Adaptive NN Consensus Control for Second-Order Nonlinear Multi-Agent Systems Against Sparse Sensor Attacks

Authors: xiao  tang
yang  yu
wei  wang
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-07-30 09:13:10

Abstract: This paper investigates the secure consensus tracking control problem of second-order nonlinear multi-agent systems against sparse sensor attacks. A secure data selector is designed to extract unattacked output data from a set of output measurements affected by sensor attacks. Subsequently, utilizing the unattacked output data, a neural network (NN) secure state observer is constructed to estimate the unavailable system states. Then, an adaptive NN consensus controller is proposed via dynamic surface control technique. The designed control method ensures that all signals of the closed-loop systems are ultimately bounded, and consensus tracking control is achieved with bounded tracking errors in the presence of sparse sensor attacks. Finally, the effectiveness of the proposed control scheme is validated through a simulation of unmanned aerial vehicle attitude control systems.

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 and sensor attacks recently. In [8] , a distributed cooperative control problem was addressed for a class of linear MASs under two types of attacks. The author in [9] developed a distributed secure consensus control with event-triggering for linear leader-following MASs under DoS attacks. The article [10] proposed an event-triggered scheme to reach consensus of systems with DoS attacks. In [11] , stability analysis of stochastic systems with DoS attacks was investigated by designing an observer-based event-triggered protocol. In [12] , observers were designed to estimate the states of encountered sensor attacks. In the article [13] , a new dynamic event-triggered strategy was introduced that could adjust the variables online, and a resilient control strategy was further devised for the paralyzed MASs under the sensor attacks [14] . In addition, the article [15] introduced two induced parameters and adaptive laws to compensate them to avoid the effect of corrupted data. However, most of the above studies are for known nonlinear systems, and there is still little research for unknown nonlinear MASs against sparse sensor attacks.Motivated by the above discussions, this paper solves the secure consensus tracking control problem of second-order nonlinear MASs against sparse sensor attacks. A secure data selector is proposed to extract the unattacked output data from a set of output measurements affected by sensor attacks, and an NN secure state observer is constructed to estimate the unavailable system states by using the unattacked output data. Then, we design an adaptive NN consensus controller by using the dynamic surface control technique, which ensures that all signals of the closed-loop systems are ultimately bounded, and consensus tracking is achieved with bounded tracking errors in the presence of sparse sensor attacks.The commmunication topology of N agents is described by the directed graph G = (V, E), where the node set V = {1, ..., N } with "1, ..., N " standing for the follows. E = {(j, i) : i, j ∈ V, i ̸ = j} is a set of edges, where the ordered edge (j, i) implies that node i can receive the information from node j. Further, define the neighbor set of node i as N i = {j ∈ V|(j, i) ∈ E}. Define an adjacency matrix as A = [a i,j ] ∈ R N ×N , where the weight a i,j > 0 if and only if j ∈ N i and a i,j = 0 otherwise. The in-degree diagonal matrix is expressed as D = diag{d 1 , ..., d N } with d i = j∈Ni a i,j . The Laplacian matrix of graph G is represented as L = D -A. The augmented graph is described as Ḡ = ( V, Ē) with V = {0, 1, ..., N } and Ē ⊆ V × V, where "0" is the leader. Similarly, define C = diag{a 1,0 , ..., a N,0 }, and a 1,0 > 0 if and only if the node i can receive the information from node 0 and a 1,0 = 0 otherwise, denotes the communication In this paper, the framework of second-ordor MASs is described