Group Hybrid Coordination Control of Multi-Agent Systems With Time-Delays and Additive Noises

I. Introduction

COORDINATION control of multi-agent systems (MASs) has become an important theme in the control community, including consensus control, containment control and so on. For these problems, fruitful achievements have been obtained. Under time-delays, many consensus results have been obtained [1]–[4]. In general, time-delays have negative impact on consensus, but Yu et al. [4] revealed that appropriate time-delays can play a positive role in delay-induced consensus problems. The effects of additive and multiplicative measurement noises on consensus behaviors were revealed in [5]–[8], [9] and [10], respectively. The consensus problems were studied for MASs with both time-delays and additive noises in [11]–[14].

As a kind of coordination control problem, the containment problem has also received extensive attention. Under a relatively ideal environment, many interesting results about containment problems have been obtained, as shown in [15]–[18]. When time-delays exist in MASs, the research results for the containment problem can be seen in [19]–[21]. The effects of additive noises on containment behaviors were revealed in [22] and the containment control of MASs with both time-delays and additive noises was studied in [23].

It can be seen that the above studies all focused on the problem where agents in a whole system achieve global coordination behavior. However, in some practical application scenarios where multiple tasks such as tracking and striking need to be performed simultaneously, global coordination control may not be able to accomplish these tasks. Hence, it is necessary to investigate group coordination control problems such as group consensus and group containment problems. These require agents in different subgroups to achieve different coordination states. In fact, many scholars have made efforts for solving group coordination control problems. Yu and Wang [24] and [25] studied group consensus under the assumption that the effect between the two subgroups is balanced. Then, the assumption was relaxed in [26] and [27], where the group consensus of MASs with generally connected topology were investigated. Some interesting results about group consensus under measurement noises can be seen in [28] and [29]. The results about the bipartite consensus problem can be found in [30]–[32].

Note that the aforementioned works about group coordination control all focused on the problem that all subgroups achieve consensus behavior. However, agents in different subgroups may be required to achieve different coordination behaviors for some multi-objective tasks such as military tasks [33]. Moreover, for opinion dynamics in social networks, there may appear agreement or fluctuation which need to be characterized by different coordination behaviors in a whole organization [34]. Hence, it is natural to think about the conditions that consensus can be achieved for one subgroup and containment be achieved for another subgroup in a whole MAS. This problem is called group hybrid coordination control problem, which has not been addressed, even in an ideal environment.

In this paper, we study the group hybrid coordination control problem of MASs with both time-delays and additive noises. We consider a whole MAS that consists of two subgroups, where communications between the two subgroups are unidirectional and only one subgroup can receive information from another subgroup. The case where a subgroup receiving information from another subgroup is required to achieve containment is called the containment-oriented case. Similarly, the case where the subgroup receiving information from another subgroup is required to achieve consensus is called the consensus-oriented case. Since there are communications between the two subgroups and different subgroups are required to achieve different coordination behaviors for a whole MAS, the system dynamics cannot simply be analyzed as a whole. This makes the existing control protocols, methods and conditions, such as those in [14], [22], and [24]–[32], fail to solve the group hybrid coordination control problem. Then, in this paper, our goal is to explore appropriate analysis methods and obtain some conditions for solving the new kind of group coordination control problem, i.e., the group hybrid coordination control problem of MASs with time-delays and additive noises.

The main contribution of this paper can be concluded as follows.

1) A new kind of group coordination control problem-group hybrid coordination control is investigated and two new group control protocols are designed to solve this problem. By developing a new analysis idea, some sufficient conditions and necessary conditions are obtained for the group hybrid coordination control problem under time-delays and additive noises. It is found that the conditions for the containment-oriented case are weaker than those for the consensus-oriented case. And for the consensus-oriented case, the conditions for weak consensus are weaker than those for strong consensus.

2) The influence mechanism of the communication impact between the two subgroups on group hybrid coordination control problem of MASs with both time-delays and additive noises is revealed: the decay of the communication impact between the two subgroups is necessary for the consensus-oriented case. This provides hints in the design of control gain functions for the similar group coordination control problem.

The structure of this paper is organized like this: In Section II, the system model, some basic concepts, and lemmas are introduced. Sections III and IV investigate the containment-oriented case and the consensus-oriented case, respectively. Some simulation results are given in Section V. Section VI summarizes this work and gives some other interesting questions for future research.

Notations: The following notations are used in the process of analysis. ${\bf{1}}_n$ and $\vartheta_{n,i}$ are two n-dimensional column vectors, where each element is one for ${\bf{1}}_n$, and the ith element is one and others are zero for $\vartheta_{n,i}$. The m-dimensional identity matrix is denoted by $I_m$. For $a, b \in \mathbb{R}$, $a \wedge b =$ min$\{a, b\}$. For any complex number λ in complex space $\mathbb{C}$, its real part is denoted by $Re(\lambda)$. For the matrix or vector P, its transpose and Euclidean norm are denoted by $P^T$ and $\Vert P\Vert$, respectively. For a given random variable or vector X, its mathematical expectation is denoted by $\mathbb EX$.

II. Preliminaries

Let ${\cal{ G}}=({\cal{ V}},{\cal{E}},{\cal{A}})$ be a weighted directed graph, where ${\cal{ V}}=\{1,\ldots,i,\ldots,n\}$ is the set of nodes with i representing the ith agent, $\mathcal{E}\subseteq {\cal{ V}}\times {\cal{ V}}$ denotes the set of edges and $(i,j)\in {\cal{E}}$ is a edge of ${\cal{ G}}$. ${\cal{A}}=[a_{ij}]\in \mathbb R^{n\times n}$ is the weighted adjacency matrix with real adjacency weighting elements $a_{ij} = 1$ or $0$ indicating whether or not there is an information flow from agent j to agent i directly. Moreover, we assume $a_{ii}=0$ for all $i\in {\cal{ V}}$. The set of neighbors of node i is denoted by $N_i$, that is, $a_{ij}=1$ for $j\in N_i$. The Laplacian matrix of ${\cal{ G}}$ is denoted by ${\cal{ L}}$. A directed path is a sequence of edges with the form $(i-1,i),\;(i,i+1),\; ...$. A directed graph contains a directed spanning tree if there exists at least one node that has a directed path to all other nodes. In the directed graph ${\cal{ G}}$, the node which has no parent node is called a root. A digraph ${\cal{ G}}$ has a spanning forest if there exists a directed forest which is the spanning subgraph of ${\cal{ G}}$. More details about the spanning subgraph, directed forest and spanning forest can be found in [22].

In this paper, $n+m\; (n \ge 1, m > 2)$ first-order continuous-time agents compose a MAS with a directed communication graph ${\cal{ G}}=({\cal{ V}},{\cal{E}},{\cal{A}})$. We divide ${\cal{ G}}=({\cal{ V}},{\cal{E}},{\cal{A}})$ into two subgraphs ${\cal{ G}}_1=({\cal{ V}}_1,{\cal{E}}_1,{\cal{A}}_1)$ and ${\cal{ G}}_2=({\cal{ V}}_2,{\cal{E}}_2,{\cal{A}}_2)$, where ${\cal{ V}}_1=\{1,\ldots, n\}$, ${\cal{ V}}_2=\{n+1,\ldots,n+m\}$ and ${\cal{E}}_1={\cal{ V}}_1\times {\cal{ V}}_1$, ${\cal{E}}_2={\cal{ V}}_2\times {\cal{ V}}_2$, that is, the first n agents belong to one subgroup, which is denoted by $\Psi_1$, and the last m agents belong to another subgroup, which is denoted by $\Psi_2$. Moreover, let $N_i=N_{1i}\cup N_{2i}$, where $N_{1i}\subseteq {\cal{ V}}_1$ and $N_{2i}\subseteq {\cal{ V}}_2$ are the sets of neighbors belonging to $\Psi_1$ and $\Psi_2$ of the ith agent respectively. Here, each agent has the following dynamics:

where $x_i(t)\in\mathbb{R}$ and $u_i(t)\in\mathbb{R}$ denote the state and control input of agent i, respectively. Denote $X_1(t)=[x_1(t),\ldots,x_n(t)]^T$, $X_2(t)=[x_{n+1}(t),\ldots,x_{n+m}(t)]^T$.

In this paper, we require the agents in subgroup $\Psi_1$ and subgroup $\Psi_2$ to achieve consensus and containment, respectively. Then, we know that there should be several leaders in subgroup $\Psi_2$. We assume that agents $n+1,\ldots,n+a\; (1<a<m)$, who have no neighbor, are stationary leaders and the other agents $n+a+1,\ldots,n+m$ are followers. Denote $X_{21}(t)=$ $[x_{n+1}(t),\ldots,x_{n+a}(t)]^T$, $X_{22}(t)=[x_{n+a+1}(t),\ldots,x_{n+m}(t)]^T$. Then, for the communication topology subgraphs ${\cal{ G}}_1$, ${\cal{ G}}_2$ and the communications between the two subgroups, we give the following assumption.

