Automatica 46 (2010) 2053–2058

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Automatica

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Brief paper

Distributed model predictive control of dynamically decoupled systems withcoupled cost✩

Chen Wang, Chong-Jin Ong ∗

Department of Mechanical Eng., National University of Singapore, EA-07-08, 9 Engineering Drive 1, 117576, SingaporeSingapore-MIT-Alliance, E4-04-10, 4 Engineering Drive 3, 117576, Singapore

a r t i c l e i n f o

Article history:Received 7 August 2009Received in revised form24 June 2010Accepted 12 July 2010Available online 16 October 2010

Keywords:Distributed controlModel predictive controlStability

a b s t r a c t

This paper considers the distributed model predictive control (DMPC) of systems with interactingsubsystems having decoupled dynamics and constraints but coupled costs. An easily-verifiable constraintis introduced to ensure asymptotic stability of the overall system in the absence of disturbance. Theconstraint introduced has a parameter which allows for the performance of the DMPC system to approachthat controlled by a centralized model predictive controller. When the subsystems are linear andadditive disturbance is present, the added constraint ensures the state of each subsystem converges toits respective minimal disturbance invariant set. The approach is demonstrated via several numericalexamples.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Model Predictive Control (MPC) has been a popular approachfor the control of constrained systems for decades (Kothare,Balakrishnan, & Morari, 1996; Mayne, Rawlings, Rao, & Scokaert,2000). More recently, there has been interest in using MPC on agroup of subsystems to achieve a common task. Such an approach,known as Distributed-MPC or DMPC, is appealing as it distributesthe computational load to each subsystem. It is also a preferredarchitecture for a system with mildly coupled subsystems since itis robust against controller failure. For example, a faulty subsystemcan be turned off with minimal effect on the performance ofthe others, a situation that is not possible with a centralizedMPC system. Past works on DMPC include the control of multipleunmanned aerial vehicles (Dunbar & Murray, 2006), control of aslice lip actuator array in the paper-making process (Borrelli et al.,2005) and others (Franco,Magni, Parisini, Polycarpou, & Raimondo,2008; Jia & Krogh, 2001; Raimondo, Magni, & Scattolini, 2007;Richards & How, 2004).

✩ This paper was not presented at any IFAC meeting. The material in this paperwas partially presented at 48th IEEE Conference on Decision and Control, December16–18, 2009, Shanghai, China. This paper was recommended for publication inrevised form by Associate Editor Lalo Magni under the direction of Editor FrankAllgöwer.∗ Corresponding author at: Department of Mechanical Eng., National University

of Singapore, EA-07-08, 9 Engineering Drive 1, 117576, Singapore. Tel.: +65 65162217; fax: +65 6779 1459.

E-mail addresses:wangchen@nus.edu.sg (C. Wang), mpeongcj@nus.edu.sg(C.-J. Ong).

0005-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.09.002

The degree of coupling among the subsystems varies. Inthe most complex situation, the dynamics or/and constraints ofsubsystems are coupled. One of the common applications of DMPCis on subsystemswhere the states are not dynamically coupled butare coupled in their objective functions. Past works on systemsunder such a setting are easily available. For example, a multi-vehicle formation stabilization problem where each vehicle is asubsystem with no external disturbances is considered by Dunbarand Murray (2006) and Keviczky, Borrelli, and Balas (2006).Asymptotic stability is achieved using a terminal state constrainton each subsystem. Raimondo et al. (2007) consider generic DMPCof dynamically decoupled disturbance-free non-linear systemswhile Franco et al. (2008) consider DMPC of the same non-linearsystem but taking into consideration the effect of informationdelay as a form of disturbance. Systems where each subsystemhas an additive disturbance are considered in (Magni & Scattolini,2006)where asymptotical stability is guaranteed by using a specialcontractive constraint on the predicted states.

This paper focuses on applications similar to the above. Itdiscusses a DMPC approach for a system where the subsystemsare constrained systems with coupled cost functions of thesubsystems. Unlike the previous works discussed above, our maincontribution is the introduction of an easily verifiable constraintin each subsystem that ensures stability of the overall system: thestate of each subsystem converges asymptotically to the origin inthe absence of disturbance, and to a neighborhood of the originwhen disturbances are present for the case of linear subsystems.

The notations used in this paper are standard. Zk denotesthe integer set {0, 1, . . . , k}, Z+

k denotes {1, . . . , k} and In is ann × n identity matrix. Similarly, R+

0 is the set of non-negative real

2054 C. Wang, C.-J. Ong / Automatica 46 (2010) 2053–2058

numbers. Given a square matrix Q , Q ≻ (≽)0 means Q is positivedefinite (semi-definite). For any Q ≻ 0, ‖x‖2

Q = xTQx. For any setX, Y ⊂ Rn, X ⊕ Y := {x + y : x ∈ X, y ∈ Y } is the Minkowski sumof X and Y and X ⊖ Y := {z : z + y ∈ X, ∀y ∈ Y } is the Pontryagindifference of the two sets. The variable xi refers to the x variable ofthe i-th subsystem.When xi ∈ Rn, its j-th element is denoted by xji.

