Adaptive dynamic event-triggered cluster consensus for second-order delayed MASs with input saturation

Abstract

In this paper, the cluster consensus problem of second-order multi-agent systems (SOMASs) is investigated in which cooperative-competitive interactions among agents, input time delay, and input saturation are involved simultaneously. A novel adaptive dynamic event-triggered control (ETC) protocol is first designed to realize cluster consensus. Since the constant in static trigger condition is replaced by a dynamic parameter, the new control scheme can reduce the trigger times and save energy consumption. Then, switching topology problem of heterogeneous multi-agent systems (MASs) is also discussed based on the dynamic ETC strategy. Some sufficient conditions for achieving the cluster consensus of SOMASs are proposed by utilizing the Lyapunov stability theory and algebraic graph theory. Especially, to solve the saturation problem, Lagrange mean value theorem is adopted. Moreover, it is also proved that no Zeno behavior occurs under the proposed control protocol. Finally, two examples justify the significance of our theoretical results.

Keywords

Adaptive dynamic event-triggered strategy second-order NMASs cluster consensus input saturation input time delay

1. Introduction

Through past research, it has been found that multi-agent systems (MASs) have gained a lot of focus because of their widely used applications in various fields, for example, electricity distribution, healthcare services, power system engineering, cooperative control of self-propelled aircraft, and so on (Vitor et al. (2017), Sujil et al. (2018) and Miao et al. (2016)). MASs have greater advantages over individual agents. The application of MASs is reflected in all aspects of our life. Many kinds of literatures (Zhu et al. (2014), Ding et al. (2017), Qiu et al. (2016), Xia et al. (2016), Zheng et al. (2018) Ma et al. (2017), Fu et al. (2018), Xin et al. (2018) and Peng et al. (2019)) focused on a variety of conditions to make agents reach consensus. The purpose of consensus is for all members to eventually reach the same state.

However, in real life, multiple tasks need to be processed simultaneously, so the concept of cluster consensus was introduced. Agents in the same cluster accomplish one goal, and those in different clusters accomplish different goals. Currently, the cluster consensus problem has been extensively studied (Ma et al. (2016), Wang and Shen (2014), Chen et al. (2011), Huang et al. (2018), Zhan and Li (2017) and Luo and Ye (2021)). For instance, in Ma et al. (2016) and Wang and Shen (2014), the cluster consensus problems for first-order MASs and SOMASs were discussed, respectively. The discrete-time cluster consensus problem has also been investigated in Chen et al. (2011). Meanwhile, on the problem of cluster consensus, Huang et al. (2018) and Zhan and Li (2017) proposed several sufficient conditions to address such problem. In addition, cluster consensus control and general partitions for linear MASs with directed topologies were given in Luo and Ye (2021). In the above literatures, all agents reached consensus with their respective clusters.

But most of above literatures adopted the continuous control methods. In some engineering applications, due to the restrictions of communication resources and computing capacity, continuous control is almost impossible to achieve. What’s more, agents usually can’t correspond with other agents in real time, and the controller can’t be immediately updated. To overcome these disadvantages and improve the resource utilization, periodic sampling control and event-triggered control (ETC) appeared. Instead of the former trigger method, the latter provides a more flexible trigger method. For ETC, only when errors reach the set threshold then the agent will change its controller, which can greatly reduce communication resources and already applied in recent studies (Xu and He (2018), Xia et al. (2023), Wang et al. (2019), Shi (2021), Yi et al. (2019), Dai and Guo (2018), Li et al. (2019) and Wu et al. (2018). Xu and He (2018) realized the cluster consensus of linear MASs by adopting ETC strategy. In Xia et al. (2023), Xia et al. employed the ETC strategy to solve the cluster consensus problem of nonlinear MASs.

While ETC strategies can decrease the frequency of controller updates as well as increase the communication bandwidth, these ETC schemes mentioned above are all static types because trigger parameters are immutable values. In the initial stages, the static trigger conditions are difficult to achieve, the communication can be effectively reduced. However, as the threshold decreases, triggering becomes more frequent, which causes undesirable trigger instants. Therefore, a dynamic ETC strategy was proposed (He et al. (2020), Yang et al. (2018) and Zuo et al. (2017)). Compared with the static event-trigger generator, a dynamic threshold is used in place of a constant threshold in the trigger function, which can be adjusted according to the adaptive law. In addition, since the next trigger time of dynamic ETC is larger than static ETC, dynamic ETC can further improve the trigger interval and reduce the controller update frequency. At present, dynamic ETC is mostly used for complete consensus. There are few results on cluster consensus of SOMASs, especially when considering time delay and saturation. However, because of the resource limitation and the actuator’s own structures, the control inputs will be finite. In other words, input saturation is also a very important factor for the actuator. Currently, there are many studies on the event-triggered consensus problem for MASs with input saturation (You et al. (2022), Li and Cao (2023), Xu et al. (2023), Chen and Yan (2023), Yan et al. (2023), and Huang et al. (2023)). For example, You et al. (2022) were concerned with periodic event-triggered consensus for multi-agent systems affected by input saturation. The event-triggered group consensus problem for MASs with input saturation in a fixed topology was investigated in Li and Cao (2023). In addition, the time delays are inevitable in actual applications. Therefore, it is crucial to research the cluster consensus problem for delayed SOMASs with input saturation.

Moreover, adaptive control has also gained a lot of attention in recent years (Sui and Tong (2023a) and Sui et al. (2023b)) because it enables the system to automatically adjust parameters to achieve a good result. For instance, Sui and Tong (2023a) studied the problem of full error-constrained adaptive control design for multi-input and multi-output nonlinear systems.

Motivated by the aforementioned discussion, this paper primarily applies the adaptive dynamic ETC method to investigate the cluster consensus problem for SOMASs which includes input saturation and input time delay. The control strategy has the advantages of low communication consumption, long controller life, and high flexibility in practical application. The key points in this paper are shown below:

(i) A novel adaptive dynamic ETC strategy is introduced to realize cluster consensus for second-order MASs. Different from Lu et al. (2022) and Wang et al. (2021), our paper provides a dynamic ETC scheme to realize cluster consensus instead of complete consensus. The proposed controller can increase the trigger interval and reduce the number of triggers, which greatly saves resources. In addition, the adaptive parameters are added in the proposed control scheme which can flexibly adjust the control parameters of each agent to derive better cluster consensus mechanism.

(ii) This paper simultaneously considers cooperative-competition interactions between agents, time delay, and saturation which are more applicable for the real-world application but make the problem more challenging. Especially, when we tackle the input saturation, an inverse tangent function is introduced in the controller and the Lagrange mean value theorem is employed to solve this problem.

(iii) Owning to external disturbances and limited communications, it is possible to dynamically change topologies of MASs in some certain situations. Therefore, we also study the cluster consensus issue for heterogeneous MASs with a switching topology. A stability criterion for cluster consensus is proposed by adaptive dynamic ETC control. Furthermore, with the ETC strategy proposed in this paper, no Zeno behavior will be generated.

Other parts of this paper are basically summarized below. Section 2 introduces the graph theory and mathematical models. Section 3 covers achieving cluster consensus in SOMASs with input saturation and input time delay using adaptive dynamic ETC method. In Section 4, switching topology problem of heterogeneous MASs is also discussed based on the dynamic ETC strategy. Section 5 provides some simulation examples. Finally, the paper concludes with a summary of the findings.

Notation. I_n denotes the identity matrix. Rⁿ refers to Euclidean space of n-dimensional. ‖⋅‖ is the 2-norm of a vector or a matrix. The superscript T represents the transpose of a matrix. For a real symmetric matrix, Q, Q >0 indicates that Q is positive definite and the others are defined similarly.

