# Decentralized Adaptive Control for Quasi-Consensus in Heterogeneous Nonlinear Multiagent Systems

Authors: Jiaju An; Wei Yang; Xiaohui Xu; Tianxiang Chen; Bin Du; Yi Tang; Quan Xu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper proposes some novel decentralized adaptive control protocols to settle the quasi-consensus problem of multiagent systems with heterogeneous nonlinear dynamics. Based on local communication with the leader and between the followers, some innovative control protocols are put forward to adapt the control gains and coupling weights simultaneously and to steer the consensus errors to some bounded areas. In particular, two new inequalities are proposed to establish the Lyapunov-based adaptive controller design approach for quasi-consensus. Some quasi-consensus criteria are derived by utilizing the designed controllers, in which the error bound can be modulated on the basis of the adaptive controller parameters. Numerical tests are conducted to show the feasibility of the theoretical derivation. Our findings highlight quasi-consensus in heterogeneous multiagent systems without adding some additional complex nonlinear control terms to cancel the dynamical differences between agents.


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## Body

## 1. Introduction

Collaborative control of multiagent systems (MASs) has become a research hotspot of distributed artificial intelligence because of its wide application in intelligent energy, multirobot formation, intelligent transportation, multiunmanned system collaboration, and other engineering systems [1–5].Among them, consensus or synchronization is a key common scientific issue of cooperative control of MASs, which has aroused great concern of multidisciplinary scholars. In brief, a core issue is to design appropriate control protocols to reach an agreement between agents. So far, many significant results have been acquired, regarding the consensus patterns, models, control algorithms, etc (please see [6–14] and some other results).However, the aforesaid works [1–14] on the consensus of MASs mainly concentrate upon the case in which an agent has the same dynamics as all the neighbors or the leader. In some real scenes, the mismatched parameters or dynamical differences among agents may be almost inevitable, which will thus result in heterogeneous (or nonidentical) multiagent systems (HMASs) [15–17]. It is remarkably that, due to heterogeneity, it is even impossible for HMASs to reach the complete consensus just by state feedback control when the coupling weights are constant. Up to now, there are few thorough research studies on complete consensus in HMASs because one has to add some additional complex nonlinear control terms or design compensators to cancel the dynamical differences [18–26]. Another alternative technique to deal with the heterogeneity is to transform HMASs to homogeneous ones [27, 28]. Unfortunately, all of the aforementioned approaches are complex and nonintuitive, which are thus not suitable for engineering applications. Nevertheless, in many practical HMASs, the consensus error may be bounded, even small enough, which is, namely, the so-called quasi-consensus (QC) [29].Instead, an immediate and natural question, then, is how to design a simple controller to reach QC in HMASs. Similar to it, work to date has considered the quasi-synchronization (QS) in heterogeneous complex dynamical networks (HCDN) [30–34]. For example, in [33, 34], some QS criteria for fractional HCDN are derived via state feedback control and impulsive control, respectively. By contrast, there are few research studies about QC of HMASs [29, 35–38]. The definition of QC for MASs was first proposed in [38], and then, the definition of QC was further broadened in [29]. After that, the QC of HMASs has been further studied. The QC problem of nonlinear HMASs is studied via sampled-data control in [35]. In [36], sufficient conditions for the QC in switched HMASs are given, considering cooperation and competition interactions simultaneously. Ye and Shao [37] attempted to prove that QC in HMASs under DOS attacks can be realized by impulsive control. It should be noted, however, that, for large scale HCDN or HMASs, the computational complexity and conservativeness of QS or QC conditions in [29–38] impede their applications. In particular, it is difficult, even impossible, to check the LMI-based conditions without adaptive schemes for large-scale HCDN or HMASs. Obviously, it is an interesting and open problem to improve the QC conditions for HMASs by exploring some new control algorithms.Motivated by the applications of decentralized adaptive control for synchronization in integer-order and fractional-order complex dynamical networks [39–43], this paper aims to design some decentralized adaptive protocols to reach QC in HMASs. Different from some previous studies [37, 44, 45], in our controller, the coupling and feedback values between agents change adaptively. The main contributions in this paper are as follows. First, two new inequalities are proposed to establish the Lyapunov-based adaptive controller design approach for QC in HMASs. Second, some innovative control protocols are introduced to accommodate the control gains and coupling weights adaptively, to steer the consensus errors to some bounded areas. Third, some QC criteria under the designed controllers are derived by the Lyapunov function method and the new inequalities.


## 2. Preliminaries and Problem Formation

### 2.1. Graph Theory

To carry out later research, we present some important concepts about graph theory in this section.G=V,E,A is defined as a weighted undirected graph with the network topology of N agents, where V=0,1,…,N and E=eiji,j⊆V×V are the separate sets of nodes and undirected edges. A=aijN×N is the weighted adjacency matrix of which the elements are nonnegative. The ith agent and the leader are modeled as the node i∈V,V=0,1,…,N and node 0, respectively. As a rule, the undirected edge i,j∈E in the weighted undirected graph G denoted by an ordered pair (Vi,Vj) represents that agent i and agent j become a pair of neighbors which can get their information from each other. There is an undirected spanning tree in an undirected graph if one agent exits an undirected path to every other distinct node.There are two matrices that are considered as the network topology, i.e., the weighted adjacency matrixA=aijN×N with aij=aji>0 if eij∈E, else aij=0 if eij∉E, and the Laplacian matrix L=LijN×N which is defined as Lii=∑j=1,j≠iNaij=degi and Lij=−aij, i≠j, noted that all of them are asymmetric for the undirected graph.Lemma 1 (see [46]).
The Laplacian matrixL is constructed from the undirected network. There are several properties in the following.(1)
Eigenvalues ofL satisfy 0=λ1L<λ2L≤⋯≤λNL and the smallest positive eigenvalue λ2L=minxT1N=0,x≠0xTLx/xTx if and only if the network is connected.(2)
For any vectorη=η1,η2,…,ηNT∈ℝN, the equation satisfies(1)ηTLη=12∑i=1∑j=1Gijηi−ηj2.


