Distributed model predictive control for tracking: a coalitional clustering approach

This is a repository copy of Distributed Model Predictive Control for Tracking: A Coalitional Clustering Approach in the Depósito de Investigación de la Universidad de Sevilla. Version: Accepted Paper. Citation: P. Chanfreut, J. M. Maestre, A. Ferramosca, F. J. Muros and E. F. Camacho, "Distributed Model Predictive Control for Tracking: A Coalitional Clustering Approach," in IEEE Transactions on Automatic Control, vol. 67, no. 12, pp. 6873-6880, Dec. 2022, doi: 10.1109/TAC.2021.3133486. To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright: © 2023 IEEE. Personal use of this material is permitted. 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Citation information: DOI 10.1109/TAC.2021.3133486, IEEE Transactions on Automatic Control SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 2021 1 Distributed Model Predictive Control for Tracking: A Coalitional Clustering Approach Paula Chanfreut , Jos´ e Mar´ ıa Maestre , Senior Member, IEEE, Antonio Ferramosca , Member, IEEE, Francisco Javier Muros , and Eduardo F. Camacho , Fellow, IEEE Abstract—A coalitional robust model predictive controller for tracking target sets is presented. The overall system is controlled by a set of local control agents that dynamically merge into cooperative coalitions or clusters so as to attain an efficient trade-off between cooperation burden and global performance optimality. Within each cluster, the agents coordinate their inputs to maximize their collective performance, while considering the coupling effect with external subsystems as uncertainty. By using a tube-based approach, the overall system state is driven to the target sets while satisfying state and input constraints despite the changes in the controllers clustering. Likewise, feasibility and stability of the closed-loop system are guaranteed by tracking techniques. The applicability of the proposed approach is illustrated by an academic example. Index Terms—Coalitional model predictive control, control by clustering, tracking, robust control. I. INTRODUCTION IN the last decades, distributed control schemes have gained attention due to the increasing size, complexity, and modularity requirements of many systems [1]. In essence, the overall control problem is divided into smaller pieces that are assigned to local controllers, also known as agents. Conversely to decentralized control, where no inter-controller communication is used, in distributed approaches the agents share information through a network to coordinate their control actions. In this context, the classical assumption is to consider a fixed communication network with static neighborhoods. By contrast, some recent works explore dynamic neighbourhoods by considering switching communication topologies [2], [3], time-varying partitioning [4], plug and play schemes [5], [6], and clustering [7]. In this work, we will focus on coalitional control [8], a clustering approach where the use of the communication resources is penalized to deter the control scheme from exchanging unnecessary information, leading to a time-varying degree of coordination. In this context, the local agents dynamically merge into disjoint clusters or coalitions whose members coordinate their control actions. The underlying goal of these schemes is to enable communication only among highly coupled agents so as to obtain a performance close to a fully coordinated system, P. Chanfreut, J. M. Maestre, F. J. Muros and E. F. Camacho are with the Department of Systems and Automation Engineering, University of Seville, Spain (e-mails: {pchanfreut, pepemaestre, franmuros, efcamacho}@us.es) A. Ferramosca is with the Department of Management, Information and Production Engineering, University of Bergamo, Italy (e-mail: [email protected]). This work is supported by the Spanish Training Program for Academic Staff under Grant (FPU17/02653), the European Research Council Advanced Grant OCONTSOLAR (SI-1838/24/2018), and the Spanish MINECO Project C3PO (DPI2017-86918-R). but reducing the complexity of the control problem and the cooperation efforts, thus improving the scalability. This idea has interesting connections with cooperative game theory [9], [10] and has been extended to the well-known framework of model predictive control (MPC) [11], considering applications where the coupling varies in time and space, such as irrigation canals [12] and traffic freeways [13]. Similar approaches have also been recently proposed for dealing with large-scale power systems, e.g., in [14], a distribution network is split into control areas to solve a voltage regulation problem in an admissible time; in [2], the communication topology switches according to the oscillation modes; and in [15], a wind farm