Assumption 1: ${\cal{ G}}_1$ contains a spanning tree and ${\cal{ G}}_2$ has a spanning forest whose roots are exactly the leaders in MASs. Moreover, the communications between the two subgroups are unidirectional and only one subgroup can receive information from another subgroup.

In the communication environment considered in this paper, the exchange of state information between agents can not be performed accurately. It is often assumed that the agent i can receive information from its neighbors as follows:

where $y_{ji}(t)$ denotes the measured value of the jth agent’s state by the ith agent, $\tau \ge 0$ is the time-delay, $\eta_{ji}(t) \in \mathbb{R}$ denotes the measurement noise and $\sigma_{ji}$ is the corresponding intensity function. Here, let the initial states be constants, $X_1(t)=\Delta_1 \in \mathbb R^n$, $X_2(t) = \Delta_2 \in \mathbb R^m$, $X_{21}(t) = \Delta_{21} \in \mathbb R^a$ and $X_{22}(t)=\Delta_{22} \in \mathbb R^{m-a}$, $t\in [-\tau,0]$.

In this paper, independent Gaussian white noises are used to model the measurement noises.

Assumption 2: The stochastic process $\eta_{ji}(t)$ satisfies $\int_{0}^{t} \eta_{ji}(s)ds = \omega_{ji}(t),\;t\ge 0,\;i,j = 1,\ldots,n+m$, where $\{\omega_{ji}(t),\;i,j = 1,\ldots,n+m\}$ are independent Brownian motions.

In this paper, the goal of group hybrid coordination control is to achieve consensus for subgroup $\Psi_1$ and containment for subgroup $\Psi_2$. Below, we give the definitions of consensus and containment, respectively.

Definition 1 [14]: For all distinct agents $i,j$ in a subgroup Ψ, if $\lim_{t\to \infty} \mathbb E{|x_i(t) - x_j(t)|}^2= 0$ (or $\lim_{t\to \infty} |x_i(t) - x_j(t)|= 0$ almost surely (a.s.)), then the agents in this subgroup are said to achieve mean square weak consensus (or almost sure weak consensus).

Definition 2 [14]: For all agents i in a subgroup Ψ, if there is a random variable $\bar x^*$ such that $\mathbb E|\bar x^*| < \infty$ (or $\mathbb P\{|\bar x^*| < \infty\} = 1$) and $\lim_{t\to \infty} \mathbb E{|x_i(t)-\bar x^*|}^2 = 0$ (or $\lim_{t\to \infty}{|x_i(t) - \bar x^*|} = 0$, a.s.), then the agents in this subgroup are said to achieve mean square strong consensus (or almost sure strong consensus).

Definition 3 [22]: For all followers i in a subgroup Ψ, if there exist deterministic variables $\bar x_i^*\in co_L \triangleq co$ $\{x_{(n+1)f},\ldots,$ $x_{(n+a)f}\}= \{x_f|x_f = \sum_{q=1}^{a} k_qx_{(n+q)f}, k_q\ge 0,\sum_{q=1}^{a} k_q = 1\}$ ($co_L$ denotes the convex hull spanned by the leaders’ final positions $x_{jf}=\lim_{t\to \infty}x_j(t), \;j=n+1,\ldots,n+a$) such that $\mathbb E|\bar x_i^*| < \infty$ (or $\mathbb P\{|\bar x_i^*| < \infty\} = 1$) and $\lim_{t\to \infty}\mathbb E|x_i(t)-\bar x_i^*|^2 = 0$ (or $\lim_{t\to \infty}|x_i(t)- \bar x_i^*|= 0$, a.s.), $i=n+a+1,\ldots,n+m$, then the agents in this subgroup are said to achieve mean square containment (or almost sure containment).

Remark 1: In general, under a stochastic background system, the coordination control problems of MASs are studied in the mean square and almost sure senses. Compared with mean square coordination behaviors, almost sure coordination behaviors are more intuitive and acceptable. This paper will study group hybrid coordination control problem in both senses.

Based on above definitions of consensus and containment, we make the following statement: the agents in a subgroup Ψ are said to achieve stochastic consensus (or stochastic containment) if the agents in this subgroup achieve consensus (or containment) in both the mean square and almost sure senses. Then, in this paper, the following two types of group hybrid coordination behavior will be investigated for both the containment-oriented case and consensus-oriented case: 1) Agents in subgroup $\Psi_1$ achieve stochastic weak consensus and agents in subgroup $\Psi_2$ achieve stochastic containment; 2) Agents in subgroup $\Psi_1$ achieve stochastic strong consensus and agents in subgroup $\Psi_2$ achieve stochastic containment. As mentioned above, we will study group hybrid coordination control in both the above two senses. These studies will enrich the group coordination control problem.

To study group hybrid coordination control under time-delays and additive noises, we need two important lemmas as follows.

Lemma 1 [14]: Suppose that the directed graph contains a spanning tree. For the Laplacian matrix ${\cal{ L}}'$, we have the following statements:

1) There exists a probability measure π such that $\pi^T {\cal{ L}}' = 0$.

2) There exists a nonsingular matrix $Q = (({1}/{\sqrt n})\mathbf 1_n,\widetilde{Q})$ and

where n is the number of nodes, $\overline{Q}\; \in \;\mathbb{R}^{(n-1)\times n}$, $\widetilde {{\cal{ L}}}' \in$ $\mathbb{R}^{(n-1)\times (n-1)}$. For $\upsilon$, one has $\upsilon^T{\cal{ L}}'=0$ and $({1}/{\sqrt n})\upsilon^T\mathbf 1_n=1$.

3) The directed graph contains a spanning tree if and only if each eigenvalue of $\widetilde{{\cal{ L}}}'$ has positive real part. Moreover, if the directed graph contains a spanning tree, then the probability measure π is unique and $\upsilon=\sqrt n\pi$.

Lemma 2 [14]: For the linear scalar equation

$\bar X(t)=\xi(t)$ for $t\in[-\tau,0]$, where $Re(\lambda)>0, \tau \ge 0$ and $\xi \in C([-\tau, 0],\mathbb{C})$, $C([-\tau,0],\mathbb{C})$ is the space of all continuous $\mathbb{C}$-valued functions defined on $[-\tau,0]$. The solution has the form $\bar X(t)=\Gamma(t,s) \bar X(s),\;\; \forall t\ge s\ge 0$, where $\Gamma(t,s)$ is the differential resolvent function, satisfying $\Gamma(t,t)=1$ for $t>0$, $\Gamma(t,s)=0$ for $t<s$ and

If there is a constant $t_0\ge 0$ such that $\tau \bar c_{t_0}({\vert \lambda\vert^2}/{Re(\lambda)} )< 1$, then

here, $\bar c_{t_0}:={\rm{sup}}_{t\ge t_0}c(t), t_0\ge 0$. $b(\lambda)$ is a positive constant depending on λ and $\varrho(\lambda):=\rho_1(\lambda)\wedge \rho_2(\lambda)$, where $\rho_1(\lambda)$ is the unique root of the equation $3\rho \vert\lambda\vert^2 \tau^2 \bar c_{t_0}^2 e^{\rho \bar c_{t_0}\tau}+2\rho- 2(Re(\lambda)- \vert\lambda\vert^2\tau\bar c_{t_0})=0$ and $\rho_2(\lambda)=({1}/{\bar c_{t_0}\tau}){\rm {log}}({1}/{\vert\lambda\vert \bar c_{t_0}\tau})$.

Remark 2: Lemmas 1 and 2 are powerful for analyzing the group hybrid coordination control problem of MASs with time-delays and additive noises. In fact, after fully mining the properties of $\mathcal{L}'$, Lemma 1 is naturally obtained. Moreover, with the help of Lemma 2, we can not only analyze the stability of stochastic systems with time-delays, but also reveal the decay rate.

In this paper, according to the topology structure of ${\cal{ G}}$, the Laplacian matrix ${\cal{ L}}$ can be written as

where ${\cal{ L}}=\left[l_{ij}\right]$ is defined as

and ${\cal{ L}}_{11}\in \mathbb R^{n\times n}$ is the Laplacian matrix of ${\cal{ G}}_1$, ${\cal{ L}}_{22}\in$ $\mathbb R^{m\times m}, \;{\cal{ L}}_{12}\in \mathbb R^{n\times m}$ and ${\cal{ L}}_{21}\in \mathbb R^{m\times n}$. Here, we define

where ${\cal{{\widehat{L}}}}_{21}\in \mathbb R^{(m-a)\times n},\; {\cal{\widetilde{L}}}_{22}\in \mathbb R^{(m-a)\times a}$ and ${\cal{{\widehat{L}}}}_{22}\in \mathbb R^{(m-a)\times (m-a)}$.