### 2.2. Problem Formation

In this paper, we consider that there areN follower multiagent systems, which can be described by(2)ω˙it=Aiωit+Bifωit,t+uit,where ωit∈Rn can be regarded as the position vector of the agent i.f:Rn×Rn×R+⟶Rn represents a continuous nonlinear vector function. Ai, Bi∈ℝn×n represent the system matrices of the agent i, respectively. uit∈Rn denotes the control protocol to be designed.The leader agent can be described as follows:(3)ω˙0t=Aω0t+Bfω0t,t,where ω0t∈Rn represents the leader’s state vector and A, B∈ℝn×n represent the leader’s system matrices, respectively.For the above heterogeneous leader-follower multiagent system, we give the following assumptions.Assumption 1.
Suppose there is a normal quantityl so that the vector function f for any vector λ,ν∈Rn satisfies(4)fλ,t−fν,t≤lλ−ν.Lemma 2 (see [47]).
For any vectorx,y∈Rn, the following holds:(5)xTy≤12xTx+12yTy.Lemma 3.
Any two continuous functions satisfy(6)v˙t+w˙t≤−γvt,where γ>0. Then, there is a t≥0 so that the following holds:(7)vt≤v0+w0e−γt−wt+γe−γt∗wt,t≥0,where ∗ is the convolution.Proof.
Sincevt+w˙t≤−γvt, the existence of zt≥0 makes the following equation:(8)v˙t+w˙t+zt=−γvt.
We calculate the Laplace transform of (8) to obtain(9)sVs+sWs−v0+w0+Zs=−γVs.
Then, we can obtain(10)Vs=v0+w01s+γ−ss+γWs−1s+γZs.
The inverse Laplace transform of (10) can be obtained as(11)vt=v0+w0e−γt−wt+γe−γt∗wt−e−γt∗zt.
Through (10), we can obtain(12)vt≤v0+w0e−γt−wt+γe−γt∗wt.Lemma 4.
Any two continuous functions satisfy(13)v˙t+w˙t≤−γvt+ε,where γ>0 and ε>0. Then, there is a t≥0 so that the following formula holds:(14)vt<v0+w0e−γt−wt+γe−γt∗wt+εγ,t≥0.Proof.
Sincev˙t+w˙t≤−γvt+ε, then vt−ε/γ˙+w˙t≤−γvt−ε/γ.
By Lemma3, we can obtain(15)vt−εγ≤v0−εγ+w0e−γt−wt+γe−γt∗wt.
We can further obtain(16)vt≤v0+w0e−γt−wt+γe−γt∗wt+εγ.Lemma 5 (see [47]).
IfA, B, C, and D represent four different matrices, respectively, and the matrix products AC and BD makes sense, the Kronecker product ⊗ satisfies(17)1A⊗B+C=A⊗B+A⊗C,2A⊗BC⊗D=AC⊗BD.Definition 1 (see [29]).
The leader-follower HMASs are decided to reach QC if(18)limt⟶∞ωit−ω0t≤ξ,i=1,2,…,N,where ξ is a nonnegative constant.Assumption 2.
The network topology between agents is undirected and connected, and each agent can acquire the status information of the agent that has a connection relationship with it and the leader agent at any time.


### 2.3. Our Controller

To obtain QC between HMASs (2) and (3), we design the control input for all follower agents as(19)uit=−c∑j=1NLijtωjt−ritωit−ω0t,where c is a positive constant.The adaptive law for the control gains is described as(20)r˙it=μωit−ωjtTωit−ω0t,where μ is a positive constant to be selected.The adaptive law for the coupling weights is described as(21)L˙ijt=−αijωit−ωjtTωit−ωjt,Lij0=Lji0>0,i,j∈E,where αij=αji are the positive constants to be selected.Remark 1.
It should be noted that, in controller (19), adaptive laws (20) and (21), the coupling weights Lijt and the control gains rit are adjusted adaptively based on local communication with the leader and between the followers. Combining adaptation of the coupling weights and control gains, adaptive law (20) ensures the QC of the follower agents, while adaptive law (21) drives the follower agents to the leader agent.
Let the QC error vector beeit=ωit−ω0t. Then, the error model with the controller iscom(22)e˙it=Aieit+Bif˜eit,t+hiω0t,t−c∑j=1NLijtejt−riteit,where f˜eit,t=fωit,t−fω0t,t and hiω0t,t=Ai−Aω0t+Bi−Bfω0t,t.
hiω0t,t represents the difference between different agents. It can be obtained by Assumption 1:(23)hiω0t,t=Ai−Aω0t+Bi−Bfω0t,t≤Ai−Aω0t+Bi−Bfω0t,t≤Ai−Aω0t+Bi−Blmaxω0t.
Sinceω0t is bounded, it can be obtained that hiω0t,t is bounded.


## 2.1. Graph Theory

To carry out later research, we present some important concepts about graph theory in this section.G=V,E,A is defined as a weighted undirected graph with the network topology of N agents, where V=0,1,…,N and E=eiji,j⊆V×V are the separate sets of nodes and undirected edges. A=aijN×N is the weighted adjacency matrix of which the elements are nonnegative. The ith agent and the leader are modeled as the node i∈V,V=0,1,…,N and node 0, respectively. As a rule, the undirected edge i,j∈E in the weighted undirected graph G denoted by an ordered pair (Vi,Vj) represents that agent i and agent j become a pair of neighbors which can get their information from each other. There is an undirected spanning tree in an undirected graph if one agent exits an undirected path to every other distinct node.There are two matrices that are considered as the network topology, i.e., the weighted adjacency matrixA=aijN×N with aij=aji>0 if eij∈E, else aij=0 if eij∉E, and the Laplacian matrix L=LijN×N which is defined as Lii=∑j=1,j≠iNaij=degi and Lij=−aij, i≠j, noted that all of them are asymmetric for the undirected graph.Lemma 1 (see [46]).
The Laplacian matrixL is constructed from the undirected network. There are several properties in the following.(1)
Eigenvalues ofL satisfy 0=λ1L<λ2L≤⋯≤λNL and the smallest positive eigenvalue λ2L=minxT1N=0,x≠0xTLx/xTx if and only if the network is connected.(2)
For any vectorη=η1,η2,…,ηNT∈ℝN, the equation satisfies(1)ηTLη=12∑i=1∑j=1Gijηi−ηj2.


