Distributed Model Predictive Control with Actuator Saturation for Markovian Jump Linear System

Transcription

1 374 IEEE/CAA JOURNAL OF AUOMAICA SINICA VOL 2 NO 4 OCOBER 2015 Dstrbuted Model Predctve Control wth Actuator Saturaton for Markovan Jump Lnear System Yan Song Hafeng Lou and Shua Lu Abstract hs paper s concerned wth the dstrbuted model predctve control MPC) problem for a class of dscrete-tme Markovan jump lnear systems MJLSs) subject to actuator saturaton and polytopc uncertanty n system matrces he global system s decomposed nto several subsystems whch coordnate wth each other A set of dstrbuted controllers s desgned by solvng a mn-max optmzaton problem n terms of the solutons of lnear matrx nequaltes LMIs) An teratve algorthm s developed to acheve the onlne computaton Fnally a smulaton example s employed to show the effectveness of the proposed algorthm Index erms Dstrbuted model predctve control MPC) actuator saturaton Markovan jump lnear system MJLS) lnear matrx nequalty LMI) I INRODUCION MANY real systems such as solar thermal central recevers economc systems and manufacturng systems are subject to random abrupt changes n ther nputs nternal varables and other system parameters 1 It s dffcult to descrbe the above systems usng conventonal lnear tmenvarant systems However Markovan jump lnear systems MJLSs) 2 4 have been put forward to successfully solve ths problem herefore n the past decades MJLSs have receved consderable nterest and we can fnd examples n 5 7 Snce t s often dffcult to dentfy exact model parameters the control of MJLSs wth uncertanty n model parameters has been taken an mportant consderaton of 1 Wth the ncreasng development of computer scence and technology dscrete-tme MJLSs has been appled n a recedng horzon control scheme 8 Model predctve control MPC) whch s also called recedng horzon control has drawn extensve attenton from both theoretcal and ndustral communtes wth ts advantages n processng large-scale constrant mult-varable pro- Manuscrpt receved October ; accepted February hs work was supported n part by Natonal Natural Scence Foundaton of Chna ) Shangha Pujang Program 13PJ ) Shangha Natural Scence Foundaton of Chna 13ZR ) Innovaton Program of Shangha Muncpal Educaton Commsson 14YZ083) and Hujang Foundaton of Chna C14002 B1402/D1402) Recommended by Assocate Edtor Yugeng X Ctaton: Yan Song Hafeng Lou Shua Lu Dstrbuted model predctve control wth actuator saturaton for Markovan jump lnear system IEEE/CAA Journal of Automatca Snca ): Yan Song and Hafeng Lou are wth the Department of Control Scence and Engneerng Unversty of Shangha for Scence and echnology Key Laboratory of Modern Optcal System Engneerng Research Center of Optcal Instrument and System Mnstry of Educaton Shangha Chna e-mal: sonya@ussteducn; lhfbalance@163com) Shua Lu s wth the Busness School Unversty of Shangha for Scence and echnology Shangha Chna e-mal: lushua871030@163 com) cesses Nowadays the systems to be handled are ncreasngly complex 9 It s pretty dffcult to control these systems wth a centralzed MPC control strategy due to the computatonal complexty and communcaton bandwdth lmtatons 10 Recently the dstrbuted MPC framework where each subsystem s controlled by an ndependent controller has been ncreasngly focused on for ts flexblty of system structure error-tolerance less computatonal efforts and no requrements for global nformaton 11 In dstrbuted MPC algorthm we dvde the global system nto several subsystems and each subsystem exchanges nformaton wth