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1 Systems & Control Letters Contents lists available at ScienceDirect Systems & Control Letters journal homepage: wwwelseviercom/locate/sysconle Cooperative distributed model predictive control Brett T Stewart a, Aswin N Venkat c, James B Rawlings a,, Stephen J Wright b, Gabriele Pannocchia d a Chemical and Biological Engineering Department, University of Wisconsin, Madison, WI, USA b Computer Science Department, University of Wisconsin, Madison, WI, USA c Shell Global Solutions US Inc, Westhollow Technology Center, 3333 Hwy 6 S, Houston, TX 77, USA d Dip Ing Chim, Chim Ind e Sc Mat DICCISM, University of Pisa, Pisa, Italy a r t i c l e i n f o a b s t r a c t Article history: Received 8 July 9 Received in revised form 5 June Accepted 7 June Available online xxxx Keywords: Distributed control Cooperative control Large-scale control Model predictive control In this paper we propose a cooperative distributed linear model predictive control strategy applicable to any finite number of subsystems satisfying a stabilizability condition The control strategy has the following features: hard input constraints are satisfied; terminating the iteration of the distributed controllers prior to convergence retains closed-loop stability; in the limit of iterating to convergence, the control feedback is plantwide Pareto optimal and equivalent to the centralized control solution; no coordination layer is employed We provide guidance in how to partition the subsystems within the plant We first establish exponential stability of suboptimal model predictive control and show that the proposed cooperative control strategy is in this class We also establish that under perturbation from a stable state estimator, the origin remains exponentially stable For plants with sparsely coupled input constraints, we provide an extension in which the decision variable space of each suboptimization is augmented to achieve Pareto optimality We conclude with a simple example showing the performance advantage of cooperative control compared to noncooperative and decentralized control strategies Published by Elsevier BV Introduction Model predictive control MPC has been widely adopted in the petrochemical industry for controlling large, multi-variable processes MPC solves an online optimization to determine inputs, taking into account the current conditions of the plant, any disturbances affecting operation, and imposed safety and physical constraints Over the last several decades, MPC technology has reached a mature stage Closed-loop properties are well understood, and nominal stability has been demonstrated for many industrial applications Chemical plants usually consist of linked unit operations and can be subdivided into a number of subsystems These subsystems are connected through a network of material, energy, and information streams Because plants often take advantage of the economic savings available in material recycle and energy integration, the plantwide interactions of the network are difficult to elucidate Plantwide control has traditionally been implemented in a decentralized fashion, in which each subsystem is controlled This paper was not presented at any IFAC meeting Corresponding author Tel: ; fax: addresses: btstewart@wiscedu BT Stewart, aswinvenkat@shellcom AN Venkat, rawlings@engrwiscedu JB Rawlings, swright@cswiscedu SJ Wright, gpannocchia@ingunipiit G Pannocchia independently and network interactions are treated as local subsystem disturbances 3,4 It is well known, however, that when the inter-subsystem interactions are strong, decentralized control is unreliable 5 Centralized control, in which all subsystems are controlled via a single agent, can account for the plantwide interactions Indeed, increased computational power, faster optimization software, and algorithms designed specifically for large-scale plantwide control have made centralized control more practical 6,7 Objections to centralized control are often not computational, however, but organizational All subsystems rely upon the central agent, making plantwide control difficult to coordinate and maintain These obstacles deter implementation of centralized control for large-scale plants As a middle ground between the decentralized and centralized strategies, distributed control preserves the topology and flexibility of decentralized control yet offers a nominal closed-loop stability guarantee Stability is achieved by two features: the network interactions between subsystems are explicitly modeled and openloop information, usually input trajectories, is exchanged between subsystem controllers Distributed control strategies differ in the utilization of the open-loop information In noncooperative distributed control, each subsystem controller anticipates the effect of network interactions only locally 8,9 These strategies are described as noncooperative dynamic games, and the plantwide 67-69/$ see front matter Published by Elsevier BV

