Automatica 99 (2019) 246–252

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Automatica
journal homepage: www.elsevier.com/locate/automatica

Brief paper

Distributed algorithms for aggregative games of multiple
heterogeneous Euler–Lagrange systems✩
Zhenhua Deng a, *, Shu Liang b
a
b

School of Information Science and Engineering, Central South University, Changsha, 410075, China
School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China

article

info

Article history:
Received 9 November 2017
Received in revised form 2 July 2018
Accepted 18 September 2018
Available online 13 November 2018
Keywords:
Aggregative games
Nash equilibrium
Cyber–physical systems
Euler–Lagrange systems

a b s t r a c t
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.
© 2018 Elsevier Ltd. All rights reserved.

1. 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 consensusbased 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,
Yi, and Hong (2017a, 2017b) exploited distributed continuoustime 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
✩ 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.
Corresponding author.
E-mail addresses: zhdeng@amss.ac.cn (Z. Deng), sliang@amss.ac.cn (S. Liang).

*

https://doi.org/10.1016/j.automatica.2018.10.041
0005-1098/© 2018 Elsevier Ltd. All rights reserved.

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, 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:

Z. Deng, S. Liang / Automatica 99 (2019) 246–252

(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. Rn presents the ndimensional Euclidean space. ⊗ and ∥·∥ 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.
In is a n × n identity matrix. xi is the ith element of vector x, and
col(x1 , . . . , xn ) = [xT1 , . . . , xTn ]T . 1n and 0n are the column vectors
of n ones and zeros, respectively. 0 denotes a matrix of all 0s with
appropriate dimensions.

∑N
1

247

, where ϕi (qi ) : Rm → Rn is a linear function
for the local contribution to the aggregate and q := col(q1 , . . . , qN )
is the strategy profile of the game. The aggregate function σ specifies the cost function as Ji (qi , q−i ) = ϑi (qi , σ (q)), ∀ i ∈ V , where
ϑi (·, ·) : Rm × Rn → R. To be strict, player i faces the following

σ (q) := N

ϕ

i=1 i (qi )

parameterized optimization problem:
min Ji (qi , q−i ).

(1)

qi ∈Rm

The Nash equilibrium of the game is defined as follows (referring to Liang et al. (2017a, 2017b), Paccagnan et al. (2016) and Ye
and Hu (2017)).
Definition 1. A strategy profile q∗ := col(q∗1 , . . . , q∗N ) is said to
be a Nash equilibrium of the aggregative game (1) if Ji (q∗i , q∗−i ) ≤
Ji (qi , q∗−i ), ∀ qi ∈ Rm , i ∈ V .
By Definition 1, a Nash equilibrium is a strategy profile on
which the cost of player i is not decreased if this player changes
its decision unilaterally.
In preparation for our development, define following maps for
our game.
Fi (qi , σ ) := ∇qi Ji (qi , q−i ),

(2a)

Gi (qi , yi ) := Fi (qi , σ )|σ =yi ,

(2b)

Φε (q, y) :=

G(q, y)
,
ε (y − ϕ (q))

[

]

(2c)

2. Preliminaries and formulation

where G(q, y) = col(G1 (q1 , y1 ), . . . , GN (qN , yN )), y = col(y1 , . . . ,
yN ) ∈ RNn , ϕ (q) = col(ϕ1 (q1 ), . . . , ϕN (qN )), and ε > 0. Obviously,
Gi (qi , yi ) = Fi (qi , σ ) if yi = σ .
In addition, the following assumptions were used in aggregative games (e.g., Koshal et al. (2016), Liang et al. (2017a, 2017b),
Paccagnan et al. (2016) and Ye and Hu (2017)).

2.1. Preliminaries

Assumption 1. The undirected graph G is connected.

Consider a weighted undirected graph G := {V , E , A}, where
V = {1, . . . , N } is the node set, E ∈ V × V is the edge set, and
A is the adjacency matrix. An edge of G is denoted by a pair of
nodes (i, j) ⊂ E if[ j is
] a neighbor of i. The adjacency matrix is
defined by A := aij N ×N with aij being the weighting of (i, j),

Assumption 2. The cost function Ji (qi , q−i ) is continuously differentiable in q and convex in qi for every fixed q−i and i ∈ V .

