1. ISA Transactions 40 (2001) 341±351 www.elsevier.com/locate/isatransGuide lines for the tuning and the evaluation of decentralized and decoupling controllers for processes with recirculation Dominique Pomerleau a,*, Andre Pomerleau b Â a Âs Âbec, Canada, J3H 6C3 Breton, Banville et associe s.e.n.c., 375 Boul. Laurier, Mont-St-Hilaire, Que b Á Ârale), Department of Electrical and Computer GRAIIM (Groupe de recherche sur les applications de linformatique a lindustrie mine Â Engineering, Universite Laval, Ste-Foy, Que Âbec, Canada, G1K 7P4Abstract This paper gives guidelines for the pairing, the time response speci®cation, and the tuning for processes with recir-culation when decentralized controllers are used. This selection is based on the condition number, which is an indicatorof the process directionality, and on the generalized dynamic relative gain (GDRG), which is a measure of the inter-action. Simple tuning rules are developed and results are compared to algebraic controllers with decouplers. Perfor-mances are evaluated for set-point changes as well as disturbance rejection using the generalized step response (GSR).The GSR gives a 3D graphic of the system response as a function of the input direction. # 2001 Published by ElsevierScience Ltd. All rights reserved.Keywords: Pairing; PID tuning; Decentralized control1. Introduction main conclusion is a decrease in the understanding of the circuit. This leads to important problems in Mineralurgical and chemical industries have a operating the process in manual. A lack of propermultitude of multivariable processes which, for choice of control structure and ecient tuningreasons of eectiveness, have circulating loads. methods are also major reasons. The addition ofBlakey et al.  made a study of the advantage of circulating loads creates zeros in the processrecirculating loads on a ¯otation circuit. They transfer functions and requires tuning rules takingdemonstrated that a signi®cantly higher grade± into account these zeros . It is thus importantrecovery relationship is possible for rougher-sca- to be able to anticipate the eect of the open-venger circuit designs that incorporate circulating loop system characteristics on the closed-looploads. A recent trend in Canadian mineral industry, system response and to develop simple rules forthough, has been the reduction of the number of the tuning of controller for such multivariablerecirculating loads in processing ¯ow sheet design. processes.This philosophy results from diculties observed Good tuning of decentralized PI controllers forin day-to-day plant operability. Stowe  and multivariable processes is relatively complex. InEdwards and Flinto  discussed the operation particular, the design of single-input single-outputproblems of ¯otation circuits with recycle. The (SISO) controllers for highly coupled multivariable processes often leads to poor performance because * Corresponding author. Fax: +1-418-656-3159 of a bad choice of manipulated variables, poor E-mail address: email@example.com (D. Pomerleau). speci®cations and poor tuning of the controllers.0019-0578/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved.PII: S0019-0578(00)00040-9
2. 342 D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351Despite considerable work on decoupling con- mathematical measure of directionality is given bytrollers, decentralized PI controllers remain the the singular value decomposition (SVD) . Thestandard for most industries. According to Skoges- singular values give, for each frequency, the max-tad and Morari , they have fewer tuning para- imum [ j!] and the minimum [ j!] values of themeters, are easier to understand, and are more easily gain of the process and the singular vectors give themade failure tolerant. Furthermore, decoupling directions of theses maximum and minimum. Thecontrollers are complex, require excessive engi- gain of a multivariable process, at a given fre-neering manpower, have a lack of integrity, have a quency, is not limited to a single value but a rangelack of robustness and often result in operator non of possible values between j! and j!. A pro-acceptance, according to Luyben . cess with a wide range of possible gains has a large For decentralized controllers, Desbiens et al.  directionality and a process with a small range ofhave proposed a method where the time speci®ca- possible gains has a low directionality. This char-tions in closed loop have to be given and the con- acteristic is important because processes with largetrollers are evaluated by solving two quadratic directionality can show control problems [13±15].equations. Some authors have proposed tuning The ratio j!= j! is called the condition num-methods which take into account the process uncer- ber. It is an indicator of the directionality or howtainty. Skogestad and Morari  have proposed an ill-conditioned the process is.independent tuning method for decentralized con- Here, a more intuitive representation, based ontrollers based on individual loop conditions. They the step response, is given for measuring twohave derived their conditions from the global inputs±two outputs (TITO) processes direction-robust performance condition of the m-synthesis ality and for the evaluation of the closed-loopenvironment. Chiu and Arkun  and Ito et al. system characteristics. The process input u t or have proposed sequential design methods for the disturbance d t can be represented at a time tdecentralized controllers. Gagnon et al.  have in a condensed form by a vector d t with ampli-also use the robust performance concept de®ned in tude given by its L2-norm. Similarly, the outputsthe m-synthesis environment. can also be represented by a vector, y t, with an In this paper, the condition number, which is a amplitude given by its L2-norm. The method con-measure of directionality, and the generalized sists in simulating the TITO process when thedynamic relative gain (GDRG), which is a measure inputs di t are step functions. Keeping the ampli-of interaction, are used to determine the most tude of d t constant (Fig. 1) and simulating y tappropriate control structure (decentralized ordecoupling controllers). They also give the possibi-lity to determine the pairing and the time responsespeci®cations. From there, in decentralized con-trol, the tuning of the SISO controllers based onan approximation of the transfer functions seen byeach one is given. The controllers obtained arecompared to the corresponding controllers wherea decoupler is inserted between the process and thecontrollers. Both control structures are comparedfor set-point changes as well as in regulation usingthe generalized step input (GSR).2. Process characteristics Multivariable processes are mainly character-ized by their directionality and interaction. A Fig. 1. Input vector for TITO process.
