Distributed constrained aggregative games of uncertain Euler-Lagrange systems under unbalanced digraphs

Abstract

In this paper, the constrained Nash equilibrium seeking problem of aggregative games is investigated for uncertain nonlinear Euler-Lagrange (EL) systems under unbalanced digraphs, where the cost function for each agent depends on its own decision variable and the aggregate of all other decisions. By embedding a distributed estimator of the left eigenvector associated with zero eigenvalue of the digraph Laplacian matrix, a dynamic adaptive average consensus protocol is employed to estimate the aggregate function in the unbalanced case. To solve the constrained Nash equilibrium seeking problem, an integrated distributed protocol based on output-constrained nonlinear control and projected dynamics is proposed for uncertain EL players to reach the Nash equilibrium. The convergence analysis is established by using variational inequality technique and Lyapunov stability analysis. Finally, a numerical example in electricity market is provided to validate the effectiveness of the proposed method.

1 Introduction

With an effective integration of sensing, communication, computation, optimization and decision making with physical dynamics, flourishing research activities on cyber-physical systems have been observed in the last decade. In particular, the distributed strategies combined with physical dynamics have drawn increasing research attention in resource allocation [1], smart grid [2, 3], mobile robot network [4], and autonomous vehicles coordination [5–7]. Due to the flexibility and unification in the modeling, EL systems as one typical class of physical systems have been extensively investigated to describe many mechanical systems, such as manipulator robots, rigid bodies, underwater vehicles and spacecrafts (see [8, 9] and references therein). Leaderless and leader-following consensus problems for EL systems were investigated in [10–12], distributed optimization problems for EL systems were considered in [5, 13], and distributed aggregative game strategies for EL systems were exploited in [3, 4].

Due to broad potential applications in demand response management of power systems [14], congestion control of communication networks [15], public environmental models [16], electricity market games of smart grids [3], routing planing of electric vehicles [17], just to name a few, the aggregative game has become a hot research issue in distributed games recently. In ordinary non-cooperative games, the objective function of each player is determined by its own decisions, together with the other players’ decisions, which makes each individual need to estimate all the decision variables of its non-neighbor players in the design of distributed game algorithms. Especially as the number of players increases, the designed algorithm will contain more estimated variables, which increases the complexity and computation cost of the algorithm. In comparison, the loss function of each player in aggregative games is not only dependent on its own decision variables, but also related to an unknown aggregate function defined by decision variables of all players, an example being the well-known Cournot model where the aggregate supply of opponents that matters rather than their individual strategies(see [18] and the references therein). Therefore, in aggregative game algorithms, all players only need to estimate the aggregate terms rather than all the decision variables, which can reduce the complexity and computation cost of the game playing algorithm on one hand; on the other hand, for each player, the influence of uncertain factors in the game environment on the estimated value of aggregate term may be less than the total influence on the estimated values of all individual decision variables, which can improve the predictability of aggregate games to some extent. As one of the effective game model, aggregation game have been widely utilized to predict and analysis strategies in many economic and social fields [14, 16, 18, 19], such as the prediction and analysis of public resources, market prices, elections and voting, etc.

Although remarkable achievements have been made on this topic during the past few years, most of the existing studies focus on the first-order integrator system [14, 20, 21], and the complicated dynamic of physical systems may have an effect on the implementation of game strategies in practical applications [3, 4]. There are still many problems involving with the uncertainties, directed or switching topology, and constraints for the complicated high-order nonlinear dynamics. However, relatively few results have been obtained for these promising topics, especially for the integration of these cases mentioned above. In [4], Deng made the first step to solve the unconstrained aggregative games of EL systems, and then extended the results to the equality constraint case in [3]. The unconstrained aggregative game for high-order nonlinear dynamics with the external disturbance was investigated in [7]. However, the communication graphs among most of the existing literatures were assumed to be undirected [3, 4, 7, 14, 21] or weight-balanced directed [22, 23]. Additionally, some prior knowledge of the global parameters is required in the implementation of most of the above algorithms, such as the Laplacian matrix [3, 4]. Furthermore, nearly all practical physical systems are subject to output constraints coming from specifications of operation and considerations of safety [24]. However, most of the existing results are about the unconstrained game or equality constraint game [3, 4, 7, 23].

Motivated by the above discussion, in this paper, we consider the distributed constrained aggregative game for uncertain El systems under unbalanced directed communication topology. The main contributions are summarized as follows:

(i) Different from most of the existing literatures about the distributed aggregative games of the first-order integrator systems [14, 20, 21], the distributed optimization of EL systems [5], and the distributed unconstrained aggregative game of EL systems [3, 4], we consider the distributed aggregative game of uncertain EL systems with set constraints. This problem is more challenging under the concurrent influence of the uncertain nonlinear EL dynamics and the output constraints. Motivated by the output-constrained control design techniques in [25], we propose a fault compensation scheme to conquer the above difficulties, which is totally different from the tracking controller design in [3–5].

(ii) The communication topology among agents is described by a weight-unbalanced digraph, which is different from most of the existing results [3, 4, 7, 14, 21–23]. In order to tackle the difficulties caused by unbalanced communication topology, we design a distributed estimator to estimate the left eigenvector of the Laplacian matrix associated with the zero eigenvalue.

(iii) Different from [3, 4], we proposed a fully distributed protocol for EL systems to solve the constrained aggregative games under unbalanced digraphs, and the selection of control parameters does not depend on any global information.

This paper is organized as follows. In Sect. 2, we formulate the considered problem along with basic concepts. In Sect. 3, we present a distributed algorithm for EL systems to reach the Nash equilibrium of aggregative games with a prescribed accuracy, and then we give the main results and convergence analysis under unbalanced directed interaction topologies. Following that, we carry out simulation studies to verify our theoretical results in Sect. 4. Finally, we give concluding remarks in Sect. 5.

Notations: $\|\cdot \|$ and $\|\cdot \|_{\infty}$ denote the Euclidean norm and $L_{\infty}$ of a vector or matrix, respectively. For $x_{i}\in \mathbb{R}^{m}$, $i=1,2,\ldots,N$, $x_{i}^{j}$ is the j-th element of $x_{i}$. Let $\textbf{1}_{n}$ and $\textbf{0}_{n}$ be the n-dimensional vectors of all entries as 1 and 0, respectively.

2 Problem formulation

In this section, we first review some related preliminaries in graph theory [26], convex analysis [27] and variational inequality [28, 29].

2.1 Preliminaries

Consider a directed graph $\mathcal{G}= (\mathcal{V}, \mathcal{E}, A)$, where $\mathcal{V}= \{1,2, \ldots, N\}$ is the node set, $\mathcal{E} \subseteq \mathcal{V}\times \mathcal{V}$ is the edge set, and $A=[a_{ij}]\in \mathbb{R}^{N\times N}$ is the weighted adjacency matrix. If $(i,j)\in \mathcal{E}$, then $a_{ij}>0$, which means that the agent i can receive information from agent j; otherwise, $a_{ij}=0$. Denote $\mathcal{N}_{i}=\{j \in \mathcal{V} \lvert (i,j)\in \mathcal{E} \}$ as the set of neighbors of agent i. The degree matrix of the graph $\mathcal{G}$ is defined as $D = \operatorname{diag} ( {d_{1}^{\mathrm{in}},\ldots,d_{N}^{\mathrm{in}}} )$, where $d_{i}^{\mathrm{in}} = \sum_{j = 1}^{N} {{a_{ij}}}$ means the inward degree of agent i. Denote $d_{i}^{\mathrm{out}} = \sum_{j = 1}^{N} {{a_{ji}}}$ as the outward degree of agent i. The directed graph is weight-balanced if $\sum_{j=1}^{N} a_{ji}=\sum_{j=1}^{N} a_{ij}$, $\forall i \in \mathcal{V}$; and unbalanced, otherwise. The Laplace matrix is defined as $L=D-A$. $\mathcal{G}$ is weight balanced if $1_{N}^{T}L = 0_{N}^{T}$; unbalanced, otherwise. A directed graph $\mathcal{G}$ is strongly connected if there exists a directed path between any two nodes in $\mathcal{V}$. If $\mathcal{G}$ is strongly connected, then zero is the simple eigenvalue of L, and all the other eigenvalues have positive real parts.