others va network In 12 Ja et al provded a communcaton based on MPC scheme where each subsystem optmzed a local cost functon and made coordnaton va communcaton Mercangöz and Doyle 13 put forward an teratve mplementaton of a resemblng dstrbuted MPC scheme and appled t to a fourtank system hese communcaton based on dstrbuted MPC algorthm only acheved a Nash-optmzaton 14 A robust dstrbuted MPC algorthm was proposed n n whch the model uncertan problem was converted nto solvng a lnear matrx nequalty LMI) optmzaton problem In 17 Dng addressed the dstrbuted MPC of a set of polytopc uncertan local systems wth decoupled dynamcs In ths paper we propose a dstrbuted MPC algorthm for a class of uncertan dscrete-tme MJLSs subject to actuator saturaton We use state feedback to ensure the stablty of system hus we assume that system states are measurable We decompose the global system nto several subsystems and a mn-max dstrbuted MPC strategy s proposed for the polytopc uncertan dstrbuted systems In order to better reflect the practcal system under the networked envronment we consder the MJLSs subject to actuator saturaton At each tme step an MPC algorthm optmzes a cost functon whch s assocated wth the future states and manpulated varables 9 to obtan a control nput In next tme step a new cost functon s formulated and solved based on the new measurements An teratve algorthm s developed to get suboptmzatons of the neghbor subsystems he rest of ths paper s organzed as follows Secton I descrbes the problem statement In Secton II based on the state feedback control law a dstrbuted MPC algorthm for nput-saturated polytopc uncertan systems s proposed he effectveness of the proposed approach s llustrated n Secton III wth the help of numercal examples Fnally we gve a concluson to our work n Secton IV Notatons he followng notatons are used throughout the paper R n and R nm denote the n-dmensonal Eucldean space and the set of all nm real matrces respectvely x k

2 SONG et al: DISRIBUED MODEL PREDICIVE CONROL WIH ACUAOR SAURAION FOR MARKOVIAN JUMP LINEAR SYSEM 375 n k) denotes the state of subsystem at tme kn predcted based on the measurements at tme k u k n k) refers to the predcted control move at tme step k n and u k k) s the control move to be mplemented at tme step k x 2 denotes the Eucldean norm of the vector x I and 0 denote the dentty matrx and zero matrx of compatble dmenson respectvely E{x} stands for the expectaton of stochastc varable x G > 0 means matrx G s real symmetrc and postve defnte he symbol denotes the symmetrc part n a symmetrc matrx II PROBLEM SAEMEN Consder a certan dscrete-tme MJLS subject to actuator saturaton descrbed by { xk 1) = A ξ) r k )xk) B ξ) r k )σuk)) 1) yk) = C ξ) r k )xk) where xk) R nx s the system state; uk) R nu s the control nput; yk) R ny s the controlled output; r k s the system mode he ntal state s x 0 and the ntal mode s r 0 Accordng to the defnton n 6 we can see the true state nvolves n two parts: the contnuous parts e xk) and the dscrete part e the mode r k For a fxed mode r k the MJLS s a lnear tme-varyng system n essence In another word mode r k e the present mode s known whle the next mode r k1 s stochastc and uncertan Let r k k 0 N) be a Markov chan takng values n a fnte state space M = {1 2 S} wth transton probablty gven by pg h) = prob {r k1 = h r k = g} g h M 2) where pg h) 0 g h M) s the transton probablty from g to h and S pg h) = 1 g M It s assumed that for r k M A ξ) r k ) B ξ) r k ) and C ξ) r k ) are unknown matrces