Predefined-Time Adaptive Neural-Network-Based Consensus Tracking Control for Nonlinear Multiagent Systems With Zero Tracking Error

This paper is dedicated to addressing the predefined-time consensus tracking control problem for unknown high-order nonlinear multiagent systems. The prominent difference compared with some existing papers is that the follower is modeled in the form of non-strict-feedback structure. Meanwhile, instead of being constants, the real control gains are unknown functions. A distinct advantage in our work is that the outputs of followers are able to track the output of leader within the time specified in advance. In order to get our desired predefined-time controller, radial basis function (RBF) neural networks (NNS) are applied to compensate those unknown nonlinearities. Then in the design framework of adaptive backstepping, the predefined-time virtual control laws are presented and their derivatives are approximated by using finite-time differentiators. Under our proposed predefined-time controller, it is rigorously demonstrated that the whole closed-loop system remains stable and all outputs of the followers track the reference signal in predefined time. In the end, a simulation example is given to ulteriorly verify the efficacy of the suggested predefined-time control scheme.


I. INTRODUCTION
Consensus tracking control problem for distributed multiagent systems (MASs) has attracted a lot of attention from many great experts and scholars during the past years. The reason is that the consensus control of MASs plays a significant role in many practical engineering, for example, traffic flow control, robot team formation, sensor networks, ship navigation, cooperative surveillance [1]- [7], etc. The early literature on consensus tracking control paid more attention to linear MASs (see, e.g. [8]- [12]). It is remarkably, however, in the real world, numerous practical systems are nonlinear or include nonlinear elements. Consequently, the consensus tracking control for nonlinear MASs is a crucial control issue and it is becoming more and more challenging as nonlinear MASs become more and more intricate. Lots of interesting results about this topic have been reported. For The associate editor coordinating the review of this manuscript and approving it for publication was Dipankar Deb .
instance, the authors in [13]- [16] discussed the consensus tracking control problem of some low-order nonlinear MASs. However, the control schemes in [13]- [16] are based on the hypothesis that some part of the system's nonlinearities are known a prior. When the nonlinear system exists completely unknown nonlinearities, those control methods may not be able to get our desired control performance. Fortunately, lately advanced intelligent control algorithm which combines the adaptive control technique with the framework of fuzzy logic systems (FlSs) or NNs systems have been developed as a powerful tool to deal with nonlinear systems with completely unknown nonlinearities. For instance, [17]- [19] use FLSs and [20], [21] use NNs to deal with the unknown nonlinear functions of the systems, and the typical adaptive control scheme is further applied to develop controllers (see, e.g. [22]- [24]).
More recently, those adaptive neural network or fuzzy control schemes for common nonlinear systems have been utilized to distributed MASs (see, e.g. [25]- [29]). To just VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ name a few, the authors in [25] developed a neuralnetworks-based adaptive control protocol for a class of nonlinear MASs with time delay and external noises, and the authors in [26] focused on propose an adaptive NN controller of the nonlinear MASs that exist unknown control direction and full state constraints. In [27], a neural networks-based distributed adaptive controller was proposed for nonlinear MASs to achieve that synchronization errors satisfy a prescribed performance. However, in existing literature, most of adaptive neural network or fuzzy control methods for MASs are based on the premise that the followers are modeled by strict-feedback systems. In the real world, numerous agents in MASs appear in the form of non-strict-feedback. For example, the Helicopter model in [30] was the non-strict-feedback structure. The authors in [28] researched the consensus tracking control problem for distributed nonlinear MASs, in which each follower under consideration is modeled by a nonlinear non-strict-feedback system. But even if the existing works on MASs considered the non-strict-feedback structure, it is difficult to achieve the tracking errors converge to zero. What's more, convergence rate is also an important issue in the design of consensus scheme. In the actual industry process, lots of practical controlled systems are demanded to achieve a steady response from transient response quickly, such as power systems, robotic systems, electronic systems, and mechanic systems. Consequently, it is extremely important in practical applications to achieve the consensus control for MASs in finite time. In this context, the finite-time consensus control algorithm that is capable of ensuring that the output of each follower of the multi agent systems tracks the output of the leader in finite time has attracted extensive concern, and some interesting results on this topic have been reported (see, e.g. [33][34][35][36]). Nevertheless, the bound of the setting time of the finite-time stability multiagent systems depends on the initial conditions of agents. In order to eliminate dependence, a class of fixed-time consensus protocols for MASs were researched in [37][38][39][40]. The convergence time of the fixed-time stability systems can be upper bounded via a value that does not depend on the initial conditions and is only determined by the design parameters.
