ABCM Sympoium Serie in Mechatronic - Vol. 3 - pp.87-96 Copyright c 8 by ABCM A PLC BASE MIMO PI CONOLLE FO MULIVAIABLE INUSIAL POCESSES Joé Maria Galvez, jmgalvez@ufmg.br epartment of Mechanical Engineering Federal Univerity of Mina Gerai, Brazil Av. Antonio Carlo 667, Pampulha 3.7-9 Belo Horizonte, MG, Brazil Abtract. Indutrial procee are in general multi-input multi-output MIMO ytem. epite of that and becaue of the lack of good fine-tuning algorithm for MIMO controller, they are uually treated a ingle-input ingle-output SISO one for the purpoe of control deign and implementation. In that cae, the tuning procedure i frequently carried out taken one loop at a time and the reult are uually not a good a would be expected. hi work preent a MIMO PI tructure that lead to a fine-tuning technique that can be carried out in real time. he MIMO PI new tructure ha been implemented in a low cot PLC. he controller wa teted in a x MIMO benchmark plant with outtanding reult. hi paper preent detail of the controller tructure, the fine-tuning procedure and finally experimental reult of the controller performance. eyword: PLC Baed Control, MIMO PI controller, Multivariable Sytem.. INOUCION he improvement of power conumption efficiency of indutrial and commercial device i one of the main iue for the incoming century. It i a fact that the next decade are going to tetify a continuou and trenuou earch for new device and technologie to ave energy reource. It i already acknowledge that the olution for an efficient operation of indutrial plant relie on the proper choice and deign of control ytem. he complexity of indutrial procee ha continuouly increaed in recent year. Currently, almot every ingle indutrial proce i contituted by multiple input that affect and interact with all the proce output. hi ha created the need for new control algorithm to cope with the current high indutrial tandard. he control deign technique have alo evolved and produced advanced and ophiticated control algorithm. However, the required hardware capable to run thee new control algorithm till i relatively expenive. Currently, low cot imple controller uch a On/Off control and PI control are ued a the tandard controller in indutry. Unfortunately, they are not capable to deal with the exiting I/O cro-interaction in MIMO plant. In multiple-input multiple-output MIMO ytem, the independent and imultaneou control of the output variable i frequently a challenging tak. Beide that, model uncertaintie, time delay in the control loop and trong crocoupling interaction that uually exit between the ytem input and output are ome of the problem that might deteriorate the controller ideal performance.. Automatic control ha played an important role in the development of the modern indutry. he control community ha propoed everal new control technique during the lat decade to deal with the control problem of time varying procee, time delay and I/O cro coupling, etc. Among them, robut control, adaptive control and intelligent control are the mot important. he final choice for the control of a pecific application i uually determine by the plant characteritic, economical apect and technical pecification. A drawback of thee ophiticated alternative i that they are uually expenive and required advanced computational reource. o face time-varying load, time delay and I/O cro coupling, new low cot multi-input multi-output MIMO control trategie mut be explored. hi paper preent a MIMO control cheme that can be implemented in a low cot Programmable Logic Controller PLC. A an example of application, the propoed algorithm i applied to the control of an electrical oven ytem contituted by two heating zone, two thermocouple and two independent controlled power ource. Section preent ome comment on feedback control; Section 3 decribe the benchmark x MIMO plant; Section 4 focue on the deigning and implementation of the propoed controller cheme; Section 5 how a brief review on robut control concept that are ued to validate the propoed algorithm; Section 6 preent experimental reult; and finally, Section 7 preent final comment and concluion.. SOME COMMENS ON FEEBAC CONOL Output feedback ha been the indutrial tandard for control purpoe not only to hape the plant repone, fulfilling performance pecification, but alo to deal with output diturbance and model uncertaintie. raditionally, the indutrial control community ha relied on the intrinic robutne of output feedback controller to face the control deign problem for SISO plant. A diverity of controller tuning algorithm ha been uccefully developed and applied to SISO indutrial plant. Behind thi ucce, there ha alway been a property that exit for all phyical ytem, the dominance of the low-frequency pole in the ytem time repone. hi fact ha been the