where aij = aji > 0 if (i, j) ⊂ E , and aij = 0, otherwise. Moreover,
∑N
aii = 0 for all i ∈ V . The degree of node i is degi =
j=1 aij . The
Laplacian matrix of G is L = D − A with D = diag {deg1 , . . . degN }.
Obviously, L1N = 0. An undirected graph is connected if, for every
pair of nodes, there is a path that has the two nodes as its end nodes.
The eigenvalues of L are denoted by λ1 , . . . λN with λi ≤ λj for
i ≤ j. As shown in Godsil and Royle (2001), an undirected graph is
connected if and only if λ2 > 0. More details of graph theory can
be found in Godsil and Royle (2001).
The following definitions are given in Rockafellar and Wets
(2004). A function f : Rm → R is convex if f (α x + (1 − α )y) ≤
α f (x) + (1 − α )f (y), ∀ x, y ∈ Rm , ∀ α ∈ [0, 1]. A function f : Rm →
Rm is ω-strongly monotone (ω > 0) on Rm if (x − y)T (f (x) − f (y)) ≥
ω ∥x − y∥2 , ∀ x, y ∈ Rm . A function f : Rm → Rm is θ -Lipschitz
(θ > 0) on Rm if ∥f (x) − f (y)∥ ≤ θ ∥x − y∥ , ∀ x, y ∈ Rm .
2.2. Problem formulation
We consider an aggregative game with N players. The communication topology among players is described by an undirected
graph G with the node set V = {1, . . . , N }. Player i ∈ V has a cost
function Ji (qi , q−i ) : RNm → R, where qi ∈ Rm is the decision
of player i and q−i = col(q1 , . . . , qi−1 , qi+1 , . . . , qN ). The objective
of player i is to minimize its cost function Ji (qi , q−i ) by choosing
an appropriate decision qi . The aggregate function of the game is

Assumption 3. The map Φε is ω-strongly monotone and θ Lipschitz for some ε > 0 on RN(m+n) .
Remark 1. Φε is strongly monotone if and only if the Jacobian
matrix J Φε is uniformly positive definite (see Facchinei and Pang
(2003)). Obviously, Assumption 3 is satisfied in the aggregative
games studied in Liang et al. (2017b), Paccagnan et al. (2016) and
Ye and Hu (2017) as well as the well-known Nash–Cournot games
presented in Koshal et al. (2016).
Define F (q) as:
F (q) := col(∇q1 J1 (q1 , q−1 ), . . . , ∇qN JN (qN , q−N )).
Clearly, F (q) = G(q, σ ). Besides, Assumption 3 implies that F (q) has
the following properties.
Lemma 1. Under Assumption 3, F (q) is strongly monotone and
Lipschitz continuous in q on RNm .
Proof. Without loss of generality, let (σ ∈ R for simplicity.)Choose
y(q) = σ (q)1N , i.e., y(q) = N1 col 1TN ϕ (q), . . . , 1TN ϕ (q) . Then

[

F (q)

]

1TN ε (y(q) − ϕ (q)) = 0 and Φε (q, y(q)) = ε(y(q) − ϕ (q)) . Further, we
have

⟨

[
]⟩
q′ − q
Φε (q′ , y(q′ )) − Φε (q, y(q)),
′
y(q ) − y(q)
⟨ ′
⟩
′
= F (q ) − F (q), q − q ,

248

Z. Deng, S. Liang / Automatica 99 (2019) 246–252

which, together with the strong monotonicity of Φε , indicates the
strong monotonicity of F (q).
On the other hand, it is derived from the Lipschitz continuity of
Φε that F (q) is Lipschitz continuous in q. □
Besides, we have the following lemma about the Nash equilibrium of the aggregative game (1).

designed as follows.

τi =gi (qi ) + Ci (qi , q̇i )q̇i
− kMi (qi )q̇i − Mi (qi )Gi (qi , yi ),
ẏi =vyi ,
v̇yi = − kvyi − ε (yi − ϕi (qi ))

Lemma 2. Under Assumption 2, q∗ = col(q∗1 , . . . , q∗N } is the Nash
equilibrium of the aggregative game (1) if and only if

∇qi Ji (q∗i , q∗−i ) = 0m , i ∈ V .

