# Adaptive Fixed-Time Tracking Consensus Control for Multiagent Nonlinear Pure-Feedback Systems with Performance Constraints

Authors: Pinwei Li; Jiyang Dai; Jin Ying
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper investigates adaptive fixed-time tracking consensus control problems for multiagent nonlinear pure-feedback systems with performance constraints. Compared with existing results of first/second/high-order multiple agent systems, the studied systems have more complex nonlinear dynamics with each agent being modeled as a high-order pure-feedback form. The mean value theorem is introduced to address the problem of nonaffine structure in nonlinear pure-feedback systems. Meanwhile, radial basis function neural networks (RBFNNs) are employed to approximate unknown functions. Furthermore, a constraint variable is used to guarantee that all local tracking errors are within the prescribed boundaries. It is shown that, by utilizing the proposed consensus control protocol, each tracking consensus error can converge into a neighborhood around zero within designed fixed time, the tracking consensus performance can be ensured during the whole process, and all signals in the investigated systems are bounded. Finally, two simulations are performed and the results demonstrate the effectiveness of the proposed control strategy.


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## Body

## 1. Introduction

In the past decade, the finite-time consensus control has attracted considerable attention due to its unique advantages in faster convergence rate and higher precision [1] and its wide applications in unmanned aerial vehicles, autonomous underwater vehicles, surface vessels, and so forth [2–5].Finite-time control was first proposed in 1965, and many related research results emerged in the subsequent decades. A variety of finite-time control protocols [6–11] have developed to address the different agents’ dynamics in multiagent systems (MASs). For example, in [12, 13], finite-time consensus control for first- and second-order MASs was proposed, respectively, by using the principle of homogeneity [14]. In [15], a finite-time control for MASs with double integrator dynamics was proposed by utilizing power integrator technique. Furthermore, two classes of finite-time control protocols for MASs composed of first-order and second-order integrator agents were developed by combining the homogeneous domination method with adding a power integrator method. Thereafter, to solve the consensus control problems of high-order uncertain MASs, several finite-time control strategies were proposed in [16–18]. Even so, there is a limitation for the finite-time control results that the convergence time is strongly dependent on the initial states of the systems; in other words, once the initial states are far away from the equilibrium point, the convergence time will increase as a result. As a matter of fact, it is much more reasonable for a predicable convergence time.In 2012, to address the limitation of finite-time control, Polyakov [19] proposed fixed-time stability control, which can allow the convergence time of systems have nothing to do with its initial states. Subsequently, many related research results [20–29] for MASs have been carried out. It can be found that most of the existing fixed-time consensus control results are focused on MASs with first-order or second-order, and only a few focus on high-order MASs. Although, in [24–27], the fixed-time control strategies have considered high-order MASs; they are not general enough, since their results apply only to strict-feedback nonlinear systems. However, compared with lower-triangular systems and strict-feedback nonlinear systems, the pure-feedback nonlinear systems [30–32] are more general systems. On the other hand, in many practical industrial systems, there are some requirements for performance constraint. For example, unknown dead-zone inputs [33], output-constrained [34], preassigned transient performance [35], and other measures are taken to prevent system damage. In [36], the authors investigated the adaptive tracking control problem for a class of nonlinear time-delay systems in the presence of input and tracking error constraints.Motivated by the above observations, in this paper, we focus on addressing the problem of adaptive fixed-time tracking consensus control for a class of pure-feedback uncertain nonlinear MASs with performance constraint. In the design, to solve the problem of nonaffine structure in nonlinear pure-feedback systems, we employ mean value theorem. Meanwhile, RBF neural networks are used to compensate for uncertain nonlinear terms and some functions that are difficult to calculate induced from the controller design procedure. Furthermore, a predefined boundary function is utilized to cope up with the requirement of the prescribed performance. Finally, based on Lyapunov stability theorem, fixed-time control theory, and graph theory, a novel fixed-time consensus control strategy is designed for the considered pure-feedback nonlinear MASs, which can guarantee that all the local tracking errors are within the range of the predefined boundary function and can converge to the neighborhood of the origin in fixed time and all signals in MASs are bounded.The main contributions of this paper can be summarized as follows:(1)
To our best knowledge, this paper is the first attempt to solve the problem of adaptive fixed-time consensus control for MASs with nonlinear pure-feedback. Compared with some existing results [16–18, 24–27] on high-order MASs with nonlinear strict-feedback, the investigated pure-feedback nonlinear MASs are more general.(2)
By introducing RBF neural networks, the uncertain nonlinear iterms in each agent’s dynamics can be approximated infinitely. Therefore, the designed controller in this paper is more versatile and robust.(3)
Based on backstepping algorithm and Lyapunov theorem, a novel fixed-time consensus control protocol design method is proposed. Meanwhile, the tracking performance is governed by the predefined boundary function.The rest of this paper is arranged as follows. Section2 gives problem description and preliminaries. Section 3 presents the detailed design process of fixed-time controller. In Section 4, the stability analysis will be given. In Section 5, the validity of the proposed control scheme is proven by two simulations. Finally, the conclusion of this work is summarized in Section 6.


## 2. Problem Description and Preliminaries

### 2.1. Problem Statement

The considered nonlinear MASs consist ofM followers, labeled as agents 1 to M, and a leader, labeled as an agent 0 under a directed communication graph topology. The dynamic models of M followers can be described as the form of the following pure-feedback nonlinear systems:(1)x˙i,kt=fi,kx¯i,kt,xi,k+1t+di,kx¯i,k,t,x˙i,nit=fi,nix¯i,nit,uit+di,nix¯i,ni,t,yit=xi,1t,where i=1,…,M, k=1,…,ni−1, x¯i,k=xi,1t,…,xi,ktT∈Rk with k=1,…,ni, uit∈R, and yit∈R are system state variables, system input, and system output, respectively; fi,k⋅ are unknown smooth nonaffine functions; di,k⋅ are unknown but bounded disturbances.We utilize the mean value theorem to solve the nonaffine structures in pure-feedback systems:(2)fi,kx¯i,k,xi,k+1=fi,kx¯i,k,xi,k0+gi,k⋅xi,k+1−xi,k0,fi,nix¯i,ni,ui=fi,nix¯i,ni,xi,ni0+gi,ni⋅ui−xi,ni0,where gi,k≔gi,kx¯i,k,xi,μk=∂fi,kx¯i,k,xi,μk/∂xi,μk with k=1,2,…,ni, xi,ni+1=ui, xi,μk=μkxi,k+1+1−μkxi,k0 with 0<μk<1, and xi,k0 are known at the given time t0.Therefore, system (1) can be rewritten as(3)x˙i,k=fi,kx¯i,kt,xi,k0+gi,k⋅xi,k+1−xi,k0+di,kx¯i,k,t,x˙i,ni=fi,nix¯i,nit,xi,ni,0+gi,ni⋅ui−xi,ni,0+di,nix¯i,ni,t,yi=xi,1.Assumption 1 (see [37]).
gi,k⋅ are unknown nonlinear functions and bounded, and there exists a positive constant bi such that bi≤gi,k⋅.
For ease of description, we defined vector functionsy¯0,i=y0,y01,…,y0iT, where y0 is the leader’s output and y0i is its ith derivative.Assumption 2.
Suppose that the leader’s vector functionsy¯0,i are known smooth continuous functions and bounded. y¯0,i∈Ω0,i⊂Ri+1 with i=1,…,ni, where Ω0,i being known compact sets and y0 is an ni-order differentiable function.


### 2.2. Graph Theory

The communication between agents can be represented by a directed graphG=V,E,A, where V=N1,N2,…,NM denotes the set of nodes, and E⊆V×V denotes the set of edges. An edge j,i∈E means that agent i can obtain information from agent j, but not vice versa, where j and i are the parent node and child node, respectively. Ni=j|j,i∈E is the set of neighbors of a node i, which is the set of nodes with edges incoming to node i. A=ai,j∈ℝM×M is the related adjacency matrix, if j,i∈E, aij>0; otherwise, ai,j=0. Self-edges are not allowed, i.e., ai,j=0. The Laplacian matrix for graph G is denoted by ℒ=li,jN×N with li,i=∑j=1,j≠iNai,j and li,j=−ai,j for i≠j, and ℒ=D−A∈ℝM×M where D=diagd1,…,dM; di=∑j=1,j≠iMaij is the diagonal element of the degree matrix D. A directed graph G has a directed spanning tree if there exists at least one agent that has directed paths to all other agents.If including a leader agentN0, the communication topology for the M+1 agents is described by a directed graph G¯=V¯,E¯,A¯, where V¯=V∪N0 and E¯=V¯×V¯. The connection matrix between the followers and the leader can be expressed as H=diagh1,…,hN where hi>0 if and only if Ni can directly obtain the leader’s information; otherwise, hi=0.


