# Distributed Fuzzy Adaptive Control for Heterogeneous Nonlinear Multiagent Systems with Similar Composite Structure

Authors: Yongqing Fan; Tiantian Xiao; Zhen Li
Journal: Complexity
 (2020)

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## Abstract

A distributed fuzzy adaptive control with similar parameters is constructed for a class of heterogeneous multiagent systems. Unlike many existing works, the dimensions of each multiagent dynamic system are considered to be nonidentical in this paper. Firstly, similar properties for different dimensions of multiagent systems are introduced, and some similar parameters among multiagent systems are also proposed. Secondly, a distributed fuzzy adaptive control on the basis of similar parameters is designed for the consensus of leader-follower multiagent systems. Following the graph theory and Lyapunov stability approach, it is concluded that UUB (uniformly ultimately bounded) of all signals in the closed-loop system can be guaranteed, and the consensus tacking error converges to a small compact zero set. Finally, a simulation example with different dimensions is provided to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Multiagent systems have been widely utilized in various fields such as remedial actions [1, 2], social engineering systems [3], satellites engineering [4], and robots cooperative [5, 6]. More and more researchers inclined to design the fundamental collective controls for multiagent systems to make sure that the consensus or synchronization of leader-follower can be guaranteed, and many excellent controls for linear and nonlinear multiagent systems were proposed in recent years [7–10]. Generally speaking, the main work in designing controls is that all agents in the entire dynamic network must reach an agreement, and the information of each agent only can be shared locally. Unfortunately, in lots of actual engineering systems, uncertain nonlinear components existed such as electrical control systems and mechanical control systems; hence, it is a challenge to project appropriate control with limited information.Fuzzy logic system (FLS) and neural network (NN) are two universal approximations to compensate uncertain terms in all sorts of complexity fields [11–15], and a great quantity of corresponding research works was derived by scholars [16–21]. For example, aiming at the high-order multiagent systems with unknown nonlinearities in [22], an observer-based distributed fuzzy adaptive control was designed to deal with the unknown nonlinear functions. For a class of strict feedback form of multiagent systems, a novel event-triggered control was presented for the consensus tracking in [23]. Adaptive NN event-triggered control plan was investigated for the nonstrict feedback multiagent systems with sensor faults and input saturation in [24]. In [25], a fuzzy observer was designed to evaluate the unmeasurable states of nonlinear multiagent systems, and an event-triggered control approach was studied to make the followers synchronize with leader’s trajectory. However, these existing results only researched on the identity of each agent, which means that the states of every agent have same dynamical behaviors [22–25]. In order to break this limitation, different dynamic behaviors of each agent that can be called as heterogeneous multiagent systems have been studied [26–28]. For instance, a distributed adaptive fuzzy control combining with the backstepping technique was addressed for a class of second-order heterogeneous multiagent systems in [29]. In [30], the output consensus of multiagent systems was guaranteed by using the devised fuzzy adaptive control. A robust consensus protocol was designed for essential heterogeneous multiagent systems in [31]. In order to ensure the consensus of heterogeneous multiagent systems, a distributed proportional integral control based on sufficient conditions was derived in [32]. It should be noted that the proposed control schemes in [8, 10, 26–28] were only valid for linear multiagent systems. These abundant research achievements provided well guidance for some new design algorithm controls of heterogeneous multiagent systems. Nevertheless, the dimensions of every agent are completely congruent in these literatures [8, 26–32], and the raised control schemes will be invalided to settle the consensus or synchronization of multiagent systems with different dimensions. Consequently, it is necessary to exploit other original control approaches to tackle the consensus of multiagent systems with distinct dimensions.Motivated by the similar properties of large-scale systems in [33–38], the definition of similar nodes was introduced for large-scale composite systems with different dimensions, and some effective controls with similar parameters were addressed. From the viewpoint of mathematics, every agent can be defined as a series of nodes in a network; hence, the character of similar nodes in these excellent research works can be drawn to develop consensus control with similar parameters.This paper attempts to investigate a novel consensus fuzzy adaptive control for a class of multiagent systems with different dimensions, in which the dimensions of each agent are unequal, and the similar parameters of agents are used for devising consistency control. Compared to recent existing works on the consensus of heterogeneous multiagent systems, the principal contributions are three aspects: first, the dimensions of follower systems are different with the dimensions of leader, and the similar definition among multiagent systems is explored. Second, a distributed fuzzy adaptive control methodology with similar parameters is provided. Last, the control matrix gain can be solved by the condition of proposed linear matrix inequality (LMI).The remaining parts of this paper are organized as follows. Interaction topology, the property of similar composite structure, and FLS are displayed in Section2. Section 3 presents the fuzzy adaptive control and stability analysis. A simulation example is given for the consensus of multiagent system with nonidentical dimension in Section 4. Finally, Section 5summarizes conclusions.Throughout this paper, the following notations are hired.Rn×n denotes the n×n dimensional Euclidean space; diagA1A2⋯AN expresses the block-diagonal matrix with matrices A1A2⋯AN on its principal diagonal; the notation ⋅ refers to the vector-2-norm. AΤ and A−1 denote the transpose matrix and inverse matrix of A, respectively; In represents the identity matrix with n appropriate dimensions; and P>0<0 means that P is a positive (negative) definite matrix. The Kronecker product of matrices A and P is symbolized by A⊗P; the maximum and minimum eigenvalues matrix A are denoted corresponding to λmaxA and λminA.


