# Distributed Adaptive Fixed-Time Tracking Consensus Control for Multiple Uncertain Nonlinear Strict-Feedback Systems under a Directed Graph

Authors: Pinwei Li; Jiyang Dai; Jin Ying; Zhe Zhang; Cheng He
Journal: Complexity
 (2020)

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## Abstract

In this brief, we study the distributed adaptive fixed-time tracking consensus control problem for multiple strict-feedback systems with uncertain nonlinearities under a directed graph topology. It is assumed that the leader’s output is time varying and has been accessed by only a small fraction of followers in a group. The distributed fixed-time tracking consensus control is proposed to design local consensus controllers in order to guarantee the consensus tracking between the followers and the leader and ensure the error convergence time is independent of the systems’ initial state. The function approximation technique using radial basis function neural networks (RBFNNs) is employed to compensate for unknown nonlinear terms induced from the controller design procedure. From the Lyapunov stability theorem and graph theory, it is shown that, by using the proposed fixed-time control strategy, all signals in the closed-loop system and the consensus tracking errors are cooperatively semiglobally uniformly bounded and the errors converge to a neighborhood of the origin within a fixed time. Finally, the effectiveness of the proposed control strategy has been proved by rigorous stability analysis and two simulation examples.


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## Body

## 1. Introduction

In the past decade, the finite-time consensus control of the multiagent has become one of the hot topics, since the finite-time control has a faster convergence rate, higher precision, and more robustness. However, there is a limitation for finite-time control that the convergence time increases accordingly when the initial state of the system is far from the equilibrium point. To overcome this limitation, fixed-time stability control [1] was proposed in 2012 by Polyakov et al.In 1965, finite-time control was first proposed, and many related research results emerged in the following decades. Due to the different agent dynamics in multiagent systems (MASs), there are various finite-time control developed [2–7]. To list a few, in [8, 9], by utilizing the principle of homogeneity [10], the authors proposed finite-time consensus control protocols for first-order and second-order MASs, respectively. In [11], power integrator technique was employed to solve the problem of finite-time consensus control for MASs with double integrator dynamics. Furthermore, in [12], by combining the homogeneous domination method with the power integrator method, the authors proposed two classes of finite-time consensus protocols for MASs composed of first-order and second-order integrator agents. Thereafter, by using the Lyapunov-function-based method, several finite-time consensus control strategies were proposed for high-order uncertain MASs [13–15]. Even so, there is a limitation for the finite-time consensus control results that the convergence time is seriously dependent on the MASs’ initial states. That is to say, once the initial state is far away from the equilibrium point, the convergence time will increase as a result. In practice, it is much more reasonable for a predicable convergence time.To overcome the limitation of finite-time control, a serial of research results [16–26] on fixed-time consensus control for MASs has been carried out since its setting time can be bounded and predefined. Note that most of the fixed-time control results [16–20] were designed for MASs with first order or second order. Only a few results [21–26] solved the fixed-time consensus problems for high-order MASs. On the contrary, strict-feedback or lower triangular systems have been widely regarded as a control target due to the description of various physical systems [27]. In fact, the strict-feedback systems require the recursive and systematic control design procedure since the control input is not matched with system nonlinearities. In [28], the adaptive backstepping design was proposed to satisfy this requirement. Furthermore, in [29, 30], the adaptive neural network backstepping controllers were proposed for a class of strict-feedback nonlinear systems. Nevertheless, there is a problem with using backstepping to design a controller for the strict-feedback systems; it can cause “explosion of complexity.” To overcome this problem, in [31], a dynamic surface design technique was proposed for strict-feedback systems. Furthermore, in [32], neural network-based adaptive dynamic surface control was proposed. Subsequently, this method was extended to MASs in [33–35], and a distributed adaptive dynamic surface design technique was developed.Motivated by the above observations, in this paper, a distributed adaptive fixed-time tracking consensus control is proposed for multiple strict-feedback systems with unknown nonlinearities under a directed graph topology. In the design, we employed RBF neural networks to compensate for unknown nonlinear terms and some functions that are difficult to calculate induced from the controller design procedure. From Lyapunov stability theorem and fixed-time control theory, by using distributed coordinate conversion, virtual controllers and actual controllers were designed. Finally, a novel distributed fixed-time consensus control protocol is designed for the considered strict-feedback nonlinear MASs, which can guarantee that all signals in the total closed-loop systems are bounded and the tracking errors can quickly converge to the neighborhood of the origin within the fixed time. The principal contributions of this paper are as follows. (1) A novel method to solve distributed adaptive fixed-time tracking consensus control for strict-feedback nonlinear MASs has been proposed. Compared with some existing results [16–26], our control strategy, by using adaptive RBF neural network control algorithm, is applicable to high-order MASs with different unknown nonlinear functions and order of dynamics. (2) In order to solve the “complexity explosion” problem caused by the repeated differentiation in the controller design process, it is different from the traditional dynamic surface technique; this paper constructs a smooth function MZderived from Lyapunov stability to compensate the part of ∂νi/∂θ^iθ^⋅i in the differential of the virtual controller vi, while the other part of the differential is approximated by RBF neural networks. This method greatly simplifies the design process of the controller while ensuring Lyapunov stability. (3) In [36–39], their controller designed have a power function similar to z2k−1, 0<k<1. However, incorrect selection of k will lead to singularity. In this paper, a novel fixed-time controller is proposed to solve this problem. (4). RBF neural networks are introduced to approximate the unknown functions fi⋅ in MASs to make our designed fixed-time control can suitable more systems with complex dynamics.The organization for the remaining part of this paper is given below. Section2 presents preliminaries and problem description. Sections 3 and 4 give the detailed process of distributed fixed-time control protocol and stability analysis, respectively. In Section 5, the proposed control scheme is proved to be effective through a simulation experiment. Finally, conclusions are summarized in Section 6.


## 2. Preliminaries and Problem Description

### 2.1. Graph Theory

LetG=υ,ε be a directed graph with the set of nodes or vertices υ=1,…,M and the set of edges or arcs ε⊆υ×υ. An edge j,i∈ε means that agent i can obtain information from agent j, but agent j cannot obtain agent i’s. The set of neighbors of a node i is Ni=jj,i∈ε, which is the set of nodes with edges incoming to node i. The weighted adjacency matrix A=aij∈RM×Mis defined as aij>0 if j,i∈ε and aij=0 otherwise. Self-loop is not allowed, i.e., aii=0. The Laplacian matrix L is defined as 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 path from nodei1 to node ik is a sequence of edges of the form i1,i2,i2,i3,…,ik−1,ik in a directed graph. A directed tree is a directed graph where every node has exactly one parent except for the root and the root has directed paths to every other node. A directed graph has a directed spanning tree if there exists at least a node having a directed path to all the other nodes.


### 2.2. RBFNNs

An RBF neural network [40] is applied in this paper to approximate arbitrary continuous functions. The RBF neural network is defined as follows:(1)f^Z=WTSZ,where W=w1,…,wlT∈Rl is the weight vector, l>0 is the number of nodes of the neural network, Z∈ΩZ⊂Rq is the input of the RBF neural network, q is the input dimension, SZ=s1Z,…,slZT∈Rl is the basis vector function, and siZ is the output of the i th neural node. A Gaussian function is always chosen as siZ, i.e., siZ=exp−Z−ξiTZ−ξi/ri2, i=1,…,l, where ri is the width of the base function and ξi=ξi1,…,ξiqT is the center of the basis function. With a sufficient number l of neural nodes selected, an RBF neural network can approximate arbitrary continuous function fZ in a compact set ΩZ∈Rq with arbitrary accuracy ε:(2)fZ=W∗TSZ+δZ,∀Z∈ΩZ∈Rq,where δZ is the approximation error with δZ≤ε and W∗ is the given ideal constant weight vector, which is defined as follows:(3)W∗=argminW∈RlsupZ∈ΩZfZ−f^Z.In this paper, letθ=maxW∗/b with θ˜=θ−θ^, where θ^ are the estimates of the unknown constants θ, W∗ is the ideal weight vector of the RBF neural network, b is a positive design parameter, and ⋅ is the norm.Lemma 1.
(see [41]). Consider the Gaussian function (1); SZ has an upper bound such that(4)SZ≤∑k=0∞3qk+2q−1e−2p2k2/r2≔s,where p=1/2mini≠jξi−ξj.Remark 1.
Lemma1 has been proved in [41, 42]. Since ∑k=0∞3qk+2q−1e−2p2k2/r2 is convergent, s is a limited value. In addition, s is independent of the neural network node numbers l and the neural network inputs Z.Assumption 1.
(see [36]). There are unknown constants Qi that make θi˜≤Qi<∞, i=1,2,…,n.


