Ind. Eng. Chem. Res. 1994,33, 1708-1717
1708
Autotuning of Multiloop Control Systems Mats Friman and Kurt V. Waller' Process Control Laboratory, Department of Chemical Engineering, Abo Akademi, FIN 20500 Abo, Finland
Autotuning by relay feedback is studied with emphasis on multiloop PID control of two-input two-output systems. For the purpose of control design many chemical engineering processes can be modeled by integrator-plus-dead-time or gain-plus-dead-time models. Relay feedback is a convenient method t o identify the model parameters, the form of the output response indicating which model is preferable. For a multi-input multi-output process, the off-diagonal elements can be determined in the same experiment as the diagonal elements. The usefulness of the approach is illustrated by a number of systems, including processes with inverse response. Two experimental two-input two-output processes, a mixing tank and a distillation column are used, both with and without decoupling. The approach results in decouplers of simple forms. Autotuning is useful also for tuning the decoupled system.
Introduction Controller tuning by trial and error is boring and time consuming even for SISO systems. However, since the underlying principles are simple, this procedure is suitable for automation. An efficient method is to use relay feedback and determine the approximate values for the ultimate gain and for the critical frequency (Astrom and Hagglund, 1984,1988),from which reasonable controller settings can be determined by various procedures, such as Ziegler-Nichols (1942). Astrom and Hiigglund (1990) describe an interesting industrial application of this method. The problem is a sluggish distillation control loop, previously thought to be impossible to be put on automatic. After a 6-h autotuning experiment reasonable controller settings were obtained and the loop was successfully put on automatic. A loop as slow as this is naturally very difficult to tune manually. The appealing usefulness of relay autotuning has led to a number of commercial autotuners (Astrom and Hagglund, 1988). From the two parameters determined in an autotune test (e.g., critical frequency and ultimate gain), two parameters in a process model can be calculated. If one wants a three-parameter model, such as a first-order-plusdead-time model, one parameter has to be determined in some other way. Luyben (1987) used autotuning to determine firstsecond- and third-order process models when the steadystate gain was determined by other means. The procedure was later modified (Li et al., 1991) so as to remove the condition that the gain must be known. The modified procedure uses two relay experiments. The first is a straightforward one; the second is run with a dead time added. Thus additional parameters can be determined. Chiang et al. (1992) suggested further modifications to increase the accuracy of the method for identifying process models. Chiang and Yu (1993) used a similar idea to get more information. They varied the amount of hysteresis in the relay so as to obtain several points on the frequency response curve to get a more solid basis for controller design. Astrom and Hligglund (1988) also discussed various means of identifying several points on the Nyquist curve, such as varying the amplitude and hysteresis of the relay or adding a filter with known characteristics to the loop.
* Author
to whom correspondence should be addressed.
E-mail:
[email protected].
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Several other modifications or extensions of the AstromHiigglund autotuner have also been suggested for SISO systems. One example is given by Shei (1992), who connected the relay between the process output and the controller reference signal. The feedback controller (PID) is then iteratively tuned during the relay-setpoint experiment. Many modifications of the Ziegler-Nicholstunings have been suggested over the years. Recently Astrom and coworkers (Hang et al., 1991) found it especially important in PI control of processes with large dead time to increase the integral action so as to become larger than it is in the Ziegler-Nichols settings. The same authors (Hang et al., 1993) also recently suggested means of increasing the accuracy in relay feedback identification for the case when the system is subject to certain constant disturbances. SISO tuning, e.g., by Ziegler-Nichols or modified rules, is used also for multiloop systems. A common procedure is to tune each loop with all other loops open and then detune the loops as they are simultaneously put on automatic. As a rule, detuning is necessary because the interaction between the feedback loops usually decreases the stability margin. This detuning is often made by trial and error-a common approach for PI control being to decrease the gain and increase the integral time by the same factor (e.g., Toijala and Fagervik, 1972). A more systematic procedure, called BLT tuning, was suggested by Luyben (1986) for multiloop PI tuning. The method was later extended by Luyben and co-workers (Monica et al., 1988) so as to include derivative action in the controller as well as weighted detuning. The procedure depends on the availability of a process model, also concerning the off-diagonal elements in the process. The most used dynamic model for chemical engineering processes is the first-order-plus-dead-time model, which has three parameters. In many cases the time constant is very large; in some cases the dead time is dominating. For feedback control the time constant per se is not very interesting in either case, the control properties being mainly determined by the high-frequency behavior of the system. In both situations these properties are well captured by two parameters. They are the dead time and the ratio between gain and time constant for systems with large time constants, resulting in an integrator-plus-deadtime model. For systems dominated by dead time the two important parameters for feedback control are the dead time and the gain. For both models the two model parameters are conveniently identified by relay feedback. 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1709 i.e., Kl T for a first-order-plus-dead-timesystem expressed as I
0
10
20
lime
0
200
400
lime
Figure 1. Open-loopresponse of y of system y/u = exp(-s)/(s + a) to a step change in u from 0 to 1 at time = 0 (lower part of figure) and closed-loopresponse of y with feedback controller u = y m - y (upper part of figure) to a step change in setpoint ymtfrom 0 to 1 at time = 0.
