RPN TUNING STRATEGY FOR MODEL PREDICTIVE CONTROL

Transcription

1 RPN TUNING TRATEGY FOR MODEL PREDICTIVE CONTROL J. O. Trierweiler #, L. A. Farina, and R. G. Duraiki Laboraory of Proce Conrol and Inegraion (LACIP) Deparmen of Chemical Engineering, Federal Univeriy of Rio Grande do ul (UFRG) Rua Marechal Floriano, 5, CEP: Poro Alegre - R - BRAZIL, jorge@enq.ufrg.br Abrac: A novel uning raegy baed on RPN for MIMO MPC i preened. The RPN indicae how poenially difficul i i for a given yem o achieve he deired performance robuly. I reflec boh he aainable performance of a yem and i degree of direcionaliy. Thee yem' properie are he bai of he propoed RPN-MPC uning raegy, which i applied in he conroller deign of an air eparaion plan. Alhough i wa only ued a linear nominal model, he reul can alo be applied a lea a ome exen for nonlinear yem wih uncerainie. Copyrigh IFAC Keyword: - model predicive conrol, conrollabiliy meaure, uning raegy, RPN INTRODUCTION Model Predicive Conrol (MPC) i a dicree-ime echnique in which he conrol acion i obained by olving open loop opimizaion problem a each ime ep. The flexibiliy of hi ype of implemenaion ha been ueful in addreing variou implemenaion iue ha radiionally have been problemaic. From a pracical viewpoin, an aracive feaure of MPC i i abiliy o naurally and explicily handle boh mulivariable inpu and oupu conrain by direc incorporaion ino he opimizaion. The MPC raegy wa fir exploied and uccefully employed on linear plan, epecially in he proce indurie, where relaively low ample ime made exenive on-line inerample compuaion feaible. Recen improvemen in compuer power have made MPC a viable alernaive approach in a variey of addiional applicaion a well. Dynamic marix conrol (DMC) (Culer and Raemaker, 98) i conidered he mo popular MPC algorihm currenly ued in he chemical proce indury. I i no urpriing why DMC, one of he earlie formulaion of MPC, repreen he indury andard oday. A large par of DMC appeal i drawn from an inuiive ue of a finie ep repone model of he proce, a quadraic performance objecive over a finie predicion horizon, and opimal manipulaed inpu move compued a he oluion o a lea quare problem. Anoher form of MPC ha ha rapidly gained accepance in he conrol communiy i Generalized Predicive Conrol (GPC) (Clarke e al. 987). I differ from DMC in ha i employ a conrolled auoregreive and inegraed moving average (CARIMA) model of he proce which allow a rigorou mahemaical reamen of he predicive conrol paradigm. The GPC performance objecive i very imilar o ha of DMC. Neverhele, GPC reduce o he DMC algorihm when he weighing polynomial ha modifie he prediced oupu rajecory i aumed o be uniy (Camacho and Bordon, 995). The uning raegy propoed in hi paper i direcly applicable o DMC and GPC. Moreover, i can be eaily exend o include all oher MPC conrol raegie. The full dimenion of he conrol deign ak coni of wo par: conrol rucure deign and conroller # Auhor o whom he correpondence hould be addreed. Preprin of DYCOP-, June 4-6,, Jejudo Iland, Korea 83

2 deign. In (Trierweiler, 997) and (Trierweiler and Engell, 997a) a new index, he Robu Performance Number (RPN), wa inroduced. RPN i a conrollabiliy meaure ha can be ued boh for conrol rucure deign and for conroller deign. In hi paper, we preen a uning raegy for MPC algorihm baed on RPN. The main idea of he propoed uning raegy i caling he yem and weighing marice correcly. To do ha, we applied a caling procedure baed on RPN. The paper i rucured a follow: in ecion, he neceary background abou MPC i preened. In ecion 3, he RPN i horly dicued. ecion 4 preen he RPN-uning raegy for MPC. In ecion 5, he RPN-MPC uning raegy i analyzed uing an air eparaion plan a example. BACKGROUND A MPC