## 2.2. Problem Formation

In this paper, we consider that there areN follower multiagent systems, which can be described by(2)ω˙it=Aiωit+Bifωit,t+uit,where ωit∈Rn can be regarded as the position vector of the agent i.f:Rn×Rn×R+⟶Rn represents a continuous nonlinear vector function. Ai, Bi∈ℝn×n represent the system matrices of the agent i, respectively. uit∈Rn denotes the control protocol to be designed.The leader agent can be described as follows:(3)ω˙0t=Aω0t+Bfω0t,t,where ω0t∈Rn represents the leader’s state vector and A, B∈ℝn×n represent the leader’s system matrices, respectively.For the above heterogeneous leader-follower multiagent system, we give the following assumptions.Assumption 1.
Suppose there is a normal quantityl so that the vector function f for any vector λ,ν∈Rn satisfies(4)fλ,t−fν,t≤lλ−ν.Lemma 2 (see [47]).
For any vectorx,y∈Rn, the following holds:(5)xTy≤12xTx+12yTy.Lemma 3.
Any two continuous functions satisfy(6)v˙t+w˙t≤−γvt,where γ>0. Then, there is a t≥0 so that the following holds:(7)vt≤v0+w0e−γt−wt+γe−γt∗wt,t≥0,where ∗ is the convolution.Proof.
Sincevt+w˙t≤−γvt, the existence of zt≥0 makes the following equation:(8)v˙t+w˙t+zt=−γvt.
We calculate the Laplace transform of (8) to obtain(9)sVs+sWs−v0+w0+Zs=−γVs.
Then, we can obtain(10)Vs=v0+w01s+γ−ss+γWs−1s+γZs.
The inverse Laplace transform of (10) can be obtained as(11)vt=v0+w0e−γt−wt+γe−γt∗wt−e−γt∗zt.
Through (10), we can obtain(12)vt≤v0+w0e−γt−wt+γe−γt∗wt.Lemma 4.
Any two continuous functions satisfy(13)v˙t+w˙t≤−γvt+ε,where γ>0 and ε>0. Then, there is a t≥0 so that the following formula holds:(14)vt<v0+w0e−γt−wt+γe−γt∗wt+εγ,t≥0.Proof.
Sincev˙t+w˙t≤−γvt+ε, then vt−ε/γ˙+w˙t≤−γvt−ε/γ.
By Lemma3, we can obtain(15)vt−εγ≤v0−εγ+w0e−γt−wt+γe−γt∗wt.
We can further obtain(16)vt≤v0+w0e−γt−wt+γe−γt∗wt+εγ.Lemma 5 (see [47]).
IfA, B, C, and D represent four different matrices, respectively, and the matrix products AC and BD makes sense, the Kronecker product ⊗ satisfies(17)1A⊗B+C=A⊗B+A⊗C,2A⊗BC⊗D=AC⊗BD.Definition 1 (see [29]).
The leader-follower HMASs are decided to reach QC if(18)limt⟶∞ωit−ω0t≤ξ,i=1,2,…,N,where ξ is a nonnegative constant.Assumption 2.
The network topology between agents is undirected and connected, and each agent can acquire the status information of the agent that has a connection relationship with it and the leader agent at any time.


## 2.3. Our Controller

To obtain QC between HMASs (2) and (3), we design the control input for all follower agents as(19)uit=−c∑j=1NLijtωjt−ritωit−ω0t,where c is a positive constant.The adaptive law for the control gains is described as(20)r˙it=μωit−ωjtTωit−ω0t,where μ is a positive constant to be selected.The adaptive law for the coupling weights is described as(21)L˙ijt=−αijωit−ωjtTωit−ωjt,Lij0=Lji0>0,i,j∈E,where αij=αji are the positive constants to be selected.Remark 1.
It should be noted that, in controller (19), adaptive laws (20) and (21), the coupling weights Lijt and the control gains rit are adjusted adaptively based on local communication with the leader and between the followers. Combining adaptation of the coupling weights and control gains, adaptive law (20) ensures the QC of the follower agents, while adaptive law (21) drives the follower agents to the leader agent.
Let the QC error vector beeit=ωit−ω0t. Then, the error model with the controller iscom(22)e˙it=Aieit+Bif˜eit,t+hiω0t,t−c∑j=1NLijtejt−riteit,where f˜eit,t=fωit,t−fω0t,t and hiω0t,t=Ai−Aω0t+Bi−Bfω0t,t.
hiω0t,t represents the difference between different agents. It can be obtained by Assumption 1:(23)hiω0t,t=Ai−Aω0t+Bi−Bfω0t,t≤Ai−Aω0t+Bi−Bfω0t,t≤Ai−Aω0t+Bi−Blmaxω0t.
Sinceω0t is bounded, it can be obtained that hiω0t,t is bounded.


## 3. Main Results

In this section, we present some theorems for achieving QC in HMASs.