Then, in next two sections, we will study the group hybrid coordination control problem in detail. By examining the containment-oriented case and consensus-oriented case respectively, we will obtain some useful results and find some interesting phenomena for group hybrid coordination behaviors under time-delays and additive noises.

III. The Containment-Oriented Case

For the containment-oriented case, we give the following assumption for the communications between the two subgroups.

Assumption 3: The followers in subgroup $\Psi_2$ can measure the state information from the agents in subgroup $\Psi_1$.

Then, under this communication model, we assume that each agent can receive its neighbor information in the following form:

where $\bar y_{ji}(t)$ and $\bar{\bar y}_{ji}(t)$ denote the state measurement of agent j by the ith agent in subgroup $\Psi_1$ and $\Psi_2$, respectively. $\bar{\bar y}_{ji1}(t)=x_j(t-\tau)+\sigma_{ji}\eta_{ji}(t), \;i\in {\cal{ V}}_2,\;j\in N_{2i}$, is the state measurement of agent j belonging to subgroup $\Psi_2$ by agent i, and $\bar{\bar y}_{ji2}(t)=\kappa(t)(x_j(t-\tau)+\sigma_{ji}\eta_{ji}(t)), \;i\in {\cal{ V}}_2,\;j\in N_{1i}$, is the state measurement of agent j belonging to subgroup $\Psi_1$ by agent i. $\kappa(t)$ is a time-varying function, which reflects the communication intensity between subgroup $\Psi_1$ and subgroup $\Psi_2$.

Remark 3: Note that there are communications between the two subgroups. This means that the MAS is still a whole system. Under the assumption that the in-degrees from the other subgroups are equal at any time for each node in one subgroup, Shang [28] concluded that the group consensus behavior can be reached under appropriate time-delay and multiplicative noise intensity. Under time-delays and additive noises, we determine to study the group hybrid coordination control problem when the assumption mentioned above is released. This is a key problem that will be explored in this paper.

Based on interactions between any two agents in the environment with both time-delays and additive noises, we propose the control protocol as follows:

where $t>0$, $c_1(t)$, $c_2(t)\in C((0,\infty);[0,\infty))$ are two control gain functions to be designed.

Remark 4: Obviously, the communications between agents in the same subgroup and in different subgroups are considered in this protocol. In fact, the group control protocol (2), which considers time-delay, noise and the communication intensity $\kappa(t)$, is the generalization of the protocol in [24]. Also, the control protocol (2) is different from protocols in [14], [22], [23], and [28]. In fact, these protocols fail to solve the group hybrid coordination control problem. Moreover, we can see that there may be some coupling relationships between the time-varying function $\kappa(t)$ and control gain functions $c_1(t), c_2(t)$ from protocol (2). This may make it more difficult to analyze the group hybrid coordination control problem.

Under protocol (2), from (1) we can obtain the following equations:

and

where $\overline {{\cal{ L}}}_{22}=\left[\bar l_{ij}\right]$ is the Laplacian matrix of ${\cal{ G}}_2$, which is defined as

Moreover, we rewrite $\overline {{\cal{ L}}}_{22}$ as

where ${\cal{\widetilde{L}}}_{22}\in \mathbb R^{(m-a)\times a},\; \widehat{{\cal{L}}}_{22}'\in \mathbb R^{(m-a)\times (m-a)}$.

From Assumption 1 we know that each eigenvalue of $\widehat{{\cal{L}}}_{22}'$ has positive real part [22]. Let $Q_1 = (({1}/{\sqrt n})\mathbf 1_n,\widetilde Q_1)$ and $Q_1^{-1} = \begin{pmatrix} \upsilon_1^T \\ \overline Q_1 \end{pmatrix},\;\;Q_1^{-1} {\cal{ L}}_{11}Q_1 = \begin{pmatrix} 0 & 0 \\ 0 & \mathcal{\widetilde L}_{11} \end{pmatrix}$, where $\upsilon_1=\sqrt n\pi_1$ $=[\upsilon_{11},\ldots, \upsilon_{1n}]^T$, $\pi_1^T{\cal{ L}}_{11}=0$. Let $\{\lambda_j\}{}_{j=2}^{n}$ and $\{\lambda_k\}{}_{k=1}^{m-a}$ be the eigenvalues of $\mathcal{\widetilde L}_{11}$ and $\widehat{{\cal{L}}}_{22}'$, respectively. Then, we give the following assumption.

Assumption 4: There exists a constant $t_0\ge 0$ such that $\tau \tilde c_{t_0}$max$_{2\le j\le n}{\vert \lambda_j\vert^2}/{Re(\lambda_j)} < 1$ and $\tau \tilde c_{t_0}'$ max$_{1\le k\le m-a}{\vert \lambda_k\vert^2}/{Re(\lambda_k)} <$ 1, where $\tilde c_{t_0}={\rm{sup}}_{t\ge t_0}c_1(t)$ and $\tilde c_{t_0}'={\rm{sup}}_{t\ge t_0}c_2(t)$.

A Stochastic Weak Consensus + Stochastic Containment

According to the interactions between agents in subgroup $\Psi_1$, we know that the consensus problem of subgroup $\Psi_1$ is the general global consensus problem. Then, the following conditions on $c_1(t)$ were explored to solve the global stochastic weak consensus problem [5], [14]:

(C1) $\int_{0}^{\infty} c_1(t)dt = \infty$;

(C2) $\lim_{t\to \infty} c_1(t) = 0$;

(C3) $\lim_{t\to \infty}c_1(t)\log \int_{0}^{t}c_1(s)ds=0$.

Remark 5: For MASs with both time-delays and additive noises, conditions (C1)−(C3) have been explored for the weak consensus problem in both the two senses mentioned above [14]. It can be seen that the form of condition (C3) is more complex, but it can reflect the decay rate of the control gain function accurately and help us to design a more suitable control protocol.

However, for subgroup $\Psi_2$, since the anticipant behavior is different from subgroup $\Psi_1$ and the state information from subgroup $\Psi_1$ may be available for the followers, we also need to consider the effect of the information from subgroup $\Psi_1$. Note that the existing conditions are not sufficient to analyze the containment control problem of subgroup $\Psi_2$. Then, we will analyze the effect of communications between the two subgroups and redesign control gain functions such that the MASs can achieve group hybrid coordination behaviors under time-delays and additive noises. We first give some conditions for $c_2(t)$ and introduce a new condition (C7) on $c_1(t)$ and $\kappa(t)$ as follows:

(C4) $\int_{0}^{\infty} c_2(t)dt = \infty$;

(C5) $\lim_{t\to \infty} c_2(t) = 0$;

(C6) $\lim_{t\to \infty}c_2(t)\log \int_{0}^{t}c_2(s)ds=0$;

(C7) $\lim_{t\to \infty} \kappa^2(t)\int_{0}^{t}c_1^2(s)ds=0$.

Remark 6: Conditions (C4)−(C6) are sufficient for both the global stochastic weak consensus and stochastic containment. For solving the group hybrid coordination control problem of MASs with both time-delays and additive noises, condition (C7) related to the communication intensity between the two subgroups is given. It reveals the influence of the communication impact between the two subgroups and gives a hint that we can impose the joint condition on $\kappa(t)$ and $c_1(t)$ to solve the group hybrid coordination control problem for the containment-oriented case.

Theorem 1: For the MAS (1), suppose that Assumptions 1−4 hold. Then, under protocol (2), the MAS (1) can achieve group hybrid coordination behavior with stochastic weak consensus and stochastic containment if (C1)−(C7) hold.