background of nearly every robut control deign technique. Conidering thi concept in the controller deign, there i no need for olving the modeling problem a rigorouly a it could be required without the pole dominance property. Several attempt have been made to extend the SISO deign technique to the MIMO cae. In thi context, the uually trong input-output cro-coupling exiting in MIMO ytem become a important a the model uncertaintie due to the ize order of large-cale ytem. With ome exception, the ucce of MIMO control deign alo depend on the pole dominance property. In recent year, the reearch ha been focued in new uncoupling technique. It i worth to mention the pioneer contribution from Britol 966, ouvaritaki 979, Mee 98, McAvoy, 983 and Grodidier and Morari 986. Some characteritic of the ue of thee technique are: he deign procedure i uually carried out in the frequency domain. Model uncertaintie are eaily repreented in the frequency domain particularly, non-tructural uncertaintie. Low frequency model are, in general, accurate enough for control deign in thi environment. he tandard PI controller that repond for more de 9% of the indutrial controller i deigned to perform in the low frequency range. Output diturbance are uually low frequency ignal. 3. HE MIMO PLAN Large indutrial oven uually include everal heating zone with the objective of improving the internal temperature profile. However, due to the heat flow there i trong iteration among thee zone. A in other cae of multiple-input multiple-output ytem, thi iteration may become relevant and caue eriou difficultie to the control ytem []. he MIMO ytem, an electrical oven, ha two input and two output Figure. he ytem input are the voltage applied to the heating reitor. he ytem output are the temperature at each heating zone. he control goal i to regulate the temperature profile with a minimum error. ue to the unidirectional and aturation characteritic of the power ource a well a to the two different time contant for cooling and heating, the ytem in thi cae i nonlinear, however, it can be hown that for ome temperature profile the ytem behave linearly allowing a linear analyi and deign. In addition, a trong cro-coupling interaction among input and output characterize thi type of ytem. A major challenge in thi cae i that the thermal pecification work againt the control performance, i.e., to optimize the moothne of the temperature profile it i neceary to increae the iteration among the heating zone that caue the undeirable cro-coupling among input and output. Actually, each of the output i a function of both input a hown in Figure. u t y t U G Electrical G u t Oven y t G U G Figure. he Open Loop MIMO Plant. Baed on experimental reult, a matrix tranfer function model wa built uch that G G G G U U [ ] [ G ][ U ] he model identification procedure led to x matrix tranfer function of the form:.8 48 S.5 5 S e e G G 5 34 G G.5 5 S.8 48 S e e 34 5 [ G ]
4. HE POPOSE CONOLLE he propoed trategy i a frequency-domain procedure. In thi cae, the MIMO controller deign i carried out in two tep. Firt a MIMO pre-compenator,, i deigned to cale the ytem and reach diagonal dominance at low frequencie and then a MIMO controller,, i deigned to meet performance pecification. he advantage of thi procedure it that for deign purpoe, i diagonal and can be treated a a multiple SISO deign ince G i trongly diagonal dominant at low frequencie and diagonal at ω. Additionally, exact modeling i only required at teady tate ω or at mot at low frequencie. he MIMO control law ha the form: [ ] [ ][ ][ ] [ ][ ] E U 3 with [ ] [ ] [ ] [ ] & ; ; U U U E E E 4 and [ ] [ ][ ] 5 with G - decoupling at ω, thu, a x MIMO PI controller can be written a [ ] [ ][ ] [ ] N N N N f S I P f I P 6 It hould be noticed that, all entrie of have the general form of a SISO PI controller, however, the engineer only ha to determine the diagonal element of. he propoed deigning technique lead to a cloed loop tranfer function that can be approximated at low frequencie by a diagonal matrix tranfer function, Equation. [ ] [ ] 7 Becaue of that, the independent control of the heating zone i tangible a it i hown in the next ection. Figure 4 how the MIMO controller tructure. Figure 5 how the block diagram for the cloed loop ytem. Electrical Oven Control Algorithm - - E E U U E E U U Figure. he MIMO Controller.