is split up into subsets of wind turbines according to the wake coupling effect. Undoubtedly, MPC has become the accepted standard for complex control problems in the processes industry [16]. One of its most appealing features is the ability to drive the system state to a pre-fixed target fulfilling constraints. Nevertheless, if the desired setpoint changes, then the feasibility of the controller may be lost and the controller can fail to track the reference [17]. Several solutions are proposed in the literature to deal with this tracking problem [18], based on switching strategies [19], reference governors [20]–[22], or treating the target change as a disturbance to be rejected [17], [23]. In this article, we focus on the methodology presented in [24]– [26], which guarantees recursive feasibility and convergence to the admissible setpoints while providing an enlarged domain of attraction. In particular, the distributed strategies in [27], [28] are enhanced here in a coalitional framework. The main goal is to design a robust coalitional MPC for tracking, where both coupling and external disturbances are uncertain from a local standpoint. To this end, the robust MPC in [29] is employed. In this regard, the uncertain dynamics are addressed with a tube-based approach [30], [31], hence maintaining the real system in the neighborhood of the nominal predictions, and also allowing an independent control law computation for each cluster. See also [32], [33] for some fairly recent results on coalitional MPC tube-based schemes. The global system state is driven to a given target set, while satisfying the states and inputs constraints, and assuring recursive feasibility and stability despite the dynamic formation of coalitions. In the line of [8], a double sample rate strategy is used, where the clusters composition may be periodically changed. The paper is organized as follows. In Section II, the problem setting is introduced. In Section III, the MPC for tracking sets approach is implemented in a coalitional fashion. The control scheme that assures feasibility and stability is presented in Section IV. Finally, simulation results and concluding remarks are respectively described in Sections V and VI. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 14,2021 at 09:09:19 UTC from IEEE Xplore. Restrictions apply. 0018-9286 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2021.3133486, IEEE Transactions on Automatic Control SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 2021 2 II. PROBLEM FORMULATION In this section, we present the model used to describe the system dynamics and the communication infrastructure. Additionally, the control objective we address is introduced. A. System dynamics Consider a class of linear systems that can be partitioned into a set N={1,2, . . . , N}of coupled subsystems, whose dynamics are modeled as1 x+ i=Aiixi+Biiui+wi, wi=X j∈Ni [Aijxj+Bijuj] + di,(1) where xi∈Rnxiand ui∈Rnuiare respectively the state and input vectors of subsystem i∈ N,x+ i∈Rnxiis its successor state, and wi∈Rnxiaggregates the coupling among the state of agent iand its set of neighbors, defined as Ni={j∈ N \ {i} | [Aij, Bij]=0}, and external disturbances di∈Rnxi. Matrices Aij and Bij are, respectively, the state transition and the input-to-state matrices for all i, j ∈ N , and 0represents a null matrix of the corresponding size. Accordingly, the global behavior can be modeled by matrices AN= (Aij)i,j∈N and BN= (Bij)i,j∈N , which aggregate (1) for the Nsubsystems, i.e., x+ N=ANxN+BNuN+wN,(2) where xN= [xi]i∈N and uN= [ui]i∈N are the global state and input vectors, respectively. Additionally, wN= [di]i∈N is the overall disturbance vector, which contains only external disturbances because all subsystems interactions are accounted for in matrices ANand BN. B. Network structure The Nsubsystems are individually governed by a set of local agents interconnected through a configurable communication network, i.e., the connections between agents can be dynamically either enabled or disabled. Let G= (N,L) be the graph describing the network, where the nodes in N represent the local agents, and set Lcontains the links, i.e., L ⊆ LN={{i, j} | {i, j} ⊆ N , i =j}. Also, let |L| be the cardinality of L, then we can derive a set Tcomposed of 2|L| communication topologies Λ⊆ L, which are defined by the sets of enabled links, i.e., T={ Λdec z}|{ Λ0,Λ1, . . . , Λcen z}| { Λ2|L|−1},(3) where Λdec and Λcen respectively refer to the decentralized and centralized configurations, i.e., Λdec =∅and Λcen =L. In this respect, the term decentralized denotes the case in which the agents do not share any data, hence their knowledge is limited to local information, while centralized refers to communication using all available resources. 