2. Problem statement

For the ease of presentation, the basic approach is illustrated forthe case where the subsystems are linear. Extensions of results tothe case of nonlinear subsystems are given in Section 4. Consideran overall system having s linear subsystems, each described byxi(t + 1) = Aixi(t) + Biui(t) + wi(t), wi(t) ∈ Wi, (1)where xi ∈ Rni , ui ∈ Rmi and wi ∈ Rni are the state, control anddisturbance of the i-th subsystem, respectively. The constraints oneach subsystem are of the form

(xi, ui) ∈ Yi ⊂ Rni+mi , (2)where Yi is an appropriate set. In addition, each subsystem isassumed to satisfy the following assumptions.

(A1) Each (Ai, Bi), i ∈ Z+s is stabilizable and xi is measurable.

(A2) Wi and Yi, i ∈ Z+s , are compact sets containing the origin

in their respective interiors.(A3) For each i ∈ Z+

s , a set X fi ⊂ Rni and feedback gain Ki ∈

Rmi×ni can be determined such that X fi is a constraint-

admissible invariant set for the i-th subsystem with ui =

Kixi.Assumption (A1) is a standard requirement of the subsystems.

(A2) is a mild assumption on the disturbance and constraint sets.(A3) refers to the existence of X f

i such that (Ai + BiKi)xi + wi ∈ X fi

and (xi, Kixi) ∈ Yi, for all xi ∈ X fi and for all wi ∈ Wi. Such a X f

i isknown to exist (Kolmanovsky & Gilbert, 1998) under assumptions(A1) and (A2) for sufficiently small Wi. The characterization of X f

ican be obtained from an iterative procedure that terminates in afinite number of steps. Additionally, the i-th subsystem with ui =

Kixi is xi(t + 1) = Φixi(t) + wi(t) with Φi := Ai + BiKi and its stateconverges to the minimal disturbance invariant set, F i

∞, defined as

F i∞

:= limj→∞

F ij ; F i

j = Wi ⊕ ΦiWi ⊕ · · · ⊕ Φj−1i Wi. (3)

Properties and computations of F i∞

have also been well studiedby Ong and Gilbert (2006) and others.

In the proposed DMPC framework, the MPC control law of eachsubsystem is obtained by solving, at every time instant, a finitehorizon optimization problem (FHOP) involving the predictedstates and predicted controls of the respective subsystem withinthe horizon. Let the k-th predicted state and predicted controlwithin the horizon at time t be denoted by xi(k|t) ∈ Rni andui(k|t) ∈ Rmi respectively for the i-th subsystem. For simplicity,suppose all subsystems have the same horizon length, N . Letthe predicted states and controls within the control horizon bexi(t) := {xi(k|t)}Nk=0, ui(t) := {ui(k|t)}N−1

k=0 and their collectionsover the s subsystem be x(t) := {xi(t)}si=1 and u(t) :=

{ui(t)}si=1 respectively. The overall system is to fulfill a commontask cooperatively, quantified by an overall cost function, (4)

ℓ(x(t),u(t), I) :=

N−1−k=0

s−

i=1

hi(xi(k|t), ui(k|t))

+

−(i,j)∈I

qij(xi(k|t), xj(k|t))

+

s−i=1

hfi (xi(N|t)), (4)

where hi : Rni × Rmi → R+

0 and hfi : X f

i → R+

0 are, respectively,the local stage and terminal costs, qij : Rni × Rnj → R+

0 is thecoupled stage cost as a function of the states of the i-th and j-thsubsystems. Additionally, these costs have the properties thathi(0, 0) = 0, hf

i (0) = 0 and qij(0, 0) = 0. The index set,I, contains pairwise indices that show the coupling among thesubsystems. Hence, if (i, j) ∈ I, subsystems i and j are coupled andthey are referred to as neighbors of each other. Clearly, (i, j) ∈ Iimplies (j, i) ∈ I and vice versa. For notational convenience, theneighbors of the i-th subsystems are collected in the set Ni := {j ∈

Z+s : (i, j) ∈ I}.In the subsequent sections, a comparison of the DMPC solution

to that obtained from a reference approach is made. The referenceapproach is anN-stage CentralizedMPC (CMPC)with (4) as the costfunction over the variable u(t) but with u(t) re-parameterized by

ui(k|t) = Kixi(k|t) + ci(k|t) (5)

where Ki is fixed and ci ∈ Rmi is the new optimization variable.Let ci(t) := {ci(k|t)}N−1

k=0 and c(t) := {ci(t)}si=1. Consequently, theoptimization problem of CMPC (Mayne et al., 2000) is

minc(t)

