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Mitigation of communication failures in distributed model predictive control strategies

2018; Institution of Engineering and Technology; Volume: 12; Issue: 18 Linguagem: Inglês

10.1049/iet-cta.2018.5044

1751-8652

Wicak Ananduta, Julián Barreiro-Gomez, Carlos Ocampo‐Martínez, Nicanor Quijano,

Distributed Control Multi-Agent Systems

IET Control Theory & ApplicationsVolume 12, Issue 18 p. 2507-2515 Research ArticleFree Access Mitigation of communication failures in distributed model predictive control strategies Wicak Ananduta, Corresponding Author Wicak Ananduta wananduta@iri.upc.edu orcid.org/0000-0002-6305-1329 Automatic Control Department, Universitat Politècnica de Catalunya, Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Barcelona, SpainSearch for more papers by this authorJulian Barreiro-Gomez, Julian Barreiro-Gomez Learning & Game Theory Laboratory, New York University in Abu Dhabi (NYUAD), Saadiyat Campus PO BOX 129188, Abu Dhabi, UAESearch for more papers by this authorCarlos Ocampo-Martinez, Carlos Ocampo-Martinez Automatic Control Department, Universitat Politècnica de Catalunya, Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Barcelona, SpainSearch for more papers by this authorNicanor Quijano, Nicanor Quijano Departamento de Ingeniería Eléctrica y Electrónica, Universidad de los Andes, Bogotá, ColombiaSearch for more papers by this author Wicak Ananduta, Corresponding Author Wicak Ananduta wananduta@iri.upc.edu orcid.org/0000-0002-6305-1329 Automatic Control Department, Universitat Politècnica de Catalunya, Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Barcelona, SpainSearch for more papers by this authorJulian Barreiro-Gomez, Julian Barreiro-Gomez Learning & Game Theory Laboratory, New York University in Abu Dhabi (NYUAD), Saadiyat Campus PO BOX 129188, Abu Dhabi, UAESearch for more papers by this authorCarlos Ocampo-Martinez, Carlos Ocampo-Martinez Automatic Control Department, Universitat Politècnica de Catalunya, Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Barcelona, SpainSearch for more papers by this authorNicanor Quijano, Nicanor Quijano Departamento de Ingeniería Eléctrica y Electrónica, Universidad de los Andes, Bogotá, ColombiaSearch for more papers by this author First published: 15 October 2018 https://doi.org/10.1049/iet-cta.2018.5044Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Information sharing among local controllers is the key feature of any distributed model predictive control (DMPC) strategy. This study addresses the problem of communication failures in DMPC strategies and proposes a distributed solution to cope with them. The proposal consists in an information-exchange protocol that is based on distributed projection dynamics. By applying this protocol as a complementary plug-in to a DMPC strategy, the controllers improve the resilience against communication failures and relax the requirements of the communication network. Furthermore, a reconfiguration algorithm, which is a contingency procedure to maintain the connectivity of the network, and a discussion on the selection criteria of the information-sharing network are also presented. In order to demonstrate the performance and advantages of the proposed approach when it is applied to a DMPC strategy, a case study of a power-network control problem is provided. 1 Introduction Research in model predictive control (MPC) has been extensively done since the past decades (see e.g. [1–4] for a review). Such studies include the application of MPC to large-scale systems, i.e. systems that have a large number of inputs, states, and/or outputs. Since an MPC controller computes control inputs by solving an optimisation problem, the problem may become intractable when the system is too large [3]. Therefore, many works in the literature have proposed non-centralised MPC approaches, in which there are multiple local controllers as opposed to the centralised approach, which has a single central controller, in order to cope with such problem [3, 4]. Furthermore, non-centralised MPC approaches could also introduce additional benefits such as better scalability, flexibility, and safety than the centralised counterpart [4]. Non-centralised MPC approaches can be classified into two broad categories: decentralised and distributed approaches. The main difference between these two schemes is the existence of communication among local controllers, which can be found in the latter category [3]. The availability of communication plays an important role in the performance of distributed MPC (DMPC) approaches, which is better than the decentralised ones when the couplings between sub-systems are not weak [4]. In fact, in certain cases, some DMPC approaches are also able to achieve similar performance as the centralised counterpart [4]. Different DMPC approaches require different communication structure as well as different ways to exchange information among local controllers [3]. Some DMPC approaches require local controllers to share information iteratively while others require that the information exchange is made only once at each time instant. In terms of information-sharing network, some approaches, such as [5–7], require a neighbour-to-neighbour communication while other approaches, e.g. [8, 9], require all controllers to exchange information with all the others. In any way, the information-sharing network, through which the exchange of information occurs, is important for systems that use a DMPC approach. Some issues of an information-sharing network may arise during the operation of the associated system. Those issues include communication failures (total loss of communication links), delays, and data packet drops [10]. This paper focuses on the failures (loss of links) in the information-sharing