### 2.3. RBF Neural Networks

RBF neural network [38] has powerful unknown function approximation ability. In this paper, we approximate uncertain nonlinear iterms in each agent’s dynamics via the following RBF neural network:(4)f^Z=WTSZ,where the input vector Z∈ΩZ⊂Rq, W=w1,w2,…,wlT∈Rl is the weight vector; SZ=s1Z,s2Z,…,slZT∈Rl is the radial basis function vector with neural network node number l>0. The basis function SZ is always chosen as the following form of Gaussian function:(5)SiZ=exp−Z−μiTZ−μiηi2,i=1,2,…,l,where ηi is the width of the Gaussian function, and μi=μi1,μi2,…,μiqT is the center of the receptive field. In theory, as long as a sufficient number l of nodes are selected, RBF neural network can approximate any continuous function fZ in a compact set ΩZ⊂Rq with arbitrary precision ε.(6)fZ=W∗TSZ+δZ,∀Z∈ΩZ∈Rq,where δZ is the approximation error and satisfies δZ≤ε, and W∗ is the ideal weight vector, which is chosen as the value of W that minimizes ε for all Z∈ΩZ, i.e.,(7)W∗=argminW∈R′supZ∈ΩZfZ−f^Z.Remark 1.
In this paper, we chooseθi,j=maxWi,j∗/bi,i=1,2,…,M,j=1,2,…,ni with θ˜i,j=θi,j−θ^i,j, where θ^i,j are the estimates of the unknown constants θi,j, bi are positive design parameters and related to Assumption 1, and ⋅ is the norm.Assumption 3 (see [39]).
There exist unknown constantsQi,j such that θ˜i,j≤Qi,j<∞, i=1,2,…,M, j=1,2,…,ni.Lemma 1 (see [40]).
Consider the Gaussian function (5). SZ has an upper bound such that SZ≤∑k=0∞3qk+2q−1e−2p2kk2/η2≔s, where p=1/2mini≠jμi−μj.


### 2.4. Fixed-Time Control

Consider the following nonlinear system,(8)x˙t=fxt,x0=x0,f0=0,where xt∈Rn is a system variable, and fx:Rn⟶Rn is a nonlinear function which may be discontinuous; the solutions of system (8) are understood in the sense of Filippov [41]. For system (8), the following definitions and lemmas are introduced.Definition 1 (see [1]).
The equilibriumx=0 of system (8) is finite-time convergent if there are an open neighborhood U⊆D of the origin and a positive definite function Tx0:U\0⟶0,∞, such that limt⟶Tx0xt,x0=0 and xt,x0=0, ∀t>Tx0 for any initial condition x0∈U, where Tx0 is the settling time function. The equilibrium x=0 of system (8) is finite-time stable if it is Lyapunov stable and finite-time convergent. Moreover, when U=D=Rn, then the equilibrium x=0 is globally finite-time stable.Definition 2 (see [19]).
The equilibrium of system (8) is said to be fixed-time stable if it is globally finite-time stable and the settling time function Tx0 is uniformly bounded for any initial states, that is, ∃Tmax such that Tx0≤Tmax, ∀x0∈Rn.Lemma 2 (see [39]).
Considering system (8), suppose that there exists a continuous differentiable positive definite function Vx:Rn⟶R such that the following inequality holds:(9)V˙x≤−ϕ1Vαx−ϕ2Vβx+τ,where ϕ1>0,ϕ2>0, α∈0,1, β∈1,∞, τ∈0,∞, and ϖ∈0,1. Then, the equilibrium of system (8) is fixed-time stable and the convergence time function T satisfies the following inequality:(10)T≤Tmax≔1ϕ1ϖ1−α+1ϕ2ϖβ−1.The residual set of the solution of system (8) is given by(11)x∈Vx≤minτ1−ϖϕ11/α,τ1−ϖϕ21/β.Lemma 3 (see [42]).
Forxi∈R, i=1,2,…,n, 0<r1≤1, r2>1, the following inequalities hold:(12)∑i=1nxir1≤∑i=1nxir1,∑i=1nxir2≤nr2−1∑i=1nxir2.Lemma 4 (see [40]) (see [43]).
Forx∈R and any positive constant ξ, the following inequality holds:(13)0≤x<ξ+x2x2+ξ2.Lemma 5.
Fora≥b>0,b,a∈R and any positive constant c, then the following inequality is satisfied:(14)aa+c≥bb+c.Proof.
(15)aa+c−bb+c=a+ca+c−b+cb+c−ca+c+cb+c=a+c−b+c+cb+c−ca+c≥0.


### 2.5. Tracking Error Constraint

In this paper, we will investigate the output feedback tracking control problem with prescribed performance. The local tracking error for theith follower is defined as(16)ei=∑j=1Maijyi−yj+hiyi−y0.To meet the requirement of output performance constraint, a positive decreasing smooth function as the following is selected:(17)Fb=ξ0e−at+ξ∞,where ξ0>0, ξ∞>0 are proper parameters, a is a positive constant, and ei0<Fb0.Defining the following state transformation as the error constraint variable of theith follower,(18)zi,1t=eitσ1Fb−eiqeit+eitσ2Fb+ei1−qeit,where σ1 and σ2 are proper design parameters. Fb satisfies the condition given in (17) and qeit satisfies(19)qeit=1,eit≥0,0,eit<0.The time derivative ofzi,1 is(20)z˙i,1=1σ1Fb−ei2e˙iσ1Fb−ei−eiσ1F˙b−e˙iq+1σ2Fb+ei2e˙iσ2Fb+ei−eiσ2F˙b+e˙i⋅1−q.For ease of writing, we obtain(21)z˙i,1=FbΦFie˙i−F˙bΦFiei,where(22)ΦFi=σ1qσ1Fb−ei2+σ21−qσ2Fb+ei2.Remark 2.
From the definition ofzi,1 and qeit, it is obvious that when tracking error approaches the boundary Fb, zi,1 increases as a result. Then, by choosing proper parameters, the tracking error will be forced within prescribed boundary, i.e., −σ2Fbt<eit<σ1Fbt.


## 2.1. Problem Statement

The considered nonlinear MASs consist ofM followers, labeled as agents 1 to M, and a leader, labeled as an agent 0 under a directed communication graph topology. The dynamic models of M followers can be described as the form of the following pure-feedback nonlinear systems:(1)x˙i,kt=fi,kx¯i,kt,xi,k+1t+di,kx¯i,k,t,x˙i,nit=fi,nix¯i,nit,uit+di,nix¯i,ni,t,yit=xi,1t,where i=1,…,M, k=1,…,ni−1, x¯i,k=xi,1t,…,xi,ktT∈Rk with k=1,…,ni, uit∈R, and yit∈R are system state variables, system input, and system output, respectively; fi,k⋅ are unknown smooth nonaffine functions; di,k⋅ are unknown but bounded disturbances.We utilize the mean value theorem to solve the nonaffine structures in pure-feedback systems:(2)fi,kx¯i,k,xi,k+1=fi,kx¯i,k,xi,k0+gi,k⋅xi,k+1−xi,k0,fi,nix¯i,ni,ui=fi,nix¯i,ni,xi,ni0+gi,ni⋅ui−xi,ni0,where gi,k≔gi,kx¯i,k,xi,μk=∂fi,kx¯i,k,xi,μk/∂xi,μk with k=1,2,…,ni, xi,ni+1=ui, xi,μk=μkxi,k+1+1−μkxi,k0 with 0<μk<1, and xi,k0 are known at the given time t0.Therefore, system (1) can be rewritten as(3)x˙i,k=fi,kx¯i,kt,xi,k0+gi,k⋅xi,k+1−xi,k0+di,kx¯i,k,t,x˙i,ni=fi,nix¯i,nit,xi,ni,0+gi,ni⋅ui−xi,ni,0+di,nix¯i,ni,t,yi=xi,1.Assumption 1 (see [37]).
gi,k⋅ are unknown nonlinear functions and bounded, and there exists a positive constant bi such that bi≤gi,k⋅.
For ease of description, we defined vector functionsy¯0,i=y0,y01,…,y0iT, where y0 is the leader’s output and y0i is its ith derivative.Assumption 2.
Suppose that the leader’s vector functionsy¯0,i are known smooth continuous functions and bounded. y¯0,i∈Ω0,i⊂Ri+1 with i=1,…,ni, where Ω0,i being known compact sets and y0 is an ni-order differentiable function.