ℓ(x(t), c(t), I) (6a)

s.t. ∀i ∈ Z+

s (6b)

xi(0|t) = xi(t), (6c)xi(k + 1|t) = Aixi(k|t) + Biui(k|t) + wi(k|t), (6d)ui(k|t) = Kixi(k|t) + ci(k|t), ∀k ∈ ZN−1, (6e)

(xi(k|t), ui(k|t)) ∈ Yi, ∀wi(k|t) ∈ Wi, ∀k ∈ ZN−1, (6f)

xi(N|t) ∈ X fi , ∀wi(k|t) ∈ Wi, ∀k ∈ ZN−1, (6g)

where X fi in (6g) is that referred to in (A2). The parametrization

of (6e) or (5) is necessary because of the presence of wi in(1). If the optimization variable of (6) is {u0(0|t), . . . , u0(N −

1|t), . . . , us(0|t), . . . , us(N − 1|t)} instead of c(t), it is well-known that the resultant closed-loop system will be conservative.Our choice of (5) stems from its popularity although otherparameterizations of ui (Löfberg, 2003; Wang, Ong, & Sim, 2009)can also be used. The Kis of (5) are assumed given in assumption(A3). It is well known that their choice can affect the conservatismof the resulting MPC system and they should be appropriatelydesigned. Approaches to their choice include standard LinearQuadratic Control, H2 or H∞ design methodologies. The numericalcomputation of (6) is similar to that under the DMPC and is, hence,deferred to after the discussion of the latter, see Remark 1.

Like all standard MPC, the first control for each i ∈ Z+s of

the solution of (6) is used as the control law ui(t) = u∗

i (0|t) =

Kixi(0|t) + c∗

i (0|t) for the i-th subsystems.

3. DMPC formulation

Using (5) on (1), the state of each subsystem can be written as

xi(k|t) = Φki xi(0|t) +

k−1−j=0

Φk−1−ji Bici(j|t) +

k−1−j=0

Φk−1−ji wi(j|t) (7)

:= xi(k|t) +

k−1−j=0

Φk−1−ji wi(j|t), (8)

where k ∈ Z+

N , Φi = Ai + BiKi and xi(k|t) is defined to be thesum of the first two terms on the right hand side of (7). Sincexi(k|t) = xi(k|t) when wi(t) ≡ 0, xi(k|t) is also called the nominalstate of (1).

The implementation of DMPC assumes that all subsystems aresynchronized in the sense that all local FHOPs are invoked at the

C. Wang, C.-J. Ong / Automatica 46 (2010) 2053–2058 2055

same instant in time. Such an implementation also means thatneither {xj(k|t) : j ∈ Ni} nor {xj(k|t) : j ∈ Ni} is available atthe start of the FHOP of the i-th subsystem. Our approach is touse an estimate, ˆxNi(t) := {ˆxj(t) : j ∈ Ni}, of the nominal statefor FHOP of the i-th subsystem. These estimates are obtained fromthe DMPC solution at time t − 1 and their expressions are givenin Algorithm 1. Using these notations, parametrization (5) and anauxiliary variable αi(t), the FHOP for the i-th subsystem, denotedby Pi(xi(t), ˆxNi(t), αi(t)), under the DMPC formulation is

minci(t)

ℓi(xi(t),ui(t), ˆxNi(t)) (9a)

s.t. xi(0|t) = xi(t), (9b)ui(k|t) = Kixi(k|t) + ci(k|t), (9c)xi(k + 1|t) = Aixi(k|t) + Biui(k|t) + wi(k|t), (9d)(xi(k|t), ui(k|t)) ∈ Yi, ∀wi(k|t) ∈ Wi, ∀k ∈ ZN−1, (9e)

xi(N|t) ∈ X fi , ∀wi(k|t) ∈ Wi, (9f)

Si(ci(t)) :=

N−1−k=0

‖ci(k|t)‖2Ψi

≤ αi(t), (9g)

where ℓi(xi(t),ui(t), ˆxNi(t)) :=∑N−1

k=0 {hi(xi(k|t), ui(k|t))+∑

j∈Ni

qij(xi(k|t), ˆxj(k|t))} + hfi (xi(N|t)). All constraints of (9) are the i-

th subsystem equivalence of those in (6) except for (9g) which isan additional constraint imposed based on stability consideration.The matrix Ψi ≻ 0 is arbitrary and αi(t) is a parameter ofPi(·, ·, ·) having value that evolves in time according to an updateformula given in Algorithm 1. Let c∗

i (t) and x∗

i (t) be the optimizerand the corresponding optimal states of Pi(xi(t), ˆxNi(t), αi(t))respectively.