A set $K\subseteq \mathbb{R}^{m}$ is convex if $ax+(1-a)y\in K$ for any $x,y \in K$ and $a \in [0,1]$. For any $x\in \mathbb{R}^{m}$, there is a unique element $P_{K}(x)\in K$ satisfying $\|x-P_{K}(x)\|= \inf_{y\in K} \|x-y\|$, which is denoted by $\lvert x\rvert _{K}$, where $K \subseteq \mathbb{R}^{m}$ is a nonempty closed convex set and $P_{K}$ denotes the projection operator onto K. Denote $\lvert x\rvert _{K}=0$ for any x if $K=\emptyset$, for convenience. Denote $N_{K}(x)=\{d \lvert \langle d,y-x\rangle \leq 0, \forall y\in K\}$ as the normal cone of K at x, and $T_{K}(x)=\{\lim_{n\rightarrow \infty}\frac{x_{n}-x}{t_{n}} \lvert x_{n} \in K, t_{n}>0, x_{n}\rightarrow x, \text{and}\; t_{n}\rightarrow 0\}$ as its tangent cone. The following are some useful results given in [30, 31] about the properties of projection operator.

Lemma 1

Let K be a closed convex set in $\mathbb{R}^{m}$. Then

• (a) $(x-P_{K}(x))^{T} (P_{K}(x)-y)\geq 0$ for any $y\in K$ and $x \in \mathbb{R}^{m}$;

• (b) $\lvert \lvert x\rvert _{K}-\lvert y\rvert _{K}\rvert \leq \|x-y\|$ for any $x,y \in \mathbb{R}^{m}$;

• (c) $\lvert x\rvert _{K}$ is a convex function and $\lvert x\rvert _{K}^{2}$ is continuously differentiable with $\nabla \lvert x\rvert _{K}^{2}=2(x-P_{K}(x))$, where ∇ stands for the gradient.

A function $f(x):\mathbb{R}^{m}\to \mathbb{R}$ is convex if $\forall \alpha \in [0,1], \forall x,y \in \mathbb{R}^{m}$, $f(\alpha x+(1-\alpha )y)\leq \alpha f(x)+(1-\alpha )f(y)$. A function $f(x):\mathbb{R}^{m}\to \mathbb{R}$ is μ-Lipschitz continuity on ${\mathbb{R}^{m}}$ if $\forall x,y\in \mathbb{R}^{m}$, $\mu > 0$, $\Vert f(x)-f(y)\Vert \leq \mu \Vert x-y\Vert$. A map $f(x):\mathbb{R}^{m}\to \mathbb{R}^{n}$ is ω-strongly monotonic on ${\mathbb{R}^{m}}$ if $(x-y)^{T}(f(x)-f(y))\geq \omega \Vert x-y\Vert ^{2}$, $\forall x,y\in \mathbb{R}^{m}$, $x\neq y$. Let $K\subset \mathbb{R}^{m}$ be a closed convex set, and $\mathcal{F}(x):K\to \mathbb{R}^{m}$ is a map, then the solution of variational inequality

$$ (y-x)^{T}\mathcal{F}(x)\geq 0,\quad \forall y \in K $$

(1)

can be equivalently reformulated via the following projection

$$ x\in \operatorname{SOL}(K,\mathcal{F}) \quad \Leftrightarrow\quad x=P_{K}\bigl(x-\mathcal{F}(x)\bigr), $$

(2)

where $\operatorname{VI}(K,\mathcal{F})$ represents the variational inequality (1) for convenience, and $\operatorname{SOL}(K,\mathcal{F})$ denotes the set of its solutions.

2.2 Problem statement

Consider a constrained aggregative game of N heterogeneous nonlinear EL players (agents) with an associated directed communication graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. The dynamics of each player $i\in \mathcal{V}$ is described as follows:

$$ M_{i}(q_{i})\ddot{q}_{i}+ C_{i}(q_{i},\dot{q}_{i}) \dot{q}_{i} + G_{i}(q_{i})= \tau _{i}, $$

(3)

where $q_{i}$, $\dot{q}_{i} \in \mathbb{R}^{m}$ denote the generalized position and velocity vectors, respectively; $M_{i}(q_{i})\in \mathbb{R}^{m \times m}$ is the positive definite inertia matrix; $C_{i}(q_{i},\dot{q}_{i})\dot{q}_{i}\in \mathbb{R}^{m}$ is the vector for Coriolis and centripetal forces; $G_{i}(q_{i})\in \mathbb{R}^{m}$ is the gravity vector; and $\tau _{i}\in \mathbb{R}^{m}$ is the control force. Note that $q_{i},\dot{q}_{i}$ are measurable, and $M_{i}(q_{i}), C_{i}(q_{i},\dot{q}_{i}), G_{i}(q_{i})$ are permitted to be unknown. The dynamics of the system (3) satisfies the following property, which was widely used in [25, 32].

Property 2

$\|C_{i}(q_{i},\dot{q}_{i})\|$ and $\|G_{i}(q_{i})\|$ are bounded if $\|q_{i}\|$ and $\|\dot{q}_{i}\|$ are bounded, $\forall i \in \mathcal{V}$.

In these EL systems, player i is endowed with a local payoff function $J_{i}(q_{i},q_{-i}): \Omega \rightarrow \mathbb{R}$, where $\Omega =\Omega _{1} \times \cdots \times \Omega _{N} \subseteq \mathbb{R}^{mN}$, $q_{i}\in \Omega _{i} \subseteq \mathbb{R}^{m}$, and $q_{-i}=\operatorname{col}(q_{1},\ldots,q_{i-1},q_{i+1},\ldots,q_{N})$. $\Omega _{i}$ is the feasible strategy set of player i, which is driven by the fact that most of mechanical systems are subject to physical constraints for safe operation. The local payoff function $J_{i}(q_{i},q_{-i})$ is only known by player i, and it is not only dependent on its own decision variable $q_{i}$, but also related to the aggregate of the decision variables of all players. We define the aggregate function $\sigma :\Omega \to \mathbb{R}^{n}$ as follows:

$$ \sigma (q)=\frac {1}{N}\sum_{i=1}^{N} \phi _{i}(q_{i}),$$

where $\phi _{i}(q_{i}):\Omega _{i}\to \mathbb{R}^{n}$ is a continuous function for the local contribution of player i to the aggregation. Then the local payoff function $J_{i}(q_{i},q_{-i})$ can be expressed as $J_{i}(q_{i},q_{-i})=\Gamma _{i}(q_{i},\sigma (q))$ with a function $\Gamma _{i}: \mathbb{R}^{m+n} \rightarrow \mathbb{R}$. Note that $\sigma (q)$ is unknown to all the players, and each player i can only obtain the partial information $\phi _{i}(q_{i})$ of the aggregate $\sigma (q)$. Moreover, players can not share their payoff functions and the local contribution functions to others for privacy.

The objective of player i is to minimize its own payoff function $J_{i}(q_{i},q_{-i})$ by choosing the appropriate decision in its safe output constraint $q_{i}\in \Omega _{i}$, which is equivalent to solve the following optimization problem:

$$ \min_{q_{i}\in \Omega _{i}}J_{i}(q_{i},q_{-i}), \quad \forall i \in \mathcal{V}. $$

(4)

Then we introduce the concept of Nash equilibrium (NE) and some related results (see [21] for details).

Definition 1

A strategy profile $q^{*}=(q_{1}^{*},\ldots ,q_{N}^{*})^{T} \in \Omega$ is a NE of the aggregative game (4) if

$$ J_{i}\bigl(q_{i}^{*},q_{-i}^{*} \bigr)\leq J_{i}\bigl(x_{i},q_{-i}^{*} \bigr),\quad \forall x_{i} \in \Omega _{i}, i\in \mathcal{V}, $$

(5)

which means NE is a profile on which no player can further decrease its local payoff function by changing its own decision unilaterally.