### 2.3. Fixed Time

Definition 1.
Consider the following nonlinear system:(5)x˙t=ft,x,x0=x0,where x∈Rn and f:R+×Rn⟶Rn, and assume that the origin is an equilibrium point.Lemma 2.
(see [22]). If there exist design constants ϕ1>0, ϕ2>0, α∈0,1, and β∈1,+∞ such that(6)V˙≤−ϕ1Vαx−ϕ2Vβx,where Vx is a continuous differentiable positive definite function; system (5) is global fixed-time stable, and the fixed convergence time satisfies(7)T≤Tmax≔1ϕ11−α+1ϕ2β−1.Remark 2.
The sufficient conditions and convergence time for finite-time and fixed-time consensus control schemes are shown in Table1.Where μ1, μ2, ρ1, and ρ2 are positive design parameters, α>1, and 0<β<1. From convergence time, it can be seen that the finite-time control is related to the initial state x0, while the fixed-time control is not.Table 1
The sufficient condition and convergence time for finite-time control and fixed-time control.
Control algorithmSufficient conditionConvergence timeFinite-time controlV˙x≤−μ2VxβT≤Tmax≔1/ρ21−βVx01−βFixed-time controlV˙x≤−μ1Vxα−μ2VxβT≤Tmax≔1/ρ1α−1+1/ρ21−βLemma 3.
(see [36]). If there exist design constants ϕ1>0, ϕ2>0, α∈0,1, β∈1,+∞, τ∈0,∞, and ϖ∈0,1 such that(8)V˙≤−ϕ1Vαx−ϕ2Vβx+τ.
Then, the origin of system (5) is practical fixed-time stable and the fixed-time Tcan be estimated by(9)T≤Tmax≔1ϕ1ϖ1−α+1ϕ2ϖβ−1.
The residual set of the solution of system (5) is given by(10)x∈Vx≤minτ1−ϖϕ21/β,τ1−ϖϕ11/α.Lemma 4.
(see [43]). Let x1,x2,…,xn≥0. Then,(11)∑i=1nxiρ≥∑i=1nxiρ,if0<ρ≤1,∑i=1nxiρ≥n1−ρ∑i=1nxiρ,if1<ρ<∞.Lemma 5.
(see [44]). For any variable x∈R and any positive constant κ, the following relationship holds:(12)0≤x<κ+x2x2+κ2.Lemma 6.
Fory≥x>0, x, y∈R and any positive constant δ, then satisfying(13)yy+δ≥xx+δ.Proof.
(14)yy+δ−xx+δ=y+δy+δ−x+δx+δ−δy+δ−δx+δ=y+δ−x+δ+δx+δ−δy+δ≥0.


### 2.4. Problem Statement

Suppose that there exist one leader agent, labeled as0, and M follower agents, labeled as 1 to M, under a directed communication graph topology. The dynamic models of M followers in the strict-feedback form are considered as follows:(15)x˙i,k=xi,k+1+fi,kx¯i,k,x˙i,ni=ui+fi,nixi,yi=xi,1,where i=1,…,M, k=1,…,ni−1, x¯i,k=xi,1T,…,xi,kTT; xi,l∈Rp with l=1,…,k, xi=xi,1T,…,xi,niTT∈Rpni, and ui∈Rp are the state vector and the control input of the i th follower, respectively, and yi∈Rp is the output of the ith follower. fi,kx¯i,k∈Rp Are C1 nonlinear function vectors. It is assumed that the leader’s motion is independent of the followers’ motion.The communication topology for theM+1 agents is described by a directed graph G=υ,ε with υ=0,1,…,M, To represent the communications among followers, we define a subgraph as G¯=υ¯,ε¯ with υ¯=1,…,M. The adjacency matrix A¯ of the subgraph G¯ is A¯=aij∈RM×M; aij>0 if j,i∈ε¯, aij=0 otherwise, and aii=0. Then, the Laplacian matrix L is defined as(16)L=001×M−hL¯+H,where h=h1,…,hMT, with hi>0 if the leader 0∈Ni and hi=0 otherwise, which denotes the communication weight from the leader to followers, H=diagh1,…,hM, and L¯=D¯−A¯ with D¯=diagd1,…,dM is the Laplacian matrix of the subgraph denoting the communication among followers.Remark 3.
For the simplicity of analysis, in the following, we only consider the case wherep=1. The analysis and main results still hold for any dimension p by using the Kronecker product.Remark 4.
If the directed graphG has a spanning tree, rankL=M [45]. Then, rankL¯+H=M from L¯+H1M=h, where 1M is an M−vector of all ones. Therefore, L¯+H is invertible.Assumption 2.
TheC1 nonlinear functions fi,k are unknown on a directed graph G.Assumption 3.
The leader output signalrt∈Rp is an ni-order differentiable and bounded function and available for the ith followers satisfying 0∈Ni, i=1,…,M.Definition 2.
(see [33]). The distributed consensus tracking errors for nonlinear followers (15) under the communication graph are said to be cooperatively semiglobally uniformly ultimately bounded (CSUUB) if there exist adjustable constants c1>0 and c2>0, and the bounds β1>0 and β2>0, independent of t0, and for every α1∈0,c1 and α2∈0,c2, there is a time T≥0, independent of t0, such that yit0−rt0≤α1⇒yit−rt≤β1 and yit0−yjt0≤α2⇒yit−yjt≤β2 for all t≥t0+T, where i=1,…,M, j=1,…,M, i≠j and k=1,…,n.
The objective of this brief is to design RBF-neural networks-based distributed fixed-time consensus control lawsui for M followers (15) with unknown nonlinearities so that, under the directed graph, the follower outputs yi synchronize to the dynamic leader output r within fixed time while all signals in the total closed-loop systems are bounded.Remark 5.
The strict-feedback system (15) can describe many state-space models of nonlinear systems, i.e., various physical systems, such as flight systems, biochemical process, jet engine, and robotic systems [28]. Therefore, system (15) under a graph topology can represent multiagent systems consisting of several practical applications with different dynamics.Remark 6.
Notice that followers (15) can have various forms. That is, a group of the followers with different nonlinear functions and order of the dynamics can be considered in this brief.Remark 7.
Compared with the previous consensus works, this brief considers the consensus problem of a group of agents consisting of nonlinear followers with nonlinearities unmatched in the control input. Besides, the nonlinearities are unknown under the total communication topology, and the tracking convergence time independent of the initial state of system (15).


## 2.1. Graph Theory

LetG=υ,ε be a directed graph with the set of nodes or vertices υ=1,…,M and the set of edges or arcs ε⊆υ×υ. An edge j,i∈ε means that agent i can obtain information from agent j, but agent j cannot obtain agent i’s. The set of neighbors of a node i is Ni=jj,i∈ε, which is the set of nodes with edges incoming to node i. The weighted adjacency matrix A=aij∈RM×Mis defined as aij>0 if j,i∈ε and aij=0 otherwise. Self-loop is not allowed, i.e., aii=0. The Laplacian matrix L is defined as 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 path from nodei1 to node ik is a sequence of edges of the form i1,i2,i2,i3,…,ik−1,ik in a directed graph. A directed tree is a directed graph where every node has exactly one parent except for the root and the root has directed paths to every other node. A directed graph has a directed spanning tree if there exists at least a node having a directed path to all the other nodes.


