# Group Consensus of Heterogeneous Multiagent Systems with Time Delay

Authors: Weixun Li; Liqiong Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a neighbour-based control algorithm of group consensus is designed for a class of hybrid-based heterogeneous multiagent systems with communication time delay. We consider the statics leaders and active leaders, respectively. The original systems are transformed into new error systems by transformation. On the basis of the systems, applying Lyapunov stability theory and adopting the linear matrix inequality method, sufficient conditions which guarantee the heterogeneous multiagent systems stability are obtained. To illustrate the validity of theoretical results, some numerical simulations are given at the end of the paper.


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## Body

## 1. Introduction

Consensus problem of multiagent systems (MAS) has always been a hot topic in the control field. In recent years, due to the huge leap in electronic technology, the application of MAS is extremely extensive, mainly involving mechanical engineering, unmanned aerial vehicles, robot formations, neural networks and so on [1–4]. In previous studies, each agent had the same dynamic system, and their own attributes and traits were the same. But in today’s ever-changing era, any electronic device is constantly improving, and the cycle of updating is becoming shorter and shorter. Therefore, in order to adapt to the development of the times, many scholars have focused on heterogeneity. On the agent system, the dynamic model of the agent in the system is not exactly the same, and consensus issues for heterogeneous MAS have been investigated in [5–7].Due to the complexity of practical network international circumstances and the diversity of control systems, it is impossible for all agents to only tend to be in a stable state. In view of this, the group consensus issue for heterogeneous MAS has attracted the favour of many academics at home and abroad, among which the most of it is common for heterogeneous systems composed of second-order MAS in continuous time [8]. In Hu et al.’s study [8], the authors considered the heterogeneous MAS with uncertain parameters. There are also heterogeneous systems which are consisted of a first-order system and a second-order MAS [9]. In Yu et al.’s study [9], investigated group consensus matters for heterogeneous multiagent networks in a rival system. In Wen et al.’s study [10], the authors discussed group consensus issues for heterogeneous MAS in the situation of input saturation. It was composed of two first-order MAS with or without nonlinear and second-order multiagent systems. In addition to continuous time, there are a large number of works dealing with the matters of consensus of cluster and group for discrete-time heterogeneous multiagent systems [11]. For example, in Shi et al.’s study [11], based on discrete time, scholars investigated dynamic interactions nonsynchronous group consistency for heterogeneous MAS in the topology of message interaction. In Jiang et al.’s study [12], the authors studied couple-group consensus for heterogeneous MAS in discrete time though adopting synergetic-competitive interaction and time-lags. In the study by Feng and Zheng [13], the issues of group consensus control for the first-order and second-order heterogeneous MAS in the situation of discrete time was discussed minutely. At present, most of the research focuses on the interactive messages of agent’s neighbours to obtain relative status information of each agent, but in the process of the information transmission often has some physical factors such as transmission channels, maybe lead to the inability of messages between agents to arrive in time. This requires scholars to refer to the important factor of communication delay in research. In Wen et al.’s study [14], researched dynamical group consensus matters for heterogeneous MAS with communication lags. Under the circumstances directional structure of message interactions, Li et al. considered group consensus issues for the MAS with sampled and quantized data in detail in [15]. However, in practical applications, many engineering problems are not a single linear system, in order to make works that are closer to the actual application, in the study of group consistency issue for heterogeneous MAS, not only time lag but also the consideration of the influence of nonlinear term. Under the situation of parametric uncertainties, Hu et al. researched group consensus for heterogeneous multiagent systems in [8]. In Liu et al.’s study [16], they investigated consensus for heterogeneous MAS under fixed and switching topologies. It enriches the content of heterogeneous multiagent systems.Inspired by the above-mentioned results, this article studies the group consensus problem for mixed-order heterogeneous MAS with time delay, which is grouped into two and three portions. Part one, in the circumstance of the presence of time lags, group consensus issues are investigated for the second-order MAS with or without nonlinear; in the second part, the group consensuses for the systems which are consisted of the second-order nonlinear multiagent system and the second-order and first-order MAS in a linear environment are mainly studied. The analysis and conclusion of three parts can be proceeded primarily via utilizing Lyapunov stability theory. The approach of linear matrix inequality is used to prove it. Finally, the results are numerically simulated by Matlab, and the validity of the conclusion is further demonstrated.The construction of the remainder of the article is described below. Section2 mainly illustrated textual preparatory work, which contains the two portions of graph theory and dynamical systems description. There are couple-part contents in Section 3, part one under the premise of active leaders in multiagent systems, a sufficient criterion is provided to implement group consensus; the other part in the presence of static leaders, we also acquire a sufficient condition in regard to group consistency of heterogeneous MAS with time-lags. In the sequel, numerical modelling and conclusion are expounded in Sections 4 and 5, separately.Notations: for simplicity of the proof process of the paper, some mathematical notations are adopted throughout this article. Supposeℝn×n denotes the real matrix with n×n. Let diag⋯ denote the matrix with the elements are all zero except the main diagonal elements. Let RT signify the transpose of matrix R. Suppose 1n denotes a column vector with all elements are 1.


## 2. Preliminaries

### 2.1. Graph Theory

SupposeG=I,T,andC is a weighted digraph which has a set of node I=1,…,n, a set of trajectory T⊆I×I, and a weighted adjacency matrix C=cij∈ℝn×n. A trajectory of Gis denoted by iandj, it means to from j to i. If element cij fulfilled the trajectory in the digraph, then it is positive, denotes cij≠0⟺i,j∈T. Let cii=0 for all agents i∈I. The set of neighbours of node i is expressed by Ni=j∈I:i,j∈T. Then, Laplacianof the weighted digraph G is defined asL=D−C∈ℝn×n, L=lij∈ℝn×n, lij=∑j=1,j≠incij,i=j−cij,i≠j.On the basis of the peculiarity of Laplacian can be aware of the all row-sums ofL are zero. Therefore, L has a zero eigenvalue corresponding to a right eigenvector 1=1,…,1T∈ℝn 1 = (1,…1) T ∈ Rn + m. Let D=diagd1,d2,…,dn be an in-degree matrix and di=degini=∑i=1ncij.