In this paper relay identification and autotuning in combination with integrator-plus-dead-time and gaindelay models is studied for various systems, including systems showing inverse response (RHP zero). Both SISO and MIMO systems are studied. In MIMO systems the emphasis is on multiloop control. In this case the relay identification gives models also for the off-diagonal elements, thus permitting systematic controller design by model-based techniques. A number of simulated SISO examples and two experimental 2 X 2 systems illustrate the approach.
Integrator-plus-Dead-TimeProcess Models The dynamics of many processes in the chemical process industries is dominatedby large time constants. Industrial examples of time constants of the order of 1 day (24 h) have been reported, e.g., for distillation (McNeill and Sacks, 1969) and for cement production (Westerlund et al., 1980). However, for control system design this large time constant is usually uninteresting as long as it is large; control properties are largely determined by highfrequency behavior. An illustration for MIMO linear quadratic control design in distillation is given by Hammarstriim et al. (1982). Another application is discussed by Sandelin et al. (1991). A simple,perspicuous illustration is provided by Figure 1, which is inspired by Astriim and Wittenmark (1989) (Astriim and Wittenmark considered a second-order system whereas the one in Figure 1 is a first-order-plus-dead-time system). The process model is given by
where a is a small value (i.e., the system has a pole close to the origin). The time constant of the system is T = l/a. In the figure this time constant has the values +100, m, and -100 time units, the last case indicating an open-loop unstable system. As seen this variation does not affect the behavior of the closed-loop system in any significant degree. For feedback control the two important parameters are the dead time and the initial nonzero slope of the step response,
(Kl T was also found to be a suitable disturbance parameter in distillation even when the first-order system was not a part of the feedback system (Sandelin et al., 1991).) Thus, when the goal of modeling is the design of a feedback control system, it is not necessary to determine both K and Tin a fiist-order-plus-dead-time system with large T (i.e., T >> L),only their ratio. A suitable process model for this purpose is the integrator-plus-dead-time model, corresponding to a = 0 in Figure 1, (3)
This model has only two parameters, the dead time L and k, the latter expressing the initial nonzero slope of the step response (equal to KIT in a first-order-plus-deadtime model). This is a model which seems ideally suited in combination with autotuning, where two parameters can be determined from one experiment. In this paper modeling by integrator-plus-dead-time models is combined with autotuning. This has several advantages. For SISO system, only two parameters have to be identified, thus eliminating the need for more than one experiment. For MIMO systems there are several additional advantages. One is that the off-diagonal elements can be quite easily determined in spite of drift during the relay experiment. Another is that the simple models make decoupler design very simple, the decouplers being gain-delay elements. Tuning of the controllers in the decoupled system is then straightforward by autotuning. The approach to use relay identification in combination with an integrator-plus-dead-time model is useful also for other than first-order-plus-dead-time models with a large ratio between time constant and dead time. This is illustrated later in this paper for second-order-plus-deadtime systems, for systems with inverse response (having a RHP zero), and for first-order-plus-dead-time systems where the ratio between time constant and dead time is not large. However, for processes that are dominated by nonminimum phase characteristics (dead time and/or RHP zeroes) and not by time constants, the gain-plus-deadtime model is a more useful two-parameter model. It is also easily identified by relay feedback.
Relay Feedback, Controller Tuning, and Controller Implementation If an ideal relay with amplitude d is connected in a feedback loop around an open-loop stable system, the system will start oscillating in a limit cycle. The relay is usually taken to have two positions at equal distance from the chosen steady state and to change position each time the system output passes the steady-state value. The ultimate gain and period of the system are approximately given by (Astriim and Hiigglund, 1988)
K, = 4d/?rh
(4)
P, = P
(5)
where h is the amplitude of the process output and P is
1710 Ind. Eng. Chem. Res., Vol. 33, No. 7 , 1994
Figure 2. Response of first-order-plus-dead-time process given in eq 2 in a relay feedback experiment for various ratios TIL. All the curves shown have the same values of K and L in eq 2.