Define a map $F: \mathbb{R}^{mN}\rightarrow \mathbb{R}^{mN}$ as

$$ F(q)=\bigl(\nabla _{q_{1}} J_{1}(q_{1},q_{-1}), \ldots , \nabla _{q_{N}}J_{N}(q_{N},q_{-N}) \bigr)^{T}, $$

a strategy profile is called a variational NE if it is a solution of $\operatorname{VI}(\Omega ,F)$. Moreover, Theorem 3.9 of [33] shows that if Ω is a convex, every solution of $\operatorname{VI}(\Omega ,F)$ is also an NE.

There are some assumptions about the communication graph and the variational NE.

Assumption 1

The communication graph $\mathcal{G}$ is directed and strongly connected.

Assumption 2

The payoff function $J_{i}(q_{i},q_{-i})$ is continuously differentiable, $\forall i\in \mathcal{V}$. $F(q)$ is μ-Lipschitz continuous and ω-strongly monotonic in q on $\Omega \subseteq \mathbb{R}^{mN}$.

Assumption 3

The strategy set $\Omega _{i}$, $i\in \mathcal{V}$ is a box constraint, i.e., $\forall q_{i} \in \Omega _{i}$, $q_{i}^{j}\in [\tilde{\nu}_{i}^{j},\hat{\nu}_{i}^{j}]$, $j=1,2,\ldots ,m$, where $\tilde{\nu}_{i}^{j}$, $\hat{\nu}_{i}^{j}$ are the lower and upper constraint boundaries, respectively.

Under Assumption 2 and 3, it is not hard to obtain the following results with a similar analysis in [21].

Lemma 3

Suppose Assumption 2and 3hold, the constrained aggregative game (4) has a unique variational NE $q^{*} \in \Omega$. Moreover, $q^{*}$ is a variational NE if and only if

$$ 0\in F\bigl(q^{*}\bigr)+N_{\Omega} \bigl(q^{*}\bigr). $$

(6)

Proof

According to Theorem 2.2.3 and Corollary 2.2.5 in [28], the existence of a variational NE can be guaranteed by Assumption 3 and the continuous differentiability of $J_{i}(q_{i},q_{-i})$ in Assumption 2; and the uniqueness can be guaranteed by the strongly monotonicity of $F(q)$ in Assumption 2.

If $q^{*}$ is a variational NE, that means $(y-q^{*})^{T} F(q^{*})\geq 0$, $\forall y \in \Omega$. Based on the definition of norm cone, we have $N_{\Omega}(q^{*})=\{d \lvert \langle d,y-q^{*}\rangle \leq 0, \forall y\in \Omega \}$, which implies $-F(q^{*}) \in N_{\Omega}(q^{*})$. And vice versa. □

Next, we formulate our considered problem.

Problem 4

Design a distributed protocol for uncertain nonlinear EL systems (3) to seek the variational NE of the constrained aggregative game (4) under unbalanced digraph such that

• the EL system outputs reach the variational NE with prescribed accuracy, i.e.,

  $$ \limsup_{t\rightarrow \infty} \bigl\Vert q-q^{*} \bigr\Vert _{\infty}\leq \epsilon ,$$

  where ϵ is a preselected arbitrary positive constant;

• the specified output constraint $q_{i} \in \Omega _{i}$ is never transgressed;

• all signals in the closed-loop system are bounded.

Remark 1

Different from the well-studied distributed aggregative games in [14, 20, 21], our formulation gets involved with the nonlinear EL dynamics of each player. The considered formulation has wide potential applications in many distributed cyber-physical systems which involve the dynamics of both scheduling system and operators, such as the demand response management of power systems [14], electricity market games of smart grids [3, 4], routing planing of electric vehicles [17], multi-robot basketball game [34], multi-robot surveillance application [35] and other fields. Note that the intrinsic nonlinear and uncertain dynamics of EL players make the constrained NE seeking problem of EL systems much more challenging than the linear integrator case. Additionally, in contrast to most of the existing NE seeking algorithms under the undirected graph in [3, 4, 21] or weight-balanced digraph in [22, 23], the communication network in this paper is unbalanced, which indicates that the considered problem enables more extensive scope of application. Furthermore, the above mentioned works did not consider the constraints on actuators, which may deter the implementation on physical systems. In our controller design, all signals in the closed-loop system are required to be bounded for safe and reliable practical implementation.

3 Main results

In this section, we propose a distributed protocol for uncertain EL systems (3) to solve the variational NE seeking Problem 4.

3.1 Algorithm design

In practical situations, the system parameters may be unavailable for each player, together with the output constraints, which makes the feedback linearization method and the path tracking method in [3–5] fail to deal with the constrained and uncertain nonlinear EL system (3). Motivated by the fault compensation technique in [25], a distributed output-constrained controller is proposed for unknown EL systems with output constraints.

The distributed controller of player $i, i\in \mathcal{V}$, which consists of three parts, is designed as follows:

• (1) the local output-constrained controller

  $$\begin{aligned} & \tau _{i}=-\gamma _{i} (\Psi _{i})^{-1} \eta _{i}, \end{aligned}$$

  (7a)
  $$\begin{aligned} & \Psi _{i}=\operatorname{diag}\bigl(\Psi _{i}^{1}, \ldots ,\Psi _{i}^{m}\bigr)\in \mathbb{R}^{m m} \\ &\quad \text{with } \Psi _{i}^{j}=\xi _{i}^{j} \cos ^{2}\biggl( \frac{\pi z_{i}^{j}}{2\xi _{i}^{j}}\biggr), \end{aligned}$$

  (7b)
  $$\begin{aligned} & \eta _{i}=\bigl(\eta _{i}^{1},\ldots , \eta _{i}^{m}\bigr)^{T}\in \mathbb{R}^{m} \\ &\quad \text{with } \eta _{i}^{j}= \tan \biggl( \frac{\pi z_{i}^{j}}{2\xi _{i}^{j}}\biggr), \end{aligned}$$

  (7c)
  $$\begin{aligned} & \xi _{i}^{j}=\bigl(\xi _{i,0}^{j}- \xi _{i,\infty}^{j}\bigr)e^{-\rho _{i} t}+ \xi _{i,\infty}^{j}, \end{aligned}$$

  (7d)
  $$\begin{aligned} & z_{i}^{j}=\dot{q}_{i}^{j}+ \ln \frac{\varsigma _{i}^{j}-\underline{k_{i}^{j}}}{\overline{k_{i}^{j}}-\varsigma _{i}^{j}}, \end{aligned}$$

  (7e)
  $$\begin{aligned} &\overline{k}_{i}^{j}=\bigl(\hat{q}_{i}^{j}-x_{i} \bigr) \bigl[ \bigl(1-\alpha _{i}^{j} \bigr)e^{- \beta t}+\alpha _{i}^{j} \bigr], \end{aligned}$$

  (7f)
  $$\begin{aligned} &\underline{k}_{i}^{j}=\bigl(\tilde{q}_{i}^{j}-x_{i} \bigr) \bigl[ \bigl(1-\alpha _{i}^{j} \bigr)e^{- \beta t}+\alpha _{i}^{j} \bigr], \end{aligned}$$

  (7g)
  $$\begin{aligned} &\varsigma _{i}= q_{i}-x_{i}, \quad i \in \mathcal{V}, j=1,\ldots ,m, \end{aligned}$$

  (7h)

where $\gamma _{i}>0$ is the constant control gain, the logarithm term in velocity error $z_{i}^{j}$ is used to ensure $\underline{k_{i}^{j}}<\varsigma _{i}^{j}<\overline{k_{i}^{j}}$ with $\underline{k_{i}^{j}}$ and $\overline{k_{i}^{j}}$ served as the bounds on the tracking error $\varsigma _{i}^{j}$. $\xi _{i}^{j}$ is the preset performance bound on $z_{i}^{j}$, $\rho _{i}>0$ and $\xi _{i,\infty}^{j}$ represent the convergence rate and ultimate bound on $z_{i}^{j}$, respectively. $\xi _{i,\infty}^{j}<\xi _{i,0}^{j}$, $\lvert z_{i}^{j}(0)\rvert <\xi _{i,0}^{j}$, $q_{i}^{j}(0)\in (\tilde{q}_{i}^{j},\hat{q}_{i}^{j})$, $\beta >0$ and $\alpha _{i}^{j}=\frac{\epsilon}{\hat{q}_{i}^{j}-\tilde{q}_{i}^{j}}$.

