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N-Step Impacted-Region Optimization based Distributed Model Predictive Control∗
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ScienceDirect IFAC-PapersOnLine 48-8 (2015) 831–836 Optimization N-Step Impacted-Region N-Step Impacted-Region Optimization based Impacted-Region Distributed ModelOptimization Predictive N-Step based Distributed Model Predictive Control based Distributed Model Predictive Control Control ∗ ∗∗
Y. Zheng S.Y. Li Y. Zheng ∗∗ S.Y. Li ∗∗ ∗∗ Y. Zheng Li ∗∗ ∗ Y.Information Zheng ∗ S.Y. S.Y. School of Electronic andLiElectrical Engineering, ∗ School of Electronic Information and Electrical Shanghai Jiao Tong University, Shanghai, 200240, Engineering, China (e-mail: ∗ ∗ SchoolJiao of Electronic Electronic Information and Electrical Electrical Engineering, Shanghai Tong University, Shanghai, 200240, Engineering, China (e-mail: School of Information and [email protected]). Shanghai Tong Shanghai, 200240, (e-mail: ∗∗ [email protected]). Shanghai Jiao Tong University, University, Shanghai, 200240, China China (e-mail: SchoolJiao of Electronic Information and Electrical Engineering, ∗∗ [email protected]). School of Electronic Information and Electrical Engineering, [email protected]). Shanghai Jiao Tong University, Shanghai, 200240, China (Tel: ∗∗ ∗∗ School of Electronic Information and Electrical Electrical Engineering, Shanghai Jiao Tong University, Shanghai, 200240,Engineering, China (Tel: School 86-021-34204011; of Electronic Information and e-mail: [email protected]). Shanghai Jiao Shanghai, 200240, e-mail: [email protected]). Shanghai 86-021-34204011; Jiao Tong Tong University, University, Shanghai, 200240, China China (Tel: (Tel: 86-021-34204011; e-mail: [email protected]). 86-021-34204011; e-mail: [email protected]). Abstract: The Distributed Model Predictive Control (DMPC) has been more and more popular Abstract: The of Distributed Model Predictive (DMPC)by hasmany been interacted more and more popular in the control distributed systems whichControl are composed subsystems. Abstract: The Distributed Model Predictive Control (DMPC) has been more more popular in control of distributed systems which are Predictive composed by interacted subsystems. Abstract: Thesubsystems Distributed Model Predictive Control (DMPC) hasmany been more and and more popular Thethe range of that each local Model Control (MPC) optimized, called in the control of distributed systems which are composed by many interacted subsystems. The range ofdegree, subsystems that each local Control optimized, in the control of distributed systems which are Predictive composed by many(MPC) interacted subsystems. coordination plays an important roleModel in improving the optimization performance of called entire The range of of subsystems that eachthe local Model Predictive Control (MPC) optimized, called coordination degree, plays important roleModel in improving the optimization performance of called entire The range subsystems that each local Predictive Control (MPC) optimized, closed-loop system. In thisan paper, N-step adjacent structure matrix based decomposition coordination degree, In plays anpaper, important role in in improving improving the optimization performance ofby entire closed-loop this the N-step adjacent matrix performance based decomposition coordination degree, plays an important role the of entire method wassystem. proposed, where the coordination degree ofstructure eachoptimization subsystem is determined the closed-loop In paper, the N-step adjacent matrix based decomposition method proposed, where the coordination degree ofstructure each subsystem determined by the closed-loop system. In this this paper, theover N-step adjacent structure matrixonis based decomposition union of was thesystem. all the adjacent matrices the predictive horizon. Based this decomposition, method proposed, where the degree each subsystem is determined by union of was the all the adjacent matrices thesubsystems predictive Based decomposition, method was proposed, where the coordination degree of of horizon. each subsystem isthis determined by the the each local MPC considers the costcoordination of over all the it impacted onon during the predictive union of the all the adjacent matrices over the predictive horizon. Based on this decomposition, each local MPC considers the cost of all the subsystems it impacted on during the predictive union of the all the adjacent matrices over the predictive horizon. Based on this decomposition, horizon, and then improves the optimization performance of entire system with reduced each local MPC considers the cost of the it on during the horizon, then improves the performance of entire withpredictive reduced each localand MPC considers thesimulation cost optimization of all allresults the subsystems subsystems it impacted impacted on during the predictive communication burdens. The show the effectiveness of system the proposed method. horizon, and then improves the optimization performance of entire system with reduced communication burdens. The simulation results show the effectiveness of system the proposed horizon, and then improves the optimization performance of entire with method. reduced © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. communication burdens. The simulation results show the effectiveness of the proposed method. communication burdens. The simulation results show the effectiveness of the proposed method. 1. INTRODUCTION MPC’s cost function covered, called coordination degree, 1. INTRODUCTION MPC’s costthe function covered, called coordination increased, optimization performance of entire degree, system 1. INTRODUCTION INTRODUCTION MPC’s cost function covered, called coordination degree, increased, the optimization performance of entire system 1. MPC’s cost function covered, called coordination degree, is improved and the communication burden increases in Consider a class of complex large-scale control systems optimization of system is improved and the communication burden increases in increased, the optimization performance of entire entire system Consider a class of ofcomplex large-scaleorcontrol systems increased, most cases the [Al-Gherwi et al.,performance 2010]. Specially, if each local which is composed many physically geographically is improved and its theown communication burden increases in Consider class of of ofEach complex large-scale control systems cases [Al-Gherwi et al., Specially, each local is improved and the communication burden increases in which composed many physically orcontrol geographically Consider aa class complex large-scale systems MPC optimizes cost2010]. function, uses if the predicdividedissubsystems. subsystem interacts with some most most cases [Al-Gherwi et al., 2010]. Specially, if each local which is composed of many physically or geographically MPC optimizes its own cost function, uses the predicmost cases [Al-Gherwi et al., 2010]. Specially, if each local divided subsystems. Each subsystem interacts with some which is composed of many physically or geographically other subsystems by their states and/or inputs. The con- tive sequence of its neighbors to estimate the interacoptimizes cost uses the predicdivided subsystems. Each subsystem interacts with some tive of its its own neighbors to estimate interacMPC optimizes its own cost function, function, uses the predicother subsystems by accomplish their subsystem states aand/or inputs. The con- MPC divided subsystems. Each interacts with some tions sequence among subsystems [Camponogara et al.,the 2002], and trol objective is to specific global perfortive sequence of its neighbors to estimate the interacother subsystems by their states and/or inputs. The conamong subsystems [Camponogara et al.,the 2002], tive sequence of its neighbors estimate interactrol objective is to accomplish specific global perforother subsystems by system their states and/or inputs. con- tions employs iterative algorithm, thetoNash optimality canand be mance of the entire or a acommon goal ofThe all subtions among subsystems [Camponogara et each al., 2002], 2002], and trol objective is to to system accomplish specificgoal global perforiterative the strategy, Nash optimality canconbe tions among et al., and mance of the entire or a aacommon of all sub- employs trol objective is accomplish specific global perforachieved[Li etsubsystems al., algorithm, 2005]. [Camponogara In this local systems. iterative the Nash optimality can be mance of of the the entire entire system system or or aa common common goal goal of of all all subsub- employs achieved[Li et al., 2005]. In this each To local employs iterative algorithm, the strategy, Nash optimality canconbe systems. mance troller connects toalgorithm, its neighboring controllers. farther achieved[Li et al., 2005]. In this strategy, each local conThe Distributed Model Predictive Control (DMPC) which systems. troller to 2005]. itsperformance, neighboring To farther achieved[Li al., In this strategy, local consystems. improveconnects theetglobal acontrollers. designeach method that The Distributed Model Predictive Controllocal (DMPC) connects to neighboring To controls each subsystem by a separated Modelwhich Pre- troller improve the global design that troller connects to its itsperformance, neighboring controllers. To farther farther each subsystem-based MPC takesacontrollers. not onlymethod the perforThe Distributed Model Predictive Predictive Control (DMPC) which controls each subsystem by abeen separated local Model Pre- each The Distributed Model Control (DMPC) which improve the global performance, a design method that dictive Control (MPC), has more and more popular subsystem-based MPC takes not only the perforimprove the global