## 2. Preliminaries and Problem Formulation

### 2.1. Graph Theory

A directed digraphG=V,E is utilized to describe the information exchange among each agent, where V=v1v2⋯vN stands for the nonempty set of nodes for each agent. The edge set E contains an edge vj,vi which means node vj is able to transfer the relative state information to node vi; then, nodes i and j are called as the neighbors when the edge vj,vi exists. Let Ni denote the set of neighbors of node ii=1,2,…,N. The directed digraph G can also be described by an adjacency matrix αij, where αij=1,vj,vi∈E0,otherwise. In addition, it is assumed that there are no repeated edges and no self-loops, i.e., αii=0. The Laplacian matrix L=lij∈RN×N is defined as lij=−αij,i≠j∑k≠i,k=1Nαik,i=j.A digraph is said to have a spanning tree, if there exists a node that is called as the root such that the node has directed paths to all other nodes in the graph. The graphG¯ consists of G, node 0 (the leader), and the directed edges from the node 0 to the followers in G, and only a small percentage of the followers can receive the information from the leader. Then, we get the following lemma.Lemma 1.
(see [8]). Let the matrix L¯=L+diagα10α20⋯αN0 is positive definite with αi0>0, the ith agent has access to the leader’s state information, whereas αi0=0 if otherwise.


### 2.2. Preliminaries and Multiagent System

Consider a group of agent system with a leader andN followers labeled as 0 and 1,2,…,N, respectively. The dynamics of the leader is described as(1)x˙0t=A0x0t+B0u0t+sx0,t,where A0t∈Rn0×n0 and B0∈Rn0×m0 are the system matrix and input matrix of leader, respectively. x0t∈Rn0×1 is the state vector of leader. u0t denotes the input vector of leader, and matrix K0∈Rm0×n0 will be given in the process of control design, which will make A0+B0K0 be Hurwitz stabilized. sx0,t is defined as an input bounded signal and satisfies sx0,t≤s¯ for all t≥t0, and s¯ is a known constant.The dynamics of the followers are defined as follows:(2)x˙it=Aixit+Biuit+gixi,i=1,2,…,N,where Ai∈Rni×ni and Bi∈Rni×mi denote the system matrix and input matrix in the ith followers system, respectively. xit∈Rni×1 and uit∈Rmi×ni are corresponding to the state vector and input vector of the ith follower, respectively. gixi represents the unknown nonlinear function.Assumption 1.
(see [38]). Consider N agent systems as given in (2), and the follower system (2) is called similar to the leader system (1), if there exists N matrices Ki∈Rmi×ni, matrix K0∈Rm0×n0, and N matrices Ti∈Rn0×ni satisfying the following condition:(3)TiAi+BiKi=A0+B0K0Ti,TiBi=B0.Definition 1.