1For simplicity, the dependence with time step has been omitted along the full manuscript. Throughout this paper, superscript ‘+’ is used for variables associated to time instant k+ 1 and no superscript refers to time k. Fig. 1: Examples of topologies and coalitions for a network with 4 agents and 4links. Letters Cand Ddenote coalitions of agents, i.e., C,D ∈ N/Λwith C ∩ D =∅. Note that in the left figure all agents can communicate (see Remark 1). Each Λ∈ T arranges the Nsubsystems into clusters or coalitions C, i.e., agents connected by a path of enabled links. In this regard, the partition induced by Λover Nis denoted as N/Λ. Note that the number of clusters in N/Λranges from one, when all agents are connected (N/Λcen =N), to N, corresponding to a fully decentralized system (N/Λdec = {{1},{2},...,{N}}). An easy example with three different topologies following this notation is depicted in Fig. 1. Remark 1: Hereafter, we consider that the communication links are bidirectional and assume indirect connectivity, i.e., any agent ican send/receive data to/from agent jif there exists a path of enabled links between iand j.□ 1) Coalitional model: Dynamically, any coalition C ∈ N/Λcan be considered as a single system modeled by x+ C=ACCxC+BCCuC+wC, wC=X D∈NC [ACDxD+BCDuD] + dC,(4) where xC= [xi]i∈C ∈RnxCand uC= [ui]i∈C ∈RnuCare respectively the aggregates of the states and inputs of subsystems i∈ C, and matrices ACC and BCC map the current coalition members states and inputs to their successor state. Similarly, wCmodels the effect of neighboring coalitions D ∈ NCand the external disturbances on subsystems i∈ C. Matrices ACD,BCD and set NCare defined analogously to the case of interacting subsystems. C. Objective statement and assumptions From now on, we consider the assumptions described below: (i) The agents in Nshould drive the global system state to a target set ΓNwhile satisfying state and input constraints. (ii) The agents can dynamically merge into disjoint clusters C to compute their control inputs. (iii) Each cluster Ccan measure xCand has full knowledge of matrices ACC and BCC, whereas the inter-coalitions coupling and external disturbances are uncertain. Also, the following considerations are explicitly assumed: Assumption 1: For any coalition C: •Pair (ACC, BCC)is stabilizable and state xCis measurable. •The state and input are constrained by xC∈ XC=O i∈C Xi, uC∈ UC=O i∈C Ui,(5) Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 14,2021 at 09:09:19 UTC from IEEE Xplore. Restrictions apply. 0018-9286 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2021.3133486, IEEE Transactions on Automatic Control SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 2021 3 where XCand UCare polytopic sets containing the origin in their interior, and Xiand Uirepresent the constraint sets of the subsystems inside the coalition. •Disturbances wCare bounded by a convex compact polytope WCcontaining the origin in its interior, i.e., wC∈ WC=M D∈NC [ACDXD+BCDUD]⊕ EC,(6) where EC=Ni∈C Ei, being Eia set containing all possible realizations of the external disturbances on subsystem i. •There exist a feedback gain Ke Csuch that matrix (ACC + BCCKe C)is Schur [34], and a robust positively invariant set RCthat satisfies (ACC +BCCKe C)RC⊕ WC⊆ RC, RC⊂int(XC), Ke CRC⊂int(UC). (7) □ Assumption 2: The global target set is defined as ΓN= Ni∈N Γi, where Γiare convex polytopic sets defining the target region for the state of subsystem i, that is, Γi={xi∈Rnxi|GΓixi≤gΓi},(8) with GΓiand gΓiproviding the H-representation of Γi. Accordingly, ΓC=Ni∈C Γiis the target set of coalition C.□ Assumption 3: For any coalition C, sets RCand ΓCare such that ΓC⊖ RC=∅.□ III. TRACKING SETS IN A COALITIONAL FRAMEWORK The robust tube-based MPC for tracking [29] is characterized by: (i) The introduction of an artificial steady state and input as optimization variables in an MPC problem based on the nominal dynamics, which allows for an extension of the domain of attraction of the MPC controller [24]. (ii) The use of an offset cost function that penalizes the deviation between the artificial and the real setpoints. (iii) The use of a control law comprising an optimal input governing the nominal dynamics and an ancillary feedback controller to keep the real system in the neighborhood of the nominal predictions [30], [31], [35]. A. Nominal steady state and input The nominal model of each cluster is derived from (4) by neglecting the effect of wC, that is, x+ C=ACCxC+BCCuC,(9) where xCand uCrepresent respectively the nominal state and input of C. Accordingly, the set of possible nominal steady states and inputs for a certain cluster Ccan be derived from ACC −InxCBCC xs C us C=0nxC×1,(10) where InxCand 0nxC×1represent respectively the identity matrix of dimensions nxC×nxCand the null matrix of dimensions nxC×1. Therefore, the