IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND … 2nd_IEEE TASE.pdf · Impacted-Region Optimization for Distributed Model Predictive Control Systems With Constraints Shaoyuan Li,

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Impacted-Region Optimization for Distributed ModelPredictive Control Systems With ConstraintsShaoyuan Li, Senior Member, IEEE, Yi Zheng, Member, IEEE, and Zongli Lin, Fellow, IEEE

Abstract—For a large-scale distributed system, distributedmodel predictive control (DMPC) is a method of choice becauseof its ability to explicitly accommodate constraints and to achievegood dynamic performance. In the design of a DMPC, guaran-teeing stability with a strong global performance is known to bea challenge. In this paper, we consider a large-scale distributedsystem whose input is constrained to given sets in their respectivespaces and propose a stabilizing DMPC design, where each sub-system-based model predictive control (MPC) optimizes the costfunction of the entire system over the region it directly impacts on.Consistency constraints and stability constraints, which bound theestimation errors of the interaction sequences among subsystems,are designed to guarantee that, if an initially feasible solution canbe found, subsequent feasibility of the algorithm is guaranteed atevery update, and that the closed-loop system is asymptoticallystable. A key feature of the proposed DMPC is that it coordinatesthe MPCs of the subsystems by redefining the impact region ofa subsystem according to the coordination strategy. Simulationresults show that the performance of the proposed DMPC is veryclose to that of a centralized MPC.

Note to Practitioners—The proposed method is designed forthe systems which are controlled in a distributed structure. Theaim of this method is to achieve a good performance of entiresystem within limited communication burden, and guarantee therecursive feasibility and asymptotical stability of entire system.To apply this method, the first step is to obtain the models ofeach subsystems. Then to configure the network information ex-changing among these subsystems and construct the optimizationproblem of each subsystem. Finally, use the existing QuadraticallyConstrained Quadratic Program solvers to calculate the optimalcontrol laws and applied these optimal control laws to the cor-responding subsystems. This method can be used in large scalechemical processes, air condition systems for multi-zone buildings,distributed energy generation systems, etc.

Manuscript received December 12, 2013; revised May 01, 2014; acceptedJune 29, 2014. This paper was recommended for publication by Associate Ed-itor Q.-S. Jia and Editor D. Tilbury upon evaluation of the reviewers’ comments.This work was supported by the National Nature Science Foundation of China(61233004, 61221003, 61374109,61304078), the National Basic Research Pro-gram of China (973 Program-2013CB035500), and partly sponsored by the In-ternational Cooperation Program of Shanghai Science and Technology Com-mission (12230709600), the Higher Education Research Fund for the DoctoralProgram of China (20120073130006,20110073110018), and the China Postdoc-toral Science Foundation (2013M540364).S. Li is with the Department of Automation, Shanghai Jiao Tong University,

Key Laboratory of System Control and Information Processing, Ministry of Ed-ucation of China, Shanghai 200240, China (e-mail: [email protected]).Y. Zheng is with the School of Electronic Information and Electrical En-

gineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail:[email protected]).Z. Lin is with the Charles L. Brown Department of Electrical and Com-

puter Engineering, University of Virginia, Charlottesville, VA 22904-4743USA(e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TASE.2014.2337259

Index Terms—Constrained control, distributedmodel predictivecontrol, impacted-region optimization, large-scale systems, modelpredictive control, plant-wide optimization.

I. INTRODUCTION

C ONSIDER a complex large-scale control system which iscomposed of many physically or geographically divided

subsystems. Each subsystem interacts with some other subsys-tems by their states and/or inputs, e.g., large-scale chemicalprocess [1], smart grid [2], [3], distributed generation systems[4], [5]. The control objective is to achieve a specific global per-formance of the entire system or a common goal of all subsys-tems.In controlling such a large-scale system, the distributed (or

decentralized) framework, where each subsystem is controlledby an independent controller, is usually adopted despite the re-sulting global performance is in general not as good as a cen-tralized solution [6], [7], since the distributed framework has theadvantages of fault tolerance, less computation and being flex-ible to system structure. [8], [9].Among the distributed solutions, the Distributed Model Pre-

dictive Control (DMPC), which controls each subsystem by aseparate local Model Predictive Control (MPC), has becomemore and more popular [1], [10] since it not only inherits MPC’sability to explicitly accommodate constraints [11], [12], but alsopossesses the advantages of the distributed framework men-tioned above. However, as pointed out in [6], the performanceof a DMPC is, in most cases, not as good as that of a centralizedMPC.Many DMPC algorithms have appeared in the literature for

different type of systems and for different problems in the de-sign of DMPC, e.g., design of DMPC for nonlinear systems[13], [14], uncertain systems [13], [15], and networked systemswith time delay [16], development of distributed optimizationalgorithms[17], [18], and design of cooperative strategies forimproving performance of DMPC [19], [20], as well as the de-sign of control structure [21]. Among them, several coordina-tion strategies focus on study how to improve the global per-formance of the DMPC, and can be classified according to theinformation exchange protocol needed (i.e., non-iterative or it-erative algorithms), and to the type of cost function which isoptimized [10]. The non-iterative algorithms which only com-municate once a control period have faster computational speedcomparing to that of iterative algorithm and the iterative algo-rithms could obtains more excellent optimization performance