performance, a design method that mance of its corresponding subsystem but also that of controls each subsystem by a separated local Model Predictive Control (MPC), has more and more controls each subsystem by abeen local Model subsystem-based MPC takes not only the perforin the control of this kinds ofseparated systems [Moro¸ sanpopular etPreal., each mance of its corresponding subsystem but also that of each subsystem-based MPC takes not only the perforthe subsystems it directly impacts on into account in its dictive Control (MPC), has been more and more popular in the since control thisonly kinds of systems [Moro¸ sanpopular et al., dictive Control (MPC), has been more and more of its corresponding subsystem but also that of 2010], it of not inherits MPC’s advantages of mance the subsystems it directly impacts on into account in its mance of its corresponding subsystem but also that of optimization index is proposed by [Li et al., 2014, Zheng in the since control ofnot thisonly kinds of systems systems [Moro¸ an et al., al., 2010], it of inherits MPC’s advantages of optimization in the control this kinds of [Moro¸ ssan et the subsystems it directly impacts on into account in its explicitly accommodating constraints and good dynamic index is proposed by [Li et al., 2014, Zheng the subsystems it directly impacts on into account in its et al., 2009]. Experiments and numeric results prove that 2010], since itbut notalso only inherits MPC’s advantages of optimization index is proposed by [Li et al., 2014, Zheng explicitly accommodating constraints and good dynamic 2010], since it not only advantages of performance, hasinherits all the MPC’s virtues of distributed et 2009]. Experiments and numeric proveZheng that optimization index is proposed by [Li etresults al., performance 2014, thisal., strategy could significantly improve the explicitly accommodating constraints and good good dynamic performance, but hasconstraints all the virtues of distributed explicitly accommodating and dynamic al., 2009]. Experiments and numeric results prove that framework[Qin andalso Badgwell, 2003, Maciejowski, 2002, et this strategy could significantly improve the performance et al., 2009]. Experiments and numeric results prove that of entire system with small increasing of network connecperformance, but also has all all the the virtues of et distributed framework[Qin and Badgwell, 2003, Maciejowski, performance, but also has virtues of distributed strategy could significantly improve the performance Sandell Jr et al., 1978, Scattolini, 2009, Leirens al., 2002, 2010, this of entire system with small increasing of network connecthis strategy could significantly improve the performance tivity, each local controller have to connect to the conframework[Qin and Badgwell, 2003, Maciejowski, 2002, Sandell Jr et et al.,al., 1978, Scattolini, 2009, Leirens2013a]. et al., 2002, 2010, tivity, framework[Qin and Badgwell, 2003, Maciejowski, of with small increasing of connecChristofides 2012, Zheng et al., 2011b, each controller have to connect to the conof entire entire system with small increasing of network network connecofsystem itslocal neighbours, and its neighbours’s neighbours. Sandell Jr Jr et et et al.,al., 1978, Scattolini, 2009, Leirens2013a]. et al., al., 2010, 2010, trollers Christofides 2012, Zheng et2009, al., 2011b, Sandell al., 1978, Scattolini, Leirens et tivity, each local controller have to connect to the controllers of its neighbours, and its neighbours’s neighbours. tivity, each local controller have to connect to the conAnother useful strategy is that each subsystem optimize However, theet of distributed implementation Christofides etperformance al., 2012, 2012, Zheng Zheng et al., al., 2011b, 2011b, 2013a]. Christofides al., et 2013a]. trollers of its neighbours, and its neighbours’s neighbours. Another useful strategy is that each subsystem optimize trollers of its neighbours, and its neighbours’s neighbours. the weighted cost of all subsystems and solves the optimal However, the performance of distributed implementation of MPC, in most of cases, is not as good as that of Another useful strategy is that each subsystem optimize However, the performance of is distributed implementation weighted cost of iteration. all subsystems and solves the optimal Another useful strategy is that subsystem solution by parallel Byeach this method, theoptimize Parato of MPC, the inMPC. most of improve cases, notglobal as good as that of the However, performance of distributed implementation centralized To the performance the weighted cost of all subsystems and solves optimal of MPC, inMPC. mostsystem, of improve cases, isthe not as good good as that that of of by parallel By this the Parato the weighted cost of iteration. all subsystems andmethod, solves the the optimal Optimality, the best optimization performance in existing centralized To global performance of MPC, in most of cases, is not as as entire closed-loop several DMPC coordination s- solution solution by parallel iteration. By this method, the Parato centralized MPC. To improve the global performance of Optimality, the best optimization performance in existing solution by parallel iteration. By this method, the et al., entire closed-loop several coordination s- DMPCs, can be achieved[Stewart et al., 2010, ZhengParato centralized MPC. system, To the DMPC global performance of trategies appeared in improve the literatures [Zheng et al., 2009, Optimality, the best optimization performance in existing entire closed-loop system, several DMPC coordination sDMPCs, can be achieved[Stewart et al., 2010, Zheng al., Optimality, the best optimization performance in existing 2011a]. However, the global information is required when trategies appeared in the literatures [Zheng et al., 2009, entire closed-loop system, several DMPC coordination sCamponogara et al., 2002, Christofides et al., 2012, Zheng DMPCs, can be achieved[Stewart et al., 2010, Zheng et et al., trategies appeared in the literatures [Zheng et al., 2009, 2011a]. However, the global information is required when DMPCs, can be achieved[Stewart et al., 2010, Zheng et al., solving each subsystem’s optimal solution in this method, Camponogara et al., 2002, Christofides et al., 2012, Zheng trategies appeared in the literatures [Zheng et al., 2009, et al., 2013b]. People found that if the subsystems of each 2011a]. However, the global information is required when Camponogara et al., 2002, Christofides et al., 2012, Zheng solving each subsystem’s optimal solution in this method, 2011a]. However, the global information is required when which is not expected. Could we find a method to exactly et al., 2013b]. People found that if the subsystems of each Camponogara et al., 2002, Christofides et al., 2012, Zheng Supported by National Basic Research Program of China (973Proeach subsystem’s optimal solution in method, et al., al., 2013b]. 2013b]. People People found found that that if if the the subsystems subsystems of of each each solving which not expected. Could wewhich find aismethod exactly solving each subsystem’s optimal solution in this this method, define is the coordination degree, able to to make the et Supported by NationalNational Basic Research of China (973Program) (2013CB035500), NaturalProgram Science Foundation of Chiwhich is not expected. Could we find a method to exactly define the coordination degree, which is able to make the which is not expected. Could we find a method to exactly resulting DMPC obtains Pareto optimality with reduced Supported by National Basic Research Program of China (973Pro gram) (2013CB035500), National Natural Science Foundation of ChinaSupported (61233004,by61221003,61304078,61473187), the Higher National Basic Research Program of ChinaEducation (973Prodefine the coordination degree, which is able to make the resulting DMPC obtains Pareto optimality with reduced define the coordination degree, which is able to make the communication burden? This stimulated this study. gram) (2013CB035500), NationalProgram Natural Science Science Foundation of ChiChina (61233004, the Foundation Higher Education Research Fund 61221003,61304078,61473187), for the Doctoral of China (20120073130006 gram) (2013CB035500), National Natural of resulting DMPC DMPCburden? obtainsThis Pareto optimality with reduced communication stimulated this study. resulting obtains Pareto optimality with reduced na (61233004, 61221003,61304078,61473187), the Higher Education Research Fund for the Doctoral Program of China (20120073130006 & 20110073110018), the China Postdoctoral Science Foundation na (61233004, 61221003,61304078,61473187), the Higher Education In this paper, an N-stepThis Impacted Region communication burden? stimulated this study. communication burden? This stimulated thisOptimization study. Research Fund the Doctoral of China (20120073130006 & 20110073110018), the China Postdoctoral Science (2013M540364), and of Program Shanghai and Foundation Technology Research Fund for for theProject Doctoral Program ofScience China (20120073130006 In thisDMPC paper, an N-step Impacted Region Optimization based (N-step IRO-DMPC) is proposed, where & 20110073110018), the China Postdoctoral Science Foundation (2013M540364), and Project of Shanghai Science and Technology Committee(12dz1200203). & 20110073110018), the China Postdoctoral Science Foundation In this thisDMPC paper, an an N-step Impacted Region Region Optimization based (N-step IRO-DMPC) is proposed, where In paper, N-step Impacted Optimization (2013M540364), and Project of Shanghai Science and Technology Committee(12dz1200203). (2013M540364), and Project of Shanghai Science and Technology based DMPC (N-step IRO-DMPC) is proposed, where based DMPC (N-step IRO-DMPC) is proposed, where Committee(12dz1200203).
