactory closed loop response. Accuracy to within 20% is adequate. - We have developed rules for setting the closed loop speed of response (i.e, Tcl) so it can be fixed. This eliminates the only tuning parameter that is required with the I M C - P I D rules.
P.S. Fruehauf et al./ ISA Transactions 33 (1994) 43-59
- ~
SloDe, R - -
/ f/
Definition
/
PI tuning rules The first and most commonly used rules are
""
/ Controlled Verlable
/f
45
1
May or May Not Level Out
Kc
(1)
2RL '
(2)
~'i = 5L,
I I:
I : ]
L, Apparent Dead Time
I 2" •,
,,,
,
Time Fig. 1. The open loop step response of a typical process.
- The most popular type of PID controller is the interacting form and the tuning settings are gain (not proportional band) and reset in minu t e s / repeat (not repeats/minutes). If one is working with tuning settings in alternative units, conversion is simple. - Open loop testing is used to determine the dynamic parameters for a process. The derivation of the simplified rules from the rigorous rules using these principles is included in Appendix A. Interestingly, the resulting rules reduce to a form that is very similar to the original Ziegler-Nichols [1] open loop tuning rules. Our rules are less aggressive and we modify the rules for some commonly encountered special cases. Before we present the tuning rules, we will briefly present the open loop procedure for process testing. With the controller in manual and the process as steady as possible, a step change in controller output is made. The process response will look something like Fig. 1. If the process levels out, or would if enough time were allowed, it is called dead time first order. If it would not, it is called dead time integrating. The tuning rules generally apply to both types of processes, though modifications to the rules are needed if the process is dead time first order and the time constant is short relative to the apparent dead time or when the apparent dead time is small (i.e., L < 0.5 minutes). As a result, we have developed three sets of rules.
where Kc is the controller proportional gain (dimensionless), ~i is the controller reset time (minutes/repeat), R is the change per minute in the process variable, (expressed as a % of the transmitter span), divided by the step change magnitude (expressed as a % of the controller output span) (1/minutes), L is the apparent dead time (minutes) These rules are derived from the IMC-PID rules for a dead time integrating process. However, as shown in [5], these rules apply to both dead time integrating and dead time first order processes. There are two types of less commonly observed responses which require modifications to the above rules. The first is when the process is dead time first order and the time constant is short relative to the apparent dead time. Fig. 2 illustrates this type of response. We have found, by performing many simulations, that the following rules give better closedloop response when the time constant divided by
Slope, R -,~,/ Final Response
63% of Final Response
_Z_~
It: :I
L, Apparent DeadTime
I Time Fig. 2. Open loop step response of a self regulating process.
46
P.S. Fruehauf et a l . / ISA Transactions 33 (1994) 43-59
the apparent dead time is 3 or less. The modified tuning rules are 1 Kc = 2 R L '
(3)
Ti =
(4)
become more conservative, thus enhancing the stability of the closed loop system. Please note that this last rule does not apply to flow loops. In summary, the simplified PI tuning rules are "t
'T,
where r is the process time constant (minutes). Notice that the only difference between this modification and the Eqs. (1) and (2) rules is that the reset time is, at times, significantly shorter which speeds up the closed-loop response. This rule is derived from the rigorous IMC-PID rules for dead time first order processes. Often when doing open loop testing it is difficult to observe the open loop time constant because of noise and other process variables that are changing that effect the dynamic response (i.e., the process never steadies out). If you cannot observe the time constant just use the first of set rules. On the occasion when you can observe the first order time constant and the ratio is small you can take advantage of the performance improvement that these modifications to the rules provides. The other special situation which requires tuning rule modification is when the apparent dead time is very small. In this case, the tuning rules in Eqs. (1) and (2) may result in excessive control action thus causing closed loop stability problems when even modest model mismatch is present. In addition, Eq. (1) can result in a very high controller gain that will produce large changes in controller output for small changes in setpoint. If the controller output is tied to a valve that adjusts the feed rate to a downstream unit operation this can cause large flow disturbances to the downstream unit. Also, when the apparent dead time is small the process is easy to control, so very tight tuning is not generally necessary. In this small dead time case, typically when L < 0.5 minutes, the tuning rules are modified to 1
Kc = ~,
(5)
r i = 4.