whch are nvolved n a convex polyhedral set Ωr k ) descrbed by L vertces { Ωr k ) Co A 1) r k ) B 1) r k ) C 1) r k ) A 2) r k ) B 2) r k ) C 2) r k ) } A L) r k ) B L) r k ) C L) r k ) where Co refers to the convex hull that s A ξ) r k ) B ξ) r k ) C ξ) r k ) = L l=1 λ l A l) r k ) B l) r k ) C l) r k ) 3) where L l=1 λ l = 1 λ l 0 Remark 1 he functon σ : R m R m s the standard saturaton functon of approprate dmensons defned as σuk)) = σu 1 k)) σu 2 k)) σu m k)) where u k) = sgnu)mn{1 u } where the notaton of sgn denotes the sgnum functon o be convenent we defne ū k) = σu k)) We decompose the global system nto M subsystems and desgn the correspondng dstrbuted MPC controllers for them In the followng we gve some assumptons for our dscusson Assumpton 1 he vector of states x = x 11 x x MM x R nx ncludes all the states of the system that s the local state x that can be measured or estmated whle the other subsystems states could be obtaned by communcaton va network hus the states of subsystems are the same to the global system that s to say x 1 = x 2 = = x we only dscuss the nputs decomposton n ths paper For smplcty we denote the state value of -th subsystem as x to stress the vector to be used for computng the local control nput u Assumpton 2 In ths paper we neglect the effect of packet loss packet dsorder tme delay etc hus the dstrbuted polytopc uncertan system models wth actuator saturaton by decomposton can be descrbed by x k 1) = A ξ) r k )x k) B ξ) r k )σu k)) M B ξ) j r k )σu j k)) 4) j=1j y k) = C ξ) r k )x k) = 1 2 M where x R nx s the system state of subsystem ; u R nu s the control nput of subsystem ; and y R ny s the controlled output of subsystem A ξ) r k ) B ξ) r k ) C ξ) L = l=1 λ l A l) r k ) r k ) B l) r k ) C l) r k ) 5) Remark 2 It s mportant to reduce the computaton complexty to apply the MPC algorthm to the practcal system In 18 by employng a scalar a an MPC algorthm subject to certan LMIs was presented In ths paper by decomposng the control nput that s n u system 4) becomes a dstrbuted MPC problem Compared wth the centralzed MPC the dstrbuted MPC strateges can reduce the computatonal complexty and relax communcaton bandwdth restrctons Besdes wth the nevtable parameter uncertantes taken nto consderaton the problem s also a robust MPC Remark 3 Owng to the lmtaton of the communcaton bandwdth the large nformaton-carry packets may be saturated at actuator node he actuator saturaton s an essental ssue n the control communty whch leads to nonlnearty of the system and has beng focused on n recent years such as n 20 In ths paper we dscuss the dstrbuted MPC problem for MJLs wth actuator saturaton he am of ths paper s to fnd a set of state feedback laws as follows: = M =1 n u u k n k) = F k r kn k )x k n k) 6) n order to stablze the system For a gven F R mnx let φf ) = {x R nx : fα 1 α = 1 m } where fα s the α-th row of matrx

3 376 IEEE/CAA JOURNAL OF AUOMAICA SINICA VOL 2 NO 4 OCOBER 2015 F In ths way t can ensure the robust stablty of certan dscrete-tme MJLS wth actuator saturaton In the followng we provde three lemmas that are used n ths paper Lemma 1 19 Let P R nn P > 0 and ρ k) > 0 An ellpsod ΩP ρ k)) = { x R nx : x P x ρ k) } s nsde φf ) f and only f fαp 1 f α ρ 1 k) α = 1 m Let Ξ be the set of m m dagonal matrces whose dagonal elements are ether 0 or 1 hen there are 2 m elements n Ξ Suppose that elements of Ξ are labeled as E β wth β {1 2 m } Denote E β = I E β obvously E β s also an element of Ξ f E β Ξ Lemma 2 20 Let F H R mnx be