The functions hi, hfi , qij of (9a) can be quite general as their

choices do not affect the stability of DMPC. For ease of comparisonwith past literature, they are chosen to be hi(xi, ui) := E[‖xi‖2

Qi+

‖ui‖2Ri], hf

i (xi) := E[‖xi‖2Pi], qij(xi, xj) :=

12E[‖xi − xj‖2

Λij] where Qi,

Ri, Pi andΛij are positive definitematrices and the expectation, E, istaken over wi within the horizon. For this choice, (9a) is a positivedefinite quadratic function of ci(t) (Wang et al., 2009).

Remark 1. Several approaches to the numerical computation ofPi exist. For example, Chisci, Rossiter, and Zappa (2001) proposedstrengthening the constraint sets to handle the effect arising fromwi(k|t) for constraints (9e) and (9f). More specifically, xi(N|t) ∈

X fi of (9f) holds if xi(N|t) ∈ X f

i ⊖ F iN , where F i

N is that givenby (3). The same idea can also be applied to (9e) yielding a newconstraint (xi(k|t), ui(k|t)) ∈ Yi ⊖ {F i

k × KiF ik}. In addition, if

each wi(t) is an independent identically distributed random noisewith zero mean, then E[‖xi‖2

Qi] = E[‖xi‖2

Qi] + ν from (8) with ν

being a constant associatedwith the covariancematrix ofwi. Usingthis assumption and the strengthened sets means that xi(k|t) andui(k|t) are replaced by xi(k|t) and ui(k|t) with wi(k|t) removedfrom the constraints of (9a)–(9f). Since xi(k|t) and ui(k|t) areaffine functions of ci(t), (9b)–(9f) and (9a) are linear and quadraticfunctions of ci(t). Since (9g) is a quadratic constraint on ci(t), Pi isa Conic Quadratic Programming problem, solvable with standardsolvers such as SDPT3-4. If other control parameterizations areused instead of (9c), the conversion to computable formulations ismore involved, see for example Goulart, Kerrigan, andMaciejowski(2006) and Wang et al. (2009).

Clearly, not all values of xi admit a solution toPi(xi, xNi , αi). Denotethe set of feasible (xi, ci, αi) that satisfy (9b)–(9g) by Ξ(xi, ci, αi).A useful set that defines the admissible states of xi for the i-thsubsystem is

XPi (αi) := {xi|∃ci such that Ξ(xi, ci, αi) = ∅}. (10)

The implementation of the DMPC controller of the overall systemand the evolution of αi(t) and xNi(t) are described in the followingalgorithm. It assumes that Ni for all i ∈ Zs are known. A logicalvariable ρij is also defined for each (i, j) ∈ I.

Algorithm 1. (1) Initialization: For each i ∈ Z+s :

Choose βi ∈ (0, 1] and solve Pi(xi(0), 0, ∞). Set ξj(−1) =

x∗

j (0) for all j ∈ Ni and ρij = 1 for all j ∈ Ni.(2) At time t where t ≥ 0 and for each i ∈ Zs, set ˆxj(t) based on

the following conditions:• if ρij = 0 and j ∈ Ni, let ˆxj(t) = {ˆxj(1|t − 1), ˆxj(2|t − 1),

. . . , ˆxj(N|t − 1), Φi ˆxj(N|t − 1)}.• if ρij = 1 and j ∈ Ni, let ˆxj(t) = ξj(t − 1).

Let, ˆxNi(t) := {ˆxj(t) : j ∈ Ni}, (11)

αi(t) =

Si(c∗

i (t − 1)) − βi‖c∗

i (0|t − 1)‖2Ψi

t ≥ 1∞ t = 0.

(12)

(3) Solve Pi(xi(t), ˆxNi(t), αi(t)) for all i ∈ Z+s and apply

ui(t) = Kixi(t) + c∗

i (0|t) (13)

to the i-th subsystem. Let

ξi(t) = {x∗

i (1|t), . . . , x∗

i (N|t), Φix∗

i (N|t)} (14)

for all i ∈ Z+s and broadcast it to all subsystems j for which

i ∈ Nj.(4) Set ρij = 0 for all j ∈ Ni. Wait for the receipt of ξj(t) from all

j ∈ Ni subsystems until t + 1 and set ρij = 1 if ξj(t) is receivedby subsystem i. Go to step (2).

Clearly, Step (1) is an initialization procedure that determines afeasible trajectory ˆxNi(0), for all i ∈ Z+

s . The choice of αi(−1) = ∞

is a relaxation for the satisfaction of (9g) at time t = −1. In Step(2), ρij is used to distinguish between success and failure in thereceipt of ξj in the preceding time period. A backup ˆxj(t) is usedin the event when failure occurs. This choice of backup does notaffect the stability of the overall system but only the performanceof subsystems. This will be seen in the next subsection.

Remark 2. Algorithm 1 proposes the broadcast (or more accu-rately, exchange) of ˆxi between the subsystems once in each timeperiod. If the time period is long, it is possible to iteratively solvethe FHOP and exchange ˆxi several times within each time period toimprove performance, see Venkat, Rawlings, and Wright (2005).