### 2.2. Dynamic Systems Description

In this subsection, a generalized graphG˜ is defined which contains followers and two active leaders. There is no absence of generality, graph G˜ is divided into two portions, the first subgroup G˜1 contains the former m followers which follow their leader l1, and the rest of the followers belong to the second group G˜2 which follows l2. Let Gj∗=Ij∗,Tj∗,Cj∗j=1,2 be a subgroup of G∗=I∗,T∗,C∗ and its leader lj for j=1,2.The dynamical dynamics of followerican be indicated as(1)p˙it=qit,q˙it=uit,i∈I1≔1,2,…,m,where pit,qit,anduit∈ℝ f (pit, qi(t)) are position status, velocity status, and control input of agent i.The dynamics of the leader of heterogeneous MAS can be constructed as(2)p˙1∗t=q1∗t,where p1∗t∈ℝ and q1∗t∈ℝ are the location state and speed status of the leader l1.The dynamical modelling of the other followeri can be depicted as(3)p˙it=qit,q˙it=uit+fipit,t,i∈I2≔m+1,…,n,where pit,qit,anduit∈ℝ and fipit,t are location status, velocity status, control input, and the inherent nonlinear term of agent i.The dynamic state of leader can be depicted as follows:(4)p˙2∗t=q2∗t,q˙2∗t=fp2∗t,t,where p2∗tandq2∗t∈ℝ and fp2∗t,t are position state, speed state, and the consecutive nonlinear function of the leader l2.On account of the presence of time lags, all agents may not receive the messages from others and their leaders. Hence, for agenti, a coupled control protocol is put forward in the following form:(5)uit=∑j=1mcijpjt−rt−pit−rt+∑j=m+1ncijpjt−rt+qjt+kbiq1∗t−qit+bip1∗t−rt−pit−rt,i=1,…,m,∑j=m+1ncijpjt−rt−pit−rt+bip2∗t−rt−pit−rt+kbiq2∗t−qit+∑j=1mcijpjt−rt+qjti=m+1,…,n.where the lag rt is ever-changing and differentiable and 0<rt<τM,r˙t<u.bi is a diagonal element of an adjacency matrix of the leader, matrix denoted as B=diagbi∈ℝn×n. And if agent i can receive a message from leader lj, then bi>0; otherwise, bi=0.k>0 is a gain constant.In this subsection, the followers can be divided into three groups as follows. And we consider static leaders. Then, the dynamical model of the second-order nonlinear agenti can be written as(6)p˙it=qit,q˙it=uit+fipit,t,i∈I1≔1,2,…,l,where pit,qit,anduit∈ℝ and fipit,t are location status, velocity status, control input, and nonlinear portion of agent i, respectively.The dynamical modality of the second-order agenti can be described as(7)p˙it=qit,q˙it=uit,i∈I2≔l+1,2,…,m,where pit,qit,anduit∈ℝ are position state, velocity state, and control input of agent i, respectively.The dynamical model of the first-order agenti can be described as(8)p˙it=uiti∈I3≔m+1,…,n,where pit∈ℝ and uit∈ℝ are position state and control input of agent i.The control protocol of systems (6)–(8) are as follows:(9)uit=k1∑j=1lcijpjt−rt−pit−rt+k1∑j=1llijpσjt−rt+∑j∉I1cijpjt−rt+qjt+k2∑j=1lcijqjt−qit−k3qit,i∈I1,k1∑j=l+1mcijpjt−rt−pit−rt+k1∑j=l+1mlijpσjt−rt+∑j∉I2cijpjt−rt+qjt+k2∑j=l+1mcijqjt−qit−k3qit,i∈I2,∑j=m+1ncijpjt−rt−pit−rt+∑j=m+1nlijpσjt−rt+∑j∉I3cijpjt−rt+qjti∈I3,where k1,k2,andk3 are all positive constants, which standing for the positive coupling strengths, severally. And pσj represents the consistent position equilibrium of the group in which the j−thagent resides, and σi=1,2,and3.In order to the simplicity of presentation, we lay emphasis on one-dimensional space. However, for any high-dimensional space, we can generalize the results through applying the characteristics of theKroneckerKronecker product, denoted as ⊗.Remark 1.
The portion of agents which are second-order agents and second-order nonlinear agents, the adjacency matrix C is partitioned as(10)C=CS1CS1S2CS2S1CS2,with CS1∈ℝm×mand CS2∈ℝn−m×n−m. Suppose LS1andLS2 signify Laplacian matrix of the second-order agents in the two situations of linearity and nonlinearity, respectively. Then, the matrix can be attained as follows:(11)L=LS1+DS1S2−CS1S2−CS2S1LS2+DS2S1,with(12)DS1S2=diag∑j∈N1,S2c1j,…,∑j∈Nm,S2cmj;DS2S1=diag∑j∈Nm+1,S1cm+1j,…,∑j∈Nn,S1cnj.Remark 2.
The portion of agents are consisted of the second-order agents with or without nonlinearity and first-order agents, then the segmentation ofC as follows:(13)C=Cs1Cs1s2Cs1fCs2s1Cs2Cs2fCfs1Cfs2Cf,with Cs1∈ℝl×l,Cs2∈ℝm−l×m−l, andCf∈ℝn−m×n−m.
PresumeLs1,Ls2,andLf are Laplacian matrix of the second-order agents with nonlinearity and linearity, and first-order agents, severally. Then, L can be depicted as(14)L=Ls1+Ds1s2+Ds1f−As1s2−As1f−As2s1Ls2+Ds2s1+Ds2f−As2f−Afs1−Afs2Lf+Dfs1+Dfs2,with Ls1=Ds1−Cs1;Ls2=Ds2−Cs2;Lf=Df−Cf(15)Ds1s2=diag∑j∈N1,s2c1j,…,∑j∈Nl,s2clj;Ds1f=diag∑j∈N1,fc1j,…,∑j∈Nl,fclj;Ds2s1=diag∑j∈N1,s1cl+1j,…,∑j∈Nl,s1cmj;Ds2f=diag∑j∈N1,fcl+1j,…,∑j∈Nl,fcmj;Dfs1=diag∑j∈Nm+1,s1cm+1j,…,∑j∈Nn,s1cnj;Dfs2=diag∑j∈Nm+1,s2cm+1j,…,∑j∈Nn,s2cnj.
Next, we will give the definition of the two-group consensus and three-group for heterogeneous multiagent systems. And the similar definition can be given for multigroup consensus problem.Definition 1.
The group consensus of heterogeneous multiagent systems can be achieved if the states of followers tend to the states of leaders in the sense of(16)limt⟶∞pit−p1∗t=0,limt⟶∞qit−q1∗t=0,∀i∈I1;limt⟶∞pit−p2∗t=0,limt⟶∞qit−q2∗t=0,∀i∈I2.Definition 2.
Group consensus of heterogeneous multiagent systems can be achieved if the states of followers tend to the states of leaders in the sense of(17)limt⟶∞pit−pσit=0,limt⟶∞qit=0,∀i∈I1;limt⟶∞pit−pσit=0,limt⟶∞qit=0,∀i∈I2;limt⟶∞pit−pσit=0,∀i∈I3.Assumption 1.
(see in [13]).(18)∑j=m+1ncij=0,i=1,…,m;∑j=1mcij=0,i=m+1,…,n.Assumption 2.
(see in [15]).(19)∑j=m+1ncij=0,∀i∈I1;∑j=1,j∉I2ncij=0,∀i∈I2;∑j=1mcij=0,∀i∈I3.
For the sake of demonstrating group consensus of heterogeneous MAS, we need to give several lemmas which are useful in the proof of the main results as follows:Lemma 1 (see in [14]).
fpit,t fpit, qit, and fpj∗t,t are continuous differentiable function vectors. In order to achieve group consensus of the heterogeneous MAS containing the nonlinear term, the following assumption is given as follows:(20)fpt,t−fp2∗t,t⋅1n≤ρpt−p2∗t⋅1n,∀pit,p2∗t∈ℝn,i=m+1,…,n;pt=pm+1t,…,pntT;1n=1,…,1T, and a nonnegative constant ρ.Lemma 2 (see in [15]).
For arbitrary constant vectorsa,b∈ℝn and a positive definite matrixR∈ℝn×n, then the following inequality holds(21)±2aTb≤aTRa+bTR−1b.Lemma 3 (see in [14]) (SchurSchur complement).
Given a matrixΔ=Δ11Δ12Δ12TΔ22. Thus the following conditions are equivalent as follows:(22)Δ<0;Δ11<0,Δ22−Δ12TΔ11−1Δ12<0;Δ22<0,Δ11−Δ12Δ22−1Δ12T<0.


## 2.1. Graph Theory

SupposeG=I,T,andC is a weighted digraph which has a set of node I=1,…,n, a set of trajectory T⊆I×I, and a weighted adjacency matrix C=cij∈ℝn×n. A trajectory of Gis denoted by iandj, it means to from j to i. If element cij fulfilled the trajectory in the digraph, then it is positive, denotes cij≠0⟺i,j∈T. Let cii=0 for all agents i∈I. The set of neighbours of node i is expressed by Ni=j∈I:i,j∈T. Then, Laplacianof the weighted digraph G is defined asL=D−C∈ℝn×n, L=lij∈ℝn×n, lij=∑j=1,j≠incij,i=j−cij,i≠j.On the basis of the peculiarity of Laplacian can be aware of the all row-sums ofL are zero. Therefore, L has a zero eigenvalue corresponding to a right eigenvector 1=1,…,1T∈ℝn 1 = (1,…1) T ∈ Rn + m. Let D=diagd1,d2,…,dn be an in-degree matrix and di=degini=∑i=1ncij.


