ARTICLE pubs.acs.org/IECR
Proportional-Integral-Derivative Tuning for Servo/Regulation Control Operation for Unstable and Integrating Processes O. Arrieta,*,† R. Vilanova,*,† and A. Visioli*,‡ † ‡
Departament de Telecomunicacio i d’Enginyeria de Sistemes, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Dipartimento di Ingegneria dell’Informazione, Facolta di Ingegneria, Universita degli Studi di Brescia, Via Branze 38, 25213 Brescia, Italy ABSTRACT: The control of stable processes using PI/PID controllers is by now a well established and understood problem (even if new methods and approaches are continuously appearing). However, when the process has integrating or unstable characteristics the problem becomes much more difficult to manage. Several proposals have appeared in the literature presenting different approaches to tackle the problem. The procedures that have been proposed heavily concentrate on tunings for servo or regulation operation. A common drawback of such approaches is the high loss of performance when the other operation mode is used. In this paper, an approach for providing a unique tuning is presented. It combines tunings for both operation modes in such a way that the performance degradation is traded-off.
’ INTRODUCTION Even in a decade where advanced control algorithms, mostly based on some kind of optimization procedure, have achieved a high degree of maturity, proportional-integral-derivative (PID) controllers are still widely used in the process industries. Their popularity is due to their simplicity—they only have three parameters—and to the satisfactory control performance shown for usual processes. However, the three adjustable PID controller parameters should be tuned appropriately. The problem of appropriate tuning has received attention from a wide spectrum of perspectives during the last decades.1-4 However, much of this effort has been concentrated on the application to stable systems, while quite a few of the important chemical processing units in industrial and chemical practices are open-loop unstable processes that are known to be difficult to control, especially when there is a time delay. Examples of these cases include continuous stirred tank reactors, polymerization reactors, and bioreactors which are inherently open-loop unstable by design.5 Clearly, the tuning of controllers to stabilize these processes and to impart adequate disturbance rejection is critical. Moreover, integrating processes are very common in process industries and many researchers have suggested that, for the purpose of designing a controller, a considerable number of chemical processes could be modeled using an integrating process with time delay. Consequently, there has been much interest in the literature in the tuning of industrially standard PID controllers for open-loop unstable systems, as well as for integrating processes. In fact, several papers can be found in the literature that deal with the tuning of unstable6-10 and integrating processes.6,7,11-14 There is, however, a common problem with the tuning of PI/PID controllers for such systems: the tunings are usually devoted to the servo or regulation operation and may exhibit a significant performance degradation when operating on the tuning mode for which they were not designed. This is also observed when operating with stable systems and becomes a serious problem for r 2011 American Chemical Society
unstable and integrating processes. A simple look at the existing literature shows that the performance is highly dependent on using the appropriate tuning mode. O’Dwyer15 presents a collection of tuning rules for PID controllers for stable, unstable, and integrating processes. On the basis of this observation, the purpose of this paper is to provide an alternative way of addressing the tuning of unstable and integral processes in order to alleviate the aforementioned situation and to provide a better overall performance. The approach constitutes an extension of the method presented in ref 16 for stable systems. As in ref 16 the idea is to find an intermediate tuning for the controller that improves the overall performance of the system, considered as a trade-off between servo and regulation operation modes. The settings are determined from the combination of the optimal ones for set-point and load-disturbance, presented in ref 7 and considering the balance between the importance of each one of the operation modes for the control system (servo or regulation). The optimization is here performed using genetic algorithms.17 It is worth stressing that the purpose of the paper is to generate the so-called trade-off tuning on the basis of previously existing optimal tunings rather than providing a new set of tuning rules. In other words, the method aims to minimize the degradation with respect to the optimal behavior provided by the extreme (servo and regulation) tunings. The paper is organized as follows. Next section introduces the control system configuration, the general problem formulation, as well as some related concepts. Then, it is presented the general approach in which we look for an intermediate tuning between the parameters of both operation modes in such a way that the weighted performance degradation (WPD) is minimized; the Received: May 3, 2010 Accepted: January 10, 2011 Revised: December 30, 2010 Published: February 22, 2011 3327
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Table 1. Tuning Rules for Optimal ISE Set-Point and LoadDisturbance for Unstable Processes7 PID parameter
Figure 1. The considered feedback control system.
results are generalized in terms of tuning procedures for unstable and integrating processes. Some comparative examples are shown, and the paper ends with the conclusions section.