Figure 4. Input u and output y in an ideal relay feedbackexperiment for the gain-plus-dead-time process G = k' exp(-L's).
also introduced into the procedure by the hysteresis that is used in the relay in order to decrease the sensitivity of the relay to noise.) The response of a gain-plus-dead-time model given by
P
G = kte-L'S
(8)
in a relay-feedback experiment is shown in Figure 4. Also here the dead time is obtained from the period of oscillation L' = PI2 Figure 3. Input u and outputy in an idealrelay feedback experiment for the integrator-plus-dead-time process G = ks-l exp(-Ls).
the period of oscillation. The relay feedback is thus conceptually quite similar to the Ziegler-Nichols ultimate sensitivity experiment, but it has no time-consuming need to find the controller gain needed to bring the system to the verge of instability and using it implies no risk of making the system unstable by using too large a controller gain. Only a reasonable relay amplitude and the sign of the process gain need to be known for the experiment. Instead of identifying the critical gain and period of the system, a simple process model could be identified as discussed above. A relay feedback experiment with a firstorder-plus-dead-time model is shown in Figure 2 for different ratios TIL between the process time constant and dead time. For large values of TIL the response is closely expressed by straight lines in a triangular wave, which is the response of the integrator-plus-dead-time model given in eq 3. For small values of TIL the response is close to a rectangular wave, which is the response of a pure dead-time process. The two parameters k and L in an integrator-plus-deadtime model, eq 3, are obtained from the relay feedback experiment illustrated in Figure 3. The dead time L is obtained from the period of oscillation as L = PI4
(6)
Looking at the initial part of they curve, where it starts out flat and then changes to a slope k when the delayed effect of the step change in the input u from zero to d takes effect, one sees that the slope k is given by
The gain k' is given by the amplitude ratio (10)
An advantage of using a process model is that the controller can be designed by some model-based technique. Chien and Fruehauf (1990) used the IMC approach to derive a PID controller for the integrator-plus-dead-time process model in eq 3. Below we have used the following of Chien and Fruehauf's suggestions for the tuning of SISO PID controllers.
+L TClL+ L2/4 Td = 2TC1+ L Ti = 2T,,
(12)
Here Tcl is a tuning parameter which has to be chosen by the user, and which indicates the speed of response of the closed-loop system (time constant of setpoint response). Tyreus and Luyben (1992)point out that the approach described above can lead to poor control unless care is taken in selectingthe closed-loop time constant Td. Tyreus and Luyben studied PI control for which Chien and Fruehauf suggested the tuning
Ti = 2T,,
Equation 4 gives an error in the order of 15-20% for a first-order-plus-dead-time system with a large time constant, as shown by Li et al. (1991),whereas the identification of L and k according to eqs 6 and 7 is exact, not an approximation. (The approximation involved in the design of the control system is caused by the difference between the model in eq 3 and the real process. An inaccuracy is
(9)
+L
(15)
Tyreus and Luyben state that trial and error is usually necessary in order to specify the closed loop time constant Tclthat will give a reasonable closed-loopdamping. They suggest the following settings for PI control of the integrator-plus-dead-time process of eq 3
K,= 0.487lkL
(16)
Disturbance
r -
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1711 Table 1. Parameters in Models G = (k/s) esp(-Ls) and G =
-+pJh
Controller
Process
k‘exp(-l’s) Identifed in Relay Feedback Simulations for Systems to G4 Described by Eqs 26-29
Y
system
T 1
Figure 6. Feedback system.
Ti = 8.75L
5 20
(17)
These settings can also be expressed by means of the ultimate gain and ultimate frequency and are given by Tyreus and Luyben as K, = Kuj3.22
(18)
Ti = 2.2Pu
(19)
This suggestion can be compared to the more agressive Ziegler-Nichols settings K, = KJ2.2 and Ti = PJ1.2. Tyreus and Luyben (1992) give no recommendation as to how the derivative time should be chosen if PID control is wanted. For the gain-plus-dead-timemodel it is generallythought that derivative action in the feedback controller is not of much use. Deriving PID controller parameters for the model by the IMC approach using the common first-order Pad6 approximation for the dead time results in that the derivative action is zero, meaning that the controller is a PI controller. The following settings are obtained for control of G = k’ exp(-L’s) (Morari and Zafiriou, 1988). K, =
L’ (L’+ 2TJk’
Ti = L‘/2
(20)
for the system in Figure 5. The error e p in the proportional part of the controller is (23)
where the factor b < 1 is chosen to reduce the overshoot and control valve saturation for setpoint changes. Astrdm and Hkigglund (1988) recommend 0.2 5 b I0.3. The error in the derivative part ed is chosen as ed = -y
(24)
By not having the derivative action respond directly to setpoint changes, “derivative kick” is avoided. The error in the integral part must be given as the true control error e=r-y
(25)
in order to avoid steady-state offset after load changes. Below the implementation given by eqs 22-25 is used. In the IMC-based settings, eqs 11-15, 20, and 21, the value b = 1 is used.