2. Preliminaries

Graph G = {W, H, A} can be used for indicating the interconnections between MASs, where $W = {w_{1}, w_{2}, \dots, w_{N}}$ denotes the set of vertices G, H means the sets of interconnected edges between two different W, and A represents the weight between two different vertices. In a directed graph, we can define its edges as (w_j, w_i), and (w_j, w_i) ∈ H which means that the edge points from w_j to w_i. For adjacency matrix A, if (w_j, w_i) ∈ H, then a_ij ≠ 0, and if a_ij > 0, this indicates the existence of a cooperative relationship among i and j, otherwise a_ij < 0 expresses the existence of a competitive relationship among i and j. The Laplacian matrix $L = {(l_{i j})}_{N \times N}$ associated with an adjacency matrix A is defined as for $l_{i i} = \sum_{j = 1, j \neq i}^{N} a_{i j}$ , l_ij = −a_ij for i ≠ j. The Laplacian matrix has the property of zero-row sum, where the elements of each row of the matrix add up to 0. In addition, the value of the main diagonal element is equal to the sum of the values of all elements in the row, and the values of the elements other than the main diagonal are equal to the opposite of the adjacency matrix. Further, if a non-negative diagonal matrix D = diag{d₁, d₂, ⋯d_N}, and if there is an exchange between the leader and node i ∈ 1, 2, 3, …, N − 1, N, then d_i > 0, otherwise d_i = 0.

Cluster consensus describes the phenomenon that the agent behavior states eventually reach the same leader. The focus of this paper is the nonlinear second-order MASs that include p leaders and N followers and will eventually reach the p cluster states. G = {G₁, G₂, …, G_p} is a partition of agent set {1, 2, …, N}, where each subset is mutually exclusive, that is, G_e ∩ G_f = 0 for $e \neq f, e, f = 1,2, \dots, p, \cup_{e = 1}^{p} G_{e} = {1,2, \dots, N}$ . In addition, we denote by $\hat{l}$ that lth agent belongs to this cluster. If $\hat{l} = \hat{k}$ , we then consider agent l and agent k are part of the identical cluster.

Suppose the dynamic equation for the ith agent in MASs is described as shown below

{\begin{cases} {\dot{x}}_{i} (t) = v_{i} (t) \\ {\dot{v}}_{i} (t) = u_{i} (t - τ) + f (x_{i} (t), v_{i} (t)) \end{cases} i = 1,2, \dots, N,

(1)

and the dynamic equation for the hth leader is presented as follows

{\begin{cases} {\dot{s}}_{h} (t) = g_{h} (t) \\ {\dot{g}}_{h} (t) = f (s_{h} (t), g_{h} (t)) \end{cases} h = 1,2, \dots, p,

(2)

where x_i(t), v_i(t) and u_i(t) in (1) represent the position, velocity, and input control information of the ith agent at time t, respectively. s_h(t) and g_h(t) are similar to the previous definition. f (x_i(t), v_i(t)) and f (s_h(t), g_h(t)) are vector-valued continuous function. τ is the time delay. One can determine if agent i belongs to the hth cluster by checking if

\hat{i} = h

Some definitions, assumptions, and lemmas that help understanding the subsequent analysis are outlined below.

Definition 1

If the following two equations hold, then we can consider that the p clusters reach consensus

{\begin{cases} \lim_{t \to \infty} ‖ x_{i} (t) - s_{\hat{i}} (t) ‖ = 0, \\ \lim_{t \to \infty} ‖ v_{i} (t) - g_{\hat{i}} (t) ‖ = 0, \end{cases} i = 1,2, \dots, N

(3)

Assumption 1

Divide the elements of Laplacian matrix L ∈ R^N×N into p clusters, thus, it can be assumed the below format holds

L = (\begin{array}{l} L_{11} & L_{12} & \dots & L_{1 p} \\ L_{21} & L_{22} & \dots & L_{2 p} \\ \dots & \dots & \dots & \dots \\ L_{p 1} & L_{p 2} & \dots & L_{p p} \end{array})

(4)

where each element of matrix (4) has the property of zero-row-sum matrices, that is, with L_mn, m, n = 1, 2, …, p for all i ∈ G_m, there are

\sum_{j \in G_{n}} a_{i j} = 0

Based on Assumption 1, let D = diag{D₁, D₂, …, D_p} = diag{d₁, d₂, …, d_N}, where D_h ∈ R^nh×nh, h = 1, 2, …, p. D_h shows the communication relationship among the leaders and the followers.

Thus, let

\overset{⌢}{L} = (\begin{array}{l} {\overset{⌢}{L}}_{11} & L_{12} & \dots & L_{1 p} \\ L_{21} & {\overset{⌢}{L}}_{22} & \dots & L_{2 p} \\ \dots & \dots & \dots & \dots \\ L_{p 1} & L_{p 2} & \dots & {\overset{⌢}{L}}_{p p} \end{array})

(5)

where

\overset{⌢}{L} = L_{h h} + D_{h}

. Let G_h, h = 1, 2, …, p represent the graph of the hth cluster, which is the graph consisting of the edges of the leader connecting agents. Define

\tilde{L} = (\begin{array}{l} c_{1} {\overset{⌢}{L}}_{11} & L_{12} & \dots & L_{1 p} \\ L_{21} & c_{2} {\overset{⌢}{L}}_{22} & \dots & L_{2 p} \\ \dots & \dots & \dots & \dots \\ L_{p 1} & L_{p 2} & \dots & c_{p} {\overset{⌢}{L}}_{p p} \end{array})

(6)

where c_h stands for the coupling strength within the hth cluster G_h.

Lemma 1

(Wang (2016)) If the following conditions hold: (i) G(x) is continuous in $[\tilde{α}, \tilde{β}]$ and (ii) G(x) is derivable in $(\tilde{α}, \tilde{β})$ , then there exists at least a point $λ \in (\tilde{α}, \tilde{β})$ such that

G (\tilde{β}) - G (\tilde{α}) = G {(λ)}^{'} (\tilde{β} - \tilde{α})

(7)

Assumption 2

There are non-negative constants k₁, k₂ and the nonlinear function F such that the Lipschitz condition holds, that is, ∃k₁, k₂ > 0

\begin{array}{l} ‖ F (m_{i} (t), n_{i} (t)) - F (p_{l} (t), q_{l} (t)) ‖ \\ < k_{1} ‖ m_{i} (t) - p_{l} (t) ‖ + k_{2} ‖ n_{i} (t) - q_{l} (t) ‖ \end{array}

(8)

Lemma 2

(Yu (2014)) If the communication network includes a directed spanning tree with the leader as the root, then exists a positive diagonal matrix ∇, $\nabla \tilde{L} + {\tilde{L}}^{T} \nabla > 0$ . Clearly, $\tilde{L}$ is invertible.

Remark 1

It is clear from Assumption 1 that the strength of coupling between clusters may be positively or negatively characterized, which describes the cooperation or competition schemes between agents. This implies that there needs to be a coupling balance between one cluster and other clusters.

Remark 2

In this paper, all the vector-valued continuous functions are nonlinear, and they all satisfied the Lipschitz condition in Assumption 2.

3. Adaptive dynamic ETC for SOMASs with input saturation and input time delay

In this part, a novel adaptive dynamic ETC protocol is proposed for SOMASs with input saturation and input time delay to realize cluster consensus.