## 2.2. Graph Theory

The communication between agents can be represented by a directed graphG=V,E,A, where V=N1,N2,…,NM denotes the set of nodes, and E⊆V×V denotes the set of edges. An edge j,i∈E means that agent i can obtain information from agent j, but not vice versa, where j and i are the parent node and child node, respectively. Ni=j|j,i∈E is the set of neighbors of a node i, which is the set of nodes with edges incoming to node i. A=ai,j∈ℝM×M is the related adjacency matrix, if j,i∈E, aij>0; otherwise, ai,j=0. Self-edges are not allowed, i.e., ai,j=0. The Laplacian matrix for graph G is denoted by ℒ=li,jN×N with li,i=∑j=1,j≠iNai,j and li,j=−ai,j for i≠j, and ℒ=D−A∈ℝM×M where D=diagd1,…,dM; di=∑j=1,j≠iMaij is the diagonal element of the degree matrix D. A directed graph G has a directed spanning tree if there exists at least one agent that has directed paths to all other agents.If including a leader agentN0, the communication topology for the M+1 agents is described by a directed graph G¯=V¯,E¯,A¯, where V¯=V∪N0 and E¯=V¯×V¯. The connection matrix between the followers and the leader can be expressed as H=diagh1,…,hN where hi>0 if and only if Ni can directly obtain the leader’s information; otherwise, hi=0.


## 2.3. RBF Neural Networks

RBF neural network [38] has powerful unknown function approximation ability. In this paper, we approximate uncertain nonlinear iterms in each agent’s dynamics via the following RBF neural network:(4)f^Z=WTSZ,where the input vector Z∈ΩZ⊂Rq, W=w1,w2,…,wlT∈Rl is the weight vector; SZ=s1Z,s2Z,…,slZT∈Rl is the radial basis function vector with neural network node number l>0. The basis function SZ is always chosen as the following form of Gaussian function:(5)SiZ=exp−Z−μiTZ−μiηi2,i=1,2,…,l,where ηi is the width of the Gaussian function, and μi=μi1,μi2,…,μiqT is the center of the receptive field. In theory, as long as a sufficient number l of nodes are selected, RBF neural network can approximate any continuous function fZ in a compact set ΩZ⊂Rq with arbitrary precision ε.(6)fZ=W∗TSZ+δZ,∀Z∈ΩZ∈Rq,where δZ is the approximation error and satisfies δZ≤ε, and W∗ is the ideal weight vector, which is chosen as the value of W that minimizes ε for all Z∈ΩZ, i.e.,(7)W∗=argminW∈R′supZ∈ΩZfZ−f^Z.Remark 1.
In this paper, we chooseθi,j=maxWi,j∗/bi,i=1,2,…,M,j=1,2,…,ni with θ˜i,j=θi,j−θ^i,j, where θ^i,j are the estimates of the unknown constants θi,j, bi are positive design parameters and related to Assumption 1, and ⋅ is the norm.Assumption 3 (see [39]).
There exist unknown constantsQi,j such that θ˜i,j≤Qi,j<∞, i=1,2,…,M, j=1,2,…,ni.Lemma 1 (see [40]).
Consider the Gaussian function (5). SZ has an upper bound such that SZ≤∑k=0∞3qk+2q−1e−2p2kk2/η2≔s, where p=1/2mini≠jμi−μj.


## 2.4. Fixed-Time Control

Consider the following nonlinear system,(8)x˙t=fxt,x0=x0,f0=0,where xt∈Rn is a system variable, and fx:Rn⟶Rn is a nonlinear function which may be discontinuous; the solutions of system (8) are understood in the sense of Filippov [41]. For system (8), the following definitions and lemmas are introduced.Definition 1 (see [1]).
The equilibriumx=0 of system (8) is finite-time convergent if there are an open neighborhood U⊆D of the origin and a positive definite function Tx0:U\0⟶0,∞, such that limt⟶Tx0xt,x0=0 and xt,x0=0, ∀t>Tx0 for any initial condition x0∈U, where Tx0 is the settling time function. The equilibrium x=0 of system (8) is finite-time stable if it is Lyapunov stable and finite-time convergent. Moreover, when U=D=Rn, then the equilibrium x=0 is globally finite-time stable.Definition 2 (see [19]).
The equilibrium of system (8) is said to be fixed-time stable if it is globally finite-time stable and the settling time function Tx0 is uniformly bounded for any initial states, that is, ∃Tmax such that Tx0≤Tmax, ∀x0∈Rn.Lemma 2 (see [39]).
Considering system (8), suppose that there exists a continuous differentiable positive definite function Vx:Rn⟶R such that the following inequality holds:(9)V˙x≤−ϕ1Vαx−ϕ2Vβx+τ,where ϕ1>0,ϕ2>0, α∈0,1, β∈1,∞, τ∈0,∞, and ϖ∈0,1. Then, the equilibrium of system (8) is fixed-time stable and the convergence time function T satisfies the following inequality:(10)T≤Tmax≔1ϕ1ϖ1−α+1ϕ2ϖβ−1.The residual set of the solution of system (8) is given by(11)x∈Vx≤minτ1−ϖϕ11/α,τ1−ϖϕ21/β.Lemma 3 (see [42]).
Forxi∈R, i=1,2,…,n, 0<r1≤1, r2>1, the following inequalities hold:(12)∑i=1nxir1≤∑i=1nxir1,∑i=1nxir2≤nr2−1∑i=1nxir2.Lemma 4 (see [40]) (see [43]).
Forx∈R and any positive constant ξ, the following inequality holds:(13)0≤x<ξ+x2x2+ξ2.Lemma 5.
Fora≥b>0,b,a∈R and any positive constant c, then the following inequality is satisfied:(14)aa+c≥bb+c.Proof.
(15)aa+c−bb+c=a+ca+c−b+cb+c−ca+c+cb+c=a+c−b+c+cb+c−ca+c≥0.


## 2.5. Tracking Error Constraint

In this paper, we will investigate the output feedback tracking control problem with prescribed performance. The local tracking error for theith follower is defined as(16)ei=∑j=1Maijyi−yj+hiyi−y0.To meet the requirement of output performance constraint, a positive decreasing smooth function as the following is selected:(17)Fb=ξ0e−at+ξ∞,where ξ0>0, ξ∞>0 are proper parameters, a is a positive constant, and ei0<Fb0.Defining the following state transformation as the error constraint variable of theith follower,(18)zi,1t=eitσ1Fb−eiqeit+eitσ2Fb+ei1−qeit,where σ1 and σ2 are proper design parameters. Fb satisfies the condition given in (17) and qeit satisfies(19)qeit=1,eit≥0,0,eit<0.The time derivative ofzi,1 is(20)z˙i,1=1σ1Fb−ei2e˙iσ1Fb−ei−eiσ1F˙b−e˙iq+1σ2Fb+ei2e˙iσ2Fb+ei−eiσ2F˙b+e˙i⋅1−q.For ease of writing, we obtain(21)z˙i,1=FbΦFie˙i−F˙bΦFiei,where(22)ΦFi=σ1qσ1Fb−ei2+σ21−qσ2Fb+ei2.Remark 2.
From the definition ofzi,1 and qeit, it is obvious that when tracking error approaches the boundary Fb, zi,1 increases as a result. Then, by choosing proper parameters, the tracking error will be forced within prescribed boundary, i.e., −σ2Fbt<eit<σ1Fbt.