• (2) the cooperative game optimizer

  $$\begin{aligned} & \dot{x}_{i}=P_{K_{i}}\bigl(x_{i}-g_{i}(x_{i},s_{i}) \bigr)-x_{i} , \end{aligned}$$

  (8a)
  $$\begin{aligned} & \begin{aligned}[b] \dot{s}_{i}={}&\nabla \phi _{i}(x_{i})- \bigl(s_{i}-\phi _{i}(x_{i})\bigr)\\ &{}- \varpi _{ii} \sum_{j\in N_{i} } a_{ij}(s_{i}-s_{j})-y_{i}, \end{aligned} \end{aligned}$$

  (8b)
  $$\begin{aligned} & \dot{y}_{i}=\varpi _{ii} \sum _{j\in N_{i} } a_{ij}(s_{i}-s_{j}), \quad \sum_{i=1}^{N} y_{i}(0)=0, \end{aligned}$$

  (8c)

where $g_{i}(x_{i},s_{i})=\nabla _{x_{i}} \Gamma _{i}(x_{i},\sigma (x)) \arrowvert _{\sigma (x)=s_{i}}$, $K_{i}=\prod_{j=1}^{m} [\tilde{q}_{i}^{j},\hat{q}_{i}^{j}]$ with $\tilde{q}_{i}^{j}=\tilde{\nu}_{i}^{j}+\epsilon$, $\hat{q}_{i}^{j}=\hat{\nu}_{i}^{j}-\epsilon$. The virtual variable x is used to estimate the variational NE $q^{*}$, and the auxiliary variables $s_{i}, y_{i} \in \mathbb{R}^{n}$ are used to estimate the aggregate function $\sigma (x)=\frac{1}{N} \sum_{i=1}^{N} \phi _{i}(x_{i})$. Denote $\tilde{\varpi}=\operatorname{col}(\varpi _{11},\ldots , \varpi _{NN})$, where $\varpi _{ii}$ is the i-th element of $\varpi _{i}$. ϖ̃ can be termed as the estimator of the left eigenvector of L associated with the zero eigenvalue.

• (3) the distributed estimator

  $$\begin{aligned} & \dot{\varpi}_{i}=-\sum _{j\in N_{i} } a_{ij}(\varpi _{i}-\varpi _{j}), \end{aligned}$$

  (9)

where $\varpi _{i}(0)=\operatorname{col}(\textbf{0}_{i-1},1,\textbf{0}_{N-i})$. Figure 1 depicts the main framework of the above algorithm. The estimator (9) is used to estimate the left eigenvector associated with zero eigenvalue of the unbalanced Laplacian matrix, which can be used to eliminate the influence of unbalance on the cooperative game optimizer (8a)–(8c); (8a)–(8c) is used to seek the variational NE and generates the reference path; and the local output-constrained controller (7a)–(7h) aims to force the EL system to track the reference path in a safe set constraint.

Figure 1

Schematic of the proposed algorithm (7a)–(7h), (8a)–(8c), (9)

Remark 2

Note that $K_{i} \subset \Omega _{i}$, $i \in \mathcal{V}$. This way of set contraction could effectively guarantee the trajectories of practical EL system never transgress the boundaries of $\Omega _{i}$. Meanwhile, the variational NE seeking accuracy can be improved by reducing the value of ϵ. Additionally, if the variational NE $q^{*}$ of aggregative game (4) is an interior point, we can conclude that the slight scaling of the boundaries does not affect the NE of (4). Furthermore, our proposed dynamic adaptive average consensus algorithm (8b), (8c) is totally different from those given in [3, 4, 36]. We design a distributed estimator to eliminate the influence of unbalance on (8b), (8c).

3.2 Convergence analysis

Denote $\omega =(\omega _{1},\ldots ,\omega _{N})^{T}$ as the left eigenvector of L associated with the zero eigenvalue, i.e., $\omega ^{T}L=\textbf{0}$. Without loss of generality, we assume $\omega ^{T} \textbf{1}=1$. Then we can obtain the following lemma.

Lemma 5

Under Assumption 1, $\tilde{\varpi}=\operatorname{col}(\varpi _{11},\ldots , \varpi _{NN})$ generated by estimator (9), is componentwise positive and exponentially converges to ω, i.e.,

$$ \textstyle\begin{cases} \varpi _{ii}(t)>0, \quad \forall t\geq 0; \\ \lim_{t\rightarrow \infty}\tilde{\varpi}(t)=\omega . \end{cases} $$

(10)

Proof

Since $\mathcal{G}$ is strongly connected, L is irreducible. Together with the essential nonnegativity of L, it follows that $e^{-Lt}, \forall t\geq 0$, is component-wise positive. Rewrite (9) in a compact form as $\dot{\varpi}=-(L\otimes I_{N})\varpi$, whose solution can be described by $\varpi (t)=(e^{-Lt}\otimes I_{N})\varpi (0)$, which implies $\varpi _{i}=e^{-Lt}\varpi _{i}(0)>0$. Since $\mathcal{G}$ is strongly connected, it follows from Theorems 1 and 3 in [37] that there exists a constant vector $d \in \mathbb{R}^{N}$ such that $\lim_{t\rightarrow \infty} \varpi (t)=d$ exponentially holds. With the change of variable $\hat{\varpi}=(\omega \otimes I_{N})^{T}\varpi$, we have $\dot{\hat{\varpi}}=-(\omega \otimes I_{N})^{T} (L\otimes I_{N}) \varpi =-(\omega ^{T} L\otimes I_{N}) \varpi = 0$, that is, $(\omega ^{T} \otimes I_{N})\varpi (t) =(\omega ^{T} \otimes I_{N}) \varpi (0)=(\omega ^{T} \otimes I_{N})(\textbf{1}_{N} \otimes d)=( \omega ^{T} \textbf{1}_{N}) d$. As a result, we have $d=\frac{\omega}{\omega ^{T} \textbf{1}_{N}}$. Therefore, $\varpi _{ii}$ exponentially converges to $\frac{\omega _{i}}{\omega ^{T} \textbf{1}_{N}}$. □

Next, we give the convergence analysis for the proposed cooperative game optimizer (8a)–(8c).

Lemma 6

Under Assumption 1, the dynamic adaptive average consensus algorithm (8b), (8c) and (9) can make $s_{i}(t), i\in \mathcal{V}$ globally exponentially converge to the aggregate function $\sigma (x)$.

Proof

Without loss of generality, we let $n=1$ for convenience. The following proof method can be extended directly to the multidimensional case.