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2 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING

than non-iterative algorithms. The DMPCs, which accommo-date same kind of cost function for each subsystem-basedMPC,can be solved either by iterative algorithm or non-iterative al-gorithm. Thus, the coordination strategies are introduced by theclassification of the type of cost function which is optimizedin each subsystem-based MPC. The simplest and most adoptedstrategy is that each local controller minimizes its own sub-system’s cost and uses the state prediction of the previous timeinstant to approximate the state sequence at the current time in-stant in computing the optimal solution [22]–[24]. Another com-monly used coordination strategy is that each subsystem-basedMPC optimizes the cost of overall system to improve the globalperformance [6], [19], [25]. In computing the optimal solution,it also uses the state prediction of the previous time instant toapproximate the state sequence at the current time instant. Thisstrategy could achieve a good global performance in some cases,but it reduces the flexibility and increases the communicationload. In an effort to achieve a trade-off between the global per-formance of the entire system and the computational burden, re-cently, an intuitively appealing strategy is proposed in [8], [26],where each subsystem-based MPC only considers the cost of itsown subsystem and those of the subsystems it directly impacts.Such a design can be referred to as Impacted-region Cost Op-timization based DMPC (ICO-DMPC). In particular, [26] ap-plies this design idea to a metallurgy system and [8] explainswhy this coordination strategy could improve the global per-formance. Numerical and practical experiments show that thiscoordination strategy could obtain a performance close to thatof a classical centralized MPC. However, neither [8] nor [26]takes constraints into consideration in the DMPC design.Control design that takes state and/or input constraints into

account, whether or not under the MPC framework, is an im-portant and challenging problem. Many methods can be foundin the literature (see, for example, [27]–[30]). Under the DMPCframework, [24] provides a design for nonlinear continuous sys-tems, which uses constraints to limit the error between the fu-ture state sequences (or called presumed sequences) of upstreamneighbors, which are calculated based on the solution in theprevious time instant, and the predictive states calculated bythe corresponding subsystem in the current time instant. Thenthe stability is ensured by judiciously integrating designs of thebound of the error [24], the terminal cost, the constraint set andthe local controllers [24], [31]. Ref. [32] gives another designfor linear system, which uses a fixed reference trajectory witha moving widow to substitute the presumed state/input of up-stream neighbors used in [24]. Both of these two methods aredesigned for DMPC in which each subsystem-based MPC op-timizes the cost of the corresponding subsystem itself. As forthe DMPC which uses the global cost function, some conver-gence conditions are deduced if using iterative algorithms, thenthe distributed problems can be reformulated into a centralizedproblem, and the stabilizing DMPC can be designed with sim-ilar method of centralized MPC [6], [25]. For the coordinationstrategy used here, there is no global model can be used. Andexcept that there are errors between the presumed state/input se-quences and predictive state sequences of upstream neighbors,the predictive state sequences of downstream neighbors calcu-

Fig. 1. An illustration of the structure of a distributed system and its distributedcontrol framework.

lated by current subsystem may not equal those calculated bythe downstream neighbors themselves; these error are hard toestimate. In the presence of constraints, the remaining part ofthe optimal control sequence calculated at the previous time in-stant may not be a feasible solution at the current time instant.All these make it difficult to design a stabilizing ICO-DMPCthat takes constraints into consideration.In this paper, the coordination strategy that optimizes the im-

pacted-region cost in each subsystem-based MPC is adopted toachieve a DMPC performance that is close to that of a central-ized MPC. The consistency constraints, which limit the errorbetween the presumed state and the state predicted at the cur-rent time instant within a prescribed bound, are designed andincluded in the optimization problem of each subsystem-basedMPC. These constraints can bound the error between the pre-sumed state and the predictive state of upstream neighbors andthe error between the predictive state of downstream neighborcalculated by current subsystem-based MPC and that of thedownstream neighbors themselves. And these constraints guar-antee that the remaining part of the solution at the current timeinstant is a feasible solution at the next time instant. In themeantime, stabilization constraints and the dual mode predic-tive control [29] strategy are adopted to result in a stabilizingICO-DMPC.The remainder of this paper is organized as follows.

