Time-varying partitioning for predictive control design: Density-games approach
Introduction
The increasing emergence of large-scale systems (LSSs), e.g., water distribution systems, smart grids, or traffic systems, have promoted the study of model predictive control (MPC) under non-centralized schemes. This fact is also motivated by two different aspects, i.e., communication issues (infrastructure) to collect and transmit data associated with the system states, and computational issues to determine the appropriate control inputs. Hence, the problem of obtaining non-centralized control formulations has become a relevant research topic. The process of making controllers non-centralized is normally addressed by dividing the whole system into m different sub-systems and by designing several local and smaller controllers. In [1], [2], [3], [4], and more recently in [5], a wide discussion related to the design of non-centralized MPC controllers is developed. Furthermore, there are several classifications within the non-centralized MPC controllers depending on their architecture and on how different local controllers share information to one another [4]. One of the non-centralized configurations corresponds to decentralized MPC controllers, where sub-systems might have a dynamical coupling among them. In the decentralized MPC architecture [6], there is a set of local MPC controllers (each one in charge of the control of a sub-system), which do not exchange information to one another. Therefore, in order to implement this control architecture, it is usually assumed that the dynamical coupling among sub-systems is weak, for which these non-centralized configurations have some limitations as studied in [7]. Therefore, the performance of the closed-loop system can be enhanced by considering that local MPC controllers can exchange information. Then, these local controllers should be coordinated to obtain a control input [8]. This modification adding available information among controllers leads to the distributed MPC (DMPC) architecture. There are two main aspects to design distributed optimization-based controllers, i.e., (i) the identification of the sub-systems composing the whole system that is desired to be controlled, and (ii) the appropriate coordination among the different local controllers associated with sub-systems. The former component consists in a partitioning problem making considerations related to dynamical coupling, existing coupled physical and/or operational constraints, and communication requirements; whereas the latter component requires the development of distributed optimization strategies given a communication structure.
This paper addresses both components involved into the design of distributed optimization-based controllers, i.e., system partitioning and distributed optimization. Regarding the former component, the system partitioning problem has gotten increasing importance in the automatic control community as systems become larger and more challenging, and as the requirements and desired closed-loop performance become more strict. Many partitioning proposals focus on specific dynamical systems, e.g., in thermal control [9], control of electric power distribution systems [10], [11], or on a particular control strategy, e.g., in decomposition structure of distributed predictive controllers [12], [13]. Regarding the second component for distributed controllers, this paper discusses the distributed optimization based on dynamic games, which have become a quite useful tool in the design of distributed controllers as it has been presented in [14], [15]. This paper focuses on the density-dependent games where variations of the population size is allowed, illustrating a situation in which death and birth, or reproduction rates, are considered as in [16], [17]. These dynamics have not been either deduced from a version of the general dynamics known as mean dynamics (Kolmogorov forward equation), and by imposing different rules on revision protocols, nor proposed in a distributed information-sharing fashion. The contribution of this paper is threefold, i.e.,
- (1)
First, a novel partitioning approach based on a non-directed graph representing information sharing inspired by the Kernighan-Lin algorithm [18] is presented.
- (2)
Secondly, it is proposed to extend the mean dynamics, which are used in the deduction of population dynamics [19], considering strategy-interaction constraints and a reproduction-rate parameter, i.e., the density-dependent mean dynamics with non-complete population-interaction structures. Then, the distributed density-dependent replicator, Smith, and projection dynamics are deduced. Afterwards, it is shown that these density dynamics may be used to solve distributed constrained optimization problems.
- (3)
As a third contribution, and taking advantage of the properties that density games have, a DMPC controller design is proposed based on the distributed DDPG, i.e., an algorithm for dynamical partitioning a communication structure and develop a distributed network control scheme is provided. This is made by means of the combination of the two previous mentioned contributions.