Denote $v_{i}=s_{i}-\frac{1}{N} \sum_{j=1}^{N} \phi _{i}(x_{i})$ and $w=y-\Pi _{N}(\dot{\phi}+\alpha \phi )$, where $\Pi _{N}=I_{N}-\frac{1}{N}\textbf{1}_{N}\textbf{1}_{N}^{T}$ and $\phi =\operatorname{col}(\phi _{1}^{T},\ldots , \phi _{N}^{T})$. We can obtain the following compact form from (8b), (8c) and (9)

$$ \textstyle\begin{cases} \dot{v}=- v -\Lambda L v -w, \\ \dot{w}= \Lambda Lv-\Pi _{N}(\ddot{\phi}+ \dot{\phi}), \\ \dot{\varpi}=-(L\otimes I_{N})\varpi , \end{cases} $$

(11)

where $\Lambda =\operatorname{diag}(\varpi _{11},\ldots , \varpi _{NN})\in \mathbb{R}^{N N}$. We first discuss the stability of zero-system of (11) as follows

$$ \textstyle\begin{cases} \dot{v}=- v -\Lambda L v -w, \\ \dot{w}= \Lambda Lv, \\ \dot{\varpi}=-(L\otimes I_{N})\varpi . \end{cases} $$

(12)

Denote $\Lambda ^{*}=\operatorname{diag}(\omega _{1},\ldots ,\omega _{N})$, it results from Assumption 1 that $\Lambda ^{*} L$ is weight-balanced and $\lambda _{1}(\Lambda ^{*}L)=0$ and $\lambda _{2}(\Lambda ^{*}L)>0$. With the singular value decomposition, there exists an orthogonal matrix $O=(r, R) \in \mathbb{R}^{N \times N}$ with $r=\frac{\textbf{1}_{N}}{\sqrt{N}}$, $r^{T} R=\textbf{0}$ and $R^{T}\Lambda ^{*}L R=\operatorname{diag}(\lambda _{2}(\Lambda ^{*}L),\ldots , \lambda _{N}(\Lambda ^{*}L))$. With the change of variables

$$ \begin{bmatrix} {p} \\ {q} \end{bmatrix} = \begin{bmatrix} O^{T}&0 \\ O^{T}&O^{T} \end{bmatrix} \begin{bmatrix} v \\ w \end{bmatrix}, $$

(13)

the zero-system (12) can be rewritten as follows

$$ \begin{bmatrix} \dot{p} \\ \dot{q} \end{bmatrix} = \begin{bmatrix} - O^{T} \Lambda L O&-I_{N} \\ 0&- I_{N} \end{bmatrix} \begin{bmatrix} p \\ q \end{bmatrix}. $$

(14)

Since $O^T\mathrm{\Lambda }LO=\left[\begin{array}{cc}0& 0\\ 0& R^T\mathrm{\Lambda }LR\end{array}\right]$, we can partition the system (14) into two subsystems

$$ \begin{bmatrix} \dot{p}_{1} \\ \dot{q}_{1} \end{bmatrix} = \begin{bmatrix} q_{1} \\ - q_{1} \end{bmatrix} $$

(15)

and

$$ \begin{bmatrix} \dot{p}_{2:N} \\ \dot{q}_{2:N} \end{bmatrix} = \begin{bmatrix} - R^{T} \Lambda L R&-I_{N-1} \\ 0&- I_{N-1} \end{bmatrix} \begin{bmatrix} p_{2:N} \\ q_{2:N} \end{bmatrix}. $$

(16)

Notice that $q_{2:N}$ in system (16) vanishes exponentially fast. Then we discuss the stability of zero-system $p_{2:N}$. With the change of variables $\hat{p}_{2:N}=R p_{2:N}$, we have

$$ \dot{\hat{p}}_{2:N}=-\Lambda L \hat{p}_{2:N}. $$

(17)

It results from Lemma 5 that there exist positive constants $\varrho _{0}$ and $\varrho _{1}$ such that $\|\Lambda L-\Lambda ^{*}L\|\leq \varrho _{0} e^{-\varrho _{1} t}$. Then we obtain that there exists $\varrho _{2}>0$ such that $\|\hat{p}_{2:N}(t)\|\leq \varrho _{2} \hat{p}_{2:N}(0)e^{-\Lambda ^{*} L}$, which implies the trajectories of (17) converge exponentially to the average of their initial value. Then we can get that $p_{2:N}\rightarrow 0$ exponentially in the zero-system of (16). Based on ISS results in [38], the trajectories of (16) converge exponentially to 0. Moreover, we can conclude from subsystem (15) that $p_{1}(t)\rightarrow p_{1}(0)$ and $q_{1}(t)\rightarrow 0$ exponentially. Recall the change of variables (13), it is straightforward to get $v_{i}(t)\rightarrow \frac{-1}{\alpha N} \sum_{j=1}^{N} w_{j}(0)$, and $w_{j}(t)\rightarrow \frac{1}{ N} \sum_{j=1}^{N} w_{j}(0)$. Since $\dot{x}_{i} \in T_{K_{i}}(x_{i})$, we have $x_{i}\in K_{i}$, $\forall t\geq 0$, which implies $\|\dot{x}_{i}\|$ is bounded. It results from Assumption 2 and 3 that $\nabla \phi _{i}$ is bounded. According to [[36], Corollary 4.1], we can obtain that $s_{i}(t)$ globally exponentially converge to $\sigma (x)$. □

Theorem 7

Suppose Assumptions 1–3hold. If control parameters satisfy $\gamma _{i}>0$, $\rho _{i}>0$, $\beta >0$, $\xi _{i,0}^{j} > \lvert z_{i}^{j}(0)\rvert$, and $\xi _{i,\infty}^{j}<\xi _{i,0}^{j}$, the proposed protocol composed of (7a)–(7h), (8a)–(8c) and (9) solves Problem 4with the initial conditions $x_{i}(0)\in K_{i}$, $q_{i}(0) \in \Omega _{i}$, $\sum_{j=1}^{N} y_{i}(0)=0$, and $\varpi _{i}(0)=\operatorname{col}(\textbf{0}_{i-1},1,\textbf{0}_{N-i})$.

Proof

First, we show that the trajectories generated by the cooperative game optimizer (8a)–(8c) converge to the variational NE $q^{*}=\operatorname{col}(q_{1}^{*},\ldots , q_{N}^{*})$ of (4), i.e.,

$$ \lim_{t\rightarrow \infty} x_{i}(t)=q_{i}^{*},\quad \forall i\in \mathcal{V}. $$

(18)

Notice that, in (8a), if $x_{i}\in K_{i}$ holds, we can conclude that $\dot{x}_{i}\in T_{K_{i}}(x_{i})$. According to the Nagumo’s theorem in [30], we obtain $x_{i}(t)\in K_{i}$, $\forall t\geq 0$ under the initial condition $x_{i}(0)\in K_{i}$. Denote $e_{i}(t)=P_{K_{i}}(x_{i}-g_{i}(x_{i},s_{i}))-P_{K_{i}}(x_{i}-g_{i}(x_{i}, \sigma (x)))$, it results from Lemma 1 and Assumption 2 that

$$\begin{aligned} \Vert e_{i} \Vert & \leq \bigl\Vert g_{i}(x_{i},s_{i})-g_{i} \bigl(x_{i},\sigma (x)\bigr) \bigr\Vert \\ &\leq \mu \bigl\Vert s_{i}-\sigma (x) \bigr\Vert . \end{aligned}$$

Together with Lemma 6, we conclude that $e_{i}$ exponentially converges to 0. With the above change of variable, we can rewrite (8a) as follows

$$ \dot{x}_{i}=P_{K_{i}} \bigl(x_{i}-g_{i}\bigl(x_{i},\sigma (x) \bigr)\bigr)-x_{i} +e_{i}, $$

(19)

which can also be described in a compact form as

$$ \dot{x}=P_{K}\bigl(x-F(x)\bigr)-x +e. $$

(20)

Define $H(x)=P_{K}(x-F(x))$, and then it results from [29] that

$$ \bigl(x-H(x)\bigr)^{T} F(x)-\frac{1}{2} \bigl\Vert x-H(x) \bigr\Vert ^{2} \geq 0.$$

Construct the following Lyapunov function

$$\begin{aligned} V =&\bigl(x-H(x)\bigr)^{T} F(x)-\frac{1}{2} \bigl\Vert x-H(x) \bigr\Vert ^{2} \\ &{}+\frac{1}{2} \bigl\Vert x-q^{*} \bigr\Vert ^{2}, \end{aligned}$$

(21)

then differentiating it along (20) leads to

$$\begin{aligned} \dot{V}= {}& \bigl(F(x)+\bigl(\mathcal{J}F(x)-I_{N} \bigr) \bigl(x-H(x)\bigr)+\bigl(x-q^{*}\bigr)\bigr)^{T}\\ &{}\times \bigl(H(x)-x+e\bigr) \\ \leq {}& \bigl(F(x)+\bigl(\mathcal{J}F(x)-I_{N}\bigr) \bigl(x-H(x) \bigr)+\bigl(x-q^{*}\bigr)\bigr)^{T}\\ &{}\times\bigl(H(x)-x\bigr)+ \hat{e}, \end{aligned}$$

where $\hat{e}=(F(x)+(\mathcal{J}F(x)-I_{N})(x-H(x))+(x-q^{*}))^{T}e$. With the boundedness of $x_{i}\in K_{i}$ and the exponential convergence of e, we conclude that

$$ \int _{0}^{\infty} \bigl\Vert \hat{e}(t) \bigr\Vert \, dt< +\infty . $$

(22)