## 2.2. Dynamic Systems Description

In this subsection, a generalized graphG˜ is defined which contains followers and two active leaders. There is no absence of generality, graph G˜ is divided into two portions, the first subgroup G˜1 contains the former m followers which follow their leader l1, and the rest of the followers belong to the second group G˜2 which follows l2. Let Gj∗=Ij∗,Tj∗,Cj∗j=1,2 be a subgroup of G∗=I∗,T∗,C∗ and its leader lj for j=1,2.The dynamical dynamics of followerican be indicated as(1)p˙it=qit,q˙it=uit,i∈I1≔1,2,…,m,where pit,qit,anduit∈ℝ f (pit, qi(t)) are position status, velocity status, and control input of agent i.The dynamics of the leader of heterogeneous MAS can be constructed as(2)p˙1∗t=q1∗t,where p1∗t∈ℝ and q1∗t∈ℝ are the location state and speed status of the leader l1.The dynamical modelling of the other followeri can be depicted as(3)p˙it=qit,q˙it=uit+fipit,t,i∈I2≔m+1,…,n,where pit,qit,anduit∈ℝ and fipit,t are location status, velocity status, control input, and the inherent nonlinear term of agent i.The dynamic state of leader can be depicted as follows:(4)p˙2∗t=q2∗t,q˙2∗t=fp2∗t,t,where p2∗tandq2∗t∈ℝ and fp2∗t,t are position state, speed state, and the consecutive nonlinear function of the leader l2.On account of the presence of time lags, all agents may not receive the messages from others and their leaders. Hence, for agenti, a coupled control protocol is put forward in the following form:(5)uit=∑j=1mcijpjt−rt−pit−rt+∑j=m+1ncijpjt−rt+qjt+kbiq1∗t−qit+bip1∗t−rt−pit−rt,i=1,…,m,∑j=m+1ncijpjt−rt−pit−rt+bip2∗t−rt−pit−rt+kbiq2∗t−qit+∑j=1mcijpjt−rt+qjti=m+1,…,n.where the lag rt is ever-changing and differentiable and 0<rt<τM,r˙t<u.bi is a diagonal element of an adjacency matrix of the leader, matrix denoted as B=diagbi∈ℝn×n. And if agent i can receive a message from leader lj, then bi>0; otherwise, bi=0.k>0 is a gain constant.In this subsection, the followers can be divided into three groups as follows. And we consider static leaders. Then, the dynamical model of the second-order nonlinear agenti can be written as(6)p˙it=qit,q˙it=uit+fipit,t,i∈I1≔1,2,…,l,where pit,qit,anduit∈ℝ and fipit,t are location status, velocity status, control input, and nonlinear portion of agent i, respectively.The dynamical modality of the second-order agenti can be described as(7)p˙it=qit,q˙it=uit,i∈I2≔l+1,2,…,m,where pit,qit,anduit∈ℝ are position state, velocity state, and control input of agent i, respectively.The dynamical model of the first-order agenti can be described as(8)p˙it=uiti∈I3≔m+1,…,n,where pit∈ℝ and uit∈ℝ are position state and control input of agent i.The control protocol of systems (6)–(8) are as follows:(9)uit=k1∑j=1lcijpjt−rt−pit−rt+k1∑j=1llijpσjt−rt+∑j∉I1cijpjt−rt+qjt+k2∑j=1lcijqjt−qit−k3qit,i∈I1,k1∑j=l+1mcijpjt−rt−pit−rt+k1∑j=l+1mlijpσjt−rt+∑j∉I2cijpjt−rt+qjt+k2∑j=l+1mcijqjt−qit−k3qit,i∈I2,∑j=m+1ncijpjt−rt−pit−rt+∑j=m+1nlijpσjt−rt+∑j∉I3cijpjt−rt+qjti∈I3,where k1,k2,andk3 are all positive constants, which standing for the positive coupling strengths, severally. And pσj represents the consistent position equilibrium of the group in which the j−thagent resides, and σi=1,2,and3.In order to the simplicity of presentation, we lay emphasis on one-dimensional space. However, for any high-dimensional space, we can generalize the results through applying the characteristics of theKroneckerKronecker product, denoted as ⊗.Remark 1.
The portion of agents which are second-order agents and second-order nonlinear agents, the adjacency matrix C is partitioned as(10)C=CS1CS1S2CS2S1CS2,with CS1∈ℝm×mand CS2∈ℝn−m×n−m. Suppose LS1andLS2 signify Laplacian matrix of the second-order agents in the two situations of linearity and nonlinearity, respectively. Then, the matrix can be attained as follows:(11)L=LS1+DS1S2−CS1S2−CS2S1LS2+DS2S1,with(12)DS1S2=diag∑j∈N1,S2c1j,…,∑j∈Nm,S2cmj;DS2S1=diag∑j∈Nm+1,S1cm+1j,…,∑j∈Nn,S1cnj.Remark 2.
The portion of agents are consisted of the second-order agents with or without nonlinearity and first-order agents, then the segmentation ofC as follows:(13)C=Cs1Cs1s2Cs1fCs2s1Cs2Cs2fCfs1Cfs2Cf,with Cs1∈ℝl×l,Cs2∈ℝm−l×m−l, andCf∈ℝn−m×n−m.
PresumeLs1,Ls2,andLf are Laplacian matrix of the second-order agents with nonlinearity and linearity, and first-order agents, severally. Then, L can be depicted as(14)L=Ls1+Ds1s2+Ds1f−As1s2−As1f−As2s1Ls2+Ds2s1+Ds2f−As2f−Afs1−Afs2Lf+Dfs1+Dfs2,with Ls1=Ds1−Cs1;Ls2=Ds2−Cs2;Lf=Df−Cf(15)Ds1s2=diag∑j∈N1,s2c1j,…,∑j∈Nl,s2clj;Ds1f=diag∑j∈N1,fc1j,…,∑j∈Nl,fclj;Ds2s1=diag∑j∈N1,s1cl+1j,…,∑j∈Nl,s1cmj;Ds2f=diag∑j∈N1,fcl+1j,…,∑j∈Nl,fcmj;Dfs1=diag∑j∈Nm+1,s1cm+1j,…,∑j∈Nn,s1cnj;Dfs2=diag∑j∈Nm+1,s2cm+1j,…,∑j∈Nn,s2cnj.
Next, we will give the definition of the two-group consensus and three-group for heterogeneous multiagent systems. And the similar definition can be given for multigroup consensus problem.Definition 1.
The group consensus of heterogeneous multiagent systems can be achieved if the states of followers tend to the states of leaders in the sense of(16)limt⟶∞pit−p1∗t=0,limt⟶∞qit−q1∗t=0,∀i∈I1;limt⟶∞pit−p2∗t=0,limt⟶∞qit−q2∗t=0,∀i∈I2.Definition 2.
Group consensus of heterogeneous multiagent systems can be achieved if the states of followers tend to the states of leaders in the sense of(17)limt⟶∞pit−pσit=0,limt⟶∞qit=0,∀i∈I1;limt⟶∞pit−pσit=0,limt⟶∞qit=0,∀i∈I2;limt⟶∞pit−pσit=0,∀i∈I3.Assumption 1.
(see in [13]).(18)∑j=m+1ncij=0,i=1,…,m;∑j=1mcij=0,i=m+1,…,n.Assumption 2.
(see in [15]).(19)∑j=m+1ncij=0,∀i∈I1;∑j=1,j∉I2ncij=0,∀i∈I2;∑j=1mcij=0,∀i∈I3.
For the sake of demonstrating group consensus of heterogeneous MAS, we need to give several lemmas which are useful in the proof of the main results as follows:Lemma 1 (see in [14]).
fpit,t fpit, qit, and fpj∗t,t are continuous differentiable function vectors. In order to achieve group consensus of the heterogeneous MAS containing the nonlinear term, the following assumption is given as follows:(20)fpt,t−fp2∗t,t⋅1n≤ρpt−p2∗t⋅1n,∀pit,p2∗t∈ℝn,i=m+1,…,n;pt=pm+1t,…,pntT;1n=1,…,1T, and a nonnegative constant ρ.Lemma 2 (see in [15]).
For arbitrary constant vectorsa,b∈ℝn and a positive definite matrixR∈ℝn×n, then the following inequality holds(21)±2aTb≤aTRa+bTR−1b.Lemma 3 (see in [14]) (SchurSchur complement).
Given a matrixΔ=Δ11Δ12Δ12TΔ22. Thus the following conditions are equivalent as follows:(22)Δ<0;Δ11<0,Δ22−Δ12TΔ11−1Δ12<0;Δ22<0,Δ11−Δ12Δ22−1Δ12T<0.


