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

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. +e 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. +e 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.


Introduction
In the past decade, the finite-time consensus control of the multiagent has become one of the hot topics, since the finitetime 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][3][4][5][6][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 secondorder integrator agents. ereafter, by using the Lyapunovfunction-based method, several finite-time consensus control strategies were proposed for high-order uncertain MASs [13][14][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. at 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][17][18][19][20][21][22][23][24][25][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][17][18][19][20] were designed for MASs with first order or second order. Only a few results [21][22][23][24][25][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 strictfeedback 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 strictfeedback nonlinear systems. Nevertheless, there is a problem with using backstepping to design a controller for the strictfeedback 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][34][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. e 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][17][18][19][20][21][22][23][24][25][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 M(Z)derived from Lyapunov stability to compensate the part of (z] i /zθ i )θ · i in the differential of the virtual controller v i , while the other part of the differential is approximated by RBF neural networks. is method greatly simplifies the design process of the controller while ensuring Lyapunov stability. (3) In [36][37][38][39], their controller designed have a power function similar to z 2k− 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 f i (·) in MASs to make our designed fixed-time control can suitable more systems with complex dynamics. e organization for the remaining part of this paper is given below. Section 2 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.

Preliminaries and Problem Description
2.1. Graph eory. Let G � (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. e set of neighbors of a node i is N i � j | (j, i) ∈ ε , which is the set of nodes with edges incoming to node i. e weighted adjacency matrix A directed path from node i 1 to node i k is a sequence of edges of the form (i 1 , i 2 ), (i 2 , i 3 ), . . . , (i k− 1 , i k ) 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. e RBF neural network is defined as follows: where W � [w 1 , . . . , w l ] T ∈ R l is the weight vector, l > 0 is the number of nodes of the neural network, Z ∈ Ω Z ⊂ R q is the input of the RBF neural network, q is the input di- , . . . , s l (Z)] T ∈ R l is the basis vector function, and s i (Z) is the output of the i th neural node. A Gaussian function is always chosen as s i (Z), i.e., where r i is the width of the base function and ξ i � [ξ i1 , . . . , ξ iq ] T 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 f(Z) in a compact set Ω Z ∈ R q with arbitrary accuracy ε: where δ(Z) is the approximation error with |δ(Z)| ≤ ε and W * is the given ideal constant weight vector, which is defined as follows: 2 Complexity In this paper, let θ � max 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); ‖S(Z)‖ has an upper bound such that Remark 1. Lemma 1 has been proved in [41,42]. Since 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.

Fixed Time
Definition 1. Consider the following nonlinear system: where x ∈ R n and f: R + × R n ⟶ R n , and assume that the origin is an equilibrium point.
Lemma 2 (see [22]). If there exist design constants ϕ 1 > 0, ϕ 2 > 0, α ∈ (0, 1), and β ∈ (1, +∞) such that where V(x) is a continuous differentiable positive definite function; system (5) is global fixed-time stable, and the fixed convergence time satisfies Remark 2. e sufficient conditions and convergence time for finite-time and fixed-time consensus control schemes are shown in Table 1.Where μ 1 , μ 2 , ρ 1 , and ρ 2 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 x(0), while the fixed-time control is not.
The residual set of the solution of system (5) is given by Lemma 4 (see [43]). Let x 1 , x 2 , . . . , x n ≥ 0. en, Lemma 5 (see [44]). For any variable x ∈ R and any positive constant κ, the following relationship holds: Lemma 6 For y ≥ x > 0, x, y ∈ R and any positive constant δ, then satisfying Proof where i � 1, . . . , M, k � 1, . . . , n i − 1, . , x T i,n i ] T ∈ R pn i , and u i ∈ R p are the state vector and the control input of the i th follower, respectively, and y i ∈ R p is the output of the ith It is assumed that the leader's motion is independent of the followers' motion. e communication topology for the M + 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 a ij � 0 otherwise, and a ii � 0. en, the Laplacian matrix L is defined as 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 where p � 1. e analysis and main results still hold for any dimension p by using the Kronecker product. e leader output signal r(t) ∈ R p is an n i -order differentiable and bounded function and available for the ith followers satisfying 0 ∈ N i , i � 1, . . . , M.
Definition 2 (see [33]). e 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 c 1 > 0 and c 2 > 0, and the bounds β 1 > 0 and β 2 > 0, independent of t 0 , and for every α 1 ∈ (0, c 1 ) and e objective of this brief is to design RBF-neural networks-based distributed fixed-time consensus control laws u i for M followers (15) with unknown nonlinearities so that, under the directed graph, the follower outputs y i synchronize to the dynamic leader output r within fixed time while all signals in the total closed-loop systems are bounded.

