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1 Delf Unversy of Technology Delf Cener for Sysems and Conrol Techncal repor a A dsrbued opmzaon-based approach for herarchcal model predcve conrol of large-scale sysems wh coupled dynamcs and consrans: Exended repor M.D. Doan, T. Kevczky, and B. De Schuer Augus 011 A shor verson of hs paper has been publshed n he Proceedngs of he h IEEE Conference on Decson and Conrol and European Conrol Conference (CDC-ECC, Orlando, Florda, pp , Dec Delf Cener for Sysems and Conrol Delf Unversy of Technology Mekelweg, 68 CD Delf The Neherlands phone: (secreary fax: URL:hp:// Ths repor can also be downloaded vahp://pub.deschuer.nfo/abs/11_039a.hml

2 A dsrbued opmzaon-based approach for herarchcal model predcve conrol of large-scale sysems wh coupled dynamcs and consrans: Exended Repor Mnh Dang Doan, Tamás Kevczky, and Bar De Schuer Absrac We presen a herarchcal model predcve conrol approach for large-scale sysems based on dual decomposon. The proposed scheme allows couplng n boh dynamcs and consrans beween he subsysems and generaes a prmal feasble soluon whn a fne number of eraons, usng prmal averagng and a consran ghenng approach. The prmal updae s performed n a dsrbued way and does no requre exac soluons, whle he dual problem uses an approxmae subgraden mehod. Sably of he scheme s esablshed usng bounded subopmaly. I. INTRODUCTION Coordnaon and conrol of neracng subsysems s an essenal requremen for opmal operaon and enforcemen of crcal operaonal consrans n large-scale ndusral processes and nfrasrucure sysems [1]. Model Predcve Conrol (MPC has become he mehod of choce when desgnng conrol sysems for such applcaons [] [4], due o s ably o handle mporan process consrans explcly. MPC reles on solvng fne-me opmal conrol problems repeaedly onlne, whch may become prohbve for largescale sysems due o he problem sze or communcaon consrans. Recen effors have been focusng on how o decompose he underlyng opmzaon problem n order o arrve a a dsrbued or herarchcal conrol sysem ha can be mplemened under he prescrbed compuaonal and communcaon lmaons [5], [6]. One common way o decompose an MPC problem wh coupled dynamcs or consrans s o use dual decomposon mehods [7] [9], whch ypcally lead o erave algorhms (n eher a dsrbued or a herarchcal framework ha converge o feasble soluons only asympocally. Implemenng such approaches whn each MPC updae perod can be problemac for some applcaons. Recenly, we have presened a dual decomposon scheme for solvng large-scale MPC problems wh couplng n boh dynamcs and consrans, where prmal feasble soluons can be obaned even afer a fne number of eraons [10]. In he curren paper we presen a novel mehod ha s movaed by he use of consran ghenng n robus Ths repor s an exended verson of he paper A dsrbued opmzaon-based approach for herarchcal MPC of large-scale sysems wh coupled dynamcs and consrans, by M.D. Doan, T. Kevczky, and B. De Schuer, Proceedngs of he h IEEE Conference on Decson and Conrol and European Conrol Conference (CDC-ECC, Orlando, Florda, pp , Dec The auhors are wh Delf Cener for Sysems and Conrol, Delf Unversy of Technology, Delf, The Neherlands {m.d.doan,.kevczky, b.deschuer}@udelf.nl MPC [11], along wh a prmal averagng scheme and dsrbued Jacob opmzaon. Snce an exac opmum of he Lagrangan s no assumed o be compuable n fnely many eraons, an approxmae scheme s needed for solvng he MPC opmzaon problem a each me sep. We presen a soluon approach ha requres a nesed wo-layer eraon srucure and he sharng of a few crucal parameers n a herarchcal fashon. The proposed framework guaranees prmal feasble