2 BT Stewart et al / Systems & Control Letters performance converges to the Nash equilibrium If network interactions are strong, however, noncooperative control can destabilize the plant and performance may be, in these cases, poorer than decentralized control A more extensive and detailed comparison of cooperative and noncooperative approaches is provided in, pp Alternatively, cooperative distributed control improves performance by requiring each subsystem to consider the effect of local control actions on all subsystems in the network 3 Each controller optimizes a plantwide objective function, eg, the centralized controller objective Distributed optimization algorithms are used to ensure a decrease in the plantwide objective at each intermediate iterate Under cooperative control, plantwide performance converges to the Pareto optimum, providing centralizedlike performance Because the optimization may be terminated before convergence, however, cooperative control is a form of suboptimal control for the plantwide control problem Hence, stability is deduced from suboptimal control theory 4 Other recent work in large-scale control has focused on coordinating an underlying MPC structure Aske et al 5 develop a coordinating MPC that controls the plant variables with the greatest impact on plant performance, then allow the other decentralized controllers to react to the coordinator MPC In a series of papers, Liu et al 6 present a controller for networked, nonlinear subsystems 7,6 A stabilizing decentralized control architecture and a control Lyapunov function are assumed to exist The performance is improved via a coordinating controller that perturbs the network controller, taking into account the closed-loop response of the network Cheng et al 8 propose a distributed MPC that relies on a centralized dual optimization This coordinator has the advantage that it can optimally handle coupling dynamics and constraints; it must wait for convergence of the plantwide problem, however, before it can provide an implementable input trajectory Cooperative distributed MPC differs from these methods in that a coordinator is not necessary and suboptimal input trajectories may be used to stabilize the plant see 9 In this paper, we state and prove the stability properties for cooperative distributed control under state and output feedback In Section, we provide relevant theory for suboptimal control Section 3 provides stability theory for cooperative control under state feedback For ease of exposition, we introduce the theorems for the case of two controllers only Section 4 extends these results to the output feedback case The results are modified to handle coupled input constraints in Section 5 We then show how the theory extends to cover any finite number of controllers We conclude with an example comparing the performance of cooperative control with other plantwide control strategies Notation Given a vector x R n the symbol x denotes the Euclidean -norm; given a positive scalar r the symbol B r denotes a closed ball of radius r centered at the origin, ie B r {x R n, x r} Given two integers, l m, we define the set I l:m {l, l+,, m, m} The set of positive reals is denoted R + The symbol indicates the matrix transpose Suboptimal model predictive control Requiring distributed MPC strategies to converge is equivalent to implementing centralized MPC with the optimization distributed over many processors Alternatively, we allow the subsystems to inject suboptimal inputs This property increases the flexibility of distributed MPC, and the plantwide control strategy can be treated as a suboptimal MPC In this section, we provide the definitions and theory of suboptimal MPC and draw upon these results in the sequel to establish stability of cooperative MPC We define the current state of the system as x R n, the trajectory of inputs u {u, u,, un } R Nm, and the state and input at time k as xk, uk For the latter, we often abbreviate the notation as x, u Denote the input constraints as u U N in which U is compact, convex, and contains the origin in its interior Denote X N as the set of all x for which there exists a feasible u Initialized with a feasible input trajectory ũ, the controller performs p iterations of a feasible path algorithm and computes u such that some performance metric is improved At each sample time, the first input in the suboptimal trajectory is applied, u u The state is updated by the state evolution equation x + f x, u, in which x + is the state at the next iterate For any initial state x, we initialize the suboptimal MPC with a feasible input sequence ũ hx with h continuous For subsequent decision times, we denote ũ + as the warm start, a feasible input sequence for x + used to initialize the suboptimal MPC algorithm Here, we set ũ + {u,, un, } This sequence is obtained by discarding the first input, shifting the rest of the sequence forward one step and setting the last input to zero We observe that the input sequence at termination u + is