## 2.2. RBFNNs

An RBF neural network [40] is applied in this paper to approximate arbitrary continuous functions. The RBF neural network is defined as follows:(1)f^Z=WTSZ,where W=w1,…,wlT∈Rl is the weight vector, l>0 is the number of nodes of the neural network, Z∈ΩZ⊂Rq is the input of the RBF neural network, q is the input dimension, SZ=s1Z,…,slZT∈Rl is the basis vector function, and siZ is the output of the i th neural node. A Gaussian function is always chosen as siZ, i.e., siZ=exp−Z−ξiTZ−ξi/ri2, i=1,…,l, where ri is the width of the base function and ξi=ξi1,…,ξiqT is the center of the basis function. With a sufficient number l of neural nodes selected, an RBF neural network can approximate arbitrary continuous function fZ in a compact set ΩZ∈Rq with arbitrary accuracy ε:(2)fZ=W∗TSZ+δZ,∀Z∈ΩZ∈Rq,where δZ is the approximation error with δZ≤ε and W∗ is the given ideal constant weight vector, which is defined as follows:(3)W∗=argminW∈RlsupZ∈ΩZfZ−f^Z.In this paper, letθ=maxW∗/b with θ˜=θ−θ^, where θ^ are the estimates of the unknown constants θ, W∗ is the ideal weight vector of the RBF neural network, b is a positive design parameter, and ⋅ is the norm.Lemma 1.
(see [41]). Consider the Gaussian function (1); SZ has an upper bound such that(4)SZ≤∑k=0∞3qk+2q−1e−2p2k2/r2≔s,where p=1/2mini≠jξi−ξj.Remark 1.
Lemma1 has been proved in [41, 42]. Since ∑k=0∞3qk+2q−1e−2p2k2/r2 is convergent, s is a limited value. In addition, s is independent of the neural network node numbers l and the neural network inputs Z.Assumption 1.
(see [36]). There are unknown constants Qi that make θi˜≤Qi<∞, i=1,2,…,n.


## 2.3. Fixed Time

Definition 1.
Consider the following nonlinear system:(5)x˙t=ft,x,x0=x0,where x∈Rn and f:R+×Rn⟶Rn, and assume that the origin is an equilibrium point.Lemma 2.
(see [22]). If there exist design constants ϕ1>0, ϕ2>0, α∈0,1, and β∈1,+∞ such that(6)V˙≤−ϕ1Vαx−ϕ2Vβx,where Vx is a continuous differentiable positive definite function; system (5) is global fixed-time stable, and the fixed convergence time satisfies(7)T≤Tmax≔1ϕ11−α+1ϕ2β−1.Remark 2.
The sufficient conditions and convergence time for finite-time and fixed-time consensus control schemes are shown in Table1.Where μ1, μ2, ρ1, and ρ2 are positive design parameters, α>1, and 0<β<1. From convergence time, it can be seen that the finite-time control is related to the initial state x0, while the fixed-time control is not.Table 1
The sufficient condition and convergence time for finite-time control and fixed-time control.
Control algorithmSufficient conditionConvergence timeFinite-time controlV˙x≤−μ2VxβT≤Tmax≔1/ρ21−βVx01−βFixed-time controlV˙x≤−μ1Vxα−μ2VxβT≤Tmax≔1/ρ1α−1+1/ρ21−βLemma 3.
(see [36]). If there exist design constants ϕ1>0, ϕ2>0, α∈0,1, β∈1,+∞, τ∈0,∞, and ϖ∈0,1 such that(8)V˙≤−ϕ1Vαx−ϕ2Vβx+τ.
Then, the origin of system (5) is practical fixed-time stable and the fixed-time Tcan be estimated by(9)T≤Tmax≔1ϕ1ϖ1−α+1ϕ2ϖβ−1.
The residual set of the solution of system (5) is given by(10)x∈Vx≤minτ1−ϖϕ21/β,τ1−ϖϕ11/α.Lemma 4.
(see [43]). Let x1,x2,…,xn≥0. Then,(11)∑i=1nxiρ≥∑i=1nxiρ,if0<ρ≤1,∑i=1nxiρ≥n1−ρ∑i=1nxiρ,if1<ρ<∞.Lemma 5.
(see [44]). For any variable x∈R and any positive constant κ, the following relationship holds:(12)0≤x<κ+x2x2+κ2.Lemma 6.
Fory≥x>0, x, y∈R and any positive constant δ, then satisfying(13)yy+δ≥xx+δ.Proof.
(14)yy+δ−xx+δ=y+δy+δ−x+δx+δ−δy+δ−δx+δ=y+δ−x+δ+δx+δ−δy+δ≥0.


## 2.4. Problem Statement

Suppose that there exist one leader agent, labeled as0, and M follower agents, labeled as 1 to M, under a directed communication graph topology. The dynamic models of M followers in the strict-feedback form are considered as follows:(15)x˙i,k=xi,k+1+fi,kx¯i,k,x˙i,ni=ui+fi,nixi,yi=xi,1,where i=1,…,M, k=1,…,ni−1, x¯i,k=xi,1T,…,xi,kTT; xi,l∈Rp with l=1,…,k, xi=xi,1T,…,xi,niTT∈Rpni, and ui∈Rp are the state vector and the control input of the i th follower, respectively, and yi∈Rp is the output of the ith follower. fi,kx¯i,k∈Rp Are C1 nonlinear function vectors. It is assumed that the leader’s motion is independent of the followers’ motion.The communication topology for theM+1 agents is described by a directed graph G=υ,ε with υ=0,1,…,M, To represent the communications among followers, we define a subgraph as G¯=υ¯,ε¯ with υ¯=1,…,M. The adjacency matrix A¯ of the subgraph G¯ is A¯=aij∈RM×M; aij>0 if j,i∈ε¯, aij=0 otherwise, and aii=0. Then, the Laplacian matrix L is defined as(16)L=001×M−hL¯+H,where h=h1,…,hMT, with hi>0 if the leader 0∈Ni and hi=0 otherwise, which denotes the communication weight from the leader to followers, H=diagh1,…,hM, and L¯=D¯−A¯ with D¯=diagd1,…,dM is the Laplacian matrix of the subgraph denoting the communication among followers.Remark 3.
For the simplicity of analysis, in the following, we only consider the case wherep=1. The analysis and main results still hold for any dimension p by using the Kronecker product.Remark 4.
If the directed graphG has a spanning tree, rankL=M [45]. Then, rankL¯+H=M from L¯+H1M=h, where 1M is an M−vector of all ones. Therefore, L¯+H is invertible.Assumption 2.
TheC1 nonlinear functions fi,k are unknown on a directed graph G.Assumption 3.
The leader output signalrt∈Rp is an ni-order differentiable and bounded function and available for the ith followers satisfying 0∈Ni, i=1,…,M.Definition 2.
(see [33]). The distributed consensus tracking errors for nonlinear followers (15) under the communication graph are said to be cooperatively semiglobally uniformly ultimately bounded (CSUUB) if there exist adjustable constants c1>0 and c2>0, and the bounds β1>0 and β2>0, independent of t0, and for every α1∈0,c1 and α2∈0,c2, there is a time T≥0, independent of t0, such that yit0−rt0≤α1⇒yit−rt≤β1 and yit0−yjt0≤α2⇒yit−yjt≤β2 for all t≥t0+T, where i=1,…,M, j=1,…,M, i≠j and k=1,…,n.
The objective of this brief is to design RBF-neural networks-based distributed fixed-time consensus control lawsui for M followers (15) with unknown nonlinearities so that, under the directed graph, the follower outputs yi synchronize to the dynamic leader output r within fixed time while all signals in the total closed-loop systems are bounded.Remark 5.
The strict-feedback system (15) can describe many state-space models of nonlinear systems, i.e., various physical systems, such as flight systems, biochemical process, jet engine, and robotic systems [28]. Therefore, system (15) under a graph topology can represent multiagent systems consisting of several practical applications with different dynamics.Remark 6.
Notice that followers (15) can have various forms. That is, a group of the followers with different nonlinear functions and order of the dynamics can be considered in this brief.Remark 7.
Compared with the previous consensus works, this brief considers the consensus problem of a group of agents consisting of nonlinear followers with nonlinearities unmatched in the control input. Besides, the nonlinearities are unknown under the total communication topology, and the tracking convergence time independent of the initial state of system (15).