(3)

Proof. From Facchinei and Kanzow (2010, Theorem 3.9 and Theorem 4.8), the solution of the variational inequality VI(RNm , F ), satisfying (3), coincides with the Nash equilibrium of the aggregative
game (1). □
Remark 2. Based on (Facchinei & Pang, 2003, Theorem 2.2.3),
Assumptions 2 and 3 indicate that the Nash equilibrium of the
aggregative game (1) is unique. Here we consider the case of 0 ≤
∥q∗ ∥ < +∞.
On the other hand, the dynamics of player i ∈ V is described by
the following EL equation.
Mi (qi )q̈i + Ci (qi , q̇i )q̇i + gi (qi ) = τi ,

(4)

where Mi (qi ) ∈ Rm×m is the positive definite inertia matrix;
Ci (qi , q̇i )q̇i ∈ Rm is the Coriolis and centripetal force vector;
gi (qi ) ∈ Rm represents the gravitational effect; and τi ∈ Rm is the
control input. It is well-known that EL systems have the following
properties (Spong et al., 2006):
(i) Ṁi (qi ) − 2Ci (qi , q̇i ) is skew symmetric.
(ii) For any x, y ∈ Rm , Mi (qi )x + Ci (qi , q̇i )y + gi (qi ) =
Ωi (qi , q̇i , x, y)
ψi , where Ωi (qi , q̇i , x, y) ∈ Rm×p is a known regression
matrix and ψi ∈ Rp is a constant vector consisted of the
uncertain parameters of (4).
Our task is to design distributed algorithms for EL players with
dynamics (4) to achieve the Nash equilibrium of the aggregative
game (1).
Remark 3. Different from the well-studied aggregative games
(Gharesifard et al., 2016; Koshal et al., 2016; Liang et al., 2017a, b;
Paccagnan et al., 2016; Ye & Hu, 2017), our formulation involves the
EL dynamics for every player, though the considered aggregative
game is static. Without considering physical systems, most of
distributed Nash equilibrium seeking algorithms cannot be applied
to our problem. Besides, the nonlinearity of EL systems and cost
functions lead to the difficulty in algorithm design and analysis.
3. 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.

−

N
∑

aij (yi − yj ) −

N
∑

j=1

żi =

N
∑

(5a)
(5b)

aij (zi − zj ),

(5c)

aij (vyi − vyj ),

(5d)

j=1

aij (yi − yj ) +

j=1

N
∑
j=1

where,
k>

1
θ2
+ λN + 1 .
ω
4

(6)

The algorithm (5) consists of two parts: game part (5a) and estimation part (5b)–(5d). To be exact, there are three items in (5a):
gi (qi ) + Ci (qi , q̇i )q̇i for nonlinear elimination, kMi (qi )q̇i for damping,
and Mi (qi )Gi (qi , yi ) for game. In estimation part (5b)–(5d), yi is the
estimation of σ (q), and vyi , zi are auxiliary variables for ensuring
exact estimation.
Let x̃ = col(q, y), ṽ = col(q̇, vy ), z = col(z1 , . . . , zN ), q̇ =
col(q̇1 , . . . , q̇N ), and vy = col(vy1 , . . . , vyN ). Then, by combining (4)
and (5), we have

⎧
⎨ x̃˙ = ṽ,
ṽ˙ = −kṽ − φε (x̃) − H1 x̃ − H2 z ,
⎩
ż = H3 (x̃ + ṽ ),

(7)

where,

[
H1 =

0
0

0
L ⊗ In

]

, H2 =

[

0

L ⊗ In

]

[
, H3 = 0

L ⊗ In .

]

We have the following lemma about the equilibrium point of
(7).
Lemma 3. Under Assumptions 1 and 2 , if (q∗ , y∗ , vy∗ , z ∗ ) is an
equilibrium point of (7), q∗ is a Nash equilibrium of the aggregative
game (1). Conversely, if q∗ is a Nash equilibrium of the aggregative
game (1), there exist y∗ ∈ RNn , vy∗ ∈ RNn and z ∗ ∈ RNn such that
(q∗ , y∗ , vy∗ , z ∗ ) is an equilibrium point of (7).
Proof. See Appendix A. □
Lemma 3 reveals that the multi-agent system (4) with the
algorithm (5) approaches the Nash equilibrium of the aggregative
game (1) if (7) converges to its equilibrium points. By analyzing the
convergence of (7), the following result is obtained.
Theorem 1. Under Assumptions 1–3, the multi-agent system (4) with
the algorithm (5) exponentially converges to the Nash equilibrium of
the aggregative game (1).
Proof. See Appendix B. □
Remark 4. The algorithm (5) is designed via feedback linearization,
where the accurate parameters of EL systems are used. Sequentially, the algorithm (5) achieves exponential convergence.