algorihm employ a diincly idenifiable model of he proce o predic i fuure behavior over an exended predicion horizon. A performance objecive o be minimized i defined over he predicion horizon, uually a um of quadraic e poin racking error and conrol effor erm. Thi co funcion i minimized by evaluaing a profile of manipulaed inpu move o be implemened a ucceive ampling inan over he conrol horizon. The feedback behavior i achieved by implemening only he fir manipulaed inpu move and repeaing he complee equence of ep a he ubequen ample ime. The variou MPC algorihm propoe differen co funcion for obaining he conrol law. A quie general expreion for he objecive funcion i: J P P M yˆ ( j ) r( j) u ( j ) Q () where y^( j ) i he prediced oupu j ep ino he fuure baed upon informaion available a ime, r(j) i he reference ignal j ep ino he fuure, u() (-z - ) u() u()-u(-), and weighed Euclidean norm of x x T x n n n x i he x R defined a wih R poiive definie. The uning parameer are he minimum coing horizon (P ), he maximum coing horizon (P), he conrol horizon (M), he ampling ime ( ), he conrolled variable weigh (Q), and he move uppreion weigh (). The weighing marice Q and can be choen a ime-varying (i.e., funcion of j). Here, for impliciy, hey are aumed o be ime-invarian. Each of he above parameer ha a pecific role in uning of MPC algorihm. Uing he accumulaed experience of applying predicive algorihm, a number of engineering rule have been idenified o obain appropriae value of he parameer for good performance in differen applicaion, uch a: P and P. The meaning of P and P i raher inuiive. They mark he limi of he ime in which i i deirable for he oupu o follow he reference. Thu, if a high value of P i aken, i i becaue i i unimporan if here are error in he fir inan which will provoke a mooh repone of he proce. Noe ha proce wih dead ime here i no reaon for P o be le han hi ime delay, ince he oupu will only begin o evolve paed hi ime. Equivalenly, proce wih invere repone, he oupu will only o go o he final repone direcion afer paed invere repone effec. Of coure, he correponding invere repone ime can alo be ued o e P. The maximum coing horizon P hould be equal or maller han he open-loop eling ime of he proce in ample, ince nohing i gained by coing fuure error in () ha canno be influenced by fuure conrol acion. M. The conrol horizon M hould be M P. The ineger M pecifie he degree of freedom in elecing fuure conrol, o ha, afer M fuure ampling inerval, he conrol incremen are aumed o be zero, giving a conan conrol ignal. A baic rule for elecing M in GPC algorihm i o e i a lea equal o he number of unable pole in he plan (Rawling and Muke, 993). I mean ha for able yem, M can be ued. Alhough hi i a feaible choice, i hould be avoided, ince he conroller will uually preen a very poor performance. On he oher ide, pecial care mu be aken if MP and, ince in hi cae he MPC reduce o a Minimum Variance conroller (Grimble, 99), which i known o be unable on nonminimum phae procee.. Of coure, he choice of predicion horizon P canno be made independen of he ampling ime. I i appropriae o relae he ampling rae o he cloed-loop bandwidh of he feedback yem, f c, ince f c i relaed o he peed a which he feedback yem hould rack he command inpu. Alo, he bandwidh f c i relaed o he amoun of aenuaion he feedback yem mu provide in face of plan diurbance. A a general rule of humb, he ampling period hould be choen in he range (anina e al. 996) < <. () 3 f 5 f c c Preprin of DYCOP-, June 4-6,, Jejudo Iland, Korea 84