## 3. Distributed Fixed-Time Consensus Controller Design

The design procedure on theith follower contains ni steps. The distributed backstepping design coordinate transformation is as follows:(17)zi,1=∑j=1Maijyi−yj+hiyi−y0,(18)zi,k=xi,k−νi,k−1,where i=1,…,M and k=2,…,ni and vi,k is the virtual controller of the kth subsystem of ithfollower.An RBF neural network is used in this paper to approximate the unknown functionsf^i.kZi,k:(19)f^i.kZi,k=Wi,k∗TSi,kZi,k+δi,kZi,k,and the inequalities involved in the following text are as below:(20)zi,kδi,kZi,k≤biςi,k3zi,k2+εi,k24biςi,k3,(21)zi,kWi,k∗TSi,kZi,k≤biθi,k2ηi,k2Si,kTZi,kSi,kZi,kzi,k2+ηi,k22,(22)−biςi,k221+β/212zi,k21+β/2+biθ˜i,k22γi1+β/2≤−2biςi,k212zi,k2+biθ˜i,k22γi1+β/2,(23)−biςi,k121+α/212zi,k21+α/2+biθ˜i,k22γi1+α/2≤−2biςi,k112zi,k2+biθ˜i,k22γi1+α/2,where Zi,k=xi,1,…,xi,k,θ^i,1,…,θ^i,k,hiy0,hiy˙0,…,hiy0k,ai1y1,1,…,aiMyM,1,…,ai1y1,k,…,aiMyM,k with i=1,…,M, k=1,…,ni is the input vector, and bi, ςi,k3, ςi,k2, ςi,k1, εi,k, ηi,k, and γi are positive design parameters.Step 1.
: according to (15), (17), and (18), we have(24)z˙i,1=di+hizi,2+di+hiνi,1+f^i,1Zi,1−∑j=1Maijxj,2,where f^i,1Zi,1=di+hifi,1−∑j=1Maijfj,1−hiy˙0.
Construct a Lyapunov function as(25)Vi,1=12zi,12+biθ˜i,122γi.
Time differentiation ofVi yields(26)V˙i,1=di+hizi,1zi,2+di+hizi,1νi,1+zi,1f^i,1Zi,1−zi,1∑j=1Maijxj,2−biθ˜i,1θ^⋅i,1γi.
The virtual controllerνi,1 is defined as(27)νi,1=1di+hi−biθ^i,12ηi,12Si,1TZi,1Si,1Zi,1zi,1+∑j=1Maijxj,2−biςi,13zi,1−ςi,11Szi,1,1−ςi,12Szi,1,2,where ςi,11, ςi,12, ςi,13, and ηi,1 are positive design parameters.
In (27), Szi,1,1 and Szi,1,2 are defined as(28)Szi,k,1=zi,k21+α/2zi,k,zi,k≥εk0,∑j=1niajzi,k2jεk02−j+1+α/2,zi,k<εk0,(29)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, and εk0 is a positive design parameter, and coefficients aj, j=1,…,ni are calculated using the following equation:(30)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 8.
One of the main contributions of this paper is to design suitable virtual controllersνi,k so that nonlinear strict-feedback systems meet the requirement of fixed-time control. In [36-39], virtual controllers designed exhibit similar power functions z2α−1, where 0<α<1. If αis not appropriately selected, it will make zunsolvable at the origin and in the negative domain. As 0<α<1, therefore, power exponents 2α−1∈−1,1, which leads to the possibility of negative power exponents. For example, if 2α−1=−1/3, z−1/3 is unsolvable at z=0, that is, 0−1/3 does not exist. Suppose 2α−1=q1/q2∈−1,1, where q2 is an even number, then zq1/q2 is unsolvable at the negative domain. For example, if q1/q2=1/2, z1/2 is unsolvable at the negative domain, that is, −11/2 does not exist. The controller designed in this paper overcomes the abovementioned defect and promotes the application of fixed-time control in more common nonlinear systems.
Substituting (27) into (26) yields(31)V˙i,1≤di+hizi,1zi,2+zi,1f^i,1Zi,1biθ^i,12ηi,12Si,1TZi,1Si,1Zi,1zi,12−biςi,13zi,12−biθ˜i,1θ^⋅i,1γi−ςi,11zi,1Szi,1,1−ςi,12zi,1Szi,1,2.
According to (19) and inequalities (20)-(21), we have(32)V˙i,1≤biθ˜i,1γiγi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−θ^⋅i,1+di+hizi,1zi,2+σi,1−ςi,11zi,1Szi,1,1−ςi,12zi,1Szi,1,2,where σi,1=1/2ηi,12+εi,12/4biςi,13.
The adaptive lawθ^i,1 is then defined as(33)θ^⋅i,1=γi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−λiθ^i,1,where λi with i=1,…,M are positive design parameters.
Combining with Assumption1 and substituting (33) into (32), we have(34)V˙i,1≤−ςi,11zi,1Szi,1,1−ςi,12zi,1Szi,1,2−ςi,1121+α/2biθ˜i,122γi1+α/2−ςi,1221+β/2biθ˜i,122γi1+β/2+di+hizi,1zi,2+Ci,1,where Ci,1=σi,1+βi,1+λibi/2γiθi,12, βi,1=ςi,1121+α/2biQi,12/2γi1+α/2+ςi,1221+β/2biQi,12/2γi1+β/2, and λibiθ˜i,1θ^i,1/γi≤λibi/γiθi,12/2−θ˜i,12/2≤λibi/2γiθi,12.
According to (22), (23), (28), and (29), if zi,1≥ε10, then (34) is rewritten as(35)V˙i,1≤−ςi,1121+α/212zi,12+biθ˜i,122γi1+α/2−2ςi,1212zi,12+biθ˜i,122γi1+β/2+di+hizi,1zi,2+Ci,1.
According to (22), (23), (28), and (29), if zi,1<ε10, then (34) is rewritten as(36)V˙i,1≤ςi,11zi,121+α/2−ςi,11zi,1∑j=1niajzi,12jε102−j+1+α/2+ςi,12zi,121+β/2−ςi,12zi,1∑j=1niajzi,12jε102−j+1+β/2−ςi,1121+α/212zi,12+biθ˜i,122γi1+α/2−2ςi,1212zi,12+biθ˜i,122γi1+β/2+di+hizi,1zi,2+Ci,1.Remark 9.