3.1. EL players without uncertain parameters
3.2. EL players with uncertain parameters
In this subsection, we consider the case that the parameters of
EL players are available.
Since EL systems are nonlinear second-order systems, the distributed Nash equilibrium seeking algorithm for player i ∈ V is

In this subsection, a distributed Nash equilibrium seeking algorithm is developed for the EL multi-agent system (4) with uncertain parameters.

Z. Deng, S. Liang / Automatica 99 (2019) 246–252

Before giving the algorithm, the following lemma is presented.
Lemma 4. Consider the following second-order multi-agent system.
ẍi = ui ,

i ∈ V,

(8)

where xi ∈ Rm and ui ∈ Rm are the output and the control input
of player i, respectively. Suppose Assumptions 1–3 hold, and then
the outputs of multi-agent system (8) under the following algorithm
exponentially converge to the Nash equilibrium of the aggregative
game (1).
ui = − kẋi − Gi (xi , yi ),

(9a)

ẏi =vyi ,

(9b)

v̇yi = − kvyi − ε(yi − ϕi (xi ))
−

N
∑

aij (yi − yj ) −

N
∑

żi =

N
∑

aij (zi − zj ),

(9c)

aij (vyi − vyj ),

(9d)

j=1

j=1

aij (yi − yj ) +

j=1

N
∑
j=1

where i ∈ V and k satisfies (6).
Proof. The proof of this lemma is similar to that of Theorem 1, and
here we omit it. □
Lemma 4 shows that the second-order multi-agent system
(8) with the controller (9) can seek the Nash equilibrium of the
aggregative game (1). If there is a controller for the multi-agent
system (4) to track the second-order multi-agent system (8) with
the controller (9), the EL players approach the Nash equilibrium.
Therefore, in what follows, a tracking controller is designed for the
multi-agent system (4) to track the path generated by the secondorder multi-agent system (8).
Define the following auxiliary variables as done in Spong et al.
(2006):
q̂˙ i := ẋi − (qi − xi ), i ∈ V ,

(10)

which can be viewed as the estimation of ẋi . Then we have
q̂¨ i = ui − (q̇i − ẋi ).

(11)

Based on Property (ii) of EL systems, there are a known matrix

Ωi = Ωi (qi , q̇i , q̂˙ i , q̂¨ i ) ∈ Rm×p and an unknown constant vector
ψi ∈ Rp such that
Mi (qi )q̂¨ i + Ci (qi , q̇i )q̂˙ i + gi (qi ) = Ωi ψi .

(12)

The tracking error between EL player i and its second-order system
(8) is defined as
ei := q̇i − q̂˙ i = (q̇i − ẋi ) + (qi − xi ).
The distributed Nash equilibrium seeking algorithm for the
multi-agent system (4) is described as follows.

τi = − ei + Ωi ξi ,

(13a)

ξ̇i = − ΩiT ei ,
ei =(q̇i − ẋi ) + (qi − xi ),
ẍi = − kẋi − Gi (xi , yi ),
ẏi =vyi ,
v̇yi = − kvyi − ε (yi − ϕi (xi ))

(13b)

−

N
∑

aij (yi − yj ) −

N
∑

N
∑
j=1

aij (yi − yj ) +

(13d)
(13e)

aij (zi − zj ),

(13f)

aij (vyi − vyj ),

(13g)

j=1

j=1

żi =

(13c)

N
∑
j=1

249

where k satisfies (6) and ξi is used to estimate ψi . The algorithm (13) consists of tracking part (13a)–(13c) and game part
(13d)–(13g).
The following result is about the multi-agent system (4) with
the algorithm (13).
Theorem 2. Under Assumptions 1–3, the multi-agent system (4) with
the algorithm (13) asymptotically converges to the Nash equilibrium
of the aggregative game (1).
Proof. See Appendix C. □
Remark 5. The algorithm (13) is based on the tracking control,
which results that the algorithm is independent of the parameters
of EL systems and is with asymptotic convergence.
4. 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 ∈ V is Ji (Pi , P−i ) = ci (Pi ) − p(σ )Pi , where Pi ∈ R
is the output power of the generation system i, in p.u., P−i =
col(P1 , . . . , Pi−1 , Pi+1 , . . . , PN ), ci (Pi ) is the generation cost and p(σ )
is the electricity price. Referring to Binetti, Davoudi, Lewis, Naso,
and Turchiano (2014), ci (xi ) = αi + βi Pi + γi Pi2 , where αi , βi and
γi are the characteristics of the generation system i.∑
Furthermore,
N
p(σ ) = p0 − aN σ with the aggregate function σ = N1 i=1 Pi , where
p0 , a are constants.
Besides, when the mechanical and electromagnetic losses are
neglected, the dynamics of the ith turbine-generator system can
be written as follows (referring to Guo, Hill, and Wang (2000)).

