Practical Time-Varying Formation Control for Nonlinear Multiagent Systems with Event-triggered Mechanism

Authors: shuxing  xuan
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-16 10:30:34

Abstract: Abstract—This paper addresses the problem of practical timevarying formation control for nonlinear multi-agent systems with
an event-triggered mechanism. In dynamic and complex multiagent systems, maintaining a desired formation while adapting
to nonlinearities and varying communication constraints presents
significant challenges. We propose a novel control strategy that
incorporates a time-varying formation criterion and utilizes
an event-triggered mechanism to optimize communication efficiency.Our approach begins with the design of a practical timevarying formation control law tailored to handle the nonlinear
dynamics inherent in multi-agent systems. The event-triggered
mechanism is employed to reduce the frequency of control
updates and communication overhead by activating the control
updates only when necessary, based on predefined criteria. This
mechanism ensures that the agents adjust their control actions in
response to significant deviations from the desired formation, thus
maintaining formation integrity while conserving communication
resources.We derive stability conditions for the proposed control
strategy and demonstrate its effectiveness through rigorous theoretical analysis and simulations. Theoretical results confirm that
the system can achieve and maintain the desired time-varying
formation despite the nonlinearities and dynamic communication
constraints. Simulation experiments validate the practicality and
robustness of the proposed method, showing that it effectively
maintains the formation and reduces communication load.