network, in which some local controllers are no longer able to communicate with others. As mentioned in [10] and later discussed, communication failures may lead to severe problems such as the inability of the controllers to compute control inputs or the sub-optimality of the solutions. Some recent literature has addressed the problem of communication failures in a distributed control strategy, in particular DMPC. Li et al. [11] analyse the performance degradation of a DMPC strategy that is based on Nash optimality during such failures while assuming that the algorithm is convergent. Heidarinejad et al. [12] propose a scheme where the sub-systems assume that their neighbours take null control actions during communication failures. In [13], the authors develop a methodology to extend the DMPC strategy that is proposed in [14] such that it can cope with communication failures. The methodology involves substituting the coupled constraints with tube-based constraints that restrict the control inputs. Furthermore, Kumar et al. [15] proposed to add an observer for a robust DMPC strategy. Hence, during a communication failure, the state bounds are estimated by the observer and are posed as extra constraints into the DMPC design. Additionally, a resilient information-sharing network architecture for distributed frequency regulation is proposed in [16]. Moreover, the controller in [16] adopts a zero-bias control strategy and allows other sub-systems to stabilise themselves. The aforementioned contributions only improve some DMPC strategies to tackle communication failures in a way that they specifically add or modify the algorithms. Therefore, they limit the application of the solution only to the DMPC strategies that are discussed in those papers. In this regard, different from the works previously discussed, this paper proposes a communication protocol that can be applied regardless DMPC strategy that is used to control the system. The protocol is an iterative algorithm that requires local controllers to communicate at each iteration until the information that is received converges to the correct value. It is based on the distributed projection dynamics (DPD) [17, 18] and can also be perceived as a distributed consensus protocol [19]. For an extensive treatment of the notion of distributed population dynamics, the reader is referred to [17, 18]. In this paper, an application of the proposed approach is illustrated in a power-allocation problem of interconnected microgrids. These kinds of systems are of large nature and spread over large geographical areas. Furthermore, in the current development of these systems, the power generation is shifted towards distributed generation, particularly renewable energy sources [20]. For these reasons, distributed controllers have been viewed as suitable control approaches [20–23]. In this regard and highlighting the fact that power networks are considered to be a critical infrastructure, it is important to ensure the resiliency of the control approach against failures, in particular communication failures. In summary, the main contribution of this paper is the information-exchange protocol that is based on DPD, as presented in Section 3. This protocol can be applied to any DMPC strategy and improves the resilience of the DMPC strategy against communication failures as well as relaxes the communication requirements. Criteria to select a suitable information-sharing network for the proposed protocol and a network reconfiguration algorithm that supplements the protocol are also proposed. Moreover, prior to proposing the protocol, Section 2 provides an analysis of the impact of communication failures in a DMPC strategy. Furthermore, the advantages of the proposed protocol in a power allocation problem of microgrids are shown in Section 4. Notations The statements and imply A and B are positive semi-definite and positive definite, respectively. The set of real numbers is denoted by , whereas the set of integers is denoted by . Moreover, denotes all real numbers in the set { } and, similarly, denotes all integers in the set { }. In addition, is the cardinality operator and is the Euclidean norm. The vector denotes and, similarly, . The subscript denotes discrete-time instants while the subscript denotes continuous-time instants. For the vectors with , the operator denotes the column-wise concatenation that results in a column-wise vector. Moreover, consider a time-dependent vector and a prediction horizon . Then, denotes the trajectory of over the prediction horizon , i.e. . Finally, is the round operator that approximates the argument to the nearest integer. In the case there are two nearest integers, the argument is approximated to the larger integer. 2 Distributed MPC and communication failures In this section, a general description of the system that is considered and the DMPC approach are presented. Afterwards, a discussion of the impact of communication failures in a DMPC strategy is provided. 