⎧
Kmi
1
⎪
⎪
Pi +
Xei ,
Ṗi = −
⎪
⎪
⎪
Tmi
Tmi
⎪
⎨
Kei
1
1
Ẋei = −
ωi − Xei + ui ,
⎪
Tei Ri ω0
Tei
Tei
⎪
⎪
⎪
⎪
⎪
⎩ ω̇i = − Di ωi , i ∈ V ,
2Hi

where Xei ∈ R and ωi ∈ R are the valve opening and the relative
speed of the ith generation system, in p.u. and rad/s, respectively;
Kmi is the gain of the ith machine’s turbine; Tmi and Tei are the
time constants of the ith machines’s turbine and speed governor,
in s, respectively; Ri is the regulation constant of the ith machines’s
turbine, in p.u.; Di is the per unit damping constant; Hi is the inertia
constant, in s; ω0 is the synchronous machine speed, in rad/s; and
ui ∈ R is the control input of the ith generation system.
The parameters of six generation systems are shown in Table 1.
Besides, ω0 = 314.159, p0 = 200, and a = 0.1. By taking
ε = 0.6, we obtain that Φε is ω-strongly monotone and θ -Lipschitz
continuous, where ω = 1.2 and θ = 5.6. Then, we choose k =
30 according to (6). Moreover, xi (0), yi (0), vyi (0) and zi (0) are all
zeros. The simulation results of the algorithms (5) and (13) are
presented in Figs. 1 and 2, respectively, where (P1∗ , . . . , P6∗ ) is the
Nash equilibrium of the game.
As shown in Figs. 1 and 2, the output powers of these six
generation systems converge to the Nash equilibrium of the game
under the algorithms (5) or (13), which verifies the effectiveness of
the two algorithms. By comparing Fig. 1 with Fig. 2, it is clear that
the simulation results of the algorithm (5) are better than those
of the algorithm (13), since the output powers of six generation
systems under the algorithm (13) are more fluctuant than those
under the algorithm (5).

250

Z. Deng, S. Liang / Automatica 99 (2019) 246–252
Table 1
System parameters.

Generator #1
Generator #2
Generator #3
Generator #4
Generator #5
Generator #6

Tmi

Tei

Kmi

Kei

Di

Hi

Ri

αi

βi

γi

Pi (0)

Xei (0)

ωi (0)

0.35
0.30
0.28
0.40
0.43
0.35

0.10
0.12
0.08
0.11
0.90
0.10

1.0
1.1
0.9
1.2
0.8
1.0

1.0
1.1
0.9
1.2
0.8
1.0

5.0
4.0
3.0
4.5
3.5
5.0

4.0
3.5
2.8
4.2
3.0
4.0

0.05
0.04
0.03
0.06
0.04
0.05

5
8
6
9
7
8

12
10
11
11
13
14

1.0
0.5
0.8
0.7
1.1
0.6

30
25
20
35
28
37

6
5
4
7
5
8

4.3
3.5
3.0
4.8
4.0
5.0

Appendix A. Proof of Lemma 3
When (7) is at its equilibrium points, we have

ṽ ∗ =0N(m+n) ,
− kṽ ∗ − φε (x̃∗ ) − H1 x̃∗ − H2 z ∗ =0N(m+n) ,
H3 (x̃∗ + ṽ ∗ ) =0N(m+n) .

Fig. 1. The evolutions of strategy profile under the algorithm (5).