(see [46]). Based on (28) and (29), when zi,1<ε10, there is an additional term in (36): ςi,11zi,121+α/2−ςi,11zi,1∑j=1niajzi,12jε102−j+1+α/2+ςi,12zi,121+β/2−ςi,12zi,1∑j=1niajzi,12jε102−j+1+β/2. Note that if zi,1<ε10, this additional term is obviously limited by some smaller constant ε11, so the structure of (35) is retained, while the constant term Ci,1 only slightly increases. Owing to page limitations and to avoid repetitive discussions, we will omit this part in the rest of the analysis.Step 2.
: according to (15) and (18), we have(37)z˙i,2=xi,3+fi,2x¯i,2−ν˙i,1,where ν˙i,1=∂νi,1/∂xi,1x˙i,1+∑j=1Maij∂νi,1/∂xj,1x˙j,1+∂νi,1/∂xj,2x˙j,2+hi∂νi,1/∂x0,1x˙0,1+∂νi,1/∂x0,2x˙0,2+∂νi,1/∂θ^i,1θ^⋅i,1.
Construct a Lyapunov function as(38)Vi,2=Vi,1+12zi,22+biθ˜i,222γi.
The derivative ofVi,2 is written as(39)V˙i,2≤−ςi,1121+α/212zi,12+biθ˜i,122γi1+α/2−2ςi,1212zi,12+biθ˜i,122γi1+β/2+zi,2f^i,2Zi,2+zi,2zi,3+zi,2νi,2−zi,2∂νi,1∂θ^i,1θ^⋅i,1−biθ˜i,2θ^⋅i,2γi+zi,2Mi,1Zi,2+Ci,1,where(40)f^i,2Zi,2=fi,2+zi,1di+hi−∂νi,1∂xi,1x˙i,1−∑j=1Maij∂νi,1∂xj,1x˙j,1+∂νi,1∂xj,2x˙j,2−hi∂νi,1∂x0,1x˙0,1+∂νi,1∂x0,2x˙0,2−Mi,1Zi,2.Remark 10.
Mi,1Zi,2 is a smooth function that is used to overcome the design difficulty of ∂νi,1/∂θ^i,1θ^⋅i,1.
The virtual controllerνi,2 is defined as(41)vi,2=−ςi,21Szi,2,1−ςi,22Szi,2,2−biθ^i,22ηi,22Si,2TZi,2Si,2Zi,2zi,2−biςi,23zi,2,where ςi,21, ςi,22, ςi,23, and ηi,2 are positive design parameters.
Choose the adaptive lawθ^i,2 as(42)θ^⋅i,2=γi2ηi,22Si,2TZi,2Si,2Zi,2zi,22−λiθ^i,2.
Combining (19)–(21) and Assumption 1 and substituting (41) and (42) into (38) and adopting the same design method as in Step 1 yields(43)V˙i,2≤−ςi,21zi,2Szi,2,1−ςi,22zi,2Szi,2,2−ςi,2121+α/2biθ˜i,222γi1+α/2−ςi,2221+β/2biθ˜i,222γi1+β/2−ςi,1121+α/212zi,12+biθ˜i,122γi1+α/2−2ςi,1212zi,12+biθ˜i,122γi1+β/2∑j=12Ci,j+zi,2zi,3+zi,2Mi,1Zi,2−∂νi,1∂θ^i,1θ^⋅i,1,where Ci,2=σi,2+βi,2+λibi/2γiθi,22, βi,2=ςi,2121+α/2biQi,22/2γi1+α/2+ςi,2221+β/2biQi,22/2γi1+β/2, σi,2=1/2ηi,22+εi,22/4biςi,23, and λibiθ˜i,2θ^i,2/γi≤λibi/γiθi,22/2−θ˜i,22/2≤λibi/2γiθi,22.
According to (22), (23), and Remark 9, we consider only zi,2≥ε20. Then, (43) can be written as(44)V˙i,2≤−21+α/2∑j=12ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=12ςi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=12Cj+zi,2zi,3+zi,2Mi,1Zi,2−∂νi,1∂θ^i,1θ^⋅i,1.
It can be seen from (44) that defining the design smooth function Mi,1Zi,2 to overcome the design difficulty of ∂νi,1/∂θ^i,1θ^⋅i,1 is one of the difficulties of designing the controllers in this paper.
From Lemma1, Lemma 5, Lemma 6, and (33), it follows that(45)−zi,2∂νi,1∂θ^i,1θ^⋅i,1≤−zi,2∂νi,1∂θ^i,1γi2ηi,12Si,1TZi,1Si,1Zi,1zi,12−λiθ^i,1≤zi,22∂νi,1/∂θ^i,12γi/2ηi,12Si,1TZi,1Si,1Zi,1zi,122zi,22∂νi,1/∂θ^i,12γi/2ηi,12Si,1TZi,1Si,1Zi,1zi,122+κ2+zi,2∂νi,1∂θ^i,1λiθ^i,1≤zi,22∂νi,1/∂θ^i,12γi/2ηi,12S2zi,122zi,22∂νi,1/∂θ^i,12γi/2ηi,12S2zi,122+κ2+zi,2∂νi,1∂θ^i,1λiθ^i,1.
Therefore,Mi,1Zi,2 can be defined as(46)Mi,1Zi,2=−zi,2∂νi,1/∂θ^i,12γi/2ηi,12S2zi,122zi,22∂νi,1/∂θ^i,12γi/2ηi,12S2zi,122+κ2−∂νi,1∂θ^i,1λiθ^i,1,with the result that(47)zi,2Mi,1Zi,2−∂νi,1∂θ^i,1θ^⋅i,1≤0.
Substituting (47) into (44) yields(48)V˙i,1≤−21+α/2∑j=12ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=12ςi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=12Ci,j+zi,2zi,3.
Stepk3≤k≤ni−1: according to (15) and (18), we have(49)z˙i,k=xi,k+1+fi,kx¯i,k−ν˙i,k−1,where ν˙i,k−1=∑j=1k−1∂νi,k−1/∂xi,jx˙i,j+∑j=1Maij∑q=1k∂νi,k−1/∂xi,qx˙j,q+hi∑j=1k∂νi,k−1/∂x0,jx˙0,j+∑j=1k−1∂νi,k−1/∂θ^i,jθ^⋅i,j.
Construct a Lyapunov function as(50)Vi,k=Vi,k−1+12zi,k2+biθ˜i,k22γi.
The derivative ofVi,k is written as(51)V˙i,k≤−21+α/2∑j=1k−1ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1k−1ςi,j212zi,j2+biθ˜i,j22γi1+α/2+zi,kf^i,kZi,k+zi,kzi,k+1+zi,kνi,k−zi,k∑j=1k−1∂νi,k−1∂θ^i,jθ^⋅i,j−biθ˜i,kθ^⋅i,kγi+zi,kMi,k−1Zi,k+∑j=1k−1Ci,j,where(52)f^i,kZi,k=fi,kx¯i,k−∑j=1k−1∂νi,k−1∂xi,jx˙i,j−∑j=1Maij∑q=1k∂νi,k−1∂xj,qx˙j,q−hi∑j=1k∂νi,k−1∂x0,jx˙0,j+zi,k−1−Mi,k−1Zi,k.
Defining virtual controllerνi,k as(53)vi,k=−ςi,k1Szi,k,1−ςi,k2Szi,k,2−biθ^i,k2ηi,k2Si,kTZi,kSi,kZi,kzi,k−biςi,k3zi,k,where ςi,k1, ςi,k2, ςi,k3, and ηi,k are positive design parameters.
Choose the adaptive lawθ^i,k as(54)θ^⋅i,k=γi2ηi,k2Si,kTZi,kSi,kZi,kzi,k2−λiθ^i,k.
Combining (19)–(21) and Assumption 1 and substituting (53) and (54) into (51) yields(55)V˙i,k≤−21+α/2∑j=1k−1ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1k−1ςi,j212zi,j2+biθ˜i,j22γi1+β/2−ςi,k1zi,kSzi,k,1−ςi,k2zi,kSzi,k,2−ςi,k121+α/2biθ˜i,k22γi1+α/2+zi,kzi,k+1+∑j=1kCi,j−ςi,k221+β/2biθ˜i,k22γi1+β/2+zi,kMi,k−1Zi,k−∑j=1k−1∂νi,k−1∂θ^i,jθ^⋅i,j,where Ci,k=σi,k+βi,k+λibi/2γiθi,k2, βi,k=ςi,k121+α/2biQi,k2/2γi1+α/2+ςi,k221+β/2biQi,k2/2γi1+β/2, σi,k=1/2ηi,k2+εi,k2/4biςi,k3, and λibiθ˜i,kθ^i,k/γi≤λibi/γiθi,k2/2−θ˜i,k2/2≤λibi/2γiθi,k2.
