Constrained Robust Model Predictive Control Based on Polyhedral Invariant Sets by Off-line Optimization

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1 A pblcaton of 47 CHEMICA ENGINEERING TRANSACTIONS VO 3, 3 Cef Edtors: Saro Percc, Jří J Klemeš Copyrgt 3, AIDIC Servz Srl, ISBN ; ISSN Te Italan Assocaton of Cemcal Engneerng Onlne at: wwwadct/cet Constraned Robst Model Predctve Control Based on Polyedral Invarant Sets by Off-lne Optmzaton Sooratep Keawom*, Pornca Bmroongsr Department of Cemcal Engneerng, Faclty of Engneerng, Clalongkorn Unversty, Payata Rd, Patmwan, Bangkok, 33, Taland sooratepk@claact Ts paper proposes a fast robst model predctve control sng polyedral nvarant sets for ncertan polytopc dscrete-tme systems A seqence of nested polyedral nvarant sets correspondng to a seqence of state feedback gans s constrcted off-lne Ts, most of te comptatonal brdens are moved off-lne At eac samplng tme, wen te measred state les between two adjacent polyedral nvarant sets, a state feedback gan s calclated by solvng a lnear programmng based on lnear nterpolaton between two pre-compted state feedback gans Te controller desgn s llstrated wt an example Te smlaton reslts sowed tat te proposed algortm provdes a better control performance wle on-lne comptaton s stll tractable as compared to prevosly reported algortms Introdcton Model predctve control (MPC) s a control tecnqe tat optmzes ftre beavor of a process by sng a process model Robst model predctve control (RMPC) s a specfc type of MPC wc explctly ncldes model ncertanty n te problem formlaton RMPC as been appled n a wde varety of applcaton areas sc as control of a tablar eat excanger (Bakošová and Oravec, ) and dstllaton colmn (Martn et al, 3) In RMPC, all possble state trajectores are restrcted to le n te nvarant set constrcted, so robst stablty of te system can be garanteed Altog te polyedral nvarant set s well-known to ave some advantages over te ellpsodal nvarant sets sc as better andlng of asymmetrc constrants and enlargement of stablzable regon (Plymers et al, 5), te ellpsodal nvarant set s sally sed n RMPC formlaton de to ts relatvely low on-lne comptatonal complexty Recently, an off-lne RMPC algortm sng polyedral nvarant sets as been developed by Bmroongsr and Keawom (b) Te on-lne comptatonal complexty s redced by constrctng off-lne a seqence of polyedral nvarant sets correspondng to a seqence of pre-compted state feedback gans At eac samplng nstant, te smallest polyedral nvarant set contanng te measred state s determned and te correspondng state feedback gan s mplemented to te process Ts, all of te comptatonal brdens are moved off-lne However, te conservatveness s obtaned becase te control law mplemented at eac tme step s only an approxmaton of te tre optmal control law Moreover, te npt dscontntes cased by a swtcng between state feedback control laws are occrred Terefore, te algortm reqres constrctng a large nmber of polyedral nvarant sets, ence large data storage, n order to mprove te control performance and redce te npt dscontntes In ts paper, we present a fast RMPC sng polyedral nvarant sets tat reqres very small comptaton complexty and data storage A seqence of nested polyedral nvarant sets correspondng to a seqence of state feedback gans s constrcted off-lne At eac samplng nstant, wen te measred state les between two adjacent polyedral nvarant sets, te real-tme control law s calclated by solvng a comptatonally low-demandng lnear programmng tat s based on lnear nterpolaton between two precompted state feedback gans