’ MATERIALS AND METHODS This section presents the control system configuration, as well as the problem formulation and general aspects. Control System Configuration. We consider the unity-feedback system shown in Figure 1, where P is the process and C is the controller. In this system, r is the set-point, u is the controller output signal, d is the load-disturbance, e is the control error (e = r - y) and y is the controlled process variable. The closed-loop system output, y, to a change in its inputs, r and d is given by
The system can operate in two different modes, known as servo control or regulatory control. In the first case, the control objective is to provide a good tracking of the signal reference r, whereas in the second case the objective is to maintain the output variable at the desired value, despite possible disturbances in d. For the design of the control system, it is necessary to consider both operation modes; however depending on the controller’s structure (e.g., 1-DoF PID), it is not always possible to specify different performance behaviors for changes in the set-point and load-disturbances. If we consider an unstable system, the process P is assumed to be modeled by PðsÞ ¼
K -Ls e Ts - 1
ð2Þ
or if we have an integrating process, the model will be PðsÞ ¼
K -Ls e s
ð3Þ
In both cases, K is the process gain and L is the dead-time. For the unstable system (2), T is the time constant. These models are commonly used because they are capable of satisfactorily modeling the dynamics of unstable and integrating processes. Let us consider for controller C, the ideal one-degree-offreedom (1-DoF) PID controller, that is considered as 1 ð4Þ CðsÞ ¼ Kp 1 þ þ Td s Ti s where Kp is the proportional gain and Ti and Td are the integral and derivative time constants, respectively. Performance Degradation of the Control System. If the control-loop has always to operate on one of the two possible
set-point -0.92
Kp
1.32/K(L/T)
Ti Td
0.47
4.00(L/T) T 3.78T(1-0.84(L/T)-0.02)/(L/T)-0.95
load-disturbance 1.37/K(L/T)-1 2.42(L/T)1.18T 0.60(L/T)T
Table 2. Tuning Rules for Optimal ISE Set-Point and LoadDisturbance for Integrating Processes7 PID parameter
set-point
load-disturbance
Kp
1.03/KL
1.37/KL
Ti
-
1.49L
Td
0.49L
0.59L
operation modes (servo or regulator) the tuning choice will be clear. However, when both situations occur, the most appropriate controller settings may not be so evident. The performance degradation concept for set-point and loaddisturbance tunings depending on the operation mode was previously presented and developed in ref 18 for stable systems. Therein, the performance of the control system is measured in terms of a performance index that takes into account the possibility of an operation mode different from the selected one. Performance will not be optimal for both situations. The performance degradation measure helps in the evaluation of the loss of performance with respect to their optimal value. Tuning Formulas for Unstable and Integrating Processes. The analysis presented here is an extension of the performance degradation idea, adapting all the aspects and considerations to the cases of unstable and integrating Rsystems. We rely on the 2 integral square error (ISE) criteria, J = ¥ 0 e(t) dt, which is one of the most well-known and most often used;19 however, the general analysis could be developed in terms of any other performance criterion. Table 1 and Table 2 show the tuning formulas for unstable and integrating systems, respectively, where the resulting settings are optimal to the ISE criteria.
’ GENERAL APPROACH FOR SERVO/REGULATION OPERATION Controller’s Search Space. The tuning approaches presented in the previous section can be considered extremal situations. The controller settings are obtained by considering exclusively one mode of operation. This may generate poor performance if the non-considered situation happens. This fact suggests to analyze if, by loosing some degree of optimality with respect to the tuning mode, the performance degradation can be reduced when the operation is different to the selected one for tuning. On the basis of this observation we suggest to look for an intermediate controller. To define this exploration, we need to define the search-space and the overall performance degradation index to be minimized. The search of the controller settings that provide a trade-off performance for both operating modes could be stated in terms of a completely new optimization procedure. However, we would like to take advantage of the tuning formulas, in order to keep the procedure, as well as the resulting controller expression, in the 3328
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tions is possible by introducing weighting factors associated to each operating mode16 as WPDðγ;RÞ ¼ RPDld ðγÞ þ ð1 - RÞPDsp ðγÞ Figure 2. γ h-Tuning procedure for the search of the intermediate controller.
