Neural-network Distributed Event-triggered Consensus Tracking Control for High-Order Nonlinear Strict-Feedback Multiagent Systems

The problem of communication constraints for multiagent systems has attracted considerable attention, but there still lacks the result to solve communication constraints between neighbor agents. This signiﬁcant and practical issue shall be addressed by establishing event-triggered mechanisms between neighbor agents, such that neighbor information can be transmitted only when the preset events are activated. Subsequently, this paper proposes a novel consensus control strategy for nonlinear strict-feedback multiagent systems carrying with triggered mechanisms and uncertain functions. These uncertainties are going to be handled by using radial basis function neural network, whose unknown weight vector can be estimated by only one required adaptive law. The stability of the control algorithm is strictly proved by Lyapunov functional theory, while the boundedness of all the closed-loop signals are guaranteed and the consensus errors are ensured to be exponentially converged to an adjustable domain. Simulation studies eventually present the e ﬀ ectiveness of this method.


Introduction
In a long period of history, natural biological population movement of many animals have received extensive observations and kinds of studies, such as bird migration [2], ant and insect aggregation [7], fungus community [29].With the development of artificial intelligence technology, biological population movement can be abstracted as consensus movement of multiagent systems.For this reason, consensus control problems gradually turn into research focus in the area of distributed artificial intelligence and information interactive technology, for example, in AlphaGo [27], formation control [37,45], distributed sensor network [18,35].Originally, consensus control technology focuses on investigating the asymptotic consistency of distributed decision systems and parallel computing [3,34].The surveys then develop to address the consistency problem by constructing a theoretical framework for network dynamic systems [17].Furthermore, their theoretical results are extended to solve the issue of information consensus, where every vehicle communicate with its neighbors [20].Note that these aforementioned control theories are limited since they can not apply to track more complex nonlinear or high-order multiagent systems.To remove these constraints, consensus control schemes for high order multiagents are proposed by [5,12,26].These schemes are so constructive that lots of complex engineering systems can be eventually supported by theoretical methods.
However, limited bandwidth resources of communication network are not enough to support application from theory to engineering.To remove this constraint, event-triggered control (ETC) [1,10] attracts considerable attention for its remarkable property of improving communication rate.The advantage of this strategy is that it gives an effective way to transmit signals only when a specific event occurs [33].Recently, corresponding results set up a triggering event for state variable based on the rigorous input-to-state stable (ISS) assumption [9,19], or design controller and triggering event simultaneously [30,40,41].Moreover, event-triggered strategy is applied to first-order MASs [38], second-order MASs [46,48], general linear MASs [42,44], uncertain nonlinear MASs [11,23,47], and so on.However, limitations of these existing literature mainly include that: limited network resources between neighbor agents are not taken into account, communication rate and system performance cannot both guarantee, steady-state and transient performance do not explored at the same time.To be concluded, it is of practically and theoretically significant to find an efficient strategy to resolve these limitations.
In order to achieve the control objectives, intelligent technologies need to be utilized, which are treated as a research hot spot in the control field.Actually, isolating communication constraints, neural-network (NN) based intelligent control methods have been greatly enriched, which focus on autonomous underwater vehicle (AUV) control [24], flexible robot manipulators control [21,32], intelligent flight control [22,31], and so on.Nevertheless, most of the aforementioned results handle the unknown RBFNN via estimating every weight vectors, hence numerous estimators are required.To release calculation burden, the paper [4] proposes an effective method to reduce parameter estimators by addressing the square of the norm of neural networked weight vectors.But the application of neural network control methods always lead to conservative and uncertainty in terms of the boundary of tracking error.The limitation has been concerned by many scholars recently, and a series of solutions have been developed [8,36].However, the solutions are not available for nonlinear multiagent systems, since communication constraints need to be considered and considerable parameter estimators are still required.
Inspired by the above observations, the consensus tracking problem of high-order nonlinear strict-feedback multiagent systems is investigated.The systems contain inherent uncertain functions and the transmission of neighbor signals occur in every sample time, resulting in nontrivial control task and tremendous communication burden.To handle the existing problems, a novel neural-network distributed event-triggered control approach is established, where the contributions are well summarized as below.
1. Creatively, the time-sampled transmission mode between neighbor agents is replaced with event-triggered transmission, such that a new communication protocol is set with respect to distributed consensus errors.Besides, the established ETC possesses physical realizability for it can avoid Zeno behavior [13].
2. More generally, the considered multiagent systems contain unknown and unavailable functions, which can be approximated by utilizing RBFNN.Unlike the existing RBFNN technology, this paper proposes an effective method that involves less computation and estimators via handling only the norm of neural network weight vector for the whole multiagent systems.
3. A tradeoff inevitably occurs when the problems of communication network resources and system performance are both under considered.To handle this contradictory relation, the paper provides a solution to keep balance between communication resources and system performance.Furthermore, both steady-state and transient performance can be guaranteed with the proposed method.
The outline of the paper is organized as below.Section II describes a problem formulation and some preliminaries of graph theory, event-triggered strategy and radial basis function neural network.Section III performs how to establish a series of distributed event-triggered consensus controllers, accompany by stability analysis to prove its theoretical correctness.Section IV applies the proposed method to a practical simulation to verify its feasibility and effectiveness.Finally, Section V summarizes the contributions of this paper and looks forward with the future work.