soluons and MPC sably usng a fne number of eraons wh bounded subopmaly. The paper s organzed as follows. In Secon II, we descrbe he MPC opmzaon problem and s ghened verson, whch wll be used o guaranee feasbly of he orgnal problem even wh a subopmal prmal soluon. Secon III descrbes he man elemens of he algorhm used o solve he dual verson of he ghened opmzaon problem: he approxmae subgraden mehod and he dsrbued Jacob updaes. In Secon IV, we show ha he prmal average soluon generaed by he approxmae subgraden algorhm s a feasble soluon of he orgnal opmzaon problem, and ha he cos funcon decreases hrough he MPC updaes. Ths allows o be used as a Lyapunov funcon for showng closed-loop MPC sably. Secon VI concludes he paper and oulnes fuure research. A. MPC problem II. PROBLEM DESCRIPTION We consder M nerconneced subsysems wh coupled dscree-me lnear me-nvaran dynamcs: x k+1 = M j=1 A j x j k +Bj u j k, = 1,...,M (1 and he correspondng cenralzed sae-space model: x k+1 = Ax k +Bu k ( wh x k = [(x 1 k T (x k T...(x M k T ] T,u k = [(u 1 k T (u k T...(u M k T ] T, A = [A j ],j {1,...,M} and B = [B j ],j {1,...,M}. The MPC problem a me sep s formed usng a convex cos funcon and convex consrans: mn u,x +N 1 k= (x TkQx k +u TkRu k +x T +NPx +N (3 1

3 s.. x k+1 = j N A j x j k +Bj u j k, = 1,...,M, k =,...,+N 1 (4 x k X,k = +1,...,+N 1 (5 x +N X f X (6 u k U,k =,...,+N 1 (7 u k Ω, = 1,...,M, k =,...,+N 1 (8 x = x( X (9 where u = [u T,...,u T +N 1 ]T, x = [x T +1,...,x T +N ]T, he marces Q, P, and R are block-dagonal and posve defne, he consran ses U, X and X f are polyopes and have nonempy nerors, and each local consran se Ω s a hyperbox. Each subsysem s assgned a neghborhood, denoed N, conanng subsysems ha have drec dynamcal neracons wh subsysem, ncludng self. The nal sae x s he curren sae a me sep. As U, X and X f are polyopes, he consrans (5 and (6 are represened by lnear nequales. Moreover, he sae vecor x s affnely dependen on u. Hence, we can elmnae sae varables x +1,...,x +N and ransform he consrans (4, (5, and (6 no lnear nequales of he npu varable u. Elmnang he sae varables n (3 (9 leads o an opmzaon problem n he followng form: f = mn u f(u,x (10 s.. g(u,x 0 (11 u Ω (1 where f and g = [g 1,...,g m ] T are convex funcons, and Ω = M =1 Ω wh each Ω = N 1 k=0 Ω s a hyperbox. Noe ha f(u,x > 0, u 0,x 0, due o he posve defneness of Q, P, and R. We wll use (u, x o denoe a feasble soluon generaed by he conroller for problem (3 (9 a me sep. Ths soluon s requred o be feasble bu no necessarly opmal.we wll make use of he followng assumpons: Assumpon.1: There exss a block-dagonal feedback gan K such ha he marx A + BK s Schur (.e., a decenralzed sablzng conrol law for he unconsraned aggregae sysem. Assumpon.: The ermnal consran se X f s posvely nvaran for he closed-loop x k+1 = (A + BKx k (x n(x f (A+BKx n(x f. Assumpon.3: The Slaer condon holds for problem (10 (1,.e., here exss a vecor ha sasfes src nequaly consrans [1]. I s also assumed ha pror o each me sep, a Slaer vecor ū s avalable, such ha g j (ū,x < 0,j = 1,...,m (13 Remark.4: Snce g(u,x 0 has a nonempy neror, so do s componens g j (u,x 0,j = 1,...,m. Hence, here wll always be a vecor ha sasfes he Slaer condon (13. In fac, we wll only need o fnd he Slaer vecor ū 0 for he frs me sep, whch can be compued off-lne. In Secon V-A we wll show ha a new Slaer vecor can hen be obaned for each 1, usng Assumpon.. Assumpon.5: A each me sep, he followng holds f(u 1,x 1 f(ū,x > x T 1Qx 1 +u T 1Ru 1 (14 For laer reference, we defne > 0 whch can be compued before me sep as follows: = x T 1Qx 1 +u T 1Ru 1 (15 Remark.6: Assumpon.5 s ofen sasfed wh an approprae ermnal penaly marxp. A mehod o consruc a block-dagonal P wh a gven decenralzed sablzng conrol law s provded n [13]. Assumpon.7: For each x X, he Eucldean norm of g(u,x s bounded: L g(u,x, u Ω (16 Remark.8: In he frs me sep, wh gven x 0, we can fnd L 0 by evaluang g(u,x 0 a he verces of Ω, he maxmum wll hen sasfy (16 for = 0, due o he convexy of g and Ω. For he subsequen me seps, we wll presen a smple mehod o updae L n Secon V-B. B. The ghened problem We wll no solve problem (10 (1 drecly. Insead, we wll make use of an erave algorhm based on a ghened verson of (10 (1. Consder he ghened consran: g (u,x g(u,x +1 m c 0 (17 wh g (u,x = [g 1,...,g m] T, 0 < c < mn j=1,...,m { g j (ū,x }, and 1 m he column vecor wh every enry equal o 1. Due o (13, we have max j=1,...,m {g j(ū,x } = max {g j(ū,x }+c < 0 (18 j=1,...,m Hence g j (ū,x < 0,j = 1,...,m. Moreover, usng (16 and he rangle nequaly of he -norm, we wll ge L = L +c as he norm bound forg,.e.l g (u,x, u Ω. Noe ha L mplcly depends on x, as ū and c are updaed based on he curren sae x. Usng he ghened consran (17, we formulae he ghened problem: f = mn u f(u,x (19 s.. g (u,x 0 (0 u Ω (1 Remark.9: Only he coupled consrans (11 are ghened, whle he local npu consrans (1 are unchanged. The Slaer condon also holds for he ghened problem (19 (1, wh ū beng he Slaer vecor. III. THE PROPOSED OPTIMIZATION ALGORITHM Our objecve s o calculae a feasble soluon for problem (3 (9 usng a mehod ha s favorable for dsrbued compuaon. The man dea s o use dual decomposon for he ghened problem (19 (1 nsead of he orgnal one, such ha afer a fne number of eraons he consran volaons n he ghened problem wll be less han he dfference beween he ghened and he orgnal consrans.

4 Thus, even afer a fne number of eraons, we wll oban a prmal feasble soluon for he orgnal MPC opmzaon problem. A. The dual problem We wll ackle he dual problem of (19 (1, n order o deal wh coupled consran g (u,x 0 n a dsrbued way. In hs secon, we defne he dual problem and s subgraden. For smplcy, n hs secon he dependence of funcons on he nal condon x s no ndcaed explcly. The Lagrangan of problem (19 (1 s defned as: L (u,µ = f(u+µ T g (u ( n whch u Ω,µ R m +. The dual funcon for (19 (1: q (µ = mn u Ω L (u,µ (3 s a concave funcon on R m +, and s non-smooh when f and g are no srcly convex funcons [1]. Gven he assumpon ha Slaer condon holds for (19 (1, dualy heory [1] shows ha: q = f (4 wh q = max µ R m + q (µ and f he mnmum of (19 (1. Thanks o hs resul, nsead of mnmzng he prmal problem, we may maxmze he dual problem, whch s ofen more amenable o decomposon due o smpler consrans. Snce we may no have he graden of q n all pons of R m +, we wll use a mehod based on he subgraden. Defnon 3.1: A vecor d s called a subgraden of a convex funcon f over X a he pon x X f: f(y f(x+(y x T d, y X (5 The se of all subgradens of f a he pon x s called he subdfferenal of f a x, denoed f(x. For each Lagrange mulpler µ R m +, frs assume we have u( µ = argmn u Ω L (u, µ. Then a subgraden of he dual funcon s drecly avalable, snce [1]: q (µ q ( µ+(µ µ T g (u( µ, µ R m + (6 In case an opmum of he Lagrangan s no aaned due o ermnaon of he opmzaon algorhm afer a fne number of seps, a value ũ( µ ha sasfes L (ũ( µ, µ mn u Ω L (u, µ+δ (7 wll lead o he followng nequaly: q (µ q ( µ+δ +(µ µ T g (ũ( µ, µ R m + (8 where g (ũ( µ s called δ-subgraden of he dual funcon q a he pon µ. The se of all δ-subgradens of q a µ s called δ-subdfferenal of q a µ. Ths means we do no have o look for a subgraden (or δ-subgraden of he dual funcon, s avalable by jus evaluang he consran funcon a he prmal value u( µ (or ũ( µ. B. The man algorhm We organze our algorhm for solvng (10 (1 a me sep n a nesed eraon of an ouer and nner loop. The man procedure s descrbed as follows: Algorhm 3.: Approxmae subgraden mehod wh nesed Jacob eraons 1 Gven a Slaer vecor ū of (10 (1, deermne c and consruc he ghened problem (19 (1. Deermne sep sze α and subopmaly ε, see laer n Secon III-C.1. 