null space of matrix ACC −InxCBCCallows a parametrization of the set of steady states in terms of a single variable in RnuC×1(see [24]). B. Invariant set for tracking Consider that the nominal coalitional dynamics (9) are controlled by a linear feedback control law, defined as uC=us C+KC(xC−xs C),(11) where KCis a constant matrix that yields closed-loop stability of system (9) and satisfies PC−(ACC+BCCKC)TPC(ACC+BCCKC)=QC+KT CRCKC,(12) being PCa positive definite matrix. Also, let us introduce the augmented system   xC xs C us C   + =  ACC+BCCKC−BCCKCBCC 0nxC InxC 0nuC 0nxC 0nxC InuC    xC xs C us C  ,(13) whose state is formed by the nominal coalitional state and its steady state and input. Considering this, in the tracking optimization problem the terminal set is defined as an invariant set Ωa Cof augmented system (13), i.e., the terminal set is an invariant set for any possible nominal equilibrium point, and contains all nominal initial states (and steady states) that can be stabilized by (11) while satisfying constraints (5) (see [24]). Remark 2: The global invariant set for the nominal dynamics of each topology Λcan be computed as Ωa Λ=NC∈N /ΛΩa C. Also, notice that the nominal state of any coalition Clies in the projection of Ωa Conto the axis corresponding to xC. The equivalent conclusion holds for xNand Ωa Λ.□ C. Objective function The objective function used in the computation of u∗ C and x∗ Cis: VC(xC(0),uC, xs C, us C, xt C)= Np−1 X n=0 ∥xC(n)−xs C∥2 QC+∥uC(n)−us C∥2 RC +∥xC(Np)−xs C∥2 PC+VC O(xs C, xt C), (14) where the time index between brackets indicates the step time of the prediction horizon, QC= (Qi)i∈C ≥0and RC= (Ri)i∈C >0are weighting matrices, with Qiand Ribeing the weighting matrices of agent i,uC= [uC(n)]Np−1 n=0 is the nominal sequence of inputs for a prediction horizon of Np time steps, and (xs C, us C)is the setpoint, which becomes a variable that will be optimized each time instant. Note that this artificial setpoint remains constant during the prediction horizon. Additionally, ∥xC(Np)−xs C∥2 PCis the terminal cost, being PCa properly chosen matrix, and VC O(·)is the offset cost function, which penalizes the deviation of xs Cfrom the coalitional target set ΓC. In particular, we use VC O(xs C, xt C) = αX i∈C ∥xs i−xt i∥∞,(15) where α > 0is a weighting parameter, and xt C= [xt i]i∈C is any point contained in the robust target set, i.e., ψC= ΓC⊖ RC. Accordingly, the global objective function for any topology Λis defined by adding (14) for all C ∈ N/Λ, i.e., VN(xN(0),uN, xs N, us N, xt N,Λ)=X C∈N /Λ VC(xC(0),uC, xs C, us C, xt C), (16) Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 14,2021 at 09:09:19 UTC from IEEE Xplore. Restrictions apply. 0018-9286 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2021.3133486, IEEE Transactions on Automatic Control SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 2021 4 where uN= [uC]C∈N /Λ,xs N= [xs C]C∈N /Λ,us N= [us C]C∈N /Λ and xt N= [xt C]C∈N /Λ. Note that VN(·)depends on Λ, i.e., the overall objective function changes with the topology. Remark 3: The offset cost function (15) will be zero for all xs Ccontained in ψC. As a consequence, any artificial steady state placed on the border of ψCcancels the offset cost. □ D. Optimization problem Every coalition solves the following problem at each time step: min uC,xC(0),xs C,us C,xt C VC(xC(0),uC, xs C, us C, xt C) s.t. xC(0) ∈xC⊕(−RC), xC(n+ 1) = ACCxC(n) + BCCuC(n), uC(n)∈ UC=UC⊖Ke CRC, xC(n)∈ XC=XC⊖ RC, xt C∈ψC, (xC(Np), xs C, us C)∈Ωa C, ∀n= 0, ..., Np−1, (17) where the state and input constraints are shrunk to account for the possible future realizations of the disturbances. The minimizer of (17) is formed by the optimal coalitional sequence of inputs u∗ Cand initial state x∗ C(0), together with the optimal artificial states and inputs (xs∗ C, us∗ C), and a certain xt∗ C. Note that if Λdec is imposed, then each agent considers only its local objective and variables, thus minimizing the communication burden and the local problems complexity. Conversely, if Λcen is used, then all agents jointly optimize the overall objective, thus maximizing the global performance. Remark 4: Let the system be controlled using a static topology Λand let uC=u∗ C+Ke C(xC−x∗ C)be the control law of any cluster C ∈ N/Λ, where u∗ C, and x∗ Care respectively the first component of u∗ Cand the initial state obtained from (17). Then, the global state trajectory will evolve in the tube defined by RΛ=⊗C∈N /ΛRC[31, Proposition 1], i.e., the deviation between the predicted and real trajectories is bounded. □ E. Topology Switching Features We consider a double sample rate strategy where every Ttop steps a network topology is selected through a comparative assessment of different configurations. 