The feasibility of Pi(t) := Pi(xi(t), ˆxNi(t), αi(t)) for increasing thevalue of t and the stability of the overall system are stated formallyin the following theorem.

Theorem 3. If xi(0) ∈ XPi (∞), for all i ∈ Z+

s and assumptions(A1)–(A3) are satisfied, then (i) αi(t) and ˆxNi(t) in Algorithm 1 aredefined for all i ∈ Z+

s and t ≥ 0 and Pi(xi(t), ˆxNi(t), αi(t)) isfeasible for all i ∈ Z+

s and t ≥ 0; (ii) (xi(t), ui(t)) ∈ Yi, for allt ≥ 0 and for all i ∈ Z+

s ; (iii) the system state converges to theminimal disturbance invariant set F i

∞of xi(t + 1) = Φixi(t)+wi(t)

for all subsystems; (iv) the overall system is asymptotically stable ifwi(t) ≡ 0 for all i ∈ Z+

s .

Proof. (i) To prove the stated result, one needs only to showthat Pi(xi(t + 1), ˆxNi(t + 1), αi(t + 1)) is feasible following thefeasibility of Pi(xi(t), ˆxNi(t), αi(t)). To this end, let c∗

i (t) be theoptimal solution of Pi(xi(t), ˆxNi(t), αi(t)) and u∗

i (t) := {u∗

i (0|t),u∗

i (1|t), . . . , u∗

i (N − 1|t)}, x∗

i (t) := {x∗

i (0|t), x∗

i (1|t), . . . , x∗

i (N|t)}be the corresponding optimal state and control sequence. Define

2056 C. Wang, C.-J. Ong / Automatica 46 (2010) 2053–2058

ci(t + 1) as ci(t + 1) := [c∗

i (1|t), . . . , c∗

i (N − 1|t), 0]. Henceci(t + 1) defines the following control sequence ui(t + 1) :=

{u∗

i (1|t), . . . , u∗

i (N − 1|t), Kix∗

i (N|t)}. Since x∗

i (N|t) ∈ X fi , under

(A2) we have (x∗

i (N|t), Kix∗

i (N|t)) ∈ Yi, (Ai + BiKi)x∗

i (N|t) ∈ X fi .

Therefore the only condition for ci(t + 1) to be a feasible solutionof Pi(xi(t + 1), ˆxNi(t + 1), αi(t + 1)) is Si(ci(t + 1)) ≤ αi(t + 1).This holds since Si(ci(t + 1)) =

∑N−1k=1 ‖c∗

i (k|t)‖2Ψi

= Si(c∗

i (t)) −

‖c∗

i (0|t)‖2Ψi

≤ Si(c∗

i (t))−βi‖c∗

i (0|t)‖2Ψi

= αi(t+1). Hence ci(t+1)

is a feasible solution of Pi(xi(t + 1), ˆxNi(t + 1), αi(t + 1)). SincePi(xi(t), ˆxNi(t), αi(t)) is always feasible, αi(t + 1) and ˆxNi(t + 1)are always defined.(ii) This result follows directly from (i).(iii) Following (i), constraint (9g) holds for all t . Since αi(t) isconstructed in (12), the inequality Si(c∗

i (t + 1)) ≤ αi(t + 1) =

Si(c∗

i (t)) − βi‖c∗

i (0|t)‖2Ψi

holds. The above inequality impliesβi

∑∞

t=0 ‖c∗

i (0|t)‖2Ψi

≤ Si(c∗

i (0)) < ∞. Since Ψi ≻ 0, this implieslimt→∞ c∗

i (0|t) = 0. Hence, the controller in (13) converges to Kixiand the closed-loop system state defined in (7) converges to F i

∞as

defined in (3). As t tends to infinity, the state converges to the F i∞

set.(iv) With the absence of the disturbance, the closed-loop systemstate converges asymptotically to the origin as shown in (7). �

Remark 4. As shown in the proof of Theorem 3, feasibility of theFHOP depends only on the choice of the control parametrizationwhile convergence of system states depends on the satisfactionof the stability constraint at every instant of time. These twoproperties are not affected by communication delay or loss.The information exchange among subsystems affects the optimalvalues of the cost function which, in turn, affects the transientbehavior of the overall system.

4. Extension to non-linear case

This section discusses the generalization of Theorem 3 to non-linear systems. Let the subsystems be

xi(t + 1) = fi(xi(t), ui(t)), ∀i ∈ Zs+

(15)

where xi, ui have the samemeanings and samedimensions as thosein the previous sections, fi is continuous and fi(0, 0) = 0 ∀i ∈ Zs

+.