similar simple terms. Therefore, the resulting controller settings could be considered as an extension of the (extreme and) optimal ones. On this basis, we define a controller settings family parametrized in terms of a vector as γ ¼ ½γ1 ,γ2 ,γ3
ð5Þ
where γi is a variable for each controller parameter (Kp, Ti, Td) that allows searching for the intermediate tuning. The values for this factor are restricted to γi ∈ [0,1] i = 1,2,3. Figure 2 shows graphically the procedure and the application for the 1-DoF PID controller tuning. Criterion of Overall Operating Mode. To include a consideration about the operation mode of the system, it is necessary to redefine the usual form of the ISE performance index as Z ¥ Jx ðzÞ ¼ eðt, x, zÞ2 dt ð6Þ
that we call weighted performance degradation (WPD) index, where R ∈ [0,1] is the weight factor and indicates which of the two possible operation modes is preferred or more important. One way to express the importance between both operation modes could be the total time that the system operates in each one of them. For example, for a system that operates 75% of the time as a regulator (or vice versa, 25% as a servo), it is R = 0.75. However, the R parameter allows one to make a more general choice for the preference of the system operation (not only taking into account the time for each operation mode). The intermediate tuning will be determined by proper selection of γ h = [γ1,γ2,γ3]. This choice will correspond to the solution of the following optimization problem, γop :¼ ½γ1op ,γ2op ,γ3op ¼ arg½min WPDðγ;RÞ γ
PDld(γ h) will represent the performance degradation of the γ htuning on regulation operating mode: J ðγÞ - J ðldÞ ld ld ð8Þ PDld ðγÞ ¼ Jld ðldÞ From these performance degradation definitions, the overall performance degradation is introduced and interpreted as a function of γh. There may be different ways to define the PD(γ h) function, depending on the importance associated to every operating mode (e.g., applying weighting factors to each component). However, every definition must satisfy the following contour constraints ( PDld ðspÞ f or γ ¼ ½0 , 0 , 0 PDðγÞ ¼ PDsp ðldÞ f or γ ¼ ½1 , 1 , 1 A balanced reduction of PD(γ h) from both performance degrada-
ð10Þ
It is obvious that R = 0 means WPDðγ;0Þ ¼ PDsp ðγÞ
0
where x denotes the operating mode of the control system and z is the selected operating mode for tuning, that is, the tuning mode. Thus, we have x ∈ {sp,ld} and z ∈ {sp,ld}, where sp states for set-point (servo) tuning and ld for load-disturbance (regulator) tuning. Performance degradation, PDx(z), will be associated to the tuning mode, z, and tested on the opposite, operating mode, x. Now, for every combination of γh the performance degradation needs to be measured with respect to both operating modes (because the corresponding γ h-tuning does not necessarily corresponds to an operating mode). Hence PDsp(γ h) will represent the performance degradation of the γ htuning on servo operating mode: J ðγÞ - J ðspÞ sp sp ð7Þ PDsp ðγÞ ¼ Jsp ðspÞ
ð9Þ
ð11Þ
and of course the γ hop that minimizes the performance degradation for servo operation mode (11), is the one that corresponds to the set-point tuning (γ h = [0,0,0]). On the other side, R = 1 is equivalent to WPDðγ;1Þ ¼ PDld ðγÞ
ð12Þ
and the tuning that minimizes the performance degradation for regulation operation (12) is the load-disturbance tuning that equals to γ h = [1,1,1]. The optimal values (10) give a tuning formula that provides a worse performance than the optimal settings operating in the same way, but also a lower degradation in the performance when the operating mode is different from the tuning mode. Remark: It is important to note that the presented procedure has just considered the performance with respect to the proposed performance degradation index. Other closed-loop characteristics such as control effort or control system robustness are not taken into account explicitly. They can be taken into account implicitly by considering different performance indexes.7 Indeed, it is clear that in order to include such characteristics into consideration they need to be part of the original extreme tunings. Optimization. To provide the possibility to specify any possible combination between both operation modes, the index (9), with an appropriated weight factor R and subjected to the optimization (10), gives the suitable γi values that provide the PID tuning according to the selected controller’s family. However, from a more practical point of view it is unusual and very difficult to say for example, that the regulation mode, in a control system, has the 63% of the importance (that means 37% for the servo). As a consequence, we can establish a categorization in order to make the analysis simpler and also to help the choice of the weight factor. Therefore, depending on the operation for the control system, we can identify the following general cases: i. operation only as a servo that means R = 0 3329
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ii. operation only as a regulator that means R = 1 iii. same importance for both system operation modes, servo and regulation, that is equivalent to R = 0.50 iv. more importance for the servo than the regulation operation, that can be expressed by R = 0.25 v. more importance for regulator than servo, that can be indicated as R = 0.75 This broad classification allows a qualitative specification of the control system operation. Here, the optimization was performed using genetic algorithms,17 taking problem (10) as the fitness function. The implementation was using MATLAB 7.6.0(R2008a) for a population size of 20 and a maximum number of generations of 50. The optimal solution was found for R = {0.25,0.50,0.75}. As we said before for R = {0,1}, as extreme situations, the optimal tunings are the related to set-point and load-disturbance.