Committee(12dz1200203). Copyright © 2015 IFAC 832 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 832Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 832 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 832Control. 10.1016/j.ifacol.2015.09.072
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the range of each local MPC’s cost function covered is determined by the union of the all the adjacent matrices over the predictive horizon. Each local MPC optimized the cost of all subsystems it impacted on over the prediction horizon to cooperate with each other. In addition, an iterative algorithm is developed to resolve each local MPC. Through these ways, the Parato optimality was achieved with reduced communication resource comparing to the global cost optimization based DMPC. The remainder of this paper is organized as follows. Section 2 describes the problem to be solved in this paper. Section 3 presents the design of the proposed distributed MPC. Section 4 presents the simulation results to demonstrate the effectiveness of the proposed method. Finally, a brief conclusion to the paper is drawn in Section 5. 2. PROBLEM 2.1 Distributed system A distributed system is composed of many interacting subsystems, each of which is controlled by an independent controller, which in turn is able to exchange information with other controllers. Suppose that the distributed system S is composed of m discrete-time linear subsystems Si , i ∈ P = {1, 2, · · · , m} and m controllers Ci , i ∈ P. If subsystem Si is directly affected by Sj , for any i ∈ P and j ∈ P, subsystem Si is said to be a directly (or one-step) downstream system of subsystem Sj , and subsystem Sj is a directly (or onestep) upstream system of Si . Let P+i denote the set of the subscripts of the one-step upstream systems of Si , P−i is the set of the subscripts of the one-step downstream systems of Si . Let the subsystems interact with each other through their states. Then, subsystem Si can be expressed as xi,k+1 = Aii xi,k + Bii ui,k + Aij xj,k , (1) j∈P+i yi,k = Cii xi,k ,
where xi ∈ Xi ⊂ Rnxi , ui ∈ Ui ⊂ Rnui and yi ∈ Yi ⊂ Rnyi are respectively the local state, input and output vectors, and Xi , Ui and Yi are respectively the feasible set of the state xi , input ui and output yi which are used to bound the state, input and output according to the physical constraints on the actuators, the control requirements or the characteristics of the plant. A nonzero matrix Aij , j ∈ P+i , indicates that Si is directly affected by Sj . In the concatenated vector form, the system dynamics can be written as xa,k+1 = Axa,k + Bua,k , (2) ya,k = Cxa,k , T where xa = [xT xT · · · xT ∈ R nx , u a = m] 2 1 T T T T nu T T T [u1 u2 · · · um ] ∈ R and ya = [y1 y2T . . . ym ] ∈ Rny are respectively the concatenated state, control input and output vectors of the overall system S, and A, B and C are constant matrices of appropriate dimensions. Also, xa ∈ X = X1 × X2 × · · · × Xm , ua ∈ U = U1 × U2 × · · · × Um and ya ∈ Y = Y1 × Y2 × · · · × Ym .
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2.2 Control objective The control objective is to find a control method under the distributed framework, which could obtain the optimal solution of entire system with a reduced communication burden. The performance index of the whole system can be expressed as Ji (k) (3) J(k) = i∈P
where Ji (k) =
N Cii xi,k+l|k − ysp 2 i
l=1
Qi,l
2 + ∆ui,k+l−1|k R ,(4) i,l
yisp is the set point of the ith subsystem and ∆ui,k+l|k = ui,k+l|k − ui,k+l−1|k is the input increment of the ith subsystem at the time instant k. Constant matrix Qi,l , Ri,l > 0, l = 1, 2, . . . , N , is weighting coefficients for the ith subsystem, and let the weighting matrices for Si be Qi = block − diag{Qi,1 , Qi,2 , . . . , Qi,N } > 0 Ri = block − diag{Ri,1 , Ri,2 , . . . , Ri,N } > 0. 3. N-STEP IMPACTED-REGION OPTIMIZATION BASED DISTRIBUTED MPC
Consider that each local MPC Ci , i ∈ P, only optimize N (predictive horizon) step ahead performance, thus the solution of each local MPC only impacts performance of the subsystems which are interacted with Si during the predictive horizon. To take this interaction into account, a strategy that each local MPC optimize its N-step impacted region’s performance was proposed, and which is detailed as follows. 3.1 N-step impacted-region To proceed, we need the following definitions. Definition 1. Adjacent Matrix: Consider a system xk+1 = Axk , A ∈ Rnx ×nx which is composed of m subsystem, ¯ and its the adjacent matrix refers to a m × m matrix A, th th i , j element 1, when Aij = 0 a ¯ij = (5) 0, when Aij = 0 It can be seen from (5) that the adjacent matrix only reflects the directly interaction among subsystems. In fact, we usually concern that if one unit indirectly impacts another unit through some intermediate units. In this case, ¯ can the structure matrix corresponding to the k power of A be used to express the interaction in k step ahead. Then, following N-Step Accessible Matrix is defined to express the relationship among subsystem during N step ahead. Definition 2. N-Step Accessible Matrix: Consider a system xk+1 = Axk , A ∈ Rnx ×nx , which consists of m unit Si , i = 1, 2, ..., m, define that each unit Si is accessible to itself, the all accessible relationship can be described by a so called N-Step Accessible Matrix R, and ¯N ¯ ∪A ¯2 ∪ ... ∪ A R=I∪A It is also a kind of structure matrix, where the ith row and j th column element equals zero expresses that subsystem
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Sj is un-accessible to subsystem Sj . By logic operating rules, the above equation can be simply rewritten as ¯ N (6) R = (I ∪ A) In fact, for the control of system (1), the relationship among the states, inputs and outputs are very important. However, (5) can not exactly reflect this relationship. Consider that AB0 xa,k xa,k+1 ua,k+1 = 0 0 0 ua,k , (7) ya,k C 0 0 ya,k−1 for system (2), the adjacent matrix can be defined as ¯ B ¯ 0 A ¯d = 0 0 0 . (8) A ¯ 0 0 C From (6), the N-step Accessible Matrix can be expressed as Rxx Rxu Rxy N −1 = Rux Ruu Ruy R = (I ∪ Ad ) Ryx Ryu Ryy (9) ¯ + I)N ¯ A ¯ + I)N −1 0 (A B( 0 I 0. = N −1 ¯ ¯ N −2 ¯ ¯ ¯ C(A + I) C(A + I) B I
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The predictive model can be expressed as yi,k+l|k = Cii Alii xi,k +
l
Cii Al−h ii Bii ui,k+h−1|k
h=1
+
l
Cii Al−h ˆj,k+h−1|k ii Aij x
(13)
j∈P+i h=1
Consider the physical limitations on the outputs, the input and the input increment, we can get following optimization problem for Si in each control period. Problem 1. For all subsystem Si , i ∈ {1, 2, . . . , m}, provided that xi,k , x ˆj,k+l|k , j ∈ P+h P+i , h ∈ P−iN and ∆ˆ ui,k+l−1|k−1 ,j ∈ P−iN , l = 1, 2, . . . , N , find the control sequence ∆ui,k:k+N −1|k , which minimize the performance index min J¯i (k) ui,k:k+N −1|k
Subject to the constraints: Equation(13), yi,L ≤ yi,k+l|k ≤ yi,U ,
yj,L ≤ yj,k+l|k ≤ yj,U , j ∈ P−iN ,
ui,L ≤ ui,k+l−1|k ≤ ui,U ,
(14) (15) (16)
Thus, the N-step input to output accessible matrix is ¯ A ¯ + I)N −2 B. ¯ Ryu = C( (10)
∆ui,L ≤ ∆ui,k+l−1|k ≤ ∆ui,U ,
(17)
¯ is a unit matrix in (10), the N-step Consider that B accessible matrix can be redefined as ¯ A ¯ + I)N −2 . (11) Ryu = C(
||yi,k+N |k − yisp ||2Qi,N < ε2 .