Control of SISO Systems Modeled as Integrator-plus-Dead-Timeor Gain-plus-Dead-TimeModels The basic idea of this study-to use either an integratorplus-dead-time model or a gain-plus-dead-time model in
k’
L’
0.62 0.17 0.045 0.77 0.24 0.067 1.5 0.30 0.076 1.1 0.41 0.12
1.5 1.8 1.9 1.8 3.7 7.4 1.4 3.7 7.7 3.4 4.7 5.5
combination with relay feedback-is based on the observation that many chemical engineering processes are dominated by a large time constant and that some are dominated by dead time. We have, however, found the approach quite useful also for other processes. To illustrate this, the folllowingfour process models are used for various values of T:
GI=-Tse++ 1 G, =
e+ (0.57’5 + 1)2
G, =
-5 + 1 (0.52% 1),
G, =
(27)
+
(21)
For the implementation of a PID controller Astr6m and Hiigglund (1988) suggest the following
ep=br-y
1 5 20 1 5 20 1 5 20
k 0.84 0.19 0.049 0.85 0.13 0.018 2.2 0.16 0.020 0.65 0.17 0.044
parametera L 0.75 0.93 0.97 0.90 1.9 3.7 0.69 1.8 3.8 1.7 2.4 2.8
(-5
+ 1)e”
(Ts + l)(s + 1)
(29)
These four models have in turn been used as the process in Figure 5. Replacing the controller by a relay, the ultimate gain and frequency have been determined according to eqs 4 and 5. Also integrator-plus-dead-time models and gain-plusdead-time models have been determined from the relay feedback simulations as is described above in eqs 6,7,9, and 10. Table 1 gives the model parameters obtained. Figure 6 shows two examples of the responses in the relay feedback simulation. For systems with inverse response, eqs 28 and 29, the step response may differ considerably from the step responses of an integrator-plus-dead-time system. The relay responses, however, seem to be reasonably similar as is illustrated in Figure 6. The RHP zero has the effect of increasing the apparent gain k in the model in eq 3, which is a quite natural result considering the inverse response. Otherwise there seem to be no simple relations between the parameters in the models 26-29 on the one hand and those of the models in eqs 3 and 8 on the other. (For the models in eqs 3 and 8 the relation L‘ = 2L. is valid.) The Ziegler-Nichols controller settings have been calculated from the values of the ultimate gain and frequency obtained in the relay feedback experiment and applied to the process in eqs 26 and 29. Simulated responses to step disturbances according to Figure 5 are shown in Figures 7-10. Controller settings have been calculated according to eqs 11-15,20, and 21 from the integrator-plus-dead-time models and the gain-plus-time-delay models in Table 1
1712 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
'
-2
I
I
I
,
- 1
02 r
1
'
I
-0.2
Time
Figure 9. As Figure 7 but for inverse response process given in eq 28.
10
t
40
0
Time
I
0
Y
40
0
I
I
40
0
Time
40
0 Time
Figure 10. As Figure 7 but for inverse response process given in eq 29. 0
40
t
Figure 6. Input u and output y in relay-feedback identification of Gs = (-5 + 1)/(0.5s 1)2 (upper figure) and Gq = (-9 + 1) exp(-s)/ [(20e + l)(s + I)].
+
I
I
20
0
20
0 Time
Time
Figure 7. Response of first-order process in eq 26 to a unit step disturbance according to Figure 5. PID control is shown to the left, PI control to the right. The controllers are tuned according to the Ziegler-Nichols ultimate sensitivity settings (dashed line) determined from a relay feedback experiment, according to the settings given in eqs 11-15 for the integrator-plus-dead-time model (full line) and according to the settings in eqs 20 and 21 for the gain-plus-dead-time model (dash-dot line).
I
I
40
0 lime
0
40 Time
Figure 8. As Figure 7 but for second-order process in eq 27.
and applied to the processes in eqs 26-29. The tuning parameter Tcl has in all cases been chosen as L d 1 0 or L'dlO, respectively-avalue suggested for integrator-plusdead-time models by Tyreus and Luyben (19921, and in this study used also for the gain-plus-dead-time models. A significantly smaller value for Tclcould be used for pure dead-time processes, but that would also decrease the robustness. For example, the general Ziegler-Nichols recommendation for PI control, i.e., K,k' = 0.45, implies Tcl = 0.6L' in eq 20. A general recommendation for processes modeled by pure dead time L' could be to use a value for Tclbetween L' and 3L'. The values for L and L' used are those identified in the relay feedback experi-
ment, given in Table 1. With the choice Tcl = L d l O or L'dlO, respectively, the settings in eqs 14 and 15 for PI control closely resemble the settings in eqs 18 and 19 suggested by Tyreus and Luyben. Simulated responses to step disturbances according to Figure 5 are shown in Figures 7-10. Figures 7-10 illustrate that the approach to identify a system as an integrator-plus-dead-time model to be used for controller design is efficient not only for first-orderplus-dead time processes with a large ratio between time constant and dead time. It works well also for other processes, such as second-order-plus-dead-time processes with equal time constants, and for processes with inverse response (RHP zero). (A number of other processes than those shown here have also been studied.) However, when the nonminimum part of the process becomes dominating (low values for T in Figures 7-10), the response of the controlled system becomes very sluggish with the controller tunings according to eqs 11-15. For such systems the model identified in a relay feedback experiment should preferably be chosen as a gain-plusdead-time model and controller settings should be chosen according to eqs 20 and 21. Then one question remains: If nothing is known a priori, how should the model form be chosen, integrator plus dead time or gain plus dead time? The answer seems to be to choose from the form of the response of the output in the relay experiment, as illustrated in Figures 3 and 4 for the two model types, and in Figure 2 for a number of intermediate cases.