’ TUNING RULES FOR UNSTABLE PROCESSES By following the explained general procedure in the section above, for the unstable processes, it can be said that the controller settings family [Kp(γ1),Ti(γ2),Td(γ3)] will be generated by a linear evolution of the parameters from the setpoint tuning to the load-disturbance one and the other way
constant
a
b
γ1
0.544 -1.631
γ2
0.807
γ3
0.660 -0.019
R = 0.50 c
a
b
2.194 0.629 -0.801
0.100 -1.491 0.787 0.293 0.718
Kp ðγ1 Þ ¼ γ1 Kpld þ ð1 - γ1 ÞKpsp sp
Ti ðγ2 Þ ¼ γ2 Tild þ ð1 - γ2 ÞTi sp Td ðγ3 Þ ¼ γ3 Tdld þ ð1 - γ3 ÞTd
To pursue the previous idea, by repeating the problem optimization posed in eq 10 for the three weighting factors and different values of the normalized dead-time τ = L/T, we can find an optimal set for each γi parameter. For each one of these groups, it is possible to approximate a function to determine a general procedure that allows the user to find the suitable values for the γi’s that provide the best intermediate tuning. Results are fitted to the general expression as γi ðτÞ ¼ a þ bτ þ cτ2
a
ð14Þ
where a, b, and c are given in Table 3, according to the weighting factor R and for each γi. Equation 14 for each γi along with the settings (13) provide what we call here γhuR-tuning for unstable processes offering a weighted servo/regulation operation. It is worth to note that the validity of the provided settings is determined by the range of application of the original extreme optimal tunings: τ e 0.7.
Kp
Ti
Td
set-point (sp)
5.803
1.877
0.109
load-disturbance (ld)
6.850
0.362
0.120
u γ hR=0.25-tuning u γ hR=0.50-tuning
6.123
0.715
0.116
6.336
0.526
0.117
6.547
0.442
0.117
tuning
R = 0.75 c
ð13Þ
Table 4. Unstable Process P1—PID Controller Parameters
u Table 3. γ hR-Tuning Settings for Unstable Systems
R = 0.25
around. Therefore,
b
c
1.009 0.711 0.061 -0.324
1.026 -2.513 0.687 2.173 -4.026
u γ hR=0.75-tuning
0.268 -0.580 0.547 1.265 -2.116
Figure 3. Unstable process: servo and regulation control responses for system P1. 3330
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Table 5. Unstable Process: PD and WPD Values for the System P1 and the Improvement Obtained with γhuR-Tuning tuning
PDsp
set-point (sp)
PDld 1.8829
load-disturbance (ld) 0.6788 u γ hR=0.25-tuning u γ hR=0.50-tuning u γ hR=0.75-tuning
0.1207
0.3190
0.2524
0.0993
0.4201
0.0088
i Table 6. γ hR-Tuning Values for Integrating Systems
WPDR=0.25 WPDR=0.50 WPDR=0.75
γ1
0.5591
0.7064
0.8274
1.4122
γ2
0.3906
0.5702
0.7532
0.5091
0.3394
0.1697
γ3
0.5903
0.7421
0.8731
0.1703 0.1759
Then, the original PID parameters can be obtained by applying an inverse concept of (16), as
0.1116
82.21% (ld) 83.06% (sp) 63.82% (sp)
u γ hR=0.50-tuning
62.81% (ld) 94.73% (sp)
Kp ¼ Kc ,
66.55% (ld) 81.32% (sp) 48.18% (ld) 38.10% (ld) 99.54% (sp)
92.10% (sp) 34.23% (ld)
Illustrative example. Consider the following unstable system represented by
1 -0:2s e s-1
ð15Þ
u The application of the ISE tuning formulas,7 as well as, the γ h Rtuning, provides the PID parameters showing in Table 4. Figure 3 shows the control system performance for the two possible operation modes for the above tuning methods. It can be seen that the proposed γhuR-tuning gives lower performance than the optimum settings when the system operates in the same way as it was tuned. However, higher performance can be obtained for the whole system operation (regulatory-control and servo-control), when the intermediate controller is used. Table 5 shows the PD and WPD indices and the improvement, u in percentage, that can be achieved for each case of the γ hR-tuning with respect to the extreme tunings (set-point and