Distributed Synthesis and Stability of Cooperative Distributed Model Predictive Control for Linear Systems

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1 Distributed Synthesis and Stability o Cooperative Distributed Model Predictive Control or Linear Systems Christian Conte a, Colin N. Jones b, Manred Morari a, Melanie N. Zeilinger c a Automatic Control Laboratory, Department o Inormation Technology and Electrical Engineering, ETH Zurich, 8092 Zurich, Switzerland b Automatic Control Laboratory, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland c Institute or Dynamic Systems and Control, Department o Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland Abstract This paper presents a new ormulation and synthesis approach or stabilizing cooperative distributed model predictive control (MPC) or networks o linear systems, which are coupled in their dynamics. The controller is deined by a network-wide constrained optimal control problem, which is solved online by distributed optimization. The main challenge is the deinition o a global MPC problem, which both deines a stabilizing control law and is amenable to distributed optimization, i.e., can be split into a number o appropriately coupled subproblems. For such a combination o stability and structure, we propose the use o a separable terminal cost unction, combined with novel time-varying local terminal sets. For synthesis, we introduce a method that allows or constructing these components in a completely distributed way, without central coordination. The paper covers the nominal case in detail and discusses the extension o the methodology to reerence tracking. Closed-loop unctionality o the controller is illustrated by a numerical example, which highlights the eectiveness o the proposed controller and its time-varying local terminal sets. Key words: Distributed control; predictive control; large-scale systems. 1 Introduction Control o large-scale networks o dynamic systems is a challenging problem, in particular i the systems in the network are subject to communication constraints as well as constraints on states and inputs. MPC is a well-established methodology or the control o constrained systems. Its application under communication constraints has been a ield o active research in recent years, with applications in ields such as power networks (Venkat, Hiskens, Rawlings & Wright 2008) and building automation (Ma, Richter & Borrelli 2012). Two key challenges in distributed MPC are closed-loop stability and controller synthesis under distributed computations. This paper addresses these challenges and proposes a less restrictive solution approach compared to methods currently available in the literature. In order to obtain a distributed MPC ormulation with sta- This paper was not presented at any IFAC meeting. Corresponding author C. Conte. Tel addresses: christian.conte@siemens.com (Christian Conte), colin.jones@epl.ch (Colin N. Jones), morari@control.ee.ethz.ch (Manred Morari), mzeilinger@ethz.ch (Melanie N. Zeilinger). bility guarantees, results rom unconstrained decentralized and distributed control can be used. In decentralized control, the controllers in the network do not exchange inormation, while in distributed control they do. Important indings related to the analysis o decentralized systems are summarized in (Šiljak 1991), where especially vector Lyapunov unctions are used or stability analysis. Synthesis approaches or distributed control laws based on linear matrix inequalities have been proposed, e.g., in (Langbort, Chandra & D Andrea 2004) and (Zečević & Šiljak 2010). The literature on distributed MPC mainly distinguishes between non-cooperative and cooperative approaches. In non-cooperative distributed MPC, e.g. (Farina & Scattolini 2012), neighboring systems typically communicate once per time-step and each system is equipped with a local MPC controller that acts selishly and is robust against coupling to neighboring systems. While requiring less communication, non-cooperative approaches can become very conservative or even ineasible in presence o strong dynamic coupling. In cooperative distributed MPC, as e.g. in (Venkat, Rawlings & Wright 2005) or more recently (Giselsson & Rantzer 2013), neighboring systems typically communicate several times per time-step in order to solve a globally deined MPC problem by distributed optimization. Preprint submitted to Automatica 25 January 2016