2 48 T Notaton: For a matrx A, A denotes ts transpose, A denotes ts nverse I denotes te dentty matrx For a vector x, x ( k / denotes te state measred at real tme k, x ( k + / k ) denotes te state at predcton tme k + predcted at real tme k Te symbol denotes te correspondng transpose of te lower block part of symmetrc matrces Problem formlaton Te model consdered ere s te followng lnear tme varyng (TV) system wt polytopc ncertanty x( k + ) A( x( + B( ( y( Cx( () were x ( s te state of te plant, ( s te control npt and y ( s te plant otpt We assme tat [ A (, B( ] Ω, Ω Co{[ A, B ],[ A, B ],,[ A, B ]} () were Ω s te polytope, Co denotes convex ll,, B ] [ j j [ A (, B( ] wtn te polytope s a lnear combnaton of te vertces sc tat A are vertces of te convex ll Any [ A(, B( ] λ j ( [ Aj, B j ], λ j(, λ j ( (3) j j were λ ( [ λ (, λ(,, λ ( ] s te ncertan parameter vector Te am of ts researc s to fnd te state feedback control law ( k + / Kx( k + / (4) tat stablzes () and aceves te followng performance cost nder te nomnal model assmpton x( k / T Θ x( k / ( ( k + / R ( k + / mn J (, J ( k /, (5) were Θ > and R > ( k /,max are symmetrc wegtng matrces, sbject to npt and otpt constrants +,,,3,, n (6) y r( k / yr,max +, r,,3,, ny (7) 3 Te proposed algortm In ts secton, a fast RMPC sng polyedral nvarant sets s presented A seqence of polyedral nvarant sets correspondng to a seqence of state feedback gans s constrcted off-lne At eac samplng nstant, wen te measred state les between two adjacent polyedral nvarant sets, te realtme state feedback gan s calclated by lnear nterpolaton between two pre-compted state feedback gans Te dea of te proposed algortm s based on lnear nterpolaton between two pre-compted state feedback gans to get te real-tme state feedback gan tat s as large as possble Algortm 3 Off-lne step : Coose a seqence of states x {,,,N} te correspondng state feedback gans K YG, and solve te followng problem to obtan

3 49 mn γ Y, G, Qj, st, j,,, x Q j, T G + G Qj, ^ ^ AG + BY Q l, Θ G γi RY γi, j,,,, l,,, (8) (9) () T G + G Qj,, j,,,, l,,, AG j + BY j Ql X, j,,,, X,,,, n Y G G Q T T, max + j, Note tat te followng condton mst be satsfed Q ( A + BK ) Q ( A + BK ) >, j,,,,,,, N, k,,, (3) T j, k k + j, k k + Te optmzaton problem sed to derve te state feedback gans n ts step s based on te onlne RMPC controller proposed by Bmroongsr and Keawom (a) It s te mnmzaton of pper bond of nfnte orzon nomnal cost performance However, te otpt constrants are relaxed n ts step n order to enlarge te stablzable regon and to redce conservatveness Te otpt constrants and also npt constrants are ten properly taken nto accont drng polyedral nvarant set constrcton n off-lne-step Te condton (4) s sed to assre robst stablty satsfacton of a convex combnaton of K and K + / Off-lne Step : Constrct a seqence of polyedral nvarant sets S { x M x d },, N correspondng to a seqence of pre-compted state feedback gans () (),, K,,,, N by followng te procedres of Bmroongsr and Keawom (b) On-lne Step : At eac samplng nstant, f te measred state les between S and S +,,,, N, mplement K λ K + ( λ) K + to te process were λ s calclated by solvng te followng optmzaton problem mn λ (4) λ st mn ( λk + ( λ ) K+ ) xk max (5) M ( A + B ( λ K + ( λ) K )) x d, j,,, (6) j j + k λ (7) If te state les n S N, mplement K N to te process Snce npt and otpt constrants mpose lesser and lesser lmts on state feedback gans as te state converges to te orgn, te norm of pre-compted state feedback gan ncrease from oter to nner polyedral nvarant sets ( K < K +,,,, N ) By mnmzng λ at eac control teraton, te realtme state feedback gan tat s as large as possble s mplemented to te process

4 4 4 An example Consder te applcaton of or approac to te nonlnear two-tank system wc s descrbed by te followng eqaton ρ S ρa g + (8) ρs ρa g ρa g (9) were s te water level n tank, s te water level n tank and s te water flow Te operatng parameters are sown n table Table : Te operatng parameters of nonlnear two-tank system Operatng parameters Vales S,5 cm S,6 cm A 9 cm A 4 cm g 98 cm/s ρ 3 kg/cm et,, eq, eq and eq were sbscrpt eq s sed to denote te correspondng varable at eqlbrm condton, te objectve s to reglate to te orgn by manplatng Te npt and otpt constrants are 5 kg/s, 3 cm, cm 5 By evalatng te Jacoban matrx of (7) and (8) along te vertces of te constrants set, we ave tat all te soltons are also te solton of te followng dfferental nclson S 4 ρ p j A j + () S j ρ were A j, j,, 4 are gven by ρa A ρa ρa A ρa g,mn g,mn g,max g,max ρa ρa g,mn g,mn