Although having a significant advantage than the finite-time stability control method, the fixed-time stability control scheme is always very hard to find a direct relation between the convergence time upper bound and controller parameters, which leads to a great difficulty in designing and tuning parameters to meet the requirement of convergence time. To overcome those shortcomings mentioned above, a stronger form of stability named predefined-time stability was investigated in [41]. For predefined-time stability systems, the upper bound of the fixed stabilization time appears explicitly in tuning parameters, which means that the convergence time bound of the controlled system is able to be predefined by user requirement and the design parameters of the controller are capable of being decided directly from the convergence time bound. Due to this attractive property, the predefined-time stability has become a hot research topic in the field of control and some interesting results have been reported. For example, [42] focused on Lyapunov-like conditions for a class of dynamical systems to achieve predefined-time stability. The authors in [43] presented a global predefined-time control scheme for a class of nonlinear strict-feedback systems to achieve that the tracking error converge to a predefined accuracy and the authors in [44] proposed a predefined-time control method for high-order integrator systems to achieve predefined-time convergence. In [45], the predefined-time consensus tracking control problem was researched for second-order multiagent systems. Whereas, to our knowledge, there are no predefined-time results for high-order nonlinear non-strict-feedback multiagent systems so far. The reason is that the complexity of such systems makes it very hard to get the corresponding results.
Inspired by the above discussions, we propose a novel predefined-time adaptive neural-networks-based control scheme to solve the leader-following consensus tracking problem of the high-order nonlinear non-strict-feedback MASs. To sum up, the main contributions in our paper can be listed as follows: (i) Firstly, under our proposed controller, the outputs of the followers are able to track the output of leader within the time specified in advance, which addresses a momentous design obstacle in the field of control. Since a great deal of practical applications are characterized for demanding hard time response constraints. (ii) Secondly, in spite of having the non-strict-feedback structure of each follower, the considered predefined-time adaptive control scheme still ensures that the tracking errors converge to zero instead of a small neighborhood of zero. This means our proposed control scheme has an even bigger superiority than the ones in [43], [44]. (iii) We apply finite-time differentiators to approximate the derivatives of the virtual control laws instead of directly using the differentiators of the virtual control laws in the design procedure of backstepping, which successfully avoid the problem of '' explosion of complexity''.

II. PRELIMINARIES AND PROBLEM STATEMENT A. GRAPH THEORY
This paper focuses on directed graphs (digraphs). We usually denote the digraph as . v N } stand for the edge set and the vertex set, respectively. e ij = v j , v i ∈ E is the edge of the digraphs G, which means that the agent i is able to gain information from the agent j and also means the agent j is one of agent i's neighbor, but not vice versa. Therefore, N i = v j | v j , v i ∈ E denotes the neighbor set of the agent i. This graph is called a weight graph when the weight of the edge is taken into account. We usually employ the adjacency matrix A = a ij ∈ R N ×N to represent the graphic topology. The element a ij > 0 only if e ji = v j , v i ∈ E, otherwise, a ij = 0. Self-loop are not usually considered, that is a ii = 0. Defined D = diag d 1 , d 2, . . . , d N ∈ R N ×N as the in-degree matrix with d i = N j=1 a ij of D denoting the in-degree of the agent i. The Laplacian matrix L = l ij ∈ R N ×N in connection with the digraph G is defined as L = D − A and the row sum of L is 0. If an edge sequence exists in the form of B. PREDEFINED-TIME STABILITY Consider the follow system: where x ∈ R n , ρ ∈ R b stand for the state and parameter of system (1), respectively. g : R + × R n → R n represents the nonlinear function and the origin is assumed as the equilibrium point of system (1). In addition, x 0 = x (0) is the initial condition of systems (1).