2 A key requirement is or the MPC problem to be structured such that distributed optimization methods are applicable. While some cooperative distributed MPC approaches derive stability guarantees based on long horizons (Giselsson & Rantzer 2013), most approaches rely on terminal costs and terminal invariant sets (Mayne, Rawlings, Rao & Scokaert 2000). However, standard terminal costs and invariant sets are based on global Lyapunov stability and invariance concepts and do not exhibit a structure which is amenable to distributed optimization. In order to obtain such a structure or the terminal cost, vector Lyapunov unctions (Šiljak 1991) or linear matrix inequality (LMI) based methods (Langbort et al. 2004), (Zečević & Šiljak 2010) can be used. As or a structured terminal cost, however, the available methods are limited. One possibility is the use o a trivial terminal set, i.e. a point, as suggested in (Stewart, Venkat, Rawlings, Wright & Pannocchia 2010), which will, however, reduce the size o the region o attraction o the resulting MPC controller. Another option is to resort to robust positively invariant sets (Maestre, Muñoz de la Peña, Camacho & Alamo 2011), considering dynamic coupling as a disturbance. In presence o strong coupling however, the resulting terminal sets tend to be small or may even be empty. Another possibility is the use o time-varying local sets, as suggested in (Raković, Kern & Findeisen 2010), the synthesis o which is however non-obvious and not discussed in the paper. The contribution o this paper is twoold and builds on (Conte, Voellmy, Zeilinger, Morari & Jones 2012). The irst contribution, based on (Jokić & Lazar 2009), is a novel concept or time-varying local terminal sets leading to a cooperative distributed MPC controller with closed-loop stability guarantee. The proposed methodology can be used to construct terminal sets or both regulation and reerence tracking MPC. As opposed to the concept presented in (Raković et al. 2010), the set dynamics advocated in this paper are directly linked to the system dynamics and stability properties. The second contribution is a practical distributed synthesis method or networks o linear systems with quadratic costs and polytopic constraints. This method can be executed in a completely distributed way, a eature which is particularly beneicial in case o changing network topologies, where new controllers have to be synthesized on the ly without central coordination. This case has received considerable attention in the context o plug-and-play MPC (Riverso, Farina & Trecate 2013), (Zeilinger, Pu, Riverso, Ferrari-Trecate & Jones 2013). In Section 2, preliminaries on distributed systems and MPC are introduced. In Section 3, the ormulation o the nominal cooperative distributed MPC controller is presented and in Section 4, its distributed synthesis is discussed. Section 5 summarizes the nominal case and Section 6 covers the extension to reerence tracking. In Section 7, a numerical example is provided and Section 8 concludes the paper. 2 Preliminaries 2.1 Notation The set {1,..., M} N is denoted as M. A block-diagonal matrix S with blocks S i, where i M, is denoted as S = diag i I (S i ) or S = diag(s 1,..., S M ), depending on the context. Similarly, a vector which consists o the stacked subvectors x i, i I N, is denoted as col i I (x i ) or col(x 1,..., x M ). I a matrix S is positive deinite, we write S > 0 and i it is positive semi-deinite, we write S 0. The n-dimensional identity matrix is denoted as I n. A unction β( ) : R + R + is o class K i it is continuous, strictly increasing and i β(0) = 0. It is o class K i additionally it holds that lim s β(s) =. 2.2 Distributed Linear Time-Invariant (LTI) Systems We consider a network o M linear time-invariant systems, where each system i M has a state x i R ni, an input u i R mi and an output y i R pi. We consider systems that are coupled in the state and in the output. The dynamics o the local systems can thus be written as x + i = A ij x j + B i u i, y i = C ij x j i M, (1) j=1 j=1 where A ij R ni nj, B i R ni mi and C ij R pi nj. Note that additional coupling in the inputs could be easily reormulated into the orm in (1) by deining the original inputs as additional states and the changes in the original inputs as the new inputs. The coupling in states and outputs is used to deine the notion o neighboring systems. Deinition 1 (Neighboring Systems) System j is a neighbor o system i i A ij 0 or C ij 0. The set o all neighbors o i, including i itsel, is denoted as N i. The states o all systems j N i are denoted as x Ni = col j Ni (x j ) R n N i. The local systems (1) can thus, with matrices o appropriate dimensions, be equivalently written as x + i = A Ni x Ni + B i u i, y i = C Ni x Ni i M. (2) Throughout the paper, it is assumed that neighboring systems can communicate with each other. Assumption 2 (Communication) Two systems i and j can communicate, in a bidirectional way, i i N j or j N i. Both the local states and inputs are subject to constraints x i X i, u i U i i M, (3) where or each i M, X i R ni and U i R mi are convex sets which contain the origin in their interior. 2