a function of the state initial condition x + and of the warm start ũ + Noting that x + and ũ + are both functions of x and u, the input sequence u + can be expressed as a function of only x, u by u + gx, u We refer to the function g as the iterate update Given a system x + f x, with equilibrium point at the origin f, denote φk, x as the solution xk given the initial state x We consider the following definition Definition Exponential Stability on a Set X The origin is exponentially stable on the set X if for all x X, the solution φk, x X and there exists α > and < γ < such that φk, x α x γ k for all k The following lemma is an extension of 4, Theorem for exponential stability Lemma Exponential Stability of Suboptimal MPC Consider a system x + Fx, u f x, u u + x, u given gx, u gx, u with a steady-state solution, f,, g, Assume that the function V : R n R Nm R + and input trajectory u satisfy a x, u Vx, u b x, u a Vx +, u + Vx, u c x, u b u d x x B r c in which a, b, c, d, r > If X N is forward invariant for the system x + f x, u, the origin is exponentially stable for all x X N Notice in the second inequality b, only the first input appears in the norm x, u Note also that the last inequality applies only for x in a ball of radius r, which may be chosen arbitrarily small Proof of Lemma First we establish that the origin of the extended system is exponentially stable for all x, u X N U N For x B r, we have u d x Consider the optimization s max u U N u The solution exists by the Weierstrass theorem since U N is compact and by definition we have that s > Then we have u s/r x for x B r Therefore, for all x X N, we have u d x in which d maxd, s/r, and x, u x + u + d x + d x, u

3 BT Stewart et al / Systems & Control Letters 3 which gives x, u c x, u with c / + d > Therefore the extended state x, u satisfies Vx +, u + Vx, u c x, u x, u X N U N 3 in which c c c Together with, 3 establishes that V is a Lyapunov function of the extended state x, u for all x X N and u U N Hence for all x, u X N U N and k, we have xk, uk α x, u γ k in which α > and < γ < Notice that X N U N is forward invariant for the extended system Finally, we remove the input sequence and establish that the origin is exponentially stable for the closed-loop system We have for all x X N and k φk, x xk xk, uk α x, u γ k α x + u γ k α + d x γ k ᾱ x γ k in which ᾱ α + d >, and we have established exponential stability of the origin by observing that X N is forward invariant for the closed-loop system φk, x Remark For Lemma, we use the fact that U is compact For unbounded U exponential stability may instead be established by compactness of X N 3 Cooperative model predictive control We now show that cooperative MPC is a form of suboptimal MPC and establish stability To simplify the exposition and proofs, in Sections 3 5 we assume that the plant consists of only two subsystems We then establish in Section 6 that the results extend to any finite number of subsystems 3 Definitions 3 Models We assume for each subsystem i that there exist a collection of linear models denoting the effects of inputs of subsystem j on the states of subsystem i for all i, j I : I : x + A x + B u j in which x R n, u j R m j, A R n n, and B R n m j For a discussion of identification of this model choice, see The Appendix shows how these subsystem models are related to the centralized model Considering subsystem, we collect the states to form + x A x x + A x B u + u B which denotes the model for subsystem To simplify the notation, we define the equivalent model x + A x + B u + B u for which x x x A A A B B B B in which x R n, A R n n, and Bj R n m j with n n + n Forming a similar model for subsystem, the plantwide model is + x A x B B + u x A x + u B B We further simplify the plantwide model notation to x + Ax + B u + B u for which x x x A A A B B B B B B 3 Objective functions Consider subsystem, for which we define the quadratic stage cost and terminal penalty, respectively l x, u x Q x + u R u V f x x P f x 3a 3b in which Q R n n, R R m m, and P f R n n We define the objective function for subsystem N V x, u, u l x k, u k + V f x N k Notice V is implicitly a function of both u and u because x is a function of both u and u For subsystem, we similarly define an objective function V We define the plantwide objective function V x, x, u, u ρ V x, u, u + ρ V x, u, u in which ρ, ρ > are relative weights For notational simplicity, we write Vx, u for the plant objective 33 Constraints We require that the inputs satisfy u k U u k U k I :N in which U and U are compact and convex such that is in the interior of U i i I : Remark The constraints are termed uncoupled because the feasible region of u is not affected by u, and vice versa 34 Assumptions For every i I :, let A i diaga i, A i and B i B i B i following assumptions are used to establish stability The Assumption 3 For all i I : a The systems A i, B i are stabilizable b The input penalties R i > c The state penalties Q i d The systems A i, Q i are detectable e N max i I: n u i, in which nu i is the number of unstable modes of A i, ie, the number of λ eiga i such that λ The Assumption 3e is required so that the horizon N is sufficiently large to zero the unstable modes 35 Unstable modes For an unstable plant, we constrain the unstable modes to be zero at the end of the horizon to maintain closed-loop stability To construct this constraint, consider the real Schur decomposition of A for each i, j I : I : A S s S u A s A u S s S u in which A s is stable and Au has all unstable eigenvalues 3 36 Terminal penalty Given the definition of the Schur decomposition 3, we define the matrices