E E U U G G G G E E PI PI U U _ G _ G _ G _ G Figure 3. he MIMO Controller from the Fine uning View Point. he MIMO controller wa implemented in a low-cot high-performance PLC. Figure how the PLC ZAP5 from HI-ecnologia ued in thi work. Figure 3 preent the internal architecture of the ZAP5 PLC. Figure 4a. he ZAP5 PLC. Figure 4b. he ZAP5 PLC Internal Architecture. Several technique for multivariable loop haping can be found in the literature Maciejowki, 989, Skogetad, 996, Ho & Xu, 998. In thi work, a robut performance wa achieved uing the following control law with and [ ] [ ] [ ] 8.8.5.59.683 G 9.5.8.683.59 [ ] [ ] [ ].75.5.3.5.75.5.3.5 Finally, the MIMO controller i given by
.3757.753.53.3.45.85.5.5.3.45.85.3757.753.53.5.5 [ ] [ ][ ] Notice that the cloed-loop ytem i given by: [ I G ] G 5. OBUS CONOL ASSESMEN A BIEF EVIEW hi ection preent ome baic concept on multivariable robut control ytem that are going to be ued in the next ection to validate the propoed deign procedure and to ae the cloed loop ytem robutne. he following i baed on the book from Maciejowki 989 and Skogetad et al 996. he ytem output, y, can be written a y P r S d m 3 where r i the reference input, d repreent the diturbance and m i the meaurement noie. the function S i known a the output enitivity function and i defined a [ I G ] S 4 and the ytem cloed loop tranfer function or complementary enitivity,, i then given by S G 5 the input enitivity function i defined a [ I G ] S i 6 and it correponding complementary function a i G Si 7 a multiplicative model for the plant uncertainty can be written a G G [ I W ] 8 o i Hence, the following criteria to ae the ytem performance and tability can be etablihed: a he criterion for nominal performance i defined by S W p < ; W p wp [ I ] 9a where W p i a performance-weighting matrix, thu, the nominal performance criterion can be re-written a σ [ S ] < ; σ [ ] i the greatet ingular value of [ ] w p 9b b he criterion for robut performance non-tructured uncertainty i given by γ σ W S σ W ; γ min plant - controller conditioning number p i i i
c he criterion for robut tability non-tructured uncertainty i defined by [ I ] Wi < ; Wi wi a where W i i an uncertainty weighting matrix, then, the criterion for robut tability can be re-written a σ [ ] < b wi d Structured-ingular-value baed robutne analyi wa formerly introduced by oyle et al 98. In thi cae, a robut performance condition for tructured uncertainty i given by Q < ω µ a where, the matrix Q i defined a Q Q Q Q Q W p So Wi So Wp So Go W S G i o o b and S o I Go c where w p and w I are defined in the frequency domain. thu, a robut tability condition for tructured uncertainty can be written a < ω µ Q 3 Equation from 9 to 3 are applied in Section 6 to ae the cloed loop ytem robutne and to validate the controller deign. 6. EXPEIMENAL ESULS From the previou ection, the nominal performance criterion wa pecified a σ [ S ] < 4 w p and the criterion for robut tability wa choen a σ [ ] < 5 w i Simulation reult are preented next to illutrate the controller performance. Figure 5 preent the open loop frequency repone of the plant. Figure 6 how the ytem decoupling in the frequency domain due to the incluion of. Figure 7 to 9 graphically diplay the controller deigning procedure in the time domain. Figure 7 preent the open loop ytem tep repone; it can be oberved in quadrant I and III the trong effect of the I/O cro coupling in an ideal cae, the time repone hown in quadrant I and III hould remain at zero for all time or at leat return to zero at teady tate. Figure 8 preent the effect of the decoupling pre-compenator ; it how the ytem open loop time repone to a unit tep. It can alo be oberved in quadrant I and III that the teady tate effect of the I/O cro coupling were eliminated by the incluion of. Figure 9 preent the ytem cloed loop performance. Finally, Figure and diplay the robutne analyi correponding to Equation 9 to 3.