Assume that SOMASs are described as (1) and (2). The measurement error of all agents can be depicted below

{\begin{cases} e_{i x} (t) = x_{i} (t_{k}^{i} - τ) - x_{i} (t) \\ e_{i v} (t) = v_{i} (t_{k}^{i} - τ) - v_{i} (t) \end{cases}

(9)

The error between followers and leaders can be written as

δ_{i x} (t) = x_{i} (t) - s_{i} (t), δ_{i v} (t) = v_{i} (t) - g_{i} (t)

(10)

For convenience, we employ following notations

δ_{i} (t) = {(δ_{i x}^{T} (t), δ_{i v}^{T} (t))}^{T}, e_{i} (t) = {(e_{i x}^{T} (t), e_{i v}^{T} (t))}^{T}

(11)

To achieve cluster consensus, the adaptive saturated ETC strategy is written as

\begin{array}{l} u_{i} (t - τ) \\ = - m_{i x} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i}))] \\ - m_{i v} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i}))] \end{array}

(12)

Remark 3

Note that the controller considers saturation and time delay, but fault tolerance is not involved. However, the security of agents has received a lot of attention in recent years, especially, when agents are attacked, which may cause controller failure or even change communication topology. Thus, controller fault tolerance is worth considering, which is ignored in this paper.

Where $t_{k}^{i}$ is kth trigger instant in the normal situation for the ith agent and τ is time delay. m_ix(t) and m_iv(t) are the adaptive variables satisfying the following equation

{\dot{m}}_{i x} (t) = γ_{i x} z_{i x}^{T} (t) z_{i x} (t), {\dot{m}}_{i v} (t) = γ_{i v} z_{i v}^{T} (t) z_{i v} (t)

(13)

where γ_ix and γ_iv are the given positive constants. The equations for z_ix(t) and z_iv(t) are defined below

\begin{array}{l} z_{i x} (t) & = \sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} (x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i})) \\ z_{i v} (t) & = \sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} (v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i})) \end{array}

(14)

For convenience, let $z_{i} (t) = {(z_{i x}^{T} (t), z_{i v}^{T} (t))}^{T}$ .

Assumption 3

Assume that the adaptive variable is bounded, that is, there exist constants M_ix _max and M_iv _max which satisfy ‖m_ix(t)‖ ≤ M_ix _max, ‖m_iv(t)‖ ≤ M_iv _max.

Remark 4

Note that m_ix(t) and m_iv(t) are the adaptive parameters, which are adjusted according to equation (13). Therefore, when γ_ix and γ_iv are selected larger, the adaptive parameters change faster and suitable parameters are determined. The cluster consensus can be achieved faster. According to (13), the value of m_ix(t) and m_iv(t) will increase rapidly at the beginning because of the large error between leader and agents. As time goes, they will tend to equilibrium states when the error becomes smaller and smaller. Therefore, we assume m_ix(t) and m_iv(t) are bounded. In addition, considering that the leader has no trigger time, one has $s_{\hat{i}} (t_{k}^{i}) = s_{\hat{i}} (t), g_{\hat{i}} (t_{k}^{i}) = g_{\hat{i}} (t)$ .

Note that

\begin{array}{l} | u_{i} (t - τ) | \\ \leq π (\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} + \sum_{j \notin G_{\hat{i}}} a_{i j} + c_{\hat{i}} d_{i}) (M_{i x \max} + M_{i v \max}) \end{array}

(15)

which will be limited to a range and is associated with the agent’s topology and adaptive parameters.

Assumption 4

In real life, there are limits to the input of agents and the state and speed of leaders, that is, there exist constants S_max, G_max, and U_max which satisfy ‖s_i(t)‖ ≤ S_max, ‖v_i(t)‖ ≤ V_max, ‖u_i(t)‖ ≤ U_max.

Remark 5

According to (15), it is known that the actuator is saturated. For convenience, we assume ‖u_i(t)‖ ≤ U_max here. In addition, we consider leaders’ states and velocity are bounded.

The triggering time sequence $t_{k}^{i} - τ$ for the agent i is determined by

t_{1}^{i} - τ = 0, t_{k + 1}^{i} = t_{k}^{i} + Δ_{k}^{i}

(16)

where

Δ_{k}^{i}

denotes the triggering time interval between trigger instant

t_{k}^{i} - τ

and

t_{k + 1}^{i} - τ

for agent i. The

Δ_{k}^{i}

can be described as

Δ_{k}^{i} = \inf_{t > t_{k}^{i} - τ} {t - t_{k}^{i} + τ | θ_{i} (e_{i}^{T} (t) e_{i} (t) - β_{i} z_{i}^{T} (t) z_{i} (t) \geq ψ_{i} (t)}

(17)

β_i is a positive constant to be designed, and ψ_i(t) is a variable satisfying

{\dot{ψ}}_{i} (t) = - σ_{i} ψ_{i} (t) + ε_{i} (β_{i} z_{i}^{T} (t) z_{i} (t) - e_{i}^{T} (t) e_{i} (t))

(18)

with ψ_i (0) > 0, θ_i > 0, ɛ_i > 0 and σ_i > 0.

From (18), it is clear that the dynamic parameters satisfy ψ_i(t) > 0, i = 1, 2⋯, N − 1, N. The proof is given as follows.

Proof

For $t \in \cup_{k = 1}^{\infty} (t_{k}^{i} - τ, t_{k + 1}^{i} - τ] = [0, + \infty)$ , when using the dynamic ETC strategy, no triggering will occur between consecutive instants $t_{k}^{i} - τ$ and $t_{k + 1}^{i} - τ$ , one has

θ_{i} (e_{i}^{T} (t) e_{i} (t) - β_{i} z_{i}^{T} (t) z_{i} (t) < ψ_{i} (t)

(19)

Substituting (19) into (18)

{\dot{ψ}}_{i} (t) > - σ_{i} ψ_{i} (t) - \frac{ε_{i}}{θ_{i}} ψ_{i} (t)

(20)

Then

ψ_{i} (t) \geq ψ_{i} (t_{k}^{i} - τ) e^{- (σ_{i} + \frac{ε_{i}}{θ_{i}}) (t - t_{k}^{i} + τ)}, t \in (t_{k}^{i} - τ, t_{k + 1}^{i} - τ]

(21)

Using mathematical induction, one has

\begin{array}{l} ψ_{i} (t) \geq ψ_{i} (t_{k}^{i} - τ) e^{- (σ_{i} + \frac{ε_{i}}{θ_{i}}) (t - t_{k}^{i} + τ)} \\ \geq ψ_{i} (t_{k - 1}^{i} - τ) e^{- (σ_{i} + \frac{ε_{i}}{θ_{i}}) (t - t_{k - 1}^{i} + τ)} \\ \geq \dots \geq ψ_{i} (0) e^{- (σ_{i} + \frac{ε_{i}}{θ_{i}}) t} > 0, t \in (t_{k}^{i} - τ, t_{k + 1}^{i} - τ] \end{array}

(22)

Remark 6

Here, assume ψ_i(t) remains unchanged within the interval (0, τ) which equals to the initial value. In (18), ψ_i(t) changes according to time, and its transformation pattern is related to the relative error between agents and the measurement error. Compared with static ETC strategy, the introduction of ψ_i(t) in (17) is a key point to reduce the number of event triggers. If ψ_i(t) = 0, the ETC condition in (17) becomes the static strategy. In addition, according to (17), the parameters β_i and θ_i have a significant impact on the number of triggers. As β_i increases and θ_i decreases, the trigger condition becomes more difficult to occur and the controller updates will decrease.