(6)
Notice that K c does not have L in the denominator and thus is smaller and '/'i is larger compared to Eqs. (1) and (2). The controller settings
"g
-->3 L 1
Kc ri
2RL 5L
--<3 L 1
2RL r
L<0.5 1
R 4
The above tuning rules work very well for the vast majority of the control loops encountered in the chemical industry. For specific control loops like flow and level we provide other recommendations later in this paper. For rare occasions when the open loop step response exhibits overshoot, oscillatory, or inverse behavior, we recommend modeling the process dynamics as a more complicated second order response and refer to Table 1 in [5] for controller settings. 2.1. F i l t e r a n d d e r i v a t i v e t u n i n g rules
Before we discuss the filter and derivative tuning rules it will be helpful to discuss filter action, derivative action and the relationship between the two. Filter action
We use signal filtering in some loops to improve performance. One common example is in level loops where the measurement is often noisy. Without filter action the noise is passed by the proportional action of the controller directly to the control valve causing it to move unnecessarily. Reducing this movement by decreasing the controller gain is undesirable because of the destabilizing effect this has on the level control loop. Filter action is a superior solution. We are not aware of any rules on how to set filter time constant. We believe that filtering is now used more often because it is available in many digital controllers and smart field instruments. It has become a fourth controller action. The task of setting the filter time is one of using as much as you dare without degrading the performance of the loop. The goal is to specify a
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
filter time constant that provides the benefits of filtering, without adversely impacting closed loop response. We used time domain and frequency analysis to develop a general rule. A dead time first order process and a dead time integrating process were simulated, with the process feedback signal being filtered in each case. The tuning rules were used to provide the PI controller settings. Fig. 3 illustrates one of the plots used for this analysis. It shows the time domain responses for a dead time
47
first order process subjected to a step change in setpoint. A filter time constant less than or equal to 50% of the apparent dead time was selected as the general rule because it provides the desired filtering action without severely impacting loop performance. Derivative action
Derivative action can improve the performance of a control loop by shortening the closed loop natural period. The performance improve-
Response to Step Change in Setpoint Deadtime First-Order Lag Process
1.2
I
/
Tf
.T£
:
2*L
-
M
e a s u r e d
4*L
ffi
L
0.8
0.6
f
-
0.5*L
f
= 0.2*L
0.4
0.2
t
I
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
90
100
~me Fig. 3. S e t p o i n t r e s p o n s e s ~ r d i f f e r e n t v a l u e s o f f i l t e r t i m e c o n s t a n t .
48
P.S. Fruehauf et al./1SA Transactions 33 (1994) 43-59
ment is not dramatic; however, in some situations it can help enough to warrant using it. We have used the rigorous I M C - P I D tuning rules to derive rules for setting derivative time which say to make it equal to one half the apparent dead time. In order to limit the number of rules, we elected not to incorporate the slight gain increase and integral decrease given by the rigorous rules. These changes are small relative to the required accuracy for setting these parameters.
Relationship between filter and derivative action Filter and derivative action are essentially the mathematical inverse of each other. They are in series in the feedback circuit so they cancel each other. Therefore, we propose rules that specify one or the other but not both. Derivative action causes the controller output to bounce more than when it is not used, but improves control performance. Filter action dampens this bouncing but degrades performance. Often the output of one loop is a disturbance to another loop. Derivative action could improve one loop's performance but could hurt another. Filter action does the opposite. The decision to use filter or derivative action becomes a compromise between performance of an individual loop versus the system as a whole. Often the best compromise is when neither is used. This is one reason why we feel the PI only controller is so popular. In summary, the rules for setting filter or derivative time are Tf : 0.5L or "/'d= 0.5L, where % is the filter time constant (minutes), ~'d is the derivative time (minutes).
robustness of the simplified I M C - P I D rules to other tuning rules in two different ways. The first section uses time response plots to show the effect of model mismatch for PI tuning rules. There is a model mismatch when the process model parameters change from what they originally were. This mimics what can happen when the operating conditions of a chemical process are changed, because almost all processes are non-linear to some degree. When we tune we approximate the dynamics at a particular operating condition with a linear model. The second section compares PID tuning rules using stability plots. This comparison is also different from the first, in that we are looking at mismatch between what the actual process dynamics are and what we estimate them to be. This measure of robustness tells us how accurately we must estimate the process dynamics when doing an open loop step test to prevent unstable closed loop response.