3 Noe ha he parameer P, P, and M have a direc influence on he ize of marice required o compue he opimal conrol, and hu on he amoun of compuaion involved. Uually he weighing marice Q and are diagonal marice, whoe he elemen are uned o achieve he deired performance in cloed loop and o cale he inpu and oupu making he uni of meauremen and manipulaed variable comparable. Addiionally, hey are alo ued for: Q. I i poible o achieve igher conrol of a paricular meaured oupu by elecively increaing he relaive weighing elemen.. The role of i o penalize exceive incremenal conrol acion. The larger he value of, he more luggih he conrol will become. The parameer and M are rongly relaed o each oher. Regardle, M mu be in he range Number of RHP-polo M P, i coninue o be an imporan uning parameer. For able proce, M and can produce accepable reul, bu quie beer performance can be achieve wih M > and >. Q,, and M can be een a he main parameer o be manipulaed o improve he conrol performance. Of coure, he oher parameer are alo imporan, bu hey are well deermined by he proce dynamic. In ecion 4, i i hown how he MPC uning parameer hould be e o achieve a given aainable performance. The uning mehodology i baed on he Robu Performance Number (RPN) which will now be inroduced. 3 RPN ROBUT PERFORMANCE NUMBER The Robu Performance Number (RPN) wa inroduced in (Trierweiler 997; Trierweiler and Engell, 997a) a a meaure o characerize he conrollabiliy of a yem. The RPN indicae how poenially difficul i i for a given yem o achieve he deired performance robuly. The RPN i influenced boh by he deired performance of a yem and i degree of direcionaliy. 3. Definiion The Robu Performance Number (RPN, Γ) of a mulivariable plan wih ranfer marix G() i defined a ( G, T, ω ) up{ ( G, T )} RPN Γ up Γ ω Γ( G, T) σ( [ I T( jω )] T( jω) * ) γ ( G( jω) ) * γ (3a) ( G( jω) ) (3b) where γ * (G(jω)) i he minimized condiion number of G(jω) and σ _ ([I-T] T) i he maximal ingular value of he ranfer funcion marix [I-T] T, being T he (aainable) deired oupu complemenary eniiviy marix, which i deermined for he nominal model G(). 3. Aainable Performance In hi ecion, i i dicued how he aainable cloed loop performance can be characerized for yem wih RHP-zero. pecificaion of he deired performance e pecify he deired performance by he (oupu) complemenary eniiviy funcion T which relae he reference ignal r and he oupu ignal y in he degree of freedom (DOF) conrol configuraion (ee fig. ). For he IO cae, pecificaion a eling ime, rie ime, maximal overhoo, and eady-ae error can be mapped ino he choice of a ranfer funcion of he form ε Td ζ ωn ωn (4) where ε i he oleraed offe (eady-ae error). The parameer ω n (undamped naural frequency) and ζ (damping raio) of (4) can be eaily calculaed from he ime domain pecificaion, a i can be een in Trierweiler (997). Fig. : andard feedback configuraion For he MIMO cae, a raighforward exenion of uch a pecificaion i o precribe a decoupled or almo decoupled repone, wih poibly differen parameer for each oupu, i.e., T d diag(t d,,...,t d,no ), where each T d,i correpond o a IO ime domain pecificaion. Facorizaion of yem wih RHP-zero and RHP-pole To aify he RHP-zero and pole conrain i i poible o make ue of he Blachke inpu and oupu facorizaion (for he definiion of he facorizaion and an algorihm o calculae i, ee, e.g., Havre and kogead (996) or Trierweiler (997)). The aainable performance for he cae when he plan G() ha boh RHP-zero and pole can eaily be obained (Trierweiler, 997) and i given by T ( ) B ( ) B () I ( I T ( ) [ B () B ( ) ] O, z O, z d I, p I, p. (5) where B O,z () and B I,p () are he zero oupu and pole inpu Blachke facorizaion, repecively, and he Preprin of DYCOP-, June 4-6,, Jejudo Iland, Korea 85