Substituting (12) into (1), the dynamics of agent satisfy the following equations

{\begin{cases} {\dot{x}}_{i} (t) = v_{i} (t) \\ {\dot{v}}_{i} (t) = f (x_{i} (t), v_{i} (t)) \\ - m_{i x} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i}))] \\ - m_{i v} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i}))] \end{cases}

(23)

According to formula (10), the expression of the error system is listed as

{\begin{cases} {\dot{δ}}_{i x} (t) = δ_{i v} (t) \\ {\dot{δ}}_{i v} (t) = f (x_{i} (t), v_{i} (t)) - f (s_{h} (t), g_{h} (t)) + u_{i} (t - τ) \end{cases}

(24)

With the control protocol (12), we can solve the cluster consensus as Theorem 1.

Theorem 1

If Assumptions 1–4 are satisfied, for given scalars σ_i, ɛ_i, θ_i, β_i, the inequalities satisfy the following

ζ_{1 i} \leq 0, ζ_{2 i} \leq 0, \frac{κ_{1 i} + κ_{2 i}}{θ_{i}} - σ_{i} \leq 0

where

ζ_{1 i} = m_{i x} (t) - α_{1} / 2 + {‖ H ‖}^{2} ω_{1 m a x} + ε_{i} β_{i} + κ_{1 i} β_{i} + κ_{2 i} β_{i}

ζ_{2 i} = m_{i v} (t) - α_{2} / 2 + {‖ H ‖}^{2} ω_{2 m a x} + ε_{i} β_{i} + κ_{1 i} β_{i} + κ_{2 i} β_{i}

, κ_1i = ω_1max + p_i − ɛ_i ≥ 0, κ_2i = ω_2max + p_i − ɛ_i ≥ 0,

p_{i} = \sum_{j \in G_{\hat{i}}} c_{\hat{i}} | a_{i j} | + \sum_{j \in G_{\hat{i}}} c_{\hat{i}} | a_{j i} | + \sum_{j \notin G_{\hat{i}}} | a_{i j} | + \sum_{j \notin G_{\hat{i}}} | a_{j i} | + c_{\hat{i}} d_{i} > 0

n_{i} = (8 \sum_{j \in G_{\hat{i}}} c_{\hat{i}} | a_{i j} | + 8 \sum_{j \notin G_{\hat{i}}} | a_{i j} | + 4 c_{\hat{i}} d_{i}) (m_{i x}^{2} (t) + m_{i v}^{2} (t)) \geq 0

, then the cluster consensus problem of SOMASs can be addressed through the control protocol (12).

Proof

Consider the time interval $[t_{k}^{i} - τ, t_{k + 1}^{i} - τ)$ , the triggered condition $θ_{i} (e_{i}^{T} (t) e_{i} (t) - β_{i} z_{i}^{T} (t) z_{i} (t) < ψ_{i} (t)$ holds. Taking

V (t) = V^{1} (t) + V^{2} (t)

(25)

where V¹(t) and V²(t) are selected as

V^{1} (t) = \sum_{i = 1}^{N} δ_{i}^{T} (t) δ_{i} (t)

(26)

\begin{array}{l} V^{2} (t) & = \sum_{i = 1}^{N} \frac{{(m_{i x} (t) - α_{1})}^{2}}{2 γ_{i x}} \\ + \sum_{i = 1}^{N} \frac{{(m_{i v} (t) - α_{2})}^{2}}{2 γ_{i v}} + \sum_{i = 1}^{N} ψ_{i} (t) \end{array}

(27)

of which α₁, α₂ are positive constants.

Calculating the derivative for V¹ (t), we have

\begin{array}{l} {\dot{V}}^{1} (t) = 2 \sum_{i = 1}^{N} δ_{i}^{T} (t) {\dot{δ}}_{i} (t) \\ = 2 \sum_{i = 1}^{N} δ_{i x}^{T} (t) {\dot{δ}}_{i x} (t) + 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) {\dot{δ}}_{i v} (t) \\ = 2 \sum_{i = 1}^{N} δ_{i x}^{T} (t) δ_{i v} (t) + 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) u_{i} (t - τ) \\ + 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) (f (x_{i} (t), v_{i} (t)) - f (s_{l} (t), g_{l} (t))) \end{array}

(28)

and

\begin{array}{l} 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) u_{i} (t - τ) \\ = - 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) m_{i x} (t) \\ [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i}))] \\ - 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) m_{i v} (t) \\ [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i}))] \end{array}

(29)

Let $Ω_{1 i} = e_{i x} (t) - e_{j x} (t) + δ_{i x} (t) - δ_{j x} (t)$ and $Ω_{2 i} = e_{i v} (t) - e_{j v} (t) + δ_{i v} (t) - δ_{j v} (t)$ . According to Lemma 1 and Assumption 1, there exist $ζ_{1 i x} \in (0, x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ))$ , $ζ_{2 i x} \in (0, x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i}))$ , $ζ_{1 i v} \in (0, v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ))$ , $ζ_{2 i v} \in (0, v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i}))$ such that

\begin{array}{l} 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) u_{i} (t - τ) \\ = - \sum_{i = 1}^{N} δ_{i v}^{T} (t) m_{i x} (t) \\ [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \frac{2}{1 + ζ_{1 i x}^{2}} (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \frac{2}{1 + ζ_{1 i x}^{2}} (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \frac{2}{1 + ζ_{2 i x}^{2}} (x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i}))] \\ - \sum_{i = 1}^{N} δ_{i v}^{T} (t) m_{i v} (t) \\ [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} \frac{2}{1 + ζ_{1 i v}^{2}} (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j} \frac{2}{1 + ζ_{1 i v}^{2}} (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \frac{2}{1 + ζ_{2 i v}^{2}} (v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i}))] \\ = - \sum_{i = 1}^{N} δ_{i v}^{T} (t) m_{i x} (t) \\ [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} Ω_{1 i} \frac{2}{1 + ζ_{1 i x}^{2}} + \sum_{j \notin G_{\hat{i}}} a_{i j} Ω_{1 i} \frac{2}{1 + ζ_{1 i x}^{2}} \\ + c_{\hat{i}} d_{i} \frac{2}{1 + ζ_{2 i x}^{2}} (δ_{i x} (t) + e_{i x} (t))] \\ - \sum_{i = 1}^{N} δ_{i v}^{T} (t) m_{i v} (t) \\ [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} Ω_{2 i} \frac{2}{1 + ζ_{1 i v}^{2}} + \sum_{j \notin G_{\hat{i}}} a_{i j} Ω_{2 i} \frac{2}{1 + ζ_{1 i v}^{2}} \\ + c_{\hat{i}} d_{i} \frac{2}{1 + ζ_{2 i v}^{2}} (δ_{i v} (t) + e_{i v} (t))] \end{array}

(30)

Then, according to (30), we have

\begin{array}{l} - \sum_{i = 1}^{N} δ_{i v}^{T} (t) m_{i x} (t) \\ [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j} Ω_{1 i} \frac{2}{1 + ζ_{1 i x}^{2}} + \sum_{j \notin G_{\hat{i}}} a_{i j} Ω_{1 i} \frac{2}{1 + ζ_{1 i x}^{2}} \\ + c_{\hat{i}} d_{i} \frac{2}{1 + ζ_{2 i x}^{2}} (δ_{i x} (t) + e_{i x} (t))] \\ \leq 2 \sum_{i = 1}^{N} \sum_{j \in G_{\hat{i}}} c_{\hat{i}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ e_{i x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} \sum_{j \in G_{\hat{i}}} c_{\hat{i}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ e_{j x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} \sum_{j \in G_{\hat{i}}} c_{\hat{i}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ δ_{i x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} \sum_{j \in G_{\hat{i}}} c_{\hat{i}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ δ_{j x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} \sum_{j \notin G_{\hat{i}}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ e_{i x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} \sum_{j \notin G_{\hat{i}}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ e_{j x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} \sum_{j \notin G_{\hat{i}}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ δ_{i x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} \sum_{j \notin G_{\hat{i}}} m_{i x} (t) | a_{i j} | ‖ δ_{i v}^{T} (t) ‖ ‖ δ_{j x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} m_{i x} (t) c_{\hat{i}} d_{i} ‖ δ_{i v}^{T} (t) ‖ ‖ δ_{i x} (t) ‖ \\ + 2 \sum_{i = 1}^{N} m_{i x} (t) c_{\hat{i}} d_{i} ‖ δ_{i v}^{T} (t) ‖ ‖ e_{i x} (t) ‖ \end{array}