### 3.1. Adaptive Control Protocol

Theorem 1.
It is assumed thatfωit,t satisfies Assumption 1 and the follower system (2) satisfies Assumption 2. Under the action of controller (19) and adaptive laws (20) and (21), HMASs (2) and (3) can achieve QC.The Lyapunov function we constructed is(24)V1t=12∑i=1NeiTteit+∑i=1N∑j=1,j≠iNc4αijLijt+kij2+∑i=1N12μrit−di∗2,where kij=kjii≠j is a nonnegative constant if and only if Lijt=0 is kij=0. di∗ is the normal constant waiting for the value.We take the derivative ofV1t along (22) together with controller (19) and adaptive laws (20) and (21), and we can obtain(25)V˙1t=∑i=1NeiTte˙it+∑i=1N∑j=1,j≠iNc2αijLijt+kijL˙ijt+1μ∑i=1Nrit−di∗r˙it=∑i=1NeiTtAieit+Bif˜eit,t+hiω0t,t−c∑j=1NLijtejt−riteit−∑i=1N∑j=1,j≠iNc2Lijt+kijωit−ωjtTωit−ωjt+∑i=1Nrit−di∗eiTteit=∑i=1NeiTtAieit+∑i=1NeiTtBif˜eit,t+∑i=1NeiTthiω0t,t−∑i=1NeiTtc∑j=1NLijtejt−∑i=1Ndi∗eiTteit−c2∑i=1N∑j=1,j≠iNLijt+kijωit−ωjtTωit−ωjt.Through Assumption1 and Lemma 2, one has(26)∑i=1NeiTtBif˜eit,t=∑i=1NeiTtBifωit,t−fω0t,t≤12∑i=1NeiTtBiBiTeit+12∑i=1Nfωit,t−fω0t,t2≤12∑i=1NeiTtBiBiT+l2Ineit,(27)∑i=1NeiTthiω0t,t≤12∑i=1NeiTteit+12∑i=1Nhiω0t,t2.From (25)–(27), we can obtain(28)V˙1t≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2−∑i=1NeiTtc∑j=1NLijtejt+12∑i=1NeiTtBiBiT+l2Ineit−∑i=1Ndi∗eiTteit−c2∑i=1N∑j=1,j≠iNLijt+kijωit−ωjtTωit−ωjt.We can define the Laplacian matrixΩ=τijN×N, where τij=kij,i≠j and τii=−∑j=1j≠iNτij. By Lemma 1, we can obtain(29)c2∑i=1N∑j=1,j≠iNLijt+kijeit−ejtTeit−ejt=−c∑i=1N∑j=1NLijteiTtejt+c∑i=1N∑j=1NτijeiTtejt.Combining with (28) and (29), we have(30)V˙1t≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2−∑i=1NeiTtc∑i=1NLijtejt+12∑i=1NeiTtBiBiT+l2Ineit+c∑i=1N∑i=1NLijteiTtejt−c∑i=1N∑i=1NτijeiTtejt−∑i=1Ndi∗eiTteit=∑i=1NeiTtAieit+12∑i=1NeiTtBiBiT+l2+1In−2di∗⋅Ineit+12∑i=1Nhiω0t,t|2−c∑i=1N∑i=1NτijeiTtejt.Then, there is a unitary matrixU=u1,…,uN so that UTΩU=Δ, mt=UT⊗Inet,(31)V˙1t≤eTtA+12BBT+IN⊗l2+1In−2D∗⊗In−cΩ⊗Inet+12hω0t,t2=eTtA+12BBT+IN⊗l2+1In−2D∗⊗In−cU⊗InΔ⊗InUT⊗Inet+12hω0t,t2=eTtA+12BBT+IN⊗l2+1In−2D∗⊗Inet−cmTtΔ⊗Inmt+12hω0t,t2,where D∗=diagd1∗,d2∗,…,dN∗, B=diagB1,B2,…,BN, and A=diagA1,A2,…,AN.SinceIn is positive definite, by Lemma 1, we can get the following:(32)mTtΔ⊗Inmt≥λ2ΩmTtIN⊗Inmt.Then, we can obtain(33)V˙1t≤eTtA+12BBT+IN⊗l2+1In−2D∗⊗Inet+12hω0t,t2−cλ2ΩmTtIN⊗Inmt=eTtA+12BBT+IN⊗l2+1In−2D∗⊗Inet+12hω0t,t2−cλ2ΩeTtP⊗InIN⊗InPT⊗Inet=eTtA+12BBT+IN⊗l2+1In−2D∗⊗In−cλ2ΩIN⊗Inet+12hω0t,t2.We can choose large enoughkij and di∗ so that(34)λmaxA+12BBT+IN⊗l2+1In−2D∗⊗In−cλ2ΩIN⊗In+θ≤0,where θ>0 is a positive constant.(35)V˙1t≤−θeTtet+12hω0t,t2,where et=e1T,e2T,e3T,…,eNTT.By Lemma4, it yields(36)eTtet≤∑i=1NeiT0ei0+∑i=1N∑j=1,j≠iNc2αijLij0−kij2+∑i=1N1μri0−di∗2e−2θt+2θe−2θt∗∑∑i=1N1μri0−di∗2+∑i=1N∑j=1,j≠iNc2αijLij0−kij2+hω0t,t22θ.The error will eventually converge to finite regionΞ=et∈Rn|t≤hω0t,t2/2θ as t⟶+∞. This completes the proof.