It is shown that the population-interaction structure can be modified dynamically along the time by adding conditions over the optimization problem constraints depending on the current system state, leading this fact to a time-varying information-sharing network for control purposes. The remainder of this paper is organized as follows. Section 2 presents the addressed problem statement showing a general scheme and the optimization problem corresponding to the distributed LSSs. Therefore, Sections 3 and 4 present the contribution of the paper, i.e., the distributed system partitioning algorithm, the distributed density-dependent population games with their features, and how a distributed constrained optimization problem can be solved by using the proposed game-theoretical approach. Section 5 introduces different control approaches that can be designed by using the proposed system partitioning and distributed density games. Then, in Section 6, a case study is introduced, the proposed approach is implemented and results are discussed. Finally, conclusions are drawn in Section 7.
Notation: Column vectors are denoted by bold style, e.g., p. Matrices are denoted by bold upper case, e.g., A. Differently, scalars are denoted by non-bold style, e.g., n. The sets are denoted by calligraphic upper case, e.g., . The norm ||x||Q is defined as ||x||Q = x⊤Qx. The function [·]+ = max(0, ·) is used to simplify the notation. The identity matrix of size n × n is denoted by , is the column vector with n unitary entries, i.e., , similarly 0n is the column vector with n null entries, i.e., , in addition, 0n×l is the matrix of null entries and dimension n × l, and diag(x) is the diagonal matrix of the vector x. Let A = [aij] and B = [bij] be matrices with the same dimension, i.e., , then let be the Hadamard product, i.e., [cij] = [aij][bij], for all i = {1, …, n} and j = {1, …, l}. If matrix A is positive (negative) semi-definite, it is denoted by A ⪰ 0 (A ⪯ 0). Throughout this paper, both continuous-time and discrete-time systems are treated. Continuous time is denoted by t and it is mostly omitted throughout the paper in order to simplify the notation. Moreover, denotes the derivative with respect to continuous time, i.e., . In contrast, denotes the discrete time.
Section snippets
Problem statement
Consider a state-space discrete-time system with a sampling time Δt and represented by the following model:where is the system state vector, is the vector of control inputs, denotes the vector of disturbances that affect the system, and the state-space matrices are given by , , and . The states and control inputs are subject to physical and operational constraints, which define feasible sets denoted by , and
Distributed LSS partitioning algorithm
Consider an information-sharing network whose topology is represented by an undirected connected graph , where represents the set of nodes associated to the control-strategy variables, and denoted the set of edges of representing the possible information sharing among nodes . Notice that the graph is undirected assuming that the edges represent bidirectional-information channels. Hence, A ∈ {0, 1}n×n denotes the adjacency matrix whose elements aij = 1
Density-dependent games
Throughout this section, a novel game-theoretical approach to solve optimization problems, which can deal with constraints and time-varying information-sharing networks, is presented. To this end, consider an undirected connected graph , where corresponds to the set of strategies in the population. The set of edges representing the information sharing and/or interaction among agents selecting different strategies is given by ; and A = [aij] denotes the n × n
Control approaches
Three alternative control approaches (three different Scenarios) may be designed by using the proposed methodology since the system partitioning algorithm and the consideration of time-varying information-sharing network can be combined in different ways (see the summary in Table 3).
Case study, results and discussion
The Barcelona Water Supply Network (BWSN) is an LSS composed by nx tanks, nu control inputs (valves and pumps), ns drinking water sources, and nd water demands as reported in [13]. State vector is associated to the volumes in tanks, the vector of control inputs is associated to the manipulated flows throughout valves and pumps, and the vector of disturbances is associated to the water-demanded flows. The corresponding discrete-time model is the one presented in (1) and its
Concluding remarks
A multi-objective partitioning procedure considering several aspects such as the amount of links connecting different partitions, the size of partitions, the distance among elements, and the importance of links has been presented in order to determine the appropriate partitions in an LSS. As one of the most relevant features of the proposed partitioning is that it can be performed in a distributed manner. Therefore, the DMPC controller based on DDPG is combined with the distributed partitioning
Acknowledgements
Julian Barreiro-Gomez gratefully acknowledges support from U.S. Air Force Office of Scientific Research under Grant Number FA9550-17-1-0259. The work of Carlos Ocampo-Martinez is partially supported by the project DEOCS (Ref. DPI2016-76493-C3-3-R) from the Spanish MINECO/FEDER. Authors also thank the second stage of the project Pavco-Mexichem-Colciencias.
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