INPUT-OUTPUT PAIRING OF MULTIVARIABLE PREDICTIVE CONTROL

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1 INPUT-OUTPUT PAIRING OF MULTIVARIABLE PREDICTIVE CONTROL Lng-Cong Chen #, Pu Yuan*, Gu-L Zhang* *Unversty of Petroleu, P.O. Box 902 Beng 00083, Chna # GAIN Tech Co., P.O. Box 902ext.79, Beng 00083, Chna Abstract: Regardless of what predctve control strategy s used, the predctve horzon s the an desgn paraeter. The stablty, control perforance and robustness of predctve control syste are anly depended on t. For ultvarable predctve controller, selecton of predctve horzon s an nput-output parng proble. In ths paper, Response Index Array, Dynac Interacton Index Array and Relatve Steady-State Index Array are proposed as the crtera for the selecton of predctve horzon and parng. The desgn procedure for ultvarable predctve controller s sued up. As an exaple, the parng of a heavy ol fractonator s gven. The desgn has been successfully pleented on several ndustral fractonators. Copyrght 2002 IFAC Keywords: Predctve control, Input-output parng, MIMO Syste. INTRODUCTION Durng the last two decades, odel predctve control (MPC) has becoe an attractve control strategy wthn the area of process ndustres. MPC s a successful strategy for handlng ultvarable and/or constraned control probles (Garca and Morar, 989). Generally, the ultvarable controller does not need nput-output parng, whch s a an desgn proble n the ult-loop control, such as conventonal PID control. If the predctve horzon and control horzon of MPC are deterned, there s no nput-output parng proble. But, parng proble wll rse durng MPC desgn to deterne predctve horzon. So far, the MPC presented n the lteratures ay be classfed nto two strateges:. MPC based on the nput(anpulated varable, MV)-output (controlled varable, CV) odel, such as MAC (Rchalet, J et.al. 978; Rouhan,R. and R.K. Mehra 982), DMC (Cutler, C.R. and B.L. Reaker, 980), GPC (Clarke, D.W. et.al. 987,989), IMC (Garca and Morar,982,985). Soeterboek (992) proposed predctve control: a unfed approach for such knd of MPC strateges. 2. MPC based on the state space odel and state varable feedback (Yuan, 993). Sun and Yuan (993, 997) proposed Unfed Predctve Control, whch s based on Polynoal Matrx Descrpton (PMD), for all knds of the MPC strateges. Yu and Yuan (2002) proved theoretcally that all knds of MPC are equvalent,.e., the sae control perforance, depends on predcton horzon P, wll be acheved by dfferent MPC strateges as long as there s no odel satch and no dsturbance. In real world, there are unknown dsturbance and odel satch. So dfferent MPC are dfferent n robustness and dsturbance reecton. Ths topc wll not be dscussed n ths paper. For ultvarable process, RGA (Brstol, 966) s usually used to easure the nteracton and the desgn of ult-loop control. RGA, based on steady-state gan of controlled process, s not sutable for the MPC desgn, whch s based on the dynac response. In the lteratures, contrbutons on the desgn of MPC are presented as well as the dfferent MPC strateges entoned above. The an desgn ssue s how to deterne the predctve horzon. MPC has been wdely used on ultvarable systes, yet, by the author s knowledge, the dscusson n lteratures of how to deterne the predctve horzon for ultvarable systes s uch less than that of SISO systes. In ths paper, the relatonshp between predctve horzon and stablty, control perforance and robustness of MPC syste, as the bass of syste desgn, are revewed n second secton. The desgn of ultvarable MPC s an nput-output parng proble and dynac response ndex, nteracton ndex and relatve steady-state ndex are proposed as parng crtera n thrd secton. MPC syste desgn procedure was sued up n secton IV. As an exaple, desgn of MPC for a heavy ol fractonator s llustrated. 2. PREDICTIVE HORIZON For ultvarable MPC, dfferent CV has dfferent