4 operaor B denoe he peudo-invere of B, in uch way ha B() B () I. T() i differen from he original deired ranfer funcion T d (), bu ha exacly he ame ingular value, being alo he pecified robune properie preerved a he plan oupu. The facor B O,z () and B I,p () enure ha T() T d () o ha he eadyae characeriic (uually T d () I) are preerved. 3.3 RPN-caling Procedure The caling of he ranfer marix i very imporan for he correc analyi of he conrollabiliy of a yem and for conroller deign. In he definiion of γ * (G(jω)), L and R are frequency dependen, however, in he deign L and R uually are conan. The following procedure baed on he RPN i recommended o be ued o opimally cale a yem G. RPN-caling procedure:. Deerminaion of he frequency ω up where Γ(G,T,ω) achieve i maximal value.. Calculae he caling marice L and R, uch ha γ(l G(jω up )R ) achieve i minimal value γ * (G(jω up )). 3. cale he yem wih he caling marice L and R, i.e., G () L G() R The analyi and conroller deign hould hen be performed wih he caled yem G. 4 RPN-TUNING TRATEGY Before aring o deign he conroller, i i neceary o deermine how difficul he conrol problem i. For i, we calculae he RPN for he yem uing an aainable deired performance T, which i a funcion boh of nonminimum-phae behavior of he yem and of he cloed loop deired performance. The RPN i a meaure of how poenially difficul i i for a given yem o achieve he deired performance robuly. The eaie way o deign a conroller i o ue he proce invere. An invere-baed conroller will have poenially good performance robune only when he RPN i mall. Then, he fir ep of our uning raegy, which i ummarized in Table, coni of pecifying he deired performance. The econd ep i he facorizaion of G() and deerminaion of an aainable performance T. The hird ep correpond o he applicaion of RPN-caling procedure o calculae he caling marice L and R. The forh ep i he choice of ampling ime. Baed on () and on he deired performance (4), The ampling ime,, can be expreed a a funcion of he rie ime τ r a follow (anina e al., 996):.6τ r < <. 4τ r. Table : MPC uning procedure for MIMO-yem baed on RPN. pecificaion of he deired performance T d. Deerminaion of an aainable performance by he facorizaion of he nominal model 3. RPN-caling procedure where he caling marice L and R are deermined a hown in ecion ampling ime:.6τ r < <. 4τ r 5. Two poibiliie for coing horizon: 8% ( A) P, P, and r ( ) L T( ) r ( ) 9% % ( B) P, P, and r ( ) L r ( ) 6. Conrol horizon: M P 4 7. The conrol acion u hould be calculaed uing he following caled objecive funcion: J M j P j P u yˆ ( j ) r ( j) ( j ) where y^( j ) i he prediced oupu j ep ino he fuure baed upon he caled model G (), i.e., G () L G() R. 8. Inpu and oupu weighing marice: Q and y Z,, ( u, ) log( RPN ) mean( g ( ωup ) ) i j Z where y Z and u Z are repecively he oupu and inpu zero direcion of he RHP-zero cloe o he origin. 9. Back o he original uni of he manipulaed variable, i.e., u R u In ep 5, P, P, and r are deermined. The minimum (P ) and maximum (P) coing horizon are relaed o cloed loop yem' dynamic. For able proce, i can be expreed a a funcion of he open loop yem' dynamic. A already menioned, he maximum coing horizon P hould be equal or maller han he open-loop eling ime of he proce in ample, ince nohing i gained by coing fuure error. If he reference ignal r i baed on he aainable performance (i.e., r L T()r d (), d d Q Preprin of DYCOP-, June 4-6,, Jejudo Iland, Korea 86