4 4 BT Stewart et al / Systems & Control Letters S s i diags s i, Ss i As i diaga s i, As i i I : 33a S u i diags u i, Su i Au i diaga u i, Au i i I : 33b Lemma 4 The matrices 33 satisfy the Schur decompositions A i S s i S u A s i S s i A u i i I : i S u i Let Σ and Σ denote the solution of the Lyapunov equations A s Σ A s Σ S s Q S s A s Σ A s Σ S s Q S s 34 We then choose the terminal penalty for each subsystem to be the cost to go under zero control, such that P f S s Σ S s P f S s Σ S s Cooperative model predictive control algorithm Let υ be the initial condition for the cooperative MPC algorithm see Section 3 for the discussion of initialization At each iterate p, the following optimization problem is solved for subsystem i, i I : min υ i Vx, x, υ, υ 36a subject to + x x υ i U N i A x B B + υ A x + υ B B 36b 36c S u ji xji N j I : 36d υ i d xji i if x ji B r j I : 36e j I : υ j υ p j j I : \ i 36f in which we include the hard input constraints, the stabilizing constraint on the unstable modes, and the Lyapunov stability constraint We denote the solutions to these problems as υ x, x, υ p, υ x, x, υ p Given the prior, feasible iterate υ p, υp, the next iterate is defined to be υ p+, υ p+ w υ υp, υp + w υ p, υ υp 37 w + w, w, w > for which we omit the state dependence of υ and υ to reduce notation This distributed optimization is of the Gauss Jacobi type see, pp 9 3 At the last iterate p, we set u υ p, υ p and inject u into the plant The following properties follow immediately Lemma 5 Feasibility Given a feasible initial guess, the iterates satisfy υ p, υp UN UN for all p Lemma 6 Convergence The cost Vx, υ p is nonincreasing for each iterate p and converges as p Lemma 7 Optimality As p the cost Vx, υ p converges to the optimal value V x, and the iterates υ p, υp converge to u, u in which u u, u is the Pareto centralized optimal solution The proofs are in the Appendix Remark 3 This paper presents the distributed optimization algorithm with subproblem 36 and iterate update 37 so that the Lemmas 5 7 are satisfied This choice is nonunique and other optimization methods may exist satisfying these properties 3 Stability of cooperative model predictive control We define the steerable set X N as the set of all x such that there exists a u U N satisfying 36d Assumption 8 Given r >, for all i I :, d i is chosen large enough such that there exists a u i U N satisfying u i d x i j I : and 36d for all x B r j I : Remark 4 Given Assumption 8, X N is forward invariant We now establish stability of the closed-loop system by treating cooperative MPC as a form of suboptimal MPC We define the warm start for each subsystem as ũ + {u, u,, u N, } ũ + {u, u,, u N, } The warm start ũ + i is used as the initial condition for the cooperative MPC problem in each subsystem i We define the functions g p and g p as the outcome of applying the cooperative control iteration 37 p times u + g p x, x, u, u u + g p x, x, u, u The system evolution is then given by x + A x + B u + B u x + u + A x + B u + B u g p u + x, x, u, u g p x, x, u, u which we simplify to x + u + Ax + B u + B u g p x, u Theorem 9 Exponential Stability Given Assumptions 3 and 8, the origin x of the closed-loop system x + Ax + B u + B u is exponentially stable on the set X N Proof By eliminating the states in l i, we can write V in the form Vx, u /x Qx + /u Ru + x S u Defining H Q S S >, V satisfies a by choosing a R / min i λ i H and b / max i λ i H Next we show that V satisfies b Using the warm start at the next sample time, we have the following cost Vx +, ũ + Vx, u ρ l x, u ρ l x, u + ρ x N A P f A P f + Q x N + ρ x N A P f A P f + Q x N 38 Using the Schur decomposition defined in Lemma 4, and the constraints 36d and 35, the last two terms of 38 can be written as