2 control deand and dfferent response to MV. A reasonable desgn s that every CV has ts own predctve horzon p. The predctve horzon of the syste P s a vector: [ ] P! (2-) p p2 p r where: p s the predctve horzon (nuber of dscrete nterval) of th controlled varable. For llustraton and wthout loss of gernalzaton, MPC wth sngle predcton algorth (Yuan, 992) s used n the followng dscusson. The optal control ove was deduced as: uk ( ) S ( P)[ Y( k) Y( k)] (2-2) where: u R Manpulated varable (MV); r Y R Controlled varable (CV); uk ( ) uk ( ) uk ( ) S[(P)] S S ( p ) S ( p )! S ( p ) 2 r S ( p )!! S ( p ) 2 2 2r 2 " " " " S ( p ) s th CV response at S ( p )!! S ( p ) r r rr r T p (2-3) th p nterval nstant after th MV unt step. YS ( k ) Set pont of controlled varable; Y ( k) Y( k) F ( z + ) X( k) + F ( z ) u( k) p X u (Predcton of CV whle uk ( + ) 0, 0) X R X( k) X( k) X( k ) n Measurable state varable (nclude CV); F ( z ) F + Fz +! + F z 0 q q (Feedback polynoal atrx) X (993), Yuan (992, 993, 994, and 997) and others proved soe theoretcal results (assung no odel satch and r) for stablty and control perforance of MPC syste related to predctve horzon: Theore : det[ SP ( )] 0 (2-4) s a necessary stablty condton for MPC syste. Theore 2: If the controlled process s stable and functonally controllable, then: det[ S(P) ] det[ S( ) ] > 0 (2-5) s a necessary stablty condton for MPC syste, where: [ S( )] s the steady-state gan atrx of controlled process. Theore 3: If the controlled process s stable and p (,2,!, r ) s tuned suffcently large, then the MPC syste s stable. Theore 4: If: p δ + ; and Theore and Theore2 are satsfed, then: the th CV reaches to perfect control. If p δ + ;, 2,!, r (2-6) And both Theore and Theore 2 are satsfed; then: the MPC syste reaches to perfect control (all CV d n reach to perfect control), where: δ δ δ, δ d and n δ are the orders of denonator and nonator of th row n pulse transfer functon atrx, respectvely. Perfect Control s defned as: f CV reaches to ts set-pont at every control (saplng) nstant after nu te delay of set-pont or dsturbance step change. It s obvous that perfect control s decoupled between CV and CV to dsturbance. In real world, perfect control can be reached only for a class of controlled process wth specal dynac property. In ost cases, t s dffcult to reach, not only lted by the above condton, but also lted by odel satch and robustness. The control (MV) ove s usually another lt. For sae CV s devaton, large control ove usually lead to fast response and weaker robustness. If ncreasng predcton horzon p akes saller control ove, then, the sluggsh response and the better robustness; otherwse, f ncreasng predctve horzon leads to larger control ove (ay be constraned by lt), then, the contrary. Accordng to above analyss, Yuan (992) proposed to use Relatve Predctve Horzon (RPH) β for SISO syste to select predctve horzon and trade-off the control perforance and control ove constrants. RPH s defned as: S( p) β (2-7) S( ) Where: S( p) s the value of step response at predctve horzon; S( ) s the steady-state value of step response. β s recoended. Large β leads to a stronger robustness, less control ove and sluggsh response. If β s specfed, predctve horzon P can be calculated fro eq.(2-7). Snce β s a float varable and P s an nteger, Sn ( ) Sn ( ) If S( ) 0, < β S( ) S( ), then: p n ; If S( ) 0, then: p. (2-8) Ths result s extended to ultvarable syste n ths paper. 3. INPUT-OUTPUT PAIRING CRITERIA For MIMO syste, every CV s related to

3 anpulated varables, and dfferent MV has dfferent dynac response. If β s specfed, dfferent MV has dfferent predctve horzon. Whch MV should be used to deterne the correspondng predctve horzon? In ths pont, the nput-output parng s stll a proble for ultvarable predctve control syste as well as ult-loop control syste, but n dfferent content. For MIMO syste, better control perforance s desred as well as SISO syste and fast response MV should be selected. The dstncton s the nteracton between CV and MV, and decouplng or less nteracton s always requred. More MV than CV or ore CV than MV ade the syste ore coplcated. The