5 where r d i he deired reference ignal), P can be made, ince he nonminimum-phae behavior i conidered in r auomaically. In hi cae, P 8% / will uually give very good reul. 8% correpond o he ime when he repone of G() o an inpu ep reache 8% of i final value. For mulivariable yem he maximal value of 8% for all oupu and inpu hould be ued. hen r i no an aainable performance, P and P hould approximaely be equal o % / and 9% /, repecively, where % and 9% correpond o he ime when he repone of G() o an inpu ep reache % and 9% of i final value. In ep 6, he conrol horizon M i choen approximaely equal o P/4. Thi value i a good compromie beween performance and abiliy. ep 7 calculae he conrol acion u uing he caled objecive funcion: J P P M yˆ u ( j ) r ( j) ( j ) Q (6) where y^( j ) i he prediced oupu j ep ino he fuure baed upon he caled model G (), i.e., G () L G() R, uing Q and y Z,, ( u ) log ( RPN ) mean( g ( ωup ) ) i j Z, (7) a weighing marice. Here y Z and u Z are he oupu and inpu zero direcion of he RHP-zero cloe o he origin calculaed for he caled yem G (). Thee direcion give an idea how he RHP-zero effec i diribued on he oupu (y Z ) and inpu (u Z ), repecively. The idea i o apply in le exenion he manipulaed variable where he RHP-zero effec i concenraed. In principle, if we conider y Z and u Z equal o, i will produce almo he ame reul. Therefore, we recommend he uer o ue y Z and u Z ju a guideline. The RPN facor i included in o penalize exceive incremenal conrol acion. The larger he value of RPN, he more luggih he ( g up ) i, j conrol will become. The facor mean ( ω ) i included o make he econd erm independen of he caling marice L and R. Finally, o apply he conrol acion o he yem, he caled conrol acion u hould be reored o he original uni, i.e., u R u. 5 A HEAT INTEGRATED AIR EPARATION PLANT Here, i i analyzed he conrol rucure T_63 for air eparaion uni udied in (Trierweiler and Engell, ). Thi conrol rucure ha everal RHP-zero cloe o he origin and RPN6.3. ince hi conrol rucure i a difficul conrol problem and, herefore, more indicae o how he benefi of he propoed MPC uning procedure. Here, all imulaion were made uing he funcion MPCCON and MPCIM of he MATLAB MPC-oolbox (Morari and Ricker,994). Thee funcion ue a model in ep forma a he DMC algorihm. imilar reul are alo obained by he correponding aepace funcion, i.e., MPCCON and MPCIM. The deired performance ued in he imulaion wa: y- u6 (5 min rie ime and % overhoo), y-u3 (3 min rie ime and % overhoo), and y3-u (3 min rie ime and % overhoo). The aainable performance T i coniderable differen from T d for T_63, ince hi conrol rucure ha RHP-zero cloe o origin..5 Conrolled variable Manipulaed variable y y y 3 u 6 u 3 u 5 5 Time [min] Fig. : epoin ep in y, P, M6, Q and a in Table, and r L T()r d () applied o G ().5 Conrolled variable x 4 Manipulaed variable Time [min] Fig. 3: epoin ep in y uing conroller parameer P, M6, QI,, and G() Fig. how he imulaion of a MPC conroller wih he uning parameer baed on RPN-MPC-uning procedure (ee Table ) o a epoin ep in y and.5 min. The calculaed conroller parameer wa P, M 6, caling marice L diag([4, 47, 8]) and R diag([, 8.6, 8.]) and weighing marice Q diag([.97,.77,.77]) and diag([.5,.95,.97]). Fig. 3 how he ame imulaion for QI and, and he uncaled yem. The MPC uned by RPN-uning procedure ha a coniderably beer performance. Moreover, he y y y 3 u 6 u 3 u Preprin of DYCOP-, June 4-6,, Jejudo Iland, Korea 87

6 conrol acion i much maller han for he ununed cae (cf. Fig and 3). Of coure, he quie beer performance of RPN uned MPC i rongly relaed o he reference ignal T which include he nonminimum-phae behavior of T_63 auomaically. In Fig. 4, we can ee he cloed loop performance for he cae when T i no ued a reference ignal. Oberve ha already he correc caling can coniderably improve he nominal yem performance..5.5 Conrolled variable Manipulaed variable y y y 3 u 6 u 3 u 5 5 Time [min] Fig. 4: epoin ep in y uned a in Fig., bu wihou reference rajecory. 