(31)

Here, Young’s inequality is introduced: $x^{T} y \leq \frac{ρ}{2} x^{T} x + \frac{1}{2 ρ} y^{T} y, x, y \in R^{n}, ρ \in R^{+}$ . Then

\begin{array}{l} (31) \leq \sum_{i = 1}^{N} p_{i} e_{i x}^{T} (t) e_{i x} (t) + \sum_{i = 1}^{N} p_{i} δ_{i x}^{T} (t) δ_{i x} (t) + \\ (4 \sum_{i = 1}^{N} \sum_{j \in G_{\hat{i}}} c_{\hat{i}} | a_{i j} | + 4 \sum_{i = 1}^{N} \sum_{j \notin G_{\hat{i}}} | a_{i j} | + 2 \sum_{i = 1}^{N} c_{\hat{i}} d_{i}) m_{i x}^{2} (t) δ_{i v}^{T} (t) δ_{i v} (t) \end{array}

(32)

According to formula (8), we can get

\begin{array}{l} 2 \sum_{i = 1}^{N} δ_{i x}^{T} (t) δ_{i v} (t) \\ + 2 \sum_{i = 1}^{N} δ_{i v}^{T} (t) (f (x_{i} (t), v_{i} (t)) - f (s_{l} (t), g_{l} (t))) \\ \leq \sum_{i = 1}^{N} δ_{i x}^{T} (t) δ_{i x} (t) + \sum_{i = 1}^{N} δ_{i v}^{T} (t) δ_{i v} (t) \\ + 2 \sum_{i = 1}^{N} ‖ δ_{i v}^{T} (t) ‖ ‖ f (x_{i} (t), v_{i} (t)) - f (s_{l} (t), g_{l} (t)) ‖ \\ \leq \sum_{i = 1}^{N} δ_{i x}^{T} (t) δ_{i x} (t) + \sum_{i = 1}^{N} δ_{i v}^{T} (t) δ_{i v} (t) \\ + 2 \sum_{i = 1}^{N} ‖ δ_{i v}^{T} (t) ‖ (r_{1} ‖ δ_{i x} (t) ‖ + r_{2} ‖ δ_{i v} (t) ‖) \\ \leq \sum_{i = 1}^{N} (1 + r_{1}) δ_{i x}^{T} (t) δ_{i x} (t) \\ + \sum_{i = 1}^{N} (1 + r_{1} + 2 r_{2}) δ_{i v}^{T} (t) δ_{i v} (t) \end{array}

(33)

Finally, we have

\begin{array}{l} {\dot{V}}^{1} (t) \leq \sum_{i = 1}^{N} (1 + r_{1} + p_{i}) δ_{i x}^{T} (t) δ_{i x} (t) \\ + \sum_{i = 1}^{N} (1 + r_{1} + 2 r_{2} + p_{i}) δ_{i v}^{T} (t) δ_{i v} (t) \\ + \sum_{i = 1}^{N} \frac{n_{i}}{2} δ_{i v}^{T} (t) δ_{i v} (t) \\ + \sum_{i = 1}^{N} p_{i} e_{i v}^{T} (t) e_{i v} (t) + \sum_{i = 1}^{N} p_{i} e_{i x}^{T} (t) e_{i x} (t) \end{array}

(34)

It follows from equation (25) that

\begin{array}{l} \dot{V} (t) = {\dot{V}}^{1} (t) + {\dot{V}}^{2} (t) \\ \leq \sum_{i = 1}^{N} (m_{i x} (t) - α_{1}) z_{i x}^{T} (t) z_{i x} (t) \\ + \sum_{i = 1}^{N} (m_{i v} (t) - α_{2}) z_{i v}^{T} (t) z_{i v} (t) \\ + \sum_{i = 1}^{N} p_{i} e_{i x}^{T} (t) e_{i x} (t) + \sum_{i = 1}^{N} p_{i} e_{i v}^{T} (t) e_{i v} (t) \\ + \sum_{i = 1}^{N} (1 + r_{1} + 2 r_{2} + p_{i} + \frac{n_{i}}{2}) δ_{i v}^{T} (t) δ_{i v} (t) \\ + \sum_{i = 1}^{N} (1 + r_{1} + p_{i}) δ_{i x}^{T} (t) δ_{i x} (t) + \sum_{i = 1}^{N} {\dot{ψ}}_{i} (t) \end{array}

(35)

Define

δ_{x} (t) = {[δ_{1 x}^{T} (t), δ_{2 x}^{T} (t), \dots, δ_{N x}^{T} (t)]}^{T} \in R^{N n},

δ_{v} (t) = {[δ_{1 v}^{T} (t), δ_{2 v}^{T} (t), \dots, δ_{N v}^{T} (t)]}^{T} \in R^{N n},

e_{x} (t) = {[e_{1 x}^{T} (t), e_{2 x}^{T} (t), \dots, e_{N x}^{T} (t)]}^{T} \in R^{N n},

e_{v} (t) = {[e_{1 v}^{T} (t), e_{2 v}^{T} (t), \dots, e_{N v}^{T} (t)]}^{T} \in R^{N n},

z_{x} (t) = {[z_{1 x}^{T} (t), z_{2 x}^{T} (t), \dots, z_{N x}^{T} (t)]}^{T} \in R^{N n},

z_{v} (t) = {[z_{1 v}^{T} (t), z_{2 v}^{T} (t), \dots, z_{N v}^{T} (t)]}^{T} \in R^{N n} .

From the above, we can obtain $z_{x} (t) = - (\tilde{L} \otimes I_{n}) (δ_{x} (t) + e_{x} (t))$ , $z_{v} (t) = - (\tilde{L} \otimes I_{n}) (δ_{v} (t) + e_{v} (t))$ .

Then, let $H = {\tilde{L}}^{- 1} \otimes I_{n}$ , we have δ_x(t) = −Hz_x(t) − e_x(t), δ_v(t) = −Hz_v(t) − e_v(t). Hence

\begin{array}{l} {‖ δ_{x} (t) ‖}^{2} \leq {(‖ H ‖ ‖ z_{x} (t) ‖ + ‖ e_{x} (t) ‖)}^{2} \\ = {‖ H ‖}^{2} {‖ z_{x} (t) ‖}^{2} + {‖ e_{x} (t) ‖}^{2} \\ + 2 ‖ H ‖ ‖ z_{x} (t) ‖ ‖ e_{x} (t) ‖ \\ \leq 2 ‖ H ‖^{2} {‖ z_{x} (t) ‖}^{2} + 2 {‖ e_{x} (t) ‖}^{2} \end{array}

(36)