as follows. Suppose that there exist N followers, labeled as agaents 1 to N , the i th agent is described as a class of strict-feedback nonlinear systems with sensor attacks:EQUATIONwhere i = 1, 2, ..., N , xi = [x i,1 , x i,2 ] T ∈ R 2 is the unavailable state vector, f i,1 (x i,1 ) and f i,2 (x i ) are the unknown continuous nonlinear functions, u i is the controller to be designed. y ζ,i ∈ R pi is the attacked output vector with p i being the number of the output sensors. ζ i (t) = [ζ i,1 (t), ..., ζ i,pi (t)] T denotes the injected data caused by the adversarial attacker, i.e., if the sensor k (k ∈ {1, 2, ..., p i }) is compromised, then ζ i,k (t) ̸ = 0 for some t ≥ 0; otherwise ζ i,k (t) = 0 for all t ≥ 0. The output distribution matrix is defined asC i = [E 1 , ..., E 1 ] T ∈ R pi×2 , E 1 ∈ R 2 isa vector with the first th element being one and the rest being zero. The unattacked output vector of the ith agent is ȳi =C i xi = [x i,1 , ..., x i,1 ] T ∈ R pi .The control objective of this paper is to design an adaptive NN consensus controller u i for N followers (1), such that under the sparse sensor attacks, the unattacked outputs of all followers ⌣ y i converge to a small neighborhood of the virtual leader's output y 0 .Consider that the attack vector ζ i (t) is s i -sparse, which means that at most s i (s i < p i ) elements of the vector ζ i (t) are simultaneously nonzero. To design an effective data selector, the following assumption is introduced.Assumption 1: For each agent in MASs (1), the number of simultaneously attacked sensors s i satisfies 2s i < p i .Based on Assumption 1, a median value operator M ed[•] is designed to pick out the sensor data that is not attacked. Given a vector y ζ,i (t), rearrange its elements in increasing numerical order to produce a new vector ℏEQUATIONwhere M i = 0.5(p i +1) if p i is an odd integer, and M i = 0.5p i otherwise. Under Assumption 1, a secure data selector of the ith (i = 1, 2) agent is given asEQUATIONProposition 1: For the system (1) under sparse sensor attacks, if Assumption 1 is satisfied, then the output of data selector ( 3) is equal to the unattacked output, i.e.,y i (t) = ⌣ y i (t), ∀t ≥ 0 with y i (t) = [ȳ i ] 1,1 .With the help of secure data selector and adaptive NN, a secure state observer is designed to estimate the unavailbale states. The system (1) can be rewritten asEQUATIONwhereA i0 = 0 I 1 0 0 , F i (x i ) = [f i,1 (x i,1 ), f i,2 (x i )] T .The NN secure state state observer is constructed asEQUATIONEQUATION)] T . Accroding to (4) and ( 5), the dynamics of xi are derived asEQUATIONwhereA i = A i0 -L i E T 1 .Based on the outputs of the secure data selector and NN secure state observer, the error surfaces are defined asEQUATIONandEQUATIONwhere i = 1, 2, ..., N , α i,2 and ᾱi,2 are the virtual control law and the filterd virtual control law, respectively.Step 1: The derivative of z i,1 along (1), ( 7) and ( 8), isEQUATIONwherek i = d i + b i and Γ i,1 (x j,1 ) = -1 ki M j=1 a ij f j,1 (x j,1 ).Use two RBF NNs to approximate unknown nonlinear function f i,1 (x i,1 ) and Γ j,1 (x j,1 ). They can be described as following forms:f i,1 (x i,1 ) = θ * T i,1 ϕ i,1 (x i,1 ) + ε i,1 Γ j,1 (x j,1 ) = θ * T j,1 ϕ j,1 (x j,1 ) + ε j,1where θ * i,1 , θ * j,1 are the optimal NN weight vectors, ϕ i,1 (x i,1 ), ϕ j,1 (x j,1 ) are the Gaussian basis functions, and ε i,1 , ε j,1 are the minimum approximation errors, respevtively.To stabilize (9) , the first virtual control law α i,2 for the ith follower is designed asEQUATIONwhere c i,1 is a positive parameter. Choose the Lyapunov function V i,1 asEQUATIONThe derivative of V i,1 with respect to time isEQUATION) Design the adaptive law asEQUATIONwhere σ i,1 and σ j,1 are positive parameters. From ( 10)-( 13), Vi,1 follows that:EQUATION(14) Then, in order to obtain the filtered virtual control law ᾱi,2 , we pass α i,2 through a first order filter with a small time constant τ i,2 > 0.EQUATIONStep 2: According to (1), ( 8), we get żi,2 = ẋi,2 -αi,2EQUATIONChoose the controller