5 4 ρa A3 ρa g,mn g,mn ρa g,max ρa A4 ρa g,max g,max ρa g,max () Te dscrete-tme model s obtaned by dscretzaton of (8) and (9) sng Eler frst-order approxmaton wt a samplng perod of s and t s omtted ere for brevty Te tnng parameters are Θ and R Te proposed algortm s compared wt ts onlne conterpart RMPC algortm proposed by Bmroongsr and Keawom (a) and te off-lne RMPC algortm proposed by Bmroongsr and Keawom (b) Fgre sows a seqence of for polyedral nvarant sets S { x / M x d },,,, 4 constrcted off- S wle te norm of state feedback gans lne Te szes of polyedral nvarant sets decrease from S to 4 ncrease from S to S 4 Fgre : A seqence of for polyedral nvarant sets constrcted off-lne Fgre sows profles of te water level n tank (reglated otpt) and te water flow (control npt) obtaned by eac algortm Algortm 3 can steer te state to te orgn faster tat oter algortms Ts s de to te fact tat n te algortm of Bmroongsr and Keawom (b), tere s no nterpolaton between pre-compted state feedback gans Conseqently, te control law mplemented at eac tme step s only an approxmaton of te tre optmal control law

6 4 Fgre : a) Te water level n tank (reglated otpt) and b) te water flow (control npt) Te algortm of Bmroongsr and Keawom (a) ses onlne optmzaton to compte a state feedback gan However, te algortm tlzes an ellpsodal nvarant set tat s more conservatve tan a polyedral nvarant set It s seen tat by nterpolaton between pre-compted state feedback gans as proposed n algortm 3, te control npt varable becomes contnos, and te smoot response s obtaned For eac control teraton, te average on-lne comptatonal tme reqred for algortm 3 s as low as s, becase all of te on-lne optmzaton problems are formlated n te form of lnear programmng Te algortm of Bmroongsr and Keawom (a) ses onlne optmzaton to compte a state feedback gan, and comptatonal tme of 3s s reqred for eac samplng tme 5 Conclsons In ts paper, we ave presented a fast RMPC sng polyedral nvarant sets A seqence of polyedral nvarant sets correspondng to a seqence of pre-compted state feedback gans s constrcted off-lne At eac samplng nstant, te smallest nvarant set contanng te crrently state measred s determned A state feedback gan s calclated by solvng a lnear programmng based on lnear nterpolaton between two pre-compted state feedback gans assocated wt crrent nvarant set determned and te adjacent smaller nvarant set Te smlaton reslts sow tat te proposed algortms can aceve better control performance tan exstng algortms ncldng te off-lne RMPC algortm based on polyedral nvarant sets wtot nterpolaton, and ts onlne conterpart RMPC algortm based on ellpsodal nvarant sets Acknowledgement Ts work was spport by te Hger Edcaton Researc Promoton and Natonal Researc Unversty Project of Taland, Offce of te Hger Edcaton Commsson (EN636A) and Integrated Innovaton Academc Center: IIAC Clalongkorn Unversty Centenary Academc Development Project (CU56- EN8), and Specal Task Force for Actvatng Researc (STAR), Clalongkorn Unversty Centenary Academc Development Project References Bakošová M, Oravec J,, Robst model predctve control of eat excangers, Cem Eng Trans 9, Bmroongsr P, Keawom S, a, Robst constraned MPC based on nomnal performance cost wt applcatons n cemcal processes, Proceda Eng 4, Bmroongsr P, Keawom S, b, An off-lne robst MPC algortm for ncertan polytopc dscretetme systems sng polyedral nvarant sets, J Process Contr, Martn P, Odloak D, Kassab F, 3, Robst model predctve control of a plot plant dstllaton colmn, Control Eng Pract,, 3-4 Plymers B, Rosster JA, Sykens JAK, Moor BD, 5, Te effcent comptaton of polyedral nvarant sets for lnear systems wt polytopc ncertanty, Proc Amercan Control Conf, 84-89

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