Then

\begin{array}{l} {‖ δ_{x} (t) ‖}^{2} = \sum_{i = 1}^{N} δ_{i x}^{T} (t) δ_{i x} (t) \\ \leq 2 ‖ H ‖^{2} ‖ z_{x} (t) ‖^{2} + 2 ‖ e_{x} (t) ‖^{2} \\ = 2 {‖ H ‖}^{2} \sum_{i = 1}^{N} z_{i x}^{T} (t) z_{i x} (t) + 2 \sum_{i = 1}^{N} e_{i x}^{T} (t) e_{i x} (t) \end{array}

(37)

As the same thing

‖ δ_{v} (t) ‖^{2} \leq 2 ‖ H ‖^{2} \sum_{i = 1}^{N} z_{i v}^{T} (t) z_{i v} (t) + 2 \sum_{i = 1}^{N} e_{i v}^{T} (t) e_{i v} (t)

(38)

Then, combined with equation (18), we have

\begin{array}{l} \dot{V} (t) = {\dot{V}}^{1} (t) + {\dot{V}}^{2} (t) \\ \leq \sum_{i = 1}^{N} (m_{i x} (t) - \frac{α_{1}}{2} + {‖ H ‖}^{2} ω_{1 m a x} + ε_{i} β_{i}) z_{i x}^{T} (t) z_{i x} (t) \\ + \sum_{i = 1}^{N} (m_{i v} (t) - \frac{α_{2}}{2} + {‖ H ‖}^{2} ω_{2 m a x} + ε_{i} β_{i}) z_{i v}^{T} (t) z_{i v} (t) \\ - \sum_{i = 1}^{N} \frac{α_{1}}{2} z_{i x}^{T} (t) z_{i x} (t) - \sum_{i = 1}^{N} \frac{α_{2}}{2} z_{i v}^{T} (t) z_{i v} (t) \\ - \sum_{i = 1}^{N} σ_{i} ψ_{i} (t) + \sum_{i = 1}^{N} (ω_{1 m a x} + p_{i} - ε_{i}) e_{i x}^{T} (t) e_{i x} (t) \\ + \sum_{i = 1}^{N} (ω_{2 m a x} + p_{i} - ε_{i}) e_{i v}^{T} (t) e_{i v} (t) \end{array}

(39)

where

ω_{1 m a x} = \underset{i}{m a x} {2 + 2 r_{1} + 2 p_{i}}, ω_{2 m a x} = \underset{i}{m a x} {2 + 2 r_{1} + 4 r_{2} + 2 p_{i} + n_{i}}

, ω_1max + p_i − ɛ_i ≥ 0, and ω_2max + p_i − ɛ_i ≥ 0. Then, we have

\begin{array}{l} \sum_{i = 1}^{N} (ω_{1 m a x} + p_{i} - ε_{i}) e_{i x}^{T} (t) e_{i x} (t) + \sum_{i = 1}^{N} (ω_{2 m a x} + p_{i} - ε_{i}) e_{i v}^{T} (t) e_{i v} (t) \\ \leq \sum_{i = 1}^{N} (κ_{1 i} + κ_{2 i}) (e_{i x}^{T} (t) e_{i x} (t) + e_{i v}^{T} (t) e_{i v} (t)) \\ \leq \sum_{i = 1}^{N} (κ_{1 i} + κ_{2 i}) (\frac{ψ_{i} (t)}{θ_{i}} + β_{i} z_{i x}^{T} (t) z_{i x} (t) + β_{i} z_{i v}^{T} (t) z_{i v} (t)) \\ \leq \sum_{i = 1}^{N} (\frac{κ_{1 i} + κ_{2 i}}{θ_{i}}) ψ_{i} (t) + \sum_{i = 1}^{N} (κ_{1 i} β_{i} + κ_{2 i} β_{i}) z_{i x}^{T} (t) z_{i x} (t) \\ + \sum_{i = 1}^{N} (κ_{1 i} β_{i} + κ_{2 i} β_{i}) z_{i v}^{T} (t) z_{i v} (t) \end{array}

(40)

where κ_1i = ω_1max + p_i − ɛ_i and κ_2i = ω_2max + p_i − ɛ_i, then

\begin{array}{l} \dot{V} (t) = {\dot{V}}^{1} (t) + {\dot{V}}^{2} (t) \\ \leq \sum_{i = 1}^{N} ζ_{1 i} z_{i x}^{T} (t) z_{i x} (t) + \sum_{i = 1}^{N} ζ_{2 i} z_{i v}^{T} (t) z_{i v} (t) \\ - \sum_{i = 1}^{N} \frac{α_{1}}{2} z_{i x}^{T} (t) z_{i x} (t) - \sum_{i = 1}^{N} \frac{α_{2}}{2} z_{i v}^{T} (t) z_{i v} (t) \end{array}

+ \sum_{i = 1}^{N} (\frac{κ_{1 i} + κ_{2 i}}{θ_{i}} - σ_{i}) ψ_{i} (t) < 0

(41)

where

ζ_{1 i} = m_{i x} (t) - α_{1} / 2 + {‖ H ‖}^{2} ω_{1 m a x} + ε_{i} β_{i} + κ_{1 i} β_{i} + κ_{2 i} β_{i}

and

ζ_{2 i} = m_{i v} (t) - α_{2} / 2 + {‖ H ‖}^{2} ω_{2 m a x} + ε_{i} β_{i} + κ_{1 i} β_{i} + κ_{2 i} β_{i}

. Under the sufficient condition of Theorem 1, the proof is accomplished and the cluster consensus is achieved.

Considering the design of the distributed ETC scheme, the following will demonstrate that no Zeno behavior occurs in this system.

Proof

Since ${t_{k}^{i} - τ}_{k = 1}^{\infty}$ is the triggered instant sequence of agent i. From (9), we have $e_{i v} (t_{k}^{i} - τ) = 0$ and $e_{i x} (t_{k}^{i} - τ) = 0$

\begin{array}{l} ‖ {\dot{e}}_{i} (t) ‖ = ‖ (\begin{array}{l} {\dot{e}}_{i x}^{T} (t) \\ {\dot{e}}_{i v}^{T} (t) \end{array}) ‖ \\ \leq ‖ {\dot{e}}_{i x}^{T} (t) ‖ + ‖ {\dot{e}}_{i v}^{T} (t) ‖ \\ = ‖ v_{i} (t) ‖ + ‖ f (x_{i} (t), v_{i} (t)) + u_{i} (t - τ) ‖ \\ \leq ‖ v_{i} (t) - g_{i} (t) + g_{i} (t) ‖ \\ + z_{1} ‖ x_{i} (t) ‖ + z_{2} ‖ v_{i} (t) ‖ + ‖ U_{\max} ‖ \\ \leq (1 + z_{2}) ‖ δ_{i v} (t) ‖ + (1 + z_{2}) ‖ g_{i} (t) ‖ \\ + z_{1} ‖ δ_{i x} (t) ‖ + z_{1} ‖ s_{i} (t) ‖ + ‖ U_{\max} ‖ \\ \leq (1 + z_{1} + z_{2}) (‖ δ_{i v} (t) ‖ + ‖ δ_{i x} (t) ‖) \\ + (1 + z_{2}) ‖ G_{\max} (t) ‖ + z_{1} ‖ S_{\max} (t) ‖ + ‖ U_{\max} ‖ \end{array}

(42)

where G_max, S_max, and U_max represent the upper bound of functions g_i(t), s_i(t), and u_i (t − τ) as mentioned in Assumption 4.