## 3. Fixed-Time Controller Design

In this section, an adaptive RBF neural network fixed-time controller is designed for MASs with uncertain nonlinear pure-feedback in the framework of backstepping method. The coordinate transformation is as follows:(23)zi,k=xi,k−vi,k−1,where i=1,2,…,M, k=2,3,…,ni, and vi,k represents the virtual controller of the kth subsystem of the ith follower, and zi,1 is given in (18).In this paper, we employ RBF neural network to approximate uncertain functionsf^i,kZi,k,(24)f^i,kZi,k=Wi,k∗TSi,kZi,k+δi,kZi,k.From Young’s inequality, complete square formula, and Lemma3, the following inequalities hold:(25)zi,kδi,kZi,k≤biζi,k3zi,k2+εi,k24biζi,k3,(26)zi,kWi,k∗TSi,kZi,k≤biθi,k2ηi,k2Si,kTZi,kSi,kZi,kzi,k2+ηi,k22,(27)−biζi,k121+α/212zi,k21+α/2+biθ˜i,k22γi1+α/2≤−biζi,k121+α/212zi,k2+biθ˜i,k22γi1+α/2,(28)−biζi,k221+β/212zi,k21+β/2+biθ˜i,k22γi1+β/2≤−2biζi,k212zi,k2+biθ˜i,k22γi1+β/2,where Zi,k=xi,1,…,xi,k,θ^i,1,…,θ^i,k,hiy¯0,kT,ai1y1,1,…,aiMyM,1,…,ai1y1,k,…,aiMyM,k,Fb,F˙b,…,Fbk with i=1,…,M, k=1,…,ni is the input vector, and bi, ζi,k3, ζi,k2, ζi,k1, εi,k, ηi,k, and γi are positive parameters.Step 1.
Choosing the Lyapunov function as(29)Vi,1=12zi,12+biθ˜i,122γi.
From (3), (16), and (18)–(23), we can obtain the time derivative of Vi,1:(30)V˙i,1=di+hizi,1FbΦFifi,1+gi,1zi,2+gi,1vi,1−gi,1xi,10+di,1−zi,1FbΦFi∑j=1Maijx˙j,1+hiy˙0−zi,1F˙bΦFiei−biθ˜i,1θ^i,1γi.
The virtual controllervi,1 is defined as(31)vi,1=1di+hiFbΦFi−θ^i,12ηi,12Si,1TZi,1Si,1Zi,1zi,1−ζi,11Szi,1,1−ζi,12Szi,1,2−ζi,13zi,1+xi,10,where ζi,11, ζi,12, ζi,13, and ηi,1 are positive design parameters. Szi,1,1 and Szi,1,2 are defined by the following equalities:(32)Szi,k,1=zi,k21+α/2zi,k,zi,k≥εk0,∑j=1niajzi,k2jεk02−j+1+α/2,zi,k<εk0,(33)Szi,k,2=zi,k21+β/2zi,k,zi,k≥εk0,∑j=1niajzi,k2jεk02−j+1+β/2,zi,k<εk0,where k=1,…,ni, εk0 is the positive design parameter, and coefficients aj with j=1,…,ni are calculated using the following equation:(34)11⋯1112⋯ni−1ni02×1⋯ni−1ni−2nini−1⋮⋮⋱⋮⋮00⋯∏j=0ni−2ni−1−j∏j=0ni−2ni−j×a1a2a3⋮ani=b1b2b3⋮bni,where b1=1, b2=3/4, b3=3/43/4−1, …, and bn=∏j=0ni−23/4−j.Remark 3.
The above process is to eliminate the problem of fixed-time controller’s singularity. In this paper, we just consider the situation of|zi,k|≥εk0, since when |zi,k|<εk0, there we just add an additional term and it clearly does not affect the final conclusion.
Substituting (31) into (30) yields(35)V˙i,1≤di+hizi,1FbΦFigi,1zi,2+zi,1f^i,1Zi,1−biθ^i,12ηi,12Si,1TZi,1Si,1Zi,1zi,12−biζi,11zi,1Szi,1,1−biζi,12zi,1Szi,1,2−biζi,13zi,12−biθ˜i,1θ^.i,1γi,where f^i,1Zi,1=di+hiFbΦFifi,1+di,1−FbΦFi∑j=1Maijx˙j,1+hiy˙0−F˙bΦFiei.
From (24)–(26), we have(36)V˙i,1≤biθ˜i,1γiγi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−θ^.i,1+di+hizi,1FbΦFigi,1zi,2+σi,1−biζi,11zi,1Szi,1,1−biζi,12zi,1Szi,1,2,where σi,1=1/2ηi,12+εi,12/4biζi,13.
The adaptive lawθ^.i,1 is defined as(37)θ^.i,1=γi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−λiθ^i,1,where λi with i=1,…,M is the positive parameter.
From Assumption3 and (37), we have(38)V˙i,1≤−biζi,11zi,1Szi,1,1−biζi,12zi,1Szi,1,2−biζi,1121+α/2biθ˜i,122γi1+α/2−biζi,1221+β/2biθ˜i,122γi1+β/2+di+hizi,1FbΦFigi,1zi,2+Ci,1,where Ci,1=σi,1+βi,1+λibi/2γiθi,12, βi,1=ζi,1121+α/2biQi,12/2γi1+α/2+ζi,1221+β/2biQi,12/2γi1+β/2, and λ1biθ˜i,1θ^i,1/γi≤λibi/γiθi,12/2−θ˜i,12/2≤λibi/2γiθi,12.
According to (27), (28), (32), and (33), and when zi,1≥ε10,(39)V˙i,1≤−biζi,1121+α/212zi,12+biθ˜i,122γi1+α/2−2biζi,1212zi,12+biθ˜i,122γi1+β/2+di+hizi,1FbΦFigi,1zi,2+Ci,1.Step 2.
From (3) and (23), we can obtain(40)z˙i,2=x˙i,2−v˙i,1=fi,2x¯i,2,xi,20+gi,2xi,3−xi,20+di,2−v˙i,1,where v˙i,1=∂vi1/∂xi1,1x˙ii,1+∑j=1Maij∂vi1,1/∂xjjxj,1+∂vi,1/∂xj,2x˙j,2+hi∂vi,1/∂x0,1x˙0,1+∂vi,1/∂x0,2x˙0,2+∂vi,1/∂θ^i,1θ^.i,1+∂vi,1/∂FbF˙b+∂vi,1/∂F⟶bF¨b.
Construct Lyapunov function as(41)Vi,2=Vi,1+12zi,22+biθ˜i,222γi.
The derivative ofVi,2 is written as(42)V˙i,2≤−biζi,1121+α/212zi,12+biθ˜i,122γi1+α/2+zi,2f^i,2Zi,2−2biζi,1212zi,12+biθ˜i,122γi1+β/2+zi,2gi,2zi,3+zi,2gi,2vi,2−zi,2gi,2xi,20−zi,2∂vi,1∂θ^i,1θ^.i,1−biθ˜i,2θ^.i,2γi+zi,2Mi,1Zi,2+Ci,1,where f^i,2Zi,2=fi,2x¯i,2,xi,20+di,2+di+hizi,1FbΦFigi,1−∂vi1,1/∂xi,1x˙i,1−∑j=1Maij∂vi,1/∂xj,1x˙j,1+∂vi,1/∂xj,2x˙j,2−hi∂vi,1/∂x0,1x˙0,1+∂vi,1/∂x0,2x˙0,2−Mi,1Zi,2.Remark 4.
Mi,1Zi,2 is a smooth function that is used to compensate ∂vi,1/∂θ^i1θ^.i,1.
The virtual controllervi,2 is defined as(43)vi,2=−θ^i,22ηi,22Si,2TZi,2Si,2Zi,2zi,2+xi,20−ζi,21Szi,2,1−ζi,22Szi,2,2−ζi,23zi,2,where ζi,21, ζi,22, ζi,23, and ηi,2 are positive design parameters.
According to (24)–(26) and substituting (43) into (42), we have(44)V˙i,2≤biθ˜i,2γiγi2ηi,22Si,2TZi,2Si,2Zi,2zi,22−θ^.i,2+zi,2gi,2zi,3+σi,2−biζi,21zi,2Szi,2,1−biζi,22zi,2Szi,2,2+Ci,1−biζi,1121+α/212zi,12+biθ˜i,122γi1+α/2−2biζi,1212zi,12+biθ˜i,122γi1+β/2+zi,2Mi,1Zi,2−∂vi,1∂θ^i,1θ^.i,1,where σi,2=1/2ηi,22+εi,22/4biζi,23.
Then, the adaptive lawθ^i,2 is defined as(45)θ^.i,2=γi2ηi,22Si,2TZi,2Si,2Zi,2zi,22−λiθ^i,2.
From Assumption3 and (45), we have(46)V˙i,2≤−biζi,21zi,2Szi,2,1−biζi,22zi,2Szi,2,2−biζi,2121+α/2biθ˜i,222γi1+α/2−biζi,2221+β/2biθ˜i,222γi1+β/2−biζi,1121+α/212zi,12+biθ˜i,122γi1+α/2−2biζi,1212zi,12+biθ˜i,122γi1+β/2+∑j=12Ci,j+zi,2gi,2zi,3+zi,2Mi,1Zi,2−∂vii,1∂θ^i,1θ^.i,1,where Ci,2=σi,2+βi,2+λibi/2γiθi,22, βi,2=biζi,2121+α/2biQi,22/2γi1+α/2+biζi,2221+β/2biQi,22/2γi1+β/2, and λ1biθ˜i,2θ^i,2/γi≤λibi/γiθi,22/2−θ˜i,22/2≤λibi/2γiθi,22.
According to (27), (28), (32), and (33) and when zi,2≥ε20,(47)V˙i,2≤−bi21+α/2∑j=12ζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=12ζi,j212zi,j2+biθ˜i,j22γi1+β/2+zi,2Mi,1Zi,2−∂vi,1∂θ^i,1θ^.i,1+∑j=12Cj+zi,2gi,2zi,3.
From Lemmas1, 4, and 5 and (37), we have(48)−zi,2∂vi,1∂θ^i,1θ^.i,1≤−zi,2∂vi,1∂θ^i,1γi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−λiθ^i,1≤zi,22∂vi,1/∂θ^i,12γi/2ηi,12Si,1TZi,1Si,1Zi,1zi,122zi,22∂vi,1/∂θ^i,12γi/2ηi,12Si,1TZi,1Si,1Zi,1zi,122+κ2+zi,2∂vi,1∂θ^i,1λiθ^i,1≤zi,22∂vi,1/∂θ^i,122γi/2ηi,12S2zi,122zi,22∂vi,1/∂θ^i,12γi/2ηi,12S2zi,122+κ2+zi,2∂vi,1∂θ^i,1λiθ^i,1.
Therefore,Mi,1Zi,2 can be defined as(49)Mi,1Zi,2=−zi,2∂vi,1/∂θ^i,12γi/2ηi,12S2zi,122zi,22∂vi,1/∂θ^i,12γi/2ηi,12S2zi,122+κ2−∂vi,1∂θ^i,1λiθ^i,1,with the result that(50)zi,2Mi,1Zi,2−∂vi,1∂θ^i,1iθ^i,1≤0.
