Distributed algorithms for aggregative games of multiple heterogeneous Euler–Lagrange systems

doi:10.1016/j.automatica.2018.10.041

Automatica

Volume 99, January 2019, Pages 246-252

https://doi.org/10.1016/j.automatica.2018.10.041 Get rights and content

Abstract

In this paper, an aggregative game of Euler–Lagrange (EL) systems is investigated, where the cost functions of all players depend on not only their own decisions but also the aggregate of all decisions. Two distributed algorithms are designed for these heterogeneous EL players to reach the Nash equilibrium of aggregative games. By constructing suitable Lyapunov functions, the convergence of the two algorithms are analyzed. The first algorithm achieves globally exponential convergence without parameter uncertainty, and the other achieves globally asymptotic convergence, even in the presence of uncertain parameters. Numerical examples are given to illustrate the effectiveness of the methods.

Introduction

Aggregative games arise in a variety of applications, such as communication networks and smart grids (see Barrera and Garcia (2015) and Ye and Hu (2017)). In aggregative games, every player has its own cost function, depending on its decision variable and the aggregate of the decisions of all players. The aim of players is to seek the Nash equilibrium of the game to minimize their cost functions. To this end, many distributed algorithms have been developed. For instance, Gharesifard, Basar, and Dominguez-Garcia (2016) and Ye and Hu (2017) designed distributed consensus-based algorithms for unconstrained and box-constrained aggregative games, respectively. Paccagnan, Gentile, Parise, Kamgarpour, and Lygeros (2016) presented a distributed algorithm for quadratic aggregative games with affine coupling constraints. Besides, Liang et al., 2017a, Liang et al., 2017b exploited distributed continuous-time algorithms for constrained aggregative games with nonlinear aggregates. Moreover, Koshal, Nedić, and Shanbhag (2016) proposed distributed synchronous and asynchronous algorithms for aggregative games with time-varying and static undirected graphs, respectively. Furthermore, Deng and Nian (2018) provided distributed projection-based algorithms for aggregative games with weight-balanced digraphs. Additionally, Grammatico (2017) studied the dynamical control of aggregative games.

With the development of cyber–physical systems, distributed strategies united with physical systems have attracted more and more research attention in recent years, referring to Deng and Hong (2016), Deng, Liang, and Yu (2018), Wang, Deng, Ma, and Du (2017), Zhang, Deng, and Hong (2017) and Zhang,Papachristodoulou, and Li (2018). In particular, Euler–Lagrange (EL) systems have been extensively considered due to the flexibility and unification in the modeling and design for many systems such as mobile robots, spacecrafts, and autonomous vehicles (see Spong, Hutchinson, and Vidyasagar (2006) and references therein). For example, Deng and Hong (2016) and Zhang et al. (2017) studied the distributed optimization problems of EL systems, and Cai and Huang (2016) investigated the consensus problems of EL systems. However, to the best of our knowledge, few results about distributed Nash equilibrium seeking with EL systems have been reported. Moreover, existing distributed Nash equilibrium seeking algorithms, such as Gharesifard et al. (2016), Koshal et al. (2016), Liang et al., 2017a, Liang et al., 2017b, Paccagnan et al. (2016) and Ye and Hu (2017), cannot be applied to the problem directly without further integrating some control of EL dynamics. These observations motivate us to study aggregative games of EL systems.

The objective of this paper is to investigate the aggregative games of multiple heterogeneous nonlinear EL systems and design distributed algorithms for these EL players to autonomously seek the Nash equilibrium of the game. The contributions of this paper are summarized as:

(i)
We formulate an aggregative game of multiple heterogeneous EL systems. The problem can be viewed as extensions of the aggregative games discussed in Ye and Hu (2017) by adding EL dynamics and the distributed optimization of EL systems studied in Deng and Hong (2016), Wang et al. (2017) and Zhang et al. (2017) by considering aggregative games.
(ii)
We firstly consider the case of EL systems with accurate parameters. Based on feedback linearization, we design a distributed algorithm for the case, under which EL players exponentially converge to the Nash equilibrium of aggregative games. Then we consider the case of EL players with parameter uncertainty. With tracking control idea, we develop another distributed algorithm. The algorithm achieves asymptotic convergence to the Nash equilibrium, though the accurate parameters of EL systems are unknown.

The organization of the paper is as follows. In Section 2, preliminaries are introduced and the considered problem is formulated. In Section 3, two distributed Nash equilibrium seeking algorithms are designed, and their convergence is analyzed. In Section 4, simulation examples are presented. Finally, in Section 5, the conclusion is given.