3. D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351 343for all possible directions of d t generates the is preferred to the RGA, which only considers thegeneralized step response (GSR). steady-state. The GSR gives information about directionality,but it is not a measure of the interaction. Indeed, aprocess with a large directionality can have no 4. Tuninginteraction. A TITO process with zero gains in thecross-coupled transfer functions and a large and a 4.1. Decentralized controllow gain in the direct branches is an example of amultivariable process with high directionality. As for SISO processes, the tuning of decen-Directionality can come from the intrinsic properties tralized controllers consists in opening the loopof a process (ex. system with an even number of under study and evaluating the transfer functionpositive gain for a TITO process) or a system seen by the controller as presented in Fig. 2. Thewhere the actuators are badly sized. transfer function seen by controller Gc1 s is G1 s and the one seen by controller Gc2 s is G2 s where G1 s and G2 s are given by:3. Interaction and pairing G12 sG21 sGc2 s G1 s G11 s À 2 An interesting interaction measure is the gen- 1 Gc2 sG22 seralized relative dynamic gains (GRDG) of Huanget al. . The GRDG takes into account the G12 sG21 sGc1 s G2 s G22 s À 3dynamics of the closed-loops. The GRDG l11 s 1 Gc1 sG11 sfor a TITO process is de®ned as follows: Eqs. (2) and (3) show that the tuning of one R 2 s controller depends on the other one controller. The Gp11 sGp22 s system can then be separated into two SISO Y 2 sl11 s 1 systems as seen in Fig. 3. A set-point change on one R 2 s Gp11 sGp22 s À Gp12 sGp21 s loop is seen as a disturbance by the other loop. Y 2 s Dierent approximations can be used to evalu- ate G1 s and G2 s. Since the controllers includewhere R2 s=Y2 s is the desired dynamics of the an integrator to prevent static errors, a possiblesecond loop. The variables R2 s and Y2 s are the approximation at frequencies lower than theset point and the process output of the other loop cross-over frequency (!co ) is:respectively. The transfer functions Gp11 s, Gp12 s,Gp21 s and Gp22 s are the elements of the process G12 sG21 s G1 s G11 s À for Gc2 sG22 s 1transfer matrix Gp s. In this paper, a representa- G22 stion of the GRDG is given as a function of both 4closed-loop bandwidths . For easier controland tuning, the speci®cations on the closed-loopset point responses have to be chosen in a frequencyband where interaction is reduced so the systembehaves more like SISO systems. In order to do so,the closed-loop response speci®cations are chosen infrequency band where the GRDG is close to onesince, as for relative gain array (RGA), it meansthat the interaction is low. Because the zeros in a transfer function aectthe process dynamic, the GRDG, which takes inaccount the dynamic part of the transfer function, Fig. 2. Decentralized control.