Proof: Here, we will examine the two subgroups respectively. First, we consider the agents in subgroup $\Psi_1$. Let $J_n= ({1}/{\sqrt n})\mathbf 1_n \upsilon_1^T$. By the properties of matrix ${\cal{ L}}_{11}$ and Lemma 1, we have ${\cal{ L}}_{11}\mathbf 1_n = 0$, $\upsilon_1^T{\cal{ L}}_{11} = 0$ and $(I_n-J_n){\cal{ L}}_{11} = {\cal{ L}}_{11} = {\cal{ L}}_{11}(I_n-$ $J_n)$. Let $\psi(t)=(I_n-J_n)X_1(t)=[\psi_1(t),\psi_2(t),\ldots,\psi_n(t)]^T$, where $\psi_i(t)\in \mathbb{R},\;i=1,\ldots,n$. Define $\tilde\psi(t)=Q_1^{-1}\psi(t)=[\tilde\psi_1(t),\tilde\psi_2(t),\ldots, \tilde\psi_n(t)]^T$, $\bar\psi(t)=[\tilde\psi_2(t),\ldots,\tilde\psi_n(t)]^T$, $\tilde \psi_i(t)\in \mathbb R, \;i=1,\ldots,n$. By the definition of $Q_1^{-1}$ in Lemma 1, we can get $\tilde\psi_1(t)=\upsilon_1^T\psi(t)= \upsilon_1^T(I_n-J_n)X_1(t)=0$ and

where $M(t) = \sum_{i,j=1}^{n}a_{ij}\sigma_{ji}\overline Q_1(I_n-J_n)\vartheta_{n,i}$$\int_{0}^{t}c_1(s)d\omega_{ji}(s)$. Then, by the semi-decoupled skill, variation of constants formula, and some important estimation methods, we can get $\lim_{t\to \infty} \mathbb E{\Vert \bar \psi(t)\Vert}^2 = 0$ and $\lim_{t\to \infty} \Vert \bar \psi(t)\Vert = 0$ from (C1)−(C3) directly [14]. Then we can come to the conclusion that the agents in subgroup $\Psi_1$ can achieve stochastic weak consensus. Now we consider the stochastic containment problem of subgroup $\Psi_2$. Let $\gamma(t) =Q_1^{-1}X_1(t)$ $=[\gamma_1(t),\gamma_2(t),\ldots,\gamma_n(t)]^T$, $\bar \gamma(t) = [\gamma_2(t),\ldots,\gamma_n(t)]^T$, where $\gamma_i(t)\in \mathbb{R},\;i=1,\ldots,n$. We can obtain $d\gamma_1(t)=\upsilon_1^T c_1(t) \sum_{i,j=1}^{n}a_{ij}\sigma_{ji}\vartheta_{n,i}d\omega_{ji}(t)$. Then, $\gamma_1(t) = \gamma_1^*+ \upsilon_1^T$ $\sum_{i,j=1}^{n}a_{ij}\sigma_{ji}\vartheta_{n,i}\int_{0}^{t} c_1(s)d\omega_{ji}(s)$, where $\gamma_1^*=\upsilon_1^T \Delta_1$. We also have

where $M'(t)=\sum_{i,j=1}^{n}a_{ij}\sigma_{ji}\overline Q_1\vartheta_{n,i}\int_{0}^{t}c_1(s)d\omega_{ji}(s)$. From [7], $\lim_{t\to \infty}\bar\gamma(t)= [0,\ldots,0]^T$ can be proved easily. From (4), we have

and

where $V_1(t)=\sum_{i=n+a+1}^{n+m}\sum_{j=n+1}^{n+m} a_{ij}\sigma_{ji}\vartheta_{m-a,i-n-a}d{\omega_{ji}} (t)$, and $V_2(t)=\sum_{i=n+a+1}^{n+m}\sum_{j=1}^{n} a_{ij}\sigma_{ji}\vartheta_{m-a,i-n-a}d\omega_{ji}(t)$. Letting $\overline X_{22}(t)= X_{22}(t)+\widehat{{\cal{L}}}_{22}'^{-1}{\cal{\widetilde{L}}}_{22}X_{21}(t)$, we obtain

From the above analysis, we have

where $P={\cal{{\widehat{L}}}}_{21}Q_1$. From the matrix theorem we have $R\widehat{{\cal{L}}}_{22}'R^{-1} = J$, where R is a complex invertible matrix, and J is the Jordan normal form of $\widehat{{\cal{L}}}_{22}'$. And we know $J =$ $diag(J_{\lambda_1,n_1},\ldots,J_{\lambda_l,n_l})$, $\sum_{k=1}^{l} n_k = m-a$, where $\lambda_1,\ldots,\lambda_l$ are all the eigenvalues of $\widehat{{\cal{L}}}_{22}'$ and $J_{\lambda_k,n_k}$ is the Jordan block corresponding to eigenvalue $\lambda_k$, and its dimension is $n_k$. Letting $Y(t)=R\overline X_{22}(t)=[Y_1(t),\ldots,Y_{m-a}(t)]^T$ with $Y_j(t)\in \mathbb{R},\;j=1,\ldots, m-a$, then we have $dY(t) = -c_2(t)JY(t-\tau)dt -c_2(t)\kappa(t)\times$ $D\gamma(t\;-\;\tau)dt \;+ \; Rd\bar M(t)\;+\; Rd\bar M'(t)$, where $D=RP$, $\bar M(t)=$ $\sum_{i=n+a+1}^{n+m}\sum_{j=n+1}^{n+m}a_{ij}\sigma_{ji}\vartheta_{m-a,i-n-a}\int_{0}^{t}c_2(s)d\omega_{ji}(s)$, and $\bar M'(t)=$ $\sum_{i=n+a+1}^{n+m}\sum_{j=1}^{n}a_{ij}\sigma_{ji}\vartheta_{m-a,i-n-a}\int_{0}^{t}c_2(s)\kappa(s)d \omega_{ji}(s)$. Considering the kth Jordan block, and letting $\varphi_k(t)\;=\;[\varphi_{k,1}(t),\ldots,$ $\varphi_{k,n_k}(t)]^T$, $D(k) = [D_{k,1}^T,\ldots,D_{k,n_k}^T]^T$, $R(k) = [R_{k,1}^T,\ldots,R_{k,n_k}^T]^T$, where $\varphi_{k,j}(t)= Y_{k_j}(t)$, $D_{k,j}$ and $R_{k,j}=R_{k_j}$ are $k_j$th row of D and R with $k_j=\sum_{l=1}^{k-1} n_l+j$, respectively, one has

From (10) we obtain the following semi-decoupled equations:

and

where $\bar M_{k,j}(t)\;=\;\sum_{i=n+a+1}^{n+m}\;\sum_{q=n+1}^{n+m} r_{k_j,i}\;a_{iq}\sigma_{qi}\;\;\int_{0}^{t}c_2(s)d \omega_{qi}(s)$, $\bar M_{k,j}'(t)\;= \; \sum_{i=n+a+1}^{n+m}\;\sum_{q=1}^{n} r_{k_j,i}a_{iq}\sigma_{qi}\;\int_{0}^{t}c_2(s)\kappa(s) d\omega_{qi}(s)$, $r_{k_j,i}= R_{k_j}\vartheta_{m-a,i-n-a}$, $j=1,\ldots,n_k$. Denote $\Gamma_k(t,t_0)$ as the differential resolvent function defined in Lemma 2 with λ being replaced with $\lambda_k$. We let $D_{k,n_k}=[d_1,\ldots,d_n]$. Then, from (11), we have

where $Z_{k,\;n_k}(t)\;=\;\int_{t_0}^{t}\Gamma_k(t,\;s)d\bar M_{k,\;n_k}(s)\,, \;Z_{k,n_k}'(t)\;=\;\int_{t_0}^{t}\Gamma_k(t,s) \times$ $d\bar M_{k,n_k}'(s)$. Taking the expectation for (13) and using the Jensen inequality tool, we obtain

where $C_{k,n_k} =\sum_{i=n+a+1}^{n+m}\sum_{q=n+1}^{n+m}|r_{k_{n_k},i}|^2 a^2_{iq}\sigma^2_{qi}$, $C_{k,n_k}' = \sum_{i=n+a+1}^{n+m}\times$ $\sum_{q=1}^{n}|r_{k_{n_k},i}|^2 a^2_{iq}\sigma^2_{qi}$. From Lemma 2, we obtain

We now need to prove that the second term, the third term and the last term on the right hand side of (15) vanish in infinite time because other terms go toward zero, which can be obtained from [5] and [14]. We first consider the second term mentioned above. Fix the k and write $S_{k,n_k}(t)=$ $\int_{t_0}^{t} e^{-0.5\varrho(\lambda_k) \int_{s}^{t}c_2(u)du}c_2(s)\kappa(s)d_1\gamma_1(s)ds$. Then, $\mathop {{\rm{lim}}}\limits_{t\to \infty}\mathbb{E}{| S_{k,n_k}(t)|}^2 \le \lim_{t\to\infty}\mathbb{E}(\int_{0}^{t}e^{-0.5\varrho(\lambda_k)\int_{s}^{t}c_2(u)du}$ $c_2(s)\kappa(s)|d_1\gamma_1(s)|ds)^2$. Let $U(t)=$ $\int_{0}^{t} e^{-0.5\varrho(\lambda_k) \int_{s}^{t}c_2(u)du}$ $c_2(s)\kappa(s)|d_1\gamma_1(s)|ds$. Then, we can obtain $\sqrt{\mathbb{E}(U(t))^2}\;\le \; \int_{0}^{t}e^{-0.5\varrho(\lambda_k)\int_{s}^{t}c_2(u)du}c_2(s)\kappa(s)\sqrt{\mathbb{E}|d_1\gamma_1(s)|^2}ds.$ By means of L’Hôpital’s rule, we can get