Section II describes the problem to be solved in this paper.Section III presents the design of the stabilizing ICO-DMPC.The feasibility of the proposed ICO-DMPC and the stabilityof the resulting closed-loop system are analyzed in Section IV.Section V discusses DPMC formulations under other coordi-nation strategies. Section VI presents the simulation resultsto demonstrate the effectiveness of the proposed DMPC al-gorithm. Finally, a brief conclusion to the paper is given inSection VII.

II. PROBLEM DESCRIPTION

A. Distributed Systems

A distributed system, as illustrated in Fig. 1, is composed ofmany interacting subsystems, each of which is controlled by asubsystem-based controller, which in turn is able to exchangeinformation with other subsystem-based controllers.


LI et al.: IMPACTED-REGION OPTIMIZATION FOR DISTRIBUTED MODEL PREDICTIVE CONTROL SYSTEMS WITH CONSTRAINTS 3

Suppose that the distributed system is composed of dis-crete-time linear subsystems , andcontrollers , . Let the subsystems

interact with each other through their states. If subsystem isaffected by , for any and , subsystem is said tobe a downstream system of subsystem , and subsystem isan upstream system of . Let denote the set of the subscriptsof the upstream systems of , is the set of the subscripts ofthe downstream systems of . Then, subsystem can be ex-pressed as

(1)

where , and are re-spectively the local state, input and output vectors, and is thefeasible set of the input , which is used to bound the input ac-cording to the physical constraints on the actuators, the controlrequirements or the characteristics of the plant. A non-zero ma-trix , that is, , indicates that is affected by . In theconcatenated vector form, the system dynamics can be writtenas

(2)

where ,and

are respectively the concatenated state, control input andoutput vectors of the overall system , and , andare constant matrices of appropriate dimensions. Also,

.The control objective is to stabilize the overall system in

an DMPC framework.

B. Existing Methods and Motivations

The works on DMPC can be roughly divided in optimiza-tion-based frameworks which focus on the solution of the op-timization problems and control-based frameworks which areconcerned mostly with stability and control performance [18],[20]. The control-based frameworks can be classified accordingto the type of cost function which is optimized. We briefly re-view these methods as motivations for the problem we have justformulated and its solution to be presented later in the paper.1) Distributed algorithms where each subsystem-based con-troller minimizes the cost function of its own subsystemwere proposed in [22], [23]:

(3)

When computing the optimal solution, each local con-troller exchanges state estimation with the neighboringsubsystems to improve the performance of the local sub-system. This method is simple and very convenient forimplementation. An extension of this stabilizing DMPCwith input constraint for nonlinear continuous systems is

given in [24], and a stabilizing DMPC with inputs andstates constraints is given in [32].

2) To improve the global performance, distributed algo-rithms, where each local controller minimizes a globalcost function

(4)

were proposed in [6], [19]. In this method, each subsystem-based MPC exchanges information with all other subsys-tems. This strategy may result in a better performance butconsumes much more communication resources, in com-parison with the method in described in 1).

3) To balance the performance, communication cost, andcomplexity of the DMPC algorithm, the strategy thateach subsystem-based controller only minimizes its owncost function and those of the subsystems that its ownsubsystem directly impacts was recently proposed in [8],[26], that is

(5)

where or is the set of subscriptsof the downstream subsystems of subsystem , that is theregion impacted on by subsystem . The resulting controlalgorithm is termed as an Impacted-region Cost Optimiza-tion based DMPC, or ICO-DMPC. It could achieve betterperformance than the first method, and its communicationburden is much less than the second method [21], [33].

Clearly, this coordination strategy as proposed in [8], [26] anddescribed in 3) is a preferable method to trade off the communi-cation burden and the global performance. However, the DMPCunder this coordination strategy does not take the constraintsinto consideration. A stabilizing DMPC that takes constraintsinto consideration remains to be developed under this coordi-nation strategy. The objective of this paper is to develop such aDMPC design.