## 3. Results

### 3.1. Active Leaders

In systems with active leaders, the models change as follows:Let(23)p¯i1t=pit−p1∗t;q¯i1t=qit−q1∗t;i=1,…,m,p¯i2t=pit−p2∗t;q¯i2t=qit−q2∗t;i=m+1,…,n,p¯1t=p¯11t,…,p¯m1tT;p¯2t=p¯m+12t,…,p¯n2tT;q¯1t=q¯11t,…,q¯m1tT;q¯1t=q¯m+12t,…,q¯n2tT;p¯t=p¯1Tt,p¯2TtT;q¯t=q¯1t,q¯2t.p∗t=p1∗t,p2∗tT;q∗t=q1∗t,q2∗t.Then, according to the characteristic ofL, (1) and (2) can be substituted as(24)p¯˙1t=q¯1t,q¯˙1t=−L¯11+B1p¯1t−rt−L¯12p¯2t−rt−kB1q¯1t,(25)p¯˙2t=q¯2t,q¯˙2t=−L¯21p¯1t−rt−L¯22+B2p¯2t−rt−kB2q¯2t+f¯p¯2t,t,where f¯p¯2t,t=fpt,t−1fp2∗t,t,fpt,t=fm+1pm+1t,t,…,fnpnt,t;L¯11=LS1+DS1S2;L¯21=−CS2S1;L¯22=LS2+DS2S1;B=diagB1,B2,B1=diagb1,…,bm,andB2=diagbm+1,…,bn.Then, letχ1t=p¯1Tt,q¯1TtT;χ2t=p¯2Tt,q¯2TtT(26)χ˙1t=M¯1χ1t−rt+M˜1χ2t−rt+N1χ1t,χ˙2t=M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+F,where(27)M¯1=00−M10,M˜1=00−L¯120,N1=0Im0−kB1Im,M¯2=00−L¯210,M˜2=00−M20,N2=0In−m0−kB2In−m,F=0f¯p¯2t,t,M1=L¯11+B1,M2=L¯22+B2.Theorem 1.
Group consensus of systems (24) and (25) can be achieved under control protocol (5) if the following condition holds:(28)Θ1τMR1τMR2∗−τMR0∗∗−τMR<0,where R1andR2 both are arbitrary matrices; τM is a constant; ∗ denotes symmetry elements of the matrix.(29)Θ1=N1TP1+P1N1+τMN1TRN1M¯1TP1+τMM¯1TRN10M˜1TP1+τMM˜1TRN1P1M¯1+τMN1TRM¯1τMM¯1TRM¯1P2M¯2τMM˜1TRM¯10M¯2TP2N2TP2+P2N2+P2R−1P2+ρ2R+QM˜2TP2P1M˜1+τMN1TRM˜1τMM¯1TRM˜1P2M˜2−1−uQ+τMM˜1TRM˜1.Proof.
According to Lyapunov functional theory, we construct Lyapunov functions as follows:(30)V=V1+V2+V3+V4.
whereV1t=χ1TtP1χ1t;V2t=χ2TtP2χ2t;V3t=∫t−rttχ2TsQχ2sds;V4t=∫−τM0∫t+stχ˙1TvRχ˙1vdvds.
AndP1,P2,Q,andRare all positive definite matrices. Take the derivation of Vt along (26), we have(31)V˙1t=χ˙1TtP1χ1t+χ1TtP1χ˙1t,=M¯1χ1t−rt+M˜1χ2t−rt+N1χ1tTP1χ1t+χ1TtP1M¯1χ1t−rt+M˜1χ2t−rt+N1χ1t,=χ1Tt−rtM¯1TP1χ1t+χ2Tt−rtM˜1TP1χ1t+χ1TtN1TP1χ1t+χ1TtP1M˜1χ2t−rt+χ1TtP1N1χ1t+χ1TtP1M¯1χ1t−rt,V˙2t=χ˙2TtP2χ2t+χ2TtP2χ˙2t,=M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+FTP2χ2t+χ2TtP2M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+F,=χ1Tt−rtM¯2TP2χ2t+χ2Tt−rtM˜2TP2χ2t+χ2TtN2TP2χ2t+χ2TtP2M˜2χ2t−rt+χ2TtP2N2χ2t+2χ2TtP2F+χ2TtP2M¯2χ1t−rt,V˙3=∫t−rttχ2TsQχ2sds′=χ2TtQχ2t−1−r˙tχ2Tt−rtQχ2t−rt≤χ2TtQχ2t−1−uχ2Tt−rtQχ2t−rt,V˙4=∫−τM0∫t+stχ˙1TvRχ˙1vdvds′=τMχ˙1TtRχ˙1t−τMχ˙1Tt+sRχ˙1t+s,=τMχ˙1TtRχ˙1t+∫tt−τMχ˙1TsRχ˙1sds,=τMM¯1χ1t−rt+M˜1χ2t−rt+N1χ1tT⋅RM¯1χ1t−rt+M˜1χ2t−rt+N1χ1t+∫tt−τMχ˙1TsRχ˙1sds,=τMχ1Tt−rtM¯1TRM¯1χ1t−rt+τMχ1Tt−rtM¯1TRM˜1χ2t−rt+τMχ1Tt−rtM¯1TRN1χ1t+τMχ2Tt−rtM˜1TRM¯1χ1t−rt+τMχ2Tt−rtM˜1TRM˜1χ2t−rt+τMχ2Tt−rtM˜1TRN1χ1t+τMχ1TtN1TRM¯1χ1t−rt+τMχ1TtN1TRM˜1χ2t−rt+τMχ1TtN1TRN1χ1t+∫tt−τMχ˙1TsRχ˙1sds.
By Lemma2, we get(32)2χ2TtP2F≤χ2TtP2R−1P2χ2t+FTRF.
According to Lemma1, (32) become(33)2χ2TtP2F≤χ2TtP2R−1P2χ2t+ρ2χ2TtRχ2t=χ2TtP2R−1P2+ρ2Rχ2t.
Thus, the derivative ofV2t can be written as(34)V˙2t=χ˙2TtP2χ2t+χ2TtP2χ˙2t,=M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+FTP2χ2t+χ2TtP2M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+F,=χ1Tt−rtM¯2TP2χ2t+χ2Tt−rtM˜2TP2χ2t+χ2TtN2TP2χ2t+χ2TtP2M¯2χ1t−rt+χ2TtP2R−1P2+ρ2Rχ2t+χ2TtP2M˜2χ2t−rt+χ2TtP2N2χ2t.
LetZ1t=χ1Tt,χ1Tt−rt,χ2Tt,χ2Tt−rtT, then one can get(35)V˙t≤Z1TtΘ1Z1t+∫tt−τMχ˙1TsRχ˙1sds+2Z1TtR1χ1t−χ1t−rt−∫t−rttχ˙1sds+2Z1TtR2χ1t−rt−χ1t−τM−∫t−τMt−rtχ˙1sds.
As known by Lemma2,(36)−2Z1TtR1∫t−rttχ˙1sds≤rtZ1TtR1R−1R1TZ1t+∫t−rttχ˙1TsRχ˙1sds,−2Z1TtR2∫t−τMt−rtχ˙1sds≤τM−rtZ1TtR2R−1R2TZ1t+∫t−τMt−rtχ˙1TsRχ˙1sds.
Thus, the derivative ofVt can become(37)V˙t≤Z1TtΘ1Z1t+rtZ1TtR1R−1R1TZ1t+τM−rtZ1TtR2R−1R2TZ1t≤Z1TtΘ1+τMR1R−1R1T+τMR2R−1R2TZ1t.
Therefore, we can able to see the second-order heterogeneous MAS can attain group consensus if the above-mentioned systems satisfy the condition which has been given in Theorem1.