where $c_3 = d_1^2 \sum_{k=1}^{n} \upsilon_{1k}^2 \sum_{i,j=1}^{n}a_{ij}^2\sigma_{ji}^2$， $c_4$ is a constant. From (C7), we can get $\mathop {{\rm{lim}}}\limits_{t\to \infty}{\mathbb{E}(U(t))^2}=0$, after which $\mathop {{\rm{lim}}}\limits_{t\to \infty}\mathbb{E}{| S_{k,n_k}(t)|}^2= 0$ can be obtained directly. Now we consider the remaining terms mentioned above. Note that $\lim_{t\to \infty}\sum_{p=2}^{n}d_p\gamma_p(t)=0$. By similar skills used in estimating the second term, we can get $\lim_{t\to \infty}\mathbb{E} \sum_{p=2}^{n}$ $(\int_{t_0}^{t}e^{-0.5\varrho(\lambda_k)\int_{s}^{t}c_2(u)du}c_2(s)\kappa(s)|d_p\gamma_p(s)|ds)^2= 0$ directly. From (C7), we have $\lim_{t\to \infty}\kappa(t)=0$. By L’Hôpital’s rule, combining (C5) and (C7), we can similarly get

then $\lim_{t\to \infty}\mathbb{E}{| \varphi_{k,n_k}(t)|}^2=0$ can be obtained for the fixed k. Similarly, we can obtain $\mathbb{E}{| \varphi_{k,j}(t)|}^2=0, \;j=1,\ldots,n_k-1$ for the fixed k. Repeating the above process, similar induction yields $\lim_{t\to \infty} \mathbb{E}{\Vert \varphi_{k}(t)\Vert}^2=0$ for all $k=1,\ldots,l$. Hence,

We still consider the kth Jordan block. Similar to the definition of $\varphi_k(t)$, let $\delta_k(t)=[\delta_{k,1}(t),\ldots,\delta_{k,n_k}(t)]^T$. When $\tau=0$, we can get $d\delta_{k,n_k}(t)\;=\;-\;c_2(t)\lambda_k\delta_{k,n_k}(t)dt\;- \; c_2(t)\kappa(t)D_{k,n_k}\gamma(t)dt+ d\bar M_{k,n_k}(t)\;+\; d\bar M_{k,n_k}'(t)$ and $d\delta_{k,j}(t)\;=\;-\;c_2(t)\lambda_k\delta_{k,j}(t)dt\;-$ $c_2(t)\delta_{k,j+1}(t)dt- c_2(t)\kappa(t)D_{k,j}\gamma(t)dt+ d\bar M_{k,j}(t)+ d\bar M_{k,j}'(t),\;\,j=1,$ $\ldots,n_k - 1$. According to [35] and above analyses, it is easy to obtain $\lim_{t\to \infty}| \delta_{k,j}(t)|\;=\;0, \;j\;=\;1,\ldots,n_k$. Letting $\theta_{k,n_k}(t)=$ $\delta_{k,n_k}(t)- \varphi_{k,n_k}(t)$, we have

where $g_{k,n_k}(t)=\lambda_k(\delta_{k,n_k}(t-\tau)-\delta_{k,n_k}(t))$ and $h_{k,n_k}(t)=D_{k,n_k}\times$ $(\gamma(t-\tau)-\gamma(t))$. From the above analyses we can obtain $\lim_{t\to \infty}| g_{k,n_k}(t)|=0$ and $\lim_{t\to \infty}| h_{k,n_k}(t)|=0$ easily. Then, we obtain $\theta_{k,n_k}(t)\;=\;\Gamma_k(t,t_0)\theta_{k,n_k}(t_0)+\int_{t_0}^{t} \Gamma_k(t,s)c_2(s)g_{k,n_k}(s)ds+\ $ $\int_{t_0}^{t} \Gamma_k(t,s)c_2(s)\kappa(s)h_{k,n_k}(s)ds$. By the similar skills used in above analysis, we can obtain $\lim_{t\to \infty}| \theta_{k,n_k}(t)|=0$. From this and $\lim_{t\to \infty}| \delta_{k,n_k}(t)|=0$, we have $\lim_{t\to \infty}| \varphi_{k,n_k}(t)|=0$ for the fixed k. We now assume that $\lim_{t\to \infty}| \varphi_{k,j+1}(t)|=0$ for some fixed $j<n_k$. Letting $\tilde g_{k,j+1}(t)=\delta_{k,j+1}(t)-\varphi_{k,j+1}(t-\tau)$, then we have $\lim_{t\to \infty} | \tilde g_{k,j+1}(t)|=0$. We also have $d\theta_{k,j}(t)=-c_2(t)\lambda_k\theta_{k,j}(t- \tau)dt+c_2(t)g_{k,j}(t)dt+c_2(t)\kappa(t)h_{k,j}(t)dt\;-\;c_2(t)\tilde g_{k,j+1}(t)dt$, where $g_{k,j}(t)\;=\;\lambda_k(\delta_{k,j}(t-\tau)-\delta_{k,j}(t))$, $h_{k,j}(t)\;=\;D_{k,j}(\gamma(t-\tau)-\gamma(t))$, $\lim_{t\to \infty} |g_{k,j}(t)|\;=\;0$ and $\lim_{t\to \infty}|h_{k,j}(t)|\;=\;0$. Also, we have $\theta_{k,j}(t)=\Gamma_k(t,t_0)\theta_{k,j}(t_0)+\int_{t_0}^{t} \Gamma_k(t,s)c_2(s)g_{k,j}(s)ds\;+\; \int_{t_0}^{t} \Gamma_k(t,s)\;\times$ $c_2(s)k(s)h_{k,j}(s)ds-\int_{t_0}^{t} \Gamma_k(t,s)c_2(s)\tilde g_{k,j+1}(s)ds$. Similarly, we can get $\lim_{t\to \infty}| \theta_{k,j}(t)|=0$. From this and $\lim_{t\to \infty} |\delta_{k,j}(t)|=0$ one has $\lim_{t\to \infty}| \varphi_{k,j}(t)|=0$ for the fixed k. Furthermore, one can obtain $\lim_{t\to \infty} {\Vert\varphi_k(t)\Vert}=0$ for all $k=1,\ldots,l$. Then,

From (17), (19) and the definition of $\overline X_{22}(t)$, we have $\lim_{t\to \infty} \mathbb{E}\Vert X_{22}(t)-\bar X^*\Vert^2=0$ and $\lim_{t\to \infty}\Vert X_{22}(t)-\bar X^*\Vert=0$, where $\bar X^*=-\widehat{{\cal{L}}}_{22}'^{-1}{\cal{\widetilde{L}}}_{22}\Delta_{21}$, and the ith element of $-\widehat{{\cal{L}}}_{22}'^{-1}{\cal{\widetilde{L}}}_{22}\Delta_{21}$ belongs to $co_L$ in Definition 3. Hence, the agents in subgroup $\Psi_2$ achieve stochastic containment. That is, the MAS (1) achieve group hybrid coordination behavior with stochastic weak consensus and stochastic containment. ■

B Stochastic Strong Consensus + Stochastic Containment

For agents in subgroup $\Psi_1$, we know that in addition to conditions (C1)−(C3), the following condition is also needed to explore the global stochastic strong consensus:

(C8) $\int_{0}^{\infty} c_1^2(t)dt < \infty$.

Then, we have the following theorem.

Theorem 2: For the MAS (1), suppose that Assumptions 1−4 hold. Then, under protocol (2), the MAS (1) can achieve group hybrid coordination behavior with stochastic strong consensus and stochastic containment if (C1)−(C8) hold, and only if (C1) and (C8) hold.

Proof: We first prove the “if ” part. We now consider the agents in subgroup $\Psi_1$. Combining condition (C8) and the proof of Theorem 1, we can get that $\int_{0}^{\infty} c_1(s)d\omega_{ji}(s)$ is well defined [5], $\lim_{t\to \infty} \mathbb E{\Vert\bar\gamma(t)\Vert}^2=0$ and $\lim_{t\to \infty}\Vert\bar\gamma(t)\Vert= 0$. Then, we have $\lim_{t\to \infty}\mathbb{E}\Vert X_1(t)-\bar x^*\mathbf{1}_n\Vert^2=0$ and $\lim_{t\to \infty}\Vert X_1(t)-\bar x^*\mathbf{1}_n\Vert= 0$, where $\bar x^*=\gamma_1(\infty)$. For the stochastic containment problem of subgroup $\Psi_2$, the proof is similar to the proof of Theorem 1, and is omitted here. Also, the proof of the “only if ” part can be found in [14], [35] and we omit it. ■

Above, we explored the group hybrid coordination control problem for the containment-oriented case and obtained some interesting results. Then, for the consensus-oriented case, we need to determine if the above conditions sufficient. We will explore this problem in Section IV.