3 Deermne k (he suffcen number of ouer eraons, see laer n Secon III-C.. 4 Ouer loop: Se µ (0 = 0 1 m. For k = 0,..., k, fnd u (k,µ (k+1 such ha: L (u (k,µ (k mn u Ω L (u,µ (k +ε (9 { } µ (k+1 = P R m + µ (k +α d (k (30 wherep R m + denoes he projecon ono he nonnegave orhan, d (k = g ( u (k,x. Inner loop: Deermne p k (he suffcen number of nner eraons, see laer n Secon III-D.1. Solve problem (9 n a dsrbued way wh a Jacob algorhm. For p = 0,..., p k, every subsysem compues: u (p+1 =arg mn u Ω L (u 1 (p,...,u 1 (p, u, u +1 (p,...,u M (p,µ (k (31 where Ω s he local consran se for conrol varables of subsysem. Defne u (k [u 1 ( p k T,...,u M ( p k T ] T, whch s guaraneed o sasfy (9. 5 Compue û ( k = 1 k k u(l, ake u = û ( k as he soluon of (10 (1. Remark 3.3: Algorhm 3. s suable for mplemenaon n a herarchcal fashon where he man compuaons occur n he Jacob eraons and are execued n parallel by local conrollers, whle he updaes of dual varables and common parameers are carred ou by a hgher-level coordnang conroller. In he nner loop, each subsysem only needs o communcae wh s neghbors, whch wll be dscussed n Secon IV-A. Ths algorhm s also amenable o mplemenaon n dsrbued sengs, where here are communcaon lnks avalable o help deermne and propagae he common parameers α,ε, k, and p k. In he followng secons, we wll descrbe n deal how he compuaons are derved, and wha he resulng properes are. C. Ouer loop: Approxmae subgraden mehod The ouer loop a eraon k uses an approxmae subgraden mehod. The prmal average sequence û (k = 1 k k u(l has he followng properes: 3

5 For k 1 : [ g ( ] + û (k,x 1 ( 3 [f(ū,x q kα γ + α L +α L γ f (û (k,x f µ (0 + + α L kα ] (3 +ε (33 where g + denoes he consran volaon,.e. g + = max{g,0 1 m }. The proof of (3 can be found n [14], and he proof of (33 s gven n Appendx VII-A. 1 Deermnng α and ε : Usng he lower bound of he cos reducon (14 and he upper bound of he subopmaly (33 for he ghened problem (19 (1, we wll choose α and ε such ha f(u,x < f(u 1,x 1. The sep sze α and subopmaly ε should sasfy: α L +ε (34 where s defned n (15, and L s he norm bound for g. Ths condon allows us o show he decreasng propery of he cos funcon n problem (3 (9, whch can hen be used as a Lyapunov funcon. Noe ha a larger α wll lead o a smaller number of ouer eraons, whle a larger ε wll lead o a smaller number of nner eraons. For he remander of he paper we choose her values accordng o α = L (35 ε = (36 Deermnng k : Usng he consran volaon bound (3, we wll choose k such ha a he end of he algorhm, we wll ge a feasble soluon for problem (10 (1, whch s he average of prmal eraes generaed by (9: k û ( k = 1 k u (l (37 The subgraden eraon (9 (30 s performed for k = 1,..., k, wh he neger ( 1 3 k = f(ū,x + α L +α L (38 α c γ γ defned a pror, where s he celng operaor whch gves he closes neger equal o or above a real value, γ = mn j=1,...,m { g j (ū,x } = mn j=1,...,m { g j (ū,x } c, and ū s he Slaer vecor of (19 (1. D. Inner loop: Jacob mehod The nner eraon (31 performs parallel local opmzaons based on a sandard Jacob dsrbued opmzaon mehod for a convex funcon L (u,µ (k over a Caresan produc, as descrbed n [15, Secon 3.3]. In order o fnd he suffcen soppng condon of hs Jacob eraon, we need o