III. THE ICO-DMPC FORMULATION

In this section, separate optimal control problems, one foreach subsystem, and the ICO-DMPC algorithm are defined. Ineach of these optimal control problems, the same constant pre-diction horizon , , is used. The resulting separatesubsystem-basedMPC laws are updated synchronously. At eachupdate, every subsystem-based MPC optimizes only over itsown predicted open-loop control, given its impacted region’scurrent states, and their upstream subsystems’ estimated inputsand states.To proceed, we need the following assumption.Assumption 1: For every subsystem , , there exists a

decoupled static feedback such thatis Shur stable, and the closed-loop systemis asymptotically stable, where and

.This assumption is usually used in the design of stabilizing

DMPC, see [24], [32]. It presumes that each subsystem is ableto be stabilized by a decentralized control , . and the



TABLE INOTATION

decentralized control gain based on LMI have been proposedin [34] for continuous time systems. These techniques can beeasily adapted to the discrete-time case here considered.We also define the necessary notation in Table I.As the state evolution of the downstream subsystems of is

affected by the optimal control decision of , the performanceof these downstream subsystems may be affected negativelyby the control decision of . Thus, in the ICO-DMPC, eachsubsystem-based MPC takes into account the cost functions ofits downstream subsystems. More specifically, the performanceindex is defined as

(6)

where , and . Thematrix is chosen to satisfy the Lyapunov equation

where . Denote

Then, it follows that

where .To get the predicted state sequence of subsystemunder the control decision sequence of in (6),

the system evolution model should be deduced first. Since everysubsystem-based controller is updated synchronously, the stateand control sequences of other subsystems are unknown to sub-system . Thus, at the time instant , the presumed state se-quence of and thepresumed control sequence

of are used in the predictive model of the MPC in ,which is given as

(7)

where

...

......

. . ....

Given , , the pre-sumed control sequence for subsystem is given by

(8)

and

(9)



Set each presumed state sequence to be the remainderof the sequence predicted at time instant , concatenatedwith the closed-loop response under the state feedback control

, that is

(10)

and is calculated by substi-tuting into (7) based on the presumed statesequence obtained at time .In addition, a decoupled terminal state constraint will be in-

cluded in each subsystem-based MPC, which guarantees thatthe terminal controllers are stabilizing inside a terminal set. Todefine this terminal state set, we need to make an assumptionand establish a technical lemma.Assumption 2: The block-diagonal matrix

and the off-diagonal matrixsatisfy the following inequality:

It, along with Assumption 1, helps with the design of theterminal set. This assumption quantifies how strengthening thecoupling among subsystems is sufficient such that the overallsystem can be stabilized by the proposed DMPC here.Lemma 3.1: Under Assumptions 1 and 2, for any positive

scalar , the set

is a positive invariant region of attraction for the closed-loopsystem . Additionally, there exists a smallenough positive scalar such that is in the feasible input set

for all .Proof: Consider the function . The time

difference of along the trajectories of the closed-loopsystem can be evaluated as

which holds for all . This implies that alltrajectories of the closed-loop system that starts inside willremain inside and converge to the origin.The existence of an such that for all

follows from the fact that is positive definite, which impliesthat the set shrinks to the origin as decreases to zero. Thiscompletes the proof.In the optimization problem of each subsystem-based MPC,

the terminal state constraint set for each can then be set to be

Clearly, if , then the decoupled con-trollers will stabilize the system at the origin, since

implies that

which in turn implies that . Suppose that at some time, for every subsystem. Then, by Lemma 3.1,

stabilization can be achieved if every employs its decoupledstatic feedback controller after time instant .Thus, the objective of each subsystem-based MPC law is to

drive the state of each subsystem to the set . Once allsubsystems have reached these sets, they switch to their decou-pled controllers for stabilization. Such switching from an MPClaw to a terminal controller once the state reaches a suitableneighborhood of the origin is referred to as the dual-mode MPC[29]. For this reason, the DMPC algorithm we propose in thispaper is a dual-mode DMPC algorithm.In what follows, we formulate the optimization problem for

each subsystem-based MPC.1) Problem 1: Consider subsystem . Let be as

specified in Lemma 3.1. Let the update time be . Given, and , , and ,

, find the control sequencethat minimizes

subject to the following constraints:

(11)

(12)

(13)

(14)

(15)

(16)

In the constraints above,

(17)

(18)



(19)

(20)

where

(21)and where , and are respectively the th,

th and th subsystem in the downstreamregion of . Finally, the constants andare design parameters whose values will be chosen in the sequel.Equations (11)–(13) are referred to as the consistency con-

straints, which require that each predictive sequence and con-trol variables remain close to their presumed values. These con-straints are key to proving that is a feasible state sequenceat each update.Eqaution (14) will be utilized to prove that the ICO-DMPC

algorithm is stabilizing, where is a design parameterwhose value will be specified later to satisfy (33),

is a feasible state sequence, andequals to the solution of (7) under the initial state of andthe feasible control sequence , , is definedby

,.