According to (22), (23), and Remark 9, we consider only zi,k≥εk0. Then, (55) can be written as(56)V˙i,k≤−21+α/2∑j=1k−1ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1k−1ςi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=1kCj+zi,kzi,k+1+zi,kMi,k−1Zi,k−∑j=1k−1∂νi,k−1∂θ^i,jθ^⋅i,j.
From Lemma1, Lemma 5, Lemma 6, and (54), it follows that(57)−zi,k∑j=1k−1∂νi,k−1∂θ^i,jθ^⋅i,j≤−zi,k∑j=1k−1∂νi,k−1∂θ^i,jγi2ηi,j2Si,jTZi,jSi,jZi,jzi,j2−λiθ^i,j≤∑j=1k−1zi,k2∂νi,k−1/∂θ^i,j2γi/2ηi,j2Si,jTZi,jSi,jZi,jzi,j22zi,k2∂νi,k−1/∂θ^i,j2γi/2ηi,j2Si,jTZi,jSi,jZi,jzi,j22+κ2+λizi,k∑j=1k−1∂νi,k−1∂θ^i,jθ^i,j≤∑j=1k−1zi,k2∂νi,k−1/∂θ^i,j2γi/2ηi,j2S2zi,j22zi,k2∂νi,k−1/∂θ^i,j2γi/2ηi,j2S2zi,j22+κ2+λizi,k∑j=1k−1∂νi,k−1∂θ^i,jθ^i,j.
Therefore,Mi,k−1Zi,k can be defined as(58)Mi,k−1Zi,k=−∑j=1k−1zi,k2∂νi,k−1/∂θ^i,j2γi/2ηi,12S2zi,122zi,k2∂νi,k−1/∂θ^i,j2γi/2ηi,j2S2zi,j22+κ2+λizi,k∑j=1k−1∂νi,k−1∂θ^i,jθ^i,j,with the result that(59)zi,kMi,k−1Zi,k−∑j=1k−1∂νi,k−1∂θ^i,jθ^⋅i,j≤0.
Substituting (59) into (56) yields(60)V˙i,k≤−21+α/2∑j=1k−1ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1k−1ςi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=1kCj+zi,kzi,k+1.
Stepni: according to (15) and (18), we have(61)z˙i,ni=ui+fi,nixi−ν˙i,ni−1,where ν˙i,ni−1=∑j=1ni−1∂νi,ni−1/∂xi,jx˙i,j+∑j=1Maij∑q=1ni∂νi,ni−1/∂xj,qx˙j,q+hi∑j=1ni∂νi,ni−1/∂x0,jx˙0,j+∑j=1ni−1∂νi,ni−1/∂θ^i,jθ^⋅i,j.
Construct a Lyapunov function as(62)Vi,ni=Vi,ni−1+12zi,ni2+biθ˜i,ni22γi.
The derivative ofVi,ni is written as(63)V˙i,ni≤−21+α/2∑j=1ni−1ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1ni−1ςi,j212zi,j2+biθ˜i,j22γi1+β/2+zi,nif^i,niZi,ni+zi,niui−zi,ni∑j=1ni−1∂νi,ni−1∂θ^⋅i,jθ^˙i,j−biθ˜i,niθ^⋅i,niγi+zi,niMi,ni−1Zi,ni+∑j=1ni−1Ci,j,where(64)f^i,kZi,ni=fi,nixi−∑j=1ni−1∂νi,ni−1∂xi,jx˙i,j−∑j=1Maij∑q=1ni∂νi,ni−1∂xj,qx˙j,q−hi∑j=1ni∂νi,ni−1∂x0,jx˙0,j+zi,ni−1−Mi,ni−1Zi,ni.
Defining virtual controllerui as(65)ui=−ςi,ni1Szi,ni,1−ςi,ni2Szi,ni,2−biθ^i,ni2ηi,ni2Si,niTZi,niSi,niZi,nizi,ni−biςi,ni3zi,ni,where ςi,ni1, ςi,ni2, ςi,ni3, and ηi,ni are positive design parameters.
Choose the adaptive lawθ^i,ni as(66)θ^⋅i,ni=γi2ηi,ni2Si,niTZi,niSi,niZi,nizi,ni2−λiθ^i,ni.
Combining (19)–(21) and Assumption 1 and substituting (65) and (66) into (63) yields(67)V˙i,ni≤−21+α/2∑j=1ni−1ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1ni−1ςi,j212zi,j2+biθ˜i,j22γi1+β/2−ςi,ni1zi,niSzi,ni,1−ςi,ni2zi,niSzi,ni,2−ςi,ni121+α/2biθ˜i,ni22γi1+α/2+∑j=1kCi,j−ςi,ni221+β/2biθ˜i,ni22γi1+β/2+zi,niMi,ni−1Zi,ni−∑j=1ni−1∂νi,ni−1∂θ^i,jθ^⋅i,j,where Ci,ni=σi,ni+βi,ni+λibi/2γiθi,ni2, βi,ni,=ςi,ni121+α/2biQi,ni2/2γi1+α/2+ςi,ni221+β/2biQi,ni2/2γi1+β/2, σi,ni=1/2ηi,ni2+εi,ni2/4biςi,ni3, and λibiθ˜i,niθ^i,ni/γi≤λibi/γiθi,ni2/2−θ˜i,ni2/2≤λibi/2γiθi,ni2.
According to (22), (23), and Remark 9, we consider only zi,ni≥εni0. Then, (67) can be written as(68)V˙i,ni≤−21+α/2∑j=1ni−1ςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1ni−1ςi,j212zi,j2+biθ˜i,j22γi1+β/2+∑j=1niCi,j+zi,niMi,ni−1Zi,ni−∑j=1ni−1∂νi,ni−1∂θ^i,jθ^⋅i,j.
From Lemma1, Lemma 5, Lemma 6, and (54), it follows that(69)−zi,ni∑j=1ni−1∂νi,ni−1∂θ^i,jθ^⋅i,j≤−zi,ni∑j=1ni−1∂νi,ni−1∂θ^i,jγi2ηi,j2Si,jTZi,jSi,jZi,jzi,j2−λiθ^i,j≤∑j=1ni−1zi,ni2∂νi,ni−1/∂θ^i,j2γi/2ηi,j2Si,jTZi,jSi,jZi,jzi,j22zi,ni2∂νi,ni−1/∂θ^i,j2γi/2ηi,j2Si,jTZi,jSi,jZi,jzi,j22+κ2+λizi,ni∑j=1ni−1∂νi,ni−1∂θ^i,jθ^i,j≤∑j=1ni−1zi,ni2∂νi,ni−1/∂θ^i,j2γi/2ηi,j2S2zi,j22zi,k2∂νi,ni−1/∂θ^i,j2γi/2ηi,j2S2zi,j22+κ2+λizi,ni∑j=1ni−1∂νi,ni−1∂θ^i,jθ^i,j.
Therefore,Mi,ni−1Zi,ni can be defined as(70)Mi,ni−1Zi,ni=−∑j=1ni−1zi,ni2∂νi,ni−1/∂θ^i,j2γi/2ηi,j2S2zi,j22zi,k2∂νi,ni−1/∂θ^i,j2γi/2ηi,j2S2zi,j22+κ2−λizi,ni∑j=1ni−1∂νi,ni−1∂θ^i,jθ^i,j,with the result that(71)zi,niMi,ni−1Zi,ni−∑j=1ni−1∂νi,ni−1∂θ^i,jθ^⋅i,j≤0.
Substituting (71) into(68) yields(72)V˙i,ni≤−21+α/2∑j=1niςi,j112zi,j2+biθ˜i,j22γi1+α/2−2∑j=1niςi,j212zi,j2+biθ˜i,j22γi1+β/2+τi,where τi=∑j=1niCi,j.
Makingϕi,1=min21+α/2ςi,11,ςi,21,…,ςi,ni1and ϕi,2=min2ςi,12,ςi,22,…,ςi,ni2, in accordance with Lemma 4, yields(73)−21+α/2∑j=1niςi,j112zi,j2+biθ˜i,j22γi1+α/2≤−ϕi,1∑j=1ni12zi,j2+biθ˜i,j22γi1+α/2,(74)−2∑j=1niςi,j212zi,j2+biθ˜i,j22γi1+β/2≤−ni1+β/2ϕi,2∑j=1ni12zi,j2+biθ˜i,j22γi1+β/2,and then substituting (73) and (74) into (72) yields(75)V˙i,ni≤−ϕi,1Vi,ni1+α/2−ni1+β/2ϕi,2Vi,ni1+β/2+τi.
Up to this point, the design of theith follower’s controller is finished.