PI tuning rule sensitivity to model mismatch In this section, the simplified I M C - P I D tuning rules are demonstrated using three numerical examples with different process dynamic characteristics. These tuning rules will be compared to the popular tuning methods of Ziegler and Nichols [1] and Cohen and Coon [2]. Fig. 4 shows the feedback control structure for all the simulations in this section. y is the controlled variable, u is the controller output, Ys is the setpoint, d is the disturbance, G c is the feedback controller (PI), Gp is the process transfer function, G L is the disturbance transfer function.
3. Robustness
Robustness is the term used to describe the sensitivity of closed loop response to changes. A set of tuning rules are more robust if they can tolerate larger changes in the dynamics parameters (e.g., L and R) and still maintain adequate response. The next two sections compare the
YS
~
~
~_*
Y
Fig. 4. Feedback control structure used for simulations illustrated in Figs. 5-10.
P.S. Fruehauf et al./ ISA Transactions 33 (1994) 43-59
Example 1. Time-constant-dominant process:
sponse for both controlled variable and controller output. On the contrary, both Ziegler-Nichols and C o h e n - C o o n methods result in overshoot and an oscillatory closed-loop response. In order to test the robustness of the proposed tuning rules, + 50% model mismatches in process gain (related to R) and dead time were simulated. Fig. 6 illustrates the effect of the process gain mismatch. The figure clearly shows the superior closedloop behavior of the simplified IMC-PID settings under severe model mismatch conditions.
10e-2S
Gp=G/= 5 0 0 s + l " From the step response, R, L and T are calculated to be R = 0 . 0 2 , L = 2 and T = 5 0 0 . The resulting PI controller settings are
Kc ri
Simplified IMC-PID
Ziegler-Nichols
Cohen-Coon
12.5 10
22.5 6.7
22.5 6.6
Fig. 5 shows the disturbance closed-loop response for the simplified IMC-PID, ZieglerNichols and C o h e n - C o o n rules. In the simulation, d is changed from 0 to 1 at time equal to zero. Notice that the proposed tuning rules give very smooth closed loop re-
0.08
49
Example 2. Process with short time constant and significant dead time: e-4S Gp=Gt-
2s+l"
IMC-PID
0.06 0.04,
/ /
0.02
Both Z-N and C-C
\\\ , ~"
0 -0.02
.
.
.
.
!
.
.
.
.
10
!
•
•
•
•
I
.
30
20 Time
.
.
.
I
.
.
4,0
.
.
I
50
(minutes)
0
•I .I ""0":]
~
_.. IMC-PID
\ 'X
",
-is
/f
"
0
f
// /
X
-20
"-
Bolh Z-N
"
"
I
i0
"
I
"
"
"
•
20 Time
I
30 (minutes)
Fig. 5. Load response for ExampLe 1.
"
|
40
•
I
5O
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
50
IMC-PID
o--1 /
Both Z-N and C-C
',,
/ ', ",,l
-0.05
-0.I
•
|
ii
i
|
•
•
•
'
I
"
'
"
IMC-PID,..,
/"
/
I
/*%, / /
',\
\
"
"
\£ "
\ \ \ \
I
'
/ 1 I II
\
'
"
"
I
4,0
50
(minutes)
Both Z-N and C-C.
f
// %
/,~, I
0 =
-10
/
,
"
30
Time
i0--
'
20
10
0
/
/ / /7
/
\
1
~'
=
-20"
-30
-
'\\/
\\ /I' I
.
.
.
~
\\
.
I
10
.
.
.
.
I
20
T1hae
I
\\
~
I
"
t
\ "
30
"
I
I
.
.
.
.
40
I 50
(minutes)
Fig. 6. Load response for Example 1 for a + 50% gain mismatch.