2.1 Distributed MPC Consider a time-invariant linear large-scale system (LSS) that consists of n sub-systems. Let it be described by an undirected graph , where the set of nodes, denotes its sub-systems and the set of unweighted links, , represents the existence of coupling between the sub-systems, i.e. implies that the i th sub-system is physically coupled to the j th sub-system. Furthermore, denotes the set of the neighbours of the i th sub-system, i.e. . Let the dynamics of this LSS be described by the discrete-time state-space model of each sub-system, i.e. for each (1)where the states and the control inputs the i th sub-system are denoted by the vectors and , respectively. Moreover, and are the state-space matrices with the appropriate sizes that define the dynamics in (1). Note that the dynamics of each sub-system might also depend on the states, inputs, and/or disturbances of the neighbouring sub-systems. However, for simplicity of the exposition in Section 2.2, such dynamical couplings are omitted. Instead, some constraints that couple the inputs will be introduced. Moreover, is undirected and unweighted since the interest of this paper is in the existence of communication, as in [13]. The control inputs are constrained by local polytopic constraints described by , for each , where and are constant matrix and vector, respectively. Furthermore, there also exist some hard coupled constraints on the control inputs that are described as , where , for all , and are constant matrices and vector, respectively. Moreover, , for all , denote the elements of input that appear in the coupled constraints. In other words, can be arranged such that , where denotes the local control inputs. Notice that the coupled constraints may be constraints in which the sub-systems share a limited common input source. Hence, the optimisation problem that should be solved in an MPC scheme at each time instant is (2a) (2b) (2c) (2d)for all , where the trajectory of control input sequence throughout at time instant k for each sub-system is denoted by . Notice that the cost function over is separable and it is defined as where , and it is assumed that Problem (2) is feasible. Moreover, , which denotes a convex terminal cost function and , which denotes a terminal set that is convex, compact, and contains the origin, are introduced and defined in order to guarantee stability [4, 6]. Therefore, since the cost function is strictly convex and by definition the constraints are also convex, Problem (2) is convex. In a DMPC scheme, sub-problems are derived by decomposing (2) and they are solved by local controllers assigned to the sub-systems. When solving the sub-problems, the local controllers need to share some information among each other. Therefore, a DMPC strategy requires an information-sharing network. In this regard, let the information-sharing network of system be described by an undirected connected graph , where the set shows how the local controllers share information in this network, i.e. the link infers that there exists a bidirectional information flow between the i th and j th sub-systems through a communication channel. Many distributed MPC methods, e.g. [5–7], require that local controllers share information with their neighbours in . This means that . Other methods, e.g. the method presented in [8, 9], require full information exchange, i.e. should be a complete graph. 2.2 Impacts of communication failures The availability of communication between sub-systems is the main feature of DMPC strategies. Therefore, this section is dedicated to show, by an example, what could happen when there are failures in the information-sharing network, i.e. at least one communication link fails for a certain time slot. During this period, the affected sub-systems cannot communicate with each other while employing a DMPC strategy. For this example, without loss of generality, consider a system that consists of two sub-systems, i.e. . Moreover, consider that the system has dynamics and constraints as described in the previous subsection, implying that Problem (2) must be solved by the DMPC controllers. Furthermore, let , for all , be the optimal solution of this problem. Now, a DMPC strategy that is based on dual decomposition is considered to be implemented and the control inputs obtained by this strategy are analysed. In principle, the DMPC strategies that are based on dual decomposition are iterative and use the concept of duality in convex optimisation [24]. The iterations, which are called the distributed dual ascent, are intended to optimise the Lagrange multipliers that are associated with the coupled constraints in order to solve its dual problem [25]. The solution of this algorithm converges to the global optimal value if the primal problem is convex and the cost function is strictly convex [25]. The iteration steps of the distributed dual ascent algorithm that is applied by the i th sub-system for problem (2) are as follows. The update of the decision at the iteration is (3)in which , for all and , are the Lagrange multipliers that are associated with the input-coupled constraints (2d). The update of the Lagrange multipliers is for all , where , . Note that all steps of the dual ascent algorithm can be done locally and each sub-system requires the information of the decision of its neighbourhood in order to update the Lagrange multipliers. When exchanging information, consider the information sharing network , where . During a communication failure, it is assumed that each sub-system uses the information of the neighbourhood from the previous time instant and considers the unknown information to be null, as in [12, 16]. This is precisely stated in Assumption 1. Assumption 1.Consider that the communication link fails. During the period of the failure, the i th sub-system considers the input trajectory of the the j th sub-system (), denoted by , as follows: and vice versa. Note that based on the structure of , , where and correspond to the local inputs and the coupled inputs, respectively. In this case, during a communication failure, both sub-system cannot communicate and receive the necessary information, which is the coupled inputs of their neighbour. It implies that the steps of updating the Lagrange multipliers for both sub-systems become (4)for all and , where . The modification of the dual-ascent algorithm implies that instead of solving Problem (2), the DMPC controllers solve the following optimisation problem: (5a) (5b)for all and . The difference between Problem (5) and Problem (2) is in the existence of coupled constrains (2d) in Problem (2), which are replaced by (5b). Notice