gven Supposng that h α x 1 for all α = 1 m then we have } σf x ) Co {E β F x E βh x β = 1 2 m where h α s the αth row of the matrx H In ths way the saturated feedback σf x ) can be transformed nto the convex hull of a group of lnear feedbacks Lemma 3 11 Let Aλ) Bλ) be polytopc uncertan as Aλ) = L =1 λ A Bλ) = L =1 λ B and let F be a constant vector Aλ) Bλ) and F are of approprate dmensons hen Sλ) = Aλ) Bλ)F s polytopc uncertan A Controller Desgn III MAIN RESULS In ths secton the stablty of the dstrbuted polytopc certan dscrete-tme MJLS subject to actuator saturaton s dscussed and the correspondng dstrbuted MPC controllers are desgned o dscuss a dstrbuted MPC problem under the state feedback control an upper bound on a robust performance objectve s mnmzed for every subsystem defned by 4)-6) he mn-max problem on the objectve s solved as follows: wth mn u kn k) n 0 =1M max A ξ) r kn k )B ξ) r kn k ) C ξ) r kn k ) Ωr k ) =1M r kn k M J k) = E x k n k) 2 Q r kn k ) n=0 ) ū k n k) 2 R r kn k) =1 J k) 7) where Q > 0 and R > 0 are symmetrc weghtng matrces Wth the overall cost functon taken nto consderaton n 7) t becomes a cooperatve dstrbuted MPC algorthm subject to actuator saturaton Applyng the state feedback law 6) to 4) results n the followng closed-loop system: x k n 1 k) = A ξ) r kn k )x k n k) B ξ) r kn k )σf k r kn k )x k n k)) j=1j σf j k r kn k )x j k n k)) 8) where the term M j=1j σf jk r kn k )x j k n k)) represents the effects of neghbor subsystems he coupled nputs are saturated and cannot be handled drectly In order to derve the closed-loop system we ntroduce an unsaturated feedback law Γ j k r kn k ) whch s restructured as follows: F jα k r kn k ) F jα k r kn k )x j k n k) 1 Γ jα k r kn k ) = F jp k r kn k ) Fjp k r kn k )x j k) F jα k r kn k )x j k n k) > 1 9) where α = 1 m Γ jα k r kn k ) stands for the α-th row of Γ j k r kn k ) F jα k r kn k ) refers to the α-th row of F j k r kn k ) hen we have ū j k n k) = σu j k n k)) = σf j k r kn k )x j k n k)) = Γ j k r kn k )x j k n k) 10) In the sequel of the paper for presentaton convenence we denote each possble r kn k = p p M) he closedloop system 8) by applyng unsaturated-nput couplngs can be rewrtten as follows: x k n 1 k) where ˆB ξ) = A ξ) = p)x k n k) B ξ) p) j=1j Âξ) p) Â ξ) E β F k p) E β H k p) B ξ) j p)γ j k p)x k n k) ) x k n k) ξ) ˆB p) x k n k) 11) p) = A ξ) p) M j=1j Bξ) p)γ j k p) p)e β F k p) E β H k p)) By Lemma p) = B ξ) 3 we can conclude Âξ) p) s a polytopc uncertanty Consder the followng quadratc functon: V k n k) = x k n k)p k p)x k n k) where = 1 M n 0 p M Accordng to the stochastc swtchng of the dscussed system 4) n certan gven modes the system should be a stochastc one and then the quadratc functons V kn k) should be stochastc hus durng the stablty analyss we should take the expectaton of the quadratc functons V k n k) o derve an upper bound on 7) we mpose the followng robust stablty constrants E V k n 1 k) V k n k) E x k n k)q p)x k n k)

4 SONG et al: DISRIBUED MODEL PREDICIVE CONROL WIH ACUAOR SAURAION FOR MARKOVIAN JUMP LINEAR SYSEM 377 ū k n k)r p)ū k n k) ū j k n k)r j p)ū j k n k) j=1j 12) heorem 1 For each subsystem robust stablty constrants 12) are satsfed f there exst a postve scalar ρ k) postve defnte matrces W k p) = ρ k)p 1 k p) > 0 and any matrces Y k p) = F k p)w k p) such that the followng LMIs W k p) Φ k p) p 1 p 1)W k 1) Φ k p) 0 Ψ p) 1 2 W k p) ) 0 R 1 2 p) E β Y k p) E β Z k p) 0 < 0 p 1 p S)W k S) 0 ρ k)i 0 0 ρ k)i hold hen the gan matrx 13) ) x k) W r