1) Topologies evaluation: The following index is used as a surrogate to evaluate all Λ∈ T [8]: r(Λ, xN) = (xN−xΓ)TPΛ(xN−xΓ) + c|Λ|,(18) where xΓis the Chebyshev center of the target set, c > 0 is the cost of enabling a link and |Λ|is the number of enabled links in Λ. The first term of index (18) provides an upper bound estimate of the unconstrained performance costs associated with topology Λ, while the second term weights the coordination costs and promotes the use of sparse topologies. The procedure to obtain PΛis detailed in [8]. 2) Condition on the switchings: A change of topology entails a switching of both the nominal control model and the state and input constraints in (17). In particular, notice that splitting any coalition increases the uncertainty of the corresponding subsystems, leading to greater reductions of the constraints sets in (17). For stability and feasibility purposes, we impose two conditions on the switchings: (i) A transition to Λ+takes place iff the system state at the switching instant fulfills xN∈ProjxΩa Λ+, where ProjxΩa Λ+is the projection of the augmented invariant set of topology Λ+onto xN. (ii) The topologies remain active for a period that guarantees a decreasing evolution of nominal costs (16). Note that if sets ProjxΩa Λ+are computed offline for all possible Λ+, then (i) translates into checking if a matrix inequality is satisfied. Alternatively, it can be checked if there exists some (xs N, us N)such that (xN, xs N, us N)∈Ωa Λ+. The latter can be formulated as a simple optimization that would be solved every Ttop instants with minimum computation burden. IV. CONTROL SCHEME The robust coalitional control scheme for tracking target sets introduced in this work is presented next: Control Scheme 1: Let us consider that the communication topologies may be switched every Ttop time instants with the set K={pTtop :p∈N>0}containing the switching instants. Also, set the topology Λto an initial configuration, say Λini, and the initial value of Vto infinity. Then, at each sample time kproceed as follows: 1: if k∈ K then 2: Coalitions share their state and current costs. 3: if V∗ N< V then 4: Topology Λ+∈ T that minimizes (18) is selected. 5: If xN∈ProjxΩa Λ+, update V=V∗ Nand set Λ=Λ+. 6: end if 7: end if 8: Each cluster C ∈N/Λcomputes u∗ Cand x∗ Cthrough (17). 9: Each cluster C ∈ N /Λimplements tube-based control law uC=u∗ C+Ke C(xC−x∗ C), where u∗ C, and x∗ Care obtained from (17). Note that Control Scheme 1 uses the same tracking methodology as the distributed MPC in [27], which is also extended in [28] by exploiting the subsystems interactions to reduce the communication overhead. However, there are notable differences, among which are the following: in [27], [28], all agents are cooperative and all (interactive) subsystems communicate to optimize a plant-wide objective, while here each cluster optimizes the aggregate benefit of its members and there is no inter-coalition communication; [27], [28] guarantees convergence to the centralized optimum, while the coalitional controller balances the coordination effort and system performance; also, [27] is based on a warm-start initialization that is not used in this article. Remark 5: Set Tin Control Scheme 1 may not contain all possible topologies and may be adapted in real time using a reasonably-sized subset of candidate topologies [12]. Also, notice that the minimum and maximum topologies in Tare Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 14,2021 at 09:09:19 UTC from IEEE Xplore. Restrictions apply. 0018-9286 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2021.3133486, IEEE Transactions on Automatic Control SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 2021 5 not necessarily Λdec and Λcen. Hence, Λcen may be replaced by some other configuration involving smaller clusters, thus simplifying the computations at expense of a certain loss of performance. Similarly, if the coupling conditions are such that Assumptions 1 or 3 do not hold for Λdec, then other topology can be used as sparsest viable configuration. Finally, note that it is possible to check offline if some topology involves clusters that do not satisfy these assumptions, and hence to remove it from the set of configurations that may be chosen. □ Remark 6: By using homothetic,elastic and polytopic tube-based approaches [36]–[38], conservatism in Control Scheme 1 can be reduced. Another possibility is introduced in [39], where the worst-case constraint tightening, as used in (17), is replaced by an optimization-based strategy where these constraints reduction depends on decision variables. Similarly, [40] proposes a tube-based DMPC where agents share data so as to reduce the size of the coupling uncertainty. □ Let SNpbe the domain of attraction of the initial topology Λini. Then, the following two theorems hold: Theorem 1 (Recursive