by R.R. Negenborn, J.M. Maestre

In this chapter the motivation for developing a comprehensive overview of distributed MPC techniques such as presented in this book is discussed. Understanding the wide range of techniques available becomes easier when a common structure and notation is adopted. Therefore, a list of questions is proposed that can be used to obtain a structured way in which such techniques can be described, and a preferred notation is suggested. This chapter concludes with an extensive categorization of the techniques described in this book, and compact representations of the properties of each individual technique. As such, this chapter serves as a starting point for further developing understanding of the various particularities of the different techniques.

by J.M. Maestre, D. Muñoz de la Peña, E.F. Camacho

In this chapter we propose a distributed model predictive control scheme based on agent negotiation. In particular, we consider the control of several subsystems coupled through the inputs by a set of independent agents that are able to communicate and we assume that each agent has access only to the model and the state of one of the subsystems. This implies that in order to take a cooperative decision, i.e. for the whole system, the agents must negotiate. At each sampling time, following a given protocol, agents make proposals to improve an initial feasible solution on behalf of their local cost function, state and model. These proposals are accepted if the global cost improves the cost corresponding to the current solution. In addition, we study the stability properties of the proposed distributed controller and provide precise conditions that guarantee that the closed-loop system is practically stable along with an optimization based controller and invariant design procedure.

by J.L. Nabais, R.R. Negenborn, R.B. Carmona-Benítez, L.F. Mendonça, M.A. Botto

Transportation networks are large scale complex systems spatially distributed whose objective is to deliver commodities at the agreed time and at the agreed location. These networks appear in different domain fields, such as communication, water distribution, traffic, logistics and transportation. A transportation network has at the macroscopic level storage capability (located in the nodes) and transport delay (along each connection) as main features. Operations management at transportation networks can be seen as a flow assignment problem. The problem dimension to solve grows exponentially with the number of existing commodities, nodes and connections. In this work we present a Hierarchical Model Predictive Control (H-MPC) architecture to determine flow assignments in transportation networks, while minimizing exogenous inputs effects. This approach has the capacity to keep track of commodity types while solving the flow assignment problem. A flow decomposition of the main system into subsystems is proposed to diminish the problem dimension to solve in each time step. Each subsystem is managed by a control agent. Control agents solve their problems in a hierarchical way, using a so-called push-pull flow perspective. Further problem dimension reduction is achieved using contracted projection sets. The framework proposed can be easily scaled to network topologies in which hundreds of commodities and connections are present.

by R.R. Negenborn

In this chapter we described two distributed MPC schemes for control of interconnected time-invariant discrete-time linear systems: a scheme with serial iterations, and a scheme with parallel iterations. Under the given assumptions, the schemes converge to a solution that a centralized controller would obtain. The schemes have originally been derived froman overall augmented Lagrange formulation in combination with either a block coordinate descent or the auxiliary problem principle. The chapter describes the characteristics of the type of system and control architecture for which the distributed MPC schemes can be used, as well as the actual steps of the schemes, availability of more theoretically oriented extensions, application oriented results, and emerging potential new applications.

by A. Ferramosca

In this chapter, a cooperative distributed MPC is presented. The main features of this control strategy are: constraints satisfaction; cooperation between agents to achieve an agreement; closed-loop stability that is always ensured, even in the case of just one iteration; achieved control actions that are plantwide Pareto optimal and equivalent to the centralized solution; Pareto optimality is achieved also in case of coupled constraints; a coordination layer is not needed. It is proved that cooperative MPC is a particular case of suboptimal MPC; exponential stability is then proved, based on exponential stability of suboptimal centralized MPC.

by A. Zafra-Cabeza and, J.M. Maestre

This chapter presents a hierarchical distributed model predictive control algorithm. Two levels in the problem optimization are presented. At the lower level, a distributed model predictive controller optimizes the operation of the plant manipulating the control variables in order to follow the set-points. The higher level implements a risk management strategy based on the execution of mitigation actions if risk occurrences are expected. In this way it is possible to take into account additional relevant information so that better results are achieved in the optimization of the system.

by M.D. Doan, T. Keviczky, and B. De Schutter

In this chapter we describe an iterative two-layer hierarchical approach to MPC of large-scale linear systems subject to coupled linear constraints. The algorithm uses constraint tightening and applies a primal-dual iterative averaging procedure to provide feasible solutions in every sampling step.  This helps overcome typical practical issues related to the asymptotic  convergence of dual decomposition based distributed MPC approaches.  Bounds on constraint violation and level of suboptimality are provided. The method can be applied to large-scale MPC problems that are feasible in the first sampling step and for which the Slater condition holds (i.e., there exists a solution that strictly satisfies the inequality constraints). Using this method, the controller can generate feasible solutions of the MPC problem even when the dual solution does not reach optimality, and closed-loop stability is also ensured using bounded suboptimality.

by J.M. Maestre, F.J. Muros, F. Fele, D. Munoz de la Pena, and E. F. Camacho

In this chapter we present a distributed scheme based on a team game for the particular case in which the system is controlled by two agents. The main features of the proposed scheme are the limited amount of global information that the agents share and the low communication burden that it requires. For this reason, this scheme is a good candidate to be  implemented in systems with reduced capabilities, for example in wireless sensor and actuator networks.

by R. Marti, D. Sarabia, and C. de Prada

This chapter presents a distributed coordinated control algorithm based on a hierarchical scheme for systems consisting of nonlinear subsystems coupled by input constraints: the botton layer is composed of several non-linear model predictive controllers (NMPC) working in parallel, and in a top layer, a price-driven coordination technique is used to coordinate these controllers. The price coordination problem is formulated as a feedback control law to fulfill the global constraints that affect all NMPC controllers. To illustrate this approach, the price-driven coordination method is used to control a four-tank process in a distributed manner and is compared with centralized and fully decentralized approaches.

by P. Giselsson, A. Rantzer

We consider distributed model predictive control (DMPC) where a sparse centralized optimization problem without a terminal cost or a terminal constraint set is solved in distributed fashion. Distribution of the optimization algorithm is enabled by dual decomposition. Gradient methods are usually used to solve the dual problem resulting from dual decomposition. However, gradient methods are known for their slow convergence rate, especially for ill-conditioned problems. This is not  desirable in DMPC where the amount of communication should be kept as low as possible. In this chapter, we present a distributed optimization algorithm applied to solve optimization problems arising in DMPC that has significantly better convergence rate than the classical gradient method. This improved convergence rate is achieved by using accelerated gradient methods instead of standard gradient methods and by in a well-defined manner, incorporating Hessian information into the gradient-iterations. We also present a stopping condition to the distributed optimization algorithm that ensures feasibility, stability and closed loop performance of the DMPC-scheme, without using a stabilizing terminal cost or terminal constraint set.