k ) Y k p) = F k p)w k p) Z k p) = H k p)w k p) and the upper bound of the optmzaton problem s ρ k) where Φ k p) = Âξ) p)w k p) B l) p) Ψ p) = Qp) E β Y k p) E β Z k p) j=1j ) Γ j k p)r j p)γ j k p) Proof he dfference of V k n k) along system 11) can be calculated as follows: E V k n k) = E { V k n 1 k) V k n k) } = E { x k n 1 k)p k r kn1 k )x k n 1 k) } x k n k)p k p)x k n k) = x k n k) { Âξ) p) B ξ) p) E β F k p) E β H k p) ) E P k r kn1 k ) Âξ) p) B ξ) p) E β F k p) E β H k p) ) } P k p) x k n k) 15) Furthermore 12) can be reformulated as follows: E V k n k) E x k n k)q p)x k n k) ū k n k)r p)ū k n k) ū j k n k)r j p)ū j k n k) 16) j=1j By applyng 9) nto 16) we can get followng nequalty: E V k n k) { E x k n k) Q p) E β F k p) E β H k p) ) ) R p) E β F k p) E β H k p) j=1j Accordng to 2) t s clear that } Γ j k p)r j p)γ j k p) x k n k) 17) EP k r kn1 k ) = pg h)p k h) hen combnng 15) and 17) we can obtan the followng nequalty: x k n k) {  ξ) ) p) B ξ) p) E β F k p) E β H k p) pg h)p k h)  ξ) ) p) B ξ) p) E β F k p) E β H k p) P k p) Q p) j=1j Γ j k p)r j p)γ j k p) E β F k p) E β H k p)) R p) E β F k p) E β H k p) )} x k n k) 0 18) Accordng to 5) the above nequalty s satsfed f and only f  l) ) p) B l) p) E β F k p) E β H k p) pg h)p k h)  l) ) p) B l) p) E β F k p) E β H k p)

5 378 IEEE/CAA JOURNAL OF AUOMAICA SINICA VOL 2 NO 4 OCOBER 2015 P k p) Q p) j=1j Γ j k p)r j p)γ j k p) E β F k p) E β H k p)) R p) E β F k p) E β H k p) ) 0 19) Pre- and post-multplyng 19) wth P 1 k p) result n  l) p)p 1 k p) ) B l) p) E β F k p) E β H k p) P 1 k p) pg h)p k h)  l) p)p 1 k p) ) B l) p) E β F k p) E β H k p) P 1 k p) P 1 j=1j k p) P 1 k p)q p)p 1 k p) Γj k p)p 1 k p) ) Rj p) Γ j k p)p 1 k p) ) E β F k p)p 1 k p) E β H k p)p 1 k p)) R p) E β F k p)p 1 k p) E β H k p)p 1 k p) ) 0 20) hen nequaltes 20) can be further smplfed by multplyng wth ρ k)  l) ) p)w k p) B l) p) E β Y k p) E β Z k p)  l) pg h)w 1 k h) ) p)w k p)b l) p) E β Y k p) E β Z k p) W k p) W k p)ρ 1 k)q p)w k p) W k p)ρ 1 k) Γ j k p)r j p)γ j k p)w k p) j=1j ) E β Y k p) E β Z k p) ρ 1 k)r p) E β Y k p) E β Z k p)) } 0 21) By usng the Schur complement f LMI 13) holds the above nequalty wll be satsfed In the followng based on the robust stablty constrants 12) an upper bound on the worst-case nfnte horzon 7) can be obtaned Frstly takng the expected values on the both sdes of 12) condtonal on the nformaton avalable at tme nstant k the followng nequalty holds E V k n 1 k) V k n k) E x k n k)q p)x k n k) ū k n k)r p)ū k n k) j=1j ū j k n k)r j p)ū j k n k) hen summng up the above nequalty from n = 0 to on both sdes we have E V k) V k) { E x k n k)q p)x k n k) n=0 ū k n k)r p)ū k n k) j=1j ū j k n k)r j p)ū j k n k) } Because the robust performance objectve functon J k) should to be fnte we must have x k) = 0 whch leads to V k) = 0 J k) = E x k n k) 2 Q r kn k ) that s n=0 ) ū k n k) 2 R r kn k) =1 E V k) J k) E V k) = x k)p k r k )x k) By defnng an upper bound x k)p k r k )x k) ρ k) then by Schur complement t can be wrtten as 14) and we can obtan J k) ρ k) We can see that ρ k) s an upper bound on the expected cost functon Usng Lemma 1 condton h α k)x k) 1 can be wrtten as follows: 1 zα k) 0 α = 1 m 22) z α k) W k) where z α k) = W k)h α k) Next we wll mnmze upper bound to approxmately mnmze the worst-case nfnte horzon expected cost functon Now we are ready to present the unconstraned dstrbuted MPC of MJLSs wth polytopc uncertantes n terms of a mnmzaton problem at each tme nstant k as follows: mn ρ k) st 13) 14) 22) 23) W kp)y kp)p M In the followng theorem we wll dscuss the feasblty and stablty of 23) heorem 2 Consder the MJLSs wth actuator saturaton descrbed by 3)-5) If there s a feasble soluton to optmzaton problem 23) at tme nstant k for ntal state xk) and ntal mode r k there wll also exst a feasble soluton at tme