4. 344 D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351 with G21 s. The error made by the approximation has then a reduced importance for the transfer function seen by the controller. If the time con- stants of G12 s and G21 s are smaller than the crossover frequency !co , the ®ltering eect will be reduced but, in this case, the gains K12 and K21 should be much smaller than K11 and the transfer function seen by the controller will depend mostly on G11 s. If this is not the case, the wrong pairing has been used. Fig. 3. Equivalent system (decentralized control). 4.2. Decoupling controllersand For the tuning of the controllers when a decou- pler is inserted between the process and the con- G12 sG21 s troller, as seen in Fig. 4, one has:G2 s G22 s À for Gc1 sG11 s 1 G11 s 5 ÀG12 sD22 s D12 s 8 G11 s This facilitates the tuning since the transferfunction seen by one controller is independent of ÀG21 sD11 s D21 s 9the other controller. For the other output variable, G22 sthe system is in regulation. The process dynamicson the regulated variable depends primary on the The transfer functions seen by each controllerdynamic of the manipulated variable where the are then given by:set-point change occurred. From Eqs. (4) and (5),one can expect a slow response if the transfer G12 sG21 sD11 s G1 s G11 s Àfunctions in the direct branches contain a large G22 stime constant in the numerator since it is trans- G12 sG21 sD22 s G2 s G22 s À 10lated as a pole in the controller. G11 s This relation cannot be applied if the transferfunctions of G11 s or G22 s contain an unstablezero or a delay longer than 12 s 21 s, where It is observed that the transfer functions seen byrepresents the process delay. In these cases, the each controller are the same as the ones seen byapproximation given by Eqs. (6) and (7) can be the decentralized controllers when Eqs. (4) and (5)used. are used. G12 sG21 sG1 s G11 s À 6 K22and G12 sG21 sG2 s G22 s À 7 K11where K11 and K22 are, respectively, the gains ofG11 s and G22 s. Generally, the transfer function Gc2 s1Gc2 sG22 s is low pass ®ltered by G12 s in series Fig. 4. Control with decouplers.
5. D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351 3455. Evaluation Gc sGp s U s L s 12 1 Gc sGp s It is always dicult to make a valuable evalua-tion of dierent controllers. Here, since the speci-®cations are given for set-point changes, similar Where L s is the disturbance and where Gc sdynamics will be taken as a reference point for represents the controller. When a decoupler isboth control structures and the controllers will be used for the system, Gc s includes the decoupler.evaluated for both set-point changes and in reg- Here, a limited number of cases will be studiedulation for a GSR at the process inputs. For sym- and we will try to generalize the results. The dif-metrical process, the GSR [Y s] to set point ferent processes under consideration are given inchanges is equivalent to the manipulated variables Table 1. System A and B are only dierent in the[U s] in regulation for a disturbance at the pro- signs of the gain of G12 s. Only two dierent signscess input, as illustrated by Eqs. (11) and (12). of the gain are studied, since all the other cases can be deduced from these two as seen in Table 2. Gc sGp s Cases 1, 6, 7, 8, 9, 10, 11 and 16 are similar toY s R s 11 1 Gc sGp s system A where the number of positive gain sign isTable 1Dierent processes under considerations System A System B 4 4Initial process G11 G22 G11 G22 1 10s 1 10s 3 À3 3 G12 G21 G12 G21 1 10s 1 10s 1 10s 4 1 À 10s 4 4 1 À 10s 4Process 1 G11 G22 G11 G22 1 10s2 1 10s 1 10s2 1 10s 3 À3 3 G12 G21 G12 G21 1 10s 1 10s 1 10s 4 4Process 2 G11 G22 G11 G22 1 10s 1 10s 3 1 À 10s 3 À3 1 À 10s 3 G12 G21 G12 G21 1 10s2 1 10s 1 10s2 1 10s 4 1 50s 4 4 1 50s 4Process 3 G11 G22 G11 G22 1 10s2 1 10s 1 10s2 1 10s 3 À3 3 G12 G21 G12 G21 1 10s 1 10s 1 10s 4 4Process 4 G11 G22 G11 G22 1 10s 1 10s 3 1 50s 3 À3 1 50s 3 G12 G21 G12 G21 1 10s2 1 10s 1 10s2 1 10s