u i and the adaptive law θi,2 as:EQUATIONThen żi,2 can be rewritten asEQUATIONChoose the Lyapunov funtionEQUATIONThe derivative of V i,2 with respect to time isEQUATIONAccording to the following the Young's inequalityEQUATIONThen, (20) follows thatVi,2 ≤ -c i,2 z 2 i,2 -EQUATIONTheorem 3.1: For nonlinear second-order systems (1), if we consider the controller (17), virtual control law (10) , and the adaptive laws ( 13), (17), we can get the overall control scheme has the following performance:• Close-loop system is semi-globally uniformly asymptotically stable; • Consensus tracking errors between followers and leader converge to a small neighborhood of the virtual leader's output y 0 .Proof: Choosing the total Lyapunov candidate funtion V asEQUATIONThen, the time derivative of V along ( 11)-( 13), ( 21), ( 23) is computed asV ≤ N i=1 -c i,1 z 2 i,1 - σ i,1 2λ i,1 θi,1 2 - σ j,1 2λ j,1 θj,1 2 + k i z i,1 z i,2 + k i z i,1 s i,2 + k i z 2 i,1 + k i ( 1 2 ε 2 i,1 + 1 2 ε 2 j,1 ) + σ i,1 2λ i,1 θ * i,1 2 + σ j,1 2λ j,1 θ * j,1 2 -c i,2 z 2 i,2 - 1 τ i,2 s 2 i,2 - σ i,2 λ i,2 θT i,2 θi,2 + 1 2 z 2 i,2 + 1 2 θi,2 2 + σ i,2 λ i,2 θ * T i,2 θ * i,2 + s 2 i,2 B2 i,2 2δ i,2 + 2δ i,2(24) Using the Young's inequality, we haveEQUATIONChoosing the parameters satisfyc i,1 -2k i > 0, c i,2 -ki+1 2 > 0, 1τi,2 -ki 2 > 0 and from the known above σ i,1 , σ j,1 , λ i,1 , λ j,1 , λ i,2 , τ i,1 , τ i,2 , δ i,2 are positive parameters, we can obtain thatEQUATION)whereη = min[(c i,1 -2k i , c i,2 -ki+1 2 , 1 τi,2 -ki 2 ] > 0, and D = N i=1 ki 2 ε 2 i,1 + ki 2 ε 2 j,1 + σi,1 2λi,1 θ * i,1 2 + σj,1 2λj,1 θ * j,1 2 + 1 2 θi,2 2 + σi,2 λi,2 θ * i,2 2 + s 2 i,2 B2 i,2 2δi,2 + 2δ i,2 . According to (27), we have V (t) ≤ e -ηt V (t 0 ) + ( D η )[1 -e -ηt ].Then, the signals in closed-loop systems are SGUUB. Moreover, ultilizing 1 2 ∥z 1 ∥ 2 ≤ V (t) withz 1 = [z T 1,1 , ..., z T N,1 ] T , we get ∥z i,1 ∥ 2 ≤ 2V (t 0 )e -ηt + ( 2D η )[1 -e -ηt ]. According to z 1 = (L + B)(y -(1 N ⊗ y 0 )) where y = [ ⌣ y 1 , ..., ⌣y N ] T and ⊗ stand for the Kronecker product. Therefore, the tracking errors can be made as small as possible by choosing appropriate design parameters.Based on the above analysis, in this section we Desige a system which has a virtual leader and four UAVs with the C = diag{1, 0, 0, 0}, and the adjacency matrix A = [0, 0, 0, 0; 1, 0, 0, 0; 1, 0, 0, 0; 1, 0, 0, 0]. The ithEQUATIONwherex i,1 = [ρ i , β i , ψ i ]T are the roll, pitch, yaw angles of UAV, x i,2 = [ω ρ i , ω β i , ω ψ i ] T are the angular velocities where -π 2 < β i < π 2 . Design the coefficient matrix asγ i = diag{I -1 x , I -1 y , I -1 z } which has I x = 0.0027kg • m 2 , I y = 0.0027kg • m 2 , I z = 0.0047kg • m 2 , and u i = [u ρ i , u β i , u ψ i ]T represent the roll, pitch and yaw torques. Then the output distribution matrix isC i = [I 3 , 0] ∈ R 3×6 , define x ′ i = [x T i,1 , x T i,2 ] T with xi = [x ′ i , ..., x ′ i ] ∈ R 6×pi . Therefore the real output matrix is ȳi = C i xi = [x i,1 , ..., x i,1 ] T ∈ R 3×pi .We choose the number of the output sensors as p 1 = 5, p 2 = 6, p 3 = 7, p 4 = 7. Besides, the unknown functions f i,1 and f i,2 are given asEQUATIONWe choose the adaptive NN membership function asµ F 1 i,k (x i ) = exp[- (x i,k -8) 2 2 ], µ F 2 i,k (x i ) = exp[- (x i,k -6) 2 2 ] µ F 3 i,k (x i ) = exp[- (x i,k -4) 2 2 ], µ F 4 i,k (x i ) = exp[- (x i,k -2) 2 2 ] µ F 5 i,k (x i ) = exp[- (x i,k -0) 2 2 ], µ F 6 i,k (x i ) = exp[- (x i,k +2) 2 2 ] µ F 7 i,k (x i ) = exp[- (x i,k +4) 2 2 ], µ F 8 i,k (x i ) = exp[- (x i,k +6) 2 2 ] µ F 9 i,k (x i ) = exp[- (x i,k +8) 2 2] (30) and order thatEQUATIONEQUATIONAs for the injected data matrices, we design ζ i (t) to describe them asEQUATIONThe attitude of the virtual leader is determinded by y 0 = [y ρ 0 , y β 0 , y ψ 0 ] T = [0.1π sin(0.1πt), -0.1π cos(0.1πt), 0] T , and we designed other parameters as follow c i,1 = diag[2, 2, 1], c i,2 = diag[20, 20, 50], λ i,1 = λ i,2 = 2, σ i,1 = σ i,2 = 20, τ i,2 = 0.005, l i,1 = diag[(0) = [0.02, -0.3, -0.01] T , x 2,1 (0) = x 