(18)
And the N-step down stream neighbour of Si can be defined as the subsystems Sj where the j th row, and ith collum element of Ryu equals 1. And denote P−iN as the set of the subscript of all the N-step downstream neighbor of Si . 3.2 Distributed MPC design
The performance of local MPC for subsystem Si is defined as N Cii xi,k+l|k − ysp 2 i
l=1
+
Qi,l
2 + ∆ui,k+l−1|k R i,l
N 2 Cjj x ˆj,k+l|k − ysp j
j∈P−iN l=1
where, [yi,L , yi,U ], [ui,L , ui,U ] and [∆ui,L , ∆ui,U ] are the bounds of outputs, inputs and the increment of inputs respectively. Equation (18) is a final constraint for improve the stability of each subsystem-based MPC, and ε > 0. To solve problem (1) efficiently, following iterative algorithm is given for ∀Si , i ∈ P. Algorithm 1. (N-step IRO-DMPC Algorithm). Step 1: Initialization.
Consider that the control law of current subsystem Si effects the performance of its N-step downstream neighboring subsystems Sj , j ∈ P−iN , in the N-Step IRO-DMPC, the performance of Sj , j ∈ P−iN is added into the performance index of the MPC which controls Si based on a approximation of the updated state sequence of Sj . In this way, the coordination degree is expanded and is equivalent to that of global cost optimization based DMPC.
J¯i (k) =
l = 1, 2, ..., N ;
Qj,l
(12)
Define that x ˆi,k+l|k , u ˆi,k+l|k and ∆ˆ ui,k+l|k be the assumed states, the assumed input and the assumed input increment which are calculated in the previous calculation, respectively. 834
• Initialize xi,k0 , xi,k0 +l|k0 , l = 1, 2, . . . , N , which satisfy the constraints of Problem 1. Step 2: Update control law at time k > k0 . • Step 2.1 Set iteration t = 1, and set x ˆi,k+l|k = xi,k+l|k−1 . • Step 2.2 Measure xi (k), transmit x ˆi,k+l|k to its N-step down stream neighbors and u ˆ i,k+l|k upstream neighbors; And receive u ˆ i,k+l|k from its N-step down stream neighbors and x ˆj,k+l|k from its upstream neighbors. • Step 2.3 Solving Problem 1 to obtain the optimal solution ∆uti,k+l|k , and predict the future state xi,k+l|k based on the solution ∆uti,k+l|k . • Step 2.4 If 2 ||∆uti,k+l−1|k − ∆ut−1 i,k+l−1|k ||2 ≤ ε0 or t > tmax then set
u∗i,k = ui,k−1 + ∆u∗i,k+l−1|k ,
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1
x1
2
6
Y. Zheng et al. / IFAC-PapersOnLine 48-8 (2015) 831–836
x6
7
x2
x4 3
x3
4
x5 x5
x8
5
x8
8
9
Ryu
x9
Fig. 1. Distributed MPC configuration and goto Step 3; Else set x ˆi,k+l|k = xi,k+l|k , t = t + 1, and goto Step 2.2.
0 1 0 1 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0
0 1 0 1 0 0 0 0 0
0 1 0 1 1 1 1 0 0
0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 1 1
0 0 0 0 0 0 1 1
(28)
1
From (28) the network connectivity is 15 when using proposed N-step ICO-DMPC, which is most less than 72 when using global cost optimization based DMPC. The network connectivity is dramatically deduced.
Step 3: Update control at time k + 1. • Let k + 1 → k, repeat Step 2. It should be noticed that although an iterative algorithm is presented, the Problem 1 can also be solved by a noniterative algorithm through setting tmax = 1. Since the communication burden will increase with the increasing of iteration, tmax should not be set too large in practice. So far the N-step impacted-region optimization based DMPC for distributed system is introduced, some simulation results will be presented in the next section to show the effectiveness of the proposed method. 4. SIMULATION For simplicity, a 9 nodes distributed network is taken as example, and the relationship among these notes is shown in Fig. 1 where the arrow from subsystem Si to Sj expresses that Sj is directly effected by Si .
The dynamic models of these nodes are respectively by x1,k+1 = 0.57x1,k + 0.38u1,k S1 : y1,k = x1,k , x2,k+1 = 0.53x2,k + 0.38u2,k + 0.16x1,k +0.16x4,k S2 : y2,k = x2,k , x3,k+1 = 0.55x3,k + 0.38u3,k S3 : y3,k = x3,k , x4,k+1 = 0.61x4,k + 0.39u4,k + 0.18x2,k +0.18x3,k + 0.18x5,k S4 : y4,k = x4,k , x5,k+1 = 0.68x5,k + 0.42u5,k S5 : y5,k = x5,k , x6,k+1 = 0.55x6,k + 0.38u6,k + 0.16x5,k S6 : y6,k = x6,k , x7,k+1 = 0.71x7,k + 0.42u7,k + 0.21x6,k +0.21x8,k S7 : y7,k = x7,k , x8,k+1 = 0.57x8,k + 0.38u8,k + 0.17x9,k S8 : y8,k = x8,k , x9,k+1 = 0.66x9,k + 0.41u9,k + 0.20x8,k S9 : y9,k = x9,k .
1 1 0 1 = 0 0 0 0
given
(19) (20) (21) (22) (23) (24) (25) (26) (27)
For the purpose of comparison, both the Centralized MPC and the N-step IRO-DMPC are applied to this system. Let the constraint on the input be [ui,L , ui,U ] = [−2, 2] and the constraint on the increment of input be [∆ui,L , ∆ui,U ] = [−1.5, 1.5]. Set the all controllers’ (both centrialized MPC and N-step IRO-DMPC) parameters of control horizon be N = 10, the weighting matrices be Qi = [1, 1, 1, 1, 1, 1, 1, 1, 1, 5], Ri = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], where i ∈ {1, 2, ..., 9}.
The state responses and the inputs of the closed-loop system under the control of the centralized MPC and LCO-DMPC are shown in Figs. 2 and 3, respectively. The shape of the state response curves under the control of N-step IRO-DMPC are almost equals to those under the Centralized MPC. Under the N-step IRO-DMPC control design, when set point changed, there is no significant overshooting, but some fluctuation exists in the trajectories of states of the interacting subsystems.
From these simulation results, it can be seen that the proposed N-step IRO-DMPC could obtain a global performance almost equal to that of using centralized MPC and the global information is not necessary for every local MPC, which keeps the characteristics of good error tolerance and high flexibility of the Distributed Control Framework. 5. CONCLUSION In this paper, an N-step Impact-Region Optimization based DMPC is provided for distributed systems. The simulation results of the control of a distributed network composed by nine first-order systems shows the efficiency of the proposed method. With the proposed method, the closed-loop system could obtain a global performance almost equivalent to that of with the centralized MPC. In addtion, the global information is not necessary for each local MPC in the N-Step IRO-DMPC comparing to the global cost optimization based DMPC, which could significantly deduce the network-connectivity for sparse systems, increase the capability of error tolerance of control system. The stabilizing implementation of proposed DMPC subject to pdecoupled constraints maybe a extension of this work and will be done in the near future. REFERENCES W. Al-Gherwi, H. Budman, and A. Elkamel. Selection of control structure for distributed model predictive
According to (11), the N-step accessible matrix is: 835
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Fig. 2. The evolution of the states under the centralized MPC, and N-step Impated-region optimization based DMPC.
Fig. 3. The evolution of the inputs under the centralized MPC, and N-step Impated-region optimization based DMPC.
control in the presence of model errors. Journal of Process Control, 20(3):270–284, 2010. E. Camponogara, D. Jia, B.H. Krogh, and S. Talukdar. Distributed model predictive control. Control Systems Magazine, IEEE, 22(1):44–52, 2002. P.D Christofides, R. Scattolini, D.M. de la Pe˜ na, and J. Liu. Distributed model predictive control: A tutorial review and future research directions. Computers & Chemical Engineering, 2012. S. Leirens, C. Zamora, RR Negenborn, and B. De Schutter. Coordination in urban water supply networks using distributed model predictive control. In American Control Conference (ACC), 2010, pages 3957–3962. IEEE, 2010. S. Li, Y. Zhang, and Q. Zhu. Nash-optimization enhanced distributed model predictive control applied to the shell benchmark problem. Information Sciences, 170(2-4): 329–349, 2005. S. Li, Y. Zheng, and Z. Lin. Impacted-region optimization for distributed model predictive control systems with constraints. IEEE Transactions on Automation Science and Engineering, 2014. J.M. Maciejowski. Predictive control: with constraints. Pearson education, 2002. P. Moro¸san, R. Bourdais, D. Dumur, and J. Buisson. Building temperature regulation using a distributed model predictive control. Energy and Buildings, 42(9): 1445–1452, 2010.