6 SONG et al: DISRIBUED MODEL PREDICIVE CONROL WIH ACUAOR SAURAION FOR MARKOVIAN JUMP LINEAR SYSEM 379 nstant t k; and the dstrbuted MPC control law F k r k ) = Y k r k )W 1 k r k ) based on 23) guarantees closed-loop stablty of the system n the mean-square sense Proof he proof of ths theorem s smlar to 11 and 15 If the soluton to problem 23) at tme k s feasble then t s also feasble at all future samplng steps k n n > 0 hs s because the only constrant that depends on the states n problem 23) s constrant 14) e 1 x k)w r k )x k) 0 where the states are gven by x k n) = Âξ) p) ˆB ξ) p)x kn 1) hs constrant can be feasble by usng the defnton of nvarant set that s satsfed at tme k see 11 and 15) B Iteraton Algorthm Notce that there are couplngs n LMIs 13) when solvng problem 23) We propose an teratve dstrbuted MPC algorthm n ths secton Iteratve dstrbuted MPC algorthm subject to actuator saturaton of -th subsystem s as follows: Step 1 At tme step k = 0 set an ntal feedback law F 0 r k ) = 1 2 M Step 2 At tme step k all subsystems exchange ther local states measurements and ntal feedback law F k r k ) va communcaton set teraton t = 1 and F k r k ) = F 0 r k ) Step 3 All subsystem solve LMI problems 23) n parallel to obtan the optmal Y t) k r k ) and W t) k r k ) to estmate the optmal feedback laws F t) k r k ) = Y t) k r k ) W 1t) k r k ) Each subsystem checks the convergence wth a specfed error tolerance η for all the feedback laws F t) k r k ) F t 1) k r k ) η = 1 2 M If the convergence condton or t = t max s satsfed current F t) k r k ) s taken as the optmal feedback law; Otherwse update the ntal feedback laws wth F k r k ) = F t) k r k ) and set t = t1 exchange the solutons wth other subsystems and repeat Step 3; Step 4 he optmal scheme u k) = F k r k )x k) s appled to the correspondng subsystems Set the tme nterval k = k 1 and go back to Step 2 Iteraton algorthm s solved n parallel for all subsystems thus the feedback laws can be obtaned at a same tme In ths way the computatonal tme can be saved by dstrbuted MPC algorthm Lemma 4 15 For cooperatve dstrbuted MPC algorthm the performance ndces ρ 1 k) ρ 2 k) ρ M k) of subsystems wll converge to the centralzed problem wth the ncrease of teraton tmes whch mples ρ t) 1 k) = ρt) 2 k) = = ρt) M k) = ρk) where ρk) s the upper bound of centralzed MPC at tme nterval k Proof he proof s smlar to 15 For subsystem ρ t) = ρ F t 1) 1 F t) = mn ρ F t 1) F t) F t 1) M ) 1 F t 1) F t 1) M ) {1 M} Smlarly for subsystem j ρ t) j = ρ j F t 1) 1 F t) = mn ρf t 1) F t) j 1 F t 1) j F t 1) M ) j F t 1) M ) j {1 M} j hen from the convexty of problem 13) and for any and j par of subsystems ρ t) ρ t 1) j and ρ t) j ρ t 1) hus the mnmzaton ndex of worst-case cost functon ρ contnues to decrease wth the ncrease of teraton tmes untl both nequaltes become equaltes Snce the mnmzatons gven above are convex and each s leadng to a global optmum we can consequently obtan ρ t) = ρ t) j = ρ j {1 M} j Remark 4 Lemma 4 tells us that the performance by dstrbuted MPC algorthm can be as good as the performance by centralzed MPC algorthm wthn several teratons However the more computatonal tme wll be cost wth the ncrease of teraton tmes here s a tradeoff between computatonal tme and the performance In terms of computatonal tme the dstrbuted MPC presented n ths paper can outperform centralzed MPC wth lttle performance loss when termnated after a few teratons IV NUMERICAL EXAMPLES Example 1 Consder the system as follows: { xk 1) = A ξ) r k )xk) B ξ) r k )σuk)) yk) = C ξ) r k )xk) 24) where the system matrces are represented by the followng two vertces: Mode A 1) r1) = B 1) r1) = A 2) r1) = B 2) r1) = Mode A 1) r2) = B 1) r2) = A 2) r2) = 15 01