Let $V^{1} (t) = \sum_{i = 1}^{N} δ_{i}^{T} (t) δ_{i} (t)$ , and $\dot{V} (t) < 0$ , we have

\begin{array}{l} ‖ δ_{i v} (t) ‖ + ‖ δ_{i x} (t) ‖ \leq \sqrt{2} ‖ δ_{i} (t) ‖ \\ = \sqrt{2} \sqrt{V^{1} (t)} < \sqrt{2} \sqrt{V^{1} (t_{k}^{i})} \end{array}

(43)

Integrating $‖ {\dot{e}}_{i} (t) ‖$ , we obtain

\int_{t_{k}^{i} - τ}^{t} ‖ {\dot{e}}_{i} (t) ‖ d t \leq \int_{t_{k}^{i} - τ}^{t} Δ d t

(44)

where

Δ = (1 + z_{1} + z_{2}) \sqrt{2 V^{1} (t_{k}^{i})} + (1 + z_{2}) ‖ G_{\max} (t) ‖ + z_{1} ‖ S_{\max} ‖ + ‖ U_{\max} ‖ > 0

Let $t = t_{k + 1}^{i} - τ$ , the formula (44) turns into

‖ e_{i} (t_{k + 1}^{i} - τ) ‖ - ‖ e_{i} (t_{k}^{i} - τ) ‖ \leq (t_{k + 1}^{i} - t_{k}^{i}) Δ

(45)

It follows from (17) that

{‖ e_{i} (t_{k + 1}^{i} - τ) ‖}^{2} - β_{i} {‖ z_{i} (t_{k + 1}^{i} - τ) ‖}^{2} \geq \frac{ψ_{i} (t_{k + 1}^{i} - τ)}{θ_{i}} > 0

(46)

\sqrt{\frac{ψ_{i} (t_{k + 1}^{i} - τ)}{θ_{i}}} \leq ‖ e_{i} (t_{k + 1}^{i} - τ) ‖ \leq (t_{k + 1}^{i} - t_{k}^{i}) Δ

(47)

Thus

t_{k + 1}^{i} - t_{k}^{i} \geq \frac{1}{Δ} \sqrt{\frac{ψ_{i} (t_{k + 1}^{i} - τ)}{θ_{i}}} > 0

(48)

This inequality implies that $t_{k + 1}^{i} - t_{k}^{i} > 0$ for any finite horizon. Moreover, we can show $t_{k}^{i} \to \infty$ as k → ∞. If this is not true, by denoting $t_{\infty}^{i} = \lim_{k \to \infty} t_{k}^{i}$ , then $t_{\infty}^{i} < \infty$ , and moreover $\lim_{k \to \infty} t_{k + 1}^{i} - t_{k}^{i} = 0$ . Thus, it is known from (47) that for k → ∞, $0 < \sqrt{ψ_{i} (t_{\infty}^{i}) / θ_{i}} \leq Δ (t_{k + 1}^{i} - t_{k}^{i}) = 0$ , which is a contradiction. Consequently, no Zeno behavior is exhibited.

4. Dynamic ETC for heterogeneous MASs with input saturation and time delay under switching topology

Usually, MASs are subject to interference from external signals which can be changing their connections. This is a common phenomenon. Meanwhile, agents in the different clusters may also communicate according to different protocols and have different dynamics. In this section, we discuss the cluster consensus problem of heterogeneous MASs under switching topology.

Consider MASs with p clusters, where p1 clusters are second-order linear dynamic equations and p − p1 clusters are second-order nonlinear dynamic equations. Here, let Y⁻ = {1, 2, …p1}, Y⁺ = {p1 + 1, …p}, and I_N = {1, …, N}.

Switching topology is when the system switches from structure Γ_k+1th to structure Γ_kth at time t_k − τ. Switching signal function Γ_k(t) belongs to the finite set Z = {1, 2, …, m}. The Γ_kth topology structure is executed when t ∈ (t_k−1 − τ, t_k − τ]. B = {B¹, B², …B^m} denotes available topologies.

The dynamic equations are written as

{\begin{cases} {\dot{x}}_{i} (t) = v_{i} (t), {\dot{v}}_{i} (t) = u_{i} (t - τ) & i \in Y^{-} \\ {\dot{x}}_{i} (t) = v_{i} (t), {\dot{v}}_{i} (t) = u_{i} (t - τ) + f (x_{i} (t), v_{i} (t)) & i \in Y^{+} \end{cases}

(49)

and the dynamic equations of the leader can be described as

{\begin{cases} {\dot{s}}_{h} (t) = g_{h} (t), {\dot{g}}_{h} (t) = 0 & h \in Y^{-} \\ {\dot{s}}_{h} (t) = g_{h} (t), {\dot{g}}_{h} (t) = f (s_{h} (t), g_{h} (t)) & h \in Y^{+} \end{cases}

(50)

and other symbols are the same as defined in Theorem 1 above.

Then, some cluster consensus conditions of MASs (49) and (50) under the control protocol (12) will be derived.

Substituting (12) into (49), the dynamics of agent i satisfy the following equations

{\begin{cases} {\dot{x}}_{i} (t) = v_{i} (t) \\ {\dot{v}}_{i} (t) = f (x_{i} (t), v_{i} (t)) \\ - m_{i x} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j}^{Γ_{k}} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j}^{Γ_{k}} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i}))], i \in Y^{-} \\ - m_{i v} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j}^{Γ_{k}} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j}^{Γ_{k}} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i}))] \end{cases}

{\begin{cases} {\dot{x}}_{i} (t) = v_{i} (t) \\ {\dot{v}}_{i} (t) = - m_{i x} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j}^{Γ_{k}} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j}^{Γ_{k}} \arctan (x_{i} (t_{k}^{i} - τ) - x_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (x_{i} (t_{k}^{i} - τ) - s_{\hat{i}} (t_{k}^{i}))], i \in Y^{+} \\ - m_{i v} (t) [\sum_{j \in G_{\hat{i}}} c_{\hat{i}} a_{i j}^{Γ_{k}} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + \sum_{j \notin G_{\hat{i}}} a_{i j}^{Γ_{k}} \arctan (v_{i} (t_{k}^{i} - τ) - v_{j} (t_{k}^{i} - τ)) \\ + c_{\hat{i}} d_{i} \arctan (v_{i} (t_{k}^{i} - τ) - g_{\hat{i}} (t_{k}^{i}))] \end{cases}

(51)

The error of MASs can be expressed as follows

\begin{array}{l} {\begin{cases} {\dot{δ}}_{i x} (t) = δ_{i v} (t) i \in Y^{-} \\ {\dot{δ}}_{i v} (t) = u_{i} (t - τ) \end{cases} \\ {\begin{cases} {\dot{δ}}_{i x} (t) = δ_{i v} (t) i \in Y^{+} \\ {\dot{δ}}_{i v} (t) = u_{i} (t - τ) + f (x_{i} (t), v_{i} (t)) - f (s_{h} (t), g_{h} (t)) \end{cases} \end{array}

(52)

Theorem 2

If Assumptions 1–4 are satisfied, for given scalars $σ_{i}^{Γ_{k}}, ε_{i}, θ_{i}, β_{i}$ , the inequalities satisfy the following

ζ_{1 i}^{Γ_{k}} \leq 0, ζ_{2 i}^{Γ_{k}} \leq 0, \frac{κ_{1 i}^{Γ_{k}} + κ_{2 i}^{Γ_{k}}}{θ_{i}} - σ_{i} \leq 0

where

ζ_{1 i}^{Γ_{k}} = m_{i x} (t) - \frac{α_{1}}{2} + ‖ H^{Γ_{k}} ‖ ω_{1 m a x}^{Γ_{k}} + ε_{i} β_{i} + κ_{1 i}^{Γ_{k}} β_{i} + κ_{2 i}^{Γ_{k}} β_{i}, ζ_{2 i}^{Γ_{k}} = m_{i v} (t) - \frac{α_{2}}{2} + ‖ H^{Γ_{k}} ‖ ω_{2 m a x}^{Γ_{k}} + ε_{i} β_{i} + κ_{1 i}^{Γ_{k}} β_{i} + κ_{2 i}^{Γ_{k}} β_{i},