Definition 1 [37]: The origin of system (1) is said to be fixed-time stable if the system (1) is asymptotically convergent, meanwhile, there is a setting function T (x 0 ) as well as Remark 1: It is important to note that for a fixed-time stable system, to find an explicit relationship between the convergence time bound T M and the design parameter ρ is incredibly difficult. In addition, in some situations, even if by adjusting the system parameters, the convergence time bound is not able to be less than a fixed-constant.
Definition 2 (Predefined-Time Stability [38]- [40]): For the system (1), if it is fixed-time stability and exists a predefined constant T C such that the setting time function T (x 0 ) ≤ T C , the original point is named predefined-time stable, where constant T C is a function concerned with the system parameter ρ.

C. PROBLEM STATEMENT
Assume that there are N followers that are numbered as agents 1 to N as well as a leader (labeled L), meanwhile, the communication topology of followers is described by a direct graph G. The i-th (i = 1, 2, . . . , N ) follower has the following form: the entire state variables, output and input of the i-th follower. f i,k (x i ) and g i,k (x i ) stand for unknown nonlinear functions. The dynamic of the leader is considered as follows: where y L ∈ R represents the output of the leader. f L is a piecewise continuous function concerned with t. Remark 2: When the leader is taken into account, we employ the augmented graphḠ to stand for the communication topology of the leader and followers.Ḡ = V ,Ē represents an augmented graph of the aforementioned graph withV denoting the corresponding node set ofḠ,Ē denoting the corresponding edge set ofḠ. In addition, forḠ = V ,Ē , V is composed of the leader and all follower andĒ displays the flow of information between leaders and followers.
In this paper, we aim to develop a predefined-time adaptive consensus tracking control scheme for MASs to guarantee that all the signals in the MASs remain bound and the outputs of the followers well follow the output of the leader in our predefined time. In order to realize this goal, we introduce the following assumptions.
Assumption 1: There is a continuous function F (·) as well as a positive constant X L so that for all t ≥ 0, the following inequality: Assumption 2: The sign of g i,k is known and without loss of generality we suppose that it is positive. There exist a constant g i,k > 0.
Assumption 3 [45]: (1) If we let the leader as root, thenḠ includes a directed spanning tree. (2) The i-th follower only receives the state information from its neighbors.
Remark 3: It has been demonstrated that RBF NNs is an extremely serviceable tool to approximate unknown nonlinear functions. In this work, unknown nonlinear function h (x) that is defined on some compact set is also estimated by applying RBF NN in the form of h (x) = θ * T S (Z ) + δ (Z ). Furthermore, a more detailed introduction about the theory is able to be found in the relevant literature [21], [22].
Remark 4: Apparently, a simple but significant characteristic of RBF NNs is provided by Lemma 1. Employing this Lemma, adaptive neural-network-based backstepping design method for strict-feedback MASs is capable of easily being extend to the non-strict-feedback MASs (2).

III. CONTROLLER DESIGN AND STABILITY ANALYSIS
A. PREDEFINED-TIME ADAPTIVE CONSENSUS TRACKING CONTROLLER DESIGN Inspired by [46] and [47], we describe the local tracking error for the i-th agent as where i = 1, 2, . . . , N , the paining term b i > 0 stands for the weight between the agent i and the leader. Remark 5: Due to (9) includes the weighting parameters, that is a ij and b i , the graph-based error z i,1 can be effected by the augmented graphḠ. Furthermore, the condition (1) in For the i-th follower, the backstepping-based design procedure includes n i steps and the main design procedure of this paper is given below.
Step 1: By defining the tracking error we are able to construct a sub-Lyapunov as V i,1 = 1 2 z 2 i,1 . For (9), we differentiate z i,1 aṡ We design the feasible virtual control as where T C is predefined time, q is hidden neuron number in hidden layer; λ i is positive constant and its value will be given later, 0 < ρ < 1 2 . The adaptive laws ofθ i,1 is constructed as follows: From (10), differentiating V i.1 results iṅ ing RBF NNs to approximate the unknown nonlinear function f i,1 and employing Lemma 1 and Lemma 2, we can get the following formula: where θ * i,1 ∈ R q is RBF NN weight vector, S i,1 (z i ) = S i,11 (z i ) , . . . , S i,1q (z i ) denotes the basis function vector, δ i,1 (z i ) is the estimation error satisfying δ i,1 (z i ) ≤δ i, 1 .
