by Attila Kozma, Carlo Savorgnan and Moritz Diehl.

The distributed multiple shooting method is tailored for large scale optimal control problems with decoupled structure. It can be used as a fast and distributed solver for model predictive control subproblems. The algorithm may be regarded as a generalization of the standard multiple shooting method that decomposes the original large scale optimal control problem in both the time and spatial domain to obtain high parallelizability. In each iteration, the linearization of the original problem is calculated in a parallel manner, which is then solved by a centralized structure exploiting optimizer. We demonstrate the approach on a simple mechanical example of two coupled pendula.

by João M. Lemos and José M. Igreja.

This chapter describes D-SIORHC, a distributed MPC algorithm with stability constraints based on a game approach. This controller is designed for chained linear systems, in which each local subsystem interacts only with their neighbors. At the beginning of each sampling interval, each local controller agent computes the value of the corresponding manipulated variable in an iterative process where, in each iteration, it optimizes a quadratic cost by assuming that the neighbor controllers will use for their manipulated variables the value which they have computed in the previous iteration. Therefore, in a game theory framework, if this coordination procedure converges, a Nash equilibrium is reached. The use of linear plant models and the absence of inequality operational constraints allows to compute the manipulated variables in an explicit way, in each iteration of the coordination procedure, thereby reducing the computational load. This approach differs from other distributed MPC algorithms based on linear models in the inclusion of stability constraints in the local controllers that leads to a different control law. The controller usage is illustrated through its application to a water delivery canal.

by Mohamed Yacine Lamoudi, Mazen Alamir and Patrick Béguery.

In this chapter, a hierarchical model predictive control framework is presented for a network of subsystems that are submitted to general resource sharing constraints. The method is based on a primal decomposition of the centralized openloop optimization problem over several subsystems. A coordinator is responsible of adjusting the parameters of the problems that are to be solved by each subsystem. A distributed-in-time feature is combined with a bundle method at the coordination layer that enables to enhance the performance and the real-time implementability of the proposed approach. The scheme performance is assessed using a real-life energy coordination problem in a building involving 20 zones that have to share a limited amount of total power.

by Alessandro Casavola, Emanuele Garone and Francesco Tedesco.

The term Command Governor (CG) refers to a particular class of Model Predictive Control (MPC) strategies designed to manage the reference of a pre-compensated system in such a way that set-membership constraints on relevant system variables are not violated. More precisely, a CG unit is added to a primal compensated plant, which has supposedly been designed so as to exhibit stability and good tracking performance in the absence of constraints, and is in charge of modifying the prescribed reference signal whenever its direct application would lead to constraint violations. This chapter describes a distributed CG strategy for the supervision of a network of interconnected, possibly dynamically coupled, subsystems. Such an approach could be useful in situations where the use of a centralized coordination unit may be impracticable because requiring unrealistic or unavailable communication infrastructures.

Samira Roshany-Yamchi, Rudy R. Negenborn and Avi A. Cornelio

In this chapter, a new Nash-based distributed MPC method is proposed to control large-scale multi-rate systems with linear dynamics that are coupled via inputs. These systems are multi-rate systems in the sense that either output measurements or input updates are not available at certain sampling times. Such systems can arise when the number of sensors is less than the number of variables to be controlled or when measurements of outputs cannot be completed simultaneously because of applicational limitations. The multi-rate nature gives rise to a lack of information which will cause uncertainty in the system’s performance. To compensate for the information loss due to the multi-rate nature of the systems under study, a distributed Kalman filter is proposed to provide an optimal estimate of the missing information.

C. Ocampo-Martinez, V. Puig, J.M. Grosso and S. Montes-de-Oca

In this chapter, a multi-layer decentralized model predictive control (MLDMPC) approach is proposed and designed for its application to large-scale networked systems (LSNS). This approach is based on the periodic nature of the system disturbance and the availability of both static and dynamicmodels of the LSNS. Hence, the topology of the controller is structured in two layers. First, an upper layer is in charge of achieving the global objectives from a set O of control objectives given for the LSNS. This layer works with a sampling time Dt1, corresponding to the disturbances period. Second, a lower layer, with a sampling time Dt2, Dt1 >Dt2, is in charge of computing the references for the system actuators in order to satisfy the local objectives from the set of control objectives O. A system partitioning allows to establish a hierarchical flow of information between a set C of controllers designed based on model predictive control (MPC). Therefore, the whole proposed ML-DMPC strategy results in a centralized optimization problem for considering the global control objectives, followed of a decentralized scheme for reaching the local control objectives. The proposed approach is applied to a real case study: the water transport network of Barcelona (Spain). Results obtained with selected simulation scenarios show the effectiveness of the proposedML-DMPC strategy in terms of system modularity, reduced computational burden and, at the same time, the admissible loss of performance with respect to a centralized MPC (CMPC) strategy.

Ionela Prodan, Florin Stoican, Sorin Olaru, Cristina Stoica, Silviu-Iulian Niculescu

This chapter proposes a distributed approach for the resolution of a multi-agent problem under collision and obstacle avoidance conditions. Using hyperplane arrangements and mixed integer programming, we provide an efficient description of the feasible region verifying the avoidance constraints. We exploit geometric properties of hyperplane arrangements and adapt this description to the distributed scheme in order to provide an efficient Model Predictive Control (MPC) solution. Furthermore, we prove constraint validation for a hierarchical ordering of the agents.

Daniel Axehill and Anders Hansson

In this chapter parallel implementations of hybrid MPC will be discussed. Different methods for achieving parallelism at different levels of the algorithms will be surveyed. It will be seen that there are many possible ways of obtaining parallelism for hybrid MPC, and it is by no means clear which possibilities that should be utilized to achieve the best possible performance. To answer this question is a challenge for future research.

Ralph Hermans, Mircea Lazar and Andrej Jokic

We provide an almost decentralized solution to the problem of stabilizing a network of discrete-time nonlinear systems with coupled dynamics that are subject to local state/input constraints. By “almost decentralized” we mean that each local controller is allowed to use the states of neighboring systems for feedback, whereas it is not permitted to employ iterations between the systems in the network to compute the control action. The controller synthesis method used in this work is Lyapunov-based model predictive control. The closed-loop stability conditions are decentralized via a set of structured control Lyapunov functions (CLFs) for which the maximum over all the functions in the set is a CLF for the global network of systems. However, this does not necessarily imply that each function is a CLF for its corresponding subsystem. Additionally, a solution is provided for relaxing the temporal monotonicity of the network-wide CLF. For infinity-norm based structured CLFs and input-affine dynamics, we show that the decentralized MPC algorithm can be implemented by solving a single linear program in each network node. Two application examples are provided to illustrate the effectiveness of the developed theory and to show that the proposed method can perform as well as more complex distributed, iteration-based MPC algorithms.

Alexandra Grancharova and Tor Arne Johansen

A suboptimal approach to distributed Nonlinear Model Predictive Control (NMPC) for systems consisting of nonlinear subsystems with nonlinearly coupled dynamics subject to both state and input constraints is proposed. The approach applies a dynamic dual decomposition method to reformulate the original centralized NMPC problem into a distributed quasi-NMPC problem by linearization of the nonlinear system dynamics and taking into account the couplings between the subsystems. The developed approach is based entirely on distributed on-line optimization (by gradient iterations) and can be applied to large-scale nonlinear systems. The theoretical results related to the application of the distributedMPC approach to both linear and nonlinear systems are outlined and some simulation results are provided.