characerze he convergence rae of hs algorhm. In he followng, we summarze he condon for convergence of he Jacob eraon, nong ha L (u,µ (k s a srongly convex quadrac funcon wh respec o u. Proposon 3.4: Suppose he followng condon holds: λ mn (H > j σ(h j, (39 where H j wh,j {1,...,M} denoes a submarx of he Hessan H of L w.r.. u, conanng enres of H n rows belongng o subsysem and columns belongng o subsysem j, λ mn means he smalles egenvalue, and σ denoes he maxmum sngular value. Then φ (0,1 such ha he aggregae soluon of he Jacob eraon (31 sasfes: u(p u Mφ p max u (0 u, p 1 (40 where u = argmn u Ω L (u,µ (k, and u s he componen of subsysem n u. We provde a proof for Proposon 3.4 n Appendx VII-B. Remark 3.5: Ths proposon provdes a lnear convergence rae of he Jacob eraon, under he condon of weak dynamcal couplngs beween subsysems. For he sake of llusrang condon (39, le all subsysems have he same number of npus. Consequenly, H j s a square and symmerc marx for each par (, j, hence he maxmum sngular value σ(h j equals o he maxmum egenvalue. Inequaly (39 hus reads: λ mn (H > j λ max (H j, whch mples ha he couplngs represened by H are small n comparson wh each local cos. Remark 3.6: Noe ha he srong convexy of L and he condon (39 are requred only for he convergence rae resul of he Jacob eraon n whch L s a quadrac funcon. Exensons o oher ypes of sysems, where he Lagrangan can be solved wh bounded subopmaly, are mmedae. In such cases we smply need o replace he Jacob eraon wh he new algorhm n he nner loop, whle he ouer loop wll reman nac. 1 Deermnng p k : As L (u, s connuously dfferenable n a closed bounded se Ω, s Lpschz connuous. Suppose we know he Lpschz consan Λ ofl (u, over Ω,.e. for any u 1, u Ω he followng nequaly holds: L (u 1,µ (k L (u,µ (k Λ u 1 u (41 Takng u 1 = u( p k and u = u n (41, and combnng wh (40, we oban: L (u( p k,µ (k mn u Ω L (u,µ (k Λ u( p k u ΛMφ p k max u (0 u (4 4

6 For each {1,...,M}, led denoe he dameer of he seω w.r.. he Eucldean norm, so we have u (0 u D. Hence he relaon (4 can be furher smplfed as L (u( p k,µ (k mn u Ω L (u,µ (k +ΛMφ p k maxd (43 Based on (43, n order o use u( p k as he soluon u (k ha sasfes (9, we choose he smalles neger p k such ha ΛMφ p k max D ε : ε p k = log φ (44 ΛM max D IV. PROPERTIES OF THE ALGORITHM A. Dsrbued Jacob algorhm wh guaraneed convergence The compuaons n he nner loop can be execued by subsysems n parallel. Le us defne an r-sep exended neghborhood of a subsysem, denoed by Nr, as he se conanng all subsysems ha can nfluence subsysem whn r successve me seps. Nr s he unon of subsysem ndces n he neghborhoods of all subsysems n Nr 1: Nr = N j (45 j N r 1 where N1 = N. We can see ha n order o ge updae nformaon n he Jacob eraons, each subsysem needs o communcae only wh subsysems nnn 1, wheren s he predcon horzon. Ths se ncludes all oher subsysems ha couple wh n he problem (10 (1 afer elmnang he sae varables. Ths communcaon requremen ndcaes ha we wll benef from communcaon reducon when he number of subsysems M s much larger han he horzon N, and he couplng srucure s sparse. Assume ha he weak couplng condon (39 holds, hen afer p k eraons as compued by (44, he Jacob algorhm generaes a soluon u (k u( p k ha sasfes (9 n he ouer loop. B. Feasble prmal soluon Proposon 4.1: Suppose Assumpons.1 and.3 hold. Consruc g as n (17, α as n (35. Le he ouer loop (9 (30 wh µ (0 = 0 1 m be eraed for k = 0,..., k. Then û ( k s a feasble soluon of (10 (1, where û ( k s he prmal average, compued by (37. Proof: Wh a fne number of k eraons (3 reads as [ g ( ] + û ( k,x 1 ( 3 [ f(ū,x q k ] α γ + α L +α L γ (46 Moreover, he