The coordinate transformation is designed as z i,2 = x i,2 − α i,1 and the dynamic ofż i,2 is obtained as follows: We design the feasible virtual control as where λ i,2 > 0 is a constant and its value will be given later.
Design the sub-Lyapunov as The adaptation law is given as follows: The time derivative of V i,2 is further calculated aṡ by employing RBF NN to approximate unknown nonlinear function f i,2 and employing Lemma 1 and Lemma 2, we can get the following formula: where θ * i,2 ∈ R q is RBF NN weight vector, S i,2 (z i ) = S i,21 (z i ) , . . . , S i,2q (z i ) denotes the basis function vector, δ i,2 (z i ) is the estimation error satisfying δ i,2 (z i ) ≤δ i,2 .
The coordinate transformation are designed as z i,k = x i,k − α i,(k−1) and the dynamic of z i,k is obtained as follows: where The feasible virtual control is designed as where λ i,k > 0 is a constant whose value will be determined later.
Construct the sub-Lyapunov as The adaptation law is given as follows: The time derivative of V i,k is further calculated as: by employing RBF NN to approximate unknown nonlinear function f i,k and employing Lemma 1 and Lemma 2, we can obtain the following formula: where θ * i,k ∈ R q is RBF NN weight vector, S i,k (z i ) = S i,k1 (z i ) , . . . , S i,kq (z i ) denotes the basis function vector, From (32), we can get the following inequality by employing Young's inequality: where θ i,k = θ i,k 2 and q is hidden neuron number. Apparently, θ i,k is a positive function, letθ i,k be the estimation of Submitting (28), (33) into (31), we can geṫ Step n i : Similar to Step k, we construct a finite-time differentiator to modelα i,(n−1) as follows: where (n i −1)1 as well as (n i −2)1 is the state of the differentiator. κ 1 as well as κ 2 denote the parameter of the differentiator. Same as the Step k, we getα i(n i −1) (t) = (n i −1),2 (t) + ε i,(n i −1) and the estimation error ε i,(n i −1) is bounded. Ulteriorly, We can find a constantε i,(n i −1) > 0 such that ε i,(n i −1) ≤ε i,(n i −1) .
The coordinate transformation are designed as: z i,n i = x i,n i −α i,(n i −1) and the dynamic ofż i,n i is obtained as follows: The control input u i is designed as where λ i,n i > 0 is a constant whose value will be determined later.
Construct the sub-Lyapunov as: The adaptation law is given as follows: Taking time derivative of V i,n i as: by employing RBF NN to approximate unknown nonlinear function f i,n i and using Lemma 1 and Lemma 2, we can get the following formula: where θ * i,n i ∈ R q is RBF NN weight vector, S i,n i (z i ) = S i,n i 1 (z i ) , . . . , S i,n i q (z i ) denotes the basis function vector, δ i,n i z n i is the estimation error satisfying δ i,k (z i ) ≤δ i,k .
From (41), we can get the following inequality by employing Young's inequality: where θ i,n i = θ i,n i 2 and q is hidden neuron number. Apparently, θ i,n i is a positive function, letθ i,n i be the estimation of θ i,n i andθ i,n i = θ i,n i −θ i,n i . Submitting (37), (42) into (40), we can geṫ Design the Lyapunov function as: Taking time derivative of W i as: Define the total Lyapunov candidate function W N as From (44), (45), we can get the dynamic ofẆ N aṡ Now, we have the following result. Theorem 1: For our studied MASs (2), if Assumption 1-3 hold and the gains λ i satisfy 1 2θ i,j ≤ λ i,j , the developed scheme (37) that is made up of adaptive laws (12), (21), (30), (39) adaptive control inputs (11), (19), (28) is able to ensure that all signals in MASs remain bounded and the tracking errors converge to zero in the predefined time T C .