G, Bode iagram G, Bode iagram db - -4 - -4-6 -5-4 -3 - - -6-5 -4-3 - - G, Bode iagram G, Bode iagram db - -4 - -4-6 -5-4 -3 - - Frequency rd/ -6-5 -4-3 - - Frequency rd/ Figure 5. Open Loop Frequency epone of [G]. G, Bode iagram G, Bode iagram db - -4 - -4-6 -5-4 -3 - - -6-5 -4-3 - - G, Bode iagram G, Bode iagram db - -4 - -4-6 -5-4 -3 - - Frequency rd/ -6-5 -4-3 - - Frequency rd/ Figure 6. Frequency epone of [G ].
3 G, Step epone 3 G, Step epone.5.5.5.5.5.5 5 5 5 3 5 5 5 3 3 G, Step epone 3 G, Step epone.5.5.5.5.5.5 5 5 5 3 ime ec. 5 5 5 3 ime ec. Figure 7. Open Loop Step epone of [G]. G, Step epone G, Step epone.8.6.4. -. 5 5 5 3.8.6.4. -. 5 5 5 3 G, Step epone G, Step epone.8.6.4. -. 5 5 5 3 ime ec..8.6.4. -. 5 5 5 3 ime ec. Figure 8. Open Loop Step epone of [G ].
, Step epone, Step epone.8.6.4. -. 5 5 5 3.8.6.4. -. 5 5 5 3, Step epone, Step epone.8.6.4. -. 5 5 5 3 ime ec..8.6.4. -. 5 5 5 3 ime ec. Figure 9. Cloed Loop Step epone of. Weighting Function /Wi & /Wp 3 Controller Frequency epone - - -5-4 -3 - - 4 Open Loop Principal Gain - -5-4 -3 - - Nominal Performance Condition - - - -5-4 -3 - - Frequency rd/ -3-5 -4-3 - - Frequency rd/ Figure. he eign Procedure.
obut Performance Condition NSU obut Stability Condition NSU.9.8.7 -.6-5 -4-3 - - - -5-4 -3 - -.95 µq - obut Performance Condition SU.6 µq - obut Stability Condition SU.9.5.85.8.75-5 -4-3 - - Frequency rd/.4.3.. -5-4 -3 - - Frequency rd/ 7. FINAL COMMENS AN CONCLUSION Figure. Controller obutne Validation. A MIMO controller deign procedure ha been preented. he main feature of the propoed technique are: a It ha been hown that the independent control of the heating zone in the oven ytem i a feaible tak. b A long a the controller i deigned to work in the frequency range in which the plant i diagonal dominant, the deign can be accomplihed in a SISO environment. It hould be noticed that thi i the general cae of PI controller in indutry that are deigned to perform in low frequency. c he implementation and tuning of the control law can be done conidering two independent ingle loop control. d ue to the plant pre-compenator, model accuracy i in general required only at low frequencie. e Finally, it ha been hown that due to the implicity of the propoed control cheme, it can be eaily implemented uing low cot hardware. 8. EFEENCES Britol, E.H., 966, On a New Meaure of Interaction for Multivariable Proce Control, IEEE ranaction on Automatic Control, Vol., pp.33-34. oyle & G. Stein, 98, "Multivariable Feedback eign: Concept for a Claical/Modern Synthei", IEEE ranaction on Automatic Control, Vol. 6, pp. 4-6. oyle, J.C., 98, Analyi of feedback ytem with tructured uncertaintie, Proceeding of the IEE, Part, Vol. 9, pp 4-5. Grodidier, P. and Morari, M., 986, Interaction Meaurement for Sytem under ecentralized Control, Automatica, Vol., pp. 39-39. Ho, W.. and Xu, W., 998, Multivariable PI Controller eign Baed on the irect Nyquit Array Method, Proceeding of the American Control Conference, Philadelphia, Pennylvania, pp. 354-358. ouvaritaki, B., 979, "heory and Practice of the Characteritic-Locu eign Method", IEE Proceeding, Vol. 6, pp. 54-548. Maciejowki, J.M., 989, Multivariable Feedback eign, Addion-Weley Publihing Company. McAvoy,.J., 983, Interaction Analyi - Principle and Application, Intrument Society of America. Mee, A.I., 98, Achieving iagonal ominance, Sytem and Control Letter, Vol., pp. 55-58. Skogetad, S. and Potlethwaite, I, 996, Multivariable Feedback Control - Analyi and eign, John Wiley & Son.