The first term in V̇ can be decomposed as

$$\begin{aligned} & \bigl(F(x)+\bigl(\mathcal{J}F(x)-I_{N}\bigr) \bigl(x-H(x)\bigr)+\bigl(x-q^{*}\bigr)\bigr)^{T} \bigl(H(x)-x\bigr) \\ &\quad =\bigl(\bigl(x-F(x)\bigr)-P_{K}\bigl(x-F(x)\bigr) \bigr)^{T}\bigl(q^{*}-H(x)\bigr) \\ &\qquad {}-F^{T} \bigl(q^{*}\bigr) \bigl(x-q^{*}\bigr) -\bigl(F(x)-F\bigl(q^{*}\bigr)\bigr)^{T} \bigl(x-q^{*}\bigr) \\ &\qquad {}-\bigl(x-H(x)\bigr)^{T}\mathcal{J}F(x) \bigl(x-H(x)\bigr). \end{aligned}$$

(23)

Since $F(x)$ is strongly monotonic, we have $(F(x)-F(q^{*}))^{T}\times(x-q^{*})>0$, $\forall x\neq q^{*}$ and $\mathcal{J}F(x)$ is positive semidefinite, which implies $(x-H(x))^{T}\mathcal{J}F(x)(x-H(x))\geq 0$. By using the first property in Lemma 1, we obtain $((x-F(x))-P_{K}(x-F(x)))^{T}(q^{*}-H(x))\leq 0$. Furthermore, it results from (6) that $-F(q^{*})\in N_{K}(q^{*})$, i.e., $-F^{T}(q^{*})(x-q^{*})\leq 0$. Thus, we have

$$ \dot{V}\leq -\bigl(F(x)-F\bigl(q^{*}\bigr)\bigr)^{T} \bigl(x-q^{*}\bigr)+\hat{e},$$

which yields $0 \leq \int _{0}^{\infty} (F(x)-F(q^{*}))^{T}(x-q^{*}) \,dt \leq V(0)+ \int _{0}^{\infty} \|\hat{e}(t)\|\,dt<+\infty$. With the strong monotonicity and the Lipschitz continuity of $F(x)$, we can get that $F(x)$ is uniformly continuous. And $(F(x)-F(q^{*}))^{T}(x-q^{*})=0$ if and only if $x=q^{*}$. Together with the Barbalat’ lemma in [38], we have $\lim_{t\rightarrow \infty} x(t)=q^{*}$.

Next, we prove that the reference trajectories $x_{i}$, $\forall i \in \mathcal{V}$ can be tracked by the practical EL systems with a prescribed accuracy ϵ, i.e.,

$$ \limsup_{t\rightarrow \infty}\bigl\lvert q_{i}^{j}-x_{i}^{j} \bigr\rvert \leq \epsilon , \quad \forall j\in \{1,2,\cdot ,m\}, i \in \mathcal{V}. $$

First, we need to prove that both the state error and velocity error are preserved in their respective constraint sets all the time, i.e.,

$$ \textstyle\begin{cases} \varsigma _{i}^{j}(t) \in (\underline{k}_{i}^{j}, \overline{k}_{i}^{j}), \\ \lvert z_{i}^{j}\rvert < \xi _{i}^{j} ,\end{cases}\displaystyle \forall t\geq 0, j\in \{1,2,\cdot ,m\}, i \in \mathcal{V}. $$

(24)

Suppose by contradiction that there exists $t_{1}>0$ such that the violation of the above inequalities occurs firstly. Since $\lvert z_{i}^{j}(0)\rvert <\xi _{i,0}^{j}$ and $q_{i}^{j}(0)\in (\tilde{q}_{i}^{j},\hat{q}_{i}^{j})$, we have $\varsigma _{i}^{j}(0)\in (\underline{k}_{i}^{j}(0),\overline{k}_{i}^{j}(0))$. Then

$$ \varsigma _{i}^{j}(t) \in \bigl( \underline{k}_{i}^{j},\overline{k}_{i}^{j} \bigr),\qquad \bigl\lvert z_{i}^{j}\bigr\rvert < \xi _{i}^{j}, \quad \forall t< t_{1}. $$

(25)

Choose the Lyapunov function $V_{i}^{j}=\frac{1}{2}(\ln \frac{\varsigma _{i}^{j}-\underline{k_{i}^{j}}}{\overline{k_{i}^{j}}-\varsigma _{i}^{j}})^{2}$, then the derivative of $V_{i}^{j}$ along (7a)–(7h) is

$$\begin{aligned} \dot{V}_{i}^{j}={}& \frac{\ln \frac{\varsigma _{i}^{j}-\underline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\varsigma _{i}^{j}}}{(\varsigma _{i}^{j}-\underline{k}_{i}^{j})(\overline{k}_{i}^{j}-\varsigma _{i}^{j})} \\ &{}\times\bigl(\bigl(\overline{k}_{i}^{j}- \underline{k}_{i}^{j}\bigr)\dot{\varsigma _{i}^{j}}+\bigl( \dot{\underline{k}_{i}^{j}}- \dot{\overline{k}_{i}^{j}}\bigr)\varsigma _{i}^{j}+ \dot{\overline{k}_{i}^{j}} \underline{k}_{i}^{j} - \dot{\underline{k}_{i}^{j}} \overline{k}_{i}^{j}\bigr) \\ ={}& \frac{(\overline{k}_{i}^{j}-\underline{k}_{i}^{j})\ln \frac{\varsigma _{i}^{j}-\underline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\varsigma _{i}^{j}}}{(\varsigma _{i}^{j}-\underline{k}_{i}^{j})(\overline{k}_{i}^{j}-\varsigma _{i}^{j})} \biggl(\Upsilon _{i}^{j}-\ln \frac{\varsigma _{i}^{j}-\underline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\varsigma _{i}^{j}}\biggr) \\ \leq{} & \frac{(\overline{k}_{i}^{j}-\underline{k}_{i}^{j})}{(\varsigma _{i}^{j}-\underline{k}_{i}^{j})(\overline{k}_{i}^{j}-\varsigma _{i}^{j})} \\ &{}\times \biggl(\biggl\lvert \ln \frac{\varsigma _{i}^{j}-\underline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\varsigma _{i}^{j}} \biggr\rvert \bigl\lvert \Upsilon _{i}^{j}\bigr\rvert - \biggl\lvert \ln \frac{\varsigma _{i}^{j}-\underline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\varsigma _{i}^{j}} \biggr\rvert ^{2}\biggr), \end{aligned}$$

(26)

where $\Upsilon _{i}^{j}=z_{i}^{j}-\dot{x}_{i}^{j}+ \frac{\dot{\underline{k}_{i}^{j}}-\dot{\overline{k}_{i}^{j}}}{\overline{k}_{i}^{j}-\underline{k}_{i}^{j}} \varsigma _{i}^{j} + \frac{\dot{\overline{k}_{i}^{j}} \underline{k}_{i}^{j} -\dot{\underline{k}_{i}^{j}} \overline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\underline{k}_{i}^{j}}$. Since $x_{i}\in K_{i}$, it results from (8a) that $\dot{x}_{i}^{j}$ is bounded. Then it is not hard to get the boundedness of $z_{i}^{j}$, $\varsigma _{i}^{j}$, $\dot{\underline{k}_{i}^{j}}$, $\dot{\overline{k}_{i}^{j}}$ and $\epsilon <\overline{k}_{i}^{j}-\varsigma _{i}^{j}< \hat{q}_{i}^{j}- \tilde{q}_{i}^{j}$, which implies that $\Upsilon _{i}^{j}$ is bounded, i.e., there exists a constant $\chi _{i}^{j}$ such that

$$ \bigl\lvert \Upsilon _{i}^{j}\bigr\rvert < \chi _{i}^{j}, \quad \forall t< t_{1}, j \in \{1,2,\cdot ,m\}, i \in \mathcal{V}. $$