Proof 1: (1) From (44) and (45), we can see that V i,1 , z i,j ,θ i,j are bounded. Due to the boundedness of y d , we have x i,1 is bounded. Because the fact that θ i,j are constants andθ i,j are bounded, we can get the boundedness ofθ i,j . This combined with the boundedness of z i,1 ,ẏ d , constants q, i,1 , n i , ρ, T C , λ 1 contributes to the boundedness of α i,1 . Moreover, we have that x i,2 is bounded. Because κ 2 is a constant and the differentiator is capable of giving the derivative of α i,1 within finite time, the state 1,2 of the differentiator is bounded. In view of the boundedness of z i,2 ,θ i,2 , g i,1 (x i ) and the constant q, i,2 , n i , ρ, T C , λ 2 , we have α i,2 is bounded. From the boundedness of α i,2 and z i,2 , one has x i,3 is bounded. Analogously, we obtain that x i,j , α i,j , i,2 are bounded. As a result, all signals in MASs remain bounded.
(2) Due to the boundedness of all the closed-loop signals, we can find a constant λ i,j such that 1 2θ i,j ≤ λ i,j and we havė Let From (49), (50) we can see that m 1 and m 2 are monotonously increasing. VOLUME 9, 2021 Consequently, according to Lemma 5, we rearrange m 1 z 2 i,j and m 2 z 2 i,j and we can get the following inequity: From Lemma 3, one has Due to 0 < ρ 2 < 1, according to Lemma 7, one has Therefore, (54) becomeṡ We define τ = 2V n i and we can rewrite (58) aṡ At time t, V n i (t) = 0 and τ (t) = 0 and the stabilization time bound is estimated as As a result, the output of each follower can track the output of leader within the predefined-time T C .

IV. SIMULATION EXAMPLES
In order to check the availability and the effectiveness of our proposed scheme, let us consider the following example. Fig. 1 shows a digraphḠ with one leader and three followers, which represents the communication topology of the MASs. InḠ, the node that numbered as L is the leader and the nodes labeled by the numbers from 1 to 3 are the followers. Obviously, only follower 2 can receive the signal of leader. Clearly, the adjacency matrix A and the Laplacian matrix L of the digraph G are, respectively The dynamics of the three heterogeneous followers are described as The objective of this example is to force the outputs of the followers to follow the leader's output y L = sin (t) + 0.1. From fig. 1, we can see that the leader adjacency matrix is B = diag 0, 1.5 0 . Consequently,Ḡ satisfies Assumption 3. In the simulation, the initial conditions are given as x 1 (0) = 0.5, 0 0.5 T , x 2 (0) = 1, 0.5 0.5 T , x 3 (0) = 0, 0.5 0.5 T and choose the controller parameters as g 1,1 = 0.9, g 1,2 = 0.9, g 1,3 = 1, g 2,1 = 1, g 2,2 = 1, g 2,3 = 1, g 3,1 = 1, g 3,2 = 1, g 3,3 = 2, T C = 0.6, ρ = 0.25, q = 10, λ i,j = 2.1, n = 3 The differentiator parameters are given as κ 1 = 2, κ 2 = 2.2. The simulation results are shown in figures 2-5. Figure 2 shows the output of the leader and the outputs of three followers and figure 3 shows the tracking errors between the leader and three followers. Figures 4-5 show the states of each agent. From figure. 2-3, we can see that each follower is able to track the leader within our predefined time T C .

V. CONCLUSION
This paper is committed to solving the predefined-time adaptive neural-networks-based consensus tracking control problem for high-order nonlinear MASs to achieve the tracking errors converge to zero within predefined time, in which each follower is modeled by the non-strict-feedback system. Firstly, in the design process of predefined-time controller, we employed an important structural feature of RBF NNs provided in Lemma 1 to address the difficulty caused by the non-strict-feedback structure. Then, RBF NNs were utilized to compensate the unknown nonlinear functions so that the virtual control inputs were capable of being gained and the derivatives of the virtual control inputs were estimated by the finite-time differentiators, which successfully avoids the problem of ''explosion of complexity''. It has been mathematically proven that with our presented control scheme, all the signals in the high-order nonlinear MASs remain bound and the outputs of the followers well follow the output of the leader within the predefined time T C . In the end, the simulation result testified the availability and superiority of the proposed control method. What is noteworthy is that because of environmental barriers, safety consideration and performance requirements, a large amount of practical nonlinear MASs are subjected to output or states constraints. Consequently, in the future, we will pay attention to studying predefined-time adaptive consensus tracking control problem for the nonlinear MASs with output or states constraints.