Relay Identification and Control of MIMO Systems The previous discussion forms the basis for a design method for MIMO identification and control by relay feedback and autotuning. Below the approach is illustrated by use of integrator-plus-dead-time models, but the same principles apply if gain-plus-dead-time models or mixed models are used. If all the intput-output relations of the system are described by integrator-plus-dead-time
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1713 1
-5s
.Se
1
-y,
-0.5s
se
1
0
10
5
-5
-0.25s
15
Time
Figure 12. Input u and outputs y of relay feedback experiment shown in Figure 11.
models, i.e., for a n
X
n system given by
the entire system can in principle be identified through n relay feedback experiments. Some a priori information is needed about the system, such as the sign of the gains and order-of-magnitude values for the dead times, but this information is often available, e.g., from engineering knowledge. When the relay is connected over one element in the matrix in eq 23, all the other elements in the same column can be identified from the measured outputs. The k's are given by the output slope, as in Figure 3, and the dead times from the time shift compared to the oscillations of the feedback loop in combination with the dead time determined for the feedback loop. An illustration is given by Figures 11 and 12, which show how the difference in dead times easily (in principle) can be determined. Drifting of the uncontrolled outputs seems to be no significantproblem, because the integrator-plus-dead-time model does not have a steady state in the same sense as has a first order-plus-dead-time system. Based on the identified model in eq 30 a MIMO or multiloop controller can be designed. One attractive approach is to use decoupling (for such processes where decoupling is suitable, of course). The decouplers are
especially simple in the suggested approach since they become gain-plus-delay elements. The loops in the decoupled system can then be identified by the same relay procedure as before and expressed, e.g., by integratorplus-dead-time models. The feedback controllers are then tuned for the SISO loops, e.g., according to the rules treated above. Below this approach is illustrated by two experimental examples. If one wants to use multiloop control (without decoupling) of the interacting system, various approaches can be used. A common procedure is to tune each loop having all other loops open and then detune the loops when they are simultaneously put on automatic. Detuning is usually necessary because the interaction as a rule decreases the stability margin. How much detuning that is necessary is system specific and can be determined by trial and error. This is the basis for a simple procedure we have found useful for iterative autotuning of multiloop controllers for 2 X 2 systems. The procedure goes as follows. First one of the loops is autotuned with the other loop kept open. Then the first loop is detuned, e.g., by decreasing the controller gain and increasing the integral time from, e.g., the Ziegler-Nichols values recommended for PI control. With the first loop on detuned automatic, the second is autotuned and then detuned in the same way as the first. The first loop is then autotuned again but with the second on detuned automatic, and so on. The approach has been found fast and efficient, and only a couple of iterations have been necessary (Friman, 1992). Autotuning of decentralized PID controllers for 2 X 2 systems has recently been treated also by Palmor et al. (1993). The basic idea of their approach is that during the identification phase both controllers are replaced by relays, which generate limit cycles with the same period in both loops. Thus a critical frequency and critical gains of the two loops are obtained. The approach suggested also contains means of varying the relative importance of the two loops. Another way to eliminate the trial-and-error detuning part of the multiloop controller design is to utilize the availability of the process model in eq 30. Such a systematic procedure is the BLT tuning, suggested by Luyben (1986) for multiloop PI tuning and extended by Monica et al. (1988) to include derivative action in the controller as well as weighted detuning. This approach is also used in one of the experimental examples to follow. If one wants a full MIMO controller, it can be designed from the identified model.
Experimental MIMO Example 1: Mixing Tank This process is a mixing tank (Figure 13) previously studied by Haggblom (1991,1993b) and Stromborg (1992). It has two controlled outputs, temperature and level. The two manipulators are a hot water and a cold water inflow. The process could be well controlled by some model-based technique, such as controlling the level by the sum of the inflows and the temperature by their ratio. In order to test and illustrate the discussion above experimentally, we use a multiloop SISO control system where the cold water inlet controls the level and the hot water controls the temperature. It is considered known that the dead times in both loops are roughly of the same size. At the studied steady state (h = 25 cm and T = 25 "C) the time constants are about 3 min (level) and 0.5 min (temperature). The relative gain (RGA) is of the order of 0.5. A relay identification experiment is shown in Figure 14 (first part, i.e. t < 15 min). Quite naturally, the uncon-
1714 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
30-
20.. 7
15
0
30
t [min]
Figure 15. MultiloopPIDcontrolofthemixingtank,usingcontroller settings obtained by BLT-2 tuning baaed on the process model in eq 31. The controller settings are K,1 = 0.40 Vlcm, K d = 0.72 V/OC, Til = Tiz = 1.61 min, and Tdl = T& = 0.30 min, and the controller has been implemented with b = 0.3 in eq 23.