7 380 IEEE/CAA JOURNAL OF AUOMAICA SINICA VOL 2 NO 4 OCOBER B 2) r2) = 0 08 where r 1 and r 2 denote Mode 1 and Mode 2 respectvely Assume that the transton probablty matrces wth polytopc uncertantes can be denoted as P = We decompose the global system nto two subsystems each of whch has one control nput he actuator saturaton s descrbed as σ u 1 k)) = sgn u 1 k)) mn {1 u 1 k) } and σ u 2 k)) = sgn u 2 k)) mn{1 u 2 k) } he proposed dstrbuted MPC algorthm s used to control these two subsystems wth ntal state x 0 = 1 1 and weghtng matrces Q 1 = Q 2 = I 2 R 1 = R 2 = 1 he performance of dstrbuted algorthm s compared wth centralzed MPC algorthm under the same weghtng matrces Fg 1 shows the nvarant sets obtaned by dfferent algorthms at the frst samplng step It can be seen that the proposed dstrbuted MPC algorthm reaches the same nvarant set as centralzed MPC algorthm after three teratons whch valdates the result of Lemma 4 Fg 2 a) presents the upper-bound trends of centralzed MPC subsystems 1 and 2 respectvely from whch we fnd that only after three teratons the upper bound of the subsystems 1 and 2 almost converge to the upper bound of centralzed MPC From Fg 2 b) we can see the upper bound ether by centralzed MPC or by dstrbuted MPC wll converge to zero when the tme step k tends to nfnte he comparson of upper bound and computatonal tme between centralzed MPC CMPC) and dstrbuted MPC DMPC) at the frst samplng step s shown n able I he dstrbuted MPC algorthm s appled wth one sngle teraton two teratons and three teratons From able Fg 2 Upper bounds of centralzed MPC and two subsystem I we can see the dstrbuted MPC algorthm only takes 092 s after three teratons whch s less than 123 s by centralzed MPC Dstrbuted MPC outperform centralzed MPC wth lttle performance loss whch valdate the results descrbed n Remark 4 Fg 1 Invarant sets comparson at frst samplng nterval V CONCLUSION hnkng of the uncertanty and abrupt changes of system parameters we present a dstrbuted MPC for MJLSs By decomposng the system nto several subsystems to desgn the dstrbuted controllers t can decrease the system complexty and computaton cost In each subsystem a dstrbuted MPC strategy s proposed he dstrbuted controller s obtaned by solvng an LMI optmzaton problem and an teraton algorthm s developed for makng cooperaton among subsystems Due to the consderaton of actuator saturaton the man results made n ths paper are more practcal than those wth deal network connecton In the end an example llustrates the effectveness of the proposed dstrbuted MPC algorthm ABLE I UPPER BOUND AND CPU IME COMPARISON BEWEEN CENRALIZED MPC AND DISRIBUED MPC Strategy CMPC DMPC 1 teraton) DMPC 2 teraton) DMPC 3 teraton) Subsystem 1 Subsystem 2 Subsystem 1 Subsystem 2 Subsystem 1 Subsystem 2 Upper bound CPU tme s)