that the inequalities (5b) are not coupled constraints since , for all and are known information (see Assumption 1). Hence, Problem (5) is separable by definition. An analysis regarding the performance of the controllers can be made by evaluating Problem (5). Problem (5) may be infeasible even though Problem (2) is feasible due to the changes of the constraints. In this case, the dual-ascent algorithm with the updating steps according to (3) and (4) will not obtain a solution. Suppose that Problem (5) is feasible. Since it is convex, the dual-ascent algorithm can find a solution. Consider that each sub-system obtains a solution , for all . Thus, must be examined, since this is the control input that will be applied to the system. The solution may be an infeasible solution for Problem (2), i.e. it violates some constraints of Problem (2). The DMPC scheme in [12], which apply Assumption 1 during failures, also assumes that null control input is a feasible solution. However, the latter assumption does not always hold in general. Furthermore, supposing that is feasible, if , then is the optimal solution. Otherwise, is a suboptimal solution. Thus, this DMPC strategy does not have a guarantee on the feasibility, let alone the stability and the optimality, during communication failures even though the problem is convex with a strictly convex cost function. Similar issues may also be found when the strategy is applied to a more complex and larger problem and in other DMPC strategies. Hence, increasing the resilience of communication infrastructure is important. Section 3 discusses a proposal to improve the resilience of the information-sharing network. 3 Information-exchange protocol based on distributed projection dynamics The proposal focuses on the methodology of sharing information, which is independent from the control strategy. Thus, it is a general and complementary methodology that can be applied to any DMPC method. 3.1 Description of the protocol Recall the information-sharing network as the graph , where shows how the local controllers are connected with each other in the information-sharing network. Furthermore, consider that some nodes (sub-systems) require information denoted by , from the v th node. Depending on the DMPC strategy applied to the system and the couplings in the system, may consist of either state or input information, which is required by some or all other nodes to run the DMPC algorithm. Therefore, there exists a sub-graph , where is the set that consists of node v and the nodes that require , while is the set of links that connect the nodes in . For instance, in the DMPC strategies that require neighbour-to-neighbour communication, some neighbours might be included in while the other neighbours are included in the other information-sharing subgraphs. In order to apply the protocol, the following assumptions must hold. Assumption 2.The undirected sub-graph is connected. Assumption 3.The node v, which sends the information, has prior knowledge of , while all know . Note that, following Assumption 2, it is possible that there are some nodes in that do not need the information , but they are required as intermediate nodes in order to ensure the connectivity of . The proposed information-exchange protocol is based on the following dynamics: (6a) (6b)where are the information state, the reference input, and the internal state of the i th node, respectively, is the set of nodes that are the neighbours of the i th node in , i.e. for all , and is a constant gain. Furthermore, the reference inputs of all nodes in are given by (7)and the internal states are initialised as (8)It is shown in (7) that the information is used as the reference input of the v th node. In the following subsection, the convergence of the protocol is discussed. Remark 1.The dynamics (6) are in the class of dynamic consensus, which tracks the average of the changing reference inputs. The reader is referred to [26] for further discussions on this topic. Remark 2.The vectors for all are dedicated only for the transmission of information . Thus, each local controller needs to allocate a different data storage for acquiring another information, i.e. if there are more than one source nodes. 3.2 Convergence of the protocol Theorem 1 states the convergence of the proposed information-exchange protocol. Theorem 1.Suppose that Assumptions 2 and 3 hold. Under the dynamics (6) and by providing the references as in (7) and the initial value of the auxiliary state as in (8), the equilibrium point of the information state, i.e. , for each , is asymptotically stable. Proof.First, notice that the dynamic equations (6a) correspond to DPD [17] with the fitness functions , for all , which are defined as . The protocol dynamics (6a) can be rewritten in the form of the DPD as (9)Note that , where , for each , is related to the information datum . Then, these dynamics can be rearranged into (10)where , , for all , and is the Laplacian of the graph . Hence, it is shown in (10) that the dynamics depend on the structure of the graph .Second, it is necessary to show that the set is invariant. This statement can be concluded by the fact that , i.e. , for all . Furthermore, by denoting the whole auxiliary states as and the equilibrium point as , where is the equilibrium point of the i th sub-system, and, based on Assumption 2, the equilibrium point can be characterised from (10), i.e.,since is the nullspace of when is connected.Now the solution of is derived from . Recall that is invariant. Due to the initialisation in (8), . Therefore, based on this fact and since implies for all , it is obtained that , which implies , for all and . Hence, the whole vectors , for all , have the following equilibrium point: (11)since according to (7) and Assumption 3, . Thus, by substituting into (6b) with