## 4. Stability Analysis

Theorem 1.
Consider the multiple uncertain nonlinear strict-feedback system (15) controlled by the distributed fixed-time consensus controller (65) and assume that the leader has directed paths to all followers 1 to M. Under Assumptions 1–3, virtual controllers (27), (41), (53), and (65), and the adaptive laws (33), (42), (54), and (66), the consensus tracking errors between the leader and the followers in the overall closed-loop system are CSUUB and can be made arbitrarily small within fixed-time, and the upper limit of the fixed convergence time is irrelevant to the initial state.Proof.
For the stability analysis, choose the total Lyapunov candidate functionVas(76)V=∑i=1MVi,ni.
Differentiating (76) and substituting (75) into it, we have(77)V˙≤−∑i=1Mϕi,1Vi,ni1+α/2−∑i=1Mni1+β/2ϕi,2Vi,ni1+β/2+∑i=1Mτi.
Makingρ1=minφ1,1,φ2,1,…,φM,1τ=∑i=1Mτi and ρ2=minn11+β/2M1+β/2φ1,2,…,nM1+β/2M1+β/2ϕM,2, in accordance with Lemma 4 yields(78)−∑i=1Mϕi,1Vi,ni1+α/2≤−ρ1∑i=1MVi,ni1+α/2,−∑i=1Mni1+β/2ϕi,2Vi,ni1+β/2≤−ρ2∑i=1MVi,ni1+β/2,where ρ1>0, ρ2>0, α∈0,1, and β∈1,∞.
Then, (77) can be rewritten as(79)V˙≤−ρ1V1+α/2−ρ2V1+α/2+τ.
In accordance with Lemma3, design proper parameters to make (79) satisfy the following situation.Case 1.
IfV>τ/1−ϖρ11+α/2, ϖ∈0,1, then (79) can be written as(80)V˙≤−ϖρ1V1+α/2−ρ2V1+β/2,and in that manner, the solution of system (15) converges on the compact set(81)x∈Vx≤τ1−ϖρ12/1+α.
The fixed convergence time is(82)T≤Tmax≔2ϖρ11−α+2ρ2β−1.Case 2.
IfV>τ/1−ϖρ21+β/2, ϖ∈0,1, then (79) can be written as(83)V˙≤−ρ1V1+α/2−ϖρ2V1+β/2,and in that manner, the solution of system (16) converges on the compact set(84)x∈Vx≤τ1−ϖρ22/1+β.
The fixed convergence time is(85)T≤Tmax≔2ρ11−α+2ϖρ2β−1.
When cases1 and 2 are combined, the solution of system (15) converges on(86)x∈Vx≤minτ1−ϖρ12/1+α,τ1−ϖρ22/1+β.
The fixed convergence time is(87)Ts≤Tmax≔2ϖρ11−α+2ϖρ2β−1.
It can be seen from (80) and (83) that V is bounded, with the result that zi,j and θ˜i,j are bounded, where i=1,…,M and j=1,…,ni, as θ^i,j=θi,j−θ˜i,j is also bounded. Because zi,1=∑j=1Maijyi−yj+hiyi−y0 and zi,1 and y0 are bounded, yi are bounded. Because zi,2=xi,2−νi,1, xi,2 are bounded. Similarly, νi,k−1 and xi,k, k=1,…,ni, are bounded. Therefore, all signals in the overall closed-loop system are CSUUB.Remark 11.
The distributed fixed-time consensus control algorithm in this paper is different from previous control algorithms. The principles are as follows. (1) Compared to the fixed-time control algorithm proposed in [16–26], our control strategy is applicable to higher-order agents with various forms. That is, a group of the followers with the different unknown nonlinear functions and order of the dynamics can be considered in this brief. (2) To solve the problem of “explosion of complexity”, instead of using dynamic surface design technique [33], we construct a smooth function MZ, from Lyapunov stability theorem, to compensate the part of Virtual Controller’s derivative ∂νi/∂θ^iθ^˙i and using RBFNNs to approximate the rest of it. (3) RBF neural networks are used here to approximate unknown complex nonlinear functions, obviating the designed control protocol just work for special systems.


## 5. Simulation Results

In this section, we present two simulation results to validate the proposed theoretical result.


### 5.1. Mathematical Example

We consider a group of one leader and four followers with the following nonlinear strict-feedback dynamics in the 2D space (i.e.,p=2) described by [33](88)x˙i,1=xi,2+fi,1xi,1,x˙i,2=ui+fi,2xi,yi=xi,1,where i=1,…,4, xi,k=xi,k1,xi,k2T; k=1,2, xi=xi,1T,xi,2TT, yi=yi,1,yi,2T, fi,1xi,1=sinxi,11xi.12,cosxi,12T, and fi,2xi=e−0.4xi,11xi,21cosxi,21,xi,22T.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 output rt is rt=r1t,r2t=4sint/20,cos0.5t+0.5sin1.5tT and the initial states and adaptive laws of the four followers are set to x10=1,0,1,−1T, x20=−1,−2,1,−3T, x30=0,−2,2,−2T, and x40=−1,−1,2,3T and θ^10=0,0,0,0T, θ^20=0,0,0,0T, θ^30=0,0,0,0T, and θ^40=0,0,0,0T, respectively. To display the consensus tracking errors between the leader and the followers, we use ei=xi,1t−rt, where i=1,…,4. And the design parameters are chosen as follows: α=1/3, β=5, ςi,11=ςi,12=ςi,21=ςi,22=5, ςi,13=ςi,23=0.1, bi=5, γi=5, ηi,1=ηi,2=0.25, ε10=ε20=0.001, and λi=0.04, where i=1,…,4. It is worth noting that ςi,11=ςi,12=ςi,21=ςi,22 should be at least ten times that of ςi,13=ςi,23 and bi setting of 5 is much better than setting it to 2. For the simulation, RBF neural networks are used to estimate unknown nonlinear functions, where Wi,1Si,1Zi,1 and Wi,2Si,2Zi,2 contain nine nodes, and the center of the Gaussian function are set as(89)ξi,1=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,ξi,2=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,where i=1,…,4. Figure 2 shows the outputs of the four followers and one leader in the 2D space. Figure 3 shows the consensus tracking errors ei and we can find it drop quickly in less than 0.3 second. Figure 4 shows the state xi,1 and xi,2 of the four followers. Figure 5 shows the adaptive laws θ^i. These figures reveal that the consensus tracking between the leader and followers is achieved satisfactorily under a directed network topology, although the followers have the unknown nonlinearities unmatched in the control input.Figure 1
Network topology for a group of one leader (L) and four followers F1 to F4.Figure 2
Output trajectories of the followers with one leader in the 2D space.Figure 3
The tracking errorsei between followers and leader.Figure 4
The states of the four followers. (a)x1,1 and x1,2. (b) x2,1 and x2,2. (c) x3,1 and x3,2. (d) x4,1 and x4,2.
(a)(b)(c)(d)Figure 5
The adaptive laws of the four followers. (a)θ^1,1 and θ^1,2. (b) θ^2,1 and θ^2,2. (c) θ^3,1 and θ^3,2. (d) θ^4,1 and θ^4,2.
(a)(b)(c)(d)


### 5.2. Practical Example

Consider multiple one-link robotic manipulators consisting of four followers and a leader. The dynamic of each follower manipulator is described as(90)Mq¨t+12mglsinqt=ut,yt=qt,where qt represents angular position; M denotes the instantaneous inertia; ut is the input torque; g is the gravitational torque; and m and l are the mass and the length of the rod, respectively.Perform coordinate transformation on system (90) and set xi,1t=qt and xi,2t=q˙t; then, system (90) can be rewritten as follows:(91)x˙i,1t=xi,2t,x˙i,2t=1Mut−12mglsinxi,1t,yit=xi,1t,where i=1,…,4. The directed network topology for this practical simulation is also shown in Figure 1. We choose aij=1on j∈Ni, aij=0 otherwise, h1=h2=h3=0 and h4=1. The leader output rt is y0t=0.25sint and the initial states and adaptive laws of the four followers are set to x10=0.1,−0.2T, x20=0.15,−0.1T, x30=0.2,0.2T, and x40=0.12,−0.2T and θ^10=0,0T, θ^20=0,0T, θ^30=0,0T, and θ^40=0,0T, respectively. To display the consensus tracking errors between the leader and the followers, we use ei=xi,1t−rt, where i=1,…,4. And the design parameters are chosen as follows: M=1, m=1, l=1, g=9.8α=1/3, β=5, ςi,11=ςi,12=ςi,21=ςi,22=5, ςi,13=ςi,23=0.1, bi=5, γi=5, ηi,1=ηi,2=0.25, ε10=ε20=0.001, and λi=0.04, where i=1,…,4. For the simulation, RBF neural networks are used to estimate unknown nonlinear functions, where Wi,1Si,1Zi,1 and Wi,2Si,2Zi,2 contain nine nodes, and the center of the Gaussian function are set as(92)ξi,1=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,ξi,2=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,where i=1,…,4. Then, by using the scheme in [36], simulation results are given in Figures 6(a) and 7(a). Meanwhile, by employing our method, simulation results are given in Figures 6(b), 7(b), 8, and 9.Figure 6
Output trajectories of the followers with one leader by using two methods. (a) Output trajectories by the method in [36]. (b) Output trajectories by our method.
(a)(b)Figure 7
The tracking errorsei between followers and leader by using two methods. (a) Error curves by the method in [36]. (b) Error curves by our method.
(a)(b)Figure 8
The statesxi,2 of the four followers.
(a)(b)(c)(d)Figure 9
The adaptive lawsθ^i,2 of the four followers.
(a)(b)(c)(d)It is shown in Figures6 and 7 that the tracking performance by using our method is better than [36], since our proposed scheme has faster convergence and higher precision. Figure 8 shows the states xi,2 of the four followers. Figure 9 shows the response curves of adaptive laws θ^i,2. These figures reveal that the consensus tracking between the leader and followers is achieved satisfactorily under a directed network topology.