5 ρ x N S s + ρ x N S s A s Σ A s Σ + S s Q S s ARTICLE IN PRESS A s Σ A s Σ + S s Q S s S s x N BT Stewart et al / Systems & Control Letters 5 S s x N These terms are zero because of 34 Using this result and applying the iteration of the controllers gives Vx +, u + Vx, u ρ l x, u ρ l x, u Because l i is quadratic in both arguments, there exists a c > such that Vx +, u + Vx, u c x, u The Lyapunov stability constraint 36e for x, x, x, x B r implies for x, x B r that u, u ˆd x, x in which ˆd maxd, d, satisfying c Therefore the closed-loop system satisfies Lemma Hence the closed-loop system is exponentially stable 4 Output feedback We now consider the stability of the closed-loop system with estimator error 4 Models For all i, j I : I : x + A x + B u j 4a y i C x j I : 4b in which R i diagr i, F ji S u ji Cji, f ji S u ji A N ji L ji, and C ji B ji A ji B ji A N ji B ji for all i, j I : We use h + i e as the initial condition for the control optimization 36 for all i I : Proposition The reinitialization h + h +, h+ is Lipschitz continuous on bounded sets Proof The proof follows from Prop 73, p Stability with estimate error We consider the stability properties of the extended closedloop system ˆx u e + Fˆx, u + Le g p ˆx, u, e A L e 4 in which Fˆx, u Aˆx + B u + B u and L diagl C, L C The function g p includes the reinitialization step Because A L is stable there exists a Lyapunov function J with the following properties ā e σ Je b e σ Je + Je c e σ in which σ >, ā, b >, and the constant c > can be chosen as large as desired by scaling J For the remainder of this section, we choose σ in order to match the Lipschitz continuity of the plantwide objective function V From the nominal properties of cooperative MPC, the origin of the nominal closed-loop system x + Ax + B u + B u is exponentially stable on X N if the suboptimal input trajectory u u, u is computed using the actual state x, and the cost function Vx, u satisfies We require the following feasibility assumption in which y i R p i is the output of subsystem i and C R p i n Consider subsystem As above, we collect the states to form y C C x x and use the simplified notation y C x to form the output model for subsystem Assumption For all i I :, A i, C i is detectable 4 Estimator We construct a decentralized estimator Consider subsystem, for which the local measurement y and both inputs u and u are available, but x must be estimated The estimate satisfies ˆx + A ˆx + B u + B u + L y C ˆx in which ˆx is the estimate of x and L is the Kalman filter gain Defining the estimate error as e x ˆx we have e + A L C e By Assumptions 3 and there exists an L such that A L C is stable and therefore the estimator for subsystem is stable Defining e similarly, the estimate error for the plant evolves + e AL e e A L e in which A Li A i L i C i We collect the estimate error of each subsystem together and write e + A L e 43 Reinitialization We define the reinitialization step required to recover feasibility of the warm start for the perturbed state terminal constraint For each i I :, define h + i e arg min u i { ui ũ + i F ji u i ũ + } i f ji e j j I : R i u i U N Assumption The set X N is compact, and there exist two sets Xˆ N and E both containing the origin such that the following conditions hold: i Xˆ N E X N, where indicates the Minkowski sum; ii for each ˆx Xˆ N and ê E, the solution of the extended closed-loop system 4 satisfies ˆxk X N for all k Consider the sum of the two Lyapunov functions Wˆx, u, e Vˆx, u + Je We next show that W is a Lyapunov function for the perturbed system and establish exponential stability of the extended state origin ˆx, e, From the definition of W we have a ˆx, u + ā e Wˆx, u, e b ˆx, u + b e ã ˆx, u + e Wˆx, u, e b ˆx, u + e 43 in which ã mina, ā > and b maxb, b > Next we compute the cost change Wˆx +, u +, e + Wˆx, u, e Vˆx +, u + Vˆx, u + Je + Je The Lyapunov function V is quadratic in ˆx, u and, hence, Lipschitz continuous on bounded sets By Proposition VFˆx, u + Le, h + e VFˆx, u + Le, ũ + L h L Vu e VFˆx, u + Le, ũ + VFˆx, u, ũ + L Vx Le in which L h, L Vu, and L Vx are Lipschitz constants for h + and the first and second arguments of V, respectively Combining the above inequalities VFˆx, u + Le, h + e VFˆx, u, ũ + LV e

6 6 BT Stewart et al / Systems & Control Letters in which LV L h L Vu + L Vx L Using the system evolution we have Vˆx +, h + e VFˆx, u, ũ + + LV e and by Lemma 6 Vˆx +, u + VFˆx, u, ũ + + LV e Subtracting Vˆx, u from both sides and noting that ũ + is a stabilizing input sequence for e gives Vˆx +, u + Vˆx, u c ˆx, u + LV e Wˆx +, u +, e + Wˆx, u, e c ˆx, u + LV e c e c ˆx, u c LV e Wˆx +, u +, e + Wˆx, u, e c ˆx, u + e 44 in which we choose c > LV and c minc, c LV > This choice is possible because c can be chosen arbitrarily large Notice this step is what motivated the choice of σ Lastly, we require the constraint u d ˆx, ˆx B r 45 Theorem 3 Exponential Stability of Perturbed System Given Assumptions 3, and, for each ˆx Xˆ N and e E, there exist constants α > and < γ <, such that the solution of the perturbed system 4 satisfies, for all k ˆxk, ek α ˆx, e γ k 46 Proof Using the same arguments as for Lemma, we write: Wˆx +, u +, e + Wˆx, u, e ĉ ˆx, u + e 47 in which ĉ c > Therefore W is a Lyapunov function for the extended state ˆx, u, e with mixed norm powers The standard exponential stability argument can be extended for the mixed norm power case to show that the origin of the extended closedloop system 4 is exponentially stable, p4 Hence, for all k ˆxk, uk, ek α ˆx, u, e γ k in which α > and < γ < Notice that Assumption implies that uk