In Assumption1, Ti and Ki and K0 are called as similar parameters with different dimensions.Remark 1.
Assumption1 ensures that the matrices Ai+BiKi and A0+B0K0 possess some common eigenvalues. Thus, Assumption 1 implies that the agent systems as given in (1) and (2) contain certain similar inner dynamical behavior, and these agent systems are named as similar structure agents with similar parameters.Remark 2.
From a mathematical point of view, Assumption1 admits that the state dimensions can be different or identical in multiagent systems. Especially, if Ai=A0 and ni=n0 in (1) and (2), then the agent system (1) and (2) coincides with the system in [26–32].
In order to address the unknown nonlinear function, the following fuzzy logic system (FLS) is utilized in this paper. It mainly includes four parts: fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzifier is a mapping from the input space and state space to the fuzzy sets. The fuzzy rule base contains several linguistics rules. The fuzzy rules are represented as follows:(4)pth:Ifx1isF1p,x2isF2p,…,xnisFnp,thenyisGp,p=1,2,…,h,where x=x1,x2,…,xnΤ and y are the input and output of the FLS, respectively. μFipxi and μGpy are the membership functions of fuzzy sets Fip and Gp, respectively. By employing singleton fuzzifier, center average defuzzifier, and product inference, the output of FLS can be expressed as(5)yx=∑p=1hθp∏i=1nμFipxi∑p=1h∏i=1nμFipxi,where θp=argmaxy∈RμGpy. If we denote θΤ=θ1,θ2,…,θh and ψx=ψ1x,ψ2x,…,ψnxΤ, then the fuzzy logical systems can be rewritten as(6)yx=θΤψx.Lemma 2.
(see [39]). For any given two vectors x,y∈Rn and a scalar a>0, the following inequality holds:(7)2xΤy≤axΤx+a−1yΤy.Lemma 3.
(see [11]). For any given uncertain continuous function gx on a compact set Ω and an arbitrary approximation accuracy ε>0, there exists a FLS such as (4) such that the following universal approximation holds:(8)supx∈ΩθΤψx−gx≤ε.According to Lemma3, we know that the unknown nonlinear function gixi in the ith follower system can be approximated by(9)gixi=θiΤψixi+εit,where θit=θi1t,θi2t,…,θimitΤ is unknown parameter vector that will be designed by adaptive laws and ψixi=ψi1,ψi2,…,ψimiΤ is the fuzzy basis function as shown in (6). In this paper, approximation accuracy εit is a time-varying function and satisfies εit≤ε¯i for all t≥t0, where ε¯i is a known constant.Control Purpose. The aim of this paper is to design a distributed fuzzy adaptive control by using similar parameter such that the consensus errors are UUB.