feasibility): For any initial state xN∈ SNp, feasibility is guaranteed along the closed-loop trajectory of the system. □ Proof 1: Let us first consider that the topology remains fixed. In this case, recursive feasibility is proven below following [29]. In particular, consider instant k, let Λbe the topology, and let (x∗ C(0),u∗ C, xs∗ C, us∗ C, xt∗ C)be the solution of (17) for cluster C ∈ N/Λ, with x∗ C= [x∗ C(n)]Np−1 n=0 being the corresponding sequence of predicted states. Additionally, take ˜ u+ C=      ˜u+ C(0) . . . ˜u+ C(Np−2) ˜u+ C(Np−1)      =      u∗ C(1) . . . u∗ C(Np−1) us∗ C+KC(x∗ C(Np)−xs∗ C)      , ˜x+ C(0) = x∗ C(1),˜xs+ C=xs∗ C,˜us+ C=us∗ C,˜xt+ C=xt∗ C, (19) as a candidate solution for instant k+1. Then, the following holds: (i) From the notion of tubes [31], the coalition state at k+1 is such that x+ C∈x∗ C(1)⊕RC, and hence x+ C∈˜x+ C(0)⊕RC. (ii) Admissibility of u∗ Cimplies that inputs ˜u+ C(0 : Np−2) = u∗ C(1 : Np−1) are admissible. (iii) Variable ˜x+ C(Np−1), i.e., the predicted nominal state at k+ 1 for instant k+Np, is given by x∗ C(Np), hence, (˜x+ C(Np−1),˜xs+ C,˜us+ C)∈Ωa C,(20) and, therefore, ˜u+ C(Np−1) = us∗ C+KC(˜x+ C(Np−1)−xs∗ C) is also admissible. (iv) Finally, the invariance of Ωa Centails (˜x+ C(Np),˜xs+ C,˜us+ C)∈Ωa C.(21) Therefore, for all C ∈ N /Λ,(˜x+ C(0),˜ u+ C,˜xs+ C,˜us+ C,˜xt+ C)constitutes a feasible solution of (17) for instant k+ 1. Recursive feasibility for any static Λfollows by induction. On the other hand, the condition imposed on the switchings prevents unfeasibility problems when the clusters are changed. The latter is the only condition that needs to be checked online to guarantee feasibility. In particular, a transition to a new topology Λ+takes place iff the overall system state xNis such that xN∈ProjxΩa Λ+, i.e., xC∈ProjxΩa C, for all C ∈ N/Λ+. Hence, there exists at least one feasible artificial steady state and input (ˆxs+ C,ˆus+ C)such that (xC,ˆxs+ C,ˆus+ C)∈Ωa C,∀C ∈ N/Λ+.(22) Additionally, feasible inputs can be derived from (11), i.e., ˆ u+ C=ˆu+ C(n)Np−1 n=0 =ˆus+ C+KCxC(n)−ˆxs+ CNp−1 n=0 ,(23) where xC(n+ 1) = ACCxC(n) + BCC ˆuC(n)and xC(0) = xC. Therefore, the existence of a feasible initial state, a feasible sequence of inputs, and a feasible equilibrium point (xC,ˆ u+ C,ˆxs+ C,ˆus+ C)is guaranteed. Finally, note that variable xt Ccannot itself make problem (17) unfeasible. Considering the above and that the initial topology is Λini, recursive feasibility is guaranteed for all initial states within SNp.■ Theorem 2 (Stability): For any initial state xN∈ SNp, asymptotic stability is guaranteed. □ Proof 2: Firstly, we will prove that function (16) is a decreasing function when the network topology is static. Similarly, consider time instant k, let Λbe the network topology, and let (x∗ C(0),u∗ C, xs∗ C, us∗ C, xt∗ C)be the minimizer of (17) for cluster C ∈ N/Λ. Then, V∗ C=VC(x∗ C(0),u∗ C, xs∗ C, us∗ C, xt∗ C),(24) is its optimal (nominal) performance cost attained at k. Let us move one step forward and consider candidate solution (19). By optimality, ˜ V+ C=VC(˜x+ C(0),˜ u+ C,˜xs+ C,˜us+ C,˜xt+ C)provides an upper bound on the optimal cost at time instant k+ 1. In particular, we can write: ˜ V+ C=V∗ C−∥xC−xs∗ C∥2 QC−∥u∗ C(0) −us∗ C∥2 RC +∥˜x+ C(Np−1) −˜xs+ C∥2 QC+∥˜u+ C(Np−1) −˜us+ C∥2 RC −∥x∗ C(Np)−xs∗ C∥2 PC+∥˜x+ C(Np)−˜xs+ C∥2 PC, (25) where we have considered that the offset cost remains constant since xs∗ C= ˜xs+ Cand xt∗ C= ˜xt+ C. Also, the terminal cost satisfies ∥˜x+ C(Np−1) −˜xs+ C∥2 QC+∥˜u+ C(Np−1) −˜us+ C∥2 RC −∥x∗ C(Np)−xs∗ C∥2 PC+∥˜x+ C(Np)−˜xs+ C∥2 PC≤0.(26) Then, ˜ V+ C−V∗ C≤ −∥xC−xs∗ C∥2 QC−∥u∗ C(0)−us∗ C∥2 RC, and, hence, V∗+ C−V∗ C≤ −∥xC−xs∗ C∥2 QC.(27) Since (27) holds for all clusters C ∈ N /Λ, the overall optimal (nominal) costs, i.e., V∗ N=PC∈N /ΛV∗ C, satisfy V∗+ N−V∗ N≤ −∥xN−xs∗ N∥2 QN.(28) From (28), we have that the overall nominal state tends to the artificial steady state as time goes to infinity, and following similar arguments to Lemma 1 in [29], it can be proven that the artificial steady state converges to a fixed point, for any static network topology. On the other hand, the switchings of topologies, and hence the clusters modifications, may cause temporary increases of the optimal nominal costs. That is, condition (28) does not hold generally for all time steps. However, Control Scheme 1 guarantees that function V∗ Nsampled at the instants prior to the switchings always decreases with Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 14,2021 at 09:09:19 UTC from IEEE Xplore. Restrictions apply. 