(22)

Comparing Problem 1 with the method proposed in [24], boththe optimization index and the consistent constraints are dif-ferent. In Problem 1, the constraints (11) are necessary sincethe estimation error cannot be expressed by the states sequence.In addition, the terminal constraint should bound both the finalstates of corresponding subsystem and also that of the subsys-tems it directly impacted.Before stating the ICO-DMPC algorithm, we make the fol-

lowing assumption to facilitate the initialization phase.Assumption 3: At initial time , there exists a feasible con-

trol , , for each , suchthat the solution to the full system

, denoted as , satisfiesand results in a bounded cost . Moreover, each

subsystem has access to .Assumption 3 bypasses the task of actually constructing an

initially feasible solution in a distributed way. In fact, findingan initially feasible solution for many optimization problemsis often a primary obstacle, whether or not such problems areused in a control setting. Recently, [32] has given a method toinitialize the feasible solution of distributed MPC, which can beused here with little modification.Algorithm 1 (ICO-DMPC Algorithm): The dual-mode ICO-

DMPC law for any is constructed as follows:Step 1: Initialization.• Initialize , , , to satisfyAssumption 3;

• At time , if , then apply the terminal con-troller , for all ; Else

• Compute according to (7) andtransmitand to and ;

• Receive from andfrom .

Step 2: Update of control law at time .• Measure , transmit to , and receivefrom ; If , then apply the terminal controller

; Else• Solve Problem 1 for and apply ;• Compute according to (7) and transmit

and to and ;• Receive from andfrom ;

Step 3: Update of control at time .• Let , repeat Step 2.Algorithm 1 presumes that all local controllers , ,

have access to the full state . This requirement results solelyfrom the use of the dual-mode control, in which the switchingoccurs synchronously only when , with beingas defined in Lemma 3.1. In the next section, it will be shownthat the ICO-DMPC policy drives the state to ina finite number of updates. As a result, if is chosen suffi-ciently small, then MPC can be employed for all time withoutswitching to a terminal controller, eliminating the need of thelocal controllers to access the full state. Of course, in this case,instead of asymptotic stability at the origin, we can only drivethe state toward the small set .The analysis in the next section shows that the ICO-DMPC

algorithm is feasible at every update and is stabilizing.

IV. ANALYSIS

A. Feasibility

The main result of this section is that, provided that an ini-tially feasible solution is available and Assumption 3 holds true,for any and at any time , is a feasiblecontrol solution to Problem 1. This feasibility result refers that,for any and at any update , the control and state pair

, satisfy the consistency constraints(11)–(13), the control constraint (15) and the terminal state con-straint (16).Lemma 4.1 identifies sufficient conditions that ensure

, where .Lemma 4.2 identifies sufficient conditions that ensure

for all, . Lemma 4.3 establishes that the control con-

straint is satisfied. Finally, Theorem 4.1 combines the resultsin Lemmas 4.1–4.3 to arrive at the conclusion that, for any

, , the control and state pair area feasible solution to Problem 1 at any update .Lemma 4.1: Suppose that Assumptions 1–3 hold and. For any , if Problem 1 has a solution at time and

for any , then



and

provided that and satisfy

(23)

and

(24)

Proof: Since Problem 1 has a solution at time , byconstruction, it has

Define the presumed state of , in controller as

,,(25)

and substitute (25) and the definition (9) into (7) to have

Consequently, to ensure the boundholds in all controllers , , a sufficient con-

dition is that

This completes the proof.Lemma 4.2: Suppose that Assumptions 1–3 hold and. For any , if Problem 1 has a solution at every update

time , , then

(26)

for all and all , provided that (23)and the following parametric condition hold:

(27)

where is as defined in (19). Furthermore, the feasible controland the feasible state satisfy constraints

(11)–(13).Proof: We will prove (26) first. Since a solution exists at

update time , according to (7), (9) and (22), forany , the feasible state is given by

(28)

and the presumed state is

(29)

where, according to (7),



Since ,and , it follows from the

system dynamic equation (7) that .In addition, note that . Thus, theabove equation can be rewritten as

(30)

Subtracting (29) from (28), and substituting (30) into the re-sulting equation, we obtain the discrepancy between the feasiblestate sequence and the presumed state sequence as

(31)

Let the subsystems which respectively maximize the fol-lowing two functions as and ,

Then, the following equation can be deduced from (31):

Since and satisfy constraints (11) and (12) forall times , the following equation can bededuced:

(32)

Thus, (26) holds for all .

In what follows we prove that (26) holds for . Denotethe feasible states of , used in controller , as

,.