Main paper: 
 i. introduction: 
 In the study of multi-agent systems, the problem of formation control has become a significant topic, especially in practical applications such as drone formations, autonomous vehicles, and robot fleets. Ensuring that agents can maintain a desired time-varying formation while addressing nonlinear dynamics and dynamic communication constraints is a challenging task.Traditional formation control methods often assume that the system's dynamics are linear or that communication remains stable and continuous throughout the formation process. However, in practical scenarios, interactions between agents frequently exhibit nonlinear characteristics, and communication resources may be constrained or unstable. Therefore, designing Identify applicable funding agency here. If none, delete this.an effective control strategy to address these challenges while achieving time-varying formation goals is an urgent problem.In recent years, event-triggered mechanisms have been proposed as an efficient communication strategy to optimize control performance in multi-agent systems. This mechanism significantly reduces communication overhead by triggering control updates only when necessary. However, integrating event-triggered mechanisms with time-varying formation control to address nonlinear dynamics in practical situations has not been thoroughly explored.To address these issues, this paper proposes a novel control strategy that combines practical time-varying formation control with an event-triggered mechanism to handle nonlinear dynamics and communication constraints in multi-agent systems. First, we design a control law tailored for time-varying formations and use the event-triggered mechanism to optimize the frequency of control updates. Second, we validate the effectiveness and robustness of the proposed method through theoretical analysis and simulation experiments. The results demonstrate that the proposed method effectively maintains the desired formation while significantly reducing communication overhead.The main contribution of this paper lies in providing a comprehensive approach that integrates time-varying formation control with event-triggered mechanisms, offering a new solution to practical control problems in multi-agent systems and laying a foundation for future research directions.A signed digraph G = {V, E, A} represents the communication topology of the multiagent systems, which includes N followers and M leaders. The V = {v 1 , . . . , v N +M } denotes the node set, E = {(v i , v j )|v i , v j ∈ V} represents the edge set, and A = [a ij ] ∈ R (N +M )×(N +M ) is the adjacency matrix for digraph G. If (v i , v j ) ∈ E, it indicates that information can be transmitted from j to i with a ij = 1; otherwise, a ij = 0. Additionally, both competition and cooperation are considered in the communication topology. A positive value (a ij > 0) indicates cooperation, while a negative value(a ij < 0) indicates competition. The adjacency matrix is denoted as L = D -A, where D = diag(d 1 , . . . , d N +M ) with d i = n j=1 |a ij |. Then, define B = diag{|b 1 | , |b 1 | , . . . , |b N |},where b i represents the weights between followers and leaders. If ith agent can receive information from a leader, then b i ̸ = 0; otherwise, b i = 0.Fuzzy logic systems (FLSs) are frequently used to handle unmodeled dynamics in systems. As shown in [?], a FLSs can be described asEQUATIONwherex = [x 1 , • • • , x n ]⊤ is the input variable, y is the output variable, and µV l i (x i ) (i = 1, • • • , p) is the membership function of x i , m is the number of rules.The fuzzy basis function can be described asEQUATIONConsider nonstrict-feedback multiagent systems as follows:EQUATIONwhere k = 1, . . . , n-1, xi,m = [x i,1 , . . . , x i,m ] ⊤ ∈ R m , (m = 1, . . . , n) is the state vector, f i,m (x i,m ) represents unknown smooth nonlinear function. u i ∈ R and y i ∈ R denotes the system state input and output, respectively.In this section, we investigate the problem of adaptive prescribed-time formation tracking, examining the dynamic trade-off between achieving faster stabilization and conserving communication resources for any given settling time.The time-varying formation of the multiagent systems (3) is described by a bounded continuous differentiable function ξ i , which represents the distance between ith agent and the leader.According to Definition 2, given any setting time T p , it is satisfied that limEQUATIONthe systems (3) achieves formation tracking. M represents the prescribed-time attractive set.Define the following coordinate transformation:EQUATIONwhere q = 2, ..., n, z i,1 is formation error, and η i,q-1 represents the virtual controller to be designed later.Step 1: From the systems (3) and coordinate transformation (5), the time derivative of z i,1 isżi,1 = ( N j=1 |a i,j | + b i )(z i,2 + η i,1 + f i,1 (x i,1 ) -ξi ) - N j=1 |a i,j | (sgn(a i,j ) ẏj -ξj ) -b i sgn(b i ) ẏd (6) where F i,1 (X i,1 ) = f i,1 (x i,1 ) -1 ℏi ( N j=1 |a i,j | (sgn(a i,j ) ẏj - ξj ) -b i sgn(b i ) ẏd ) with X i,1 = [x i,1 , x j,1 , y d , ẏd , ξj ] ⊤ ,EQUATIONBased on the coordinate transformation (5), the candidate Lyapunov function is selected asEQUATIONwhere θi,1 = θ i,1 -θi,1 is the estimate error, θi is the estimated value of the uncertain parameter θ i with θ i,1= ∥W i,1 ∥ 2 .According to the derivative of z i,1 in (7) and the candidate Lyapunov function (8) , one yileds Vi,1 = z i,1 (ℏ i,1 (X i,1 ) + ε i,1 (X i,1 )))EQUATIONBy