Remark 5.
e 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]. erefore, 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.
at 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).

Distributed Fixed-Time Consensus
Controller Design e design procedure on the ith follower contains n i steps. e distributed backstepping design coordinate transformation is as follows: where i � 1, . . . , M and k � 2, . . . , n i and v i,k is the virtual controller of the kth subsystem of ithfollower. An RBF neural network is used in this paper to approximate the unknown functions f i.k (Z i,k ): and the inequalities involved in the following text are as below: where and c i are positive design parameters.

Complexity
Remark 9 (see [46]). Based on (28) and (29), when ‖z i,1 ‖ < ε 10 , there is an additional term in (36): /2) . Note that if ‖z i,1 ‖ < ε 10 , this additional term is obviously limited by some smaller constant ε 11 , so the structure of (35) is retained, while the constant term C i,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 where Construct a Lyapunov function as e derivative of V i,2 is written as where is a smooth function that is used to overcome the design difficulty of ((z] i,1 )/(zθ i,1 ))θ · i,1 . e virtual controller ] i,2 is defined as where ς i,21 , ς i, 22 , ς i, 23 , and η i,2 are positive design parameters.
Choose the adaptive law θ i,2 as Combining (19)- (21) and Assumption 1 and substituting (41) and (42) into (38) and adopting the same design method as in Step 1 yields where Complexity 7 According to (22), (23), and Remark 9, we consider only ‖z i,2 ‖ ≥ ε 20 . en, (43) can be written as It can be seen from (44) that defining the design smooth function M i,1 (Z i,2 ) to overcome the design difficulty of (z] i,1 /zθ i,1 )θ · i,1 is one of the difficulties of designing the controllers in this paper.

Construct a Lyapunov function as
e derivative of V i,k is written as 8 Complexity (52) where ς i,k1 , ς i,k2 , ς i,k3 , and η i,k are positive design parameters.

Construct a Lyapunov function as
e derivative of V i,n i is written as 10 Complexity Defining virtual controller u i as where ς i,n i 1 , ς i,n i 2 , ς i,n i 3 , and η i,n i are positive design parameters. Choose the adaptive law θ i,n i as Combining (19)- (21) and Assumption 1 and substituting (65) and (66) into (63) yields According to (22), (23), and Remark 9, we consider only ‖z i,n i ‖ ≥ ε n i 0 . en, (67) can be written as Complexity 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.
It can be seen from (80) and (83) that V is bounded, with the result that z i,j and θ i,j are bounded, where i � 1, . . . , M and j � 1, . . . , n i , as Similarly, ] i,k− 1 and x i,k , k � 1, . . . , n i , are bounded. erefore, all signals in the overall closed-loop system are CSUUB.

Remark 11.
e distributed fixed-time consensus control algorithm in this paper is different from previous control algorithms. e principles are as follows. (1) Compared to the fixed-time control algorithm proposed in [16][17][18][19][20][21][22][23][24][25][26], our control strategy is applicable to higher-order agents with various forms. at 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 M(Z), from Lyapunov stability theorem, to compensate the part of Virtual Controller's derivative (z] i /zθ i ) _ θ 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.

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

Mathematical Example.
We consider a group of one leader and four followers with the following nonlinear strictfeedback dynamics in the 2D space (i.e., p � 2) described by [33] Complexity 13 where i � 1, . . . , 4, e directed network topology for the simulation is shown in Figure 1 22 should be at least ten times that of ς i,13 � ς i, 23 and b i 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 W i,1 S i,1 (Z i,1 ) and W i,2 S i,2 (Z i,2 ) contain nine nodes, and the center of the Gaussian function are set as 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 e i and we can find it drop quickly in less than 0.3 second. Figure 4 shows the state x i,1 and x i,2 of the four followers. Figure 5 shows the adaptive laws θ i . ese 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.

Practical Example.
Consider multiple one-link robotic manipulators consisting of four followers and a leader. e dynamic of each follower manipulator is described as where q(t) represents angular position; M denotes the instantaneous inertia; u(t) 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 x i,1 (t) � q(t) and x i,2 (t) � _ q(t); then, system (90) can be rewritten as follows: where i � 1, . . . , 4. e directed network topology for this practical simulation is also shown in Figure 1. We where i � 1, . . . , 4. en, 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.
It is shown in Figures 6 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 x i,2 of the four followers. Figure 9 shows the response curves of adaptive laws θ i,2 . ese figures reveal that the consensus tracking between the leader and followers is achieved satisfactorily under a directed network topology.

Conclusion
In this paper, the distributed adaptive fixed-time tracking consensus control problem is investigated for multiple strictfeedback systems with uncertain nonlinearities under a    e 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.

Data Availability
No data were used to support this study.

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