dual funconq s a concave funcon, herefore q q (0,x. Recall ha f(u,x > 0, u 0,x 0, hus q (0,x = mn u Ω f(u,x T mg (u,x = mn u Ω f(u,x > 0, hus [ g ( û ( k,x ] + < 1 k α ( 3 γ f(ū,x + α L +α L γ (47 Combnng (47 wh (38, and nocng ha k and c are all posve lead o [ g ( ] + û ( k,x < c (48 g j (û ( k,x < c, j = 1,...,m (49 ( g j û ( k,x < 0, j = 1,...,m (50 where he las nequaly mples ha û ( k s a feasble soluon of problem (10 (1, due o c < mn j=1,...,m { g j (ū,x }. C. Closed-loop sably Proposon 4.: Suppose Assumpons.3,.5, and.7 hold. Then he soluon û ( k generaed by Algorhm 3. sasfes he followng nequaly: f(u,x < f(u 1,x 1, Z + (51 Proof: Usng (33 and (34, and nong ha µ (0 = 0, we oban: f (û ( k,x f µ (0 + + α L k α +ε f + (5 Noce ha ū s also a feasble soluon of (19 (1 (due o he way we consruc he ghened problem: ū sll belongs o he neror of he ghened consran se, whle f s he opmal cos value of hs problem. As a consequence, f f(ū,x (53 Combnng (5, (53, and (14, and nong ha u = û ( k leads o: f(u,x < f(u 1,x 1, Z + (54 Noe ha besdes he decreasng propery of f(u,x, all he oher condons for Lyapunov sably of MPC [16] are sasfed. Therefore, Proposon 4. leads o closedloop MPC sably, where he cos funcon f(u,x s a Lyapunov canddae funcon. V. REALIZATION OF THE ASSUMPTIONS In hs secon, we dscuss he mehod o updae he Slaer vecor and he consran norm bound for each me sep, mplyng ha Assumpons.3 and.7 are only necessary n he frs me sep ( = 0. 5

7 A. Updang he Slaer vecor Lemma 5.1: Suppose Assumpon. holds. Le u be he soluon of he MPC problem (3 (9 a me sep, compued by Algorhm 3.. Then ũ +1 consruced by shfng u one sep ahead and addng ũ +N = Kx +N, s a Slaer vecor for consran (11 a me sep +1. Proof: Noe ha based on Proposon 4.1, û ( k s a feasble soluon of problem (10 (1. Moreover, he src nequaly (50 means ha û ( k s n he neror of he consran se of (3 (9. Ths also yelds: x +N n(x f (55 Moreover, due o Assumpon., we have (A + BKx +N n(x f. Ths means ha f we use ũ +N = Kx +N, hen he nex sae s also n he neror of he ermnal consran se X f. Noe ha U and X do no change when problem (3 (9 s shfed from o + 1, hence all he npus of ũ +1 and her subsequen saes are n he neror of he correspondng consran ses. Therefore, ũ +1 as consruced a sep 5 of Algorhm 3. s a Slaer vecor for he consran (11 a me sep +1. Ths means we can use ū +1 = ũ +1 as he qualfyng Slaer vecor for Assumpon.3 a me sep +1. B. Updang he consran norm bound In our general problem seup,g(u,x s composed of affne funcons over u and x, and hus can be wren compacly as g(u,x = Ξx+Θu+τ (56 wh consan marces Ξ,Θ and vecor τ. Then for each x 1, x, and u Ω, he followng holds: g(u,x = g(u,x 1 +Ξ(x x 1 g(u,x g(u,x 1 + Ξ(x x 1 (57 In order o fnd a bound L for g(u,x n each 1 sep, we assume o have he consran norm bound avalable from he prevous sep: L 1 g(u,x 1, u Ω (58 Hence, combnng he above nequales a norm bound updae for g(u,x can be obaned as: L = L 1 + Ξ(x x 1 (59 VI. CONCLUSIONS We have presened a consran ghenng approach for solvng an MPC opmzaon problem wh guaraneed feasbly and sably afer a fne number of eraons. The new mehod s applcable o large-scale sysems wh couplng n dynamcs and consrans, and he soluon s based on approxmae subgraden and Jacob erave mehods, whch faclae mplemenaon n a herarchcal or dsrbued way. Fuure exensons of hs scheme nclude a poseror choce of he soluon by comparng he cos funcons assocaed wh he Slaer vecor ū and he prmal average û ( k n a dsrbued way. ACKNOWLEDGMENT The auhors would lke o hank Ion Necoara for helpful dscussons on he opc of hs paper. Research suppored by he European Unon Sevenh Framework