### 3.2. Adaptive Pinning Control Protocol

In the previous section, an adaptive controller (19) and an adaptive laws (20) and (21) for all follower agents are designed. However, it is neither realistic nor economical to control all follower agents in engineering. In view of this, the adaptive pinning control schemes are considered, where a fraction of the control gains and the coupling weights are adapted.Suppose thatε˜ is a subset of E and HMASs (2) and (3) are connected by the undirected edge. ε˜ is connected. The pinning adaptive protocol is written as follows:(37)uit=−c∑j=1NLijtωjt−ϑiritωit−ω0t,where ϑi=1,fori=1,2,…,Ns,0,fori=Ns+1,…,N.The adaptive law forrit is the same as (20).The adaptive law forLijt is described as(38)L˙ijt=−αijωit−ωjtTωit−ωjt,where i,j∈ε˜ and Lij0=Lji0>0.Theorem 2.
Suppose that Assumption1 is valid and the follower system (2) satisfies Assumption 2. The HMASs (2) and (3) can achieve QC under the pinning adaptive protocol combined with (20), (21), and (37).Proof.
Take into account the Lyapunov function candidate:(39)V2t=12∑i=1NeiTteit+∑i=1N∑j=1,j≠iNc4αijLijt+kij2+∑i=1Ns12μrit−di∗2,where cij=cjii≠j is a nonnegative quantity if and only if Lijt=0 and kij=0. di∗ is the normal constant waiting for the value if i=1,2,…,Ns,Ns≥1, or di∗=0.
The derivative ofV2t along (22), controller (37), and the decentralized adaptive pinning laws (20) and (21) gives(40)V˙2t=∑i=1NeiTte˙it+∑i=1N∑j=1,j≠iNc2αijLijt+kijL˙ijt+∑i=1Ns1μrit−di∗d˙it≤∑i=1NeiTtAieit+12∑i=1NeiTtBiBiT+l2+1In−2di∗⊗Ineit+12∑i=1Nhiω0t,t2−c∑i=1N∑j=1NτijeiTtejt≤eTtA+12BBT+IN⊗l2+1In−2D˜∗⊗In−cλ2ΩIN⊗Inet+12hω0t,t2,where D˜∗=diagd1∗,d2∗,…,dNs∗,0,…,0.
The proof can be completed by using the similar analysis method in Theorem1.Theorem 3.
Suppose that Assumptions1 and 2 hold. The HMASs (2) and (3) can achieve QC under the pinning adaptive protocol combined with (19), (20), and (38).Proof.
Construct the Lyapunov functional as(41)V3t=12∑i=1NeiTteit+∑i=1N∑i,jεε˜c4αijLijt+k˜ij2+∑i=1N12μrit−di∗2,where k˜ij=k˜ji>0,i,jεε˜,andk˜ij=0i≠j, else.
LetK˜=k˜ijN×N,k˜ii=−∑i=1,i≠jNk˜ij; then,(42)Gij=Lij0,i,j∈E−ε˜,−∑j=1,j≠iNLij0,i=j,0,other.
After the derivative ofV3t along (22), controller (19), and the decentralized adaptive pinning laws (20) and (38), the following holds:(43)V˙3t=∑i=1NeiTte˙it+∑i=1N∑i,jεεc2αijLijt+k˜ijL˙ijt+1μ∑i=1Nrit−di∗d˙it=∑i=1NeiTtAieit+∑i=1NeiTtBif˜eit,t+∑i=1NeiTthiω0t,t−∑i=1NeiTtc∑i=1NLijtejt−∑i=1Ndi∗eiTteit−c2∑i=1N∑i,jεεLijt+k˜ijωit−ωjtTωit−ωjt≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2+12∑i=1NeiTtBiBiT+l2Ineit−c∑i=1N∑j=1NGijteiTtejt−c∑i=1N∑j=1Nτ˜ijeiTtejt−∑i=1Ndi∗eiTteit=eTtA+cG⊗In+12IN⊗BBT+l2+1InD∗⊗In−cλ2Ω∗IN⊗Inet+12hω0t,t2,where G=GijN×N, Ω∗=τ˜ijN×N, τ˜ij=−k˜ij,i≠j, and τ˜ii=−∑j=1j≠iNτ˜ij.
The following proof is similar to the previous derivation in Theorem1; thus, we will omit this part here.Theorem 4.
Suppose that Assumptions1 and 2 hold. The HMASs (2) and (3) can achieve QC under the pinning adaptive protocol combined with (20), (37), and (38).Proof.
The Lyapunov functional is considered as follows:(44)V4t=12∑i=1NeiTteit+∑i=1N∑i,jεε˜c4αijLijt+k˜ij2+∑i=1Ns12μrit−di∗2.
The derivative ofV4t along the error system (22), controller (37), and decentralized adaptive pinning laws (20) and (38) gives(45)V˙4t=∑i=1NeiTte˙it+∑i=1N∑i,jεεNc2αijLijt+k˜ijL˙ijt+1μ∑i=1Nsrit−di∗d˙it≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2+12∑i=1NeiTtBiBiT+l2Ineit+c∑i=1N∑j=1NGijteiTtejt−c∑i=1N∑j=1Nτ˜ijeiTtejt−∑i=1Nsdi∗eiTteit=eTtA+cG⊗In+12IN⊗BBT+l2+1In−D˜∗⊗In−cλ2Ω∗IN⊗Inet+12hω0t,t2,where G=GijN×N, Ω∗=τ˜ijN×N, τ˜ij=−k˜ij,i≠j, τ˜ii=−∑j=1j≠iNτ˜ij, and D˜∗=diagd1∗,d2∗,…,dNs∗,0,…,0.
The rest of the proof is similar to Theorem1. To save space, it is thus omitted here.Remark 2.
In Theorems2–4, some sufficient conditions are given to realize QC of HMASs (2) and (3) by using the adaptive pinning control schemes. Actually, if the follower agents (3) are connected, one can randomly choose a small fraction of coupling weights and/or the control gains to adapt. In particular, it is possible to obtain the QC by pinning one follower agent.