3 By combining the local system dynamics in (1), the linear dynamics o the global system result in x + = Ax + Bu, y = Cx, (4) where x = col i M (x i ) R n, u = col i M (u i ) R m and y = col i M (y i ) R p. At some points in the paper, the equivalent notation x t+1 = Ax t + Bu t will be used to emphasize the current time index. The global system matrix A R n n and the global output map C R p n are block-sparse with entries A ij and C ij or each (i, j) M 2, provided j N i, and the global input map B = diag i M (B i ) R n m is block-diagonal. Similarly, combining the local state and input constraints (3), the global constraints or system (4) result in x X := X 1.. X M R n, u U := U 1.. U M R m. (5) In order or the methodology in this paper to apply, we make the ollowing assumption on the stabilizability o (A, B). Assumption 3 (Structured Linear Controller) There exists a linear control law o the orm κ (x) := K x = col i M (K Ni x Ni ), (6) where K R m n and K Ni R mi n N i i M, such that the system x + = Ax + Bκ (x) is asymptotically stable. Given a stabilizing controller κ (x), the notion o a positively invariant set can be deined. Deinition 4 (Positively Invariant (PI) Set) A set X is PI or the dynamics x + = Ax+Bκ (x), i x X x + X. It is important to note that even i κ (x) is structured as in (6), any PI set according to Deinition 4 is globally deined and does not generally exhibit structure. In particular, it is generally not a Cartesian product o local sets. 2.3 Nominal Centralized MPC The nominal MPC control law or regulation is deined through the inite horizon optimal control problem V (x) = min x,u N 1 V (x(n)) + l(x(k), u(k)) k=0 (7a) s.t. x(0) = x, (7b) x(k + 1) = Ax(k) + Bu(k) k {0,.., N 1}, (7c) (x(k), u(k)) X U k {0,.., N 1}, (7d) x(n) X, (7e) which is in the ollowing reerred to as the nominal MPC problem. We deine X N R n as the set o initial states x, or which (7) is easible. Both the stage cost l(x, u) and the terminal cost V (x) are positive deinite convex unctions and the terminal set X R n is convex, compact and contains the origin in the interior. Furthermore, u = {u(0),..., u(n 1)} denotes an input sequence over the inite horizon N and u (x) = {u (x, 0),..., u (x, N 1)} denotes the sequence minimizing (7) or the initial state x. The irst element o u (x) deines the state eedback control law κ MPC (x) := u (x, 0). Suicient conditions or asymptotic stability under κ MPC (x) are given in the ollowing. Theorem 5 ((Mayne et al. 2000)) Let β 1 ( ) and β 2 ( ) be K class unctions. I X X and (8) holds x X, then the closed-loop system x + = Ax + Bκ MPC (x) is asymptotically stable on the domain X N : κ (x) U, Ax + Bκ (x) X, (8a) β 1 ( x 2 ) V (x) β 2 ( x 2 ), (8b) V (Ax + Bκ (x)) V (x) l(x, κ (x)) (8c) By (8a), X is required to be a easible PI set and by (8b) and (8c), V (x) is required to be a Lyapunov unction or x + = Ax + Bκ (x) on the domain X. 2.4 Synthesis o Nominal Centralized MPC Controllers or LTI Systems and Polytopic Constraints The main diiculty in the synthesis o stabilizing nominal MPC controllers is the construction o a terminal cost V (x) and a terminal set X satisying (8). Synthesis methods or these components are outlined in the ollowing and will orm the basis or the distributed synthesis techniques in Section Centralized Terminal Cost Synthesis For linear systems, the terminal cost is typically chosen to be quadratic, i.e. V (x) = x T P x, and the terminal controller to be linear, i.e. κ (x) = K x. Given a quadratic stage cost l(x, u) = x T Qx+u T Ru, with Q 0 and R > 0, conditions (8b) and (8c) can be reormulated as an LMI in (P, K ), i.e. E EA T + Y T B T EQ 1 2 Y T R 1 2 AE + BY E 0 0 0, (9) Q 1 2 E 0 I 0 R 1 2 Y 0 0 I where E := P 1 and Y := K E, or details see (Boyd, Ghaoui, Feron & Balakrishnan 1994). Note that (9) is obtained rom the Lyapunov decrease condition (8c) by applying Schur complement techniques. In combination with a suitable objective unction, e.g. minimization o log det E or maximizing the volume o the 1-level set ellipsoid E := {x R n x T P x 1}, P can be computed by eicient numerical tools or semi-deinite programming. 3

4 2.4.2 Centralized Terminal Set Synthesis In the ollowing, a design method or ellipsoidal PI terminal sets is given. Ellipsoidal PI sets are superior compared to polytopic sets in terms o scalability o the synthesis methods. Consider state and input constraints X = {x R n H x x h x } and U = {u R m H u u h u }, with H x R lx n, h x R lx, H u R lu m, h u R lu. Consider urthermore a linear terminal control law κ (x) = K x and a quadratic terminal cost V (x) = x T P x. I the terminal cost ulills (8), it is a Lyapunov unction and its level sets X = {x R n x T P x α} are PI or the system dynamics x + = (A + BK )x (Blanchini 1999). The problem o inding the largest easible level set o V (x) can be posed as max α (10a) s.t. σ E (Hx,i) T 2 α h 2 x,i i {1,..., l x }, (10b) σ E (K T H T u,j) 2 α h 2 u,j j {1,..., l u }, (10c) where (H x,i, h x,i ) and (H u,j, h u,j ) correspond to the ith and jth halspace constraint o the polytopes X and U respectively. Furthermore, the unction σ E (a) := max x E a T x = P 1 2 a 2, where a R n, is the support unction o the 1- level set E deined above and is available in explicit orm since E is ellipsoidal. Note that the equivalence a T x b x E σ E (a) b holds. Problem (10) is a linear program (LP) in only one variable, or which a variety o eicient solution methods are available. 