Graph Theory
A directed graph with weight G = (V, E, A) is under consideration, where V = {v 1 , v 2 , ..., v M } is the set of M nodes, E ⊆ V × V is the edge set and A = [a i, j ] ∈ ℜ M×M is an adjacent matrix with non-negative elements.The ith agent is marked by node v i , i = 1, 2, ..., M. Edge (v i , v j ) ∈ E represents that ith agent is able to receive neighbor information from jth agent, such that the weight elements satisfy a i, j > 0; otherwise a i, j = 0. Define a neighbor set of ith agent as N i = { j|(v i , v j ) ∈ E}, for agent v j is a neighbor of agent v i , i.e., (v i , v j ) ∈ E. Let the adjacent moment matrix of the leader as B = diag{b 1 , ..., b m }, where b i is treated as the communication weight between agent i and the leader.Let the degree matrix as D = diag{d 1 , ..., d m }, where d i = m j=1 a i, j is the ith sum of weight.The Laplacian matrix is then defined as L = D − A, whose properties are described in [39,28].
Assumption 1: The information transmission graph G is fixed and connected.Assumption 2: The leader's output signal y r and its first nth derivative y (k)  r , k = 1, 2, ..., n are available for the ith agent and sufficiently smooth, bounded, and continuous.
Remark 1: The graph is supposed to be connected, such that ith agent can achieve all the neighbor information.The neighbor control signal will be sampled by preset event-triggered mechanisms.Assumption 2 is so practical that almost all the physical systems are able to be abstracted as second order dynamic systems, thus only y r , ẏr , ÿr are required to be known.

Nonlinear system model
Consider the nonlinear strict-feedback systems containing m interconnected-following agents described by ẋi,k where χ i,k = [x i,1 , x i,2 , ..., x i,k ] T ∈ ℜ k is the state vector of the ith agent, y i ∈ ℜ and u i ∈ ℜ is the system output and the input, respectively.g i,k (χ i,k ) for k = 1, 2, ..., n are unknown and unavailable smooth nonlinear functions.It is worthy to point out that the multiagent model ( 1) is so valuable by which kinds of physical systems can be abstracted, such as industrial cyber-physical systems [14], multirobot systems [15], multi-flight systems [6], and so on.The control objective is to construct a NN-based consensus controller u i for each follower, such that all the outputs y i can synchronically track the dynamic reference signal y r with the cooperatively semi-globally uniformly ultimately bounded (CSUUB) tracking error.Communication constraints among agents are also under considered, hence the design procedure need to synthesize the technologies of event-triggered control, tuning function design and adaptive control.