## 3.1. Adaptive Control Protocol

Theorem 1.
It is assumed thatfωit,t satisfies Assumption 1 and the follower system (2) satisfies Assumption 2. Under the action of controller (19) and adaptive laws (20) and (21), HMASs (2) and (3) can achieve QC.The Lyapunov function we constructed is(24)V1t=12∑i=1NeiTteit+∑i=1N∑j=1,j≠iNc4αijLijt+kij2+∑i=1N12μrit−di∗2,where kij=kjii≠j is a nonnegative constant if and only if Lijt=0 is kij=0. di∗ is the normal constant waiting for the value.We take the derivative ofV1t along (22) together with controller (19) and adaptive laws (20) and (21), and we can obtain(25)V˙1t=∑i=1NeiTte˙it+∑i=1N∑j=1,j≠iNc2αijLijt+kijL˙ijt+1μ∑i=1Nrit−di∗r˙it=∑i=1NeiTtAieit+Bif˜eit,t+hiω0t,t−c∑j=1NLijtejt−riteit−∑i=1N∑j=1,j≠iNc2Lijt+kijωit−ωjtTωit−ωjt+∑i=1Nrit−di∗eiTteit=∑i=1NeiTtAieit+∑i=1NeiTtBif˜eit,t+∑i=1NeiTthiω0t,t−∑i=1NeiTtc∑j=1NLijtejt−∑i=1Ndi∗eiTteit−c2∑i=1N∑j=1,j≠iNLijt+kijωit−ωjtTωit−ωjt.Through Assumption1 and Lemma 2, one has(26)∑i=1NeiTtBif˜eit,t=∑i=1NeiTtBifωit,t−fω0t,t≤12∑i=1NeiTtBiBiTeit+12∑i=1Nfωit,t−fω0t,t2≤12∑i=1NeiTtBiBiT+l2Ineit,(27)∑i=1NeiTthiω0t,t≤12∑i=1NeiTteit+12∑i=1Nhiω0t,t2.From (25)–(27), we can obtain(28)V˙1t≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2−∑i=1NeiTtc∑j=1NLijtejt+12∑i=1NeiTtBiBiT+l2Ineit−∑i=1Ndi∗eiTteit−c2∑i=1N∑j=1,j≠iNLijt+kijωit−ωjtTωit−ωjt.We can define the Laplacian matrixΩ=τijN×N, where τij=kij,i≠j and τii=−∑j=1j≠iNτij. By Lemma 1, we can obtain(29)c2∑i=1N∑j=1,j≠iNLijt+kijeit−ejtTeit−ejt=−c∑i=1N∑j=1NLijteiTtejt+c∑i=1N∑j=1NτijeiTtejt.Combining with (28) and (29), we have(30)V˙1t≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2−∑i=1NeiTtc∑i=1NLijtejt+12∑i=1NeiTtBiBiT+l2Ineit+c∑i=1N∑i=1NLijteiTtejt−c∑i=1N∑i=1NτijeiTtejt−∑i=1Ndi∗eiTteit=∑i=1NeiTtAieit+12∑i=1NeiTtBiBiT+l2+1In−2di∗⋅Ineit+12∑i=1Nhiω0t,t|2−c∑i=1N∑i=1NτijeiTtejt.Then, there is a unitary matrixU=u1,…,uN so that UTΩU=Δ, mt=UT⊗Inet,(31)V˙1t≤eTtA+12BBT+IN⊗l2+1In−2D∗⊗In−cΩ⊗Inet+12hω0t,t2=eTtA+12BBT+IN⊗l2+1In−2D∗⊗In−cU⊗InΔ⊗InUT⊗Inet+12hω0t,t2=eTtA+12BBT+IN⊗l2+1In−2D∗⊗Inet−cmTtΔ⊗Inmt+12hω0t,t2,where D∗=diagd1∗,d2∗,…,dN∗, B=diagB1,B2,…,BN, and A=diagA1,A2,…,AN.SinceIn is positive definite, by Lemma 1, we can get the following:(32)mTtΔ⊗Inmt≥λ2ΩmTtIN⊗Inmt.Then, we can obtain(33)V˙1t≤eTtA+12BBT+IN⊗l2+1In−2D∗⊗Inet+12hω0t,t2−cλ2ΩmTtIN⊗Inmt=eTtA+12BBT+IN⊗l2+1In−2D∗⊗Inet+12hω0t,t2−cλ2ΩeTtP⊗InIN⊗InPT⊗Inet=eTtA+12BBT+IN⊗l2+1In−2D∗⊗In−cλ2ΩIN⊗Inet+12hω0t,t2.We can choose large enoughkij and di∗ so that(34)λmaxA+12BBT+IN⊗l2+1In−2D∗⊗In−cλ2ΩIN⊗In+θ≤0,where θ>0 is a positive constant.(35)V˙1t≤−θeTtet+12hω0t,t2,where et=e1T,e2T,e3T,…,eNTT.By Lemma4, it yields(36)eTtet≤∑i=1NeiT0ei0+∑i=1N∑j=1,j≠iNc2αijLij0−kij2+∑i=1N1μri0−di∗2e−2θt+2θe−2θt∗∑∑i=1N1μri0−di∗2+∑i=1N∑j=1,j≠iNc2αijLij0−kij2+hω0t,t22θ.The error will eventually converge to finite regionΞ=et∈Rn|t≤hω0t,t2/2θ as t⟶+∞. This completes the proof.


