Containment-Formation Control for Second-Order Nonlinear Multi-Agent Systems via Event-Triggering

Authors: st xinyang lan
yang  yu
wei  wang
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-07-31 16:38:02

Abstract: This paper studies the event-triggered containment-formation control problem for nonlinear multi-agent systems. Neural networks are utilized to approximate uncertain dynamics. An adaptive neural network controller is designed, and an event-triggering mechanism based on a relative threshold is constructed, which is set on the control channel. By using Lyapunov stability theory,$\thinspace$  the bounded stability of the closed-loop systems is proven, and the formation tracking error converges to a small neighborhood around zero.$\thinspace$ It is also shown that the event-triggered control method used does not cause Zeno behavior. Finally, simulation results are provided to verify the effectiveness of the proposed formation control method.

Main paper: 
 i. introduction: 
 In recent years, formation control has been widely used in various fields, such as mobile robots [1] , spacecrafts [2] and unmanned aerial vehicles [3] . And the application of formation control allows agents to share information and tasks to achieve collaborative work. The principle of formation control is to design a control strategy to make the intelligences work according to the envisaged formation [4] , and this aspect is a hot topic in control research.Nowadays, there are many formation control methods, including consensus-based formation control, rule-based formation control, behavior-based formation control, and so on. In [5] , a consensus algorithm was designed to address formation control problems by appropriately selecting information states upon which consensus was achieved. The auther in [6] considered employing a rule-based strategy to address the unique estimation issues associated with spacecraft formation flying. The aricle in [7] discussed a distributed formation control for networked multi-agent systems. The paper [8] adopted a simple leader-follower formation tracking control strategy to address the formation control problem of nonholonomic mobile robots under communication constraints.This work was supported in part by the National Natural Science Foundation of China under Grant 62273170, and in part by the National Natural Science Foundation of Liaoning Province under Grant 2023JH2/101300187, Grant JYTZD2023082, and Grant 2023-MS-300.In [9] , the authers proposed the application of behaviorbased formation control, which achieves the formation control solely by utilizing the relative position information between robots and their neighbors as well as obstacles. And in [10] proposed a communication structure design algorithm that enables followers to form a predetermined formation based on containment control. The formation control mentioned in the above article belongs to the category of formation control with no leader and with one leader. When multi leaders exist in the multi-agent systems, the formation problem is called as containment-formation control. For example, the authers in [11] studied the time-varying output formation for containment control of linear homogeneous and heterogeneous systems. In the [12] , the authers proposed a communication structure design algorithm that enables followers to form a predetermined formation based on containment control. Therefore, studying containment-formation control is meaningful.Due to the limited communication network resources in multi-agent systems, designing event-triggered mechanisms to reduce communication overhead and enhance system efficiency is crucial for ensuring optimal system performance [13] . Consequently, investigating the application of eventtriggered control