startng pont of MPC desgn s to satsfy the requred control perforance, whch s related to the Relatve Predctve Horzon RPH as entoned above. For MIMO syste, the requred control perforance of th CV and correspondng RPH β can be specfed prevously. But, the predctve horzon p s dfferent for dfferent MV. If β s specfed, to deterne s a proble of nput (MV)-output (CV) parng. For nput-output parng, three Index Arrays are defned. Defnton : Response Index Array r (RIA) For th CV, f β s specfed, correspondng predctve horzon for th MV s p (,2,!,, Let: ) p n pn Mn{ p}, γ. p γ γ2! γ γ 2 γ22 RIA { γ } $ " " $ $ " γ r!! γ r s defned as Response Index Array (RIA). (3-) RIA s a crteron of response speed of dfferent MV. The larger the p, the faster the response of th CV to th MV. In order to ake th CV has better control perforance, by the knowledge of SISO syste entoned n Secton 2, the predcton horzon P ay be selected as p Mn{ p } (,2,!,, and ) correspondngly γ. However, for ultvarable syste, the nteracton ust be taken nto account. Defnton 2: Dynac Interacton Index Array µ (DIA) For th CV, f β s specfed, t has possble CV-MV parng wth correspondng predctve horzon p. For every possble parng, the correspondng Dynac Interacton Index s defned as: p µ l S ( p ) l S ( p ) (3-2) The larger the µ, the weaker the nteracton for th CV- th MV parng. It s a possble parng canddate. If µ, t ples that th CV s affected only by the th MV and has no nteracton wth other MV n dynac. It s a pror parng canddate. However, the steady-state property ust be consdered also. Defnton 3: Steady-State Index Array λ l S ( ) S ( ) l λ (SIA) (3-3) λ, t ples that th CV s affected only by the If th MV and has no nteracton wth other MV n steady-state. Model predctve control, as showed n eq.(2-2), s a non-steady-state error control strategy for step nput and decoupled n steady-state, but the control ove ay be too large, so, the an consderaton of the SIA s the effectveness and lt of MV. The larger the λ, the saller the control ove n steady-state. If λ s near to zero, t eans that ths MV s neffectve. RIA, DIA and SIA should be consdered n MIMO syste desgn. In addton, the optzaton, safety and other requreents of MV should be also consdered. The followng parng ndex {a } s suggested. Defnton 4: Parng Index δ 0 0 ξ! ξ! A { a } 0 δ2 " $ " $ " ξ 0 r! ξr " $ $ 0! 0 δ (3-4) + q + w (3-5) ξ γ µ λ Where: q nteracton weghtng factor for th CV. w control ove weghtng factor for th CV. δ weghtng factor for th MV. For th CV, parng MV s: MV( ): { Max[ a ],(,2,!, )} (3-6) 4. MPC DESIGN PROCEDURE Accordng to the above results, the desgn procedures for predctve horzon and nput-output parng are sued up as:. Gve the prorty of each CV and correspondng β accordng to the requreent of control perforance.

4 β 0.3~0.8 s recoended. Large β leads to a stronger robustness, less control ove and sluggsh response. Usually, hgher prorty CV ay have saller β. 2. If the controlled process has ore MV than CV, gve the control prorty, optu prorty and target for each MV. If the controlled process has ore CV than MV, gve the weghtng factor of each CV. These two cases, whch are beyond the scope of ths paper, wll not be dscussed n detal. 5. EXAMPLE For llustraton, consder the parng of a heavy ol fractonator, shown n Fg.. The fractonator has top and two sde-draw products. In order to keep the product specfcaton, top and two sde-draw teperatures are an controlled varables, as CV, CV2 and CV3 n Fg.. Usually, t has three PID controllers TC to keep the teperatures at ther set-ponts. 