IV. The Consensus-Oriented Case

To examine the consensus-oriented case, we first give the following assumption for the communications between the two subgroups.

Assumption 5: The agents in subgroup $\Psi_1$ can measure the state information from the agents in subgroup $\Psi_2$.

Based on the interactions between any two agents under the communication mode mentioned above, we give the control protocol as follows:

where $t>0$, $c_1(t)$, $c_2(t)$$\in C((0,\infty);\;[0,\infty))$ are two control gain functions to be designed. $\bar y_{ji1}(t)=x_j(t-\tau)+\sigma_{ji}\eta_{ji}(t), \; i\in {\cal{ V}}_1, j\in N_{1i}$, is the state measurement of agent j belonging to subgroup $\Psi_1$ by agent i, and $\bar y_{ji2}(t)=\kappa(t)(x_j(t-\tau)+\sigma_{ji}\eta_{ji}(t)), i\in {\cal{ V}}_1,\;j\in N_{2i}$, is the state measurement of agent j belonging to subgroup $\Psi_2$ by agent i. $\kappa(t)$ is a time-varying function, which reflects the communication intensity between subgroup $\Psi_1$ and subgroup $\Psi_2$. $\bar{\bar y}_{ji}(t)=x_j(t-\tau)+\sigma_{ji}\eta_{ji}(t),\; i\in {\cal{ V}}_2,\;j\in N_i$ is the state measurement of agent j by agent i belonging to subgroup $\Psi_2$.

Under protocol (20), from (1), we obtain the following equations:

and

where $\overline {{\cal{ L}}}_{11}=\left[\bar l_{ij}\right]$ is defined as

and it is the Laplacian matrix of ${\cal{ G}}_1$. ${\cal{ L}}_{22}$ is the Laplacian matrix of ${\cal{ G}}_2$.

Similarly, from Assumption 1 we know that each eigenvalue of $\widehat{{\cal{L}}}_{22}$ has positive real part [22]. Let $Q_2 = (({1}/{\sqrt n})\mathbf 1_n,\widetilde Q_2)$ and $Q_2^{-1} = \begin{pmatrix} \upsilon_2^T \\ \overline Q_2 \end{pmatrix}$, $Q_2^{-1}\overline {{\cal{ L}}}_{11}Q_2 = \begin{pmatrix} 0 & 0 \\ 0 & \widetilde {{\cal{ L}}}'_{11} \end{pmatrix}$, where $\upsilon_2=\sqrt n\pi_2= [\upsilon_{21},\ldots,\upsilon_{2n}]^T$, $\pi_2^T\overline{{\cal{ L}}}_{11}=0$. Let $\{\lambda_j\}{}_{j=2}^{n}$ and $\{\lambda_k\}{}_{k=1}^{m-a}$ be the eigenvalues of $\widetilde{{\cal{ L}}}'_{11}$ and $\widehat{{\cal{L}}}_{22}$, respectively. Then, we give the following assumption.

Assumption 6: There exists a constant $t_0\ge 0$ such that $\tau \tilde c_{t_0}$max$_{2\le j\le n}({\vert \lambda_j\vert^2}\;/\;{Re(\lambda_j)}) \;< \; 1$ and $\tau \tilde c_{t_0}'$max$_{1\le k\le m-a}\;({\vert \lambda_k\vert^2}/ {Re(\lambda_k)} ) < 1$, where $\tilde c_{t_0}={\rm{sup}}_{t\ge t_0}c_1(t)$ and $\tilde c_{t_0}'={\rm{sup}}_{t\ge t_0}c_2(t)$.

A Stochastic Weak Consensus + Stochastic Containment

To investigate this kind of group hybrid coordination behavior for the consensus-oriented case, we need the following condition:

(C9) $\lim_{t\to \infty} \kappa^2(t)\int_{0}^{t}c_2^2(s)ds=0$.

Theorem 3: For the MAS (1), suppose that Assumptions 1, 2, 5 and 6 hold. Then, under protocol (20), the MAS (1) can achieve group hybrid coordination behavior with stochastic weak consensus and stochastic containment if (C1)−(C6) and (C9) hold.

Proof: We first consider the stochastic containment problem of subgroup $\Psi_2$. From (C4)−(C6), we can obtain the conclusion that the agents in subgroup $\Psi_2$ can achieve stochastic containment directly [14], [22]. Similar to the above analysis, from (22) we can obtain $dX_{21}(t)=0$ and

Letting $\overline X_{22}(t)=X_{22}(t)+\widehat{{\cal{L}}}_{22}^{-1}{\cal{\widetilde{L}}}_{22}X_{21}(t)$, we have

From the above analysis, we can obtain $\lim_{t\to \infty} {\Vert\overline X_{22}(t)\Vert}^2= 0$. Now we consider the consensus problem of subgroup $\Psi_1$. From the above analysis and (21), we can get

where $\bar X(t) =[X_{21}^T(t),(\bar X_{22}(t)-\widehat{{\cal{L}}}_{22}^{-1}{\cal{\widetilde{L}}}_{22}X_{21}(t))^T]^T =[\bar X_1(t),\ldots, \bar X_m(t)]^T$. Let $J_n=({1}/{\sqrt n})\mathbf 1_n \upsilon_2^T$. By the properties of matrix $\overline{{\cal{ L}}}_{11}$ and Lemma 1, we have $\overline{{\cal{ L}}}_{11}\mathbf 1_n=0$ and $\upsilon_2^T \overline{{\cal{ L}}}_{11}=0$, then $(I_n-J_n)\overline{{\cal{ L}}}_{11}=\overline{{\cal{ L}}}_{11}=\overline{{\cal{ L}}}_{11}(I_n-J_n)$. Let $\xi(t)=(I_n-J_n)X_1(t)= [\xi_1(t),\xi_2(t),\ldots,\xi_n(t)]^T$, where $\xi_i(t)\in \mathbb{R},i=1,\ldots,n$. Define $\tilde\xi(t)=Q_2^{-1}\xi(t) = [\tilde\xi_1(t),\tilde\xi_2(t),\ldots,\tilde\xi_n(t)]^T$, $\bar\xi(t)=[\tilde\xi_2(t),\ldots, \tilde\xi_n(t)]^T$, $\tilde \xi_i(t)\in \mathbb R,\;i=1,\ldots,n$. By the definition of $Q_2^{-1}$ in Lemma 1, we can get $\tilde\xi_1(t)=\upsilon_2^T\xi(t)= \upsilon_2^T(I_n-J_n)X_1(t)=0$ and

where $H\;=\;\overline Q_2(I_n\;-\;J_n)\;{\cal{ L}}_{12}$, $W(t) \;=\; \sum_{i,j=1}^{n}a_{ij}\sigma_{ji}\overline Q_2\;(I_n\;-$ $J_n)\vartheta_{n,i}\;\int_{0}^{t}c_1(s)\;d\omega_{ji}\;(s)$, $\bar W(t)\;=\; \sum_{i=1}^{n}\;\sum_{j=n+1}^{n+m} a_{ij}\sigma_{ji}\overline Q_2 \; (I_n-$ $J_n)\vartheta_{n,i}\int_{0}^{t}c_1(s)\kappa(s)d\omega_{ji}(s)$. From [14], we now need to prove that $\lim_{t\to \infty} \mathbb E$ ${\Vert \bar \xi(t)\Vert}^2 = 0$ for any initial state. Then, according to the matrix theorem, we have $\bar R\widetilde {{\cal{ L}}}'_{11}\bar R^{-1} = J$, where $\bar R$ is a complex invertible matrix, J is the Jordan normal form of $\widetilde {{\cal{ L}}}'_{11}$. And we know $J = {\rm{diag}}(J_{\lambda_1,n_1},\ldots,J_{\lambda_l,n_l})$, $\sum_{k=1}^{l} n_k = n-1$, where $\lambda_1,\ldots,\lambda_l$ are all the eigenvalues of $\widetilde {{\cal{ L}}}'_{11}$ and $J_{\lambda_k,n_k}$ is the Jordan block corresponding to eigenvalue $\lambda_k$, whose dimension is $n_k$. Letting $\bar Y(t)=\bar R\bar \xi(t)=[\bar Y_1(t),\ldots,\bar Y_{n-1}(t)]^T$ with $\bar Y_j(t)\in \mathbb{R},\;j=1,\ldots,n-1$, then we have $d\bar Y(t) = -c_1(t)J\bar Y(t-\tau)dt - c_1(t)\kappa(t)\bar D\bar X(t-\tau)dt + \bar RdW(t)+ \bar Rd\bar W(t)$, where $\bar D=\bar RH$. Here, we first consider the kth Jordan block. Let $\zeta_k(t)=[\zeta_{k,1}(t),\ldots, \zeta_{k,n_k}(t)]^T$, $\bar D(k) = [\bar D_{k,1}^T,\ldots,\bar D_{k,n_k}^T]^T$, $\bar R(k) = [\bar R_{k,1}^T,\ldots,\bar R_{k,n_k}^T]^T$, where $\zeta_{k,j}(t)=\bar Y_{k_j}(t)$, $\bar D_{k,j}$ is $k_j$th row of $\bar D$ with $k_j=\sum_{l=1}^{k-1} n_l+j$, $\bar D_{k,n_k}=[\bar d_1,\ldots,\bar d_m]$, $\bar R_{k,j}=\bar R_{k_j}$ is $k_j$th row of H with $k_j=$ $\sum_{l=1}^{k-1} n_l+j$. We obtain