As in the linear case, the constraint on each subsystem is given by(2). For each subsystem, it is assumed that there exist a continuousfunction κi : X L

i → Rmi with κi(0) = 0 and two sets X Li and

X fi containing the origin in their interiors such that the following

assumptions hold.

(B1) xi(t + 1) = fi(xi(t), κi(xi(t))) is asymptotically stable onX Li .

(B2) X fi is a constraint-admissible invariant set for system

xi(t + 1) = fi(xi(t), κi(xi(t))).

The size of X Li in assumption (B1) is reasonably large since it is

not required to be constraint admissible. The constraint admissiblecondition is needed on a smaller set X f

i ⊆ X Li in Assumption

(B2). Approaches of designing κi(·) and X fi are also available, see

for example Chen and Allgöwer (1998) and Ong, Sui, and Gilbert(2006).

For stability considerations of DMPC, the concepts of localinput-to-state stability (ISS) and local ISS-Lyapunov function areneeded. These concepts are quite standard and their definitionsand properties can be found in, for example, Jiang andWang (2001)and Goulart et al. (2006) and others. The next theorem shows theexistence of a control law such that the closed-loop system is localISS under the stated assumptions.

Fig. 1. The example system.

Theorem 5. Consider the i-th subsystem of (15) and supposeassumption (B1) holds. There exists an ϵi > 0, a set X ISS

i ⊆ X Li and

a matrix Γi ∈ Rmi×mi of continuous functions, invertible for everyxi, such that the closed-loop system xi(t + 1) = fi(xi(t), κi(xi(t)) +

Γi(xi(t))ci(t)) is local ISS on X ISSi for all ‖ci‖ ≤ ϵi.

Proof. Under (B1), there is a Lyapunov function Vi satisfying

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), (16)

where α1 and α2 are K∞-functions and Vi(fi(xi, κi(xi))) − Vi(xi) <−αi(‖xi‖), for all xi ∈ X L

i and some positive definite functionαi. Similar to the proof of Theorem 4 of Jiang and Wang (2001),define the continuous function δi : R+

× R+→ R by δi(s, r) :=

max‖xi‖=min{s,ζi},‖µi‖=r{Vi(fi(xi, κi(xi)+µi))−Vi(xi)+αi(xi)}whereζi := min{‖xi‖ : xi ∈ ∂X L

i } and ∂X Li denotes the boundary of

X Li . By the fact that δi(s, 0) < 0 for all s > 0 and Lemma 3.1 in

Sontag (1990), a K∞-function χi and a smooth function gi can beconstructed such that δi(s, gi(s)r) < 0 if s ≥ χi(r). Let Γi(xi) =

gi(‖xi‖)Imi , then Vi(fi(xi, κi(xi) + Γi(xi)ci)) − Vi(xi) + αi(xi) ≤

δi(‖xi‖, gi(‖xi‖)‖ci‖) < 0 if ‖xi‖ ≥ χi(‖ci‖) or equivalently

‖xi‖ ≥ χi(‖ci‖) ⇒ Vi(fi(xi, κi(xi) + Γi(xi)ci)) − Vi(xi)+ αi(xi) ≤ δi(‖xi‖, gi(‖xi‖)‖ci‖) < 0. (17)

Since χi is a K∞ function and X Li contains the origin in its interior,

there exists an ϵi > 0 such that (17) holds true on an invariantset X ISS

i ⊆ X Li for all ‖ci‖ ≤ ϵi. An example of X ISS

i is the level setdefined by Vi. By Remark 3.3 in Jiang andWang (2001), (17) implies

V (f (x, u)) − V (x) ≤ −α3(‖x‖) + σ(‖u‖). (18)

According to the definition of local ISS-Lyapunov function, (16) and(18) imply that Vi is an ISS-Lyapunov function defined on X ISS

i . �

Using κi, Γi and the auxiliary variable αi, the FHOP for the i-thsubsystem, denoted by Qi(xi(t), ˆxNi(t), αi(t)), is

minci(t)

ℓi(xi(t),ui(t), ˆxNi(t)) (19a)

s.t. xi(0|t) = xi(t), (19b)ui(k|t) = κi(xi(k|t)) + Γi(xi(k|t))ci(k|t), (19c)xi(k + 1|t) = fi(xi(k|t), ui(k|t)), ∀k ∈ ZN−1, (19d)

(xi(k|t), ui(k|t)) ∈ Yi, ∀k ∈ ZN−1, (19e)

xi(N|t) ∈ X fi , (19f)

Si(ci(t)) :=

N−1−k=0

‖ci(k|t)‖2Ψi

≤ αi(t), (19g)

where the cost function of (19a) is taken to be the same asthose in Pi(·, ·, ·). The DMPC controller of the overall systemis implemented using the procedure of Algorithm 1 exceptthat Pi(·, ·, ·), Kixi(t), Φix∗

i (N|t) and Φi ˆxj(N|t) are replacedrespectively by Qi(·, ·, ·), κi(xi(t)), fi(x∗

i (N|t), κi(x∗

i (N|t))) andfi(ˆxi(N|t), κ(ˆxi(N|t))). Let the admissible set of initial states of theQi(xi(t), ˆxNi(t), αi(t)) be denoted by XQ

i (αi). For the time beingwe assume XQ

i (∞) ⊆ X ISSi and for the case where XQ

i (∞) ⊆ X ISSi ,

C. Wang, C.-J. Ong / Automatica 46 (2010) 2053–2058 2057

(a) xCi — trajectories under CMPC; xDi — trajectoriesunder DMPC; solid lines — xDi with βi = 0.01; dottedlines — xDi ; dashdot lines — xCi with βi = 1.