3 Stability o Nominal Cooperative Distributed MPC 3.1 Problem Statement In cooperative distributed MPC, the global MPC problem (7) is solved online by distributed optimization. A variety o distributed optimization algorithms are available in the literature, e.g. the dual decomposition based subgradient method or the alternating direction method o multipliers (ADMM), see (Bertsekas & Tsitsiklis 1989). Most distributed optimization algorithms work conceptually similar: The network-wide optimization problem is decomposed into a number o subproblems, each o which is assigned to one system in the network. In consecutive rounds, the subproblems are solved, local solutions are shared among neighbors, and local irstorder steps towards the global optimizer are taken. In presence o communication constraints according to Assumption 2, the key requirement or the application o distributed optimization is thus, that (7) is decomposable into M subproblems, each o which involves variables (x Ni, u i ) only. Inspecting problem (7), we recognize that the constraints (7b)-(7d) ulill this property, and the stage cost l(x, u) can be chosen separable, i.e. as a sum o terms l i (x Ni, u i ). The main challenge is the construction o a separable terminal cost V (x) and a structured terminal set X, ulilling (8) or stability. In this paper, we impose the particular structure V (x) = V,i (x i ), X (α i,..., α M ) = X,1 (α 1 )... X,M (α M ), (11a) (11b) where X (α i,..., α M ) R n is a Cartesian product o parametrized local sets X (α i ). In Section 3.2 we show how a separable terminal cost (11a) can be ormulated and in Section 3.3 we show how a terminal set (11b) can be deined. 3.2 Separable Terminal Cost A simple approach to construct a terminal cost unction o orm (11a), ulilling (8b) and (8c), would be to require each unction V,i (x i ) to decrease at every time step under a given terminal control law. Such an approach would, however, be very conservative. Consider or instance a system i whose state x i rests in the origin, where V,i (x i ) = 0. I the state x j o a neighboring subsystem j N i is nonzero, x i will necessarily be driven out o the origin, causing V,i (x i ) to increase. Thereore, as proposed in (Jokić & Lazar 2009), it is desirable to allow the local terminal cost to increase, as long as at the same time the global terminal cost decreases, which is ormalized in the ollowing theorem. Theorem 6 (Theorem III.4 in (Jokić & Lazar 2009)) Let there be a structured control law κ (x) according to Assumption 3. Let X R n be a PI set as deined in Deinition 4. I there exist, i M, unctions V,i (x i ), γ i (x Ni ) and l i (x Ni, κ Ni (x Ni )), as well as unctions β 1,i ( ), β 2,i ( ) and β 3,i ( ) K, such that x = col i M (x i ) X the ollowing conditions (12) hold, then V (x) = M V,i(x i ) is a Lyapunov unction or the system x + = Ax + Bκ (x) on X. β 1,i ( x i ) V,i (x i ) β 2,i ( x i ) i M, β 3,i ( x Ni ) l i (x Ni, κ Ni (x Ni )) i M, V,i (x + i ) V,i(x i ) l i (x Ni, κ Ni (x Ni )) + γ i (x Ni ) i M, γ i (x Ni ) 0. (12a) (12b) (12c) (12d) Conditions (12c) and (12d) ensure that while single terms V,i (x i ) may increase in some time steps, where the increase is bounded by unctions γ i (x Ni ), the global unction V (x) decreases in every time step. This property can be used to deine time-varying local terminal sets, as shown in the next section. 4

5 3.3 Time-Varying Local Terminal Sets The main idea is to deine local terminal sets as level sets o the local terminal cost unctions, i.e. X,i (α i ) = {x i R ni V,i (x i ) α i }, where 0 α i, i M. For static local values α i such level sets are, however, not invariant. In particular, assuming (12), x i X,i (α i ) A Ni x Ni + B i κ Ni (x Ni ) X,i (α i ) since, as time evolves, the state o any system i might leave X,i (α i ) due to an increase in V,i (x i ). The key idea or addressing this issue is to update the size o the local terminal sets in each time step, using the previously introduced relaxation unctions γ i (x Ni ). Deinition 7 (Local time-varying terminal sets) Let κ (x t ) be a terminal control law satisying Assumption 3 and let there be a separable terminal cost V (x t ) = M V,i(x t i ), with corresponding relaxation unctions γ i (x t N i ) i M, satisying (12). Deine α such that X glob := {x R n V (x) α} X and x X glob : κ (x) U. Local terminal sets are deined as X,i (α t i) := {x i R ni V,i (x i ) α t i} i M, (13) where the sizes αi t are deined by the set dynamics α t+1 i = α t i + γ i (x t N i ) i M, (14) with M α0 i α and 0 α0 i i M. A global terminal set or the MPC problem (7) is then deined as X (α t 1,..., α t M ) := X,1 (α t 1)... X,M (α t M ). (15) As the global terminal set (15) is a Cartesian product, it is amenable to distributed optimization, it does however not obviously ulill the MPC stability conditions (8) due to its time-varying nature. Thereore, in the ollowing, it is shown that Deinition 7 provides suicient conditions or (8), i.e., (i) the local system states remain within the local timevarying terminal sets under the terminal control law, and (ii) the Cartesian product o these local terminal sets(15) is recursively easible. Property (i) is