Figure 13. Mixing tank. 30
t
T
'cold
5
[VI
3
25---
__
["CI
-
.d
20..
7t
'wan
rf'cn
30T
71
5
15
0
3
30
t [min] I
1 0
15
30
Figure 14. Experimental relay feedback experiment for process shown in Figure 13.
trolled output is subject to drift. From this experiment, the following process model was identified.
[ $1
1 3.68e"."
= 8[ -1.7ge".%
3.73edL][ 2.04e4.%
]
A U c o ~ (31)
."-"r 7
[cml
20
Au~mm
(In eq 31 the time is expressed in min, the height h is in cm, the temperature Tis in "C,and the control signals Uwld and U ,, are in V.) From the model in eq 31 the BLT-2 approach (Monica et al., 1988)is used for tuning the multiloop PID controller. The resulting control quality is illustrated in Figure 15for the mixing tank. From the process model, eq 31, decouplers can be designed. Choosing a decoupler form with elements in the diagonal equal to unity (Waller, 1974) gives
D=
Figure 16. Multiloop PID control of decoupled mixing tank. Controllersettingsdetermined by relay feedback of decoupled process and eqs 11-13 with T d = 0.63 rnin and b = 1.0.
['+0.88 -'*01] 1
After that the two loops are identified by relay feedback for the decoupled system. Such an experiment with the mixing tank is included in Figure 14 (latter part, i.e., t > 15 min). In this figure it is seen how the two valves simultaneously change in the same direction when the level but not the temperature is to be affected, and in the opposite direction when one wants to keep the level constant. The following identified model for the decoupled system was obtained (off-diagonal elements were put to
20 15
0
30
f [rnin]
Figure 17. As Figure 16 but using Ziegler-Nichols settings for the two (decoupled) loops and b = 0.3.
zero)
G=-[ 1 7.32e4'.% 0
1
(33) 4.61e4'." PID control of the decoupled system is shown in Figure 16 and 17. In Figure 16 the controller settings are given by eqs 11-13 with Tcl chosen to be 0.63 min. In Figure 17 Ziegler-Nichols settings have been used for the two decoupled loops. The results illustrate the common criticism of the Ziegler-Nichols settings, i.e., that they often are too aggressive. s o
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1715 79'7
T
["C]79.5
.
79.3
I
9"5
I
-
I
'4
-
'14 89.5
-
.
AAAr'
V " " " "
I"C1 87.5 1
T
Figure 18. Implementation of the (L,Q structure on the pilot plant distillation column.
In the experiments shown in Figures 15-17 only setpoint changes are illustrated. How the approach handles disturbances is illustrated by the next example.
Experimental MIMO Example 2 Distillation The autotuning approach has also been experimentally applied to two-point control of distillation in two series of experiments. In the first series (Friman, 1992), multiloop PID control of the 2 X 2 temperature control system has been studied for four control structures, i.e., (L,V), (O,W, ( L a ) and (OB),implemented at a pilot plant columnat Ab0 Akademi (Waller et al., 1988;Waller, 1992). Autotuning of one loop at a time with the other loop on detuned automatic as described above is used. The suitable detuning factors are between 1.5 and 3 for the different structures and are obtained by trial and error. In the second series of experiments the identification and controller design methods discussed above have been tested on the pilot column controlled by the (L,V) structure, Figure 18. The sampling time used in the distillation experiments is 0.5 min, the same as in previous experimental studies with the column. Figure 19 shows the relay feedback identification experiment made for the two temperature loops, where T4 is connected to L, and T14 to V, as in Figure 18. Enlarged parts of Figure 19 are shown in Figure 20 for the identification based on the (T4-L) loop and in Figure 21 based on the (Tl4-V) loop. From these experiments the critical frequency and the period of the oscillations were determined, and so were process models of the integrator-plus-dead-time form expressed by eq 30. The differences in dead times in the transfer functions were obtained in a rather crude way by averaging the differences between the corresponding maximum and minimum points on the curves, as shown in Figures 20 and 21. Similarly the constants k in eq 30 were obtained simply by averaging the slopes of the lines connecting the maximum and minimum points. The identified model is given by
(In eq 34 the temperature differences are expressed in time is in min, and flows L and V are in kg/h.) The model in eq 34 is quite similar to models previously obtained, identified by other methods except for the gain OC,
70
.