## 3.2. Adaptive Pinning Control Protocol

In the previous section, an adaptive controller (19) and an adaptive laws (20) and (21) for all follower agents are designed. However, it is neither realistic nor economical to control all follower agents in engineering. In view of this, the adaptive pinning control schemes are considered, where a fraction of the control gains and the coupling weights are adapted.Suppose thatε˜ is a subset of E and HMASs (2) and (3) are connected by the undirected edge. ε˜ is connected. The pinning adaptive protocol is written as follows:(37)uit=−c∑j=1NLijtωjt−ϑiritωit−ω0t,where ϑi=1,fori=1,2,…,Ns,0,fori=Ns+1,…,N.The adaptive law forrit is the same as (20).The adaptive law forLijt is described as(38)L˙ijt=−αijωit−ωjtTωit−ωjt,where i,j∈ε˜ and Lij0=Lji0>0.Theorem 2.
Suppose that Assumption1 is valid and the follower system (2) satisfies Assumption 2. The HMASs (2) and (3) can achieve QC under the pinning adaptive protocol combined with (20), (21), and (37).Proof.
Take into account the Lyapunov function candidate:(39)V2t=12∑i=1NeiTteit+∑i=1N∑j=1,j≠iNc4αijLijt+kij2+∑i=1Ns12μrit−di∗2,where cij=cjii≠j is a nonnegative quantity if and only if Lijt=0 and kij=0. di∗ is the normal constant waiting for the value if i=1,2,…,Ns,Ns≥1, or di∗=0.
The derivative ofV2t along (22), controller (37), and the decentralized adaptive pinning laws (20) and (21) gives(40)V˙2t=∑i=1NeiTte˙it+∑i=1N∑j=1,j≠iNc2αijLijt+kijL˙ijt+∑i=1Ns1μrit−di∗d˙it≤∑i=1NeiTtAieit+12∑i=1NeiTtBiBiT+l2+1In−2di∗⊗Ineit+12∑i=1Nhiω0t,t2−c∑i=1N∑j=1NτijeiTtejt≤eTtA+12BBT+IN⊗l2+1In−2D˜∗⊗In−cλ2ΩIN⊗Inet+12hω0t,t2,where D˜∗=diagd1∗,d2∗,…,dNs∗,0,…,0.
The proof can be completed by using the similar analysis method in Theorem1.Theorem 3.
Suppose that Assumptions1 and 2 hold. The HMASs (2) and (3) can achieve QC under the pinning adaptive protocol combined with (19), (20), and (38).Proof.
Construct the Lyapunov functional as(41)V3t=12∑i=1NeiTteit+∑i=1N∑i,jεε˜c4αijLijt+k˜ij2+∑i=1N12μrit−di∗2,where k˜ij=k˜ji>0,i,jεε˜,andk˜ij=0i≠j, else.
LetK˜=k˜ijN×N,k˜ii=−∑i=1,i≠jNk˜ij; then,(42)Gij=Lij0,i,j∈E−ε˜,−∑j=1,j≠iNLij0,i=j,0,other.
After the derivative ofV3t along (22), controller (19), and the decentralized adaptive pinning laws (20) and (38), the following holds:(43)V˙3t=∑i=1NeiTte˙it+∑i=1N∑i,jεεc2αijLijt+k˜ijL˙ijt+1μ∑i=1Nrit−di∗d˙it=∑i=1NeiTtAieit+∑i=1NeiTtBif˜eit,t+∑i=1NeiTthiω0t,t−∑i=1NeiTtc∑i=1NLijtejt−∑i=1Ndi∗eiTteit−c2∑i=1N∑i,jεεLijt+k˜ijωit−ωjtTωit−ωjt≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2+12∑i=1NeiTtBiBiT+l2Ineit−c∑i=1N∑j=1NGijteiTtejt−c∑i=1N∑j=1Nτ˜ijeiTtejt−∑i=1Ndi∗eiTteit=eTtA+cG⊗In+12IN⊗BBT+l2+1InD∗⊗In−cλ2Ω∗IN⊗Inet+12hω0t,t2,where G=GijN×N, Ω∗=τ˜ijN×N, τ˜ij=−k˜ij,i≠j, and τ˜ii=−∑j=1j≠iNτ˜ij.
The following proof is similar to the previous derivation in Theorem1; thus, we will omit this part here.Theorem 4.
Suppose that Assumptions1 and 2 hold. The HMASs (2) and (3) can achieve QC under the pinning adaptive protocol combined with (20), (37), and (38).Proof.
The Lyapunov functional is considered as follows:(44)V4t=12∑i=1NeiTteit+∑i=1N∑i,jεε˜c4αijLijt+k˜ij2+∑i=1Ns12μrit−di∗2.
The derivative ofV4t along the error system (22), controller (37), and decentralized adaptive pinning laws (20) and (38) gives(45)V˙4t=∑i=1NeiTte˙it+∑i=1N∑i,jεεNc2αijLijt+k˜ijL˙ijt+1μ∑i=1Nsrit−di∗d˙it≤∑i=1NeiTtAieit+12∑i=1NeiTteit+12∑i=1Nhiω0t,t2+12∑i=1NeiTtBiBiT+l2Ineit+c∑i=1N∑j=1NGijteiTtejt−c∑i=1N∑j=1Nτ˜ijeiTtejt−∑i=1Nsdi∗eiTteit=eTtA+cG⊗In+12IN⊗BBT+l2+1In−D˜∗⊗In−cλ2Ω∗IN⊗Inet+12hω0t,t2,where G=GijN×N, Ω∗=τ˜ijN×N, τ˜ij=−k˜ij,i≠j, τ˜ii=−∑j=1j≠iNτ˜ij, and D˜∗=diagd1∗,d2∗,…,dNs∗,0,…,0.
The rest of the proof is similar to Theorem1. To save space, it is thus omitted here.Remark 2.
In Theorems2–4, some sufficient conditions are given to realize QC of HMASs (2) and (3) by using the adaptive pinning control schemes. Actually, if the follower agents (3) are connected, one can randomly choose a small fraction of coupling weights and/or the control gains to adapt. In particular, it is possible to obtain the QC by pinning one follower agent.


## 4. Numerical Examples

In this section, we will confirm the theory proposed in the paper by using digital simulation experiments.We use five following agents and one leader agent, and the initial communication Laplacian matrix for follower agents is(46)L=1.5−0.3−0.3−0.4−0.5−0.30.9−0.600−0.3−0.60.900−0.4000.7−0.3−0.500−0.30.8.The leader’s system matrices can be described asA=−2.51001−110−180 and B=35/600000000, and the nonlinear function of the entire system can be assumed to be ft,x=0.5x1t+1−x1t−100. The following system matrices are, respectively, assumed to be Ai=−2.5+0.1∗i10+0.2∗i01+0.2∗i−1+0.2∗i1+0.2∗i0−18+0.2∗i0 and Bi=35/600000000i=1,2,…,5. We arbitrarily choose the initial value as x00=3.3,0.66,0.1T and xi0=11.5+1.8∗i,4.2+1.1∗i,4.9+1.5∗iTi=1,2,…,5. Figure 1 shows the change of eit2i=1,2,…,5 for HMASs without the controller.Figure 1
The changes ofeit2i=1,2,…,5 for the system without thw controller.


### 4.1. Example 1

In this case, the parameters of controller (19) and adaptive laws (20) and (21) are chosen according to Theorem 1. All agent system parameters are in accordance with Ai and Bi. We choose c=2.6, μ=0.05, α12=α21=2.7, α13=α31=2.4, α14=α41=2.1, α15=α51=1.8, α23=α32=2.1, and α45=α54=2.2. We arbitrarily choose the initial value as r0=2.3,2.1,1.8,1.8,2.1T, x00=3.3,0.66,0.1T, and xi0=11.5+1.8∗i,4.2+1.1∗i,4.9+1.5∗iTi=1,2,…,5. Figure 2 shows the change in the state of the system under the increment controller. It visibly shows that, under the action of the adaptive controller and adaptive law, the error of the leader and follower finally converges to a finite region.Figure 2
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 1.