S.J. Qin and T.A. Badgwell. A survey of industrial model predictive control technology. Control engineering practice, 11(7):733–764, 2003. N. Sandell Jr, P. Varaiya, M. Athans, and M. Safonov. Survey of decentralized control methods for large scale systems. Automatic Control, IEEE Transactions on, 23 (2):108–128, 1978. R. Scattolini. Architectures for distributed and hierarchical model predictive control-a review. Journal of Process Control, 19(5):723–731, 2009. B.T. Stewart, A.N. Venkat, J.B Rawlings, S.J Wright, and G. Pannocchia. Cooperative distributed model predictive control. Systems & Control Letters, 59(8): 460–469, 2010. Y. Zheng, S. Li, and X. Wang. Distributed model predictive control for plant-wide hot-rolled strip laminar cooling process. Journal of Process Control, 19(9):1427– 1437, 2009. Y. Zheng, S. Li, and N. Li. Distributed model predictive control over network information exchange for largescale systems. Control Engineering Practice, 19:757– 769, 2011a. Y. Zheng, S. Li, and X. Wang. Horizon-varying model predictive control for accelerated and controlled cooling process. Industrial Electronics, IEEE Transactions on, 58(1):329–336, 2011b. Y. Zheng, N. Li, and S. Li. Hot-rolled strip laminar cooling process plant-wide temperature monitoring and control.
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Control Engineering Practice, 21(1):23 – 30, 2013a. Y. Zheng, S. Li, and H. Qiu. Networked coordinationbased distributed model predictive control for largescale system. Control Systems Technology, IEEE Transactions on, 21(3):991–998, 2013b.
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ScienceDirect IFAC-PapersOnLine 48-8 (2015) 831–836 Optimization N-Step Impacted-Region N-Step Impacted-Region Optimization based Impacted-Region Distributed ModelOptimization Predictive N-Step based Distributed Model Predictive Control based Distributed Model Predictive Control Control ∗ ∗∗
Y. Zheng S.Y. Li Y. Zheng ∗∗ S.Y. Li ∗∗ ∗∗ Y. Zheng Li ∗∗ ∗ Y.Information Zheng ∗ S.Y. S.Y. School of Electronic andLiElectrical Engineering, ∗ School of Electronic Information and Electrical Shanghai Jiao Tong University, Shanghai, 200240, Engineering, China (e-mail: ∗ ∗ SchoolJiao of Electronic Electronic Information and Electrical Electrical Engineering, Shanghai Tong University, Shanghai, 200240, Engineering, China (e-mail: School of Information and [email protected]). Shanghai Tong Shanghai, 200240, (e-mail: ∗∗ [email protected]). Shanghai Jiao Tong University, University, Shanghai, 200240, China China (e-mail: SchoolJiao of Electronic Information and Electrical Engineering, ∗∗ [email protected]). School of Electronic Information and Electrical Engineering, [email protected]). Shanghai Jiao Tong University, Shanghai, 200240, China (Tel: ∗∗ ∗∗ School of Electronic Information and Electrical Electrical Engineering, Shanghai Jiao Tong University, Shanghai, 200240,Engineering, China (Tel: School 86-021-34204011; of Electronic Information and e-mail: [email protected]). Shanghai Jiao Shanghai, 200240, e-mail: [email protected]). Shanghai 86-021-34204011; Jiao Tong Tong University, University, Shanghai, 200240, China China (Tel: (Tel: 86-021-34204011; e-mail: [email protected]). 86-021-34204011; e-mail: [email protected]). Abstract: The Distributed Model Predictive Control (DMPC) has been more and more popular Abstract: The of Distributed Model Predictive (DMPC)by hasmany been interacted more and more popular in the control distributed systems whichControl are composed subsystems. Abstract: The Distributed Model Predictive Control (DMPC) has been more more popular in control of distributed systems which are Predictive composed by interacted subsystems. Abstract: Thesubsystems Distributed Model Predictive Control (DMPC) hasmany been more and and more popular Thethe range of that each local Model Control (MPC) optimized, called in the control of distributed systems which are composed by many interacted subsystems. The range ofdegree, subsystems that each local Control optimized, in the control of distributed systems which are Predictive composed by many(MPC) interacted subsystems. coordination plays an important roleModel in improving the optimization performance of called entire The range of of subsystems that eachthe local Model Predictive Control (MPC) optimized, called coordination degree, plays important roleModel in improving the optimization performance of called entire The range subsystems that each local Predictive Control (MPC) optimized, closed-loop system. In thisan paper, N-step adjacent structure matrix based decomposition coordination degree, In plays anpaper, important role in in improving improving the optimization performance ofby entire closed-loop this the N-step adjacent matrix performance based decomposition coordination degree, plays an important role the of entire method wassystem. proposed, where the coordination degree ofstructure eachoptimization subsystem is determined the closed-loop In paper, the N-step adjacent matrix based decomposition method proposed, where the coordination degree ofstructure each subsystem determined by the closed-loop system. In this this paper, theover N-step adjacent structure matrixonis based decomposition union of was thesystem. all the adjacent matrices the predictive horizon. Based this decomposition, method proposed, where the degree each subsystem is determined by union of was the all the adjacent matrices thesubsystems predictive Based decomposition, method was proposed, where the coordination degree of of horizon. each subsystem isthis determined by the the each local MPC considers the costcoordination of over all the it impacted onon during the predictive union of the all the adjacent matrices over the predictive horizon. Based on this decomposition, each local MPC considers the cost of all the subsystems it impacted on during the predictive union of the all the adjacent matrices over the predictive horizon. Based on this decomposition, horizon, and then improves the optimization performance of entire system with reduced each local MPC considers the cost of the it on during the horizon, then improves the performance of entire withpredictive reduced each localand MPC considers thesimulation cost optimization of all allresults the subsystems subsystems it impacted impacted on during the predictive communication burdens. The show the effectiveness of system the proposed method. horizon, and then improves the optimization performance of entire system with reduced communication burdens. The simulation results show the effectiveness of system the proposed horizon, and then improves the optimization performance of entire with method. reduced © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. communication burdens. The simulation results show the effectiveness of the proposed method. communication burdens. The simulation results show the effectiveness of the proposed method. 1. INTRODUCTION MPC’s cost function covered, called coordination degree, 1. INTRODUCTION MPC’s costthe function covered, called coordination increased, optimization performance of entire degree, system 1. INTRODUCTION INTRODUCTION MPC’s cost function covered, called coordination degree, increased, the optimization performance of entire system 1. MPC’s cost function covered, called coordination degree, is improved and the communication burden increases in Consider a class of complex large-scale control systems optimization of system is improved and the communication burden increases in increased, the optimization performance of entire entire system Consider a class of ofcomplex large-scaleorcontrol systems increased, most cases the [Al-Gherwi et al.,performance 2010]. Specially, if each local which is composed many physically geographically is improved and its theown communication burden increases in Consider class of of ofEach complex large-scale control systems cases [Al-Gherwi et al., Specially, each local is improved and the communication burden increases in which composed many physically orcontrol geographically Consider aa class complex large-scale systems MPC optimizes cost2010]. function, uses if the predicdividedissubsystems. subsystem interacts with some most most cases [Al-Gherwi et al., 2010]. Specially, if each local which is composed of many physically or geographically MPC optimizes its own cost function, uses the predicmost cases [Al-Gherwi et al., 2010]. Specially, if each local divided subsystems. Each subsystem interacts with some which is composed of many physically or geographically other subsystems by their states and/or inputs. The con- tive sequence of its neighbors to estimate the interacoptimizes cost uses the predicdivided subsystems. Each subsystem interacts with some tive of its its own neighbors to estimate interacMPC optimizes its own cost function, function, uses the predicother subsystems by accomplish their subsystem states aand/or inputs. The con- MPC divided subsystems. Each interacts with some tions sequence among subsystems [Camponogara et al.,the 2002], and trol objective is to specific global perfortive sequence of its neighbors to estimate the interacother subsystems by their states and/or inputs. The conamong subsystems [Camponogara et al.,the 2002], tive sequence of its neighbors estimate interactrol objective is to accomplish specific global perforother subsystems by system their states and/or inputs. con- tions employs iterative algorithm, thetoNash optimality canand be mance of the entire or a acommon goal ofThe all subtions among subsystems [Camponogara et each al., 2002], 2002], and trol objective is to to system accomplish specificgoal global perforiterative the strategy, Nash optimality canconbe tions among et al., and mance of the entire or a aacommon of all sub- employs trol objective is accomplish specific global perforachieved[Li etsubsystems al., algorithm, 2005]. [Camponogara In this local systems. iterative the Nash optimality can be mance of of the the entire entire system system or or aa common common goal goal of of all all subsub- employs achieved[Li et al., 2005]. In this each To local employs iterative algorithm, the strategy, Nash optimality canconbe systems. mance troller connects toalgorithm, its neighboring controllers. farther achieved[Li et al., 2005]. In this strategy, each local conThe Distributed Model Predictive Control (DMPC) which systems. troller to 2005]. itsperformance, neighboring To farther achieved[Li al., In this strategy, local consystems. improveconnects