Then, the discrepancy between the feasible stateand the presumed state is

Now consider

and the constraint

Then, in view of (23), we have

This completes the proof of (26).In what follows we will prove that the feasible control

and the feasible state satisfy con-straints (11)–(13).First, for any ,

. Thus, constraint (11) is satisfied.Also,

Thus, when

state , , satisfies constraint (12).Finally,

which shows constraint (13) is satisfied. This concludes theproof.In what follows we establish that, at time , if conditions (23),

(26) and (27) are satisfied, then and ,, are a feasible solution of Problem 1.



Lemma 4.3: Suppose that Assumptions 1–3 hold,, and conditions (23) and (27) are satisfied. For any , if

Problem 1 has a solution at every update time ,, then for all .Proof: Since Problem 1 has a feasible solution at

, andfor all , we only need to show that

.Since has been chosen to satisfy the conditions of Lemma

3.1, for all when . Consequently, asufficient condition for is that

.In view of Lemmas 4.1 and 4.2, using the triangle inequality,

we have

that is, . This concludes the proof.Theorem 4.1: Suppose that Assumptions 1–3 hold,and constraints (11)–(13) and (16) are satisfied at . Then,

for every , the control and state , re-spectively defined by (22) and (7), are a feasible solution ofProblem 1 at every update .

Proof: We will prove the theorem by induction.First, consider the case of . The state sequence

trivially satisfies the dynamic equation(7), the stability constraint (14) and the consistency constraints(11)–(13).Observe that

and that

Thus, . By the invariance of under theterminal controller and the conditions in Lemma 3.1, it followsthat the terminal state and control constraints are also satisfied.This completes the proof of the case of .Now suppose that is a feasible solution for

. We will show that is a feasiblesolution at update .As before, the consistency constraint (11) is trivially satisfied,

and is the corresponding state sequence that satisfiesthe dynamic equation. Since there is a solution for Problem 1 atupdates , Lemmas 4.1–4.3 can be invoked.Lemma 4.3 guarantees control constraint feasibility. In view ofLemmas 4.1 and 4.2, using the triangle inequality, we have

for each , . This shows that the terminal state con-straint is satisfied and the proof of Theorem 4.1 is completed.

B. Stability

The stability of the closed-loop system is analyzed in thissubsection.Theorem 4.2: Suppose that Assumptions 1–32 hold,, constraints (11)–(13) and (16) are satisfied, and the fol-

lowing parametric condition holds:

(33)

Then, by application of Algorithm 1, the closed-loop system (2)is asymptotically stable at the origin.

Proof: By Algorithm 1 and Lemma 3.1, when enters, the terminal controllers take over to keep it in there and

stabilize the system at the origin. Therefore, it remains to showthat if , then by the application of Algorithm 1,the state of system (2) is driven to the set in finite time.Define the non-negative function for

In what follows, we will show that, for any , if, then there exists a constant such that

. Constraint (14) implies that

Therefore,

Subtracting from and using, , gives

(34)

Assuming yields

(35)



Also, by Theorem 1 we have

(36)

and by Lemma 3, we have

(37)Using (35)–(37) in (34) then yields

(38)

which, in view of (33), implies that .Thus, for any , if , there is a constant

such that . It then followsthat there exists a finite time such that . Thisconcludes the proof.We have now established the feasibility the ICO-DMPC and

the stability of the resulting closed-loop system. That is, if aninitially feasible solution could be found, subsequent feasibilityof the algorithm is guaranteed at every update, and the resultingclosed-loop system is asymptotically stable at the origin.It should be noticed that a general mathematical formulation

is adopted in the ICO-DMPC algorithm and its analysis. TheICO-DMPC and the resulting analysis can be used for any co-ordination policy mentioned in Section II with a redefinition of. Thus, it provides a unified framework for the DMPCs which

adopts the cost function based coordination strategies. This is avery important contribution of the paper. In the next section, wewill present the formulations under other coordination strate-gies.

V. FORMULATIONS UNDER OTHER COORDINATION STRATEGIES

A. Local Cost Optimization Based DMPC

In this coordination strategy, each subsystem-based MPCminimizes its own cost. Redefine , that is, isan empty set, and . Consequently the optimizationproblem of the stabilizing DMPC, where each subsystem-basedMPC takes the local cost as the performance index, can bederived in the framework of ICO-DMPC as

subject to the constraints

It should be noted that the consistency constraints in inputs(12) do not appear here. This is because there are no inputs, ex-cept for , that appear in the predictive model. This result isconsistent with the linear version of what is presented in [24],which provides a local performance index based DMPC for con-tinuous nonlinear systems.