(A.1a)
(A.1b)
(A.1c)

which yield that (L ⊗ Im )y∗ = 0Nn , G(q∗ , y∗ ) = 0Nm , and ε (y∗ −
ϕ (q∗ )) + (L ⊗ In )z ∗ = 0Nn . Further, because 1TN L = 0TN , we have
∑N
y∗i = N1 j=1 ϕj (q∗j ) and Gi (q∗i , y∗i ) = Fi (q∗i , σ ) = 0m for any i ∈ V .
That is, ∇qi Ji (q∗i , q∗−i ) = 0m for any i ∈ V , which indicates that q∗
is the Nash equilibrium of the aggregative game (1) according to
Lemma 2.
Conversely, if q∗ is the Nash equilibrium of the aggregative
game (1), it results from (2a) that Fi (q∗i , σ ) = 0m . Choose y∗ =
col(y∗1 , . . . , y∗N ) with y∗i = σ (q∗ ) for any i ∈ V . Then, with (2b),
we have G(q∗ , y∗ ) = 0Nm . Moreover, take any v ∈ Rn , and let
ν = 1N ⊗ v . Further, based on the definition of ϕ (q), we have
(y∗ − ϕ (q∗ ))T ν = 0. Besides, because the undirected graph is
connected, L1N = 0N , which implies that (L ⊗ In )ν = 0Nn , and
hence, ν ∈ ker(L ⊗ In ). Note that ker(L ⊗ In ) and range(L ⊗ In ) form
an orthogonal decomposition of RNn by the fundamental theorem
of linear algebra (Strang, 1993). It follows from ε (y∗ −ϕ (q∗ ))T ν = 0
and ν ∈ ker(L ⊗ In ) that ε (y∗ − ϕ (q∗ )) ∈ range(L ⊗ In ), where ε > 0,
and therefore, there exists z ∗ such that ε (y∗ − ϕ (q∗ )) + (L ⊗ In )z ∗ =
0Nn is satisfied for some ε > 0. When ṽ ∗ = 0N(m+n) , (A.1) is
satisfied.
Appendix B. Proof of Theorem 1
With the following coordinate transformation,
x̄ =col(x̄q , x̄y ) = x̃ − x̃∗ ,

v̄ =col(v̄q , v̄y ) = ṽ − ṽ ∗ ,
Fig. 2. The evolutions of strategy profile under the algorithm (13).

5. 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 ensure EL players
exponentially converge to the Nash equilibrium. The other algorithm, which is independent of system parameters, achieves the
asymptotic convergence to the Nash equilibrium. The effectiveness
of our algorithms have been verified by numerical examples.

z̄ = z − z ∗ ,

(B.1)

where x̄q , v̄q , x̄y , v̄y ∈ RNm , the following system is obtained via (7)
and (A.1).

⎧
⎨ x̄˙ = v̄,
v̄˙ = −kv̄ − h − H1 x̄ − H2 z̄ ,
⎩˙
z̄ = H3 (x̄ + v̄ ),

(B.2)

where h = φε (x̃) − φε (x̃∗ ).
By (B.1), q∗ approaches the Nash equilibrium of the aggregative
game (1) if x̄ tends to the origin. Thus, the next task is to analyze
the convergence of x̄.
Take following orthogonal transformation,

[
]T
χq =col(χq1 , χq2 ) = ( r R ⊗ Im )x̄q ,
[
]T
χy =col(χy1 , χy2 ) = ( r R ⊗ In )x̄y ,
[
]T
νq =col(νq1 , νq2 ) = ( r R ⊗ Im )v̄q ,
[
]T
νy =col(νy1 , νy2 ) = ( r R ⊗ In )v̄y ,
[
]T
δ =col(δ1 , δ2 ) = ( r R ⊗ In )z̄ ,

(B.3a)
(B.3b)
(B.3c)
(B.3d)
(B.3e)

Z. Deng, S. Liang / Automatica 99 (2019) 246–252

where χx1 , νq1 ∈ Rm , χx2 , νq2 ∈ R(N −1)m , χy1 , νy1 , δ1 ∈ Rn ,
χy2 , νy2 , δ2 ∈ R(N −1)n , r = √1 1N , r T R = 0TN −1 , RT R = IN −1 and

1 2
Besides, it results from ab ≤ 2c a2 + 2c
b for c > 0 that

1 ( λ2

N

RRT = IN − N1 1N 1TN .
Let χ1 = col(χq1 , χy1 ), χ2 = col(χq2 , χy2 ), ν1 = col(νq1 , νy1 ), and
ν2 = col(νq2 , νy2 ). Then (B.2) can be rewritten as