theetglobal acontrollers. designeach method that The Distributed Model Predictive Controllocal (DMPC) connects to neighboring To controls each subsystem by a separated Modelwhich Pre- troller improve the global design that troller connects to its itsperformance, neighboring controllers. To farther farther each subsystem-based MPC takesacontrollers. not onlymethod the perforThe Distributed Model Predictive Predictive Control (DMPC) which controls each subsystem by abeen separated local Model Pre- each The Distributed Model Control (DMPC) which improve the global performance, a design method that dictive Control (MPC), has more and more popular subsystem-based MPC takes not only the perforimprove the global performance, a design method that mance of its corresponding subsystem but also that of controls each subsystem by a separated local Model Predictive Control (MPC), has more and more controls each subsystem by abeen local Model subsystem-based MPC takes not only the perforin the control of this kinds ofseparated systems [Moro¸ sanpopular etPreal., each mance of its corresponding subsystem but also that of each subsystem-based MPC takes not only the perforthe subsystems it directly impacts on into account in its dictive Control (MPC), has been more and more popular in the since control thisonly kinds of systems [Moro¸ sanpopular et al., dictive Control (MPC), has been more and more of its corresponding subsystem but also that of 2010], it of not inherits MPC’s advantages of mance the subsystems it directly impacts on into account in its mance of its corresponding subsystem but also that of optimization index is proposed by [Li et al., 2014, Zheng in the since control ofnot thisonly kinds of systems systems [Moro¸ an et al., al., 2010], it of inherits MPC’s advantages of optimization in the control this kinds of [Moro¸ ssan et the subsystems it directly impacts on into account in its explicitly accommodating constraints and good dynamic index is proposed by [Li et al., 2014, Zheng the subsystems it directly impacts on into account in its et al., 2009]. Experiments and numeric results prove that 2010], since itbut notalso only inherits MPC’s advantages of optimization index is proposed by [Li et al., 2014, Zheng explicitly accommodating constraints and good dynamic 2010], since it not only advantages of performance, hasinherits all the MPC’s virtues of distributed et 2009]. Experiments and numeric proveZheng that optimization index is proposed by [Li etresults al., performance 2014, thisal., strategy could significantly improve the explicitly accommodating constraints and good good dynamic performance, but hasconstraints all the virtues of distributed explicitly accommodating and dynamic al., 2009]. Experiments and numeric results prove that framework[Qin andalso Badgwell, 2003, Maciejowski, 2002, et this strategy could significantly improve the performance et al., 2009]. Experiments and numeric results prove that of entire system with small increasing of network connecperformance, but also has all all the the virtues of et distributed framework[Qin and Badgwell, 2003, Maciejowski, performance, but also has virtues of distributed strategy could significantly improve the performance Sandell Jr et al., 1978, Scattolini, 2009, Leirens al., 2002, 2010, this of entire system with small increasing of network connecthis strategy could significantly improve the performance tivity, each local controller have to connect to the conframework[Qin and Badgwell, 2003, Maciejowski, 2002, Sandell Jr et et al.,al., 1978, Scattolini, 2009, Leirens2013a]. et al., 2002, 2010, tivity, framework[Qin and Badgwell, 2003, Maciejowski, of with small increasing of connecChristofides 2012, Zheng et al., 2011b, each controller have to connect to the conof entire entire system with small increasing of network network connecofsystem itslocal neighbours, and its neighbours’s neighbours. Sandell Jr Jr et et et al.,al., 1978, Scattolini, 2009, Leirens2013a]. et al., al., 2010, 2010, trollers Christofides 2012, Zheng et2009, al., 2011b, Sandell al., 1978, Scattolini, Leirens et tivity, each local controller have to connect to the controllers of its neighbours, and its neighbours’s neighbours. tivity, each local controller have to connect to the conAnother useful strategy is that each subsystem optimize However, theet of distributed implementation Christofides etperformance al., 2012, 2012, Zheng Zheng et al., al., 2011b, 2011b, 2013a]. Christofides al., et 2013a]. trollers of its neighbours, and its neighbours’s neighbours. Another useful strategy is that each subsystem optimize trollers of its neighbours, and its neighbours’s neighbours. the weighted cost of all subsystems and solves the optimal However, the performance of distributed implementation of MPC, in most of cases, is not as good as that of Another useful strategy is that each subsystem optimize However, the performance of is distributed implementation weighted cost of iteration. all subsystems and solves the optimal Another useful strategy is that subsystem solution by parallel Byeach this method, theoptimize Parato of MPC, the inMPC. most of improve cases, notglobal as good as that of the However, performance of distributed implementation centralized To the performance the weighted cost of all subsystems and solves optimal of MPC, inMPC. mostsystem, of improve cases, isthe not as good good as that that of of by parallel By this the Parato the weighted cost of iteration. all subsystems andmethod, solves the the optimal Optimality, the best optimization performance in existing centralized To global performance of MPC, in most of cases, is not as as entire closed-loop several DMPC coordination s- solution solution by parallel iteration. By this method, the Parato centralized MPC. To improve the global performance of Optimality, the best optimization performance in existing solution by parallel iteration. By this method, the et al., entire closed-loop several coordination s- DMPCs, can be achieved[Stewart et al., 2010, ZhengParato centralized MPC. system, To the DMPC global performance of trategies appeared in improve the literatures [Zheng et al., 2009, Optimality, the best optimization performance in existing entire closed-loop system, several DMPC coordination sDMPCs, can be achieved[Stewart et al., 2010, Zheng al., Optimality, the best optimization performance in existing 2011a]. However, the global information is required when trategies appeared in the literatures [Zheng et al., 2009, entire closed-loop system, several DMPC coordination sCamponogara et al., 2002, Christofides et al., 2012, Zheng DMPCs, can be achieved[Stewart et al., 2010, Zheng et et al., trategies appeared in the literatures [Zheng et al., 2009, 2011a]. However, the global information is required when DMPCs, can be achieved[Stewart et al., 2010, Zheng et al., solving each subsystem’s optimal solution in this method, Camponogara et al., 2002, Christofides et al., 2012, Zheng trategies appeared in the literatures [Zheng et al., 2009, et al., 2013b]. People found that if the subsystems of each 2011a]. However, the global information is required when Camponogara et al., 2002, Christofides et al., 2012, Zheng solving each subsystem’s optimal solution in this method, 2011a]. However, the global information is required when which is not expected. Could we find a method to exactly et al., 2013b]. People found that if the subsystems of each Camponogara et al., 2002, Christofides et al., 2012, Zheng Supported by National Basic Research Program of China (973Proeach subsystem’s optimal solution in method, et al., al., 2013b]. 2013b]. People People found found that that if if the the subsystems subsystems of of each each solving which not expected. Could wewhich find aismethod exactly solving each subsystem’s optimal solution in this this method, define is the coordination degree, able to to make the et Supported by NationalNational Basic Research of China (973Program) (2013CB035500), NaturalProgram Science Foundation of Chiwhich is not expected. Could we find a method to exactly define the coordination degree, which is able to make the which is not expected. Could we find a method to exactly resulting DMPC obtains Pareto optimality with reduced Supported by National Basic Research Program of China (973Pro gram) (2013CB035500), National Natural Science Foundation of ChinaSupported (61233004,by61221003,61304078,61473187), the Higher National Basic Research Program of ChinaEducation (973Prodefine the coordination degree, which is able to make the resulting DMPC obtains Pareto optimality with reduced define the coordination degree, which is able to make the communication burden? This stimulated this study. gram) (2013CB035500), NationalProgram Natural Science Science Foundation of ChiChina (61233004, the Foundation Higher Education Research Fund 61221003,61304078,61473187), for the Doctoral of China (20120073130006 gram) (2013CB035500), National Natural of resulting DMPC DMPCburden? obtainsThis Pareto optimality with reduced communication stimulated this study. resulting obtains Pareto optimality with reduced na (61233004, 61221003,61304078,61473187), the Higher Education Research Fund for the Doctoral Program of China (20120073130006 & 20110073110018), the China Postdoctoral Science Foundation na (61233004, 61221003,61304078,61473187), the Higher Education In this paper, an N-stepThis Impacted Region communication burden? stimulated this study. communication burden? This stimulated thisOptimization study. Research Fund the Doctoral of China (20120073130006 & 20110073110018), the China Postdoctoral Science (2013M540364), and of Program Shanghai and Foundation Technology Research Fund for for theProject Doctoral Program ofScience China (20120073130006 In thisDMPC paper, an N-step Impacted Region Optimization based (N-step IRO-DMPC) is proposed, where & 20110073110018), the China Postdoctoral Science Foundation (2013M540364), and Project of Shanghai Science and Technology Committee(12dz1200203). & 20110073110018), the China Postdoctoral Science Foundation In this thisDMPC paper, an an N-step Impacted Region Region Optimization based (N-step IRO-DMPC) is proposed, where In paper, N-step Impacted Optimization (2013M540364), and Project of Shanghai Science and Technology Committee(12dz1200203). (2013M540364), and Project of Shanghai Science and Technology based DMPC (N-step IRO-DMPC) is proposed, where based DMPC (N-step IRO-DMPC) is proposed, where Committee(12dz1200203).