shown in the ollowing. Lemma 8 I there are sets X,i (αi t ) i M, as deined in Deinition 7, then the ollowing holds: x t i X,i (αi) t x t+1 i X,i (α t+1 i ) i M (16a) 0 α t+1 i i M, (16b) where x t+1 i = A Ni x t N i + B i κ Ni (x t N i ) X,i (α t+1 i ). Proo: By Deinition 7, V,i (x i ) ulills (12a) and is thereore positive deinite. Thus, x t i X,i(α t i ) implies 0 V,i(x t i ) α t i. By Deinition 7, there exist V,i(x i ), γ i (x Ni ), κ Ni (x Ni ) and l i (x Ni, κ Ni (x Ni )), such that by (12c) it holds that 0 V,i (A Ni x t N i + B i κ Ni (x t N i )) V,i (x t i) l i (x t N i, κ Ni (x t N i )) + γ i (x t N i ) V,i (x t i) + γ i (x t N i ) α t i + γ i (x t N i ) = α t+1 i, which proves properties (16). Even i, as shown above in Lemma 8, the local system states remain within the time-varying local terminal sets (13), it is not obvious that the Cartesian product (15) o these sets is recursively easible w.r.t. the global state and input constraints. It can, however, be shown that i the global terminal set in (15), at t = 0, is a subset o X glob, which, as a level set o the Lyapunov unction V (x) is PI w.r.t. x t+1 = Ax t +Bκ (x t ), then it remains in X glob at all uture time steps. This property is shown in the ollowing Lemma. Lemma 9 I there exist sets X glob and X,i (αi t ) i M, as deined in Deinition 7, then property (17) holds: X,1 (α t 1)... X,M (α t M ) X glob t 0. (17) Proo: The proo is done by induction. By Deinition 7, it holds that M α0 i α and thus X,1 (α1) 0... X,M (αm 0 ) X glob. Assuming M αt i α at any time t 0, due to (14) and (12d) it holds that α t+1 i = αi t + γ i (x t N i ) αi t α, which concludes the induction step and implies (17). The results presented in Theorem 6, Lemma 8 and Lemma 9 provide the ingredients to prove recursive easibility and asymptotic stability o the global closed-loop system x t+1 = Ax t + Bκ MPC (x t ) in X N using standard arguments (Mayne et al. 2000). Recursive easibility ollows rom Lemma 8 and Lemma 9, asymptotic stability ollows rom Theorem 5, as κ (x), V (x) and X glob together ulill (8). 4 Distributed Synthesis o Cooperative Distributed MPC Controllers or Networks o Linear Systems In the previous section, a general ormulation o a separable terminal cost and time-varying local terminal sets or the MPC problem (7) have been proposed, such that (7) is amenable to distributed optimization and stability and constraint satisaction in closed-loop are guaranteed. In this section, a synthesis approach or the proposed terminal cost and set is presented, which consists o two parts. First, a distributed procedure to construct a separable quadratic Lyapunov unction V (x) = x T P x, i.e., with P block-diagonal, is presented in Section 4.1. Second, a distributed procedure 5

6 to ind X glob as the largest easible level set o V (x) is presented in Section 4.2. Local terminal sets (13) ollow rom Deinition 7 in a straightorward way. Throughout this section, we consider linear systems, polytopic state and input constraints, as well as quadratic cost unctions. In particular, the local constraints are o the orm X i = {x i R ni H xi x i h xi } and U i = {u i R mi H ui u i h ui } or all i M, where H xi R lx,i ni, h xi R lx,i, H ui R lu,i mi and h ui R lu,i. Moreover, local stage cost unctions are deined as l i (x Ni, u i ) = x T N i Q i x Ni + u T i R iu i, and local terminal cost unctions are deined as V,i (x i ) = x T i P,ix i or all i M. We deine local relaxation unctions γ i (x Ni ) = x T N i Γ Ni x Ni or all i M. Note that the matrices Q i, R i and P,i are all positive deinite, while Γ Ni is allowed to be indeinite. Finally, we consider a structured linear terminal control law κ (x) according to Assumption 3. In order to simpliy the notation, we introduce liting matrices T i {0, 1} ni n, V i {0, 1} mi m and W i {0, 1} n N i n or each i M. These liting matrices, similar to permutation matrices, have in each row exactly one entry equal to 1, such that x i = T i x, u i = V i u and x Ni = W i x. 4.1 Distributed Synthesis o Separable Terminal Cost In this section, a distributed synthesis method or a separable terminal cost is presented, which relies on the centralized synthesis tools introduced in Section The main idea is to ormulate a global synthesis problem such that it can be solved by distributed optimization without central coordination. In Section 4.1.1, a set o LMI conditions with one global coupling constraint is derived. In Section 4.1.2, a suicient condition or this constraint is presented, exhibiting only neighbor-to-neighbor coupling Structured LMIs with Global Coupling As discussed in Section 3.2, a suicient condition or V (x) to be a Lyapunov unction, is or the local cost terms to satisy the conditions stated in (12). For the considered linear systems and quadratic unctions, conditions (12c) and (12d) are equivalent to the matrix inequalities (A Ni + B i K Ni ) T P,i (A Ni + B i K Ni ) P,i (Q i + KN T i R i K Ni ) + Γ Ni, i M, Wi T Γ Ni W i 0, (18a) (18b) where P,i := W i Ti T P,iT i Wi T is P,i lited into the space o neighboring states. In