in multi-agent systems is highly significant, as it offers a promising approach to optimizing resource utilization and achieving better overall system performance. Neural networks are used to approximate uncertain dynamics. An adaptive neural network controller is designed, incorporating an event-triggering mechanism based on a relative threshold set on the control channel. Using Lyapunov stability theory, the bounded stability of the closed-loop systems is proven, ensuring the formation tracking error converges to a small neighborhood around zero. It is also demonstrated that the event-triggered control method does not induce Zeno behavior. Simulation results confirm the effectiveness of the proposed formation control method.A. Preliminaries F = {V, W, A} represents a weighted undirected graph of A orders, where V = {v 1 , v 2 , . . . , v A } is the note set, W ⊆ {(v i , v j ) : v i , v j ∈ V} is the edge set, and A = [s ij ] ⊆ R A×A is the weighted adjacency matrix. Suppose that there are A agents, where B of them are followers and A -B > 0 are leaders. Let F = {1, 2, . . . , B} be the follower subscript set, and E = {B + 1, B + 2, . . . , A} be the leader subscript set. For anyi, j ∈ E ∪ F , we can denote s ijEQUATIONLemma 1 [14] : The following inequality holds for any ε > 0 andEQUATIONwhereta ε (x i ) = x i / x 2 i + ε, ta ε (x) = [ta ε (x 1 ), ta ε (x 2 ), . . . , ta ε (x B )] T , ε 1 =mε, and sign is the meaning of the sign function.Lemma 2 [15] : For any vector ε 3 > 0 and x ∈ R, the following inequality satisfiesEQUATIONConsider multi-agent systems consisting of A agents, where B of them are followers and A -B > 0 of them are leaders. Let F = {1, 2, . . . , B} be the follower subscript set, and E = {B + 1, B + 2, . . . , A} be the leader subscript set.The dynamics of follower i and the leader k(k ∈ E)can be modeled as followsEQUATIONEQUATIONwhere (t) = [c T 1 (t), c T 2 (t), . . . , c T B (t)] T ∈ R 2B with c i (t) = [c ix (t), c iv (t)] T (i ∈ F ) piecewise continuously differentiable.For the multi-agent system (3) and ( 4), the containment formation is achieved, if for any given bounded initial states x i (0) and v i (0) the following conditions are satisfied 1) All the states of the ith closed-loop system of the i th follower are bounded.2) There is positive scalar ε 1 and ε 2 , a finite timeT ε (∀t ≥ T ε ) such that x i (t) -c ix (t) -x k (t) ≤ ε 1 v i (t) -c iv (t) -v k (t) ≤ ε 2 (6)where ε 1 is called the tracking position error, and ε 2 is called the tracking velocity error. Assumption 1: All the following containment formation in this paper must satisfy the time-varying formation conditionEQUATIONFrom the models of system, we get that the Laplacian matrix L can be described as followsEQUATIONwhereL 1 = [s ij ] ∈ R B×B and L 2 = [s i,k ] ∈ R B×(A-B) . Let ∂ xi (t) = B j=1 s ij ((x i (t) -c ix (t)) -(x j (t) -c jx (t)))+ A k=B+1 s ik (x i (t) -c ix (t) -x k (t)), ∂ vi (t) = B j=1 s ij ((v i (t) -c iv (t)) -(v j (t) -c jv (t))) + A k=B+1 s ik (v i (t)- c iv (t) -v k (t)) be the two auxiliary variables of the multi- agent system. Let ∂ x (t) = [∂ T x1 (t), ∂ T x2 (t), . . . , ∂ T xB (t)] T , ∂ v (t) = ∂ T v1 (t), ∂ T v2 (t), . . . , ∂ T vB (t) T , u(t) = [u T 1 (t), u T 2 (t), . . . , u T B (t)] T , c v (t) = [c T 1v (t), c T 2v (t), . . . , c T Bv (t)] T , c x (t) = [c T 1x (t), c T 2x (t), . . . , c T Bx (t)] T , ċx (t) = [ ċT 1x (t), ċT 2x (t), . . . ċT Bx (t)] T , and u E (t) = [u 1 (t), u 2 (t), . . . , u B (t)] T .Then, denoting ζ(t) = [x(t), v(t)] T , multi-agent systems can be written in the following compact formEQUATION)where q(x(t), t) = [q 1 (x 1 (t), t), q 2 (x 2 (t), t) . . . , q B (xB (t), t)] T , f (ζ(t), t) = [f 1 (ζ 1 (t), t), f 2 (ζ 2 (t), t), . . . , f B (ζ B (t), t)] T - ċυ (t), and Ψ z (t) = [Ψ z1 (t), Ψ z2 (t), . . . , Ψ zB (t)] T (z = 1, 2).In this section, the containment-formation controller will be carried out based on the dynamic surface control design. And the event-triggering mechanism based on a relative threshold will be constructed on the control channel.Due to q(x(t), t) and s(ζ(t), t) are uncertain functions, they can be approximated by using the neural networks. Denote q(x(t), t)= W * T 1 (t)S 1 (t) + ε 4 and s(ζ(t), t) = W * T s (t)S 2 (t) + ε 5 , we can use Ŵ T i to approxiamte W * T i .Then, multi-agent systems (9) can be writtenEQUATIONTo design the controller, the error surface can be designedEQUATIONwhereZ 1 (t) = Z T 11 , Z T 12 , . . . , Z T 1B T is tracking error. ᾱ = ᾱT 1 , ᾱT 2 , . . . , ᾱT B Tis the virtual controller going to be designed in the following contents. Firstly, we can take the derivative of the Z 1 (t), and combine with ∂(t) in (10) . And we can denote L 1 = D+E, where D = diag[L 1,11 , L 1,22 , . . . , L 1,BB ] is the positive definite diagonal matrix of L 1 .Thus, we can obtain the following expressionsEQUATION)Since E(W * T 1 (t)S 1 (t)) and L 1 Ψ 1 (t) are bounded, we can found a positive constant γ 1 which satisfies the following inequalityEQUATION)Building the first Lyapunov function as followsEQUATIONThen, the derivative of V 1 (t) can be obtainedEQUATIONConsidering the Young's inequality, we can getEQUATIONwhere δ 0 and δ 1 are all positive scalars, and ε 4 ≤ ε 4,A . Taking ( 14) and ( 17) into (16), we can get an inequalityEQUATIONFor the above inequality, the virtual controller α 1 can be cosntructedα 1 (t) = -C 1 Z 1 (t) -D Ŵ T 1 (t)S 1 (t) -γ 1 ta ε (Z 1 (t)) (19)To avoid taking the derivative of the above virtual controller, the following low-pass filter is introduced asEQUATION)where T is a positive time contant.Denote σ(t) = ᾱ1 (t) -α 1 (t), the above equation can be rewrittenEQUATIONTaking ( 19) and ( 20) into (18), we can getEQUATIONwhereEQUATIONAccording to the Young's inequality, we can obtain thatEQUATIONwhere δ 2 is a positive scalar.Defineγ 2 = γ 1 + ε 4,A4δ1 . Then, taking (23) and ( 24) into (22), we can getV1 (t) ≤ Z T 1 (t)(Z 2 (t) -C 1 Z 1 (t) + D( W T 1 (t)S 1 (T )) + γ 2 + δ 0 Z T 1 (t)Z 1 (t) + 1 4δ 0 L 1 2 c v (t) -ċx (t) 2 + δ 2 2 Z T 1 (t)Z 1 (t) + 1 2δ 2 σ T (t)σ(t) + δ 1 Z T 1 (t)Z 1 (t) + ε 2 4,A 4δ 1 (25)Secondly, the derivative of Z 2 (t) can be obtainedŻ2 (t) = ξv (t) -α1 (t) = L 1 W * T 2 (t)S 2 (t) + ε 5 + u(t) + Ψ 2 + L 2 u E (t) -α1 (t). (26) Since ε 5 , L -1 1 L 2 u E (t), Ψ 2 and L -1 1 α1 (t) are bounded, we can obtain thatEQUATIONDue to the event-triggering mechanism shuld be set on the control channel, the event-triggering can be builtEQUATIONwhere e(t) = u(t) -o(t) represents the measurement error casued by event trigger, 0 < ϑ < 1, ε, σ, m are all positive constants, and m > m 1-ϑ .t k , k ∈ z + , represents the moment when the event is triggered.That is when (22) be sparked while this moment marks as t k , and the control signal u(t k+1 ) of this moment can apply to the system. In the gap between two event-triggered moments t ∈ [t k , t k+1 ), the control signal u(t k ) remains unchanged.Denotel(t) = -Ŵ T 2 (t)S 2 (t) -C 2 Z 2 (t) -Z 1 (t) - γ3 (t).Thus, u(t) can be rewrittenEQUATIONTriggering inerval between two events [t k , t k+1 ), (23) can be obtainedEQUATIONwhere|ρ 1 (t)| ≤ 1 and |ρ 2 (t)| ≤ 1.Then, we can obtain thatEQUATIONConsider the second Lyapunov functionEQUATIONThen, substituting u(t) into (26) yieldsEQUATIONTaking the derivative of V 2 (t) yieldsV2 (t) = Z T 2 (t)(W * T 2 (t)S 2 (t) + ε 5 + u(t) -ρ 2 (t)m 1 1 + ρ 1 (t)ϑ + Ψ 2 ) +Z T 2 (t)(L -1 1 L 2 u E -L -1 1 α1 (t)) + B i=1 (γ 3i -γ 3i ) γ3i /ς 0 + B i=1 (γ 3i -γ 3i ) Z T 2 (t) . (36) Due to λ 1 (t) ∈ [-1, 1], λ 2 (t) ∈ [-1, 1], we can obtain z 2 u 1 + ρ 1 (t)ϑ ≤ z 2 u 1 + ϑ ρ 2 (t)m 1 + ρ 1 (t)ϑ ≤ m 1 -ϑ (37)According to Lemma 2, we can getEQUATIONThen, we can obtainEQUATION)Noting that γ2i /ς 0 + |Z T 2 (t)| = 0, and substituting l(t) into (36), we can obtainEQUATIONIV. STABILITY ANALYSIS Theorem 1: For nonlinear second-order systems ( 4) and ( 5), if the controller (28), virtual controller (19), and the adaptive laws (28), ( 21), ( 46) and (47) are applied. Then, all signals in the closed-loop system are bounded and the formation tracking error will converge to a small range.To prove the Theorem 1, we can consider V (t)EQUATIONBy( 22)-( 40), taking the derivative of V (t), we can getV (t) = V1 (t) + V2 (t) + σ T (t) σ(t) +tr[ W T 1 (t)ε -1 6 ˙W1 (t)] + tr[ W T 2 (t)ε -1 7 ˙W 2 (t)] ≤ -Z T 1 (t) C 1 - δ 2 2 + δ 0 + δ 1 I 2 Z 1 (t) -Z T 2 (t)C 2 Z 2 (t) + 0.557ε 3 + 1 2δ 2 σ T (t)σ(t) +σ T (t) σ(t) + Z T 1 (t)D W T 1 (t)S 1 (t) + Z T 2 (t) W T 2 (t)S 2 (t) -tr W T 1 (t)ε -1 6 Ẇ1 (t) -tr W T 2 (t)ε -1 7 Ẇ2 (t) +γ 2 + 1 4δ 0 L 1 2 c v (t) -ċx (t) 2 .(42) From (21), it follows thatσ T (t) σ(t) = -T -1 σ(t) -σ T (t) α1 (t) ≤ -σ T (t) T -1 - δ 4 2 I σ(t) + 1 2δ 4 α1 (t) 2(43) where δ 4 is a positive scalar.By the property of the trace of a matrix, which hasZ T 1 (t)D W T 1 (t)S 1 (t) = tr[ W T 1 (t)S 1 (t)Z T 1 D] and Z T 2 (t) W T 2 (t)S 2 (t) = tr[ W T 1 (t)S 2 (t)Z T 2 (t). Substituting (38) into (37), we can obtainEQUATIONWe can getEQUATIONEQUATIONSo taking (45) and ( 46) into (44), the equation about W1 changesEQUATIONwhere 0≤ µ 1 ≤ 2λmin(Ξ1) λmax(Ξ1) , ϕ 1 = λ min (Ξ 1 ) -1 2 µ 1 λ max (Ξ 1 ). By the same reasoning, we can build W2 . From (44), we can getV (t) ≤ -Z T 1 (t) C 1 -( δ 2 2 + δ 0 + δ 1 )I 2 Z 1 (t) -Z T 2 (t)C 2 Z 2 (t) + 0.557ε 3 -σ T (t)(T -1 - 1 2δ 2 - δ 4 2 )σ(t) -ϕ 1 tr[ W T 1 (t) W1 (t)] -ϕ 2 tr W T 2 (t) W2 (t) + λ max (Ξ 1 ) 2µ 1 W * 1 (t) 2 F + λ max (Ξ 2 ) 2µ 2 W * 2 (t) 2 F + 1 2δ 4 α1 (t) 2 + γ 2 + 1 4δ 0 L 1 2 c v (t) -ċx (t) 2 ≤ -α 0 V + β 1 + 1 4δ 0 L 1 2 c v (t) -ċx (t) 2 (48) where β 1 = α1 (t) 2 /2δ 4 + γ 2 + 0.557ε 3 + λ max (Ξ 1 ) W * 1 (t) 2 F /(2µ 1 ) + λ max (Ξ 2 ) W * 2 (t) 2 F /(2µ 2 ), α 0 = min{λ 1 , λ 2 , λ 3 , λ 4 , λ 5 }, λ 1 = -λ min (C 1 -(δ 2 /2 + δ 0 + δ 1 )I), λ 2 = -λ min (C 2 ), λ 3 = -λ min (Ξ -1 -(1/2 + δ 4 /2)I), λ 4 = ϕ 1 and λ 5 = ϕ 2 .Denote Φ i (t) = (1/(4δ 0 )) L 1 2 c iv (t) -ċix (t) 2 and we can define Φ(t) = (1/(4δ 0 )) L 1 2A-1 i=1 c iv (t) -ċix (t) 2 . Then, inequality (48) can be rewritten asEQUATIONThen, we should prove the prerequisities before anlysis the system stability.Firstly, integrating (49) givesV (t) ≤ e -α0t V (0) + β 1 α 0 (1 -e -α0t ) + t 0 (e -α0(t-τ ) Φ(τ ))dτ = e -α0t V (0) + β 1 α 0 (1 -e -α0t) + e -α0t t 0 (e α0τ Φ(τ ))dτ.(50) One has lim t→∞ Φ(t) = 0 and Φ(t) is uniformly bounded, which means that there is a positive constant Φ 0 such that Φ(t) ≤ Φ 0 . Denote a positive constant > 0, there exists a fixed time T > 0, such that 0 < Φ(t) < /4, ∀t > T .Then, we can obtain thatV (t) - β 1 α 0 ≤ e -α0t V (0) - β 1 α 0 e -α0t+ e -α0t Φ 0 (e α0T -1)+ 4 e -α0t (e α0t-e α 0 T ).(51)Found a time constant T D which satisfies ∀t > T D > T . One obtains e -α0t V (0) < /4, |e -α0t β 0 /α 0 | < /4, e -α0t Φ 0 (e α0T -1) < /4, and (1 -e -α0(t-T ) ) /4 < /4. Thus, (51) can be changed to V (t) ≤ β1 α0 + , for the reason lim t→∞ = 0 and lim t→∞ V (t) ≤ β 1 /α 0 .We can easily get that from ( 6) and ( 8):x i (t) -c ix (t) -x k (t) = L -1 1 ∂ xi (t) and v i (t) -c iv (t) -v k (t) = L -1 1 ∂ vi (t). It is easy to get that the above equations are bounded.Due to the introduction of event triggers, Zeno should be prevented. From (23) we can obtain e(t) = u(t) -o(t), ∀t ∈ [t k , t k+1 ).Then, the derivitivate of e(t)EQUATIONSince u is a continuous function, and the variables in the function are all globally bounded. So