STREP projec Herarchcal and dsrbued model predcve conrol (HD-MPC, conrac number INFSO-ICT-3854, and he European Unon Sevenh Framework Programme [FP7/ ] under gran agreemen no HYCON Nework of Excellence. REFERENCES [1] J. B. Rawlngs and B. T. Sewar, Coordnang mulple opmzaonbased conrollers: New opporunes and challenges, Journal of Process Conrol, vol. 18, pp , Oc [] J. M. Macejowsk, Predcve Conrol wh Consrans. Harlow, England: Prence-Hall, 00. [3] E. F. Camacho and C. Bordons, Model Predcve Conrol. London: Sprnger, [4] J. B. Rawlngs and D. Q. Mayne, Model Predcve Conrol: Theory and Desgn. Madson, WI: Nob Hll Publshng, 009. [5] R. Scaoln, Archecures for dsrbued and herarchcal model predcve conrol - A revew, Journal of Process Conrol, vol. 19, pp , May 009. [6] A. Venka, I. Hskens, J. Rawlngs, and S. Wrgh, Dsrbued MPC sraeges wh applcaon o power sysem auomac generaon conrol, IEEE Transacons on Conrol Sysems Technology, vol. 16, pp , Nov [7] Y. Wakasa, M. Arakawa, K. Tanaka, and T. Akash, Decenralzed model predcve conrol va dual decomposon, n 47h IEEE Conference on Decson and Conrol, pp , 008. [8] I. Necoara and J. Suykens, Applcaon of a smoohng echnque o decomposon n convex opmzaon, IEEE Transacons on Auomac Conrol, vol. 53, pp , Dec [9] D. Doan, T. Kevczky, I. Necoara, M. Dehl, and B. De Schuer, A dsrbued verson of Han s mehod for DMPC usng local communcaons only, Conrol Engneerng and Appled Informacs, vol. 11, pp. 6 15, Sep [10] M. D. Doan, T. Kevczky, and B. De Schuer, A dual decomposonbased opmzaon mehod wh guaraneed prmal feasbly for herarchcal MPC problems, n 18h IFAC World Congress, (Mlan, Ialy, Aug [11] Y. Kuwaa, A. Rchards, T. Schouwenaars, and J. P. How, Dsrbued robus recedng horzon conrol for mulvehcle gudance, IEEE Transacons on Conrol Sysems Technology, vol. 15, 007. [1] D. P. Bersekas, Nonlnear programmng. Belmon, MA: Ahena Scenfc, [13] D. D. Šljak, Large-scale dynamc sysems: Sably and srucure. New York, NY: Norh Holland, [14] A. Nedc and A. Ozdaglar, Approxmae prmal soluons and rae analyss for dual subgraden mehods, SIAM Journal on Opmzaon, vol. 19, pp , Nov [15] D. P. Bersekas and J. N. Tsskls, Parallel and Dsrbued Compuaon: Numercal Mehods. Upper Saddle Rver, NJ: Prence-Hall, [16] D. Q. Mayne, J. B. Rawlngs, C. V. Rao, and P. O. M. Scokaer, Consraned model predcve conrol: Sably and opmaly, Auomaca, vol. 36, pp , June 000. VII. APPENDIX A. Proof of he upper bound on he cos funcon (33 Ths proof s an exenson of he proof of Proposon 3(b n [14], he man dfference beng he ncorporaon of he subopmaly ε n he updae of he prmal varable (9. Usng he convexy of he cos funcon, we have: ( k 1 f(û (k 1 = f u (l 1 k 1 f(u (l k k 6

8 = 1 k 1 ( f(u (l +(µ (l T g (u (l 1 k 1 (µ (l T g (u (l k k (60 Noe ha L ( u (l,µ (l = (f(u (l +g (u (l T µ (l and L ( u (l,µ (l mn u Ω L ( u (l,µ (l +ε = q ( µ (l +ε, l < k (61 Combnng he wo nequales above, we hen have: f(û (k 1 k 1 q ( µ (l +ε 1 k 1 (µ (l T g (u (l k k q +ε 1 k k 1 (µ (l T d (l (6 where d (l = g (u (l, and he las nequaly s due o q q ( µ (l, l. Usng he expresson of squared sum: µ (l+1 µ (l +α d (l = µ (l +α (µ (l T d (l + α d (l (63 we have: (µ (l T d (l 1 ( µ (l µ (l+1 +α α d (l (64 for l = 0,...,k 1. Summng sde by sde for l = 0,...,k 1, we ge: k 1 (µ (l T d (l 1 α + α ( µ (0 µ (k k 1 d (l (65 Lnkng (6 and (65, we hen have: f(û (k q +ε + 1 ( µ (0 µ (k kα + α k 1 d (l k q µ (0 + + α L kα +ε (66 n whch we ge he las nequaly by usng L as he norm bound for all g (u (l,l = 0,...,k 1. Fnally, wh he Slaer condon, here s no prmal-dual gap,.e. q = f (cf. (4, hence: f(û (k f µ (0 + + α L kα +ε B. Proof of he