the ollowing, it will be demonstrated that conditions (18) can equivalently be written as a set o M neighbor-to-neighbor coupled LMIs with one global constraint. Lemma 10 Condition (18) is equivalent to the set o LMIs Ē i + F Ni E Ni A T N i + YN T i Bi T E Ni Q 1/2 i YN T i R 1/2 i A Ni E Ni + B iy Ni E i 0 0 Q 1/2 i E Ni 0 I Ni 0 R 1/2 i Y Ni 0 0 I mi Wi T F Ni W i 0, 0 i M, (19a) (19b) where E i := P 1,i, Ē i := W i Ti T P 1,i T iwi T, E N i := W i EWi T, F N i := E Ni Γ Ni E Ni and Y Ni := K Ni E Ni i M. Proo: The equivalence (18a) (19a) is shown by let and right-multiplication o (18a) with E Ni and applying the Schur complement twice. The equivalence (18b) (19b) ollows rom the act that M W i T Γ N i W i M W i T F N i W i, highlighting the role o F Ni as a substitution or Γ Ni to achieve convexity in (19a). For details, see proo o Theorem IV.3 in (Conte et al. 2012). Remark 11 The act that distributed unconstrained controllers can be synthesized by distributed optimization methods has been recognized already in (Langbort et al. 2004). The synthesis problem presented here is however dierent in the sense that the increases in the local cost unctions are explicitly bounded by quadratic unctions γ i (x Ni ) = x T N i Γ Ni x Ni, which are synthesized together with the terminal cost. These unctions are important or the design o local time-varying terminal sets according to Deinition 7. Remark 12 Conditions (19) provide a constructive test or existence o a distributed unconstrained controller stabilizing the global system. Conservatism, compared to centralized linear controller synthesis, is introduced by (i) the choice o a block-diagonal quadratic Lyapunov unction and (ii) the structured gain matrix K. The derivation o general conditions or the existence o solutions to (19) is beyond the scope o this paper. Related literature includes (Šiljak 1991), where conditions on decentralized stabilizability are presented, and (Ho & Chu 1972), where it is shown that or networks o partially nested systems, a distributed linear controller is optimal, provided the network is stabilizable. In order to add a cost metric to the conditions in (19), we propose the maximization o the volume o the 1-level set o V (x). Maximizing this volume corresponds to minimizing the determinant o P, which is equivalent to maximizing the determinant o E = P 1, which in turn can be achieved by maximizing the concave operator log det(e). This operator, as stated in the ollowing proposition, is separable or a block-diagonal E and thus directly amenable to distributed optimization. 6

7 Proposition 13 For a matrix E = diag i M (E i ), it holds that log det(e) = log det(e i ). (20) The synthesis o a structured Lyapunov unction according to Theorem 6 can thus be posed as a decomposable convex optimization problem Structured LMIs with Neighbor-to-Neighbor Coupling The main diiculty in the synthesis LMIs (19) is constraint (19b), which couples all systems in the network in one single constraint, as similarly recognized in (Hermans, Lazar & Jokić 2010). In the ollowing, we will describe a suicient condition or (19b), which is based on neighbor-toneighbor coupling. Consider upper-bounding the matrices F Ni by block-diagonal matrices S Ni, resulting in the conditions F Ni S Ni i M, (21a) Wi T S Ni W i 0 j N i T j W T j S Nj W j T T j 0 i M, (21b) where the equivalence in (21b) is due to S Ni being blockdiagonal. The set o constraints (21) is clearly suicient or (19b). Moreover, (21a) involves only local variables and the right hand side o (21b) represents M LMIs, each o which involves variables o neighboring systems only, as exempliied in Fig. 1. Consequently, (21) provides suicient conditions or replacing (19b), which renders the terminal cost synthesis problem amenable to distributed optimization under the given communication constraints. Fig. 1. Illustration o constraint (21) or a network o three systems with N 1 = {1, 2}, N 2 = {1, 2, 3} and N 3 = {2, 3}. 4.2 Distributed Synthesis o Structured Terminal Sets Given a separable terminal cost unction V (x) = x T P x ulilling condition (12), a easible level set thereo can be used as a global set X glob = {x R n x T P x α} X, as required in Deinition 7. Once such a global level set with size α is given, the initial local set sizes αi 0 can be allocated according to M α0 i α. The largest easible level set o V (x) can be ound, analogously to the centralized case described in Section 2.4.2, by solving the linear program α max := max α α (22a) s.t. σ E (Hx T i,j) 2 α = P 1 2,i Hx T i,j 2 2α h 2 x i,j i M, j {1,..., l x,i }, (22b) σ E (KN T i Hu T i,j) 2 α = P 1 2,N i KN T i Hu T i,j 2 2α h 2 u i,j i M, j {1,..., l u,i }, (22c) where P,Ni := W i P Wi T i M. The constraints (22b) and (22c) represent conditions on the support unction o the α-level set o V (x). In particular, (22b) ensures that any point x X glob ulills the polytopic state constraints x X and (22c) ensures that the polytopic input constraints K x U are satisied. The structure in problem (22) allows or its solution by distributed optimization methods, e.g., as proposed in (Schizas, Ribeiro & Giannakis 2008). 