B. Global Cost Optimization Based DMPC

In this coordination strategy, each subsystem-basedMPC cal-culates the optimal to minimize the cost function of theentire system. The predictive model of each subsystem-basedMPC includes the state evolutions of all subsystems, and can bededuced as follows according to (1):

(39)

Also, is nonexistent, and . Con-sequently, the optimization problem of the DMPC, where theglobal cost is minimized at each subsystem-based MPC, can bededuced from the ICO-DMPC as follows:

subject to the constraints



Fig. 2. The interaction relationship among subsystems.

It can be seen that the optimization problem with global costoptimization based DMPC is much simpler than Problem 1. Theconstraints (12) in Problem 1 do not appear. This is because theassumed state sequences of the other subsystems are not used inthe predictive model of each subsystem-based MPC.

VI. SIMULATION RESULTS

A. The System

A distributed system consisting of four interacted subsys-tems is used to demonstrate the effectiveness of the proposedmethod. The relationship among these four subsystems is shownin Fig. 2, where is impacted by , is impacted by and, and is impacted by . Let be defined to reflect both

the constraint on the input and the constrainton the increment of the input .The models of these four subsystems are respectively given

by

(40)

(41)

(42)

(43)

For the purpose of comparison, both the centralizedMPC andthe ICO-DMPC are applied to this system.

B. Performance of Closed-Loop System Under the ICO-DMPC

The simulation program is developed with Matlab. And theoptimizing tool, fmincon, is used to solve each subsystem-basedMPC in every control period. The tool of fmincon has alreadybeen provided in Matlab and it is able to solve multi-variablecost function with nonlinear constraints.Some parameters of the controllers are shown in Table II.

Among these parameters, is obtained by solving theLyapnov function. The eigenvalue of each closed-loop systemunder the feedback control shown in the table is 0.5. Theeigenvalues of are

, all of which are negative.Thus, Assumption 2 is satisfied. Set . Consequently, if

, then would be less than 0.1,and the constraints on the inputs and the increments of inputs,as shown in Table II, are satisfied. Set the control horizon of allthe controllers to be . Set the initial presumed inputsand states, at time , to be the solution calculated by acentralized MPC and the corresponding predictive states.

TABLE IIPARAMETERS OF THE ICO-DMPC

Fig. 3. The evolution of the states under the ICO-DMPC.

Fig. 4. The evolution of the control inputs under the ICO-DMPC.

The state responses and the inputs of the closed-loop systemare shown in Figs. 3 and 4, respectively. The states of all foursubsystems converge close to zero in about 10 seconds. Thestate of undershoots by 0.06 before converging to zero.

C. Performance Comparison With the Centralized MPC andthe Local Cost Optimization Based MPC

To further demonstrate the performance of the proposedDMPC, a dual model centralized MPC and a local cost op-timization based DMPC are applied to the system describedby (41)–(43). In what follows, we discuss the performancecomparison with the centralized MPC and the performancecomparison with local cost optimization based MPC.In both the centralized MPC and the subsystem-based MPCs

of the local cost optimization based DMPC, the dual modestrategy is adopted, and the control horizon is set to be .The terminal constraints of the state of all subsystems are

. All MPCs switch to thefeedback control laws given in Table II when all states enterthe attractive region . The bounds of the input and theinput increment of each of the four subsystems are



Fig. 5. The evolution of the states under the centralized MPC.

Fig. 6. The evolution of the control inputs under the centralized MPC.

TABLE IIISTATE SQUARE ERRORS OF THE CLOSED-LOOP SYSTEM UNDER THE CONTROLOF THE CENTRALIZED MPC (CMPC), THE LOCAL COST OPTIMIZATION BASED

DMPC (LCO-DMPC) AND THE ICO-DMPC

and , respectively. Set the initial presumed inputs andstates of each local cost optimization based DMPC to be thesolution calculated by a centralized MPC and the correspondingpredictive state sequences at time .Figs. 5 and 6 show the state responses and the control in-

puts of the closed-loop system under the control of the central-ized MPC. Figs. 7 and 8 show the state responses and the con-trol inputs of the closed-loop system under the control of thelocal optimization based DMPC. The shape of the state responsecurves under the centralized MPC are similar to those under theICO-DMPC; all subsystems converge to near zero in 8 seconds.Under the control of local cost optimization based DMPC, allsubsystems converge to near zero in about 14 seconds. There isno significant undershooting.Figs. 9 and 10 show the differences between the absolute

value of the states and control inputs of the closed-loop systemunder the centralized MPC and those under the ICO-DMPC, re-spectively. The state responses under the ICO-DMPC are verysimilar to those under the centralized MPC. The performance

Fig. 7. The evolution of the states under the local cost optimization basedDMPC.