exists for all k because ˆxk X N We have, using the same arguments used in Lemma ˆxk, ek ˆxk, uk, ek α ˆx, u, e γ k α ˆx, e γ k in which α α + d > Corollary 4 Under the assumptions of Theorem 3, for each x and ˆx such that e x ˆx E and ˆx Xˆ N, the solution of the closed-loop state xk ˆxk + ek satisfies: xk ᾱ x γ k 48 for some ᾱ > and < γ < 5 Coupled constraints In Remark, we commented that the constraint assumptions imply uncoupled constraints, because each input is constrained by a separate feasible region so that the full feasible space is defined u, u U N U N UN This assumption, however, is not always practical Consider two subsystems sharing a scarce resource for which we control the distribution There then exists an availability constraint spanning the subsystems This constraint is coupled because each local resource constraint depends upon the amount requested by the other subsystem Remark 5 For plants with coupled constraints, implementing MPC problem 36 gives exponentially stable, yet suboptimal, feedback In this section, we relax the assumption so that u, u U N for any U compact, convex and containing the origin in its interior Consider the decomposition of the inputs u u U, u U, u C such that there exists a U U,U U, and U C for which U U U U U U C and u U U N U, u U U N U, u C U N C for which U U, U U, and U C are compact and convex We denote u Ui the uncoupled inputs for subsystem i, i I :, and u C the coupled inputs Remark 6 U U,U U, or U C may be empty, and therefore such a decomposition always exists We modify the cooperative MPC problem 36 for the above decomposition Define the augmented inputs û, û uu uu û û u C u C The implemented inputs are I u Ê û u Ê û, Ê I I Ê in which I, I are diagonal matrices with either or diagonal entries and satisfy I + I I For simplicity, we summarize the previous relations as u Êû with Ê diagê, Ê The objective function is ˆVx, x, û, û Vx, x, Ê û, Ê û 5 We solve the augmented cooperative MPC problem for i I : min ˆVx, x, ˆυ, ˆυ 5a ˆυ i subject to + x x A ˆυ i U N Ui UN C x B B + Ê A x ˆυ + Ê ˆυ B B I 5b 5c S u ji xji N j I : 5d ˆυ i d i x i if x i B r 5e ˆυ j ˆυ p j j I : \ i 5f The update 37 is used to determine the next iterate Lemma 5 As p the cost ˆVx, ˆυ p converges to the optimal value V x, and the iterates Ê ˆυ p, Ê ˆυ p converge to the Pareto optimal centralized solution u u, u Therefore, problem 5 gives optimal feedback and may be used for plants with coupled constraints 6 M Subsystems In this section, we show that the stability theory of cooperative control extends to any finite number of subsystems For M >

7 BT Stewart et al / Systems & Control Letters 7 subsystems, the plantwide variables are defined x u x u x u B i B i i I :M x M u M B Mi Vx, u ρ i V i x i, u i A diaga,, A M i I :M Each subsystem solves the optimization min υ i Vx, υ subject to x + Ax + i I :M B i υ i υ i U N i S u ji xji N j I :M υ i d i B i j I :M xji if x ji B r j I :M υ j υ p j j I :M \ i The controller iteration is given by υ p+ i I :M w i υ p,, υ i,, υp M x; υ p j, j I :M \ i After p iterates, we set in which υ i υ i u υ p and inject u into the plant The warm start is generated by purely local information ũ + i {u i, u i,, u i N, } i I :M The plantwide cost function then satisfies for any p Vx +, u + Vx, u i I :M ρ i l i x i, u i u d x x B r Generalizing Assumption 3 to all i I :M, we find that Theorem 9 applies and cooperative MPC of M subsystems is exponentially stable Moreover, expressing the M subsystem outputs as y i j I :M C x i I :M and generalizing Assumption for i I :M, cooperative MPC for M subsystems satisfies Theorem 3 Finally, for systems with coupled constraints, we can decompose the feasible space such that U i I :M U Ui U C Hence, the input augmentation scheme of Section 5 is applicable to plants of M subsystems Notice that, in general, this approach may lead to augmented inputs for each subsystem that are larger than strictly necessary to achieve optimal control The most parsimonious augmentation scheme is described elsewhere 7 Example Consider a plant consisting of two reactors and a separator A stream of pure reactant A is added to each reactor and converted to the product B by a first-order reaction The product is lost by a parallel first-order reaction to side product C The distillate of the separator is split and partially redirected to the first reactor see Fig dh dx A dx B dt dh dx A dx B dt dh 3 dx A3 dx B3 dt 3 Fig Two reactors in series with separator and recycle The model for the plant is ρa F f + F R F ρa H F f x A + F R x AR F x A k A x A ρa H F R x BR F x B + k A x A k B x B F f T + F R T R F T ρa H Q k A x A H A + k B x B H B + C p ρa C p H ρa F f + F F ρa H F f x A + F x A F x A k A x A ρa H F x B F x B + k A x A k B x B F f T + F T F T ρa H Q k A x A H A + k B x B H B + C p ρa C p H ρa 3 F F D F R F 3 ρa 3 H 3 F x A F D + F R x AR F 3 x A3 ρa 3 H 3 F x B F D + F R x BR F 3 x B3 ρa 3 H 3 F T F D + F R T R F 3 T 3 + in which for all i I :3 F i k vi H i k Ai k A exp E A RT i The recycle flow and weight percents satisfy Q 3 ρa 3 C p H 3 k Bi k B exp F D F R x AR α Ax A3 x BR α Bx B3 x 3 x 3 x 3 α A x A3 + α B x B3 + α C x C3 x C3 x A3 x B3 E B RT i The output and input are denoted, respectively y H x A x B T H x A x B T H 3 x A3 x B3 T 3 u F f Q F f Q F R Q 3 We linearize the plant model around the steady state defined by Table and derive the following linear discrete-time model with sampling time s x + Ax + Bu y x