(b) solid lines — without communication lost; dashedlines — full communication lost; dashdot lines —communication lost after 4 steps.

Fig. 2. Position trajectories of the subsystems.

Fig. 3. Velocity and control trajectories of subsystems under DMPC controller withβi = 0.01: solid lines — subsystem 1; dashed lines — subsystem 2; dashdot lines —subsystem 3.

Fig. 4. Distance between F i∞

set and xi(t) under DMPC controller with βi = 0.01.

constraints xi(k|t) ∈ X ISSi , ∀k ∈ ZN−1 can be simply imposed in

(19) so that the resultingXQi (∞) ⊆ X ISS

i . The existence of a feasiblesolution to Qi(t) := Qi(xi(t), ˆxNi(t), αi(t)) for increasing values oft and the stability of the overall nonlinear system are stated in thefollowing theorem.

Theorem 6. If Qi(0) is feasible for all i ∈ Z+s and assumption (B2) is

satisfied, then (i) Qi(xi(t), ˆxNi(t), αi(t)) is feasible for all i ∈ Z+s and

t ≥ 0; (ii) (xi(t), ui(t)) ∈ Yi, for all t ≥ 0 and for all i ∈ Z+s ;

(iii) if (B1) is satisfied additionally, each subsystem is stable in thesense that xi(t) converges to the origin as t tends to infinity.

Proof. (i)–(ii) follow a similar argument to the proof of (i)–(ii) ofTheorem 3 and additionally it can be proved that limt→∞ c∗

i (0|t)= 0. Hence, there exists a finite time t such that ‖ci(t)‖ ≤ ϵi for allt ≥ t , then from t onwards the closed-loop system is local ISS onXQ

i (∞). This, together with limt→∞ c∗

i (0|t) = 0, implies (iii). �

5. Numerical examples

This section shows experimental verification of the algorithmsand results described in the preceding sections. Consider theposition control of three point-masses on a plane shown in Fig. 1.

Two perpendicular forces (uxi along x coordinate and uy

i alongy coordinate) act on each mass and each force is subject to adisturbance. Each mass is 0.1 kg and the system is discretized

Fig. 5. Graph of J(t) versus time for the non-zero disturbance case: JC (t) is thevalue of J(t) under CMPC; JD(t) is the value of J(t) under DMPC.

using zero-order hold with a sampling period of 0.1 s, yieldingsubsystems with Ai = [1 0.1 0 0; 0 1 0 0; 0 0 1 0.1; 0 0 0 1], Bi =

[0.05 0; 1 0; 0 0.05; 0 1], i ∈ Z+

3 and the first two elements x1i andx2i of xi are the position and velocity along x coordinate and thelast two are those along y coordinate. Constraints on the positionsand velocities of each mass and the forces that can be applied aregiven by Yi = {(xi, ui)| ‖xi‖∞ ≤ 6, ‖ui‖∞ ≤ 1}, i ∈ Z+

3 , withWi = {wi| wi = Biwi, wi ∈ R2, ‖wi‖∞ ≤ 0.2}, i ∈ Z+

3 . Theterminal feedback gain Ki and the weight matrix Pi in terminal costhfi are obtained from standard LQR design with Qi = I4 and Ri = I2.The first numerical simulation involves Algorithm1 for the non-

zero disturbance case with initial state x1(0) = [5 0 − 5 0]T ,x2(0) = [5 0 5 0]T , x3(0) = [0 0 − 5 0]T , I = {(i, j), ∀i ∈ Z+

3 , ∀j ∈

Z+

3 , i = j}, N = 6, βi = 0.01, i = 1, 2, 3. The cost functions arehi(xi, ui) = ‖xi‖2

Qi+ ‖ui‖

2Ri, hf

i (xi) = ‖xi‖2Piand qij(xi, xj) =

12‖xi −

xj‖2Λij

with Λ21 = Λ12 = [5 0 0 0; 0 0 0 0; 0 0 5 0; 0 0 0 0], Λ31 =

Λ13 = [2 0 0 0; 0 0 0 0; 0 0 2 0; 0 0 0 0], Λ32 = Λ23 = [0.1 0 0 0;0 0 0 0; 0 0 0.1 0; 0 0 0 0]. The results are shown in Fig. 2(a) toFig. 5. In Fig. 2(a), position trajectories under the DMPC controllerare labeled as xDi . For easy comparison, the position trajectoriesof the subsystems under the CMPC controller are also shown inFig. 2(a), denoted by xCi . In Fig. 5, J(t) :=