5 Summary o Distributed Synthesis and Closed-Loop Operation o Nominal Cooperative Distributed MPC In this section, we briely summarize the main steps o the proposed nominal cooperative distributed MPC or networks o linear systems. The procedure or distributed controller synthesis is described in Algorithm 1. In Step 1, a separable quadratic terminal cost V (x) = M xt i P ix i, a structured linear terminal control law κ (x) = col i M (K Ni x Ni ) according to Assumption 3, and relaxation unctions γ i (x Ni ) = x T N i Γ Ni x Ni i {1,..., M} are computed, which satisy condition (12). Subsequently, in Step 2, the maximum easible level set o the global terminal cost unction V (x) is ound to serve as an outer bound X glob or the Cartesian product (15) o local terminal sets. Finally, a easible initial coniguration or the local terminal sets (13) is obtained in Step 3. The components constructed during the execution o Algorithm 1 can then directly be used in the MPC problem (7) to operate a stabilizing cooperative distributed MPC controller in closed-loop. The main steps required or distributed closed-loop operation are summarized in Algorithm 2. Note that in Algorithm 2, u i (xt, 0) and x N i (x t, N) are obtained rom the optimal solution o the global MPC problem (7), which is solved by distributed optimization in Step 2, or the current state measurement x t. In particular, given optimal inite-time input and state sequences u (x t ) = {u (x t, 0),..., u (x t, N 1)} and x (x t ) = {x (x t, 0),..., x (x t, N)}, it holds that u i (xt, 0) = V i u (x t, 0) and x N i (x t, N) = W i x (x t, N). 7

8 Algorithm 1 Oline synthesis o local terminal costs and time-varying local terminal sets as deined in Deinition 7 1: Maximize (20) subject to (19a) and (21), by distributed optimization, to ind V,i (x, i) = x T i P,ix i as well as κ Ni (x Ni ) and γ i (x Ni ) = x T N i Γ Ni x Ni ulilling (12), locally at each i in {1,..., M}. 2: Solve LP (22) by distributed optimization to ind the largest easible level set X glob = {x R n V (x) α max }, where α max is ound locally at each system. 3: Find αi 0 i {1,..., M}, such that M α0 i α max (e.g. by setting αi 0 = α/m or all i), to deine initial local terminal sets (13). Algorithm 2 Online distr. MPC, executed at every time step 1: Each system i {1,..., M} measures local state x t i. 2: Solve MPC problem (7) by distributed optimization, where systems in N i iteratively communicate. 3: Each system i {1,..., M} applies the local control input u i (0) obtained in step 2. 4: Each system i {1,..., M} updates the local terminal set to X,i (α t+1 i ), where α t+1 i = αi t + x T N i (x t, N)Γ Ni x N i (x t, N), according to (14). 6 Extension o Cooperative Distributed MPC to Reerence Tracking This section presents the main steps or extending the ideas presented in Section 3 and Section 4 to the problem o tracking piecewise constant reerences. The discussion covers common reerence tracking ormulations with ocus on convergence guarantees. Oset-ree techniques such as (Mäder, Borrelli & Morari 2009) are not considered, but could be added to the proposed problem setup. In tracking MPC, the system output is required to ollow a given reerence y t. We assume that or a given y t, there exist points (x s, u s ) satisying the global equilibrium condition [ ] [ ] [ ] A In B xs 0 = C 0 y t u s. (23) Due to the structure in the matrices A, B and C deined in (1), (23) can be used, given a reerence y t and using a separable objective unction, to ind an equilibrium pair (x s, u s ) by distributed optimization. I an equilibrium pair (x s, u s ) is given, a distributed tracking MPC can be directly reormulated as distributed regulation MPC by deining the system state as x := x x s. For stability, as in the regulation case, a separable terminal cost and a structured terminal set are required. The main diiculty is the act that the terminal set is centered around the target steady state x s. Thus, i x s changes online, a new terminal set needs to be synthesized online, which may be prohibitively time-consuming. A solution to this problem, which will be urther investigated in the ollowing, is the use o a parameterized terminal set, as proposed in (Limon, Alvarado, Alamo & Camacho 2008). Deinition 14 (Parametrized Invariant Set or Tracking) Let (x s, u s ) be an equilibrium point satisying (23). Consider a control law κ ( ) : R n R m, which stabilizes the linear system x + = Ax+Bκ (x). A set X tr R 2n+m is a easible invariant set or tracking, i the ollowing conditions hold or each ( x, x s, u s ) X tr : x + x s X, u s + κ ( x) U, ( x, x s, u s ) X tr (A x + Bκ ( x), x s, u s ) X tr. 6.1 Formulation o Parametrized Local Time-Varying Terminal Sets or Cooperative Distributed Tracking MPC A separable terminal cost V ( x) satisying (12) under the assumptions o Theorem 6 can be deined and computed as proposed in Section 3 and Section 4. A parametrized set according to Deinition 14 can then be deined