For this example, the controller settings are
K, ~'i
Simplified IMC-PID
Ziegler-Nichols
Cohen-Coon
0.25 2
0.45 13.3
0.45 2.9
Fig. 7 shows the disturbance rejection closedloop responses using the three tuning methods. Notice again that the proposed tuning rules give very smooth closed loop response for both controlled variable and controller output, while Ziegler-Nichols settings give very sluggish response. C o h e n - C o o n settings provides improvement over Ziegler-Nichols settings in this situation but are still a little oscillatory. Fig. 8 shows the closed loop responses for a + 5 0 % model mismatch in dead time. The simplified tuning rules still provide smooth closed-loop response while Ziegler-Nichols settings are still very slug-
gish and C o h e n - C o o n settings are more oscillatory. Example 3. Process with small dead time: 10e-0.16s Gp = G /
15s + 1
In this example, the process dead time is very small (i.e., about 10 seconds). The controller settings for the three tuning methods are
K,, ri
Simplified IMC-PID
Ziegler-Nichols
Cohen-Coon
1.5 4
8.43 0.53
8.44 0.52
Fig. 9 shows the closed-loop responses for the three tuning methods. Notice that Ziegler-Nichols and C o h e n - C o o n settings result in an almost identical response.
P.S. Fruehauf et a l . / ISA Transactions 33 (1994) 43-59
the closed loop response becomes unstable. We estimate these parameters as part of the open loop tuning procedure we use. A set of tuning rules are more robust if larger errors in the estimates can be tolerated. These plots were developed because we wanted a general way to assess the robustness of the simplified IMC-PID rules before we included them in our tuning course. The time response plots in the previous section only test the rules for specific values of mismatch. The plots in this section illustrate the robustness for all values of mismatch of interest. Tuning rule robustness plots have been prepared for the three example processes presented in the previous section. Fig. 11 is the plot for the example one process. The X-axis is the ratio of the estimated dead time (Lo) to the actual dead time (La). The Y-axis is the ratio of the estimated
Although their responses are much faster than the proposed tuning method, the controller output is rather oscillatory. Fig. 10 shows the response for a + 5 0 % model mismatch in dead time. We should mention that this kind of dead time mismatch (i.e., process having dead time of 15 seconds versus model having dead time of 10 seconds), is reasonable and can be expected to happen from time to time. With this small model mismatch the Ziegler-Nichols and C o h e n - C o o n settings become unstable, while the simplified IMC-PID rules still provides smooth and stable closed-loop response.
Tuning rule stability plots In this section, tuning rule stability plots are used to compare how much we can misestimate the process dead time and process slope before
f
51
IMC-PID ~ Z-N
0.5"
0
°::1 .
.
.
.
I
0
"
"
"
10
'
I
.
.
.
.
I
'
•
30
20
Time
I
"
•
I
50
40
(minutes)
O r
-0.5
-1
-1.5
-2
•
0
I
10
.
.
.
.
I
"
"
"
20
Time
I
30
(minutes)
Fig. 7. L o a d r e s p o n s e for E x a m p l e 2.
"
•
I
4o
•
I
50
52
P.S. Fruehauf et aL / ISA Transactions 33 (1994) 43-59
process slope (Ro) to the actual process slope (Ra). The curves are boundaries between the regions of stable and unstable estimates. The stable regions are above and to the right of the curves and the unstable regions are below and to the left of the curves. The curves are defined by the equations for the Bode stability criteria (i.e., magnitude ratio of unity at the resonant frequency) using the different tuning rules. These plots can be used to compare the robustness of tuning rules in the following way. Assume that we have just done an open loop step test on a process and the response looks something like Fig. 1. The process dead time is misestimated, due to signal noise, to be 1.5 minutes. Assume that the actual dead time is 2 minutes. This defines a line perpendicular to the X-axis at 0.75. This line intersects the I M C - P I D rule curve
at 0.4 and both the Ziegler-Nichols and C o h e n Coon rule curves at 1.25. If the true process slope is 1, the graph tells us that if we mis-estimate the process slope to be less than 0.4 the I M C - P I D rules will result in unstable tuning, whereas the other rules will only allow a mis-estimation of less than 1.25 before unstable tuning results. For the Ziegler-Nichols and C o h e n - C o o n rules if we estimate the process slope to be exactly what it is we will calculate tuning settings that produce unstable response! We conclude that the I M C - P I D rules are more robust because they allow for a larger mis-estimation of the slope than the other rules. More generally, the farther the stability boundary line is from the point (1, 1) (i.e., the point which corresponds to exact estimates of dead time and slope) the more robust the tuning rules are. Fig. 12 is the robustness plot for the Example
-_
.