### 3.2. Static Leaders

In systems with static leaders, the models change as follows.Let(38)p⌢i1t=pit−pσjt;q⌢i1t=qit,i=1,…,l,p⌢i2t=pit−pσjt;q⌢i2t=qit,i=l+1,…,m,p⌢i3t=pit−pσjt;q⌢i3t=qit,i=m+1,…,n,p⌢1t=p⌢11t,…,p⌢l1tT;p⌢2t=p⌢l+12t,…,p⌢m2tT;p⌢3t=p⌢m+13t,…,p⌢n3tT;p⌢t=p⌢1Tt,p⌢2Tt,p⌢3TtT⋅q⌢1t=q⌢11t,…,q⌢l1tT;q⌢2t=q⌢l+12t,…,q⌢m2tT;q⌢3t=q⌢m+13t,…,q⌢n3tT;q⌢t=q⌢1Tt,q⌢2Tt,q⌢3Tt.Then, according to the characteristic ofL, (6)–(8) can be substituted as(39)p⌢1t=q⌢1t,q⌢1t=−k1L⌢11p⌢1t−rt−k1L⌢12p⌢2t−rt−k1L⌢13p⌢3t−rt+−k2L⌢1−k3Ilq⌢1t−k2L⌢2q⌢2t+fpt,t,(40)p⌢2t=q⌢2t,q⌢2t=−k1L⌢21p⌢1t−rt−k1L⌢22p⌢2t−rt−k1L⌢23p⌢3t−rt−k2L⌢3q⌢1t+−k2L⌢4−k3Im−lq⌢2t,(41)p⌢3t=−L^31p⌢1t−rt−L^32p⌢2t−rt−L^33p⌢3t−rt,where(42)L⌢11=Ls1+Ds1s2+Ds1f;L⌢12=−Cs1s2;L⌢13=−Cs1f;Ls1=Ds1−Cs1;L⌢21=−Cs2s1;L⌢22=Ls2+Ds2s1+Ds2f;L⌢23=−Cs2f;Ls2=Ds2−Cs2;L⌢31=−Cfs1;L⌢32=−Cfs2;L⌢33=Lf+Dfs1+Dfs2;Lf=Df−Cf;L⌢1=Ls1+Ds1s2;L⌢2=−Cs1s2;L⌢3=−Cs2s1;L⌢4=Ls2+Ds2s1.Letϕ1t=p⌢1Tt,q⌢1TtTϕ2t=p⌢2Tt,q⌢2TtTϕ3t=p⌢3Tt,q⌢3TtThen, one have(43)ϕ˙1t=M5ϕ1t−rt+M¯5ϕ2t−rt+M˜5ϕ3t−rt+N¯5ϕ1t+N˜5ϕ2t+F,ϕ˙2t=M6ϕ1t−rt+M¯6ϕ2t−rt+M˜6ϕ3t−rt+N¯6ϕ1t+N˜6ϕ2t,ϕ˙3t=M7ϕ1t−rt+M¯7ϕ2t−rt+M˜7ϕ3t−rt,where(44)M5=00−k1L⌢110,M¯5=00−k1L⌢120,M˜5=00−k1L⌢130,N¯5=0Il0N5,N˜5=000−k2L⌢2,N5=−k2L⌢1−k3Il;M6=00−k1L⌢210,M¯6=00−k1L⌢220,M˜6=00−k1L⌢230,N¯6=000−k2L⌢3,N˜6=0Im−l0N6,N6=−k2L⌢4−k3Im−l;M7=00−L⌢310,M¯7=00−L⌢320,M˜7=00−L⌢330,F=0fpt,t.Theorem 2.
Group consensus of systems (39)–(41) can be attained under control protocol (9) if the following criterion holds:(45)Θ3P1τMR5τMR6∗−R00∗∗−τMR0∗∗∗−τMR<0,where R5andR6 both are arbitrary matrices; P1,P2,P3,Q1,Q2,Q3,andR are positive definite matrices; τM is a constant; ∗ denotes symmetry.(46)Θ3=N¯5TP1+P1N¯5+Q1+τMN¯6TRN¯6+ρ2RM5TP1+τMM6TRN¯6N˜5TP1+P2N¯6+τMN˜6TRN¯6P1M5+τMN¯6TRM6−1−uQ1+τMM6TRM6P2M6+τMN˜6TRM6P1N˜5+N¯6TP2+τMN¯6TRN˜6M6TP2+τMM6TRN˜6P2M6+τMN˜6TRM6P1M¯5+τMN¯6TRM¯6τMM6TRM¯6N˜6TP2+P2N˜6+Q2+τMN˜6TRN˜60M7TP30P1M˜5+τMN¯6TRM˜6τMM6TRM˜6τMN˜6TRM˜6+P2M˜6M¯5TP1+τMM¯6TRN¯60M˜5TP1+τMM˜6TRN¯6τMM¯6TRM6P3M7τMM˜6TRM6M¯6TP2+τMM¯6TRN˜60M˜6TP2+τMM˜6TRN˜6−1−uQ2+τMM¯6TRM¯6P3M¯7τMM6TRM¯6M¯7TP3Q3M7TP3τMM¯6TRM˜6P3M˜7τMM6TRM˜6.Proof:.
through Lyapunov functional theory, one set Lyapunov functionsV=V1+V2+V3+V4+V5+V6+V7 as follows:(47)V1t=ϕ1TtP1ϕ1t;V2t=ϕ2TtP2ϕ2t;V3t=ϕ3TtP3ϕ3t;V4t=∫t−rttϕ1TsQ1ϕ1sds;V5t=∫t−rttϕ2TsQ2ϕ2sds;V6t=∫t−rttϕ3TsQ3ϕ3sds;V7t=∫−τM0∫t+stϕ˙2TvRϕ˙2vdvds,where P1,P2,P3,Q1,Q2,Q3,andR are all positive definite matrices. Then, calculating the derivation of the Lyapunov functions along the system (43), we have(48)V˙3t=ϕ˙3TtP3ϕ3t+ϕ3TtP3ϕ˙3t,=M7ϕ1t−rt+M¯7ϕ2t−rt+M˜7ϕ3t−rtTP3ϕ3t+ϕ3TtP3M7ϕ1t−rt+M¯7ϕ2t−rt+M˜7ϕ3t−rt,=ϕ1Tt−rtM7TP3ϕ3t+ϕ2Tt−rtM¯7TP3ϕ3t+ϕ3Tt−rtM˜7TP3ϕ3t+ϕ3TtP3M7ϕ1t−rt+ϕ3TtP3M¯7ϕ2t−rt+ϕ3TtP3M˜7ϕ3t−rt,V˙4=∫t−rttϕ1TsQ1ϕ1sds′=ϕ1TtQ1ϕ1t−1−r˙tϕ1Tt−rtQ1ϕ1t−rt≤ϕ1TtQ1ϕ1t−1−uϕ1Tt−rtQ1ϕ1t−rt,V˙5=∫t−rttϕ2TsQ2ϕ2sds′=ϕ2TtQ2ϕ2t−1−r˙tϕ2Tt−rtQ2ϕ2t−rt≤ϕ2TtQ2ϕ2t−1−uϕ2Tt−rtQ2ϕ2t−rt,V˙6=∫t−rttϕ3TsQ3ϕ3sds′=ϕ3TtQ3ϕ3t−1−r˙tϕ3Tt−rtQ3ϕ3t−rt≤ϕ3TtQ3ϕ3t−1−uϕ3Tt−rtQ3ϕ3t−rt,V˙7=∫−τM0∫t+stϕ˙2TvRϕ˙2vdvds′,=τMϕ˙2TtRϕ˙2t−τMϕ˙2Tt+sRϕ2t+s=τMϕ˙2TtRϕ˙2t+∫tt−τMϕ˙2TsRϕ˙2sds,=τMM6ϕ1t−rt+M¯6ϕ2t−rt+M˜6ϕ3t−rt+N¯6ϕ1t+N˜6ϕ2tTRM6ϕ1t−rt+M¯6ϕ2t−rt+M˜6ϕ3t−rt+N¯6ϕ1t+N˜6ϕ2t+∫tt−τMϕ˙2TsRϕ˙2sds,=τMϕ1Tt−rtM6TRM6ϕ1t−rt+τMϕ1Tt−rtM6TRN¯6ϕ1t+τMϕ1Tt−rtM6TRN˜6ϕ2t+τMϕ1Tt−rtM6TRM¯6ϕ2t−rt+τMϕ1Tt−rtM6TRM˜6ϕ3t−rt+τMϕ2Tt−rtM¯6TRM6ϕ1t−rt+τMϕ2Tt−rtM¯6TRM¯6ϕ2t−rt+τMϕ2Tt−rtM¯6TRM˜6ϕ3t−rt+τMϕ2Tt−rtM¯6TRN¯6ϕ1t+τMϕ3Tt−rtM˜6TRM6ϕ1t−rt+τMϕ3Tt−rtM˜6TRM¯6ϕ2t−rt+τMϕ3Tt−rtM˜6TRM˜6ϕ3t−rt+∫tt−τMϕ˙2TsRϕ˙2sds+τMϕ2Tt−rtM¯6TRN˜6ϕ2t+τMϕ3Tt−rtM˜6TRN¯6ϕ1t+τMϕ3Tt−rtM˜6TRN˜6ϕ2t+τMϕ1TtN¯6TRM6ϕ1t−rt+τMϕ1TtN¯6TRM¯6ϕ2t−rt+τMϕ1TtN¯6TRM˜6ϕ3t−rt+τMϕ1TtN¯6TRN¯6ϕ1t+τMϕ1TtN¯6TRN˜6ϕ2t+τMϕ2TtN˜6TRM6ϕ1t−rt+τMϕ2TtN˜6TRM¯6ϕ2t−rt+τMϕ2TtN˜6TRM˜6ϕ3t−rt+τMϕ2TtN˜6TRN¯6ϕ1t+τMϕ2TtN˜6TRN˜6ϕ2t.
As known by Lemma2, we can obtain(49)2ϕ1TtP1F≤ϕ1TtP1R−1P1ϕ1t+FTRF.
Through Lemma1, (49) can become(50)2ϕ1TtP1F≤ϕ1TtP1R−1P1ϕ1t+ρ2ϕ1TtRϕ1t=ϕ1TtP1R−1P1+ρ2Rϕ1t.
LetZ3t=ϕ1Tt,ϕ1Tt−rt,ϕ2Tt,ϕ2Tt−rtT, then, we get(51)V˙t≤Z3TtΘ3+P1R−1P1Z3t+∫tt−τMϕ˙2TsRϕ˙2sds+2Z3TtR5ϕ2t−ϕ2t−rt−∫t−rttϕ˙2sds+2Z3TtR6ϕ2t−rt−ϕ2t−τM−∫t−τMt−rtϕ˙2sds.
From Lemma2, one can obtain(52)−2Z3TtR5∫t−rttϕ˙2sds≤rtZ3TtR5R−1R5TZ3t+∫t−rttϕ˙2TsRϕ˙2sds−2Z3TtR6∫t−τMt−rtϕ˙2sds≤τM−rtZ3TtR6R−1R6TZ3t+∫t−τMt−rtϕ˙1TsRϕ˙1sds.
Thus,V˙t becomes(53)V˙t≤Z3TtΘ3+P1R−1P1Z3t+rtZ3TtR5R−1R5TZ3t+τM−rtZ3TtR6R−1R6TZ3t≤Z3TtΘ3+P1R−1P1+τMR5R−1R5T+τMR6R−1R6TZ3t.
Therefore, if the error system (43) contents the above condition which has been given in Theorem 2, the heterogeneous MAS can reach group consensus.