Event-triggered strategy
Note that communication signal transmission among agents will occupy lots of networked resources, resulting in a heavy communication burden.In order to improve communication rate, an effective approach namely eventtriggered strategy meets considerable attention via updating communication signals based on a preset threshold.For the considered systems (1), we will establish event-triggered mechanisms between neighbor interconnected-following agents i and j, such that ȗ where ȗ j,k (t) is treated as an event sampled neighbor information, which will transform to the controller u i for ith agent.The triggering time t h , h ∈ Z + will be updated as t j,h+1 when the event triggered, such that where e j,k (t) = u j,k (t) − ȗ j,k (t) denote measurable errors between jth controller information u j and ith controller input.
The working principle of event-triggered mechanism can be concluded as: whenever the measurable error e j,k (t) exceeds the set threshold ξ, the actual neighbor information of ith agent ȗ j,k (t) will be updated as u j,k (t h+1 ), and it will transform to the controller u i , or it will hold as a constant, i.e., u j,k (t h ).
Lemma 1 [40]: During the interval [t h , t h+1 ), the actual neighbor information ȗ j,k (t) is given by where λ(t) is a continuous parameter, satisfying Remark 2: In order to overcome communication constraints, event-triggered strategy attracts lots of attention for its remarkable ability of choosing necessary information exchange with respect to state variables [9,19], or adaptive controller [1,30,40].For this reason, some papers applied event-triggered strategy to MASs to improve communication rate for single agent [42,44,46,48,16,47].However, the aforementioned papers never take neighbor communication constraint into account, regardless of ensuring system tracking performance.To handle this problem, we conceive to establish event-triggered mechanisms between neighbor agents, such that neighbor information can be selected by preset triggering event.

Radial Basis Function Neural Network
Note that the inherent uncertain functions g i,k (χ i,k ) of the system model (1) bring tremendous difficulties to our control design.RBFNNs are used to approximate these corresponding uncertainties to an arbitrary accuracy on a compact set.Defined a positive constant ε i , there always exist RBFNN expressed by for k = 1, 2, ..., n and i = 1, 2, ..., m. χ i,k ∈ Ω i,k are the input variables of RBFNN.The number of neurons is chosen as is treated as an unknown weighted column vectors, such that where is a set of radial basis function vector chosen by Gaussian membership function: for j = 1, 2, ..., ν i,k .The center of the receptive field defines as q i, j = [q i, j,1 , q i, j,2 , ..., q i, j,ν i ] T and the width of the Gaussian function is σ i, j .5) can transform to the following parametrization-like form as

Define two vectors as Wi
Remark 3: Most of the literature concerning about RBFNN estimate all the unknown weight vectors W * i,k updating online in real time.To release computation burden, an effective scheme [4] was developed by handling the norm of neural network weight vector Wi .Furthermore, this paper constructs a novel approximation equation ( 8) for multiagent systems, such that only the norm of κ remains to be estimated.In other words, this operation can minimize computation and prevent over-parametrization.

