Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 560702, 17 pages

http://dx.doi.org/10.1155/2015/560702

## Time-Varying Scheme for Noncentralized Model Predictive Control of Large-Scale Systems

^{1}Section of Railway Engineering, Delft University of Technology, Stevinweg 1, 2628 CN Delft, Netherlands^{2}Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Universitat Politècnica de Catalunya (UPC), Carrer Llorens i Artigas 4-6, 08028 Barcelona, Spain^{3}Departamento de Ingeniería de Sistemas y Automática, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain^{4}Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, Netherlands

Received 23 June 2015; Revised 17 August 2015; Accepted 19 August 2015

Academic Editor: Qingling Zhang

Copyright © 2015 Alfredo Núñez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The noncentralized model predictive control (NC-MPC) framework in this paper refers to any distributed, hierarchical, or decentralized model predictive controller (or a combination of them) the structure of which can change over time and the control actions of which are not obtained based on a centralized computation. Within this framework, we propose suitable online methods to decide which information is shared and how this information is used between the different local predictive controllers operating in a decentralized, distributed, and/or hierarchical way. Evaluating all the possible structures of the NC-MPC controller leads to a combinatorial optimization problem. Therefore, we also propose heuristic reduction methods, to keep the number of NC-MPC problems tractable to be solved. To show the benefits of the proposed framework, a case study of a set of coupled water tanks is presented.

#### 1. Introduction

During the last decades, there has been a notable increment in the size of the problems dealt by control engineers. Large-scale applications such as irrigation canals [1], transportation networks [2], urban water systems [3], or supply chains [4], among many others, are now within the scope of control theory due to the proliferation of noncentralized control techniques (see, e.g., the surveys [5, 6]). The basic idea behind these control schemes is the well-known* divide and conquer* principle. In this way, the control problem of a large-scale monolithic system is partitioned into several smaller control problems that are assigned to a set of local controllers or* agents*. A similar approach can be used to deal with the overall control problem that results from the interaction of several coupled independent dynamical systems that pursue different goals.

In the literature, most noncentralized schemes focus on the following scenarios: () the overall system is partitioned in such a way that the coupling between subsystems is weak and can be ignored; that is, the agents work in a decentralized fashion; () the coupling between the different subsystems demands coordination between the local controllers and, for this reason, a communication mechanism between the agents has to be provided. In the latter scenario, we say that the agents work in a distributed or in a hierarchial fashion. In general, distributed control schemes outperform the decentralized ones but at the price of a higher complexity from both a communication burden viewpoint and an algorithmic viewpoint. More recently, the evolution of the field has led to the development of control schemes in which the local controllers adopt a decentralized attitude when the coupling between the control tasks is low and a distributed approach when it is high. In other words, the coordination and communication structure are adapted to the coupling between the control tasks. As a result of this, the local controllers are separated dynamically into cooperative groups or* coalitions*. For example, in [7], the set of active constraints is used to modify the sets of cooperating agents; in [8, 9], the coupling structure of the plant is exploited to divide it into hierarchically coupled clusters; in [10, 11], the coalitional model predictive control (MPC) framework is used, where only the couplings with an important contribution to the overall system performance are considered. Finally, the aggregation of control nodes and the inclusion of constraints regarding the division of the benefits and costs derived from the cooperation is studied in [12].

In this work, we focus on a novel type of control schemes with time-varying communication topology, which presents several open research issues. In the first place, it is clear that in a large-scale application the control scheme cannot switch between all the possible network topologies [13, 14]. In fact, the problems derived from the resulting combinatorial explosion in this context are pointed out in several of the aforementioned works, for example, [10, 11]. How to decide on the most appropriate topology at a given time step is a difficult problem similar to that of system partitioning, for which there are relatively few results available in the literature (see, e.g., [15–18] and the references therein).

Another open issue is the optimal way to define hierarchies between local controllers [19]. Most distributed control schemes are simply based on peer-to-peer coordination, but there are also other alternatives; for example, there are schemes that implement a master-slave hierarchy in which the agents have to wait for their turn before calculating and implementing their control actions [5]. How to determine dynamically the best hierarchical relationships between the controllers is another open problem.