Substituting (50) into (47), we can obtain(51)V˙i,2≤−bi21+α/2∑j=12ζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=12ζi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=12Cj+zi,2gi,2zi,3.Step k. 3≤k≤ni−1: from (3) and (23), we have(52)z˙i,k=x˙i,k−v˙i,k−1=fi,kx¯i,k,xi,k0+gi,kxi,k+1−xi,k0+di,k−v˙i,k−1,wherev˙i,k−1=∑j=1k−1∂vi,k−1/∂xi,jx˙i,j+hi∑j=1k∂vi,k−1/∂x0,jx˙0,j+∑j=1Maij∑q=1k∂vi,k−1/∂xj,qx˙j,q+∑j=1k−1∂vi,k−1/∂θ^i,jθ^.i,j+\∑j=0k−1∂vi,k−1/∂FbjFbj+1.
Construct Lyapunov function as(53)Vi,k=Vi,k−1+12zi,k2+biθ˜i,k22γi.
The derivative ofVi,k is(54)V˙i,k≤−bi21+α/2∑j=1k−1ζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1k−1ζi,j212zi,j2+biθ˜i,j22γi1+β/2+zi,kf^i,kZi,k+zi,kgi,kzi,k+1+zi,kgi,kvi,k−zi,kgi,kxi,k0−zi,k∑j=1k−1∂vi,k−1∂θ^i,jθ^.i,j−biθ˜i,kθ^.i,kγi+zi,kMi,k−1Zi,k+∑j=1k−1Ci,j,where f^i,kZi,k=fi,kx¯i,k,xi,k0+di,k+zi,k−1gi,k−1−∑j=1k−1∂vi,k−1/∂xi,jx˙i,j−∑j=1Maij∑q=1k∂vi,k−1/∂xj,qx˙j,q−hi∑j=1k∂vi,k−1/∂x0,jx˙0,j−∑j=0k−1∂vi,k−1/∂FbjFbj+1−Mi,k−1Zi,k.
The virtual controllervi,k is defined as(55)vi,k=−θ^i,k2ηi,k2Si,kTZi,kSi,kZi,kzi,k+xi,k0−ζi,k1Szi,k,1−ζi,k2Szi,k,2−ζi,k3zi,k,where ζi,k1, ζi,k2, ζi,k3, and ηi,k are positive design parameters.
According to24)–(26) and substituting (55) into (54), we have(56)V˙i,k≤biθ˜i,kγiγi2ηi,k2Si,kTZi,kSi,kZi,kzi,k2−θ^i,k+zi,kgi,kzi,k+1+σi,k−biζi,k1zi,kSzi,k,1−biζi,k2zi,kSzi,k,2+∑j=1k−1Ci,j+zi,kMi,k−1Zi,k−∑j=1k−1∂vi,k−1∂θ^i,jθ^.i,j−bi21+α/2∑j=1k−1ζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1k−1ζi,j212zi,j2+biθ˜i,j22γi1+β/2,where σi,k=1/2ηi,k2+εi,k2/4biζi,k3.
Then, the adaptive lawθ^i,k is defined as(57)θ^.i,k=γi2ηi,k2Si,kTZi,kSi,kZi,kzi,k2−λiθ^i,k.
Substituting (57) into (56), we can obtain(58)V˙i,k≤−bi21+α/2∑j=1k−1ζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1k−1ζi,j212zi,j2+biθ˜i,j22γi1+β/2−biζi,k1zi,kSzi,k,1−biζi,k2zi,kSzi,k,2−biζi,k121+α/2biθ˜i,k22γi1+α/2−biζi,k221+β/2biθ˜i,k22γi1+β/2+zi,kMi,k−1Zi,k−∑j=1k−1∂vi,k−1∂θ^i,jθ^.i,j+zi,kgi,kzi,k+1+∑j=1kCi,j,where Ci,k=σi,k+βi,k+λibi/2γiθi,k2, βl,k=biζi,k121+α/2biQl,k2/2γi1+α/2+biζl,k221+β/2biQi,k2/2γi1+β/2, and λibiθ˜i,kθ^i,k/γi≤λibi/γiθi,k2/2−θ˜i,k2/2≤λibi/2γiθi,k2.
According to (27), (28), (32), and (33) and when zi,k≥εk0,(59)V˙i,k≤−bi21+α/2∑j=1kζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1kζi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=1kCj+zi,kgi,kzi,k+1+zi,kMi,k−1Zi,k−∑j=1k−1∂vi,k−1∂θ^i,jθ^.i,j.
From Lemmas1, 4, and 5 and (57), we have(60)−zi,k∑j=1k−1∂vi,k−1∂θ^i,jθ^.i,j≤−zi,k∑j=1k−1∂vi,k−1∂θ^i,jγi2ηi,j2Si,jTZi,jSi,jZi,jzi,j2−λiθ^i,j≤∑j=1k−1zi,k2∂vi,k−1/∂θ^i,j2γi/2ηi,j2Si,jTZi,jSi,jZi,jzi,j22zi,k2∂vi,k−1/∂θ^i,j2γi/2ηi,j2Si,jTZi,jSi,jZi,jzi,j22+κ2+λizi,k∑j=1k−1∂vi,k−1∂θ^i,jθ^i,j≤∑j=1k−1zi,k2∂vi,k−1/∂θ^i,j2γi/2ηi,j2S2zi,j22zi,k2∂vi,k−1/∂θ^i,j2γi/2ηi,j2S2zi,j22+κ2+λizi,k∑j=1k−1∂vi,k−1∂θ^i,jθ^i,j.
Therefore,Mi,k−1Zi,k is defined as(61)Mi,k−1Zi,k=−∑j=1k−1zi,k2∂vi,k−1/∂θ^i,j2γi/2ηi,j2S2zi,j22zi,k2∂vi,k−1/∂θ^i,j2γi/2ηi,j2S2zi,j22+κ2−λizi,k∑j=1k−1∂vi,k−1∂θ^i,jθ^i,j,with the result that(62)zi,kMi,k−1Zi,k−∑j=1k−1∂vi,k−1∂θ^i,jθ^.i,j≤0.
Substituting (61) into (59) yields(63)V˙i,k≤−bi21+α/2∑j=1kζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1kζi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=1kCj+zi,kgi,kzi,k+1.Stepni. From (3) and (23), we have(64)z˙i,ni=x˙i,ni−v˙i,ni−1=fi,nix¯i,ni,xi,ni0+gi,niui−xi,ni0+di,ni−v˙i,ni−1,wherev˙i,ni−1=∑j=1ni−1∂vi,ni−1/∂xi,jx˙i,j+hi∑j=1ni∂vi,ni−1/∂x0,jx˙0,j+∑j=1Maij∑q=1ni∂vi,ni−1/∂xj,qx˙j,q+∑j=1ni−1∂vi,ni−1/∂θ^i,jθ^.i,j+\\∑j=0ni−1∂vi,ni−1/∂FbjFbj+1.
Construct Lyapunov function as(65)Vi,ni=Vi,ni−1+12zi,ni2+biθ˜i,ni22γi.
The derivative ofVi,ni is(66)V˙i,ni≤−bi21+α/2∑j=1ni−1ζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1ni−1ζi,j212zi,j2+biθ˜i,j22γi1+β/2+zi,nif^i,niZi,ni+zi,nigi,niui−zi,nigi,nixi,ni0−zi,ni∑j=1ni−1∂vi,ni−1∂θ^i,jθ^.i,j−biθ˜i,niθ^.i,niγi+zi,niMi,ni−1Zi,ni+∑j=1ni−1Ci,j,where(67)f^i,niZi,ni=fi,nix¯i,ni,xi,ni0+di,ni+zi,ni−1gi,ni−1−∑j=1ni−1∂vi,ni−1∂xi,jx˙i,j−∑j=1Maij∑q=1ni∂vi,ni−1∂xj,qx˙j,q−hi∑j=1ni∂vi,ni−1∂x0,jx˙0,j−∑j=0ni−1∂vi,ni−1∂FbjFbj+1−Mi,ni−1Zi,ni.
The actual controller is defined as(68)ui=−ζi,ni1Szi,ni,1−ζi,ni2Szi,ni,2−θ^i,ni2ηi,ni2Si,niTZi,niSi,niZi,nizi,ni−ζi,ni3zi,ni+xi,ni0,where ζi,ni1, ζi,ni2, ζi,ni3, and ηi,ni are positive design parameters.
The adaptive lawθ^i,ni is defined as(69)θ^.i,ni=γi2ηi,ni2Si,niTZi,niSi,niZi,nizi,ni2−λiθ^i,ni.
Substituting (69) and (68) into (66), we can obtain(70)V˙i,ni≤−bi21+α/2∑j=1ni−1ζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1ni−1ζi,j212zi,j2+biθ˜i,j22γi1+β/2−biζi,ni1zi,niSzi,ni,1−biζi,ni2zi,niSzi,ni,2−biζi,ni121+α/2biθ˜i,ni22γi1+α/2−biζi,ni221+β/2biθ˜i,ni22γi1+β/2+zi,niMi,ni−1Zi,ni−∑j=1ni−1∂vi,ni−1∂θ^i,jθ^.i,j+∑j=1niCi,j,where Ci,ni=σi,ni+βi,ni+λibi/2γiθi,ni2, βl,ni=biζi,ni121+α/2biQl,ni2/2γi1+α/2+biζl,ni221+β/2biQi,ni2/2γi1+β/2, σi,ni=1/2ηi,ni2+εi,ni2/4biζi,ni3, and λibiθ˜i,niθ^i,ni/γi≤λibi/γiθi,ni2/2−θ˜i,ni2/2≤λibi/2γiθi,ni2.
According to (27), (28), (32), and (33) and when zi,ni≥εni0,(71)V˙i,ni≤−bi21+α/2∑j=1niζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1niζi,j212zi,j2+biθ˜i,j22γi1+β/2+zi,niMi,ni−1Zi,ni−∑j=1ni−1∂vi,ni−1∂θ^i,jθ^.i,j+∑j=1niCi,j.
Same to the process of (60), the Mi,ni−1Zi,ni can be defined as(72)Mi,ni−1Zi,ni=−∑j=1ni−1zi,ni2∂vi,ni−1/∂θ^i,j2γi/2ηi,j2S2zi,j22zi,k2∂vi,ni−1/∂θ^i,j2γi/2ηi,j2S2zi,j22+κ2−λizi,ni∑j=1ni−1∂vi,ni−1∂θ^i,jθ^i,j,with the result that(73)zi,niMi,ni−1Zi,ni−∑j=1ni−1∂vi,ni−1∂θ^i,jθ^.i,j≤0.
Substituting (73) into (71) yields(74)V˙i,ni≤−bi21+α/2∑j=1niζi,j112zi,j2+biθ˜i,j22γi1+α/2−2bi∑j=1nici,j212zi,j2+biθ˜i,j22γi1+β/2+τi,where τi=∑j=1niCi,j.
Settingϕi,1=minbi21+α/2ζi,11,ζi,21,…,ζi,n1,1 and ϕi,2=min2biζi,12,ζi,22,…,ζi,ni2 and from Lemma 3, we have(75)−bi21+α/2∑j=1niζi,j112zi,j2+biθ˜i,j22γi1+α/2≤−ϕi,1∑j=1ni12zi,j2+biθ˜i,j22γi1+α/2,−2bi∑j=1niζi,j212zi,j2+biθ˜i,j22γi1+β/2≤−ni1−β/2ϕi,2∑j=1ni12zi,j2+biθ˜i,j22γi1+β/2.
Then, (74) can be rewritten as(76)V˙i,ni≤−ϕi,1Vi,ni1+α/2−ni1−β/2ϕi,2Vi,ni1+β/2+τi.
Up to this point, the design of fixed-time controller is finished.