## 5.1. Mathematical Example

We consider a group of one leader and four followers with the following nonlinear strict-feedback dynamics in the 2D space (i.e.,p=2) described by [33](88)x˙i,1=xi,2+fi,1xi,1,x˙i,2=ui+fi,2xi,yi=xi,1,where i=1,…,4, xi,k=xi,k1,xi,k2T; k=1,2, xi=xi,1T,xi,2TT, yi=yi,1,yi,2T, fi,1xi,1=sinxi,11xi.12,cosxi,12T, and fi,2xi=e−0.4xi,11xi,21cosxi,21,xi,22T.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 output rt is rt=r1t,r2t=4sint/20,cos0.5t+0.5sin1.5tT and the initial states and adaptive laws of the four followers are set to x10=1,0,1,−1T, x20=−1,−2,1,−3T, x30=0,−2,2,−2T, and x40=−1,−1,2,3T and θ^10=0,0,0,0T, θ^20=0,0,0,0T, θ^30=0,0,0,0T, and θ^40=0,0,0,0T, respectively. To display the consensus tracking errors between the leader and the followers, we use ei=xi,1t−rt, where i=1,…,4. And the design parameters are chosen as follows: α=1/3, β=5, ςi,11=ςi,12=ςi,21=ςi,22=5, ςi,13=ςi,23=0.1, bi=5, γi=5, ηi,1=ηi,2=0.25, ε10=ε20=0.001, and λi=0.04, where i=1,…,4. It is worth noting that ςi,11=ςi,12=ςi,21=ςi,22 should be at least ten times that of ςi,13=ςi,23 and bi setting of 5 is much better than setting it to 2. For the simulation, RBF neural networks are used to estimate unknown nonlinear functions, where Wi,1Si,1Zi,1 and Wi,2Si,2Zi,2 contain nine nodes, and the center of the Gaussian function are set as(89)ξi,1=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,ξi,2=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,where i=1,…,4. Figure 2 shows the outputs of the four followers and one leader in the 2D space. Figure 3 shows the consensus tracking errors ei and we can find it drop quickly in less than 0.3 second. Figure 4 shows the state xi,1 and xi,2 of the four followers. Figure 5 shows the adaptive laws θ^i. These figures reveal that the consensus tracking between the leader and followers is achieved satisfactorily under a directed network topology, although the followers have the unknown nonlinearities unmatched in the control input.Figure 1
Network topology for a group of one leader (L) and four followers F1 to F4.Figure 2
Output trajectories of the followers with one leader in the 2D space.Figure 3
The tracking errorsei between followers and leader.Figure 4
The states of the four followers. (a)x1,1 and x1,2. (b) x2,1 and x2,2. (c) x3,1 and x3,2. (d) x4,1 and x4,2.
(a)(b)(c)(d)Figure 5
The adaptive laws of the four followers. (a)θ^1,1 and θ^1,2. (b) θ^2,1 and θ^2,2. (c) θ^3,1 and θ^3,2. (d) θ^4,1 and θ^4,2.
(a)(b)(c)(d)


## 5.2. Practical Example

Consider multiple one-link robotic manipulators consisting of four followers and a leader. The dynamic of each follower manipulator is described as(90)Mq¨t+12mglsinqt=ut,yt=qt,where qt represents angular position; M denotes the instantaneous inertia; ut is the input torque; g is the gravitational torque; and m and l are the mass and the length of the rod, respectively.Perform coordinate transformation on system (90) and set xi,1t=qt and xi,2t=q˙t; then, system (90) can be rewritten as follows:(91)x˙i,1t=xi,2t,x˙i,2t=1Mut−12mglsinxi,1t,yit=xi,1t,where i=1,…,4. The directed network topology for this practical simulation is also shown in Figure 1. We choose aij=1on j∈Ni, aij=0 otherwise, h1=h2=h3=0 and h4=1. The leader output rt is y0t=0.25sint and the initial states and adaptive laws of the four followers are set to x10=0.1,−0.2T, x20=0.15,−0.1T, x30=0.2,0.2T, and x40=0.12,−0.2T and θ^10=0,0T, θ^20=0,0T, θ^30=0,0T, and θ^40=0,0T, respectively. To display the consensus tracking errors between the leader and the followers, we use ei=xi,1t−rt, where i=1,…,4. And the design parameters are chosen as follows: M=1, m=1, l=1, g=9.8α=1/3, β=5, ςi,11=ςi,12=ςi,21=ςi,22=5, ςi,13=ςi,23=0.1, bi=5, γi=5, ηi,1=ηi,2=0.25, ε10=ε20=0.001, and λi=0.04, where i=1,…,4. For the simulation, RBF neural networks are used to estimate unknown nonlinear functions, where Wi,1Si,1Zi,1 and Wi,2Si,2Zi,2 contain nine nodes, and the center of the Gaussian function are set as(92)ξi,1=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,ξi,2=−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52−2−1.5−1−0.500.511.52,where i=1,…,4. Then, by using the scheme in [36], simulation results are given in Figures 6(a) and 7(a). Meanwhile, by employing our method, simulation results are given in Figures 6(b), 7(b), 8, and 9.Figure 6
Output trajectories of the followers with one leader by using two methods. (a) Output trajectories by the method in [36]. (b) Output trajectories by our method.
(a)(b)Figure 7
The tracking errorsei between followers and leader by using two methods. (a) Error curves by the method in [36]. (b) Error curves by our method.
(a)(b)Figure 8
The statesxi,2 of the four followers.
(a)(b)(c)(d)Figure 9
The adaptive lawsθ^i,2 of the four followers.
(a)(b)(c)(d)It is shown in Figures6 and 7 that the tracking performance by using our method is better than [36], since our proposed scheme has faster convergence and higher precision. Figure 8 shows the states xi,2 of the four followers. Figure 9 shows the response curves of adaptive laws θ^i,2. These figures reveal that the consensus tracking between the leader and followers is achieved satisfactorily under a directed network topology.


## 6. Conclusion

In this paper, the distributed adaptive fixed-time tracking consensus control problem is investigated for multiple strict-feedback systems with uncertain nonlinearities under a directed graph topology. The function approximation technique using RBF neural networks was employed to compensate for the unknown nonlinearities unmatched in the control input of followers. From Lyapunov stability and fixed-time control theory, by using distributed coordinate conversion, designed virtual controllers, and actual controllers. Based on Lyapunov stability theory, it is proved that the developed distributed consensus protocol can guarantee all the signals in multiple strict-feedback systems are bounded, and the tracking consensus errors converge to a small neighborhood of zero in fixed time. Finally, the effectiveness of the proposed approach was proved through two simulations. And in the following research, we will consider the application of fixed-time control in pure-feedback nonlinear systems with time delay.



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