## 2.1. Graph Theory

A directed digraphG=V,E is utilized to describe the information exchange among each agent, where V=v1v2⋯vN stands for the nonempty set of nodes for each agent. The edge set E contains an edge vj,vi which means node vj is able to transfer the relative state information to node vi; then, nodes i and j are called as the neighbors when the edge vj,vi exists. Let Ni denote the set of neighbors of node ii=1,2,…,N. The directed digraph G can also be described by an adjacency matrix αij, where αij=1,vj,vi∈E0,otherwise. In addition, it is assumed that there are no repeated edges and no self-loops, i.e., αii=0. The Laplacian matrix L=lij∈RN×N is defined as lij=−αij,i≠j∑k≠i,k=1Nαik,i=j.A digraph is said to have a spanning tree, if there exists a node that is called as the root such that the node has directed paths to all other nodes in the graph. The graphG¯ consists of G, node 0 (the leader), and the directed edges from the node 0 to the followers in G, and only a small percentage of the followers can receive the information from the leader. Then, we get the following lemma.Lemma 1.
(see [8]). Let the matrix L¯=L+diagα10α20⋯αN0 is positive definite with αi0>0, the ith agent has access to the leader’s state information, whereas αi0=0 if otherwise.


## 2.2. Preliminaries and Multiagent System

Consider a group of agent system with a leader andN followers labeled as 0 and 1,2,…,N, respectively. The dynamics of the leader is described as(1)x˙0t=A0x0t+B0u0t+sx0,t,where A0t∈Rn0×n0 and B0∈Rn0×m0 are the system matrix and input matrix of leader, respectively. x0t∈Rn0×1 is the state vector of leader. u0t denotes the input vector of leader, and matrix K0∈Rm0×n0 will be given in the process of control design, which will make A0+B0K0 be Hurwitz stabilized. sx0,t is defined as an input bounded signal and satisfies sx0,t≤s¯ for all t≥t0, and s¯ is a known constant.The dynamics of the followers are defined as follows:(2)x˙it=Aixit+Biuit+gixi,i=1,2,…,N,where Ai∈Rni×ni and Bi∈Rni×mi denote the system matrix and input matrix in the ith followers system, respectively. xit∈Rni×1 and uit∈Rmi×ni are corresponding to the state vector and input vector of the ith follower, respectively. gixi represents the unknown nonlinear function.Assumption 1.
(see [38]). Consider N agent systems as given in (2), and the follower system (2) is called similar to the leader system (1), if there exists N matrices Ki∈Rmi×ni, matrix K0∈Rm0×n0, and N matrices Ti∈Rn0×ni satisfying the following condition:(3)TiAi+BiKi=A0+B0K0Ti,TiBi=B0.Definition 1.
In Assumption1, Ti and Ki and K0 are called as similar parameters with different dimensions.Remark 1.
Assumption1 ensures that the matrices Ai+BiKi and A0+B0K0 possess some common eigenvalues. Thus, Assumption 1 implies that the agent systems as given in (1) and (2) contain certain similar inner dynamical behavior, and these agent systems are named as similar structure agents with similar parameters.Remark 2.
From a mathematical point of view, Assumption1 admits that the state dimensions can be different or identical in multiagent systems. Especially, if Ai=A0 and ni=n0 in (1) and (2), then the agent system (1) and (2) coincides with the system in [26–32].
In order to address the unknown nonlinear function, the following fuzzy logic system (FLS) is utilized in this paper. It mainly includes four parts: fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzifier is a mapping from the input space and state space to the fuzzy sets. The fuzzy rule base contains several linguistics rules. The fuzzy rules are represented as follows:(4)pth:Ifx1isF1p,x2isF2p,…,xnisFnp,thenyisGp,p=1,2,…,h,where x=x1,x2,…,xnΤ and y are the input and output of the FLS, respectively. μFipxi and μGpy are the membership functions of fuzzy sets Fip and Gp, respectively. By employing singleton fuzzifier, center average defuzzifier, and product inference, the output of FLS can be expressed as(5)yx=∑p=1hθp∏i=1nμFipxi∑p=1h∏i=1nμFipxi,where θp=argmaxy∈RμGpy. If we denote θΤ=θ1,θ2,…,θh and ψx=ψ1x,ψ2x,…,ψnxΤ, then the fuzzy logical systems can be rewritten as(6)yx=θΤψx.Lemma 2.
(see [39]). For any given two vectors x,y∈Rn and a scalar a>0, the following inequality holds:(7)2xΤy≤axΤx+a−1yΤy.Lemma 3.
(see [11]). For any given uncertain continuous function gx on a compact set Ω and an arbitrary approximation accuracy ε>0, there exists a FLS such as (4) such that the following universal approximation holds:(8)supx∈ΩθΤψx−gx≤ε.According to Lemma3, we know that the unknown nonlinear function gixi in the ith follower system can be approximated by(9)gixi=θiΤψixi+εit,where θit=θi1t,θi2t,…,θimitΤ is unknown parameter vector that will be designed by adaptive laws and ψixi=ψi1,ψi2,…,ψimiΤ is the fuzzy basis function as shown in (6). In this paper, approximation accuracy εit is a time-varying function and satisfies εit≤ε¯i for all t≥t0, where ε¯i is a known constant.Control Purpose. The aim of this paper is to design a distributed fuzzy adaptive control by using similar parameter such that the consensus errors are UUB.