## 3.1. Active Leaders

In systems with active leaders, the models change as follows:Let(23)p¯i1t=pit−p1∗t;q¯i1t=qit−q1∗t;i=1,…,m,p¯i2t=pit−p2∗t;q¯i2t=qit−q2∗t;i=m+1,…,n,p¯1t=p¯11t,…,p¯m1tT;p¯2t=p¯m+12t,…,p¯n2tT;q¯1t=q¯11t,…,q¯m1tT;q¯1t=q¯m+12t,…,q¯n2tT;p¯t=p¯1Tt,p¯2TtT;q¯t=q¯1t,q¯2t.p∗t=p1∗t,p2∗tT;q∗t=q1∗t,q2∗t.Then, according to the characteristic ofL, (1) and (2) can be substituted as(24)p¯˙1t=q¯1t,q¯˙1t=−L¯11+B1p¯1t−rt−L¯12p¯2t−rt−kB1q¯1t,(25)p¯˙2t=q¯2t,q¯˙2t=−L¯21p¯1t−rt−L¯22+B2p¯2t−rt−kB2q¯2t+f¯p¯2t,t,where f¯p¯2t,t=fpt,t−1fp2∗t,t,fpt,t=fm+1pm+1t,t,…,fnpnt,t;L¯11=LS1+DS1S2;L¯21=−CS2S1;L¯22=LS2+DS2S1;B=diagB1,B2,B1=diagb1,…,bm,andB2=diagbm+1,…,bn.Then, letχ1t=p¯1Tt,q¯1TtT;χ2t=p¯2Tt,q¯2TtT(26)χ˙1t=M¯1χ1t−rt+M˜1χ2t−rt+N1χ1t,χ˙2t=M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+F,where(27)M¯1=00−M10,M˜1=00−L¯120,N1=0Im0−kB1Im,M¯2=00−L¯210,M˜2=00−M20,N2=0In−m0−kB2In−m,F=0f¯p¯2t,t,M1=L¯11+B1,M2=L¯22+B2.Theorem 1.
Group consensus of systems (24) and (25) can be achieved under control protocol (5) if the following condition holds:(28)Θ1τMR1τMR2∗−τMR0∗∗−τMR<0,where R1andR2 both are arbitrary matrices; τM is a constant; ∗ denotes symmetry elements of the matrix.(29)Θ1=N1TP1+P1N1+τMN1TRN1M¯1TP1+τMM¯1TRN10M˜1TP1+τMM˜1TRN1P1M¯1+τMN1TRM¯1τMM¯1TRM¯1P2M¯2τMM˜1TRM¯10M¯2TP2N2TP2+P2N2+P2R−1P2+ρ2R+QM˜2TP2P1M˜1+τMN1TRM˜1τMM¯1TRM˜1P2M˜2−1−uQ+τMM˜1TRM˜1.Proof.
According to Lyapunov functional theory, we construct Lyapunov functions as follows:(30)V=V1+V2+V3+V4.
whereV1t=χ1TtP1χ1t;V2t=χ2TtP2χ2t;V3t=∫t−rttχ2TsQχ2sds;V4t=∫−τM0∫t+stχ˙1TvRχ˙1vdvds.
AndP1,P2,Q,andRare all positive definite matrices. Take the derivation of Vt along (26), we have(31)V˙1t=χ˙1TtP1χ1t+χ1TtP1χ˙1t,=M¯1χ1t−rt+M˜1χ2t−rt+N1χ1tTP1χ1t+χ1TtP1M¯1χ1t−rt+M˜1χ2t−rt+N1χ1t,=χ1Tt−rtM¯1TP1χ1t+χ2Tt−rtM˜1TP1χ1t+χ1TtN1TP1χ1t+χ1TtP1M˜1χ2t−rt+χ1TtP1N1χ1t+χ1TtP1M¯1χ1t−rt,V˙2t=χ˙2TtP2χ2t+χ2TtP2χ˙2t,=M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+FTP2χ2t+χ2TtP2M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+F,=χ1Tt−rtM¯2TP2χ2t+χ2Tt−rtM˜2TP2χ2t+χ2TtN2TP2χ2t+χ2TtP2M˜2χ2t−rt+χ2TtP2N2χ2t+2χ2TtP2F+χ2TtP2M¯2χ1t−rt,V˙3=∫t−rttχ2TsQχ2sds′=χ2TtQχ2t−1−r˙tχ2Tt−rtQχ2t−rt≤χ2TtQχ2t−1−uχ2Tt−rtQχ2t−rt,V˙4=∫−τM0∫t+stχ˙1TvRχ˙1vdvds′=τMχ˙1TtRχ˙1t−τMχ˙1Tt+sRχ˙1t+s,=τMχ˙1TtRχ˙1t+∫tt−τMχ˙1TsRχ˙1sds,=τMM¯1χ1t−rt+M˜1χ2t−rt+N1χ1tT⋅RM¯1χ1t−rt+M˜1χ2t−rt+N1χ1t+∫tt−τMχ˙1TsRχ˙1sds,=τMχ1Tt−rtM¯1TRM¯1χ1t−rt+τMχ1Tt−rtM¯1TRM˜1χ2t−rt+τMχ1Tt−rtM¯1TRN1χ1t+τMχ2Tt−rtM˜1TRM¯1χ1t−rt+τMχ2Tt−rtM˜1TRM˜1χ2t−rt+τMχ2Tt−rtM˜1TRN1χ1t+τMχ1TtN1TRM¯1χ1t−rt+τMχ1TtN1TRM˜1χ2t−rt+τMχ1TtN1TRN1χ1t+∫tt−τMχ˙1TsRχ˙1sds.
By Lemma2, we get(32)2χ2TtP2F≤χ2TtP2R−1P2χ2t+FTRF.
According to Lemma1, (32) become(33)2χ2TtP2F≤χ2TtP2R−1P2χ2t+ρ2χ2TtRχ2t=χ2TtP2R−1P2+ρ2Rχ2t.
Thus, the derivative ofV2t can be written as(34)V˙2t=χ˙2TtP2χ2t+χ2TtP2χ˙2t,=M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+FTP2χ2t+χ2TtP2M¯2χ1t−rt+M˜2χ2t−rt+N2χ2t+F,=χ1Tt−rtM¯2TP2χ2t+χ2Tt−rtM˜2TP2χ2t+χ2TtN2TP2χ2t+χ2TtP2M¯2χ1t−rt+χ2TtP2R−1P2+ρ2Rχ2t+χ2TtP2M˜2χ2t−rt+χ2TtP2N2χ2t.
LetZ1t=χ1Tt,χ1Tt−rt,χ2Tt,χ2Tt−rtT, then one can get(35)V˙t≤Z1TtΘ1Z1t+∫tt−τMχ˙1TsRχ˙1sds+2Z1TtR1χ1t−χ1t−rt−∫t−rttχ˙1sds+2Z1TtR2χ1t−rt−χ1t−τM−∫t−τMt−rtχ˙1sds.
As known by Lemma2,(36)−2Z1TtR1∫t−rttχ˙1sds≤rtZ1TtR1R−1R1TZ1t+∫t−rttχ˙1TsRχ˙1sds,−2Z1TtR2∫t−τMt−rtχ˙1sds≤τM−rtZ1TtR2R−1R2TZ1t+∫t−τMt−rtχ˙1TsRχ˙1sds.
Thus, the derivative ofVt can become(37)V˙t≤Z1TtΘ1Z1t+rtZ1TtR1R−1R1TZ1t+τM−rtZ1TtR2R−1R2TZ1t≤Z1TtΘ1+τMR1R−1R1T+τMR2R−1R2TZ1t.
Therefore, we can able to see the second-order heterogeneous MAS can attain group consensus if the above-mentioned systems satisfy the condition which has been given in Theorem1.