## 4. Stability Analysis

Theorem 1.
Consider pure-feedback nonlinear multiagent systems (1) satisfying Assumptions 1–3. If given initial values −σ2Fb0<ei0<σ1Fb0, then by utilizing virtual controllers (31), (43), and (52), actual controller (68) and adaptive laws (37), (45), (57), and (69) can ensure that the local consensus errors converge to a small neighborhood of the origin within a fixed time. Besides, the local tracking errors can meet the prescribed performance range, i.e., −σ2Fbt<eit<σ1Fbt.Proof.
Choose the total Lyapunov functionV as follows:(77)V=∑i=1MVi,ni.
According to the process above and (76), the derivative of V can be written as(78)V˙≤−∑i=1Mϕi,1Vi,ni1+α/2−∑i=1Mni1−β/2ϕi,2Vi,ni1+β/2+∑i=1Mτi.
Settingμ1=minϕ1,1,ϕ2,1,…,ϕM,1, τ=∑i=1Mτi, and μ2=minn11−β/2M1−β/2ϕ1,2,…,nM1−β/2M1−β/2ϕM,2 and from Lemma 3, we have(79)−∑i=1Mϕi,1Vi,ni1+α/2≤−μ1∑i=1MVi,ni1+α/2,(80)−∑i=1Mni1−β/2ϕi,2Vi,ni1+β/2≤−μ2∑i=1MVi,ni1+β/2,where μ1>0, μ2>0, α∈0,1, and β∈1,∞.
Furthermore, substituting (79) and (80) into (78), we have(81)V˙≤−μ1V1+α/2−μ2V1+β/2+τ.
According to Lemma2, choose the appropriate parameters to make (81) satisfy the following conditions.Case 1.
IfV>τ/1−ϖμ12/1+α, ϖ∈0,1, then (81) can be written as(82)V˙≤−ϖμ1V1+α/2−μ2V1+β/2.
The solution of system (1) converges on the compact set,(83)x∈Vx≤τ1−ϖμ12/1+α.
The fixed convergence time is(84)T≤Tmax≔2ϖμ11−α+2μ2β−1.Case 2.
IfV>τ/1−ϖμ22/1+β, ϖ∈0,1, then (81) can be written as(85)V˙≤−μ1V1+α/2−ϖμ2V1+β/2.
The solution of system (1) converges on the compact set,(86)x∈Vx≤τ1−ϖμ22/1+β.
The fixed convergence time is(87)T≤Tmax≔2μ11−α+2ϖμ2β−1.
Combining Cases1 and 2, the solution of system (1) converges on(88)x∈Vx≤minτ1−ϖμ12/1+α,τ1−ϖμ22/1+β.
The fixed convergence time is(89)Ts≤Tmax≔2ϖμ11−α+2ϖμ2β−1.
It can be seen from (82) and (85) that V is bounded, with the result that zi,j and θ˜i,j are bounded, where i=1,…,M, j=1,…,ni, as θ^i,j=θi,j−θ˜i,j is also bounded. Because ei=∑j=1Maijyi−yj+hiyi−y0 and ei and y0 are bounded, yi are bounded. Because zi,2=xi,2−vi,1, xi,2 are bounded. Similarly, vi,k−1 and xi,k,k=1,…,ni are bounded. Therefore, all signals in the overall closed-loop system (1) are bounded.Remark 5.
In this paper, the adaptive fixed-time tracking consensus problem has been solved for pure-feedback nonlinear MASs. Compared with existing results [16–18, 24–27] on high-order MASs with nonlinear strict-feedback, our studied system is more general. Furthermore, the introduction of tracking error constraint extends the application scope of existing fixed-time control methods.Remark 6.
In fact, when the structure of the system is too complex, the design of the controller is very difficult. In this paper, the RBF neural network is used to approximate the uncertain nonlinear function, so the precise system dynamics structure is not needed when designing the controller.


## 5. Simulation Results

In this section, two simulation examples are employed to verify the controller designed as described in the above paragraphs.