5. Influence of ome uning parameer Baed on imulaion reul, which are no hown here, we can conclude he following poin: Conrol horizon M: The conrol performance for MP for he RPN uned MPC preen he ame performance a before (hown in Fig. ), while he ununed MPC wih MP reduce o a Minimum Variance conroller producing, herefore, unlimied and unrealiic conrol acion on nonminimum phae procee. Predicion horizon P: The RPN uned MPC ha almo he ame performance a hown in Fig. for P5 and M, while he ununed MPC ha decreaed i performance aking more ime o reach he epoin. In general, we can conclude ha he RPN-MPC uning raegy i much le enible o he parameer changing. In all analyzed cae, he cloed loop performance wa almo he ame. Moreover, he RPN uned MPC ha alway produced mooh and maller conrol acion han he ununed MPC, indicaing ha he unconrained RPN-MPC uning procedure will alo produce good reul for he conrained conrol problem. 5. Applying o a commercial DMC To apply he propoed mehodology o a commercial MPC algorihm, i i ju neceary o include he caling marice ino he weighing marice a follow: U,i i /R, i and Q U,i Q i x L, i Here, U and Q U are he weighing marice for he uncaled yem G() and he ubindexe i i he elemen (i,i) of each marix. Noe ha ep 9 of Table hould no be applied in hi cae. 6 CONCLUION The paper preened a novel uning raegy for MIMO MPC. The propoed MPC uning procedure i baed on Robu Performance Number (RPN), which can meaure how difficul he conrol problem i. The RPN, becaue of he dependency on he aainable cloed-loop performance, ake he effec of nonminimum-phae behavior and he deired cloed loop performance ino accoun. In addiion, he frequency dependen direcionaliy of he yem i quanified correcly and i herefore ued o cale he yem allowing a more robu and efficien uning. The performance of he RPN-MPC uning raegy wa demonraed for a wrong eleced conrol rucure of an air eparaion plan. The conroller deign wa performed uing a linear nominal model, bu can be exend o include nonlineariie and uncerainie in he ame way a done for he conrollabiliy analyi (ee, e.g., Trierweiler and Engell, 997b and Trierweiler, 997). The RPN-MPC uning procedure wa alo uccefully applied o oher example, which are available a hp:// REFERENCE Camacho, E.F., Bordon, B.(995) Model Predicive Conrol in he Proce Indury, pringer-verlag. Clarke, D.., Mohadi, C., Tuff, P.. (987) Generalized Predicive Conrol - Par I. The Baic Algorihm, Auomaica, 3(), pp Culer, C.R., Raemaker, B.L. (98) Dynamic Marix Conrol - A compuer conrol algorihm, Proc. Auomaic Conrol Conference, an Francico, CA. Grimble, M. J. (99) Generalied predicive conrol: An inroducion o he advanage and limiaion, In. J. y. ci., 3, 85, Havre, K, kogead, (996). Effec of RHP Zero and Pole on Performance in CONTROL'96, Exeer, UK, pp Morari, M; Ricker, N.L.(994) Model Predicive Conrol Toolbox- for ue wih Malab - V.. Rawling, J. B. and Muke, K. (993). The abiliy of Conrained Receding Horizon Conrol. IEEE Tran. Auom. Conrol, 38(), pp Rohde,. (994) Producion of pure argon from air, Repor on cience and Tech. 54, Linde A.G., pp.3-7 anina, M.., ubberud, A.R., Hoeer, G.H., (996) ample-rae elecion, in he Conrol Handbook, CRC pre, pp kogead,. and Polehwaie, I.(996). Mulivariable Feedback Conrol - Analyi and Deign, John iley Trierweiler, J.O. and Engell,. (997a). The Robu Performance Number: A New Tool for Conrol rucure Deign, Comp. chem. Eng., (), uppl. (PECAPE-7), pp Trierweiler, J.O. and Engell,. (997b). Conrollabiliy Analyi via he Robu Performance Number for a CTR wih Van de Vue Reacion, Proc. ECC '97, TU- A-H3. Preprin of DYCOP-, June 4-6,, Jejudo Iland, Korea 88

7 Trierweiler, J.O. and Engell,. (). A Cae udy for Conrol rucure elecion: Air eparaion plan, Journal of Proce Conrol, pp Trierweiler, J.O. (997). A yemaic Approach o Conrol rucure Deign, Ph.D. Thei, Univeriy of Dormund. Preprin of DYCOP-, June 4-6,, Jejudo Iland, Korea 89