50
,
[Whl I
t [min]
Figure 19. Relay feedback identification experiment for the two temperature loops of the distillation column.
50
79.6
4
I
3
8
i3
is
23
28
t [min] Figure 20. Enlarged part of Figure 19 for the (T4-L) loop.
in the (2,2) element, which here is significantly larger than would be obtained from previous experiments (Waller et al., 1988; Waller, 1992). Figure 22 shows the system controlled by two SISO PID controllers,tuned by trial and error. (The loopswere tuned separately, keeping the other loop open, by the Ziegler Nichols settings and detuned by decreasing the gains K, and increasing the integral time Ti by a factor 1.5 in both loops.) The controllers were implemented as in eq 22 with b = 0.3. Setpoint changes as well as disturbances in feed flow rate and feed composition are shown. The disturbances and setpoint changes are the same as used by Hliggblom (1993a).
1716 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 79.6
79.7
T
T4
T4
(“CI
[“CI
79.5
79.3
TI 4 [“CI 88.9
4 90
-
I
0
20
V
- 34.
39
49
54
59
t [min]
Figure 21. Enlarged part of Figure 19 for the (TI,-V) loop. 79.7 I
Tq [“CI
40
60
80
100
120
140
160
180
79.5
t [min]
Figure 23. Control of decoupled distillation column. Diagonal controller tuned according to eqs 11-13 with T d = 2.5 min baaed on relay feedback experiments for decoupled column. Decoupler given by eq 35. Setpointchanges and dieturbancesintroducedin the same order and size ae in Figure 22.
79.3
90
Based on the so obtained (diagonal) model, controller settings according to eqs 11-13 were calculated and implemented in the diagonal PID controller. Tcl was chosen to be 2.5 min. Figure 23 shows the response of the controlled system to setpoint changes and disturbances in feed flow rate and feed composition.
1
T
Conclusions 30
1
90
T
I
50
I
0
20
40
60
80
100
120
140
160
180
t [min]
Figure 22. Multiloop PILI control of the distillation column. The temperature feedback controllers were tuned by trial and error. (Ziegler-Nichols rules for SISO loops based on the relay feedback experiments,anddetunedbyafador 1.5.) Inadditiontothesetpoint changes shown the following disturbances were introduced: AF = -10, +20, and -10 kg/h at t about 75,90, and 105 min, respectively, ae well ae Az = +2.5, -5, and +2.6 wt % at t about 120,140, and 160 min, respectively.
Based on the identified model in eq 34 the following decouplers were designed (for our case with 0.5 min sampling time)
D=
[’
1
(35) 0.36e4*& 1 1*40e4.b After implementing the decouplers the two loops (2’4L) and (2’14-V) were again identified through relay feedback.
The approach to use relay feedback to identify dynamic models expressed by integrator-plus-time-delay or gainplus-time-delay form has been found to be a practical and efficient basis for controller design in process control. Chemical engineering processes are often dominated by large time constants, for which the former model is very attractive. In cases where the dead time (or other nonminimum phase characteristics) are dominating, the latter model form should be chosen. Which model form that preferably should be chosen can often be determined from the form of the output response in the relay feedback experiment. Process models for MIMO systems of order R X R can be obtained from n relay feedback experiments. The simple forms of the process models make, e.g., decoupler design, very simple. Controllertunings suitablefor integrator-plus-dead-time models suggested by Chien and Fruehauf (1990) and Tyreus and Luyben (1992) have been found efficient for the SISO case. When the process is dominated by dead time (or other nonminimum phase characteristics), IMC tuning for pure dead-time processes has been found preferable and suitable. The results hold also for MIMO processes made noninteracting by decoupling. For multiloop control of the interacting process, the identified process model makes possible the use of systematic methods, such as BLT tuning.
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1717 The efficiency of the approach has been illustrated by simulation of a number of SISO systems and by two experimental 2 X 2 systems, a laboratory scale mixing process and a pilot plant distillation column.
Acknowledgment The results reported in this paper have been achieved during a long-range research project on multivariable process control. The financial support from Tekes, the Academy of Finland, Nordisk Industrifond, the Neste foundation, and Neste Oy is gratefully acknowledged.
