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.

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 [25].

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 [611] 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 [1618]. 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 [2029] 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 [2427], 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 [3032] 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 [1618, 2427] 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. Section 2 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 of followers, labeled as agents 1 to , and a leader, labeled as an agent 0 under a directed communication graph topology. The dynamic models of followers can be described as the form of the following pure-feedback nonlinear systems:where , , with , , and are system state variables, system input, and system output, respectively; are unknown smooth nonaffine functions; are unknown but bounded disturbances.

We utilize the mean value theorem to solve the nonaffine structures in pure-feedback systems:where with , , with , and are known at the given time .

Therefore, system (1) can be rewritten as

Assumption 1 (see [37]). are unknown nonlinear functions and bounded, and there exists a positive constant such that .
For ease of description, we defined vector functions , where is the leader’s output and is its derivative.

Assumption 2. Suppose that the leader’s vector functions are known smooth continuous functions and bounded. with , where being known compact sets and is an -order differentiable function.

2.2. Graph Theory

The communication between agents can be represented by a directed graph , where denotes the set of nodes, and denotes the set of edges. An edge means that agent can obtain information from agent , but not vice versa, where and are the parent node and child node, respectively. is the set of neighbors of a node , which is the set of nodes with edges incoming to node . is the related adjacency matrix, if , ; otherwise, . Self-edges are not allowed, i.e., . The Laplacian matrix for graph is denoted by with and for , and where ; is the diagonal element of the degree matrix . A directed graph has a directed spanning tree if there exists at least one agent that has directed paths to all other agents.

If including a leader agent , the communication topology for the agents is described by a directed graph , where and . The connection matrix between the followers and the leader can be expressed as where if and only if can directly obtain the leader’s information; otherwise, .

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:where the input vector , is the weight vector; is the radial basis function vector with neural network node number . The basis function is always chosen as the following form of Gaussian function:where is the width of the Gaussian function, and is the center of the receptive field. In theory, as long as a sufficient number of nodes are selected, RBF neural network can approximate any continuous function in a compact set with arbitrary precision .where is the approximation error and satisfies , and is the ideal weight vector, which is chosen as the value of that minimizes for all , i.e.,

Remark 1. In this paper, we choose with , where are the estimates of the unknown constants , are positive design parameters and related to Assumption 1, and is the norm.

Assumption 3 (see [39]). There exist unknown constants such that , , .

Lemma 1 (see [40]). Consider the Gaussian function (5). has an upper bound such that , where .

2.4. Fixed-Time Control

Consider the following nonlinear system,where is a system variable, and 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 equilibrium of system (8) is finite-time convergent if there are an open neighborhood of the origin and a positive definite function , such that and , for any initial condition , where is the settling time function. The equilibrium of system (8) is finite-time stable if it is Lyapunov stable and finite-time convergent. Moreover, when , then the equilibrium 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 is uniformly bounded for any initial states, that is, such that , .

Lemma 2 (see [39]). Considering system (8), suppose that there exists a continuous differentiable positive definite function such that the following inequality holds:where , , , , and . Then, the equilibrium of system (8) is fixed-time stable and the convergence time function satisfies the following inequality:

The residual set of the solution of system (8) is given by

Lemma 3 (see [42]). For , , , , the following inequalities hold:

Lemma 4 (see [40]) (see [43]). For and any positive constant , the following inequality holds:

Lemma 5. For and any positive constant , then the following inequality is satisfied:


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 the th follower is defined as

To meet the requirement of output performance constraint, a positive decreasing smooth function as the following is selected:where , are proper parameters, is a positive constant, and .

Defining the following state transformation as the error constraint variable of the th follower,where and are proper design parameters. satisfies the condition given in (17) and satisfies

The time derivative of is

For ease of writing, we obtainwhere

Remark 2. From the definition of and , it is obvious that when tracking error approaches the boundary , increases as a result. Then, by choosing proper parameters, the tracking error will be forced within prescribed boundary, i.e., .

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:where , , and represents the virtual controller of the th subsystem of the th follower, and is given in (18).

In this paper, we employ RBF neural network to approximate uncertain functions ,

From Young’s inequality, complete square formula, and Lemma 3, the following inequalities hold:where with , is the input vector, and , , , , , , and are positive parameters.

Step 1. Choosing the Lyapunov function asFrom (3), (16), and (18)–(23), we can obtain the time derivative of :The virtual controller is defined aswhere , , , and are positive design parameters. and are defined by the following equalities:where , is the positive design parameter, and coefficients with are calculated using the following equation:where , , , , and .

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 , since when , there we just add an additional term and it clearly does not affect the final conclusion.
Substituting (31) into (30) yieldswhere .
From (24)–(26), we havewhere .
The adaptive law is defined aswhere with is the positive parameter.
From Assumption 3 and (37), we havewhere , , and .
According to (27), (28), (32), and (33), and when ,

Step 2. From (3) and (23), we can obtainwhere .
Construct Lyapunov function asThe derivative of is written aswhere .

Remark 4. is a smooth function that is used to compensate .
The virtual controller is defined aswhere , , , and are positive design parameters.
According to (24)–(26) and substituting (43) into (42), we havewhere .
Then, the adaptive law is defined asFrom Assumption 3 and (45), we havewhere , , and .
According to (27), (28), (32), and (33) and when ,From Lemmas 1, 4, and 5 and (37), we haveTherefore, can be defined aswith the result thatSubstituting (50) into (47), we can obtainStep k. : from (3) and (23), we havewhere .Construct Lyapunov function asThe derivative of iswhere .
The virtual controller is defined aswhere , , , and are positive design parameters.
According to 24)–(26) and substituting (55) into (54), we have