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

,is 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. ,e 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.


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][3][4][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][7][8][9][10][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. ereafter, to solve the consensus control problems of high-order uncertain MASs, several finite-time control strategies were proposed in [16][17][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][21][22][23][24][25][26][27][28][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 secondorder, and only a few focus on high-order MASs. Although, in [24][25][26][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][31][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. e 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][17][18][24][25][26][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. erefore, 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. e 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.

Problem Statement.
e considered nonlinear MASs consist of M followers, labeled as agents 1 to M, and a leader, labeled as an agent 0 under a directed communication graph topology. e dynamic models of M followers can be described as the form of the following pure-feedback nonlinear systems: and y i (t) ∈ R are system state variables, system input, and system output, respectively; f i,k (·) are unknown smooth nonaffine functions; d i,k (·) are unknown but bounded disturbances.
We utilize the mean value theorem to solve the nonaffine structures in pure-feedback systems: with 0 < μ k < 1, and x i,k0 are known at the given time t 0 .
erefore, system (1) can be rewritten as Assumption 1 (see [37]). g i,k (·) are unknown nonlinear functions and bounded, and there exists a positive constant b i such that b i ≤ |g i,k (·)|. For ease of description, we defined vector functions y 0,i � [y 0 , y (1) 0 , . . . , y (i) 0 ] T , where y 0 is the leader's output and y (i) 0 is its i th derivative.
Assumption 2. Suppose that the leader's vector functions y 0,i are known smooth continuous functions and bounded.
where Ω 0,i being known compact sets and y 0 is an n i -order differentiable function.

Graph eory.
e communication between agents can be represented by a directed graph G � (V, E, A), where V � N 1 , N 2 , . . . , N M 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. N i � 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 � [a i,j ] ∈ R M×M is the related adjacency matrix, if (j, i) ∈ E, a ij > 0; otherwise, a i,j � 0. Self-edges are not allowed, i.e., a i,j � 0. e Laplacian matrix for graph G is denoted by ,j≠i a ij 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 agent N 0 , the communication topology for the M + 1 agents is described by a directed graph  [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 Z ∈ Ω Z ⊂ R q , W � [w 1 , w 2 , . . . , w l ] T ∈ R l is the weight vector; S(Z) � [s 1 (Z), s 2 (Z), . . . , s l (Z)] T ∈ R l is the radial basis function vector with neural network node number l > 0. e basis function S(Z) is always chosen as the following form of Gaussian function: where η i is the width of the Gaussian function, and μ i � [μ i1 , μ i2 , . . . , μ iq ] T 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 f(Z) in a compact set Ω Z ⊂ R q with arbitrary precision ε.
where δ(Z) is the approximation error and satisfies |δ(Z)| ≤ ε, and W * is the ideal weight vector, which is chosen as the value of W that minimizes ε for all Z ∈ Ω Z , i.e., Remark 1. In this paper, we choose are the estimates of the unknown constants θ i,j , b i are positive design parameters and related to Assumption 1, and ‖ · ‖ is the norm.

Fixed-Time Control.
Consider the following nonlinear system, _ where x(t) ∈ R n is a system variable, and f(x): R n ⟶ R n 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]). e equilibrium x � 0 of system (8) where T(x 0 ) is the settling time function. e equilibrium x � 0 of system (8) is finite-time stable if it is Lyapunov stable and finite-time convergent. Moreover, when U � D � R n , then the equilibrium x � 0 is globally finitetime stable.

Tracking Error Constraint.
In this paper, we will investigate the output feedback tracking control problem with prescribed performance. e local tracking error for the ith follower is defined as To meet the requirement of output performance constraint, a positive decreasing smooth function as the following is selected: where ξ 0 > 0, ξ ∞ > 0 are proper parameters, a is a positive constant, and Defining the following state transformation as the error constraint variable of the ith follower, where σ 1 and σ 2 are proper design parameters. F b satisfies the condition given in (17) and q(e i (t)) satisfies For ease of writing, we obtain where Remark 2. From the definition of z i,1 and q(e i (t)), it is obvious that when tracking error approaches the boundary F b , z i,1 increases as a result. en, by choosing proper parameters, the tracking error will be forced within prescribed boundary, i.e., −

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. e coordinate transformation is as follows: where i � 1, 2, . . . , M, k � 2, 3, . . . , n i , and v i,k represents the virtual controller of the kth subsystem of the ith follower, and z i,1 is given in (18). In this paper, we employ RBF neural network to approximate uncertain functions f i,k (Z i,k ), From Young's inequality, complete square formula, and Lemma 3, the following inequalities hold: where Step 1. Choosing the Lyapunov function as From (3), (16), and (18)-(23), we can obtain the time derivative of V i,1 : where ζ i, 11 , ζ i,12 , ζ i, 13 , and η i,1 are positive design parameters. S zi,1,1 and S zi,1,2 are defined by the following equalities: Mathematical Problems in Engineering where k � 1, . . . , n i , ε k0 is the positive design parameter, and coefficients a j with j � 1, . . . , n i are calculated using the following equation: Remark 3. e above process is to eliminate the problem of fixed-time controller's singularity. In this paper, we just consider the situation of |z i,k | ≥ ε k0 , since when |z i,k | < ε k0 , there we just add an additional term and it clearly does not affect the final conclusion.