.
f
.
IM@PID '
'
I
0
"
"
"
10
'
I
.
.
.
.
I
20
Time
0 ~
-0.5
r.
/
"
. "
"
"
30 { m i n u t e
I
"
'
'
40
"
.
50
B }
Z-N
=
.1
-1.5
=
•
•
"
I
10
.
.
.
.
I
"
"
20 T i m e
"
I
.
.
.
.
30 ( m i n u t
e B )
Fig. 8. Load response for Example 2 with a + 50% dead time mismatch.
I
40
"
"
5O
P.S. Fruehauf et a l . / ISA Transactions 33 (1994) 43-59
2 process where the process time constant is short relative to the dead time. The Ziegler-Nichols and C o h e n - C o o n rules are more robust for this case but are still significantly less robust than the simplified IMC-PID rules. For brevity the Example 3 process plot is not shown. The simplified IMC-PID rules are intentionally conservative for this type of process and, therefore, are much more robust than the other rules.
For general loops, we advocate that open loop testing be used. This is essentially the reaction curve testing method proposed by Ziegler and Nichols. We teach general loop tuning as a three step process. The steps are (1) to learn to run the process with controller in manual and estimate L and R, (2) to perform open loop step tests to accurately determine L, R and r (if it is possible to observe r), and to calculate tuning and (3) to test new tuning with the loop in automatic by making setpoint changes. The first step is critical and far from trivial; however, most tuning discussions we have seen do not even address this step. This method is used for general control loops, e.g. temperature, pressure, composition loops, etc. For special classes of control loops like flow and level, we provide the following controller tuning recommendations.
4. Tuning method taught in training course The tuning method we teach breaks loops up into three general categories: general loops, flow loops and level loops where averaging level control is desired.
f 0.05
m
0.03
"
53
IMC-PID
0.01
-0.01
I
.
.
.
.
I
10
.
.
.
.
20
Time
I
30
(minutes)
O~
-0.5"
j[~V
I
B o t h Z-N and C-C
-1-
-1.5
-,2
! IMC_PID? I
"
"
"
10
I
20
Time
(minut
era)
Fig. 9. Load response for Example 3.
"
"
I
30
54
P.S. Fruehauf et aL / ISA Transactions 33 (1994) 43-59
For flow loops we teach that the dynamics are generally the same for all loops and therefore so is the tuning. We recommend that the gain be set between 0.5 to 0.7 and that the reset be set between 0.2 and 0.3 minutes. Fine tuning can then be done by a closed loop trial and error method. Level loops are generally tuned so the tank will absorb flow disturbances so that they are not passed on to downstream unit operations. When this is the case, we recommend that the following equations be used: 100% × 2 Kc ALR ' 4×V zi
Kc Q '
where A L R is the allowable level range (%), V is
f
IMC-PID
the measured tank volume (gallons) and Q is the maximum flow through level control valve (gallons/minute). The allowable level range is the range of level that can be allowed to vary without any adverse effects. For a surge tank, the range is often selected to be 80%. This type of level controller tuning is referred to as averaging level control. In the rare case when tight level control is required the general loop method should be used.
5. Industrial
example
We have successfully applied these simplified I M C - P I D tuning rules to many different processes. As one example, we have applied these rules to a new 'paper' drying oven. The 'paper' is
Bolh Z-N and C-C
0.07
i I I
0.05
It
0.03
t t
I
0.01
-0.01
i
|
I 10
20 Time
1--
IMC-PID I1 It t I lt Jl F /[
t| 30
(minutes)
,,f
I
Bo'lh Z-N
m'KI C4
II I I I I I I I
-1-
-2
I I
[{ II
I ] I I I I
L
t I
f I
I lO
20
Time (m Inut ee ) Fig. 10. L o a d response for E x a m p | e 3 for + 5 0 % dead time mismatch.