(27)

From (26) and (27), $\dot{V}_{i}^{j}<0$ as $\lvert \ln \frac{\varsigma _{i}^{j}-\underline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\varsigma _{i}^{j}} \rvert >\chi _{i}^{j}$, i.e., $V_{i}^{j}> \frac{(\chi _{i}^{j})^{2}}{2}$. As a result, $V_{i}^{j}(t)\leq \max \{V_{i}^{j}(0), \frac{(\chi _{i}^{j})^{2}}{2} \}$, $t< t_{1}$, which yields

$$ \biggl\lvert \ln \frac{\varsigma _{i}^{j}-\underline{k}_{i}^{j}}{\overline{k}_{i}^{j}-\varsigma _{i}^{j}} \biggr\rvert \leq \max \biggl\{ \biggl\lvert \ln \frac{q_{i}^{j}(0)-\tilde{q}_{i}^{j}}{\hat{q}_{i}^{j}-q_{i}^{j}(0)} \biggr\rvert , \chi _{i}^{j}\biggr\} ,\quad t< t_{1}. $$

(28)

Together with the injective property of the nature logarithm function, it is essential that $\varsigma _{i}^{j}$ can not approach $\overline{k}_{i}^{j}$ and $\underline{k}_{i}^{j}$ for any $t< t_{1}$.

Consider the following Lyapunov function $V_{2}^{i}=\frac{\eta _{i}^{T} \eta _{i}}{\pi}$, the derivative of $V_{i}$ along (7a)–(7h) is as follows

$$\begin{aligned} \dot{V}_{i}&=\eta _{i}^{T} (\Psi _{i})^{-1}\bigl(-M_{i}^{-1}(C_{i}+G_{i})- \dot{\check{l}}_{i}-\dot{\breve{l}}_{i}-\gamma _{i}M_{i}^{-1}(\Psi _{i})^{-1} \eta _{i}\bigr), \\ & \leq \bigl\Vert (\Psi _{i})^{-1}\eta _{i} \bigr\Vert \Vert \mathcal{F}_{i} \Vert - \gamma _{i} \eta _{i}^{T} (\Psi _{i})^{-1}M_{i}^{-1}(\Psi _{i})^{-1}\eta _{i}, \end{aligned}$$

(29)

where $\mathcal{F}_{i}=-M_{i}^{-1}(C_{i}+G_{i})-\dot{\check{l}}_{i}- \dot{\breve{l}}_{i}$ with ${\check{l}}_{i}=\operatorname{col}(\ln \frac{\varsigma _{i}^{1}-\underline{k}_{i}^{1}}{\overline{k}_{i}^{1}-\varsigma _{i}^{1}}, \ldots , \ln \frac{\varsigma _{i}^{m}-\underline{k}_{i}^{m}}{\overline{k}_{i}^{m}-\varsigma _{i}^{m}})$ and ${\breve{l}}_{i}=\operatorname{col}( \frac{\dot{\xi}_{i}^{1} z_{i}^{1}}{{\xi}_{i}^{1}},\ldots , \frac{\dot{\xi}_{i}^{m} z_{i}^{1}}{{\xi}_{i}^{m}})$. It is not hard to derive the boundedness of $\mathcal{F}_{i}$, which means that there exists a positive constant $\tilde{\lambda}_{i}$ such that

$$ \Vert \mathcal{F}_{i} \Vert \leq \tilde{ \lambda}_{i},\quad t< t_{1}. $$

(30)

It results from the positive definiteness of $M_{i}$ that there exists a positive constant $\check{\lambda}_{i}$ such that $\eta _{i}^{T} (\Psi _{i})^{-1}M_{i}^{-1}(\Psi _{i})^{-1}\eta _{i} \geq \check{\lambda}_{i} \|(\Psi _{i})^{-1}\eta _{i}\|^{2}$. Then we have

$$\begin{aligned} \dot{V}_{i}&\leq \bigl\Vert (\Psi _{i})^{-1}\eta _{i} \bigr\Vert \Vert \mathcal{F}_{i} \Vert - \gamma _{i}\check{ \lambda}_{i} \bigl\Vert (\Psi _{i})^{-1} \eta _{i} \bigr\Vert ^{2}, \\ & = \bigl\Vert (\Psi _{i})^{-1}\eta _{i} \bigr\Vert \bigl( \Vert \mathcal{F}_{i} \Vert -\gamma _{i} \check{\lambda}_{i} \bigl\Vert (\Psi _{i})^{-1}\eta _{i} \bigr\Vert \bigr). \end{aligned}$$

(31)

It results from (7b) and (7c) that there exists $\check{\xi}_{i}$ such that $\|(\Psi _{i})^{-1}\eta _{i}\|^{2}\geq \frac{1}{(\check{\xi}_{i})^{2}} \|\eta _{i}\|^{2}$. Together with (30) and (31) that $\dot{V}_{i}\leq 0$ as $\|\eta _{i}\|\geq \frac{\tilde{\lambda}_{i} \check{\xi}_{i}}{\gamma _{i} \check{\lambda}_{i}}$. As a result, we obtain that $V_{i}(t)\leq \max \{ V_{i}(0), \frac{1}{\pi}( \frac{\tilde{\lambda}_{i} \check{\xi}_{i}}{\gamma _{i} \check{\lambda}_{i}})^{2} \}$, $t< t_{1}$, which implies

$$ \Vert \eta _{i} \Vert \leq \max \biggl\{ \sqrt{\pi V_{i}(0)}, \frac{\tilde{\lambda}_{i} \check{\xi}_{i}}{\gamma _{i} \check{\lambda}_{i}} \biggr\} ,\quad t< t_{1}. $$

(32)

Due to the injective property of the tangent function (7c), $z_{i}^{j}$ can not approach $\xi _{i}^{j}$, which means $(\Psi _{i})^{-1}$ is bounded. Therefore, the supposition that the state error and velocity error may transgress the prespecified boundary is impossible, and the expected result (24) holds. It follows from (7f) and (7g) that

$$ \bigl\lvert \varsigma _{i}^{j}(t)\bigr\rvert \leq \bigl(\hat{q}_{i}^{j}-\tilde{q}_{i}^{j} \bigr) \bigl[ \bigl(1-\alpha _{i}^{j} \bigr)e^{-\beta t}+\alpha _{i}^{j} \bigr],$$

which yields

$$\begin{aligned} &\limsup_{t\rightarrow \infty} \bigl\lvert \varsigma _{i}^{j}(t) \bigr\rvert = \limsup_{t\rightarrow \infty}\bigl\lvert q_{i}^{j}-x_{i}^{j} \bigr\rvert \leq \epsilon , \\ &\quad \forall j\in \{1,2,\cdot ,m\}, i \in \mathcal{V}. \end{aligned}$$

Additionally, with the boundedness of $\eta _{i}$ and $(\Psi _{i})^{-1}$, we can conclude that the control input $\tau _{i}$ is bounded. The proof is completed. □

4 Simulations

In this section, we give numerical simulations to illustrate the convergence performance of the proposed distributed algorithm in this paper.

Consider an energy resource competition problem among N generating firms in electricity market(see [3, 4] and references therein), which can be formulated by the following aggregative game:

$$ \min_{q_{i}\in \Omega _{i}}J_{i}(q_{i},q_{-i}), \quad \forall i \in \mathcal{V}=\{1,2,\ldots,N\}, $$

where $q_{i}$ is the output power of generation system i, $\Omega _{i}=[\tilde{\nu}_{i}, \hat{\nu}_{i}] \subset \mathbb{R}$ is the local constraint set with $\tilde{\nu}_{i}$ being the local load demand and $\hat{\nu}_{i}$ being the maximum allowed output power. $J_{i}$ is the payoff function of generation system i, which is defined as

$$ J_{i}(q_{i},q_{-i})=a_{i}+ \beta _{i} q_{i}+r_{i} q_{i}^{2} - \bigl( p_{0}-aN \sigma (q) \bigr) q_{i},$$

where $a_{i}+\beta _{i} q_{i}+r_{i} q_{i}^{2}$ is the generation cost with $a_{i}$, $\beta _{i}$, $r_{i}$ being the characteristics of generation system i, and $p_{0}-aN\sigma (q)$ is the electricity price with constants $p_{0}$, a and the aggregate term $\sigma (q)=\frac{\sum_{i=1}^{N} q_{i}}{N}$ describing the average output power.