Fig. 8. The evolution of the control inputs under the local cost optimizationbased DMPC.

Fig. 9. The errors between the absolute value of the state of each subsystemunder the centralized MPC and the absolute value of the state of each subsystemunder the ICO-DMPC.

of is slightly better under the ICO-DMPC than under thecentralized MPC, while the performance of , , and isslightly worse under the ICO-DMPC than under the centralizedMPC. Figs. 11 and 12 show the differences between the absolutevalue of the states and control inputs of the closed-loop systemunder the local cost optimization based DMPC and those underthe ICO-DMPC, respectively. The performance of all other sub-systems under the ICO-DMPC is better than that under the localcost optimization based DMPC.Table III shows the state square errors of the closed-loop

system under the control of the centralized MPC, the



Fig. 10. The difference between the input of each subsystem produced by thecentralizedMPC and the input of each subsystem calculated by the ICO-DMPC.

Fig. 11. The errors between the absolute value of the state of each subsystemunder the local cost optimization based DMPC and the absolute value of thestate of each subsystem under the ICO-DMPC.

Fig. 12. The difference between the input of each subsystem produced by thelocal cost optimization based DMPC and the input of each subsystem calculatedby the ICO-DMPC.

ICO-DMPC and the local cost optimization based DMPC,respectively. The total errors under the ICO-DMPC is 0.33(2.1%) larger than that under the centralized MPC. The totalerrors resulting from the local cost optimization based DMPC is6.55 (40.5%) larger than that results from the centralized MPC.The performance of the ICO-DMPC is significantly better thanthat of the local cost optimization based DMPC.From these simulation results, it can be seen that the proposed

algorithm is able to steer the system states to the origin if there

is a feasible solution at the initial states, and the performance ofthe closed-loop system under the ICO-DMPC is very similar tothat under the centralized MPC.

VII. CONCLUSIONS

In this paper, a distributed implementation of MPC is de-veloped for dynamically coupled systems subject to decoupledinput constraints. Each subsystem-based MPC considers notonly its own performance but also those of the subsystemsin its downstream region to improve the global performanceof the whole system. If an initially feasible solution could befound, the subsequent feasibility of the algorithm is guaranteedat every update, and the resulting closed-loop system is asymp-totically stable. Moreover, the proposed ICO-DMPC providesa unified framework for other cost function coordinating strate-gies. Under this framework, the DMPC design simply involvesa redefinition of the impacted region in each subsystem-basedMPC.

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Shaoyuan Li (SM’05) was born in Hebei, China,in 1965. He received the B.S. and M.S. degrees inautomation from Hebei University of Technology,Tianjin, China, in 1987 and 1992, respectively, andthe Ph.D. degree from the Department of Computerand System Science, Nankai University, Tianjin, in1997.Since July 1997, he has been with the Department

of Automation, Shanghai Jiao Tong University,Shanghai, China, where he is currently a Professor.His research interests include fuzzy systems, model

predictive control, dynamic system optimization, and system identification.

Yi Zheng (M’10) was born in Jilin, China, in 1978.He received the B.S. degree from Xi’an Jiao TongUniversity, China, in 2000, the M.E. degree fromShanghai University in 2006, and the Ph.D. degreein automation from Shanghai Jiao Tong Universityin 2010.He was with Shanghai Petrochemical Company,

Ltd., Shanghai, from 2000 to 2003 and was withGE-Global Research (Shanghai) from 2010 to 2012.Since 2012, he has been with Shanghai Jiao TongUniversity, Shanghai, China, where he is currently

an Assistant Professor and since 2014, he has been with University of Alberta,Edmonton, AB, Canada, where he is currently an Postdoctoral Fellow. Hisresearch interests include distributed estimation, distributed model predictivecontrol, optimization and their applications for industrial process, and smartgrids.

Zongli Lin (F’07) was born in Fujian, China, in1964. He received the B.S. degree from AmoyUniversity, Xiamen, China, the M.E. degree fromChinese Academy of Space Technology, Beijing,China, and the Ph.D. degree from Washington StateUniversity, Pullman, WA, USA, in 1983, 1989, and1994, respectively.From 1983 to 1986, he worked as a Control

Engineer at Chinese Academy of Space Technology,China. In 1994, he joined State University of NewYork, as a Visiting Assistant Professor, where he

became an Assistant Professor in 1994. Since 1997, he has been with Universityof Virginia, Charlottesville, VA, USA, where he is currently an Professor. Hiscurrent research interests include nonlinear control, robust control, and controlof magnetic bearing systems.Dr. Lin is a Fellow of IFAC.

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