loaddisturbance). Tuning Rules for Integrating Processes. Now, we analyze the case for integrating systems represented as in eq 3. For this case, it must be taken into account that the method presented by Visioli7 uses a PD (proportional-derivative) controller, when the expected operation for the system is servo-control. This kind of controller could be seen as a PID controller with the integral part disabled, that is equivalent to Ti f ¥. Regarding this consideration, the generation of the controller’s parameters family as it has been done in eq 13 has the difficulty of making a transition in the integral time between Tld i and infinite. To adjust the general procedure considering the foregoing concerns, we obtain the PID controller’s gains as
Kc ¼ Kp ,
R = 0.75
0.9414
u γ hR=0.25-tuning
P1 ðsÞ ¼
R = 0.50
0.4707
Improvement in % (respect to)
u γ hR=0.75-tuning
R = 0.25
Ki ¼ Kp =Ti , Kd ¼ Kp Td
Ti ¼ Kc =Ki ,
Td ¼ Kd =Kc
ð18Þ
With this change, it is possible to achieve a suitable easy transition in the PID parameters, maintaining the idea of γ h ∈ [0,1]. Once again, if we repeat the optimization problem posed in eq 10 for the three weighting factors and different values of dead-time L, we can find an optimal set for each γi parameter. In this case, unlike the case of unstable systems, the optimal values are practically constant, and, for that reason, the method is approximated with fixed values for each γi. Table 6 gives the corresponding constant values depending only on the weighting factor R. Summarizing, for integrating systems, the suitable values for γh along with the settings eq 17 provide the γhiR-tuning for weighted servo/regulation operation. Illustrative Example. Consider the following integrating process represented by 0:0506 -6s e ð19Þ P2 ðsÞ ¼ s Table 7 shows the PID controller parameters for the system (19) using the ISE method7 and the proposed γhiR-tuning with R = {0.25,0.50,0.75}. Figure 4 shows the control system Table 7. Integrating Process P2: PID Controller Parameters Kp
tuning
Ti
Td
-
set-point (sp)
3.393
load-disturbance (ld)
4.513
8.940
3.540
γhiR=0.25-tuning γhiR=0.50-tuning
4.019
20.384
3.363
4.184
14.536
3.448
4.319
11.361
3.552
γhiR=0.75-tuning
2.940
Table 8. Integrating Process: PD and WPD Values for the i System P2 and the Improvement Obtained with γ hR-Tuning tuning
PDsp
PDld ¥
set-point (sp) load-disturbance (ld) 0.9742 i γ hR=0.25-tuning i γ hR=0.50-tuning i γ hR=0.75-tuning
ð16Þ
WPDR=0.25
WPDR=0.50
WPDR=0.75
¥
¥
¥
0.7307
0.4871
0.2436
0.2810
0.4611 0.3260
0.4443
0.1682
0.6393
0.0468
0.3063 0.1950
Improvement in % (respect to)
after that, the controller’s family is generated with these gains according to
i γ hR=0.25-tuning
71.16% (ld) ¥ (sp) ¥ (sp)
i γ hR=0.50-tuning
54.39% (ld) ¥ (sp)
i γ hR=0.75-tuning
34.38% (ld) ¥ (sp)
55.38%(ld)
Kc ðγ1 Þ ¼ γ1 Kcld þ ð1 - γ1 ÞKcsp Ki ðγ2 Þ ¼ γ2 Kild sp Kd ðγ3 Þ ¼ γ3 Kdld þ ð1 - γ3 ÞKd
¥ (sp) 37.12% (ld)
ð17Þ
¥ (sp) 19.95% (ld)
3331
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Figure 4. Integrating process: servo and regulation control responses for system P2.
output and control variables for the servo and regulation operation modes. It can be confirmed that the γhiR-tuning gives a better performance when the system operates in both servo and regulation modes. The set-point tuning operating in regulation mode needs special attention because, being the controller of PD type (without integral action), a state error is expected and consequently the performance degradation is infinite. Table 8 shows the PD and WPD indices and the improvement that can be achieved for each case of the γhiR-tuning.