### 4.2. Example 2

In this example, we only use the coupling between followers and leaders 1 and 2 to achieve QC of the entire system. All agent system parameters are in accordance withAi and Bi. We choose c=2.6, μ=0.01, α12=α21=8.1, α13=α31=7.2, α14=α41=6.3, α15=α51=5.4, α23=α32=6.3, and α45=α54=7.2. We arbitrarily choose the 1 and 2 agents, and their initial values are selected as r0=2.3,2.1T. The initial value of each agent is selected as x00=3.3,0.66,0.1T and xi0=−6−1.6∗i,5+1.1∗i,9.4+2∗iTi=1,2,…,5. Figure 3 shows eit2i=1,2,…,5 eventually tends to a finite region.Figure 3
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 2.


### 4.3. Example 3

In this example, we use a control strategy that pinning the coupling between followers to QC of HMASs (2) and (3). All agent system parameters are in accordance with Ai and Bi. We choose c=1.2. The parameters in the adaptive law are μ=0.01, α14=α41=7.2, α15=α51=6.3, and α23=α32=5.4. We arbitrarily choose the initial value as r0=2.1,2.1,2,1.9,2.1T, x00=3.3,0.66,0.1T and xi0=−5.3−0.5∗i,3+1.6∗i,2.7+1.4∗iTi=1,2,…,5. From Figure 4, we can distinctly see that, under the control strategy designed by Theorem 3, eit2i=1,2,…,5 of HMASs (2) and (3) are concentrated in a limited area.Figure 4
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 3.


### 4.4. Example 4

In this case, by controlling the coupling between part of the follower agent and the leader and the coupling between the follower agents, we realize the QC of the whole system. All agent system parameters are in accordance withAi and Bi. We choose c=3. The parameters in the adaptive law are μ=0.01, α14=α41=4.9, α15=α51=4.2, and α23=α32=4.9. We arbitrarily choose the 1, 2, and 3 agents, and their initial values are selected as r0=1.8,2.1,1.8T. The initial value of each agent is selected as x00=3.3,0.66,0.1T and x10=x20=x30=x40=x50=12.3,5,5.1T. Figure 5 also verifies that, under the control strategy designed by Theorem 4, the entire system gradually converges to a limited area.Figure 5
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 4.


## 4.1. Example 1

In this case, the parameters of controller (19) and adaptive laws (20) and (21) are chosen according to Theorem 1. All agent system parameters are in accordance with Ai and Bi. We choose c=2.6, μ=0.05, α12=α21=2.7, α13=α31=2.4, α14=α41=2.1, α15=α51=1.8, α23=α32=2.1, and α45=α54=2.2. We arbitrarily choose the initial value as r0=2.3,2.1,1.8,1.8,2.1T, x00=3.3,0.66,0.1T, and xi0=11.5+1.8∗i,4.2+1.1∗i,4.9+1.5∗iTi=1,2,…,5. Figure 2 shows the change in the state of the system under the increment controller. It visibly shows that, under the action of the adaptive controller and adaptive law, the error of the leader and follower finally converges to a finite region.Figure 2
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 1.


## 4.2. Example 2

In this example, we only use the coupling between followers and leaders 1 and 2 to achieve QC of the entire system. All agent system parameters are in accordance withAi and Bi. We choose c=2.6, μ=0.01, α12=α21=8.1, α13=α31=7.2, α14=α41=6.3, α15=α51=5.4, α23=α32=6.3, and α45=α54=7.2. We arbitrarily choose the 1 and 2 agents, and their initial values are selected as r0=2.3,2.1T. The initial value of each agent is selected as x00=3.3,0.66,0.1T and xi0=−6−1.6∗i,5+1.1∗i,9.4+2∗iTi=1,2,…,5. Figure 3 shows eit2i=1,2,…,5 eventually tends to a finite region.Figure 3
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 2.


## 4.3. Example 3

In this example, we use a control strategy that pinning the coupling between followers to QC of HMASs (2) and (3). All agent system parameters are in accordance with Ai and Bi. We choose c=1.2. The parameters in the adaptive law are μ=0.01, α14=α41=7.2, α15=α51=6.3, and α23=α32=5.4. We arbitrarily choose the initial value as r0=2.1,2.1,2,1.9,2.1T, x00=3.3,0.66,0.1T and xi0=−5.3−0.5∗i,3+1.6∗i,2.7+1.4∗iTi=1,2,…,5. From Figure 4, we can distinctly see that, under the control strategy designed by Theorem 3, eit2i=1,2,…,5 of HMASs (2) and (3) are concentrated in a limited area.Figure 4
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 3.


## 4.4. Example 4

In this case, by controlling the coupling between part of the follower agent and the leader and the coupling between the follower agents, we realize the QC of the whole system. All agent system parameters are in accordance withAi and Bi. We choose c=3. The parameters in the adaptive law are μ=0.01, α14=α41=4.9, α15=α51=4.2, and α23=α32=4.9. We arbitrarily choose the 1, 2, and 3 agents, and their initial values are selected as r0=1.8,2.1,1.8T. The initial value of each agent is selected as x00=3.3,0.66,0.1T and x10=x20=x30=x40=x50=12.3,5,5.1T. Figure 5 also verifies that, under the control strategy designed by Theorem 4, the entire system gradually converges to a limited area.Figure 5
The changes ofeit2i=1,2,…,5 for the system under the control strategy designed by Theorem 4.


## 5. Conclusions

The decentralized adaptive control for QC of HMASs has been studied. The combined adaptation of the coupling weights and control gains allows to drive HMASs (2) and (3) to some bounded areas. In addition, some pinning schemes have been proposed to adjust a fraction of the coupling weights and control gains. To deal with the heterogeneity, two new lemmas are proposed to derive the QC criteria. It has been shown that the QC can be obtained without requiring any global. In future works, we will extend the results to more general HMASs, such as fractional-order HMASs, HMASs with time delay, and cooperative-competitive interaction. At the same time, we will attempt to optimize control protocol and extend it to some other systems, i.e., fractional-order systems [48], memristive neural networks [49, 50], and complex-valued neural networks [51, 52].



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