### 5.1. Example 1: Numerical Example

Consider the second-order nonlinear MASs consisting of four followers and one leader, and the dynamic of each agentNi is given as the following pure-feedback form:(90)x˙i,1=1−e−xi,11+e−xi,2+xi,23+xi,2e−1−xi,12+di,1x¯i,1,tx˙i,2=xi,12xi,2+2ui+ui3xi,12+xi,22+di,2x¯2,tyi=xi,1,where xi,1 and xi,2, with i=1,…,4, are the state variables, ui is the system input, and yi is the system output. di,1x¯i,1,t=0.7xi,12cos1.5t and di,2x¯i,2,t=0.5xi,12+xi,22sin3t are the external disturbance terms.The directed network topology for the simulation is shown in Figure1. We choose aij=1 on j∈Ni, aij=0; otherwise, h1=h2=h3=0, and h4=1. The leader’s output y0t is 0.5sin1.5t+cos0.5t and the initial states and adaptive laws of the four followers are set to x10=0.5,0.5T, x20=0.1,0.1T, x30=−0.3,−0.3T, and x40=0.3,0.3T, where xi0=xi,10,xi,20T with i=1,…,4, and θ^i=θ^i,1,θ^i,2=0,0T with i=1,…,4. To display the tracking errors between the leader and the followers, we use ei=yit−y0t, where i=1,…,4, and the design parameters are chosen as follows: α=1/3, β=2, ζi,11=ζi,12=ζi,21=ζi,22=0.2, ζi,13=ζi,23=10, bi=5, γi=5, ηi,1=ηi,2=0.25, ε10=ε20=0.001, λi=0.04, xi,10=0.2, and xi,20=0.5, where i=1,…,4. It is worth noting that ζi,13=ζi,23 should be at least ten times that of ζi,11=ζi,12=ζi,21=ζi,22 and bi setting of 5 is much better than setting it to 2. The performance boundary function’s design parameters ξ0, a, ξ∞, σ1, and σ2 are chosen as 0.9, 1, 0.17, 2, and 2, respectively.Figure 1
Network topology for a group of one leader and four followers F1 to F4.Then, the virtual/actual control signals and the adaptive laws are, respectively, designed as follows:(91)vi,1=1di+hiFbΦFi−θ^i,12ηi,12Si,1TZi,1Si,1Zi,1zi,1−ζi,11Szi,1,1−ζi,12Szi,1,2−ζi,13zi,1+xi,10,ui=−ζi,21Szi,2,1−ζi,22Szi,2,2−θ^i,22ηi,22Si,2TZi,2Si,2Zi,2zi,2−ζi,23zi,2+xi,20,θ^.i,1=γi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−λiθ^i,1,θ^.i,2=γi2ηi,22Si,2TZi,2Si,2Zi,2zi,22−λiθ^i,2,where i=1,…,4.In this simulation example, we set the width of the RBF neural network as four.Wi,1Si,1Zi,1 and Wi,2Si,2Zi,2 contain seven and five nodes, respectively. The center of the Gaussian function is set as(92)ξi,1=−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52,ξi,2=−2−1012−2−1012−2−1012−3−2023−3−2023−3−2023−3−2023−3−2023−3−2023−3−2023.In Figure2, the tracking curves of the output signals yi are presented. It can be seen that, by utilizing the proposed fixed-time control strategy, the followers can fast track the leader. In Figure 3, the errors between the followers and the leader are shown. It can be seen that all local tracking errors are within the prescribed boundaries. In Figure 4, the responses of adaptive laws in MASs are given, which shows that adaptive signals θ^i,ji=1,…,4,j=1,2 are all bounded for t≥0. In Figure 5, the responses of control inputs in MASs are given, which shows that control input signals uii=1,…,4 are all bounded for t≥0. All the above simulation results illustrate the validity of the proposed method.Figure 2
Response curves of the output signalsyi and y0.Figure 3
Response curves of the tracking consensus errorei.Figure 4
Adaptive laws in MASs.
(a)(b)(c)(d)Figure 5
Control inputs in MASs.
(a)(b)(c)(d)


### 5.2. Example 2: Physical Example

Consider the following multiple electromechanical systems consisting of two followers and one leader. The dynamic of each electromechanical system is described as(93)Mq¨+Bq˙+Nsinq+Δ1q˙,q,I=IL˙I˙=Ve−RI−KBq˙+Δ1q˙,q,I,where q is the angular motor position, Ve is the input control voltage, and I is the motor armature, M=J/Kτ+mL02/3Kτ+M0L02/Kτ+2M0R02/5Kτ, B=B0/Kτ, and N=mL0g/2Kτ+M0L0g/Kτ,Δ1q˙,q, and Δ2q˙,q denote the model errors. J denotes the rotor inertia, Kτ is the coefficient to characterize the electromechanical conversion of armature current to torque, L0 is the link length, m means the link mass, M0 stands for the load mass, B0 is the coefficient of viscous friction at the joint, R0 is the radius of the load, g is the gravity coefficient, R is the armature resistance, L is the armature inductance, and KB is the back-emf coefficient.Letxi,1=q, xi,2=q˙, xi,3=I, yi=xi,1, and ui=Ve. Then, the electromechanical system can be rewritten as(94)x˙i,1=xi,2,x˙i,2=e21xi,2xi,32+e22sinxi,1+e23xi,2−e23xi,22xi,33,x˙i,3=e31ui+e32xi,2+e33xi,32−e33xi,22sinxi,3,yi=xi,1,where e21=1/M, e22=−N/M, e23=−B/M, e31=1/L, e32=−KB/L, and e33=−R/L.Assume that there are three followers and one leader in MASs, and the communications areh1=1, h2=h3=0, a2,1=a3,1=1, and a1,2=a1,3=a2,3=a3,2=0. The parameters are chosen as follows: M=0.0642, N=1.1408, B=0.0181, L=0.025, KB=0.9, and R=5.0. The leader’s output y0t is sin0.5t+0.5sint. The predefined performance function’ parameters ξ0, a, ξ∞, σ1, and σ2 are chosen as 0.9, 7, 0.17, 1, and 1, respectively. The purpose of this example is to design a fixed-time controller to make each follower’s output yi track the leader’s output signal y0 within fixed time.Then, the virtual/actual control signals and the adaptive laws are, respectively, designed as follows:(95)vi,1=1di+hiFbΦFi−ζi,11Szi,1,1−ζi,12Szi,1,2−ζi,13zi,1,vi,2=−θ^i,22ηi,22Si,2TZi,2Si,2Zi,2zi,2+xi,20−ζi,21Szi,2,1−ζi,22Szi,2,2−ζi,23zi,2,ui=−ζi,31Szi,3,1−ζi,33Szi,3,3−θ^i,33ηi,33Si,3TZi,3Si,3Zi,3zi,3−ζi,33zi,3+xi,30,θ^.i,2=γi2ηi,22Si,2TZi,2Si,2Zi,2zi,22−λiθ^i,2,θ^.i,3=γi2ηi,32Si,3TZi,3Si,3Zi,3zi,32−λiθ^i,3,where i=1,2,3.To verify the effectiveness of our designed fixed-time consensus control protocol, we compared the tracking performances in system (94) of two existing control strategies that are fixed-time controller in [25] and traditional finite-time controller in [44]. For fair comparison, we choose the same design parameters as follows: ζi,11=ζi,12=10, ζi,21=ζi,22=10, ζi,31=ζi,32=10, ζi,23=ζi,33=60, xi,10=xi,20=xi,30=0.1, and bi=10. The initial conditions are x10=0.15,0.15,0.15T, x20=0.2,0.2,0.2T, and x30=0.25,0.25,0.25T, where xi0=xi,10,xi,20,xi,30, and θ^i,20,θ^i,30=0.3,0.3T with i=1,2,3. The design of RBF neural network is the same as in [44].Remark 7.
Since the control protocols in [25, 44] are not designed for MASs, we choose the performance of the agent 1 that tracks directly the leader’s output to compare, and the comparative results are shown in Figures 6 and 7.
In Figure8, the tracking curves of the output signals yi are presented. It can be seen that, by using the proposed controller in this paper, the followers can quickly follow the leader’s output. Figure 9 shows that each local error is bounded by prescribed functions. In Figures 10 and 11, the responses of adaptive laws and control inputs in MASs are given, respectively. In Figures 6 and 7, the tracking error curves between this paper and [25, 44] are presented. It can be seen that the tracking performance by using our controller is better than theirs, since the proposed scheme has higher tracking performance with higher accuracy, and the comparison results further illustrate the effectiveness and superiority of the proposed method in this paper.Figure 6
Response curves of the tracking errors between this paper and [44].Figure 7
Response curves of the tracking errors between this paper and [25].Figure 8
Response curves of the output signalsyi and y0.Figure 9
Response curves of the tracking consensus errorei.Figure 10
Adaptive laws in MASs.
(a)(b)(c)Figure 11
Control inputs in MASs.
(a)(b)(c)