## 3.2. Static Leaders

In systems with static leaders, the models change as follows.Let(38)p⌢i1t=pit−pσjt;q⌢i1t=qit,i=1,…,l,p⌢i2t=pit−pσjt;q⌢i2t=qit,i=l+1,…,m,p⌢i3t=pit−pσjt;q⌢i3t=qit,i=m+1,…,n,p⌢1t=p⌢11t,…,p⌢l1tT;p⌢2t=p⌢l+12t,…,p⌢m2tT;p⌢3t=p⌢m+13t,…,p⌢n3tT;p⌢t=p⌢1Tt,p⌢2Tt,p⌢3TtT⋅q⌢1t=q⌢11t,…,q⌢l1tT;q⌢2t=q⌢l+12t,…,q⌢m2tT;q⌢3t=q⌢m+13t,…,q⌢n3tT;q⌢t=q⌢1Tt,q⌢2Tt,q⌢3Tt.Then, according to the characteristic ofL, (6)–(8) can be substituted as(39)p⌢1t=q⌢1t,q⌢1t=−k1L⌢11p⌢1t−rt−k1L⌢12p⌢2t−rt−k1L⌢13p⌢3t−rt+−k2L⌢1−k3Ilq⌢1t−k2L⌢2q⌢2t+fpt,t,(40)p⌢2t=q⌢2t,q⌢2t=−k1L⌢21p⌢1t−rt−k1L⌢22p⌢2t−rt−k1L⌢23p⌢3t−rt−k2L⌢3q⌢1t+−k2L⌢4−k3Im−lq⌢2t,(41)p⌢3t=−L^31p⌢1t−rt−L^32p⌢2t−rt−L^33p⌢3t−rt,where(42)L⌢11=Ls1+Ds1s2+Ds1f;L⌢12=−Cs1s2;L⌢13=−Cs1f;Ls1=Ds1−Cs1;L⌢21=−Cs2s1;L⌢22=Ls2+Ds2s1+Ds2f;L⌢23=−Cs2f;Ls2=Ds2−Cs2;L⌢31=−Cfs1;L⌢32=−Cfs2;L⌢33=Lf+Dfs1+Dfs2;Lf=Df−Cf;L⌢1=Ls1+Ds1s2;L⌢2=−Cs1s2;L⌢3=−Cs2s1;L⌢4=Ls2+Ds2s1.Letϕ1t=p⌢1Tt,q⌢1TtTϕ2t=p⌢2Tt,q⌢2TtTϕ3t=p⌢3Tt,q⌢3TtThen, one have(43)ϕ˙1t=M5ϕ1t−rt+M¯5ϕ2t−rt+M˜5ϕ3t−rt+N¯5ϕ1t+N˜5ϕ2t+F,ϕ˙2t=M6ϕ1t−rt+M¯6ϕ2t−rt+M˜6ϕ3t−rt+N¯6ϕ1t+N˜6ϕ2t,ϕ˙3t=M7ϕ1t−rt+M¯7ϕ2t−rt+M˜7ϕ3t−rt,where(44)M5=00−k1L⌢110,M¯5=00−k1L⌢120,M˜5=00−k1L⌢130,N¯5=0Il0N5,N˜5=000−k2L⌢2,N5=−k2L⌢1−k3Il;M6=00−k1L⌢210,M¯6=00−k1L⌢220,M˜6=00−k1L⌢230,N¯6=000−k2L⌢3,N˜6=0Im−l0N6,N6=−k2L⌢4−k3Im−l;M7=00−L⌢310,M¯7=00−L⌢320,M˜7=00−L⌢330,F=0fpt,t.Theorem 2.
Group consensus of systems (39)–(41) can be attained under control protocol (9) if the following criterion holds:(45)Θ3P1τMR5τMR6∗−R00∗∗−τMR0∗∗∗−τMR<0,where R5andR6 both are arbitrary matrices; P1,P2,P3,Q1,Q2,Q3,andR are positive definite matrices; τM is a constant; ∗ denotes symmetry.(46)Θ3=N¯5TP1+P1N¯5+Q1+τMN¯6TRN¯6+ρ2RM5TP1+τMM6TRN¯6N˜5TP1+P2N¯6+τMN˜6TRN¯6P1M5+τMN¯6TRM6−1−uQ1+τMM6TRM6P2M6+τMN˜6TRM6P1N˜5+N¯6TP2+τMN¯6TRN˜6M6TP2+τMM6TRN˜6P2M6+τMN˜6TRM6P1M¯5+τMN¯6TRM¯6τMM6TRM¯6N˜6TP2+P2N˜6+Q2+τMN˜6TRN˜60M7TP30P1M˜5+τMN¯6TRM˜6τMM6TRM˜6τMN˜6TRM˜6+P2M˜6M¯5TP1+τMM¯6TRN¯60M˜5TP1+τMM˜6TRN¯6τMM¯6TRM6P3M7τMM˜6TRM6M¯6TP2+τMM¯6TRN˜60M˜6TP2+τMM˜6TRN˜6−1−uQ2+τMM¯6TRM¯6P3M¯7τMM6TRM¯6M¯7TP3Q3M7TP3τMM¯6TRM˜6P3M˜7τMM6TRM˜6.Proof:.
through Lyapunov functional theory, one set Lyapunov functionsV=V1+V2+V3+V4+V5+V6+V7 as follows:(47)V1t=ϕ1TtP1ϕ1t;V2t=ϕ2TtP2ϕ2t;V3t=ϕ3TtP3ϕ3t;V4t=∫t−rttϕ1TsQ1ϕ1sds;V5t=∫t−rttϕ2TsQ2ϕ2sds;V6t=∫t−rttϕ3TsQ3ϕ3sds;V7t=∫−τM0∫t+stϕ˙2TvRϕ˙2vdvds,where P1,P2,P3,Q1,Q2,Q3,andR are all positive definite matrices. Then, calculating the derivation of the Lyapunov functions along the system (43), we have(48)V˙3t=ϕ˙3TtP3ϕ3t+ϕ3TtP3ϕ˙3t,=M7ϕ1t−rt+M¯7ϕ2t−rt+M˜7ϕ3t−rtTP3ϕ3t+ϕ3TtP3M7ϕ1t−rt+M¯7ϕ2t−rt+M˜7ϕ3t−rt,=ϕ1Tt−rtM7TP3ϕ3t+ϕ2Tt−rtM¯7TP3ϕ3t+ϕ3Tt−rtM˜7TP3ϕ3t+ϕ3TtP3M7ϕ1t−rt+ϕ3TtP3M¯7ϕ2t−rt+ϕ3TtP3M˜7ϕ3t−rt,V˙4=∫t−rttϕ1TsQ1ϕ1sds′=ϕ1TtQ1ϕ1t−1−r˙tϕ1Tt−rtQ1ϕ1t−rt≤ϕ1TtQ1ϕ1t−1−uϕ1Tt−rtQ1ϕ1t−rt,V˙5=∫t−rttϕ2TsQ2ϕ2sds′=ϕ2TtQ2ϕ2t−1−r˙tϕ2Tt−rtQ2ϕ2t−rt≤ϕ2TtQ2ϕ2t−1−uϕ2Tt−rtQ2ϕ2t−rt,V˙6=∫t−rttϕ3TsQ3ϕ3sds′=ϕ3TtQ3ϕ3t−1−r˙tϕ3Tt−rtQ3ϕ3t−rt≤ϕ3TtQ3ϕ3t−1−uϕ3Tt−rtQ3ϕ3t−rt,V˙7=∫−τM0∫t+stϕ˙2TvRϕ˙2vdvds′,=τMϕ˙2TtRϕ˙2t−τMϕ˙2Tt+sRϕ2t+s=τMϕ˙2TtRϕ˙2t+∫tt−τMϕ˙2TsRϕ˙2sds,=τMM6ϕ1t−rt+M¯6ϕ2t−rt+M˜6ϕ3t−rt+N¯6ϕ1t+N˜6ϕ2tTRM6ϕ1t−rt+M¯6ϕ2t−rt+M˜6ϕ3t−rt+N¯6ϕ1t+N˜6ϕ2t+∫tt−τMϕ˙2TsRϕ˙2sds,=τMϕ1Tt−rtM6TRM6ϕ1t−rt+τMϕ1Tt−rtM6TRN¯6ϕ1t+τMϕ1Tt−rtM6TRN˜6ϕ2t+τMϕ1Tt−rtM6TRM¯6ϕ2t−rt+τMϕ1Tt−rtM6TRM˜6ϕ3t−rt+τMϕ2Tt−rtM¯6TRM6ϕ1t−rt+τMϕ2Tt−rtM¯6TRM¯6ϕ2t−rt+τMϕ2Tt−rtM¯6TRM˜6ϕ3t−rt+τMϕ2Tt−rtM¯6TRN¯6ϕ1t+τMϕ3Tt−rtM˜6TRM6ϕ1t−rt+τMϕ3Tt−rtM˜6TRM¯6ϕ2t−rt+τMϕ3Tt−rtM˜6TRM˜6ϕ3t−rt+∫tt−τMϕ˙2TsRϕ˙2sds+τMϕ2Tt−rtM¯6TRN˜6ϕ2t+τMϕ3Tt−rtM˜6TRN¯6ϕ1t+τMϕ3Tt−rtM˜6TRN˜6ϕ2t+τMϕ1TtN¯6TRM6ϕ1t−rt+τMϕ1TtN¯6TRM¯6ϕ2t−rt+τMϕ1TtN¯6TRM˜6ϕ3t−rt+τMϕ1TtN¯6TRN¯6ϕ1t+τMϕ1TtN¯6TRN˜6ϕ2t+τMϕ2TtN˜6TRM6ϕ1t−rt+τMϕ2TtN˜6TRM¯6ϕ2t−rt+τMϕ2TtN˜6TRM˜6ϕ3t−rt+τMϕ2TtN˜6TRN¯6ϕ1t+τMϕ2TtN˜6TRN˜6ϕ2t.
As known by Lemma2, we can obtain(49)2ϕ1TtP1F≤ϕ1TtP1R−1P1ϕ1t+FTRF.
Through Lemma1, (49) can become(50)2ϕ1TtP1F≤ϕ1TtP1R−1P1ϕ1t+ρ2ϕ1TtRϕ1t=ϕ1TtP1R−1P1+ρ2Rϕ1t.
LetZ3t=ϕ1Tt,ϕ1Tt−rt,ϕ2Tt,ϕ2Tt−rtT, then, we get(51)V˙t≤Z3TtΘ3+P1R−1P1Z3t+∫tt−τMϕ˙2TsRϕ˙2sds+2Z3TtR5ϕ2t−ϕ2t−rt−∫t−rttϕ˙2sds+2Z3TtR6ϕ2t−rt−ϕ2t−τM−∫t−τMt−rtϕ˙2sds.
From Lemma2, one can obtain(52)−2Z3TtR5∫t−rttϕ˙2sds≤rtZ3TtR5R−1R5TZ3t+∫t−rttϕ˙2TsRϕ˙2sds−2Z3TtR6∫t−τMt−rtϕ˙2sds≤τM−rtZ3TtR6R−1R6TZ3t+∫t−τMt−rtϕ˙1TsRϕ˙1sds.
Thus,V˙t becomes(53)V˙t≤Z3TtΘ3+P1R−1P1Z3t+rtZ3TtR5R−1R5TZ3t+τM−rtZ3TtR6R−1R6TZ3t≤Z3TtΘ3+P1R−1P1+τMR5R−1R5T+τMR6R−1R6TZ3t.
Therefore, if the error system (43) contents the above condition which has been given in Theorem 2, the heterogeneous MAS can reach group consensus.