3. Calculate p, r, µ, λ, ξ. MV2 TC MV3 CV MV 4. Fro hgher to lower prorty of CV, the MV who ade least value n ξ should be selected as the parng for control. If the selected MV has been used by hgher prorty CV, then n the reanng MVs, the one who ade ξ the least value s recoended n order 4. Check stablty by Theore, 2. If unsatsfed, tune β or p and return to step. Accordng to Theore 3, to have stronger robustness. Ths procedure results a predctve horzon for each CV and predctve horzon vector P [ p p! p ] T r for MPC. 2 larger β or p ay usually lead to a stable MPC syste. MV4 MV6 TC TC MV5 MV7 CV2 Fractonator CV3 5. Check control ove: MPC desgn should eet the requreent of control ove lt. However, the control ove depends on the set-pont change, dsturbance and status of controlled process. In order to evaluate the control ove n desgn phase, assue all set-pont has unt step and ntal state equal to zero, check the control ove at frst saplng nstant and steady-state. The control ove at frst saplng nstant after set-pont unt step s: u S ( P) (4-) The control ove at steady-state after unt step s: u S ( ) (4-2) So the axu control ove s: u ax{ S ( P), S ( P),!, S ( P) } (4-2) ax 2 (,2,!,) Where: (P) s the th eleent of th row of S (P) or S ( ); SP ( ) step response atrx [eq.(2-3)] If u ax volates the lt, then tune β or p and return to step. Large β or p usually lead to saller control ove. S 6. Sulaton. If unsatsfed, choose P agan and return to step. The desgn procedure ay be extended to the case of ore MV than CV or ore CV than MV. Fg. Heavy Ol Fractonator The fractonator ay have seven anpulated varables: MV: Top Reflux Flow rate (PID set pont) MV2: Top Heat Reove Crculaton Flow rate (PID set pont) MV3: Set Pont of Top Teperature PID Controller (Three-way valve) MV4: Frst Heat Reove Crculaton Flow rate (PID set pont) MV5: Set Pont of frst draw Teperature PID Controller (Three-way valve) MV6: Second Heat Reove Crculaton Flow rate (PID set pont) MV7: Set Pont of second draw Teperature PID Controller (Three-way valve) All of the MV has hgh and low lt as well as correspondng valve openng. If one MV s lted, the controller wll select other unlted MV. So, all of the possble CV-MV parng and correspondng predctve horzon should be gven. For the 3 CV and 7 MV of a fractonator, t has 2 possble parngs. But, f the parng has too sall value of parng ndex, t s not a sutable for control, whch wll be llustrated below. If a CV has ore sutable MV, the prorty of MV should be

5 specfed accordng to the value of requreent. a and optzaton Snce fractonator has ore MV than CV, t s able to push soe MV to ts optu value whle keep the control perforance by other sutable MVs. Usually the optzaton targets are nu heat reove flowrate or nu open of by-pass (three-way) valve of heat exchanger or stea generator. The step responses of CV, CV2 and CV3 to the 7 MVs are gven n Fg.2, Fg.3 and Fg.4 respectvely. The prorty of CV s specfed as: CV, CV2 and CV3 fro hgher to lower. The relatve predctve horzon s specfed as: β [ β β 2 β 3 ] [ ] Accordng to the unt step responses, the predctve horzon, RIA, DIA and SIA are calculated as: p γ µ Fg. 2 CV Unt Step Response λ Assung: Q W dag[] I δ 0.3, δ 2! δ 7.0 the parng ndex a s: a ξ Accordng to the value of, parng s deterned. For CV: MV, MV2, MV3, MV4 are sutable parngs. MV5, MV6, MV7 have saller parng ndex, so they are not sutable parngs. But MV4 s a better parng canddate to CV2, so the fnal parngs for CV are MV, MV2 and MV3. The prorty s: MV2, MV3, and MV fro hgher to lower. (MV has lower value of parng ndex, however t s anly requred to reach ts optu value.) Fg. 3 CV2 Unt Step Response For CV2: MV4 and MV5 are sutable parngs, and the prorty s MV4, MV5 fro hgher to lower. For CV3: MV6 and MV7 are sutable parngs, and the prorty s MV4, MV5 fro hgher to lower. These results show that aong the 2 possble parngs only seven parngs are sutable. Each CV has fewer parngs than whole MV. Nevertheless, the control syste s ultvarable accordng to the eq.(2-2). These parngs have been appled to several ndustral heavy ol fractonators. Fg. 4 CV3 Unt Step Response For heavy ol fractonator, Fnal Bolng Pont (FBP) of top product and 95% ASTM of frst draw product are ore portant controlled varables. They are depended on the top teperature and frst draw teperature respectvely. They have the sae step responses and use sae anpulated varables of teperature control, and,