## 5.1. Example 1: Numerical Example

Consider the second-order nonlinear MASs consisting of four followers and one leader, and the dynamic of each agentNi is given as the following pure-feedback form:(90)x˙i,1=1−e−xi,11+e−xi,2+xi,23+xi,2e−1−xi,12+di,1x¯i,1,tx˙i,2=xi,12xi,2+2ui+ui3xi,12+xi,22+di,2x¯2,tyi=xi,1,where xi,1 and xi,2, with i=1,…,4, are the state variables, ui is the system input, and yi is the system output. di,1x¯i,1,t=0.7xi,12cos1.5t and di,2x¯i,2,t=0.5xi,12+xi,22sin3t are the external disturbance terms.The directed network topology for the simulation is shown in Figure1. We choose aij=1 on j∈Ni, aij=0; otherwise, h1=h2=h3=0, and h4=1. The leader’s output y0t is 0.5sin1.5t+cos0.5t and the initial states and adaptive laws of the four followers are set to x10=0.5,0.5T, x20=0.1,0.1T, x30=−0.3,−0.3T, and x40=0.3,0.3T, where xi0=xi,10,xi,20T with i=1,…,4, and θ^i=θ^i,1,θ^i,2=0,0T with i=1,…,4. To display the tracking errors between the leader and the followers, we use ei=yit−y0t, where i=1,…,4, and the design parameters are chosen as follows: α=1/3, β=2, ζi,11=ζi,12=ζi,21=ζi,22=0.2, ζi,13=ζi,23=10, bi=5, γi=5, ηi,1=ηi,2=0.25, ε10=ε20=0.001, λi=0.04, xi,10=0.2, and xi,20=0.5, where i=1,…,4. It is worth noting that ζi,13=ζi,23 should be at least ten times that of ζi,11=ζi,12=ζi,21=ζi,22 and bi setting of 5 is much better than setting it to 2. The performance boundary function’s design parameters ξ0, a, ξ∞, σ1, and σ2 are chosen as 0.9, 1, 0.17, 2, and 2, respectively.Figure 1
Network topology for a group of one leader and four followers F1 to F4.Then, the virtual/actual control signals and the adaptive laws are, respectively, designed as follows:(91)vi,1=1di+hiFbΦFi−θ^i,12ηi,12Si,1TZi,1Si,1Zi,1zi,1−ζi,11Szi,1,1−ζi,12Szi,1,2−ζi,13zi,1+xi,10,ui=−ζi,21Szi,2,1−ζi,22Szi,2,2−θ^i,22ηi,22Si,2TZi,2Si,2Zi,2zi,2−ζi,23zi,2+xi,20,θ^.i,1=γi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−λiθ^i,1,θ^.i,2=γi2ηi,22Si,2TZi,2Si,2Zi,2zi,22−λiθ^i,2,where i=1,…,4.In this simulation example, we set the width of the RBF neural network as four.Wi,1Si,1Zi,1 and Wi,2Si,2Zi,2 contain seven and five nodes, respectively. The center of the Gaussian function is set as(92)ξi,1=−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52,ξi,2=−2−1012−2−1012−2−1012−3−2023−3−2023−3−2023−3−2023−3−2023−3−2023−3−2023.In Figure2, the tracking curves of the output signals yi are presented. It can be seen that, by utilizing the proposed fixed-time control strategy, the followers can fast track the leader. In Figure 3, the errors between the followers and the leader are shown. It can be seen that all local tracking errors are within the prescribed boundaries. In Figure 4, the responses of adaptive laws in MASs are given, which shows that adaptive signals θ^i,ji=1,…,4,j=1,2 are all bounded for t≥0. In Figure 5, the responses of control inputs in MASs are given, which shows that control input signals uii=1,…,4 are all bounded for t≥0. All the above simulation results illustrate the validity of the proposed method.Figure 2
Response curves of the output signalsyi and y0.Figure 3
Response curves of the tracking consensus errorei.Figure 4
Adaptive laws in MASs.
(a)(b)(c)(d)Figure 5
Control inputs in MASs.
(a)(b)(c)(d)


## 5.2. Example 2: Physical Example

Consider the following multiple electromechanical systems consisting of two followers and one leader. The dynamic of each electromechanical system is described as(93)Mq¨+Bq˙+Nsinq+Δ1q˙,q,I=IL˙I˙=Ve−RI−KBq˙+Δ1q˙,q,I,where q is the angular motor position, Ve is the input control voltage, and I is the motor armature, M=J/Kτ+mL02/3Kτ+M0L02/Kτ+2M0R02/5Kτ, B=B0/Kτ, and N=mL0g/2Kτ+M0L0g/Kτ,Δ1q˙,q, and Δ2q˙,q denote the model errors. J denotes the rotor inertia, Kτ is the coefficient to characterize the electromechanical conversion of armature current to torque, L0 is the link length, m means the link mass, M0 stands for the load mass, B0 is the coefficient of viscous friction at the joint, R0 is the radius of the load, g is the gravity coefficient, R is the armature resistance, L is the armature inductance, and KB is the back-emf coefficient.Letxi,1=q, xi,2=q˙, xi,3=I, yi=xi,1, and ui=Ve. Then, the electromechanical system can be rewritten as(94)x˙i,1=xi,2,x˙i,2=e21xi,2xi,32+e22sinxi,1+e23xi,2−e23xi,22xi,33,x˙i,3=e31ui+e32xi,2+e33xi,32−e33xi,22sinxi,3,yi=xi,1,where e21=1/M, e22=−N/M, e23=−B/M, e31=1/L, e32=−KB/L, and e33=−R/L.Assume that there are three followers and one leader in MASs, and the communications areh1=1, h2=h3=0, a2,1=a3,1=1, and a1,2=a1,3=a2,3=a3,2=0. The parameters are chosen as follows: M=0.0642, N=1.1408, B=0.0181, L=0.025, KB=0.9, and R=5.0. The leader’s output y0t is sin0.5t+0.5sint. The predefined performance function’ parameters ξ0, a, ξ∞, σ1, and σ2 are chosen as 0.9, 7, 0.17, 1, and 1, respectively. The purpose of this example is to design a fixed-time controller to make each follower’s output yi track the leader’s output signal y0 within fixed time.Then, the virtual/actual control signals and the adaptive laws are, respectively, designed as follows:(95)vi,1=1di+hiFbΦFi−ζi,11Szi,1,1−ζi,12Szi,1,2−ζi,13zi,1,vi,2=−θ^i,22ηi,22Si,2TZi,2Si,2Zi,2zi,2+xi,20−ζi,21Szi,2,1−ζi,22Szi,2,2−ζi,23zi,2,ui=−ζi,31Szi,3,1−ζi,33Szi,3,3−θ^i,33ηi,33Si,3TZi,3Si,3Zi,3zi,3−ζi,33zi,3+xi,30,θ^.i,2=γi2ηi,22Si,2TZi,2Si,2Zi,2zi,22−λiθ^i,2,θ^.i,3=γi2ηi,32Si,3TZi,3Si,3Zi,3zi,32−λiθ^i,3,where i=1,2,3.To verify the effectiveness of our designed fixed-time consensus control protocol, we compared the tracking performances in system (94) of two existing control strategies that are fixed-time controller in [25] and traditional finite-time controller in [44]. For fair comparison, we choose the same design parameters as follows: ζi,11=ζi,12=10, ζi,21=ζi,22=10, ζi,31=ζi,32=10, ζi,23=ζi,33=60, xi,10=xi,20=xi,30=0.1, and bi=10. The initial conditions are x10=0.15,0.15,0.15T, x20=0.2,0.2,0.2T, and x30=0.25,0.25,0.25T, where xi0=xi,10,xi,20,xi,30, and θ^i,20,θ^i,30=0.3,0.3T with i=1,2,3. The design of RBF neural network is the same as in [44].Remark 7.
Since the control protocols in [25, 44] are not designed for MASs, we choose the performance of the agent 1 that tracks directly the leader’s output to compare, and the comparative results are shown in Figures 6 and 7.
In Figure8, the tracking curves of the output signals yi are presented. It can be seen that, by using the proposed controller in this paper, the followers can quickly follow the leader’s output. Figure 9 shows that each local error is bounded by prescribed functions. In Figures 10 and 11, the responses of adaptive laws and control inputs in MASs are given, respectively. In Figures 6 and 7, the tracking error curves between this paper and [25, 44] are presented. It can be seen that the tracking performance by using our controller is better than theirs, since the proposed scheme has higher tracking performance with higher accuracy, and the comparison results further illustrate the effectiveness and superiority of the proposed method in this paper.Figure 6
Response curves of the tracking errors between this paper and [44].Figure 7
Response curves of the tracking errors between this paper and [25].Figure 8
Response curves of the output signalsyi and y0.Figure 9
Response curves of the tracking consensus errorei.Figure 10
Adaptive laws in MASs.
(a)(b)(c)Figure 11
Control inputs in MASs.
(a)(b)(c)


## 6. Conclusion

In this paper, the adaptive fixed-time tracking consensus problem has been investigated for multiple uncertain nonlinear pure-feedback systems with performance constraint already. Mean value theorem is employed to address the problem of nonaffine structure in pure-feedback nonlinear systems, and RBF neural networks are introduced to approximate unknown nonlinear functions. Based on Lyapunov stability theory, it is proved that the proposed control scheme can ensure all the signals in MASs are bounded, and each local tracking error can converge to a small neighborhood of zero within fixed time. Furthermore, the performance of tracking error is guaranteed to be bounded by the prescribed boundary function. The effectiveness of the work has been proved by stability analysis along with simulation results.



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