6. 346 D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351Table 2The dierent processes under considerationCase Gp11 s Gp12 s Gp21 s Gp22 s1 + + + + System A2 À + + + Identical to 3, with opposite gain sign for Gc3 + À + + System B4 + + À + Identical to 35 + + + À Identical to 3, with opposite gain sign for Gc26 À À + + Identical to 1, opposite gain sign for Gc17 À + À + Identical to 1, opposite gain sign for Gc18 À + + À Identical to 1, opposite gain sign for Gc1 and Gc29 + À À + Identical to 110 + À + À Identical to 1, opposite gain sign for Gc211 + + À À Identical to 1, opposite gain sign for Gc212 À À À + Identical to 3, opposite gain sign for Gc113 À À + À Identical to 3, opposite gain sign for Gc1 and Gc214 À + À À Identical to 3, opposite gain sign for Gc1 and Gc215 + À À À Identical to 3, with opposite gain sign for Gc216 À À À À Identical to 1, opposite gain sign for Gc 1 and Gc2even. The other cases are similar to system B where functions seen by the controllers and the approx-the number of positive gain sign is odd. The condi- imations used for controllers tuning are identical.tion number and the GRDG are given for all pro-cesses in Figs. 5 and 6 . On the basis of the condition 5.1. System Anumber, which is a measure of directionality, systemA processes 1 and 2 should be accelerated while On the basis of the condition number, systemsystem B should not be. Fig. 5 shows that the ``A, which has an even number of positive sign,condition number, at high frequencies, is lower for presents a high directionality. For the initial process,system A, and is higher for system B. On the basis which is symmetrical and has equal time con-of the GRDG, system B process 3 should also be stants, this value is constant and equal to 16.9 dB.accelerated in order to reduce interaction since it is The gain seen by the controllers is low (K11 À K12 K21near 1 at high frequencies as shown in Fig. 6. K22 1:75) since the outputs are in¯uenced by The transfer functions seen by each controller are components acting in opposite directions. For set-given in Table 3 with the corresponding tuning. The point changes, decentralized and decoupling con-tuning method proposed by Poulin et al.  has trollers give similar results on the output variablebeen used. For the process under study where for which the set-point has occurred. The GSR forG11 s G22 s for the initial system, the controllers a disturbance at the process inputs are given inare symmetrical when a zero is incorporated in one Figs. 9 and 10, respectively. It is observed that theof the cross-coupled transfer function. decentralized controller gives a response much less Fig. 7 gives the approximation used for the directional that decoupling controller. This can betransfer function seen by the controller in decen- explained by the fact that the disturbance is ®rsttralized control and the real function seen for ampli®ed in the direction of the maximum sin-initial system A while Fig. 8 gives these approx- gular vectors for both types of controllers but isimation for initial system B. The full line of the corrected very slowly with decouplers since theyBode plot refers to the approximation and the eliminate the directionality. The decentralizeddotted line refers to real function seen by the con- controllers although have a uniform directionalitytrollers for the initial process. At frequencies lower for a symmetrical process. It is also observed, forthan the cross-over frequency (!co ), the transfer the case of a symmetrical process, that the
7. D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351 347 Fig. 6. Dynamic generalized relative gain for system A and B. Fig. 5. Condition number for System A and System B. opposite directions. At the opposite, process 2 which has a non-minimal phase zero in the cross- coupled transfer function has a stable zero in themanipulated variable for a step output dis- transfer function seen by the controller. This couldturbance gives the process outputs for set point be deduced from Eqs. (4) and (5). A consequencechanges as given in Eqs. (11) and (12). As a result, of the latter is that process 2 will be easy to accel-for highly directional and coupled processes, erate and process 1 will be impossible to accel-decoupling controller will be much less robust to erate. For process 2, another advantage is thatmodelling errors. accelerating will reduce directionality as given by For process 1, which has a non-minimal phase the condition number. The GSR plots shown, forzero in the direct branch, this non-desired char- process 2 in decentralized control, con®rm this asacteristic is ampli®ed for the transfer function seen shown on Fig. 11 for Kc 0:57 (!co 0:1) and inby the controller since the action coming from the Fig. 12 for Kc 4 (!co 0:7). This explains why across-coupled transfer functions are acting in the PID has been used for the tuning of process 2.