30
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
made from a polymer. Multiple or single rolls of paper are heated using radiant heaters and then pressed to produce the product which is used in high performance composite materials. Many different gauge products are made with throughput rates varying substantially. This created some concern that the controller tuning might need to be changed for different products. The oven is divided into multiple heating zones. Infrared temperature sensors are used to measure the
55
paper temperature at various locations in the oven. PID controllers are used to control these temperatures by manipulating the upstream heater intensity. A complex control strategy, that will not be discussed, sets the setpoints of these controllers. The process is controlled with a Honeywell TDC 3000 DCS. Shortly after startup, one of the authors was asked to provide assistance in tuning these controllers. The process was not making quality
Example #1 Tp = 500 min, La = 2 min, Kp = 1
1.8
1.6
1.4
1,2 \\\x \ •
0.8
0.6
0.4
0.2
L
I
I
I
1.4
1.6
1.8
2
I 0.2
0.4
0.6
0.8
1
1.2 Lo/La
IMC Rules
-- Z-N Rules
Cohen-Coon I
Fig. 11. Stability plot for Example 1.
56
P.S. Fruehauf et al. /1SA Transactions 33 (1994) 43-59
product because there was too much variability in the p a p e r temperatures. One problem with the tuning was that both filter and derivative action were being used. In addition, they had the same tuning settings causing them to completely cancel each other. In this particular case we judged that filtering was more helpful because of the large amount of measurement noise. The derivative action was removed by setting the tuning value to zero. We applied open loop tests to each and every
loop following the procedure already outlined. From these tests we estimated the dead time and process slope and calculated tuning parameters. Fig. 13 illustrates the result of one such test. Originally all the loops were not tested because it was thought that the untested loops would have similar responses. It turned out that they did not. This is a common pitfall in tuning. Some of the loops in the middle of the oven had much larger apparent dead times than the other loops, which resulted in significantly slower tunExample #2
T p = 2 m i n , La = 4 m i n , Kp = 1
1.8
1.6
1.4
1.2
\
0.8
\ 0.6 N
\~
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Lo/La - - IMC Rules
- - Z-N Rules
Cohen-Coon I
Fig. 12. Stability plot for Example 2.
1.8
57
P.S. Fruehauf et al. / ISA Transactions 33 (1994) 43-59
LOOP T U N I N G
ADO PU, SP fiND OP TRgCES OELETE RLL TRACES
Ip
ROD
1
fiRggETER.P~
I SELECT
TINEDASE] 1247TC
DELETE | GREEN TRgCE/
I SELECT ' ADD [DATA SOURCE] I C X A N G E ZONE] RT RT RT
5
MIN
lOg.OR
-
PU 175.n t OP l g g . $ $ PU 68.5Z
J ggagigl
171.34
OEG C
PU 145.$g
OP
$.gg$
SP 158.77 OP 4g.ggg
-
i
~
I
1
L247TC g.g TO 255.g OEG C CZ ZONE 2 SECT C TERP
t
I
I
I
i
i
g .gga
I
INTEGRRL iIOERIUgTIOE
lilllillil
T3 ' g i l l '
]1
KPP l i i l i l
TS e g g l g l
KNL l i i g i l
TO Igggll
6.$$ PU OP
Q.85
liillill
171.3 ; m i n i m 4O.O , .~=i i
~ g , ,
FIT RESPONSE WITH TWO STRAIGHT LINE Fig. 13. Open loop step response test for the industrial application example.
ing. The original tuning for these loops, which was extrapolated from tests on other similar loops, produced very oscillatory response. The important lesson to remember is to test each loop. We have seen many cases where 'identical' loops had very different dynamic responses. Once the testing was complete the machine started making quality product. The tuning required about 12 hours. They had been making bad product for two weeks costing approximately $10 K / d a y . Interestingly, the tuning was found be good for all types of products. We speculate that this is partially due to the added robustness of these rules.
especially for his historical perspective on this subject. The Straight-Line Control Co., Inc. has recently published a booklet on controller tuning and control loop performance [6] that was influenced by this work. The booklet is highly recommended.