## 4. Numerical Simulations

### 4.1. Active Leaders

In the condition offipit,t=0.15sinpit and k=2,τM=0.1, when the agents belong of the second-order linear and nonlinear heterogeneous MAS, and the topology of the multiagent systems are given by Figures 1 and 2, then the error figures of the position and velocity graphs are shown as follows, respectively:Figure 1
The topology construction of followers and leaders.Figure 2
The topology of the followers and leaders.


### 4.2. Static Leaders

Whenk1=1,k2=2,k3=1,τM=0.1, and fipit,t=0.15sinpit, the agents of the first-order, second-order linear and nonlinear heterogeneous MAS, and relative state figures are as follows, respectively:From Figures3 and 4, it can be seen that all the tracks of the agents trend to balance position; in other words, each error curves all go to zero. From Figure 5, it can be observed that all trajectories of the agents which belong to the same group trend to the identical line. As you can see from Figure 6, all curves verge to a straight line; then, these graphs can illustrate that the systems achieve group consensus.Figure 3
The position error trajectory.Figure 4
The velocity error trajectory.Figure 5
The position state trajectory.Figure 6
The velocity state trajectory.


## 4.1. Active Leaders

In the condition offipit,t=0.15sinpit and k=2,τM=0.1, when the agents belong of the second-order linear and nonlinear heterogeneous MAS, and the topology of the multiagent systems are given by Figures 1 and 2, then the error figures of the position and velocity graphs are shown as follows, respectively:Figure 1
The topology construction of followers and leaders.Figure 2
The topology of the followers and leaders.


## 4.2. Static Leaders

Whenk1=1,k2=2,k3=1,τM=0.1, and fipit,t=0.15sinpit, the agents of the first-order, second-order linear and nonlinear heterogeneous MAS, and relative state figures are as follows, respectively:From Figures3 and 4, it can be seen that all the tracks of the agents trend to balance position; in other words, each error curves all go to zero. From Figure 5, it can be observed that all trajectories of the agents which belong to the same group trend to the identical line. As you can see from Figure 6, all curves verge to a straight line; then, these graphs can illustrate that the systems achieve group consensus.Figure 3
The position error trajectory.Figure 4
The velocity error trajectory.Figure 5
The position state trajectory.Figure 6
The velocity state trajectory.


## 5. Conclusions

Based on neighbour’s status information, the group consensus problem of heterogeneous MAS in the case of communication time lag was considered, and the corresponding control protocol was designed. For the active and static leaders, we considered the two-group and three-group problems for the multiagent systems. Via the methods of Lyapunov stability theory and linear matrix inequality, the sufficient condition for the heterogeneous MAS with a time delay to achieve group consensus. Finally, the given simulation examples are given to show the effectiveness of the obtained results.



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