8 8 BT Stewart et al / Systems & Control Letters Table Steady states and parameters Parameter Value Units Parameter Value Units H 98 m A 3 m x A 54 wt% A 3 m x B 393 wt% A 3 m T 35 K ρ 5 kg/m 3 H 3 m C p 5 kj/kg K x A 53 wt% k v 5 kg/m s x B 4 wt% k v 5 kg/m s T 35 K k v3 5 kg/m s H 3 37 m x A wt% x A3 38 wt% T 33 K x B3 57 wt% k A /s T 3 35 K k B 8 /s F f 833 kg/s E A /R K Q kj/s E B /R 5 K F f 5 kg/s H A 4 kj/kg Q kj/s H B 5 kj/kg F R 66 kg/s α A 35 Q 3 kj/s α B α C 5 7 Distributed control In order to control the separator and each reactor independently, we partition the plant into 3 subsystems by defining y H x A x B T u F f Q Fig Performance of reactor and separator example y H x A x B T y 3 H 3 x A3 x B3 T 3 u F f Q u 3 F R Q 3 Following the distributed model derivation in the Appendix see Appendix B, we form the distributed model for the plant 7 Simulation Consider the performance of distributed control with the partitioning defined above The tuning parameters are Q yi diag,,, i I : Q y3 diag,, 3, Q i C i Q yic i + I R i I i I :3 The input constraints are defined in Table We simulate a setpoint change in the output product weight percent x B3 at t 5 s In Fig, the performance of the distributed control strategies are compared to the centralized control benchmark For this example, noncooperative control is an improvement over decentralized control see Table 3 Cooperative control with only a single iteration is significantly better than noncooperative control, however, and approaches centralized control as more iteration is allowed 8 Conclusion In this paper we present a novel cooperative distributed controller in which the subsystem controllers optimize the same objective function in parallel without the use of a coordinator The

9 BT Stewart et al / Systems & Control Letters 9 Table Input constraints Parameter Lower bound Steady state Upper bound Units F f 833 kg/s Q 5 kj/s F f 5 kg/s Q 5 kj/s F R kg/s Q 3 5 kj/s Table 3 Performance comparison Cost Performance loss % Centralized MPC 76 Cooperative MPC iterates Cooperative MPC iterate Noncooperative MPC Decentralized MPC 85 4 e+ 4 control algorithm is equivalent to a suboptimal centralized controller, allowing the distributed optimization to be terminated at any iterate before convergence At convergence, the feedback is Pareto optimal We establish exponential stability for the nominal case and for perturbation by a stable state estimator For plants with sparsely coupled constraints, the controller can be extended by repartitioning the decision variables to maintain Pareto optimality We make no restrictions on the strength of the dynamic coupling in the network of subsystems, offering flexibility in plantwide control design Moreover, the cooperative controller can improve performance of plants over traditional decentralized control and noncooperative control, especially for plants with strong openloop interactions between subsystems A simple example is given showing this performance improvement Acknowledgements The authors gratefully acknowledge the financial support of the industrial members of the Texas-Wisconsin-California Control Consortium, and NSF through grant #CTS The authors would like to thank Prof DQ Mayne for helpful discussion of the ideas in this paper Appendix A Further proofs Proof of Lemma 5 By assumption, the initial guess is feasible Because U and U are convex, the convex combination 37 with p implies υ, υ is feasible Feasibility for p > follows by induction Proof of Lemma 6 For every p, the cost function satisfies the following Vx, υ p+ Vx, w υ, υp + w υ p, υ w Vx, υ, υp + w Vx, υ p, υ w Vx, υ p, υp + w Vx, υ p, υp Vx, υ p Aa Ab The first equality follows from 37 The inequality Aa follows from convexity of V The next inequality Ab follows from the optimality of υ i i I :, and the final line follows from w +w Because the cost is bounded below, it converges Proof of Lemma 7 We give a proof that requires only closedness not compactness of U i, i I : From Lemma 6, the cost converges, say to V Since V is quadratic and strongly convex, its sublevel sets lev a V are compact and bounded for all a Hence, all iterates belong to the compact set lev Vυ V U, so there is at least one accumulation point Let ῡ be any such accumulation point, and choose a subsequence P {,, 3, } such that {υ p } p P converges to ῡ We obviously have that Vx, ῡ V, and moreover that lim Vx, p P,p υp lim Vx, p P,p υp+ V A By strict convexity of V and compactness of U i, i I :, the minimizer of Vx, is attained at a unique point u u, u By taking