Committee(12dz1200203). Copyright © 2015 IFAC 832 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 832Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 832 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 832Control. 10.1016/j.ifacol.2015.09.072
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the range of each local MPC’s cost function covered is determined by the union of the all the adjacent matrices over the predictive horizon. Each local MPC optimized the cost of all subsystems it impacted on over the prediction horizon to cooperate with each other. In addition, an iterative algorithm is developed to resolve each local MPC. Through these ways, the Parato optimality was achieved with reduced communication resource comparing to the global cost optimization based DMPC. The remainder of this paper is organized as follows. Section 2 describes the problem to be solved in this paper. Section 3 presents the design of the proposed distributed MPC. Section 4 presents the simulation results to demonstrate the effectiveness of the proposed method. Finally, a brief conclusion to the paper is drawn in Section 5. 2. PROBLEM 2.1 Distributed system A distributed system is composed of many interacting subsystems, each of which is controlled by an independent controller, which in turn is able to exchange information with other controllers. Suppose that the distributed system S is composed of m discrete-time linear subsystems Si , i ∈ P = {1, 2, · · · , m} and m controllers Ci , i ∈ P. If subsystem Si is directly affected by Sj , for any i ∈ P and j ∈ P, subsystem Si is said to be a directly (or one-step) downstream system of subsystem Sj , and subsystem Sj is a directly (or onestep) upstream system of Si . Let P+i denote the set of the subscripts of the one-step upstream systems of Si , P−i is the set of the subscripts of the one-step downstream systems of Si . Let the subsystems interact with each other through their states. Then, subsystem Si can be expressed as xi,k+1 = Aii xi,k + Bii ui,k + Aij xj,k , (1) j∈P+i yi,k = Cii xi,k ,
where xi ∈ Xi ⊂ Rnxi , ui ∈ Ui ⊂ Rnui and yi ∈ Yi ⊂ Rnyi are respectively the local state, input and output vectors, and Xi , Ui and Yi are respectively the feasible set of the state xi , input ui and output yi which are used to bound the state, input and output according to the physical constraints on the actuators, the control requirements or the characteristics of the plant. A nonzero matrix Aij , j ∈ P+i , indicates that Si is directly affected by Sj . In the concatenated vector form, the system dynamics can be written as xa,k+1 = Axa,k + Bua,k , (2) ya,k = Cxa,k , T where xa = [xT xT · · · xT ∈ R nx , u a = m] 2 1 T T T T nu T T T [u1 u2 · · · um ] ∈ R and ya = [y1 y2T . . . ym ] ∈ Rny are respectively the concatenated state, control input and output vectors of the overall system S, and A, B and C are constant matrices of appropriate dimensions. Also, xa ∈ X = X1 × X2 × · · · × Xm , ua ∈ U = U1 × U2 × · · · × Um and ya ∈ Y = Y1 × Y2 × · · · × Ym .
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2.2 Control objective The control objective is to find a control method under the distributed framework, which could obtain the optimal solution of entire system with a reduced communication burden. The performance index of the whole system can be expressed as Ji (k) (3) J(k) = i∈P
where Ji (k) =
N Cii xi,k+l|k − ysp 2 i
l=1
Qi,l
2 + ∆ui,k+l−1|k R ,(4) i,l
yisp is the set point of the ith subsystem and ∆ui,k+l|k = ui,k+l|k − ui,k+l−1|k is the input increment of the ith subsystem at the time instant k. Constant matrix Qi,l , Ri,l > 0, l = 1, 2, . . . , N , is weighting coefficients for the ith subsystem, and let the weighting matrices for Si be Qi = block − diag{Qi,1 , Qi,2 , . . . , Qi,N } > 0 Ri = block − diag{Ri,1 , Ri,2 , . . . , Ri,N } > 0. 3. N-STEP IMPACTED-REGION OPTIMIZATION BASED DISTRIBUTED MPC
Consider that each local MPC Ci , i ∈ P, only optimize N (predictive horizon) step ahead performance, thus the solution of each local MPC only impacts performance of the subsystems which are interacted with Si during the predictive horizon. To take this interaction into account, a strategy that each local MPC optimize its N-step impacted region’s performance was proposed, and which is detailed as follows. 3.1 N-step impacted-region To proceed, we need the following definitions. Definition 1. Adjacent Matrix: Consider a system xk+1 = Axk , A ∈ Rnx ×nx which is composed of m subsystem, ¯ and its the adjacent matrix refers to a m × m matrix A, th th i , j element 1, when Aij = 0 a ¯ij = (5) 0, when Aij = 0 It can be seen from (5) that the adjacent matrix only reflects the directly interaction among subsystems. In fact, we usually concern that if one unit indirectly impacts another unit through some intermediate units. In this case, ¯ can the structure matrix corresponding to the k power of A be used to express the interaction in k step ahead. Then, following N-Step Accessible Matrix is defined to express the relationship among subsystem during N step ahead. Definition 2. N-Step Accessible Matrix: Consider a system xk+1 = Axk , A ∈ Rnx ×nx , which consists of m unit Si , i = 1, 2, ..., m, define that each unit Si is accessible to itself, the all accessible relationship can be described by a so called N-Step Accessible Matrix R, and ¯N ¯ ∪A ¯2 ∪ ... ∪ A R=I∪A It is also a kind of structure matrix, where the ith row and j th column element equals zero expresses that subsystem
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Sj is un-accessible to subsystem Sj . By logic operating rules, the above equation can be simply rewritten as ¯ N (6) R = (I ∪ A) In fact, for the control of system (1), the relationship among the states, inputs and outputs are very important. However, (5) can not exactly reflect this relationship. Consider that AB0 xa,k xa,k+1 ua,k+1 = 0 0 0 ua,k , (7) ya,k C 0 0 ya,k−1 for system (2), the adjacent matrix can be defined as ¯ B ¯ 0 A ¯d = 0 0 0 . (8) A ¯ 0 0 C From (6), the N-step Accessible Matrix can be expressed as Rxx Rxu Rxy N −1 = Rux Ruu Ruy R = (I ∪ Ad ) Ryx Ryu Ryy (9) ¯ + I)N ¯ A ¯ + I)N −1 0 (A B( 0 I 0. = N −1 ¯ ¯ N −2 ¯ ¯ ¯ C(A + I) C(A + I) B I
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The predictive model can be expressed as yi,k+l|k = Cii Alii xi,k +
l
Cii Al−h ii Bii ui,k+h−1|k
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+
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(13)
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Consider the physical limitations on the outputs, the input and the input increment, we can get following optimization problem for Si in each control period. Problem 1. For all subsystem Si , i ∈ {1, 2, . . . , m}, provided that xi,k , x ˆj,k+l|k , j ∈ P+h P+i , h ∈ P−iN and ∆ˆ ui,k+l−1|k−1 ,j ∈ P−iN , l = 1, 2, . . . , N , find the control sequence ∆ui,k:k+N −1|k , which minimize the performance index min J¯i (k) ui,k:k+N −1|k
Subject to the constraints: Equation(13), yi,L ≤ yi,k+l|k ≤ yi,U ,
yj,L ≤ yj,k+l|k ≤ yj,U , j ∈ P−iN ,
ui,L ≤ ui,k+l−1|k ≤ ui,U ,
(14) (15) (16)
Thus, the N-step input to output accessible matrix is ¯ A ¯ + I)N −2 B. ¯ Ryu = C( (10)
∆ui,L ≤ ∆ui,k+l−1|k ≤ ∆ui,U ,
(17)
¯ is a unit matrix in (10), the N-step Consider that B accessible matrix can be redefined as ¯ A ¯ + I)N −2 . (11) Ryu = C(
||yi,k+N |k − yisp ||2Qi,N < ε2 .