We further get the following semi-decoupled equations:

and

where $W_{k,j}(t)\;=\;\sum_{i,q=1}^{n} r_{k_j,i}a_{iq}\sigma_{qi}\;\int_{0}^{t}c_1(s)\;d\omega_{qi}(s), \;\bar W_{k,j}\;(t) \; =$ $\sum_{i=1}^{n}\sum_{q=n+1}^{n+m} r_{k_j,i}a_{iq}\sigma_{qi}\int_{0}^{t}c_1(s)\kappa(s)d\omega_{qi}(s)$, $r_{k_j,i}\;=\;\bar R_{k_j}\overline{Q}_2(I_n\;- J_n)\vartheta_{n,i}$, $j=1,\ldots,n_k$. From (26), we obtain

where $Z_{k,n_k} (t) = \int_{t_0}^{t} e^{-\lambda_k\int_{s}^{t} c_1(u)du}dW_{k,n_k}(s)$, $Z_{k,n_k}' (t) = \int_{t_0}^{t} e^{-\lambda_k\int_{s}^{t} c_1(u)du}$ $d\bar W_{k,n_k}(s)$. By taking the absolute value for (28), we have $|\zeta_{k,n_k}(t)| \; \le\;|\Gamma_k(t\;,t_0)| |\;\zeta_{k,n_k}\;(t_0)| \; +\;\sum_{p=1}^{m}|\;\int_{t_0}^{t} \Gamma_k\;(t,t_0)\;c_1(s)\;\kappa(s)\;\times$ $\bar d_p\bar X_p(s)ds| +|Z_{k,n_k}(t)|+|Z_{k,n_k}' (t)| \le \sqrt{b_0}e^{-0.5\varrho_0 \int_{t_0}^{t}c_1(u)du} |\zeta_{k,n_k}(t_0)| +$ $\sqrt{b_0} \sum_{p=1}^{m}$ $\int_{t_0}^{t} \;e^{-0.5\varrho_0 \int_{t_0}^{t}c_1(u)du}c_1(s)\kappa(s)|\bar d_p\bar X_p(s)| ds \;+\;|Z_{k,n_k}(t)|\;+$ $|Z_{k,n_k}'(t)|$, where $b_0=$max$_{2\le j \le n}b(\lambda_j)$, $\varrho_0=$min$_{2\le j \le n}$ $\varrho(\lambda_j)$. Then, we obtain

where $C_{k,n_k} = \sum_{i,q=1}^{n} |r_{k_{j},i}|^2 a^2_{iq}\sigma^2_{qi}$ and $C_{k,n_k}' = \sum_{i=1}^{n} \sum_{q=n+1}^{n+m} |r_{k_{j},i}|^2\times$ $a^2_{iq}\sigma^2_{qi}$. Here, let k be fixed and write $S_{k,n_k}(t) =$ $\int_{t_0}^{t} e^{-0.5\varrho(\lambda_k) \int_{s}^{t}c_1(u)du}c_1(s)\kappa(s)\bar d_1\bar X_1(s)ds$. Similar to the proof of Theorem 1, we can get $\lim_{t\to \infty} \mathbb E{|\zeta_{k,n_k}(t)|}^2=0$ and $\lim_{t\to \infty} {|\zeta_{k,n_k}(t)|}=0$ for the fixed k. Similarly, we can obtain $\lim_{t\to \infty} \mathbb E{\Vert \zeta_{k}(t) \Vert}^2=0$ and $\lim_{t\to \infty} {\Vert \zeta_{k}(t) \Vert}=0$ for all $k=1,\ldots,l$. Hence,

Then, we come to the conclusion that MAS (1) can achieve group hybrid coordination behavior with stochastic weak consensus and stochastic containment. ■

Remark 7: Note that condition (C9) holds if $\lim_{t\to \infty}$ $\kappa(t)=0$. Then, we need to determine if we can obtain necessary conditions about $\kappa(t)$. Suppose that the MAS (1) can achieve the above group hybrid coordination behavior and then $\lim_{t\to \infty} \mathbb E{| \zeta_{k,n_k}(t)|}^2=0$. From (29) and $S_{k,n_k}(t)$, we can obtain that when $\bar d_1 \neq 0$, $\lim_{t\to \infty} \kappa(t)=0$. This together with (C7) reveals the decay rate of the communication impact between the two subgroups and tells us that we can impose the joint condition on $c_1(t)$, $c_2(t)$ and $\kappa(t)$ to solve the group hybrid coordination control problem of MASs with time-delays and additive noises.

B Stochastic Strong Consensus + Stochastic Containment

To investigate this problem, the following condition is needed:

(C10) $\int_{0}^{\infty} c_1(t)\kappa(t)dt < \infty$.

Then, we give the following theorem, which is based on the above results.

Theorem 4: For the MAS (1), suppose that Assumptions 1, 2, 5 and 6 hold, and $c_1(t)\kappa(t)$ is monotonic. Then, under protocol (20), the MAS (1) can achieve group hybrid coordination behavior with stochastic strong consensus and stochastic containment if (C1)−(C6) and (C8)−(C10) hold, and only if (C1) and (C8) hold.

Proof: We first prove the “if ” parts. From the above analysis, we can get that the agents in subgroup $\Psi_2$ can achieve stochastic containment directly. Then, we consider the subgroup $\Psi_1$. Letting $\xi'(t) =Q_2^{-1}X_1(t)=[\xi'_1(t),\xi'_2(t),\ldots,\xi'_n(t)]^T$, $\bar \xi'(t) = [\xi'_2(t),\ldots,\xi'_n(t)]^T$, $\xi'_i(t)\in\mathbb{R},$ $i=1,\ldots,n$, then we can get $d\xi'_1\;(t) \;= \;-c_1(t)\;\kappa(t)\;\beta(t)dt\;+\;\upsilon_2^T c_1(t)\; \sum_{i,j=1}^{n}\;a_{ij}\;\sigma_{ji}\vartheta_{n,i} d\omega_{ji}(t)\;+ \;$ $\upsilon_2^T\; c_1(t)\;\kappa(t)$$\sum_{i=1}^{n}\;\sum_{j=n+1}^{n+m}\;a_{ij}\;\sigma_{ji}\;\vartheta_{n,i} \;d\omega_{ji}(t)$, where $\beta(t)= \sum_{q=1}^{m}d_q'$ $\bar X_q(t-\tau)$ is the first element of $Q_2^{-1}{\cal{ L}}_{12}\bar X(t-\tau)$, $[d_1',\ldots,d_m']$ is the first row of $Q_2^{-1}{\cal{ L}}_{12}$. We further obtain

where $\xi'^*_1\;=\;\upsilon_2^T \Delta_1, \upsilon_2^T \sum_{i,j=1}^{n}a_{ij}\sigma_{ji}{\vartheta_{n,i}}\;\int_{0}^{\infty}c_1(s)\;d\omega_{ji}(s)$ and $\upsilon_2^T \sum_{i=1}^{n}\sum_{j=n+1}^{n+m}a_{ij}\sigma_{ji}\vartheta_{n,i}\int_{0}^{\infty}c_1(s)\kappa(s)d\omega_{ji}(s)$ are well defined. Then, we have $\lim_{t\to \infty}\int_{0}^{t}c_1(s)\kappa(s)\beta(s)ds=\lim_{t\to \infty}\sum_{q=1}^{m} d_q'\times \int_{0}^{t}\bar X_q(s)c_1(s)\kappa(s)ds$. Noting that $\lim_{t\to \infty}$ $\sum_{q=1}^{m}d_q'\bar X_q(t)<\infty$, then from (C10) and Abel test we have $\lim_{t\to \infty}\int_{0}^{t}c_1(s)\kappa(s)\times \;\beta(s)ds < \infty$. We also have