8 SONG et al: DISRIBUED MODEL PREDICIVE CONROL WIH ACUAOR SAURAION FOR MARKOVIAN JUMP LINEAR SYSEM 381 REFERENCES 1 Park B G Kwon W H Robust one-step recedng horzon control of dscrete-tme Markovan jump uncertan systems Automatca ): Sworder D D Rogers R O An LQ-soluton to a control problem assocated wth a solar thermal central recever IEEE ransactons on Automatc Control ): Blar Jr W P Sworder D D Feedback control of a class of lnear dscrete systems wth jump parameters and quadratc cost crtera Internatonal Journal of Control ): Boukas E K Haure A Manufacturng flow control and preventve mantenance: a stochastc control approach IEEE ransactons on Automatc Control ): Costa O L V Fragoso M D Marques R P Dscrete-tme Markov Jump Lnear Systems London: Sprnger Lu J B L D W X Y G Constraned model predctve control synthess for uncertan dscrete-tme Markovan jump lnear systems IE Control heory and Applcatons ): We G L Wang Z D Shu H S Nonlnear H control of stochastc tme-delay systems wth Markovan swtchng Chaos Soltons and Fractals ): Byung-Gun P Wook H Jae-Won L E E Robust recedng horzon control of dscrete-tme Markovan jump uncertan systems IEICE ransactons on Fundamentals ): Zhang L W Wang J C L C Dstrbuted model predctve control for polytopc uncertan systems subject to actuator saturaton Journal of Process Control ): Scattoln R Archtectures for dstrbuted and herarchcal model predctve control: a revew Journal of Process Control ): Song Y Fang X Dstrbuted model predctve control for polytopc uncertan systems wth randomly occurrng actuator saturaton and packet loss IE Control heory and Applcatons ): Ja D Krogh B Mn-max feedback model predctve control for dstrbuted control wth communcaton In: Proceedngs of the 2002 Amercan Control Conference Anchorage AK USA: IEEE : Mercangöz M Doyle III F J Dstrbuted model predctve control of an expermental four-tank system Journal of Process Control ): L S Y Zhang Y Zhu Q M Nash-optmzaton enhanced dstrbuted model predctve control appled to the shell benchmark problem Informaton Scences ): Al-Gherw W Budman H Elkamel A A robust dstrbuted model predctve control algorthm Journal of Process Control ): Al-Gherw W Budman H Elkamel A Robust dstrbuted model predctve control: a revew and recent developments he Canadan Journal of Chemcal Engneerng ): Dng B C Dstrbuted robust MPC for constraned systems wth polytopc descrpton Asan Journal of Control ): Huang H L D W Ln Z L X Y G An mproved robust model predctve control desgn n the presence of actuator saturaton Automatca ): Hu S Ln Z L Control Systems wth Actuator Saturaton: Analyss and Desgn New York: Sprnger Hu S Ln Z L Chen B M Analyss and desgn for dscretetme lnear systems subject to actuator saturaton Systems and Control Letters ): Yan Song receved the B S degree from Jln Unversty Chna n 2001 M S degree from Unversty of Electronc Scence and echnology of Chna n 2005 and Ph D degree from Shangha Jao ong Unversty Chna n 2013 respectvely She s now an assocate professor at Unversty of Shangha for Scence and echnology Her research nterests nclude model predctve control networked control and sensor networks Correspondng author of ths paper Hafeng Lou receved the B S degree from Henan Polytechnc Unversty n 2009 Now he s a postgraduate student at Unversty of Shangha for Scence and echnology Hs research nterests nclude networked control system and model predctve control Shua Lu receved the B S degree from Lu Dong Unversty Yanta Chna n 2011 and the M S degree from Unversty of Shangha for Scence and echnology Shangha Chna n 2014 She s now a Ph D canddate at Unversty of Shangha for Scence and echnology Shangha Chna Her research nterests nclude model predctve control and set-membershp flterng