3,1 (0) = x 4,1 (0) = [0.03, -0.4, 0] T , x i,2 (0) = [0, 0, 0] T , x1,1 (0) = [0.02, -0.3, -0.01] T , xi,1 (0) = x i,1 (0), xi,2 (0) = [0, 0, 0] T .Simulation results are shown in Figs. 1 2 3 4 . Fig. 1 shows the leader's and UAVs' angles and their observations under the attacks above parameters. Fig. 2 shows the leader's and UAVs' angular velocities and their observations. Both figs exhibit that the followers' attitude can be consistent with the leader's under the sparse sensor attacks. The tracking errors are described in Fig. 3 , from which we can see that the tracking error converges to zero asymptotically. Fig. 4 represents the control output of four UAVs. V. CONCLUSION This paper studied the secure consensus tracking control problem of second-order nonlinear MASs against the sparse sensor attacks. By designing a secure data selector to extract a set of unattacked sensor data and using an NN secure state observer to reconstructed the unavailable system states, adaptive NN consensus controller via dynamic surface control technique is employed to ensure that all signals of the closedloop systems are ultimately bounded. Furthermore, the consensus tracking control errors between followers and the leader converge to a small neighborhood. Finally, the effectiveness of the proposed control method is proved through a simulation of multiple UAVs. 
 
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Reviews: review: title: title: The topic under consideration is interesting. This paper can be accepted after minor modifications.
main_review: main_review: This paper investigates the secure consensus tracking control problem of second-order nonlinear multi-agent systems against sparse sensor attacks. The topic under consideration is interesting. Detailed comments and suggestions are listed as follows.
1.	Some symbols for use are not clear, such as “$\bar y_i=C_i\bar x_i=[x_{i,1},\ldots,x_{i,1}]$”, “the definition of the operator $Med[\cdot]$”, and “the symbol of equation (3)” in Page 2.
2.	In Problem statement, it is mentioned that “there exist $N$ followers”. However, in the following，the virtual leader is mentioned and the actual leader is not mentioned. Is there an actual leader in the system？   
3.	The English writing of the paper needs to be further polished, and some typos should be fixed.
title: title: This paper investigates the secure consensus tracking control problem of second-order nonlinear multi-agent systems against sparse sensor attacks. The feasibility of the designed control approach is proven via the simulation example. However, the following suggestions need to be considered in the revised manuscript to further improve the quality of this paper.
main_review: main_review: 1. How does the proposed secure data selector effectively distinguish unattacked output data from measurements affected by sensor attacks? What criteria or methods are used to ensure the accuracy and reliability of the selected data? 
2. Could the authors elaborate on the construction of the NN secure state observer? How does it leverage the unattacked output data to estimate unavailable system states? 
3. What are the conditions under which the proposed method guarantees bounded tracking errors and ultimately bounded signals of the closed-loop systems?
title: title: Adaptive NN Consensus Control for Second-Order Nonlinear Multi-Agent Systems Against Sparse Sensor Attacks
main_review: main_review: 1. What is the principle of data selection for the designed secure data selector when the system is subjected to sparse attacks? In the simulation process, How does the selector perform data selection in the actual simulation process?
2. The secure state observer has been proposed to estimate unknown states. How can the accuracy of the observation mechanism be ensured, and is there any theoretical analysis of observation errors?
3. In graph theory, the definition of $0$ is repetitive, such as $a_{i,j}=0$ and the leader $0$. Please clarify the definition of the parameters to ensure their uniqueness.