there must be a constant χ which satisfies | u| ≤ χ. When e(t k ) = 0 and lim t→t k+1 e(t) = ϑu, hence a positive constant t * which satisfies t * ≥ ϑu χ can be found. Moreover, the Zeno phenomenon does not occur. From all the analysis above, it is concluded that all closed-loop signals of multi-agent systems are bounded. The containmentformation control protocols and event-triggering mechanism are designed.Consider multi-agent systems combining six followers and two leaders. The mathematical models of followers can be constructedEQUATIONwhere i = 1, 2, . . . , 6, G 1 = 1.5, G 2 = 5, Ψ 1i (t) and Ψ 2i (t) represent the disturbances. Suppose Ψ 1i (t) = 2 + 0.3 sin(0.2t) + 0.2 cos(2t) and Ψ 2i (t) = 2 cos(0.2t). And q i (x i (t), t) and s i (x i (t), v i (t), t) are the nonlinear parts, we set them as q i (x i (t), t) = d i cos(x i (t)) 2 and s i (x i (t), v i (t), t) = n i sin(x i v i ) 2 . The constants d i and n i belong to [-1; 1; 1.5; 2; -1.5; -2] and [2; 0. The follwer adjacency matrix is        0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0        , and the leader adjacency matrix is        1 1 0 0 1 1 1 1 0 0 0 0         .The desired containment-formation of followers are presentedEQUATIONThe parameters could be chosen C 1 = 50, C 2 = 50, T = 0.001, ε 6 = 0.01, ε 7 = 0.01, Ξ 1 = 0.005, Ξ 2 = 0.005, γ 1 = 2.5, m = 1, m = 2, ϑ = 0.5, ε 3 = 0.5.The simulation resluts are shown, Figs. 1-4 describe snapshots of the states of six followers and two leaders at t=0s, t=5s, t=15s, and t=20s. This paper studied the containment-formation control for second-order nonlinear multi-agent systems via eventtriggering. An adaptive neural network controller and a relative threshold event-triggered mechanism based on control signals have been designed to control multi-agent systems. In the multi-agent systems, neural networks are used to approximate unknown dynamics. In terms of stability, Lyapunov's method is adopted to prove it. Furthermore, it is demonstrated that there is no Zeno phenomenon in the multi-agent systems. Finally, simulation results are presented to validate the effectiveness of the proposed formation control method. 
 
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Reviews: review: title: title: Accept
main_review: main_review: 1. The authors should explicitly outline the motivation and contributions of this paper in comparison to existing works in the field.
2. Please clarify the reasonability and feasibility of Assumption 1 in this paper.
3. Comparative results with other references should be presented to validate the effectiveness of the proposed method.
4. Typo: Defmition. 
5. The presentation quality of this paper requires significant improvement. The authors are advised to carefully review and correct several typographical and grammatical errors within this paper.
title: title: Comments to paper 46
main_review: main_review: This paper studies the event-triggered containmentformation control problem for nonlinear multi-agent systems. Neural networks are utilized to approximate uncertain dynamics. Some comments should be considered.
1. The controller has many parameters. Please give a remark to illustrate the effect of the controller
parameters.
2. Which is the training data, which is the validation data?
3. Which is the architecture of the neural network?
4.How does the event triggering strategy proposed in this article differ from previous results?
title: title: Containment-Formation Control for Second-Order Nonlinear Multi-Agent Systems via Event-Triggering
main_review: main_review: This paper investigates the event-triggered containment control problem in multi-agent systems. The research work is substantial, and although there are some issues, it can be considered for acceptance after revisions.