κ_{1 i}^{Γ_{k}} = ω_{1 m a x}^{Γ_{k}} + p_{i}^{Γ_{k}} - ε_{i} \geq 0, κ_{2 i}^{Γ_{k}} = ω_{2 m a x}^{Γ_{k}} + p_{i}^{Γ_{k}} - ε_{i} \geq 0, p_{i}^{Γ_{k}} = \sum_{j \in G_{\hat{i}}} c_{\hat{i}} | a_{i j}^{Γ_{k}} | + \sum_{j \in G_{\hat{i}}} c_{\hat{i}} | a_{j i}^{Γ_{k}} | + \sum_{j \notin G_{\hat{i}}} | a_{i j}^{Γ_{k}} | + \sum_{j \notin G_{\hat{i}}} | a_{j i}^{Γ_{k}} | + c_{\hat{i}} d_{i} > 0, n_{i}^{Γ_{k}} = (8 \sum_{j \in G_{\hat{i}}} c_{\hat{i}} | a_{i j}^{Γ_{k}} | + 8 \sum_{j \notin G_{\hat{i}}} | a_{i j}^{Γ_{k}} | + 4 c_{\hat{i}} d_{i}) (m_{i x}^{2} (t) + m_{i v}^{2} (t)) \geq 0,

ω_{1 m a x}^{Γ_{k}} = \underset{i}{m a x} {2 + 2 \{r_{1}, 0\} + 2 p_{i}^{Γ_{k}}}, ω_{2 m a x}^{Γ_{k}} = \underset{i}{m a x} {2 + 2 \{r_{1}, 0\} + 4 \{r_{2}, 0\} + 2 p_{i}^{Γ_{k}} + n_{i}^{Γ_{k}}} .

Then, the cluster consensus can be achieved. In addition, {r₁, 0} = r₁ when i ∈ Y⁻, otherwise {r₁, 0} = 0 when i ∈ Y⁺. {r₂, 0} is similar to the definition of {r₁, 0}.

Proof

Construct function

\begin{array}{l} V (t) & = \sum_{i = 1}^{N} δ_{i}^{T} (t) δ_{i} (t) + \sum_{i = 1}^{N} \frac{{(m_{i x} (t) - α_{1})}^{2}}{2 γ_{i x}} \\ + \sum_{i = 1}^{N} \frac{{(m_{i v} (t) - α_{2})}^{2}}{2 γ_{i v}} + \sum_{i = 1}^{N} ψ_{i} (t) \end{array}

(53)

Evidently, the function V(t) is positive definite. The proof is similar to Section 3 above and then we can know that $\dot{V} (t) < 0$ on each interval (t_k − τ, t_k+1 − τ]. For this reason, V(t) is decreasing in [0, + ∞] and $\lim_{t \to \infty} V (t)$ exists. Thus, the MASs (49) will eventually realize cluster consensus. This is the end of the proof.

The proof of Zeno behavior under the ETC protocol (17) is similar to (42)–(48), which is ignored here. In addition, the implementation algorithm of the control strategy with the new trigger condition is shown in Algorithm 1.

5.Simulations

There are two numerical simulations given in this section to verify the feasibility of the above system model.

We choose a topology containing seven agents with three clusters, where nodes 1 and 2 are a cluster, nodes 3, 4, and 5 are another cluster, and the remaining two nodes are a cluster. Figure 1 shows the topology of each cluster in relation to the leader.

Example 1

Let intra-cluster coupling strength c₁ = 4, c₂ = 4, c₃ = 4. The original states of seven agents can be chosen as x₁ = 18, x₂ = 15, x₃ = −8, x₄ = −5, x₅ = −12, x₆ = 10, x₇ = 13, and the original velocities of seven agents can be chosen as v₁ = 22, v₂ = 10, v₃ = −10, v₄ = −1, v₅ = 0, v₆ = 14, v₇ = 17. The original states of three leaders can be chosen as s₁ = 15, s₂ = −10, s₃ = 8, and the original velocities of three leaders can be chosen as g₁ = 5, g₂ = −5, g₃ = 11.

Choose the nonlinear function of leader as f (s_h(t), g_h(t)) = 0.1 sin (s_h(t) ⋅ g_h(t)), and choose the nonlinear function of agent as f (x_i(t), v_i(t)) = 0.1 sin (x_i(t) ⋅ v_i(t)). In addition, set the time delay τ = 0.02s. The simulation time is T = 12s. Let ɛ_i = 20, ψ_i (0) = 10, θ_i = [1900,9600,2000,600,9200,8800,2500]^T, γ_ix = 0.001, γ_iv = 0.001, β_i = and 0.01, m_ix (0) = 0.01, m_iv (0) = 0.01. According to Lipschitz conditions, we can take r₁ = 1, r₂ = 1.

Figures 2 and 3 show the state and velocity of seven agents with the input saturation and time delay.

Figure 4 displays the error between leader-followers. It follows from Figures 2 –4 that the cluster consensus is reached as we expected. It implies that when different tasks are assigned to different clusters, they all ultimately complete their respective tasks, which has significant practical significance.

Figure 5 shows the evolution of adaptive parameters m_ix(t) and m_iv(t). We can obviously know that the values of m_ix(t) and m_iv(t) increase rapidly at the beginning because of the large error between leader and agents. As the error becomes smaller and smaller, they tend to equilibrium states.

Figure 6 illustrates the transformation of ψ_i(t). The dynamic parameter ψ_i(t) > 0 can show that the control strategy we designed can achieve the purpose of reducing the number of triggers. Finally, the comparisons between this paper and other results are given. Compared with Figure 2 in [30], this paper achieves the cluster consensus instead of complete consensus. Furthermore, although both this paper and [30] consider the input saturation, it follows from Figure 3 in [30] and Figure 7 that the control inputs in this paper are piece-wise continuous and those in [30] are updated continuously. Therefore, the control strategy proposed in this paper can further reduce the update frequency of controllers, extend their lifespan, and improve the communication bandwidth.

Figure 8 indicates the triggering time sequences of seven agents, which implies that there exists no Zeno behavior. In addition, it is observed from Table 1 that in contrast to the traditional static ETC, dynamic ETC can reduce the number of triggers significantly.

Table 1.

Trigger times of the agent (i = 1, 2, …, 7) in 12 s in Example 1.

	Agent1	Agent2	Agent3	Agent4	Agent5	Agent6	Agent7
Static ETC	635	380	698	683	372	479	672
Dynamic ETC	390	177	259	194	233	263	459

Example 2

The parameters are the same as in Example 1. Consider the switching topology in Figure 9, and suppose the original topology of MASs changes from (a) to (b) by external disturbances. Figure 10 demonstrates that the cluster consensus can be achieved under switching topology and theoretical result of this paper is feasible.

6. Conclusion

In this paper, we addressed the cluster consensus problem for SOMASs. Two novel input saturation controllers are designed to realize cluster consensus of MASs with the fixed and switching topologies. Furthermore, we also demonstrated that no Zeno behavior happened. Finally, two examples were provided to verify the efficiency of theory results. However, controller fault tolerance and other situations are not taken into consideration, which are more complicated but of more realistic significance. They will be studied further in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 61903318), Natural Science Foundation of Hunan Province 2019JJ50619, and Research Fund for the Doctoral Program of Higher Education of China (11KZ|KZ08069).

ORCID iDs

Yaping Xia

Long Tang

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