6 the sae predctve horzon as well as parngs. FBP and 95%ASTM should be keep on specfed setpont snce they are desgned as set pont controlled varable. Top and frst draw teperatures are desgned as zone controlled varables. If the predcted teperatures do not volate ther hgh or low lts, no control s requred. The nuber of CV need to control and the nuber of avalable MV are depended on the operaton stuaton. So, the structure of the fractonator as a controlled process s vared. A vared structure predctve coordnated control syste based on above desgn and control requreents for the fractonator was pleented n several ndustral plants. The applcaton shows that the parng desgn s sutable for the ultvarable control. Fg.5 s a real-te trend acqured fro the ndustral plant. Set-pont of 95% ASTM (D) has been decreased at 9:7 and frst heat reove exchanger (stea generator) by-pass valve (F) has been gradually closed to ts optu value. FBP s nearly decoupled to the set-pont change of 95% ASTM. Both FBP and 95% ASTM are runnng wth less devaton to ther setponts. MV s kept at ts optu value (not showed n Fg.5). Fg.5 Real te trend of fractonatotr,2,7: top teperature and ts set-pont(cv) 3,4: Fnal Bolng Pont and ts setpont 5,6,9: frst draw teperature and ts set-pont(cv2) 8,D: 95% ASTM of frst draw ts set-pont B: frst heat reove flowrate(mv4) C: top heat reover exchanger by-pass valve(mv3) E: top heat reover crculaton flowrate(mv2) F: frst heat reover exchanger by-pass valve(mv5) 6. CONCLUSION Input-output parng s a basc proble for ultvarable control syste desgn as well as the odel predctve control regardless of ultvarable or ult-loop structure. Parng based on dynacs of controlled process s better than that based on steady-state gan. Response ndex and nteracton ndex proposed n ths paper catch on the dynacs and an control syste desgn probles. They are effectve crtera for the desgn of ultvarable predctve control systes. The parng proble should be developed coprehensvely. for Multvarable Process Control, IEEE. Trans. Autoatc Control, AC-, No., pp Clarke, D.W., C. Mohtad and P. S. Tuffs (987). Generalzed Predctve Control, Autoatca, 23(2), pp Clarke, D.W. and C. Mohtad (989). Propertes of Generalzed Predctve Control, Autoatca, 25(6), pp Cutler, C.R. and B.L. Reaker (980). Dynac Matrx Control A Coputer Control Algorth. Jont ACC Preprnts, Paper WP5-B, San Francsco. Garca, C.E. and M. Morar (989). Model predctve control: theory and practce a survey. Autoatca, 25, pp Garca C.E. and M. Morar (982). Internal Model Control A Unfyng Revew and Soe New Results, Ind. Eng. Che. Process Des. Dev. 2, pp308. Garca C.E. and M. Morar (985). Internal Model Control 2 Desgn procedure for ultvarable systes. Ind. Eng. Che. Process Des. Dev. 24, pp Rchalet, J et.al.(978). Model predctve heurstc control.: Applcaton to ndustral processes, Autatca, 4, pp Rouhan,R. and R.K. Mehra (982). Model Algorthc Control (MAC): Basc Theoretcal Propertes. Autoatca, 8(4), pp Soeterboek, Ronald (992). Predctve Control: A Unfed Approach. Prentce Hall. Sun, D.X. and P. Yuan (993). A Unfed Predctve Control Algorth, Industral Process Modelng and Control. 6, pp05-. Sun, Y.H. and P. Yuan (994). An approach of state feedback predctve control and decouplng, J. Da-Qng Petroleun Insttute, 8(3), pp (n Chnese) X, Yu Geng (993). Predctve Control. Natonal Defence Publsher. Yu Z.J. and P. Yuan (2002). Decouplng Features of Model Predctve Control. J. Unversty of Petroleu, 26(3), pp08-2. (n Chnese) Yuan P. (992). Sngle Predcton Predctve Control, J. of Unversty of Petroleu, 6(5), pp (n Chnese) Yuan P., H.T. Zheng and X. Zuo (993). State Feedback Predctve Control, Snca Autoatca, 9(5), pp (n Chnese) Journal of Chnese Autoaton, 6(3), pp3-2, 994 (Allerton Press Inc.) Yuan P. (994). Process Dynac Model and Its On-lne Applcatons. Chna Petro-Checal Publsher. Yuan P., D. X. Sun and T. Ba (997). Unfed Predctve Control, Proceedngs 2 nd ASCC, III, pp Seoul, Korea. REFERENCES Brstol, E.H. (966). On a New Measure of Interacton