’ COMPARATIVE STUDY In this section the proposed γ hR-tuning method with R = 0.50 (balanced servo and regulation operation) is compared with other well-known PID tuning methods for unstable and integrating systems. Unstable Process Example. Let us consider the first-orderdelayed-unstable-process (FODUP) with the following transfer function 4 e-2s ð20Þ P3 ðsÞ ¼ 4s - 1 Table 9 shows the PID parameters obtained with different tuning methods for system (20). Also in Figure 5 it is possible to see the process outputs and control variables for the servo and regulation cases. Table 9. Unstable Process P3: PID Controller Parameters tuning γhuR=0.50-tuning 10
Kp
Ti
Td
0.654
6.662
1.188
Panda (2009) Sree et al. (2004)8
0.653 0.571
10.420 11.122
0.908 1.025
Lee et al. (2000)6 (λ = L)
0.606
11.732
0.840
Table 10. Unstable Process: Performance and Performance Degradation Indices for System P3 Jsp
tuning
Jld
PDspa
PDldb WPDR=0.50
γhuR=0.50-tuning Panda (2009)10
8.9776 26.3352 0.0837 0.0728 9.2178 34.3572 0.1127 0.3997
0.0783 0.2562
Sree et al. (2004)8
9.4747 46.7897 0.1437 0.9061
0.5249
Lee et al. (2000)6 (λ = L) 9.9005 44.4132 0.1951 0.8093
0.5022
a
Calculated using eq 7 with Jsp(sp) = 8.2839. b Calculated using eq 8 with Jld (ld) = 24.5470.
Table 10 gives the performance criteria (6) for servo (Jsp) and regulation (Jld) operation modes as well as the associated performance degradation indices (7-9) for each tuning. All the values confirm the fact that, in global terms, when both operating modes could appear and take into account the importance that the control-loop is operating in servo or regulation mode, the proposed γhuR-tuning is the best choice to tune the PID controller in order to get less performance degradations. Integrating Process Example. A distillation column separates a small amount of a low-boiling material from the final product. This technique is very common in chemical processes for the separation of mixed fluids. The bottom level of the distillation column is controlled by adjusting the steam flow rate. The process for the level control system is usually represented by an integrating model as13 0:2 -7:4s e P4 ðsÞ ¼ ð21Þ s The PID controller parameters, for the different tuning methods, are shown in Table 11, and in Figure 6 there are the process outputs and control variables for system (21) operating in servo and regulation modes. 3332
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Figure 5. Unstable process: servo and regulation control responses for system P3.
Figure 6. Integrating process: servo and regulation control responses for system P4.
In Table 12, there are the performance criteria (6) for servo (Jsp) and regulation (Jld) operation modes, as well as the associated performance degradation indices (7-9) for each tuning.
From all the data, it is possible to see that even if the servo performance of the proposed tuning is lower than the provided one for the other tunings, the general behavior 3333
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Table 11. Integrating Process P4: PID Controller Parameters Kp
tuning
Ti
Td
MICIT and CONICIT of the Government of the Republic of Costa Rica is greatly appreciated.