∑tk=0{

∑si=1(‖xi(k)‖

2Qi

+

‖ui(k)‖2Ri)+

∑(i,j)∈I

12‖xi(k)−xj(k)‖2

Λij} is the sumof stage cost over

all subsystems computed using the realized states and controls upto time t . Also, JC (t) denotes the value of J(t) under CMPC whileJD(t) under DMPC.

From Figs. 2(a) and 3, it can be observed that the state andcontrol constraints are satisfied at all times. In Fig. 4, dis(xi, F i

∞) :=

minxi∈F i∞‖xi− xi‖ and an approximation of F i

∞is determined using

the approach of Ong and Gilbert (2006). It can be seen that all thesubsystem states converge to the corresponding F i

∞set, a behavior

2058 C. Wang, C.-J. Ong / Automatica 46 (2010) 2053–2058

(a) With stability constraint. (b) Without stability constraint.

Fig. 6. State trajectories under DMPC: solid lines — subsystem 1; dashed lines — subsystem 2; dashdot lines — subsystem 3.

predicted in Theorem 3. It is also observed that the trajectoriesunder DMPC are close to those under CMPC in Fig. 2(a). Theperformance of the CMPC controller is expected to be better sincethe controller is obtained via optimization of the cost centrally asopposed to the local optimization of pieces of the cost in DMPC.This difference is also shown in Fig. 5. Figs. 2(a) and 5 also containthe results of a large βi value. The dashdot lines in these figuresshow the result of the DMPC system with βi = 1, i ∈ Z+

3 , since alarge value of βi yields a small feasible set of ui. Consequently, theperformance of the corresponding DMPC controller deteriorateswhen βi = 1. This is shown in Fig. 2(a) as xDi trajectories are furtheraway from xCi with increasing value of βi. Similar behaviors areobserved in Fig. 5.

The next simulation shows the results when subsystem 3 losescommunication fully or partially with the rest, a situation dis-cussed in Remark 4. The simulation results are shown in Fig. 2(b).It can be observed that with full communication loss, subsystem 3moves to the equilibrium point without any cooperation from theother two subsystems but the overall system is still stable. In theevent of partial communication loss, the performance degradationis less severe.

The last numerical simulation is designed to show the impor-tance of stability constraint (9g) and the simulation is done in adisturbance free case to highlight the results. The preceding sys-tem is discretized with a sampling period of 0.5 s. The constraintsand the weight matrices remain the same as the previous exam-ple except for Λ′

13 = 100Λ13. The initial conditions are chosen tobe x1(0) = [1.2 0 − 1.2 0]T , x2(0) = [1.2 0 1.2 0]T , x3(0) =

[0 0 − 1.2 0]T , N = 2 and βi = 0.5i = 1, 2, 3. Algorithm 1 isstimulated with and without the stability constraint (9g) and theresults are plotted respectively in Fig. 6(a) and (b). Since the systemis 4 dimensional, its state trajectories are represented on two sep-arate figures: (x1i , x

3i ) and (x2i , x

4i ). The rightmost figures of Fig. 6(a)

and (b) show ‖xi(t)‖ against time. It is quite clear that stability islost if the stability constraint is not imposed.

6. Conclusion

A new DMPC approach for the control of multiple constrainedsystems is proposed in this paper. A stability constraint isintroduced to guarantee the stability of the overall system.Compared with other works in the literature, the assumptionsin this approach are mild and can be easily verified. It is provedthat using the DMPC controller, each subsystem is asymptoticallystable in the disturbance-free case and the state of each subsystemconverges to the corresponding minimal disturbance invariant setasymptotically in the non-zero disturbance case.

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Chen Wang received his B.S. and M.S. degrees inMechatronics from the Harbin Institute of Technology,Harbin, China, in 2002 and 2004 respectively. He iscurrently a Ph.D. candidate in the Control Division ofthe Mechanical Engineering Department at the NationalUniversity of Singapore. His research interests includemodel predictive control and robust optimization.

Chong-Jin Ong received his B. Eng. (Hons) and M. Eng.degrees in mechanical engineering from the NationalUniversity of Singapore in 1986 and 1988 respectively, andhis M.S.E. and Ph.D. degrees in mechanical and appliedmechanics from the University of Michigan, Ann Arbor,in 1992 and 1993 respectively. He joined the NationalUniversity of Singapore in 1993 and is now an AssociateProfessorwith theDepartment ofMechanical Engineering.His research interests are in model predictive control,robust control and machine learning.