convergence resul of he Jacob eraon (Proposon 3.4 Accordng o Proposon 3.10 n [15, Chaper 3], he Jacob algorhm has a lnear convergence w.r.. he blockmaxmum norm, as defned below: Defnon 7.1: For each vecor x = [x T 1,...,x T M ] wh x R n, gven a norm for each, he block-maxmum norm based on s defned as: x b-m = max x (67 Defnon 7.: Wh any marx A R n nj, we assocae he nduced marx norm of he block-maxmum norm: Ax A j = max = max x 0 x Ax (68 j x j=1 In hs paper, we use he Eucldean norm as he defaul bass for block-maxmum norm,.e. =,. Proposon 3.10 n [15, Chaper 3] saes ha u(p generaed by (31 wll converge o he opmzer of L (u,x wh lnear convergence rae w.r.. block-maxmum norm (.e. u(p u b-m φ p u(0 u b-m, wh u = argmn u L (u,x and φ [0,1 f here exss a posve scalar γ such ha he mappng R : Ω R nu, defned by R(u = u γ u L (u,x, s a conracon w.r.. he blockmaxmum norm. Our focus now s o derve he condon such ha R(u s a conracon mappng. Noe ha snce f(u,x s a quadrac funcon, and g (u,x conans only lnear funcons, he funcon L (u,x s also a quadrac funcon w.r.. u, hence can be wren as: L (u,x = u T Hu+b T u+c (69 where H s a symmerc, posve defne marx, b s a consan vecor and c s a consan scalar. In order o derve he condon for R(u o be a conracon mappng, we wll make use of Proposon 1.10 n [15, Chaper 3], sang ha: If f : R nu R nu s connuously dfferenable and here exss a scalar φ [0,1 such ha I γg 1 ( F (u T + j γg 1 ( j F (u T j φ, u Ω, (70 hen he mappng T : Ω R nu defned wh each componen {1,...,M} by T (u = u γg 1 F(u s a conracon wh respec o he block-maxmum norm. The mappng T(u wll become he mappng R(u f we ake G = I n u, and F(u = u L (u,x = Hu + b. Wh such choce, and evaluang he nduced marx norm (68 n (70, he condon for conracon mappng of R(u s o fnd φ [0,1 such ha: I n u γh + j γh j φ, (71 7

9 where H j wh,j {1,...,M} denoes he submarx of H, conanng enres a rows belongng o subsysem and columns belongng o subsysem j. Noe ha he marx nsde he frs nduced marx norm s a square, symmerc marx, whle he marces H j are generally no symmerc, dependng on he number of varables of each subsysem. The scalar φ [0,1 s also he modulus of he conracon. Usng he properes of egenvalue and sngular value of marces, we ransform (71 no he followng nequaly: max γλ(h 1 +γ σ(h j φ, (7 λ j where λ means egenvalue, and σ denoes he maxmum sngular value. In order o fnd γ > 0 and φ [0,1 sasfyng (7, we need: Noe ha he closer of φ o 0, he faser he aggregae updae u(p converges o he opmzer of he Lagrange funcon. In order o ge he convergence rae w.r.. he Eucldean norm, we wll need o lnk from he Eucldean norm o he block-maxmum norm: x M =1 x M max x = M x b-m (80 Hence, he convergence rae of Jacob eraon (31 w.r.. he Eucldean norm s: u(p u Mφ p max u (0 u, p 1 (81 max γλ(h 1 +γ σ(h j < 1, (73 λ j { γλmax (H 1+γ j σ(h j < 1 1 γλ mn (H +γ j σ(h, (74 j < 1 { ( γ < 1/ λ max (H + j σ(h j λ mn (H > j σ(h, (75 j The frs nequaly of (75 shows how o choose γ, whle he second nequaly of (75 needs o be sasfed by he problem srucure, whch mples here are weak dynamcal couplngs beween subsysems. In summary, he mappng R(u sasfes (70 and hus s a conracon mappng f he followng condons hold: 1 For all : λ mn (H > j The coeffcen γ s chosen such ha: γ < σ(h j (76 1 λ max (H + j σ(h, (77 j So, when condon (76 s sasfed and wh γ chosen by (77, we can defne φ (0,1 as: { { φ = max max γ ( λ max (H + σ(h j 1, j 1 γ ( λ mn (H j σ(h j }} (78 Ths φ s he modulus of he conracon R(u, and also acs as he coeffcen of he lnear convergence rae of he Jacob eraon (31, whch means: u(p u b-m φ p u(0 u b-m, p 1 (79 where u = argmn u Ω L (u,x. 8