where $a_1$, $a_2$ are all constants and $a_1=\sum_{k=1}^{n}\upsilon_{2k}^2\sum_{i,j=1}^{n}a^2_{ij} \times$ $\sigma^2_{ji} > 0$ and $a_2\;=\;\sum_{k=1}^{n} \upsilon_{2k}^2\sum_{i=1}^{n}\;\sum_{j=n+1}^{n+m}a^2_{ij}\;\sigma^2_{ji}\; > \;0$. Similar to the proof of $\lim_{t\to \infty}\int_{0}^{t}c_1(s)\kappa(s)\beta(s)ds<\infty$, we have $\lim_{t\to \infty}\mathbb{E}|\int_{0}^{t}\;c_1(s)\kappa(s)\;\beta(s)ds|^2 \;< \;\infty$. Then, we obtain

where $\xi'^{\infty}_1\;=\;\xi'^*_1 -\int_{0}^{\infty}c_1(s)\kappa(s)\beta(s)ds\;+\; \upsilon_2^T\sum_{i,j=1}^{n}a_{ij} \sigma_{ji}\vartheta_{n,i}\times$ $\int_{0}^{\infty} c_1(s)d\omega_{ji}(s) + \upsilon_2^T \sum_{i=1}^{n}\sum_{j=n+1}^{n+m}a_{ij}\sigma_{ji}\vartheta_{n,i}\int_{0}^{\infty}$ $c_1(s)\kappa(s)d\omega_{ji}(s) < \infty$. We also have $d\bar\xi'(t)=-c_1(t) \widetilde {{\cal{ L}}}'_{11}$ $\xi'(t-\tau)dt -c_1(t)\kappa(t)P'dt+ d\tilde W(t)+d\tilde W'(t)$, where $P'=\overline Q_2{\cal{ L}}_{12}\bar X(t-\tau)$, $\tilde W(t)=\sum_{i,j=1}^{n}a_{ij}\times \sigma_{ji}\overline Q_2 {\vartheta_{n,i}} \int_{0}^{t}c_1(s)$ $d\omega_{ji}(s)$ and $\tilde W'(t) = \sum_{i=1}^{n} \sum_{j=n+1}^{n+m} a_{ij}\sigma_{ji}\overline Q_2{\vartheta_{n,i}}\times \int_{0}^{t} c_1(s)$ $\kappa(s)d\omega_{ji}(s)$. Utilizing similar skills in the above analysis, we can easily get $\lim_{t\to \infty}\mathbb{E}{\Vert \bar \xi'(t)\Vert}^2=0$ and $\lim_{t\to \infty}$ $\Vert \bar \xi'(t)\Vert=0$. This together with (33) gives the conclusion that the “if ” part of Theorem 4 holds. We now prove the “only if ” part. The necessity of (C1) can be found in the proof of Theorem 3. From (33), $\lim_{t\to \infty}\mathbb E|\xi'_1(t)|^2<\infty$, $a_1>0$ and $a_2>0$, we can get the necessity of (C8). ■

Remark 8: From the above analyses, we can see that more conditions are needed to achieve group hybrid coordination control compared to global coordination control. Since different subgroups are required to achieve different coordination behaviors, the system dynamics cannot simply be analyzed as a whole. This results in the existing methods and conditions, such as those in [14], [22], and [24]–[32], unable to solve the group hybrid coordination control problem. Hence, in this paper, we develop a new analysis idea to study this problem. Then, some sufficient conditions and necessary conditions are obtained, and the influence mechanism of the communication impact between the two subgroups is revealed for the group hybrid coordination control problem of MASs with both time-delays and additive noises.

V. Simulation

In this section, several simulation examples are presented to illustrate the validity of the protocols and conditions for the group hybrid coordination control problem.

For the containment-oriented case, we give a MAS with seven agents and the communication topology is shown in Fig. 1, where the number of subgroups is 2 and the agents 4 and 5 are two stationary leaders. We can see that Assumption 1 and Assumption 3 hold. Here, we require that agents 1, 2, 3 achieve consensus and agents 4, 5, 6, 7 achieve containment. We give the initial states $X_1(t)=\Delta_1=\left[5,-1,-3\right]^T$, $X_2(t)= \Delta_2=\left[-2,-3,7,-6\right]^T$, $t\in [-\tau,0]$, and assume that $\sigma_{ji}=1, \;i,j= 1,2,3,4,5,6,7$. We first consider $\tau=0.4$, the control gain functions $c_1(t)=(1+t)^{-0.5}$, $c_2(t)=(1+t)^{-0.8}$ and $\kappa(t)=(1+t)^{-0.4}$. It is easy to get that Assumption 4 and conditions (C1)−(C7) hold. From Theorem 1, we have the conclusion that the given MAS can achieve group hybrid coordination behavior with stochastic weak consensus and stochastic containment, which is shown in Fig. 2. Then, we consider $\tau = 0.8$, $c_1(t) = 0.6(1+t)^{-1},\; c_2(t)=0.5(1+t)^{-0.8}$ and $\kappa(t)=(1+ t)^{-0.2}$. Other conditions are the same as above. It is easy to verify that Assumption 4 and conditions (C1)−(C8) hold. From Theorem 2, we have the conclusion that the given MAS can achieve group hybrid coordination behavior with stochastic strong consensus and stochastic containment, which is shown in Fig. 3.

Figure  1.  The communication graph: The containment-oriented case.

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Figure  2.  The containment-oriented case: Stochastic weak consensus + Stochastic containment.

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Figure  3.  The containment-oriented case: Stochastic strong consensus + Stochastic containment.

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Then, for the consensus-oriented case, we give another MAS with seven agents and the interactions between agents is shown in Fig. 4, where the number of subgroups is 2 and the agents 4 and 5 are also two stationary leaders. We can also see that Assumptions 1 and 5 hold. We also require agents in the two subgroups to achieve the coordination behavior mentioned above. Now, we give $X_1(t)=\Delta_1=\left[-4,1,-2\right]^T$, $X_2(t)= \Delta_2=\left[1,2,3,-4\right]^T$, $t\in [-\tau,0]$, and assume that $\sigma_{ji}=1,\;i,j= 1,2,3,4,5,6,7$. We first consider $\tau=0.5$, $c_1(t)=0.9(1+t)^{-0.4}$, $c_2(t)=0.7(1+t)^{-1}$ and $\kappa(t)=(1+t)^{-0.7}$. It is easy to get that Assumption 6, conditions (C1)−(C6) and (C9) hold. From Theorem 3 we have the conclusion that the given MAS can achieve group hybrid coordination behavior with stochastic weak consensus and stochastic containment, which is shown in Fig. 5. We now consider $\tau=1.5$, $c_1(t)=0.3(1+t)^{-0.9},\; c_2(t)= 0.3(1+t)^{-1}$ and $\kappa(t)=(1+t)^{-0.6}$. Other conditions are the same as above. Then, it is easy to get that Assumption 6, conditions (C1)−(C6) and (C8)−(C10) hold. Theorem 4 tells us that the given MAS can achieve group hybrid coordination behavior with stochastic strong consensus and stochastic containment, which is shown in Fig. 6.

Figure  4.  The communication graph: The consensus-oriented case.

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Figure  5.  The consensus-oriented case: Stochastic weak consensus + Stochastic containment.

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Figure  6.  The consensus-oriented case: Stochastic strong consensus + Stochastic containment.

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Now, we give a counter-example to show the necessity of the decay of the communication impact between the two subgroups for the consensus-oriented case. We consider $\tau=$ 0.5, $c_1(t)=0.8(1+t)^{-0.9}, \;c_2(t)=0.9(1+t)^{-1}$ and $\kappa(t)=1$. Other conditions are the same as above. It is easy to get that Assumption 6, conditions (C1)−(C6), (C8) and (C10) hold but (C9) does not hold. Then, we obtain Fig. 7, which shows that the given MAS fails to achieve group hybrid coordination behavior.

Figure  7.  A counter-example for the consensus-oriented case.

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VI. Conclusions

This work addressed the group hybrid coordination control problem of MASs with both time-delays and additive noises. The containment-oriented case and the consensus-oriented case were discussed respectively. Using the semi-decoupled skill and some estimation methods, this work provided a new analysis idea to investigate the group hybrid coordination control problem. Then, some sufficient conditions and necessary conditions were obtained and some interesting phenomenons were found. It is concluded that to solve the group hybrid coordination control problem, more conditions are needed than to solve the global coordination control, and the decaying condition of communication impact between the two subgroups is necessary for the consensus-oriented case. Intuitively, in real networks, the communication impact between the two subgroups may gradually weaken with the increase of the distance between agents in different subgroups. This might be beneficial for designing the group control protocol. Furthermore, the following new findings are obtained: 1) The conditions for the containment-oriented case are weaker than for the consensus-oriented case; 2) For the consensus-oriented case, the conditions for weak consensus are weaker than those for strong consensus.

Recently, many important results on distributed coordination control have been obtained, such as coordination control based on dynamic event-triggered [36] and [37]. Determining how to apply these advanced control strategies to the control problem studied in this paper to improve the system communication efficiency and execution capability is a future direction of investigation.