limits in A as p for p P, and using w >, w >, we deduce directly that lim Vx, p P,p υ υp, υp V lim Vx, p P,p υp, υ υp V A3b A3a We suppose for contradiction that V Vx, u and thus ῡ u Because Vx, is convex, we have Vx, ῡ u ῡ V : Vx, u Vx, ῡ < where Vx, denotes the gradient of Vx, It follows immediately that either u Vx, ῡ ῡ / V or A4a Vx, ῡ u ῡ / V A4b Suppose first that A4a holds Applying Taylor s theorem to V Vx, υ p + ɛu υp, υp u Vx, υ p + ɛ Vx, υ p υp + u ɛ υp Vx, υ p u + γ ɛu υp, υp υp V + /4ɛ V + βɛ A5 in which γ, ɛ A5 applies for all p P sufficiently large, for some β independent of ɛ and p By fixing ɛ to a suitably small value certainly less than, we have both that the right-hand side of A5 is strictly less than V and that υ p + ɛu υp U By taking limits in A5 and using A3 and the fact that υ υp is optimal for Vx,, υ p in U, we have V < V lim Vx, p P,p υ υp, υp lim Vx, p P,p υp + ɛu υp, υp giving a contradiction By identical logic, we obtain the same contradiction from A4b We conclude that V Vx, u and thus ῡ u Since ῡ was an arbitrary accumulation point of the sequence {υ p }, and since this sequence is confined to a compact set, we conclude that the whole sequence converges to u Proof of Corollary 4 We first note that: xk ˆxk + ek ˆxk, ek From Theorem 3 we can write: xk α ˆx, e γ k ᾱ ˆx + e γ k

10 BT Stewart et al / Systems & Control Letters with ᾱ α, which concludes the proof by noticing that x ˆx + e Proof of Lemma 5 Because ˆV is convex and bounded below, the proof follows from Lemma 7 and from noticing that the point u Ê û, Ê û, with û i lim p ˆυ i, i I :, is Pareto optimal Appendix B Deriving the distributed model Consider the possibly nonminimal centralized model x + Ax + j I :M B j u j y i C i x i I :M zō c Aō co Aō c zō c B For each input/output pair u j, y i we transform the triple A, B j, C i into its Kalman canonical form 3, p7 z oc + A oc A oc c z oc zōc z o c Aōoc Aōc Aōco c Aōc c B oc zōc A o c z o c + Bōc u j y C oc C o c z oc zōc z o c zō c y i j I :M y The vector z oc captures the modes of A that are both observable by y i and controllable by u j The distributed model is then A A oc References B B oc C C oc x z oc DQ Mayne, JB Rawlings, CV Rao, POM Scokaert, Constrained model predictive control: stability and optimality, Automatica SJ Qin, TA Badgwell, A survey of industrial model predictive control technology, Control Eng Pract S Huang, KK Tan, TH Lee, Decentralized control design for large-scale systems with strong interconnections using neural networks, IEEE Trans Automat Control NR Sandell Jr, P Varaiya, M Athans, M Safonov, Survey of decentralized control methods for larger scale systems, IEEE Trans Automat Control H Cui, EW Jacobsen, Performance limitations in decentralized control, J Process Control RA Bartlett, LT Biegler, J Backstrom, V Gopal, Quadratic programming algorithms for large-scale model predictive control, J Process Control G Pannocchia, JB Rawlings, SJ Wright, Fast, large-scale model predictive control by partial enumeration, Automatica E Camponogara, D Jia, BH Krogh, S Talukdar, Distributed model predictive control, IEEE Control Syst Mag WB Dunbar, Distributed receding horizon control of dynamically coupled nonlinear systems, IEEE Trans Automat Control T Başar, GJ Olsder, Dynamic Noncooperative Game Theory, SIAM, Philadelphia, 999 JB Rawlings, BT Stewart, Coordinating multiple optimization-based controllers: new opportunities and challenges, J Process Control JB Rawlings, DQ Mayne, Model Predictive Control: Theory and Design, Nob Hill Publishing, Madison, WI, ISBN: , 9, 576 pages 3 AN Venkat, IA Hiskens, JB Rawlings, SJ Wright, Distributed MPC strategies with application to power system automatic generation control, IEEE Control Syst Tech POM Scokaert, DQ Mayne, JB Rawlings, Suboptimal model predictive control feasibility implies stability, IEEE Trans Automat Control EMB Aske, S Strand, S Skogestad, Coordinator MPC for maximizing plant throughput, Comput Chem Eng J Liu, D Muñoz de la Peña, PD Christofides, Distributed model predictive control of nonlinear process systems, AIChE J J Liu, D Muñoz de la Peña, BJ Ohran, PD Christofides, JF Davis, A two-tier architecture for networked process control, Chem Eng Sci R Cheng, J Forbes, W Yip, Price-driven coordination method for solving plantwide MPC problems, J Process Control AN Venkat, JB Rawlings, SJ Wright, Distributed model predictive control of large-scale systems, in: Assessment and Future Directions of Nonlinear Model Predictive Control, Springer, 7, pp RD Gudi, JB Rawlings, Identification for decentralized model predictive control, AIChE J DP Bertsekas, JN Tsitsiklis, Parallel and Distributed Computation, Athena Scientific, Belmont, Massachusetts, 997 G Pannocchia, SJ Wright, BT Stewart, JB Rawlings, Efficient cooperative distributed mpc using partial enumeration, in: ADCHEM 9, International Symposium on Advanced Control of Chemical Processes, Istanbul, Turkey, 9 3 PJ Antsaklis, AN Michel, Linear Systems, McGraw-Hill, New York, 997