## 3. Main Results

To solve the consensus problem of the leader-follower system (1) and (2), the following control strategy is proposed:(10)uit=cF∑j=1nαijTixit−Tjxjt+Kixit+K¯Tixit−T0x0t−θ¯iΤtψixi,where F=−0.5B0ΤP and the control gain matrix K¯=YX−1 can be proposed by solving the following LMI:(11)ΔXT∗−Q≤0,where Δ=A0X+XA0T+B¯0X+XB¯0T+B0Y+YTB0T−cλB0B0T, X>0, Q>0, and parameter 0<λ<λminL¯.In control (10), parameters θ¯it denote the estimation values of θit and θ˜it are their errors, and the relation between them is defined as θ˜it=θ¯it−θit. The estimation θ¯it can be designed as(12)θ¯˙it=−κθiθ¯it+ρθiψixiPB0Τeit,where κθi=κθi1κθi2⋯κθimi is a vector consisting of some known positive constants given by designer, θit=θi1Tt,θi2Tt,…,θimiTtT, ρθi>0.Theorem 1.
Suppose that Assumption1 is satisfied, and at least one agent system in connected graph G has access to the state information of the leader system (1). With the action of control (10), the consensus error between leader system (1) and follower system (2) is UUB and belongs to the following set:(13)D:=et=e1Τte2Τt⋯eNΤtΤ:et≤σλminIN⊗Pγ,where σ and γ will be given later.Proof.
Let consensus error aseit=Tixit−T0x0t and T0=In0×n0 is an identity matrix. By applying Assumption 1, it becomes(14)Tix˙it=A0+B0K0Tixit−B0θ¯iΤtψixi+B0gix+c∑j=1nαijB0FTixit−Tjxjt+B0K¯eit.
The leader system is transformed as(15)T0x˙0t=A0+B0K0T0x0t+cαi0B0FTixit−T0x0t+B0sx0,t,and then, the error system can be transformed as(16)e˙it=A0+B0K0+B0K¯eit+B0−θ˜iΤtψixi+εit+sx0,t+cB0F∑j=1nαijeit−ejt−αi0Tixit−T0x0t.
For brevity, (16) is equal to(17)e˙t=IN⊗A0+B0K0+B0K¯−cL¯⊗B0Fet+IN⊗B0−θ˜Τtψx+εt+sx0,t,where et=e1Tt,e2Tt,…,eNTtT, θ˜t=θ˜1Tt,θ˜2Tt,…,θ˜NTtT, ψx=ψ1Tx1,ψ2Tx2,…,ψNTxNT, εt=ε1Tt,ε2Tt,…,εNTtT, and −L¯et=∑j=1nαijeit−ejt−αi0Tixit−T0x0t
The following candidate Lyapunov function is considered:(18)Vt=12eΤtIN⊗Pet+12θ˜Τtρθ−1θ˜t.
The derivative ofVt along system (17) is(19)V˙t=12eΤtIN⊗PA0+B0K0+B0K¯+A0+B0K0+B0K¯ΤP−cL¯⊗PB0B0ΤPet+θ˜Τtρθ−1θ¯˙t+eΤtIN⊗PB0−θ˜Τtψx+εt+sx0,t≤12eΤtIN⊗PA0+B0K0+B0K¯+A0+B0K0+B0K¯ΤP−cλPB0B0ΤPet+eΤtIN⊗PB0εt+s¯−κθρθθ˜Τtθ¯t=−12eΤtIN⊗Qet−κθρθθ˜Τtθ¯t+eΤtIN⊗PB0εt+s¯.
Based on Lemma3, one obtains that(20)−θ˜Τtθ¯t=−θ˜Τtθ˜t+θt≤−12θ˜Τtθ˜t+12θΤtθt.
Combining with (19) and (20), it becomes(21)V˙t≤−12eΤtIN⊗Qet−12κθρθθ˜Τtθ˜t+12κθρθθΤtθt+eΤtIN⊗PB0εt+s¯≤−12λminIN⊗QeΤtet−12κθρθθ˜Τtθ˜t+et⋅IN⊗PB0⋅ε¯+s¯+12κθρθθΤtθt≤−γVt+σ.
If denotingγ=minλminIN⊗Q/λmaxIN⊗P,κθ, σ=et⋅IN⊗PB0⋅ε¯+s¯+1/2κθ/ρθθΤtθt, then it follows (22)Vt≤V0−σγe−γt+σγ.
Inequality (22) shows that the consensus error et can be guaranteed to be UUB with(23)et≤1λminIN⊗PV0−σγe−γt+σγ.
Accordingly, the conclusion is(24)limt⟶∞et≤σλminIN⊗Pγ.(24) means that the consensus error et converges to the set D, which is defined in (13). This completes the proof.Remark 3.
The inequalityPA0+B0K0+B0K¯+A0+B0K0+B0K¯ΤP−cλPB0B0TP≤−Q is a nonlinear matrix inequality; through multiplying by P−1 on the left and right sides of this inequality and defining B0K0=B¯0, P−1=X, and Y=K¯X, the inequality (11) can be obtained.
The multiagent system with similar composite structure and proposed control scheme is explained as the block diagram in Figure1Figure 1
The block diagram of closed-loop multiagent systems.