The dynamics of turbine-generator i can be described as follows [39]:

$$\begin{aligned} \ddot{q}_{i}={}&{-}\frac{T_{mi}+T_{ei}}{T_{mi} T_{ei}}\dot{q}_{i} - \frac{1}{T_{mi} T_{ei}}q_{i} + \frac{K_{mi}}{T_{mi} T_{ei}}\tau _{i} \\ &{}- \frac{K_{mi} K_{ei}}{T_{mi} T_{ei}R_{i} d_{0}} d_{i}, \end{aligned}$$

(33a)

$$\begin{aligned} \dot{d}_{i}={}&{-}\frac{D_{i}}{2H_{i}}d_{i}, \end{aligned}$$

(33b)

where $d_{i}$ is the relative speed of system i; $d_{0}$ is the synchronous machine speed; $R_{i}$ is the regulation constant of turbine; $K_{mi}$ is the gain of turbine i; $T_{mi}$ and $T_{ei}$ are the time constants of the ith turbine and speed governor, respectively; $D_{i}$ and $H_{i}$ are the positive per unit damping constant and inertia constant,respectively; $\tau _{i}$ is the control input of the ith generation system. The zero-system of (33a) is

$$ \ddot{q}_{i}=-\frac{T_{mi}+T_{ei}}{T_{mi} T_{ei}} \dot{q}_{i} - \frac{1}{T_{mi} T_{ei}}q_{i} + \frac{K_{mi}}{T_{mi} T_{ei}}\tau _{i}, $$

(34)

which can be rewritten as the standard EL equation (3) with $M_{i}=\frac{T_{mi} T_{ei}}{K_{mi}}$, $C_{i}=\frac{T_{mi}+ T_{ei}}{K_{mi}}$ and $G_{i}=\frac{1}{K_{mi}} q_{i}$. It follows from Theorem 7 that the zero-system (34) exponentially converges to the variational NE of (4) under the proposed algorithm (7a)–(7h), (8a)–(8c), (9). Since $D_{i}$ and $H_{i}$ are positive, $d_{i}$ vanishes exponentially, which implies that the generator system (33a)–(33b) also exponentially converges to the same variational NE via algorithm (7a)–(7h), (8a)–(8c), (9). Therefore, the considered electricity market game problem can be solved by our formulation and algorithm. Next, we present numerical simulations to verify the results.

Take the aggregative game with 5 generation systems under an unbalanced digraph, whose communication topology is described in Fig. 2, and the weighted adjacency matrix $A=[a_{ij}]$ is given by $a_{ij}=1$, if $(i,j)\in \mathcal{E}$.

Figure 2

Unbalanced communication topology

Taking $p_{0}=25$, $a=0.1$, $(a_{1},\ldots ,a_{5})=(5,8,6,9,7)$, $(\beta _{1},\ldots , \beta _{5})=(12,10,11,11,13)$, $(r_{1},\ldots ,r_{5})=(1,5,0.8,2,1.1)$, $\Omega _{i}=[-10,10]$, $K_{i}=[-0.99,0.99]$, $\forall i=1,\ldots,5$, $d_{0}=314.159$. The variational NE of aggregative game is $q^{*}=(5.2036,1.2799, 7.0162,2.9091 ,4.3163)$. The generator system and controller parameters are given in Table 1. The initial values are given as $s_{i}(0)=y_{i}(0)=\dot{q}_{i}(0)=0$, $\alpha _{i}=0.005$, $\beta =0.2$, $\xi _{i,0}=4$, $\xi _{i,\infty}=0.12$, $\gamma _{i}=0.05$, $\rho _{i}=1$, $i\in \{1,\ldots,5\}$.

Table 1 System and controller parameters

Firstly, let us study the convergence performance of our proposed algorithm compared with the distributed Nash equilibrium algorithm (13) in [4]. As show in Fig. 3 and Fig. 4, the output power trajectories of five generator systems converge precisely to the variational EN $q^{*}$ of the considered aggregative game under the unbalanced digraph Fig. 2 via the proposed algorithm, and all of the velocity trajectories converge to the zero. This is because $q^{*}$ is the inner point of the constraint set Ω, and the slight scaling of the boundaries does not affect the Nash equilibrium of the considered problem. However, it can be seen from Fig. 5 that the distributed Nash equilibrium algorithm (13) in [4] can not guarantee the output power trajectories of five generator systems converge to the variational EN $q^{*}$ of the considered problem, and some of the output power trajectories are outside the safety constraints. Therefore, the algorithm (13) in [4] is not suitable for solving set constrained games and non-balance situations. In contrast, an integrated distributed protocol based on fault compensation scheme and projected dynamics is proposed in this paper to conquer the difficulties caused by set constraints. Figure 6 gives the evolutions of control torques of five EL systems under the proposed protocol, showing that the control torque $\tau _{i}$ of each system is bounded. Moreover, from Figs. 7 and 8, it can be observed that all the trajectories of EL systems converge to the optimal reference path generated by the cooperative game optimizer under the local output-constrained controller (7a)–(7h), and the traking errors converge to 0 along with the imposed boundaries.

Figure 3

The output power trajectories of generator systems (33a)–(33b) under the proposed protocol

Figure 4

The velocity trajectories of generator systems (33a)–(33b) under the proposed protocol

Figure 5

The output power trajectories of generator systems (33a)–(33b) under the algorithm (13) in [4]

Figure 6

The control torques of generator systems (33a)–(33b) under the proposed protocol

Figure 7

The tracking errors of generator systems to the reference path generated by the cooperative game optimizer

Figure 8

The tracking errors along with the imposed boundaries

Next, we randomly choose another strongly connected communication graph Fig. 9 and select the same initialization conditions with Figs. 3 and 4. It can be seen from Figs. 10 and 11 that the output power trajectories of five EL systems also converge precisely to the variational EN $q^{*}$ under the proposed algorithm, supporting the theoretical result that our proposed algorithm can handle the unbanlanced strongly connected case.

Figure 9

Unbalanced communication topology

Figure 10

The output power trajectories of generator systems (33a)–(33b) under the proposed protocol

Figure 11

The velocity trajectories of generator systems (33a)–(33b) under the proposed protocol

To show the influence of the initial states on the convergence of the proposed algorithm, we randomly generate six different initial values, and Fig. 12 gives the evolutions of output powers in different cases, which implies that for any initial states $q_{i}(0)\in \Omega _{i}$, their values do not affect the convergence of the proposed algorithm. Figures 13 and 14 show the evolutions of output power trajectories and control torques under different control gains $\gamma _{i}=0.01,0.1,1,10,20,30$, respectively. From Fig. 14, we can observe that the control torques are bounded, and the size of the bound will decrease as the control gain increases, which implies that the proposed algorithm is also applicable to physical systems with control input constraints or limited actuators. Therefore, the above simulation results verify the effectiveness of our proposed protocol and support the theoretical results.

Figure 12

The output power trajectories with different initial values $q_{i}(0)$

Figure 13

The output power trajectories with different control gains $\gamma _{i}$

Figure 14

The control torques with different control gains $\gamma _{i}$

5 Conclusions

In this paper, we considered the distributed aggregative games for multiple heterogeneous EL systems under unbalanced digraph. We proposed a distributed protocol for uncertain nonlinear EL systems to seek the variational GNE based on the output-constrained control, projection algorithm and dynamic average consensus techniques, and gave the corresponding convergence results. Simulation results verified the effectiveness of our proposed protocol. Future works include the aggregative game for high-order nonlinear multi-agent dynamics with constraints.