⎧
⎨χ̇1 = ν1 ,
ν̇1 = −kν1 − r̂ T h,
⎩
δ̇1 = 0N −1
⎧
⎪
χ̇2 = ν2 ,
⎪
⎪
[
]
⎪
⎪
⎪
0
⎪
T
⎪
ν̇ = −kν2 − R̂ h −
χ2
⎪
⎪
⎨ 2
RT LR ⊗ In
[
]
⎪
0
⎪
⎪
−
δ2 ,
⎪
⎪
⎪
⎪
RT LR ⊗ In
⎪
[
]
[
]
⎪
⎪
⎩ δ̇ =
2
0 RT LR ⊗ In χ2 + 0 RT LR ⊗ In ν2 ,

(B.4a)

[

r ⊗ Im
0

0

]

r ⊗ In

[

,

0

R ⊗ Im
0

R̂ =

]

R ⊗ In

1
2

1

1

2

2

∥χ1 + ν1 ∥2 + (k − 1) ∥χ1 ∥2 +

+

1
2

(k − 1) ∥χ2 ∥2 +

where 0 < γ < min

{

1
2

∥χ2 + ν2 ∥2

2

,

2
2λ2 (k− θω − 14 λN −1)
k(k+1)+2λ2 λN

}
and k is given in

(6). The derivative of V1 along (B.4) is
V̇1 = − χ1T r̂ T h − χ2T R̂T h − (k − 1) ∥ν1 ∥2 − (k − 1) ∥ν2 ∥2
T
− ν1T r̂ T h − ν2T R̂T h − χy2
(RT LR ⊗ In )χy2
(
[
T
− νy2
(RT LR ⊗ In )χy2 + γ −kδ2T νy2 − δ2T 0

−δ

T
T
y2 (R LR

(B.5)

Based on Assumption 3, (B.1) and (B.3), we obtain

χ1T r̂ T h + χ2T R̂T h ≥ ω(∥χ1 ∥2 + ∥χ2 ∥2 ),

(B.6)

and

ν1T r̂ T h + ν2T R̂T h ≤

1(θ2

(∥ν1 ∥2 + ∥ν2 ∥2 )
2 ω
)
+ ω(∥χ1 ∥2 + ∥χ2 ∥2 ) .

(B.7)

According to Schur Complement Lemma, we have

⎡

T
⎢ R LR ⊗ In

F := ⎣ 1
2

T

R LR ⊗ In

1
2

4

λN I

1

2

2k νy2  +



2
With (B.5)–(B.9), we have

1
2k

)

(∥χ1 ∥2 + ∥χ2 ∥2 ) .

1 )
) (∥χ1 ∥2 + ∥χ2 ∥2 )
2
2λ2
4k
)
(
(
θ2
1
θ2
− 1 ∥ν1 ∥2 − k −
− λN − 1
− k−
ω
ω
4
k(k + 1) )
2
− γ (λ N +
) ∥ν2 ∥ ,
2λ2

(1

ω − γ(

1

(∥χ1 ∥2 + ∥χ2 ∥2 ) ,

)

(B.9b)
(B.9c)

+

(B.10)

where M(q) = diag {M1 (q1 ), . . . , MN (qN )}, C (q, q̇) = diag
{C1 (q1 , q̇1 ), . . . , CN (qN , q̇N )}, and Ω = diag {Ω1 , . . . , ΩN }.
Take V2 = 21 eT M(q)e + 12 φ T φ for (C.1). Obviously, V2 is lower
bounded. It follows from Property (i) of EL systems that V̇2 =
eT M(q)ė + 12 eT Ṁ(q)e + φ T φ̇ = −∥e∥2 ≤ 0, which indicates that e
and φ are bounded.

q̇i + qi = ei + ẋi + xi , i ∈ {1, . . . , N },

(C.2)

which can be viewed as a stable first-order linear system about
qi with a bounded input. Thus, qi and q̇i are bounded. Further,
according to (10) and (11), q̂˙ i and q̂¨ i are also bounded, which
implies Ωi in (12) is bounded. Then the boundedness of ė is obtained via (C.1). Consequentially, V̈2 = −2eT ė is bounded. Based
on Barbalat’s lemma, limt →∞ V̇2 (t) = 0, i.e., limt →∞ e(t) = 0m .
Further, it follows from (C.2) that limt →∞ (qi (t) − xi (t)) = 0m and
limt →∞ (q̇i (t) − ẋi (t)) = 0m , which indicates that qi converges
to the Nash equilibrium of the aggregative game (1) according to
Lemma 4.

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T
T
χy2
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[ ]
 2
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]

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k+1 

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.

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which indicates that q exponentially converges to q∗ , since V1 and
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where
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(B.4b)

251

(B.8)

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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, 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.