0018-9286 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2021.3133486, IEEE Transactions on Automatic Control SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 2021 6 time, which assures asymptotic stability of the overall nominal switching system. ■ From the theorems above, we can deduce that as time goes to infinity, function V∗ Ntends to a constant. Since the topology switchings are subject to a decrease of V∗ N, from a certain time instant the network configuration will remain static with some Λ∞. In this regard, given (28), the nominal system state at infinity, say xN,∞, converges to steady state xs∗ N,∞. Additionally, from the notion of tubes, it is guaranteed that the real state converges to a neighbourhood of the nominal system defined by RΛ∞, i.e., xN,∞∈xs∗ N,∞⊕ RΛ∞. Previous works on MPC for tracking [26], [29] prove that this final operating point xs∗ N,∞is defined as a minimizer of the offset cost function subject to the constraints in (17). In particular, let us define Xs C={xs C: (xs C, xs C, us C)∈Ωa C}as the set of admissible steady states consistent with the invariant set for tracking of cluster C. Under the assumption that ψC∩Xs C=∅for all C ∈ N/Λ∞, it is verified that xs∗ C,∞∈ψCfor all C ∈ N/Λ∞. Hence, from a global viewpoint, the following holds: xN,∞∈xs∗ N,∞+RΛ∞⊆ψN⊕ RΛ∞⊆ΓN,(29) where ψN=⊗C∈N /Λ∞ψC. V. SIMULATION RESULTS In this section, we apply the coalitional scheme to a modified version of the system proposed in [40], [41]. It consists of a set N={1,2, ..., 12}of trucks coupled via springs and dampers as shown in Fig. 2, with dynamics modeled by ˙si ˙vi=Aii si vi+Biiui+X j∈Ni Aij sj vj+di,(30) where Aii =  01 −1 miPj∈Nikij −1 miPj∈Nihij ,Bii =0 50, Aij =  0 0 1 miPj∈Nikij 1 miPj∈Nihij ,i=j. (31) The state xiof each subsystem is formed by its displacement sifrom the equilibrium point and its instantaneous velocity vi. Additionally, the agents can apply a longitudinal Fig. 2: 12-truck system and related communication network. TABLE I: Parameters used in the simulation Spring stiffnesses [N/m] and damping factors [N/(m×s)] k12, k45, k78, k1011 0.4 k23, k56, k89, k1112 0.8 k34, k67, k910 0.9 h12, h67, h910 0.2 h23, h45, h78, h1011 0.1 h34, h56, h89, h1112 0.15 Masses [kg] m1, m6, m8, m12 3 m2, m4, m7, m10 4 m3, m5, m9, m11 5 Other parameters Np0.5 s Ttop 0.8 s α25 c0.25 -1 0 1 -1 0 1 -4 -2 0 2 4 -1 0 1 -4 -2 0 2 4 Fig. 3: Evolution of the system to two target sets. The subsystems states at the switching instants are indicated by star marks, and the initial and final states are marked respectively with a circle and a diamond. The light blue and red regions represent the first target sets for the corresponding subsystems, and green and orange regions the second ones. Fig. 4: Topology evolution over time. It can be seen how the links are progressively disabled as the trucks approach the target sets. force Fi= 50uiN, where uiis the control action. The parameters that characterize the system are given in Table I. The continuous-time dynamics are discretized using zero-order hold and a sampling time of 0.1s. The goal is to regulate the four trucks towards a target set while allowing switchings on the network topology and satisfying |si| ≤ 4and |ui| ≤ 1. In this regard, the agents are connected by a network of 11 links as shown in Fig. 2. In addition, we have considered an external disturbance, which is defined as a bounded white noise as [−0.003,−0.03]T≤di≤[0.003,−0.03]T, for all i∈ N. Finally, the stage cost is defined by weighting matrices Qi= [1 0; 0 1.2] and Ri= 100, for all i∈ N. For the design of gains Ke Cand matrices PΛwe have followed [29] and [8], respectively. To this end, we have used the Matlab®LMI Control Toolbox [42] and specifically the solver mincx, in a 1.8 GHz quad-core Intel®CoreTM i7/8 GB RAM computer. On the other hand, feedback gains KCand matrices PChave been calculated using function dlqr from Matlab®. Sets RΛare designed following [43], where a one-step approach based on a linear program is proposed for computing minimal robust positively invariant sets. Finally, the augmented invariant sets for tracking have been computed using the Multi-Parametric Toolbox 3.0 [44], where for converge reasons a factor λ= 0.97 has Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 14,2021 at 09:09:19 UTC from IEEE Xplore. Restrictions apply. 