the expression in (11), it is obtained that the equilibrium points of the information states are , for all .Finally, the convergence of the internal state is shown since it implies the convergence of to the equilibrium point . To this end, consider the radially unbounded Lyapunov function candidate: , where as . The function and for all . The time derivative of , i.e. Since is connected by assumption, . Hence, , for all . Furthermore, notice that equality only holds when , for all , i.e. at the equilibrium point . Then, applying the LaSalle-invariance principle, convergence to the equilibrium point is concluded [27]. Hence, , for all , are also asymptotically stable. □ Remark 3.The convergence rate of the protocol depends on the structure of the network. It can be seen from the representation of the protocol in (10), which is similar to the standard consensus algorithm formulation. The convergence rate of such protocol is indicated by the second smallest eigenvalue of the Laplacian of the graph [28]. Remark 4.Although the information states asymptotically converge to , for all , in practice, sufficiently similar information can be recovered in a finite time. 3.3 Advantages of the protocol Two main advantages of the DPD-based information-exchange protocol are highlighted as follows. First, it relaxes some assumptions that are required by most of DMPC strategies. Particularly, it relaxes the requirement of the information-sharing network topology. As discussed in Section 2.1, different DMPC strategies might have different requirements regarding the information-sharing network topology supposing that the information is exchanged directly as required. However, by employing the proposed protocol, these requirements are relaxed such that does not have to fulfil certain topological structure. Instead, only the connectivity of is necessary (Assumption 2). Second, this protocol also enhances the resiliency of DMPC-type controllers against communication failures. According to Assumption 2, the information can still be exchanged although some links of the network fail as long as the network is still connected. Therefore, to some extent of link failures, a DMPC strategy that uses the proposed protocol to exchange information can still be performed. The advantages provided by the protocol also come with some costs, which are extra computation and communication, in terms of the amount of data that is exchanged. This is due to the fact that all sub-systems should reach consensus by iteratively exchanging information and applying (6) before obtaining the correct information from their neighbours. Therefore, one must ensure that the total time to exchange information using the DPD-based protocol and to compute the control inputs is smaller than the sampling time of the controlled system. In practice, the satisfaction of this assumption depends on the system complexity, i.e. the instrumentation and the other hardware as well as the software, e.g. the optimisation solver. 3.4 Selecting the information-sharing graphs Now consider that the system has an information-sharing network that connects its sub-systems in a certain way. In this subsection, a discussion on how to choose the information-sharing sub-graph, , from the available information-sharing network is provided. Two criteria of selection are the resiliency of the network against communication failures and the convergence rate of the proposed method. It is a direct implication that, when the information-sharing graph has more links (edges) connecting the sub-systems, the chance that the graph is still connected when a failure occurs is higher. Furthermore, as stated in Remark 3, the second smallest eigenvalue of indicates the convergence rate of the protocol in the sense that a larger eigenvalue implies a faster convergence rate. As noted in [28], the second smallest eigenvalue of a sparse Laplacian is relatively small compared to a dense Laplacian. Hence, these criteria lead to the fact that the network should have as many links as possible. Redundancy is required when dealing with communication failures. Thus, when selecting an information-sharing graph, it is important to consider having redundant links in the information-sharing graph. For instance, a path is not a suitable structure since once a link is disconnected, the graph is disconnected. In that sense, a cyclic graph is more redundant because the proposed method could still be applied when one link of this graph is broken. Furthermore, it is obvious that a complete graph is the most suitable one. On the other hand, the topology of an LSS usually has a sparse Laplacian matrix due to the fact that a sub-system is usually only coupled with other closest sub-systems. This may imply its information-sharing network has sparse Laplacian as well. However, this network can be decomposed into some sub-graphs that do not have sparse Laplacian, which implies they may have faster convergence rate. By also considering the redundancy criterion, then one may be able to use a smaller yet redundant information-sharing graph. As an example, consider the information-sharing graph depicted in Fig. 1. Its Laplacian has the second smallest eigenvalue of 0.23. However, now consider its sub-graphs that are depicted in Fig. 2. Notice that the sub-graphs that are formed are cyclic, in order to satisfy the redundancy requirement. Among these sub-graphs, the smallest value of the Laplacian second smallest eigenvalues is 2.00. This means that the convergence rate of the proposed method is much faster by using the smaller sub-graphs as the information-sharing graphs among the sub-systems. However, the redundancy of the sub-graphs is not as good as the overall graph, . For instance, consider sub-graph (Fig. 2 c) and suppose that the links (7,9) and (6,7) are broken, then this sub-