This work proposes a noncentralized MPC (NC-MPC) framework in which the overall system partition and the hierarchy relationship between the corresponding subsystems vary dynamically over time. The task of the NC-MPC controller is to identify the relevant regions (partitioning) and to assign to them more importance by changing the control structure. To achieve this, the amount of information exchanged between the controllers can be increased or the hierarchical level of those crucial regions/subsystems can be augmented. In particular, several possible control structures for the communication between subsystems are considered and the hierarchical control system implements the one that provides the best performance according to a set of given objectives. In this way, the control structure gains flexibility to increase its adaptability to the evolution of the system conditions and external variables. Specifically, in this paper we focus on large-scale systems in which there is a* flow* between or through the constitutive elements of the system. Water, traffic, electricity, logistic, and data networks are practical examples of this type of systems. In this context, flow is understood in the sense of movement of raw material/particles/matter related to the use or function of the system. For instance, in water networks, flow would correspond to the movement of water from point A to point B; in transportation systems, it would correspond to the movement of cars/trains/bikes within the network; in data networks, it would be related to the data packets moving within a given network.

The remainder of the paper is organized as follows. In Section 2, the control-oriented framework and a proposed partitioning method are presented. Section 3 presents the noncentralized model predictive control (NC-MPC) framework. Section 4 details the proposed rules to define the changes in the structure of the NC-MPC controller. Section 5 presents numerical results using an interconnected water tank system benchmark. Finally, the main conclusions of the paper and relevant lines for future research are given in Section 6.

#### 2. System Modelling

Given the complex nature of large-scale network systems (LSNS), from a control viewpoint it is preferable to work with control-oriented models [20, 21] that are accurate enough to capture the relevant dynamics but yet simple enough to reduce both complexity and computation burden [22].

##### 2.1. Control-Oriented Modelling Framework

In flow networks, an LSNS may be represented by a directed graph , where nodes in are compositional elements that characterize an attribute of the system [21]. This set is composed of storage elements, flow handling elements, sinks, and intersection nodes [20]. Likewise, the edge in the set indicates that the element is physically connected with the element (so there are variables from that have an influence over ).

Considering the volume as the state variable, the flow through handling elements as the controlled inputs, and flows to sinks as system disturbances, an LSNS may be generally described in a state-space form by the following linear discrete-time dynamic model:where , , and correspond to the states vector, the controlled input vector, and measured disturbances vector, respectively. Moreover, , , and are state-space system matrices for balances in storage elements, and , are matrices for static balances in nodes. Notice that there is no term in (1b) since it is supposed that all storage element outflows are controlled. Besides, is a zero vector. All vectors and matrices are dictated by the network topology. In general, states and control inputs are subject to constraints of the formwhere and are the resulting hyperboxes of the corresponding element constraints.

##### 2.2. Model Decomposition

Considering the control-oriented model (1a), (1b), when a particular partitioning methodology is applied, the resulting subsystems may be connected by* topological relations* and/or* information relations*. The former are related to the nature of the variables that different subsystems may share: states and/or control inputs. The latter are related to the information that the controllers of the corresponding subsystems might exchange.

The overall system (1a), (1b) is assumed to be decomposed in a set of nonoverlapping subsystems, which are output-decentralized and input-coupled. The model of the th subsystem , for , is stated as follows (considering the partitioning approach in [15], we assume that constraints including the state of subsystems are not coupled. The only cross-influence between subsystems is given by the established shared input variables): withwhere is the local state vector; is the local measurable disturbances vector; stands for the input vector that only affects the local dynamics; is the input vector decided by the th subsystem that affects both the local dynamics and the dynamics of the aggregated set of neighboring subsystems; and the set aggregates the neighboring subsystems whose inputs affect the th subsystem. The dimensions of the matrices in (3a), (3b), and (4) are stated in Table 1.