## 4. Simulation Example

In this section, a simulation example is given to prove the effectiveness of the proposed control. Six agent systems are considered including one leader labeled0 and five followers labeled 1,2,3,4,and5. Figure 2 shows the communication between the leader and each follower, it is easy to know that only the first agent can obtain the state information of the leader.Figure 2
The communication topology.From Figure2, the Laplacian matrix of the follower system and the degree matrix of the leader system can be calculated as follows:(25)L=2−1−100−120−10−103−1−10−1−12000−101,diagα10α20⋯α50=1000000000000000000000000.Matrices in the leader-follower system are represented by(26)A0=1−7−3−1−12−2−452−42−7−89−100−100000000−200000000−300000000−600000000−800000000−9,A1=1−7−3−1−12−2273−2−8−10200−10000000−20000000−30000000−60000000−8,B0=01O1×6,A2=1−7−3−1−1237−2−1−6−900−1000000−2000000−3000000−6,B1=01O1×5,A3=1−7−3−1−153−52300−100000−200000−3,A4=1−7−3−1132700−10000−2,A5=1−7−352100−1,B2=01O1×4Τ,B3=01O1×3Τ,B4=01O1×2Τ,B5=010Τ.By using the similar condition in Assumption1, the similar parameters can be obtained as(27)K¯=−0.5909−0.0964−12.7679−0.9654−0.0483−0.2017−0.3305−0.2047,P=0.4057−0.1665−0.1051−0.0329−0.03240.0673−0.0684−0.1369−0.16650.45040.16320.05450.0553−0.11800.12250.2487−0.10510.16320.38940.03880.0379−0.07260.07080.1398−0.03290.05450.03880.32940.0120−0.02330.02290.0453−0.03240.05530.03790.01200.4195−0.02290.02260.04490.0673−0.1180−0.0726−0.0233−0.02290.8276−0.0457−0.0914−0.06840.12250.07080.02290.0226−0.04571.11400.0923−0.13690.24870.13980.04530.0449−0.09140.09231.4012,T0=I8,T1=I7O7×1Τ,T2=I6O6×2Τ,T3=I5O5×3Τ,T4=I4O4×4Τ,T5=I3O3×5Τ,K0=3−614−2611−12−6,K1=6−1172713−5,K2=5−11121512,K3=3−715−2−4,K4=7−78−7,K5=3−69.The nonlinear functions are chosen as(28)fi=xi12sinxi2cosxi3−0.5xi23sinxi2−2xi34cosxi3.The solutions of the linear matrix inequality (11) are shown as matrices K¯ and P.The input bounded signal can be chosen as(29)sx0,t=60,0<t≤1,0,1<t≤2,60,2<t≤3,0,3<t≤4,60,4<t≤5.The initial values of the states in the leader and five followers are chosen as(30)x00=3.1,3.5,4.8,4.7,2.3,2.5,0.5,4T,x10=−4.8,−2,−0.8,−3,−2,−3.7,−1T,x20=3,2,1,2.4,3.6,4.1T,x30=−3,−2,−1,−2.2,−0.8T,x40=2,3.3,4.5,1.6T,x50=−1,−2.5,−3.8T.The initial values of adaptive parametersθ¯it are given as(31)θ¯10=0.13,0.22,0.10,0.23,0.18,0.39T,θ¯20=0.24,0.26,0.11,0.14,0.15,0.30T,θ¯30=0.16,0.29,0.17,0.21,0.35,0.27T,θ¯40=0.20,0.21,0.11,0.31,0.29,0.13T,θ¯50=0.34,0.29,0.38,0.26,0.35,0.17T.The parameters in the adaptive law (12) are chosen as(32)κθi=95786,ρθi=0.0030.0020.0020.0030.001.The simulation results of the leader system and follower systems are shown as Figures3–5.Figure 3
Trajectories of the states of leaderx0 and followers xi (i=1,2,3,4,5; h=1,2,3,4; k=1,2,3; d=1,2).
(a)(b)(c)(d)(e)(f)(g)(h)Figure 4
Trajectories of the adaptive estimation parametersθ¯it.Figure 5
Trajectories of the controlu0t and uit.As shown in Figure3, although the dimensions of leader system and follower systems are nonidentical, the trajectories of xi in follower systems can synchronize to the state of x0 in leader system with the proposed distributed fuzzy adaptive control, and it can reach a consistent state in a relatively fast time. Similarly, the norm of adaptive estimated parameters is converged to a small zero field in Figure 4, which can be updated online automatically with the given adaptive laws. From Figure 5, it is shown that the time responses of corresponding control are UUB. Finally, it is concluded that UUB of all signals in the closed-loop system can be guaranteed in Figures 3–5, and the consensus of leader-follower system can be realized by the proposed distribute fuzzy adaptive control with similar parameters whether the leader system and follower systems have the identical or nonidentical dimensions.


## 5. Conclusion

The consensus problem of leader-follower multiagent systems with different dimensions has been considered in this paper. For the unknown nonlinear functions in systems, FLSs are applied to approximate the unknown nonlinear functions, and then a distributed fuzzy adaptive control based on similar condition is designed. With the proposed fuzzy adaptive control, the states of each follower system can stably track the states of the leader system, and it is proved that all signals in the closed-loop system are UUB. The designed method has been verified by a simulation example.



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