employing Young's inequality, it can be inferredEQUATIONwhereϵ i,1 > 0 is the design parameter, |ε i,1 (X i,1 )| ≤ εi,1 with εi,1 > 0 is a constant.The virtual controller and adaptive law are designed asEQUATIONwhere k i,11 , k i,12 , and k i,13 > 0, a θi,1 , b θi,1 , and c θi,1 > 0 are design parameters. Inserting (10), ( 11) and ( 12) into (9), it can be obtainedEQUATIONStep n:Inductively, one can obtain that the derivative of z i,n satisfies żi,n =EQUATIONwhere the unknown nonlinear functionF i,n (X i,n ) = f i,n (x i,n ) -ηi,n-1 is defined, and a FLSs W ⊤ ψ (x) is utilized to approximate the function F i,n (X i,n ), such that F i,n (X i,n ) = W ⊤ i,n ψ i,n (X i,n ) + ε i,n (X i,n ) with X i,n = [x i,n , ηi,n-1 ] ⊤ . |ε i,n (X i,n )| ≤ εi,n with εi,n > 0 is a constant.By selecting the candidate Lyapunov function asEQUATIONwhere the parameter estimation error satisfies θi,q = θ i,q -θi,q , and θ i,q = ∥W i,q ∥ 2 .From (??) and (??), Vi,q satisfiesVi,q = Vi,q-1 + z i,q (z i,q+1 + W ⊤ i,q ψ i,q (X i,q ) + ε i,q (X i,q )) + z i,q η i,q -θi,q θi,q (16) By applying Young's inequality, it can be deduced z i,q (W ⊤ i,q ψ i,q (X i,q ) + ε i,q (X i,q )) ≤ 1 2 ϵ i,q z 2 i,q θ i,q ∥ψ i,q (X i,qEQUATIONwhere ϵ i,q > 0 is a design parameter, |ε i,q (X i,q )| ≤ εi,q with εi,q > 0 is a constant.The principal theory introduced in this paper are proven to be effective by the following theorem.For multiagent systems (3), if Assumptions 1-3 are satisfied, for any initial state xn (0), the proposed virtual controllers, the parameter update laws, and the event-triggered controller ensure that all signals are prescribed time stable, and the bipartite formation tracking error z i,1 satisfies:Converges to to a small neighborhood ⌣ M 1 within the preset time T p . Meanwhile, the Zeno behavior is circumvented.Note that the method proposed in this paper considers the impact of convergence time on the system. In practical system control, a very short convergence time requires a larger control input to achieve stable operation. Additionally, input saturation needs to be considered in practical control systems. For the event-triggered scheme (??), it can be seen that the dynamic threshold is composed of the prescribedtime evolution function and input saturation constraints. The parameter D k is used to balance convergence speed and system communication resource optimization. If the user sets a very short convergence time, a shorter update interval is adopted to achieve faster stabilization. When a longer convergence time is set, a longer update interval is used to better reduce communication resource consumption.In this paper, we have developed a novel control strategy for practical time-varying formation control in nonlinear multiagent systems, incorporating an event-triggered mechanism to optimize communication efficiency. Our approach addresses the dual challenges of maintaining a desired time-varying formation and managing nonlinear dynamics while reducing communication overhead. Through rigorous theoretical analysis, we derived stability conditions that guarantee the effectiveness of the proposed control strategy. Simulation results further validate the practicality and robustness of the method, demonstrating that it not only successfully maintains the desired formation but also ensures the boundedness of all signals within the closed-loop system. The proposed method effectively reduces communication load while achieving the control objectives, making it suitable for real-world applications where communication resources are limited and system dynamics are complex. In summary, this paper contributes a significant advancement in the field of multi-agent systems by providing a robust and efficient solution for time-varying formation control under nonlinear dynamics. Future work may explore extending this approach to more complex scenarios, such as systems with time delays or varying communication topologies, and further optimizing the event-triggered mechanism to enhance overall system performance. 
 
 Appendix:  



Reviews: review: title: title: Recommended rejection
main_review: main_review: The manuscript lacks sufficient novelty and does not provide a comprehensive comparison with existing formation control strategies, limiting its overall contribution to the field.  As such, I do not think this paper is suitable for ICIST 2024.
title: title: Manuscript Rejection
main_review: main_review: The author proposes a new control strategy for the nonlinear multi-agent systems with an event-triggered mechanism in this paper. However, the simulation section does not provide, and the advantages of the new control method lack depth exploration.
The author need to emphasize and describe the advantages of the proposed control scheme compared to existing achievements.
title: title: Manuscript Rejection
main_review: main_review: The paper does not sufficiently differentiate its proposed strategy from existing methods in the literature. The control strategy and event-triggered mechanism are concepts that have been explored extensively. Without a robust comparison to state-of-the-art methods, the claimed novelty and advantages of the approach are not convincingly demonstrated.

While stability conditions are derived, the theoretical analysis lacks depth in explaining how the proposed mechanism handles specific challenges posed by the time-varying constraints. More rigorous theoretical proof or a detailed exploration of edge cases would strengthen the contribution.