Construct Lyapunov function as
where Mathematical Problems in Engineering e actual controller is defined as where ζ i,n i 1 , ζ i,n i 2 , ζ i,n i 3 , and η i,n i are positive design parameters. e adaptive law θ i,n i is defined as Substituting (69) and (68) into (66), we can obtain According to (27), (28), (32), and (33) and when |z i,n i ≥ ε n i 0 |, Same to the process of (60), the M i,n i − 1 (Z i,n i ) can be defined as with the result that 12 Mathematical Problems in Engineering 2 ζ i,11 , ζ i,21 , . . . , ζ i,n 1 ,1 and 22 , . . . , ζ i,n i 2 and from Lemma 3, we have

Substituting (73) into (71) yields
en, (74) can be rewritten as Up to this point, the design of fixed-time controller is finished.
Combining Cases 1 and 2, the solution of system (1) converges on (88) e fixed convergence time is It can be seen from (82)  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][17][18][24][25][26][27] on high-order MASs with nonlinear strictfeedback, 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.

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

Example 1: Numerical Example.
Consider the secondorder nonlinear MASs consisting of four followers and one leader, and the dynamic of each agent N i is given as the following pure-feedback form: where x i,1 and x i,2 , with i � 1, . . . , 4, are the state variables, u i is the system input, and y i is the system output.
e directed network topology for the simulation is shown in Figure 1. We choose a ij � 1 on j ∈ N i , a ij � 0; otherwise, h 1 � h 2 � h 3 � 0, and h 4 � 1. e leader's output y 0 (t) is 0.5 sin(1.5t) + cos(0.5t) and the initial states and adaptive laws of the four followers are set to To display the tracking errors between the leader and the followers, we use e i � |y i (t) − y 0 (t)|, where i � 1, . . . , 4, and the design parameters are chosen as follows: 23 should be at least ten times that of ζ i,11 � ζ i,12 � ζ i,21 � ζ i, 22 and b i setting of 5 is much better than setting it to 2. e performance boundary function's design parameters ξ 0 , a, ξ ∞ , σ 1 , and σ 2 are chosen as 0.9, 1, 0.17, 2, and 2, respectively. en, the virtual/actual control signals and the adaptive laws are, respectively, designed as follows: 20 , where i � 1, . . . , 4.
In this simulation example, we set the width of the RBF neural network as four. W i,1 S i,1 (Z i,1 ) and W i,2 S i,2 (Z i,2 ) contain seven and five nodes, respectively. e center of the Gaussian function is set as In Figure 2, the tracking curves of the output signals y i 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,j (i � 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 u i (i � 1, . . . , 4) are all bounded for t ≥ 0. All the above simulation results illustrate the validity of the proposed method.

Example 2: Physical
Example. Consider the following multiple electromechanical systems consisting of two followers and one leader. e dynamic of each electromechanical system is described as where q is the angular motor position, V e is the input control voltage, and I is the motor armature, M � (J/K τ ) + (mL 2 0 /3K τ ) + (M 0 L 2 0 /K τ ) + (2M 0 R 2 0 /5K τ ), B � (B 0 /K τ ), and N � (mL 0 g/2K τ ) + (M 0 L 0 g/K τ ), Δ 1 ( _ q, q), and Δ 2 ( _ q, q) denote the model errors. J denotes the rotor inertia, K τ is the coefficient to characterize the electromechanical conversion of armature current to torque, L 0 is the link length, m means the link mass, M 0 stands for the load mass, B 0 is the coefficient of viscous friction at the joint, R 0 is the radius of the load, g is the gravity coefficient, R is the armature resistance, L is the armature inductance, and K B is the back-emf coefficient.
Leader Figure 1: Network topology for a group of one leader and four followers F1 to F4. 1 , and u i � V e . en, the electromechanical system can be rewritten as  Assume that there are three followers and one leader in MASs, and the communications are h 1 � 1, h 2 � h 3 � 0, a 2,1 � a 3,1 � 1, and a 1,2 � a 1,3 � a 2,3 � a 3,2 � 0. e parameters are chosen as follows: M � 0.0642, N � 1.1408, B � 0.0181, L � 0.025, K B � 0.9, and R � 5.0. e leader's output y 0 (t) is sin(0.5t) + 0.5 sin(t). e predefined performance function' parameters ξ 0 , a, ξ ∞ , σ 1 , and σ 2 are chosen as 0.9, 7, 0.17, 1, and 1, respectively. e purpose of    this example is to design a fixed-time controller to make each follower's output y i track the leader's output signal y 0 within fixed time.
en, the virtual/actual control signals and the adaptive laws are, respectively, designed as follows:     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: ζ i,11 � ζ i,12 � 10, e 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 Figure 8, the tracking curves of the output signals y i 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.

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. e effectiveness of the work has been proved by stability analysis along with simulation results.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Pinwei Li put forward research topics; designed the research plan; drafted the manuscript; and revised the manuscript. Jiyang Dai was involved in guidance support; obtaining research funding; and final review of the manuscript. Jin Ying checked the English grammar and guided the simulation programming.