(18)
And the N-step down stream neighbour of Si can be defined as the subsystems Sj where the j th row, and ith collum element of Ryu equals 1. And denote P−iN as the set of the subscript of all the N-step downstream neighbor of Si . 3.2 Distributed MPC design
The performance of local MPC for subsystem Si is defined as N Cii xi,k+l|k − ysp 2 i
l=1
+
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where, [yi,L , yi,U ], [ui,L , ui,U ] and [∆ui,L , ∆ui,U ] are the bounds of outputs, inputs and the increment of inputs respectively. Equation (18) is a final constraint for improve the stability of each subsystem-based MPC, and ε > 0. To solve problem (1) efficiently, following iterative algorithm is given for ∀Si , i ∈ P. Algorithm 1. (N-step IRO-DMPC Algorithm). Step 1: Initialization.
Consider that the control law of current subsystem Si effects the performance of its N-step downstream neighboring subsystems Sj , j ∈ P−iN , in the N-Step IRO-DMPC, the performance of Sj , j ∈ P−iN is added into the performance index of the MPC which controls Si based on a approximation of the updated state sequence of Sj . In this way, the coordination degree is expanded and is equivalent to that of global cost optimization based DMPC.
J¯i (k) =
l = 1, 2, ..., N ;
Qj,l
(12)
Define that x ˆi,k+l|k , u ˆi,k+l|k and ∆ˆ ui,k+l|k be the assumed states, the assumed input and the assumed input increment which are calculated in the previous calculation, respectively. 834
• Initialize xi,k0 , xi,k0 +l|k0 , l = 1, 2, . . . , N , which satisfy the constraints of Problem 1. Step 2: Update control law at time k > k0 . • Step 2.1 Set iteration t = 1, and set x ˆi,k+l|k = xi,k+l|k−1 . • Step 2.2 Measure xi (k), transmit x ˆi,k+l|k to its N-step down stream neighbors and u ˆ i,k+l|k upstream neighbors; And receive u ˆ i,k+l|k from its N-step down stream neighbors and x ˆj,k+l|k from its upstream neighbors. • Step 2.3 Solving Problem 1 to obtain the optimal solution ∆uti,k+l|k , and predict the future state xi,k+l|k based on the solution ∆uti,k+l|k . • Step 2.4 If 2 ||∆uti,k+l−1|k − ∆ut−1 i,k+l−1|k ||2 ≤ ε0 or t > tmax then set
u∗i,k = ui,k−1 + ∆u∗i,k+l−1|k ,
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0 1 0 1 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0
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From (28) the network connectivity is 15 when using proposed N-step ICO-DMPC, which is most less than 72 when using global cost optimization based DMPC. The network connectivity is dramatically deduced.
Step 3: Update control at time k + 1. • Let k + 1 → k, repeat Step 2. It should be noticed that although an iterative algorithm is presented, the Problem 1 can also be solved by a noniterative algorithm through setting tmax = 1. Since the communication burden will increase with the increasing of iteration, tmax should not be set too large in practice. So far the N-step impacted-region optimization based DMPC for distributed system is introduced, some simulation results will be presented in the next section to show the effectiveness of the proposed method. 4. SIMULATION For simplicity, a 9 nodes distributed network is taken as example, and the relationship among these notes is shown in Fig. 1 where the arrow from subsystem Si to Sj expresses that Sj is directly effected by Si .
The dynamic models of these nodes are respectively by x1,k+1 = 0.57x1,k + 0.38u1,k S1 : y1,k = x1,k , x2,k+1 = 0.53x2,k + 0.38u2,k + 0.16x1,k +0.16x4,k S2 : y2,k = x2,k , x3,k+1 = 0.55x3,k + 0.38u3,k S3 : y3,k = x3,k , x4,k+1 = 0.61x4,k + 0.39u4,k + 0.18x2,k +0.18x3,k + 0.18x5,k S4 : y4,k = x4,k , x5,k+1 = 0.68x5,k + 0.42u5,k S5 : y5,k = x5,k , x6,k+1 = 0.55x6,k + 0.38u6,k + 0.16x5,k S6 : y6,k = x6,k , x7,k+1 = 0.71x7,k + 0.42u7,k + 0.21x6,k +0.21x8,k S7 : y7,k = x7,k , x8,k+1 = 0.57x8,k + 0.38u8,k + 0.17x9,k S8 : y8,k = x8,k , x9,k+1 = 0.66x9,k + 0.41u9,k + 0.20x8,k S9 : y9,k = x9,k .
1 1 0 1 = 0 0 0 0
given
(19) (20) (21) (22) (23) (24) (25) (26) (27)
For the purpose of comparison, both the Centralized MPC and the N-step IRO-DMPC are applied to this system. Let the constraint on the input be [ui,L , ui,U ] = [−2, 2] and the constraint on the increment of input be [∆ui,L , ∆ui,U ] = [−1.5, 1.5]. Set the all controllers’ (both centrialized MPC and N-step IRO-DMPC) parameters of control horizon be N = 10, the weighting matrices be Qi = [1, 1, 1, 1, 1, 1, 1, 1, 1, 5], Ri = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], where i ∈ {1, 2, ..., 9}.
The state responses and the inputs of the closed-loop system under the control of the centralized MPC and LCO-DMPC are shown in Figs. 2 and 3, respectively. The shape of the state response curves under the control of N-step IRO-DMPC are almost equals to those under the Centralized MPC. Under the N-step IRO-DMPC control design, when set point changed, there is no significant overshooting, but some fluctuation exists in the trajectories of states of the interacting subsystems.
From these simulation results, it can be seen that the proposed N-step IRO-DMPC could obtain a global performance almost equal to that of using centralized MPC and the global information is not necessary for every local MPC, which keeps the characteristics of good error tolerance and high flexibility of the Distributed Control Framework. 5. CONCLUSION In this paper, an N-step Impact-Region Optimization based DMPC is provided for distributed systems. The simulation results of the control of a distributed network composed by nine first-order systems shows the efficiency of the proposed method. With the proposed method, the closed-loop system could obtain a global performance almost equivalent to that of with the centralized MPC. In addtion, the global information is not necessary for each local MPC in the N-Step IRO-DMPC comparing to the global cost optimization based DMPC, which could significantly deduce the network-connectivity for sparse systems, increase the capability of error tolerance of control system. The stabilizing implementation of proposed DMPC subject to pdecoupled constraints maybe a extension of this work and will be done in the near future. REFERENCES W. Al-Gherwi, H. Budman, and A. Elkamel. Selection of control structure for distributed model predictive
According to (11), the N-step accessible matrix is: 835
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Fig. 3. The evolution of the inputs under the centralized MPC, and N-step Impated-region optimization based DMPC.
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