as a level set o the unction V tr ( x,x s, u s ) := V ( x) + V x (x s ) + V u (u s ) = x T P x + x T s P x x s + u T s P u u s, (24) where P := diag i M (P,i ), P x := diag i M (P x,i ), P u := diag i M (P u,i ), and where P,i, P x,i R ni and P u,i R mi i M. Note that under the assumptions o Theorem 6 and due to the properties (12) o V ( x), any easible level set := {( x, x s, u s ) R 2n+m V tr ( x, x s, u s ) α} is invariant in the sense o Deinition 14, independent o how V x (x s ) and V u (u s ) are chosen. X glob tr Due to the block-diagonal property o P, P x and P u, local time-varying terminal sets or tracking, which are locally parametrized in (x s,i, u s,i ) i M, can be deined as X tr,i (α t i) = {( x i, x s,i, u s,i ) R 2ni+mi x T i P,i x i + x T s,ip x,i x s,i + u T s,ip u,i u s,i α t i} i M, (25) where αi t is time-varying according to (14). Similarly as in the regulation case in Lemma 8, it can be shown that i the point ( x i, x s,i, u s,i ) is contained in X tr,i (αi t ), then the point (A Ni x + B i κ Ni ( x Ni ), x s,i, u s,i ) will be contained in X tr,i (α t+1 i ). Furthermore, as in the regulation case in Lemma 9, given a easible global invariant set or tracking X glob tr, it can be shown that i α αm 0 α, then the Cartesian product X tr,1 (α1) t... X tr,m (αm t ) o local time-varying terminal sets will remain within X glob tr at all times t 0. Thus, the main nominal results rom Section 3 extend to the reerence tracking case, see also (Conte, Zeilinger, Morari & Jones 2013) or more details. 7 Numerical Examples In this section, the methodology or cooperative distributed MPC presented in this paper is illustrated by a numerical 8

9 example simulating the behavior o a chain o masses m = 1 kg, connected by springs with k = 3 N/m and dampers with b = 3 Ns/m, see Fig. 2. The continuous-time ODEs describing the dynamics are discretized by the orward Euler method with a sampling time o 0.2 s. The resulting linear discrete-time system is o orm (1), with x i containing the position and velocity o mass i and u i being a orce applied to mass i. Constraints are imposed as x i 10 and u i 1, Q Ni and R i are identity, and the MPC horizon is N = 5. The local terminal cost terms V,i (x i ) as well as the local relaxation unctions γ i (x Ni ) are designed to be quadratic unctions as introduced in Section 4. Position Mass 1 (centr.) Mass 2 (centr.) Mass 1 (distr.) Mass 2 (distr.) Mass 1 ( ) Mass 2 ( ) Simulation step (a) Mass position trajectories. Fig. 2. Chain o masses, connected by spring and damper elements. The closed-loop behavior or basic regulation to the origin is investigated on a chain o M = 5 masses, comparing three controller setups: (i) centralized MPC with a terminal set deined by the maximum volume easible ellipsoid characterized by a dense Lyapunov matrix P, (ii) distributed MPC as proposed in this paper and (iii) distributed MPC with the origin as a trivial terminal set. The initial states are zero or all systems except or the irst one, i.e., x 1 = [0.27 m, 0 m/s] T. The system is simulated or 15 steps, its closed-loop behavior is illustrated in Fig. 3. Note that the trajectories or setups (i) and (ii) are quite similar, illustrating the proximity in perormance between (i) and (ii). Furthermore, Fig. 3(c) depicts the terminal set sizes o masses 1, 2 and 5 under distributed MPC as proposed in this paper. Note that the local terminal set sizes change considerably during convergence o the closed-loop system (b) Input trajectories. Mass 1 (distr.) Mass 2 (distr.) Mass 5 (distr.) The region o attraction (RoA) o the closed-loop system is analyzed in Fig. 4, showing that or short prediction horizons, there is a signiicant advantage in terms o RoA size when non-trivial, locally time-varying terminal sets are used instead o single-point terminal sets. For long prediction horizons, the RoAs o both controllers converge to the maximum RoA or the given constrained system. Note that each data point in Fig. 4 is a relative estimate o the extension o the RoA, either or controller (ii) or (iii) under a given horizon length N. Each estimate is obtained as the average o RoA extensions in 100 random directions. The estimate is relative as the averages or setup (ii) and (iii) is normalized by the respective average or controller setup (i). 8 Conclusions In this paper, two contributions regarding stability and synthesis o cooperative distributed MPC were presented: (i) the ormulation o a cooperative distributed MPC controller, which is stabilizing and suitable or distributed optimization in closed-loop and (ii) the synthesis procedure or such a controller, which can be executed in a purely distributed Simulation step (c) Local terminal set sizes. Fig. 3. Trajectories under nominal cooperative distributed MPC or regulation. relative RoA size non-trivial terminal set trivial terminal set MPC horizon length Fig. 4. Estimates o the relative RoA size or cooperative distributed MPC with non-trivial and trivial terminal conditions. 9

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