Distributed Event-triggered Consensus Control Design
This section presents a complete control scheme by synthesizing event-triggered and tuning function design.The recursive designed procedure contains n steps, which will be elaborated as below.To begin with, the following definition is needed.
Definition 1: The graph-based consensus error of ith agnet is given by Combining with the definition, error variables are defined as for i = 1, 2, ..., m and k = 2, ..., n.
Step 2: Given żi,2 as żi,2 = z i,3 for αi,1 By utilizing RBFNN, G i,2 is approximated by for χi,2 = [χ 1,2 , ..., χ j,2 ] T , j ∈ N i .Define the second virtual controller α i,2 as where Υ i,2 = ψ i,2 , and The second continuous Lyapunov candidate is chosen as Cooperating the definition in ( 20), ( 22) and ( 23), Vi,2 is expressed by where Step k(k = 3, ..., n − 1): The derivative of z i is able to be defined as żi,k =z i,k+1 where With the aid of RBFNN, G i,k can be approximated as The ith virtual controller α i,k is defined as where Define kth Lyapunov candidate as Combining with ( 27)-( 31), Vi,k is then computed by with Step n: Based on Lemma 1, event-triggered mechanisms are established between jth neighbor control signal u j and ith controller input, that is From ( 1) and ( 10), we have żi,n =u i + g i,n − αi,n−1 where G i,n is approximated by RBFNN, that is where χi,n = [ χ1,n , ..., χ j,n ] T , j ∈ N i .
Given the nth Lyapunov candidate V i,n by Define the final virtual controller α i,n by where ξ ≥ ξ, The consensus controller of ith agent u i is eventually defined as Accompany by the adaptive law as Remark 4: Computation burden will be further released by estimating only one parameter θ for the whole multiagent systems, rather than estimating every parameter for single agent.

Stability analysis
Theorem 1: Considering the nonlinear strict-feedback MASs (1) consist of event-triggered mechanisms among agents (2), distributed consensus controller (40) and adaptive law of NN weight (41), under Assumption 1.For any initial condition V(0) < µ, the MASs then possess the following properties: (1) All the signals of the considered systems are CSUUB.
(2) The consensus tracking errors between the leader and the followers converges to an arbitrarily compact set.
(3) Zeno behavior can be avoided.Proof: The total Lyapunov function V of the multiagent systems is finally defined as For stability analysis, V can be computed by combing with ( 35) and ( 36) as Substituting the final controller u i , ȗ j and Vi,n−1 into (43), we have where δ i,n = δ i,n−1 + 0.2785ǫ i,n + 0.2785ǫ.With the NN weight updated law θ in (41), we have where δ = m i=1 δ i,n .Depended on Young's inequality, the following inequation is given as Thus ( 44) is able to be performed as where ρ = min{c i, j , γ −1 } and ∆ = δ + 1 2 θ 2 .Note that V < 0 can be satisfied when ρ > ∆/µ, for V = µ.Therefore V ≤ µ is treated as an invariant set.That is to say, V ≤ µ can be held for all t > 0 when V(0) ≤ µ is chosen.Following the standard analysis in [25,43], a conclusion can be made that all the signals are CSUUB.Furthermore, the following inequation is derived as Combining with the definition of V in (42), yields that This implies that there exist T for all t > T , the graph-based consensus error e(t) = [e 1 (t), ..., e m (t)] T converges exponentially towards a compact set, for e(t) ≤ 2∆/ρ (50) The approximation precision can be improved by tuning the user-defined parameter c i, j and γ.
To prove that the proposed approach can avoid Zeno behavior, we suppose that a t * > 0 satisfying the condition of {t h+1 − t h } ≥ t * is existing.Based on the previous analysis, e j,k (t) = u j,k (t) − ȗ j,k (t), ∀t ∈ [t h , t h+1 ) is bounded since u j,k (t) and u j,k (t) are all bounded.Via the operation of derivative, it yields that Note that ui,1 is a function with α i,1 , hence ui,1 must be continuous and bounded, such that Now we can get the lower bound of the inter-execution time as t * ≥ (|ȗ j (t)| + b)/ζ.To be concluded, the events can be triggered in infinite times within a finite time interval.The number i = 1, 2, 3, 4 represents the ith agent. 2 The number j = 1, 2 represents the order of ith agent.