γhiR=0.50-tuning Ali and Majhi (2010)13
0.858
17.928
4.253
’ REFERENCES
0.696
23.458
3.626
Chidambaram and Sree (2003)12
0.834
33.300
3.330
Lee et al. (2000)6 (λ = L)
0.675
27.099
2.620
(1) Ziegler, J.; Nichols, N. Optimum settings for automatic controllers. ASME Trans. 1942, 759–768. (2) Åstr€om, K.; H€agglund, T. Revisiting the Ziegler-Nichols step response method for PID control. J. Process Control 2004, 14, 635–650. (3) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291–309. (4) Kristiansson, B.; Lennartson, B. Evaluation and simple tuning of PID controllers with high frequency robustness. J. Process Control 2006, 16, 91–102. (5) Sree, R. P.; Chidambaram, M. Control of Unstable Systems; Alpha Science International, Ltd.: UK, 2006. (6) Lee, Y.; Lee, J.; Park, S. PID controller tuning for integrating and unstable processes with time delay. Chem. Eng. Sci. 2000, 55, 3481–3493. (7) Visioli, A. Optimal tuning of PID controllers for integral and unstable processes. IEE Proc. Control Theory Appl. 2001, 148, 180–184. (8) Sree, R. P.; Srinivas, M. N.; Chidambaram, M. A simple method of tuning PID controllers for stable and unstable FOPTD systems. Comput. Chem. Eng. 2004, 28, 2201–2218. (9) Vivek, S.; Chidambaram, M. An improved relay auto tuning of PID controllers for unstable FOPTD systems. Comput. Chem. Eng. 2005, 29, 2060–2068. (10) Panda, R. C. Synthesis of PID controller for unstable and integrating processes. Chem. Eng. Sci. 2009, 64, 2807–2816. (11) Chen, D.; Seborg, D. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 2002, 41, 4807– 4822. (12) Chidambaram, M.; Sree, R. P. A simple method of tuning PID controllers for integrator/deadtime processes. Comput. Chem. Eng. 2003, 27, 211–215. (13) Ali, A.; Majhi, S. PID controller tuning for integrating processes. ISA Trans. 2010, 49, 70–78. (14) Visioli, A.; Zhong, Q. Control of Integral Processes with Dead Time; Springer Verlag Advances in Industrial Control Series; Springer Verlag: London, 2011. (15) O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules; Imperial College Press: London, UK, 2003. (16) Arrieta, O.; Visioli, A.; Vilanova, R. PID autotuning for weighted servo/regulation control operation. J. Process Control 2010, 20, 472–480. (17) Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: Cambridge, MA, 1998. (18) Arrieta, O.; Vilanova, R. Performance degradation analysis of controller tuning modes: Application to an optimal PID tuning. Int. J. Innovative Comput. Informat. Control 2010, 6, 4719–4729. (19) Åstr€om, K.; H€agglund, T. Advanced PID Control; ISA: Research Triangle Park, NC, 2006.
Table 12. Integrating process: Performance and Performance Degradation Indices for System P4 tuning
Jsp
Jld
PDspa
PDldb
WPDR=0.50
γhiR=0.50-tuning
14.4299 22.4365 0.4443 0.1684
0.3063
Ali and Majhi (2010)13 13.0946 36.8297 0.3106 0.9179
0.6142
Chidambaram and
12.3686 33.7810 0.2380 0.7591
0.4985
13.3858 44.9949 0.3398 1.3430
0.8414
Sree (2003)12 Lee et al. (2000)6 (λ = L) a Calculated using eq 7 with Jsp (sp) = 9.9912. b Calculated using eq 8 with Jld (ld) = 19.2036.
(taking into account both operation modes) of the control system, tuned with γhiR-tuning, is better. This means achieving the best value for the weighted performance degradation index (WPD).
’ CONCLUSIONS Methods for tuning PID controllers for unstable and integrating processes have been actively studied in the last years. These procedures are usually related to servo or regulation control problems; however, in process control it is very usual to have changes in the set-point as well as in the disturbance. When the controller operates in a different mode from the one it was designed for, a performance degradation is expected. Tuning formulas for unstable and integrating processes, depending on the importance given to the system operations in servo and regulation modes, are presented in this paper as the main contribution. It is a novel feature that allows the user to select the tuning according to a general qualitative specification of the control system operation and to achieve in this context a reduction in the degradation of the performance. Results are given for PID controllers, in order to get results closer to industrial applications. Even if the results were presented and exemplified using the ISE performance criteria, it could be possible to reproduce the described general procedure for other performance objectives. Further, the developed tuning formulas can be applied easily in an automatic tuning context. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected];
[email protected];
[email protected].
’ ACKNOWLEDGMENT This work has received financial support from the Spanish CICYT program under Grant DPI2007-63356. Also, the financial support from the University of Costa Rica and from the 3334
dx.doi.org/10.1021/ie101012z |Ind. Eng. Chem. Res. 2011, 50, 3327–3334