0018-9286 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2021.3133486, IEEE Transactions on Automatic Control SUBMITTED TO IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 2021 7 been used (see [24]). Due to the high computational burden of operating with high-dimensional sets, we have limited the coalitions size to |C| ≤ 4. The latter pre-selects a subset of 1450 topologies among the 211 = 2048 combinations of enabled and disabled links, and allows to manage the system without requiring great efforts to compute the constraints sets of (17) for any coalition, at expense of limiting the maximum degree of coordination that can be attained. Nevertheless, notice that using a zonotope representation of the polytopes involved in (17) can facilitate the constraints sets computation for larger clusters. Likewise, the methodology in [29], which scales poorly for high dimensional disturbance sets, can be replaced by any other method providing matrices Ke Csatisfying Assumption 1. The results obtained from the simulation of Control Scheme 1 with the parameters in Table I and using as initial topology Λini the one shown in Fig. 4 are summarized below. Fig. 3 illustrates the subsystems’ state trajectories on planes (si, vi)with the coalitional controller, and Fig. 4 shows the network topologies used over time. The simulation lasts for 18s, with a change of target set at 9s. When the target is changed, topology Λini is imposed and Vis set to ∞. As shown in Fig. 4, when the system is in the vicinity of the target, a decentralized control suffices to drive and maintain the overall state within this region. The use of topologies with one or more enabled links reduces the coupling uncertainty of the merged agents in comparison to a static decentralized configuration, which leads to less restrictive constraints during part of the simulation (see the projections of sets XΛdec and XΛini onto plane (s9, v9)in Fig. 5). Note that the decentralized controller refers to the case in which Λdec is permanently imposed, that is, each local agent performs as the robust MPC for tracking introduced in [29], but considering that the disturbances account for both coupling effect and external perturbations. For the sake of comparison, we show the results obtained with the centralized MPC in [26] with perfect external disturbances forecast and without shrinking constraint sets XNand UN. Note that the latter provides a lower bound on the performance costs that can be attained and uses the centralized model (2) derived from (30) as prediction model. -4 -3 -2 -1 0 1 2 3 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fig. 5: Trajectory of truck 9 on plane (s9, v9)when using the decentralized, coalitional and centralized controllers. The figure also shows how the coupling uncertainty reduces the decentralized topology constraint set XΛdec in comparison with XΛini . Additionally, it shows that the projection of the augmented invariant set for tracking Ωa Λdec contains practically all points in XΛdec . 0 5 10 15 Time (s) 2000 4000 6000 8000 10000 12000 14000 16000 18000 Cumulative performane cost Decentralized Coalitional Centralized (disturb. perfectly estim.) 0 5 10 15 Time (s) 0 200 400 600 Offset cost (a) Performance cost. 0 5 10 15 Time (s) 0 100 200 300 400 500 Cumulative communication cost Decentralized Coalitional All links enabled (b) Communication costs. Fig. 6: Comparison between the centralized, coalitional and decentralized cumulative costs. In (a), the evolution of the offset cost is also shown, i.e., the value of αPi∈N ∥xs i−xt i∥∞attained each time instant. -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fig. 7: Invariant tube, nominal initial states, and real evolution of subsystem 9 when using the decentralized topology. For the sake of clarity, the tube is just plotted every eight time steps. A comparison of the overall performance and communication cumulative costs is provided in Fig. 6. In particular, the cumulative performance costs are measured through the following stage index: X i∈N ∥xi−xs i∥2 Qi+∥ui−us i∥2 Ri+α∥xs i−xt i∥∞,(32) while for the communication costs we use c|Λ|, being Λthe active topology each time step. Finally, Fig. 7 shows the invariant tube that characterizes the coupling uncertainty of subsystem 9when the decentralized topology is imposed. VI. CONCLUSIONS A robust tube-based MPC strategy for tracking target sets has been introduced in a coalitional MPC framework. In this context, the overall system is controlled through a set of local agents that dynamically merge into clusters to take coordinated decisions. The proposed formulation drives the global system state to the target set in an admissible way, while assuring recursive feasibility and stability regardless the topology switching. The use of the tracking strategy in conjunction with the coalitional approach allows enlarging the MPC controller domain of attraction without renouncing to the use of decentralized or sparse structures when they suffice for reaching the goals. This fact may be interesting for applications where communications between controllers can fail. In this regard, the dynamic formation of disconnected coalitions limits the information available to the local agents to compute the control Authorized licensed use limited to: Universidad de Sevilla. Downloaded on December 14,2021 at 09:09:19 UTC from IEEE Xplore. Restrictions apply. 0018-9286 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. 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