Simulation results
This section set out to evaluate the feasibility and effectiveness of the proposed consensus tracking control scheme.Therefore, the multiagent systems consisting of four continuous torsional pendulums are applied as below.
where β i and ω i denote the current angle and the angular velocity of ith agent, respectively.M i and l i are the inherent parameters of the pendulums, which corresponds to the mass and the length.J i , g d,i and g i represent the rotary inertia, the frictional factor and the acceleration of gravity, respectively.To transform (53) into a strict-feedback form, we introduce x i,1 = β i and x i,2 = ω i .The leader's model is defined as y r = cos(0.5t).Giving the neighbor matrix as B = diag{1 0 0 0}, and the adjacency matrix A d and Laplacian matrix L a as More vividly, the connection graph between agents and the leader is drew by Fig. 1.With the aid of event-triggered mechanisms, exchange information among neighbors is going to be instead of  Angular velocity x i 2.
x 12 x 22 x 32 x 42   triggering transmission mode.For simulation, the event-triggered strategy for (53) are set by xi,1 (t) = x j,1 (t j,h ) where ξ = 0.08, j ∈ N i .Suppose that the inherent nonlinear functions g i,2 (•) = J −1 i M i g i l i sin(x i,1 )+g d,i x i,2 are unknown.To remove these constraints, the unknown function T and choose the number of neurons as ν i,2 = 3, then  where is a set of radial basis function vector chosen by Gaussian membership function: for i = 1, 2, 3, 4, j = 1, 2, and q i, j = −2, 0, 2 are the center of the receptive field defines.For simulation, the initialization parameters and the inherent parameters are given by Table I.The control objective can be simply concluded as: a.With the aid of RBFNN, the controllers u i (t) is able to be designed with the proposed scheme, such that all the agent's outputs y i (t) track the leader's signal y r (t) to a preset accuracy; b.Based on the event-triggered strategy (54), communication rate among neighbor agents shall be improved.
The simulation studies can be seen by Fig. 2-Fig.7. The tracking performance between system outputs y i and the leader's signal y r is showed by Fig. 2, and the angular velocities x i2 are given in Fig. 3.The present result in Fig. 4 shows that consensus errors can exponentially converge to an adjustable domain.Furthermore, the adaptive estimator for the whole systems θ is presented as Fig. 5.The findings from Fig. 6 and Fig. 7 have demonstrated, for the first time, that jth neighbor control signal ȗ j can be transmitted based on event-triggered mechanism.Respectively, the thresholds and the number of events for ith agent are shown in Fig. 8. Besides, Fig. 9 shows the triggering time for ith agent, which can be found that the lower bound of the inter-execution time exists.
The presented simulation provides strong empirical confirmation that: a.With the proposed recursive design approach, all the agent's outputs y i (t) are able to track the leader y r (t) to a small user-defined accuracy, seen as Fig. 1-Fig.3; b.Only one parameter estimator is needed for the whole systems, seen as Fig. 5; c.The event-triggered mechanisms between neighbor information can obviously improve the utilization of communication bandwidth, seen as Fig. 6-Fig.7. Furthermore, all the events do not infinite triggered at one time, as seem from Fig. 8-Fig.9.

Conclusion
This paper was undertaken to design a novel event-based tracking approach for high-order nonlinear multiagent systems.One of the innovation of this study is that event-triggered mechanisms can be established between neighbor agents, such that communication exchange information can be sampled by preset events.Furthermore, the containing uncertainties are successfully approximated by using the technique of RBFNN, to which whole weight vectors are estimated by only one update law.Via stability analysis, the proposed scheme has been proved that all the signals and the consensus errors are guaranteed to be bounded.Moreover, the errors can be proved to exponentially converge to a designed domain.The final simulation studies verify the established theorem.Future work possibly investigates the problem of non-smooth actuator constraints occurring in high-order multiagent systems, which shall be treated as a significant and practical research for theoretical and industrial applications.

Figure 1 :
Figure 1: Communication network of the multiagent systems.

Funding Information:
This work is supported partially by the National Natural Science Foundation of China under Grant 6210021076, partially by the Guangzhou Municipal Science and Technology Project under Grant 202201010381.

Table 1 :
INITIAL VALUES AND DESIGN PARAMETERS FOR SIMULATION