Nomenclature BLT tuning = controller tuning procedure suggested by Luyben (1986) BLT-2 tuning = controller tuning procedure suggested by Monica et al. (1988) IMC = internal model control (Morari and Zdiriou, 1989) (L,V)= distillation control structure (see Figure 18) MIMO = multi-input multi-output PID controller = controller with proportional, integral, and derivative action RHP = right half plane SISO = single-input single-output 2 x 2 system = system with two inputs and two outputs Literature Cited Astrtim, K. J.; Higglund, T. Automatic tuning of simple regulators with specifications on phase and amplitude margins. Autornatica 1984,20,645-651. Astrtim, K. J.; HQglund, T. Automatic Tuning of PZD Controllers; Instrument Society of America: Research Triangle Park, 1988. Astrtim, K. J.; Wittenmark, B. Adaptive Control; Addison-Wesley: Reading, MA, 1989. Astrtim, K. J.; Higglund, T. Practical experiences of adaptive techniques. Proc. ACC 1990,1599. Chiang, R.-C.; Yu, C. C. Monitoring procedure for intelligent control: On-line identification of maximum closed-loop log modulus. Znd. Eng. Chem. Res. 1993,32,9049. Chiang, R.-C.; Shen, S.-H.; Yu, C.4. Derivation of transfer function from relay feedback systems. Znd.Eng. Chem.Res. 1992,31,855860. Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve performance. Chem. Eng. B o g . 1990 Oct, 33-41. Friman, M. An experimental investigation of automatic controller tuning for distillation columns. MSc. Thesis (in Swedish), Ab0 Akademi, Abo, Finland, 1992. Higgblom, K. E.'Conventional and model based control of a mixing tank"; Report 91-14,Process Control Laboratory, Ab0 Akademi, Abo, Finland, 1991. Higgblom, K. E. Combined internal model and inferential control of a pilot-scale distillation column. Proc. ECC 1993s,643-648. Higgblom, K. E. Experimental comparison of conventional and nonlinear model-based control of amixingtank. Znd. Eng. Chem. Res. 1993b,32,2653-2661.
Hammarstrtim, L. G.; Waller, K. V.; Fagewik, K. C. On modeling accuracy for multivariable distillation control. Chem. Eng. Commun. 1992,19,77-90. Hang, C. C.; AstrBm, K. J.; Ho, W. K. Refinements of the ZieglerNichols tuning formula. ZEE Proc. D 1991,138(No. 2,March), 111-118. Hang, C. C.; Astrtim, K. J.; Ho, W. K.Relay auto-tuning in the presence of static load disturbances. Autornatica 1993,29,563-564. Li, W.; Eskinat, E.; Luyben, W. L. An improved autotune identification method. Znd. Eng. Chem. Res. 1991,30,1530-1541. Luyben, W. L. A simple method for tuning SISO controllers in multivariable systems. Znd. Eng. Chem. Process Des. Deu. 1986, 25,654. Luyben, W. L. Derivation of transfer functions for highly nonlinear distillation columns. Znd. Eng. Chem. Res. 1987,26,2490-2495. McNeill, G. A.; Sacks,J. D. High performance column control. Chem. Eng. Prog. 1969,65 (3),33-39. Monica, T. J.; Yu, C.-C.; Luyben, W. L. Improved multiloop singleinput/single output (SISO) controllers for multivariable processes. Znd. Eng. Chem. Res. 1988,27,969-973. Morari, M.; Zafkiou, E. Robust Process Control; Prentice-Hall International: Englewood Cliffs, NJ, 1989. Palmor, Z. J.; Halevi, Y.;Krasney, N. Automatic tuning of decentralized PID controllers for TIT0 processes. Proc. ZFAC World Congr. 1993,2,311-314. Sandelin, P. M.; Hiiggblom, K. E.; Waller, K. V. A disturbance sensitivity parameter and its application to distillation control. Znd. Eng. Chem. Res. 1991,29,1182-1186. Shei, T. S. A method for closed loop automatic tuning of PID controllers. Automatica 1992,28,587-591. Strtimborg, K.B.'Multivariable self-tuning PID control of a mixing tank"; Report 92-7. Process Control Laboratory, Abo Akademi, Abo, Finland, 1992. Toijala (Waller), K.;Fagervik, K. A digital simulation study of twopoint control of distillation columns. Kem. Teollisuw 1972,29, 5-16. Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/ dead time processes. Ind. Eng. Chem. Res. 1992,31,2625-2628. Waller, K.V. Decoupling in distillation. MChE J. 1974,20,592594. Waller, K. V. Experimental comparison of control structures. In Practical distillation control; Luyben, W. L., Ed.; Van Nostrand Reinhold New York, 1992;pp 313-330. Waller, K. V.; Finnerman, D. H.; Sandelin, P. M.; Higgblom, K. E.; Gustafsson, S. E. An experimental comparison of four control structures for two-point control of distillation. Znd. Eng. Chem. Res. 1988,27,624-630. Westerlund, T.; Toivonen, H.; Nyman, K. E. Stochastic modelling and self-tuning control of a continuous cement raw material mixing system. Model. Zdentif. Control 1980,1, 17-37. Ziegler, J. G.;Nichols, N. B. Optimum settings for automatic controllers. Trans. ASME 1942,64,759-765.
Received for reuiew October 25, 1993 Reuised manuscript received March 25, 1994 Accepted April 1,1994O 0
Abstract published in Advance ACSAbstracts, May 15,1994.