Notations: $R$ is the set of real numbers. $R^{n}$ presents the $n$ -dimensional Euclidean space. $\otimes$ and $‖ \cdot ‖$ denote the Kronecker product and the standard Euclidean norm, respectively. $X^{T}$ and $‖ X ‖$ are the transpose and the spectral norm of matrix $X$ , respectively. $I_{n}$ is a $n \times n$ identity matrix. $x_{i}$ is the $i$ th element of vector $x$ , and $c o l (x_{1}, \dots, x_{n}) = {[x_{1}^{T}, \dots, x_{n}^{T}]}^{T}$ . $1_{n}$ and $0_{n}$ are the column vectors of $n$ ones and zeros, respectively. $0$ denotes a matrix of all $0$ s with appropriate dimensions.

Section snippets

Preliminaries

Consider a weighted undirected graph $G ≔ {V, E, A}$ , where $V = {1, \dots, N}$ is the node set, $E \in V \times V$ is the edge set, and $A$ is the adjacency matrix. An edge of $G$ is denoted by a pair of nodes $(i, j) \subset E$ if $j$ is a neighbor of $i$ . The adjacency matrix is defined by $A ≔ {[a_{i j}]}_{N \times N}$ with $a_{i j}$ being the weighting of $(i, j)$ , where $a_{i j} = a_{j i} > 0$ if $(i, j) \subset E$ , and $a_{i j} = 0$ , otherwise. Moreover, $a_{i i} = 0$ for all $i \in V$ . The degree of node $i$ is $\deg_{i} = \sum_{j = 1}^{N} a_{i j}$ . The Laplacian matrix of $G$ is $L = D - A$ with $D = d i a g {\deg_{1}, \dots \deg_{N}}$ . Obviously, $L 1_{N} = 0$ . An

Main results

In this section, two distributed Nash equilibrium seeking algorithms are developed for the multi-agent system (4) without and with uncertain parameters, respectively. Then their convergence are analyzed.

Simulations

In this section, numerical examples are given to illustrate our results.

In electricity market, the competition among distributed energy resources can be described by aggregative games (see Hobbs and Pang (2007) and Liu et al. (2018)). Consider an aggregative game with six generation systems, whose communication topology is an undirected ring graph. The cost function of the generation system $i \in V$ is $J_{i} (P_{i}, P_{- i}) = c_{i} (P_{i}) - p (σ) P_{i},$ where $P_{i} \in R$ is the output power of the generation system $i$ , in $p . u .$ , $P_{- i} = c$

Conclusions

The aggregative games of multi-agent systems have been studied in this paper, where the dynamics of all players is described by nonlinear EL equations. Based on the estimation for the aggregate of the decisions of all players, two distributed algorithms have been proposed for these heterogeneous EL players to seek the Nash equilibrium of the game, and their convergence have been analyzed via Lyapunov stability theory. One of the two algorithms, which is dependent on system parameters, can

Zhenhua Deng received the B.S. degree in automation from Dalian Maritime University, Dalian, China, the M.S. degrees in control science and engineering from Central South University, Changsha, China and the Ph.D. degree in operational research and cybernetics from University of Chinese Academy of Sciences, Beijing, China, in 2011, 2014 and 2017, respectively.

He is currently a Lecturer with Central South University. His research interests include multi-agent systems, distributed optimization,

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He is currently a Lecturer with Central South University. His research interests include multi-agent systems, distributed optimization, game theory, smart grids, and control of electrical machines.

He was a recipient of the outstanding student paper award at the 2015 Chinese Intelligent Systems Conference.

Shu Liang received the B.S. degree in automatic control and the Ph.D. degree in engineering from the University of Science and Technology of China, Hefei, China, in 2010 and 2015, respectively.

From 2015 to 2017, he was a Post-Doctoral Fellow with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. He is currently a Lecturer with the University of Science and Technology Beijing. His research interests include nonsmooth systems and control, distributed optimizations, game theory, and fractional order systems.

^☆: This work was supported by the National Key Research and Development Program of China (2016YFB0901902), NSFC, China (61733018, 61333001, 61573344, 61803385), Fundamental Research Funds for the China Central Universities of USTB (FRF-TP-17-088A1). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Vijay Gupta under the direction of Editor Christos G. Cassandras.

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Brief paperDistributed algorithms for aggregative games of multiple heterogeneous Euler–Lagrange systems☆

Abstract

Introduction

Section snippets

Preliminaries

Main results

Simulations

Conclusions

Automatica

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Brief paper
Distributed algorithms for aggregative games of multiple heterogeneous Euler–Lagrange systems☆