Conclusions We have greatly simplified the IMC-PID tuning rules. These rules cover most loops encountered in the chemical industry. The tuning rules are For general loops: "g
We would like to acknowledge the help of Dave W. St. Clair, formerly of DuPont, and now with Straight-Line Control Co., Inc., for his help with developing the simplified tuning rules and
T
-->3 L
Acknowledgments Kc ri
,rf
1
1
2RL
2RL
5L =
--<3 L
0.5L or ~'d = 0.5L.
r
L<0.5 1 4
P.S. F r u e h a u f et a l . / ISA Transactions 33 (1994) 4 3 - 5 9
58
For flow loops: K c
= 0.5 to 0.7,
From numerous simulations and industrial application, we find setting rcj = 2 L provides very good closed-loop performance, the resulting Kc and T i are
ri = 0.3 to 0.2.
For level loops: 100% × 2 K c -
ALR
4× V ,
Ti
5 9RL ' which we elected
K~
K c × Q "
We have shown that the simplified I M C - P I D tuning rules are more robust then the rules of Ziegler-Nichols and C o h e n - C o o n . We have shown that filter and derivative action cancel each other and should not be used together and we have proposed rules for setting these actions. We have developed rules for setting filter action so that maximum filtering can be obtained without adversely affecting closed loop performance. We have shown an example where the rules have been successfully applied to an industrial process.
1 to simplify to K~ - 2RL
(A.3) (A.4)
and r i = 5 L . Tuning rule when r / L < 3
When self regulating process behavior is observed and the dead time is relatively large compared to the process time constant, the above tuning rules will be too conservative. A better set of tuning rules is to tune the control loop based on first order plus dead time response. From [5], the PI tuning rules for process model of first order plus dead time are Kp
Appendix A. Derivation of the simplified tuning rules from the rigorous IMC-PID rules
for a process model of K c
e
(rs +
Ls
1) '
T
(A.5)
K c -
Kv(rcl + L)
'
Tuning rule when r / L > 3
(A.6)
Ti :T.
From our experience, a large portion of Chemical Plant control loops are time constant dominant processes, thus the initial portion of the dynamic step response can be approximated as a system with integrator. The importance of the initial response for controller tuning purposes has been pointed out by Chien and Fruehauf [5], thus controller tuning based on a system with integrator can provide excellent closed loop performance. The original tuning rules for a system with integrator and dead time are Re -
Ls
for process model of - S
Kc
2rcl + L R(.rcl + L)2 ,
r i = 2r d +L.
(A.1) (A.2).
It can easily be shown that R = K p / T choosing rot = L from experience, then
1 2RL
Kc r i = r.
and by
(A.7) (A.8)
Tuning rule when L < 0.5
For a process with negligible dead time, typically when L < 0.5 minute, by choosing rc~ = 2 and L = 0 in Eqs. (A.1) and (A.2), we enhance the closed-loop stability. The resulting K c and rg are
1 K c = ~, "J'i =
4.
(A.9) (A.10)
P.S. Fruehauf et aL /1SA Transactions 33 (1994) 43-59
References [1] J.G. Ziegler and N.B. Nichols, "Optimum settings for automatic controllers", ASME Trans. 64 (1942) 759. [2] G.H. Cohen and G.A. Coon, "Theoretical consideration of retarded control", ASME Trans. 75 (1953) 827. [3] D.E. Rivera, M. Morari and S. Skogestad, "Internal model control, 4. PID Controller Design", Ind. Eng. Chem. Proc. Des. Dev. 25 (1986) 252.
59
[4] I-L. Chien, "IMC-PID controller design - An extension", IFAC Proc. on Adaptive Control of Chemical Processes 147, Copenhagen, Denmark, August 1988. [5] I-L. Chien and P.S. Fruehauf, "Consider IMC tuning to improve controller performance", Chemical Engineering Progress 33 (October 1990). [6] D.W. St. Clair, Controller Tuning and Control Loop Performance, 2nd Edition, © (1993) by the Straight-Line Control Co., Inc., 3 Bridle Brook Lane, Newark, DE 19711-2003.