8. 348 D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351 Processes 3 and 4 have a stable zero. For a stable zero in the direct branch, it means that the transfer function seen by the controller will also have a stable zero. The tuning will be easy and the process can be easily accelerated. However, the controller will contain a large pole to satisfy set-point chan- ges speci®cations. As a result of this large pole, one might expect slow time response for the out- put in regulation in decentralized control. For process 4, as one might expect, the stable zero in the cross-coupled transfer function is seen as a non-minimal phase system by the controller. This will limit the system response in both types of control structures. Fig. 7. Bode plot for system A initial process. 5.2. System B System B presents no directionality on the initial system. The components coming from the direct and the cross-coupled transfer functions are acting on the same directions (K11 À K122221 6:25). It K K means that for the systems, which have a zero, the eect of the zero will be reduced for the transfer function seen by the controller. This is con®rmed by the results shown in Table 3. For process 3, according to the GRDG, it should be accelerated in order to reduce the interaction. This is shown in Figs. 13 and 14 where the devia- tion of the regulated variable is reduced from 0.25 to 0.15 for a gain of 0.16 and 0.64 of the controller, respectively. It also shows that the presence of an Fig. 8. Bode plot for system B initial process. important zero in G11 s has a determinant eectFig. 9. (a) GSR output for system A, initial process (decentralized control). (b) GSR input for system A, initial process (decentralizedcontrol).
9. D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351 349Fig. 10. (a) GSR output for system A, initial process (control with decouplers). (b) GSR input for system A, initial process (controlwith decouplers). Fig. 11. GSR for system A process 2 (Kc 0:57). Fig. 13. Step response for process 3 of system B (Kc 0:16). Fig. 12. GSR for system A process 2 (Kc 4). Fig. 14. Step response for process 3 of system B (Kc 0:64).
10. 350 D. Pomerleau, A. Pomerleau / ISA Transactions 40 (2001) 341±351Table 3Transfer functions seen by each controller and the corresponding tuning Initial process Process 1 Process 2 Process 3 Process 4System A À0:75 1 À 10s À0:75 1 10s À0:75 1 50sD12 (decoupler) À0.75 0 1 10s 1 50s 1 10sD21 (Decoupler) À0.75 À0.75 À0.75 À0.75 À0.75 1:75 1:75 1 À 36s 1:75 1 36s 1:75 1 101s 1:75 1 À 42sG1 1 10s 1 10s2 1 10s2 1 10s2 1 10s2 1:75 4 1:75 1 36s 1:75 1 84s 1:75 1 À 42sG2 1 10s 1 10s 1 10s2 1 23s2 1 10s2 0:57 1 10s 0:127 1 15s 0:57 1 10s2 0:57 1 15s 0:11 1 15sGC1 10s 15s 15s 1 36s 15s 1 101s 15s 0:57 1 10s 0:25 1 10s 0:57 1 10s2 0:57 1 15s 0:11 1 15sGC2 10s 10s 15s 1 36s 15s 1 84s 15sSystem B 0:75 1 À 10s 0:75 1 10s 0:75 1 50sD12 (decoupler) 0.75 0 1 10s 1 50s 1 10sD21 (decoupler) À0.75 À0.75 À0.75 À0.75 À0.75 6:25 6:25 1 À 2:8s 6:25 1 2:8s 6:25 1 35:6s 6:25 1 24:4sG1 1 10s 1 10s2 1 10s2 1 10s2 1 10s2 6:25 4 6:25 1 2:8s 6:25 1 36s 6:25 1 24:4sG2 1 10s 1 10s 1 10s2 1 10s 1 50s 1 10s2 0:16 1 10s 0:125 1 15s 0:16 1 15s 0:16 1 10s2 0:16 1 15sGC1 10s 15s 15s 1 2:8s 15s 1 35:6s 15s 1:24:4s 0:16 1 10s 0:25 1 10s 0:16 1 15s 0:16 1 52:6s 0:16 1 15sGC2 10s 10s 15s 1 2:8s 52:6s 1 36s 15s 1 24:4son the time response of the regulated variable y2 t observed the transfer functions seen by the decen-for a set point change on y1 t. In both cases, a PID tralized controllers are the same as the ones seenwith a pole zero cancellation method has been used when a decoupler is used. For system that have ain order to be able to accelerate the process. non-minimal zero in the direct transfer functions another approximation has been used since the formed cannot be inverted and fully decoupled6. Conclusion systems are impossible. For processes which have high directionality Simple tuning rules have been developed for (even number of positive sign) and are highlydecentralized controllers. The approximations coupled, decentralized control should be used inused remain valid for most systems since the order to reduce this directionality in regulation fortransfer functions seen by the controllers are low process input disturbances. For these processes, sincepass ®ltered by the cross-coupled functions. It is the components of the direct and cross-coupled
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