Security control method for non-integrable constraint chain multi-agent system under dos attack

Abstract

The application relates to a security control method for a non-integrity constraint chain multi-agent system under a Dos attack. 1) A communication topology graph of a multi-agent system is defined based on graph theory, an adjacency matrix and a Laplacian matrix of the communication topology graph are given, and the input assumption of a leader is considered under the Dos attack. 2) A detection mechanism of a Dos attack model is established, a high-order chain multi-agent system dynamic equation is considered, and an observer is developed to estimate the leader information in a limited time. 3) By establishing an elastic distributed observer nonlinear protocol, the consistency tracking control problem of a chain system is converted into the tracking control problem of a single chain system. 4) Based on the backstepping control, it is proved that, for the given leader and follower chain dynamic equations, the safe consistency tracking in a limited time is realized under the Dos attack.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of secure control of nonholonomic high-order chained multi-agent systems, and particularly relates to a secure control method for nonholonomic chained multi-agent systems under denial-of-service attacks, which can maintain resilience and achieve consistent tracking in a limited time. BACKGROUND

[0002] Network security is of vital importance for the distributed control of multi-agent networks that exchange information over an interactive network. However, in a real environment, it is extremely challenging to build a completely secure network environment, because many attackers may intentionally tamper with or destroy the data transmitted between agents. What is particularly noteworthy is that, unlike the external disturbances (such as noise, faults, etc.) commonly seen in typical control systems, information-physical attacks are malicious behaviors deliberately and unpredictably designed and injected into the system, aiming to weaken or even destroy the performance stability of the entire system, and in severe cases, even lead to system out-of-control.

[0003] Moreover, with the frequent occurrence of network attacks, the secure control method of traditional linear multi-agent systems is not up to the challenge when facing nonholonomic constraints. Nonholonomic multi-agent systems, such as mobile robots, have complex dynamic characteristics and are easily affected by network attacks, especially in scenarios requiring limited-time convergence, the challenge is even more severe.

[0004] The security control of multi-agent systems includes any means specifically designed to prevent data attacks, and defend against common network attacks such as denial of service attacks and deception attacks. From an implementation perspective, network attacks can be roughly divided into two categories. The first category is deception attacks, including false information injection and data replay. The second category is denial of service attacks, in which the attacker prohibits the transmission of measurement or control signals between agents. Reference 1 (Li Y, Tong S. Bumpless transfer distributed adaptive backstepping control of nonlinear multi-agent systems with circular filtering under DoS attacks [J]. Automatica, 2023, 157: 111250.) studies the problem of resilient distributed collaborative control of a class of uncertain nonlinear multi-agent systems under denial of service attacks. Reference 2 (Liu ZW, Shi YL, Yan H, et al. Secure consensus of multiagent systems via impulsive control subject to deception attacks[J]. IEEE Transactions on Circuits and Systems II: Express Briefs, 2022, 70(1): 166-170.) studies the security consistency problem of multi-agent systems with impulsive control under deception attacks.

[0005] Current research focuses on linear systems, while exploration of nonlinear, nonholonomic, high-order chain systems and finite-time convergence control strategies is relatively insufficient. This situation highlights the urgency and importance of the problem, and it is urgent to further explore the safety issues in the consensus control of multi-agent systems. Summary of the Invention

[0006] The purpose of the present invention is to provide an innovative safety control method for non-holonomic constraint chain multi-agent systems under DoS attacks based on an observer to achieve consistency tracking within a finite time, so as to ensure that the system can converge rapidly in a short time.

[0007] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] The present invention develops a novel elastic distributed observer that can effectively estimate the leader's information within a finite time, thereby effectively resisting the impact of denial of service attacks on the system. Subsequently, based on the precise information estimation provided by the observer, a finite-time control protocol is carefully designed using the backstepping control principle. This protocol aims to enable followers to accurately complete tracking and control of the leader within a preset finite time period, significantly improving the system's response speed and stability.

[0009] The security control method of the non-holonomic constraint chain multi-agent system under DoS attack of the present invention specifically comprises the following steps:

[0010] Step 1: Define the communication topology of the multi-agent system based on graph theory. Give the adjacency matrix and Laplace matrix of the communication topology. Consider the changes in the Laplace matrix of the system communication topology under DoS attacks, and make assumptions about the leader's input.

[0011] In the first step, without loss of generality, we assume that the multi-agent system has n followers and one leader, numbered 0, 1, ..., n; based on graph theory, we define the system communication topology, adjacency matrix and Laplace matrix; the leader's input satisfies Where d1, d2, d3, τ1 are positive constants, and u 2,0 (t) has a finite upper bound; the leader's state satisfies |x k,0 |≤τ k ,k=2,3,…,m, where τ k is a known positive constant;

[0012] In the second step, we assume that the communication topology of the agents is undirected. The communication between agents may be cut off due to a DoS attack. When there is no attack, the communication topology of the system is an undirected graph, and at least one follower can obtain the leader's information.

[0013] Step 2: Establish a detection mechanism for the DoS attack model. Considering the dynamics equations of the high-order chain multi-agent system, an observer is developed to estimate the leader information in a finite time under the DoS attack scenario.

[0014] In the first step, the detection mechanism under DoS attack can be summarized as follows: the information of agent i is transmitted to its neighbor j at time t; then, when the neighbor of agent i receives the information, it sends a confirmation report to agent i; assuming that there are multiple secure channels with different protocols from the communication channel between agents, each confirmation message can be successfully transmitted; if agent i is in the time interval [t, t+Δ maxIf this message is not received within ], it can be detected as a DoS attack, where Δ max It is defined as the maximum time to receive an acknowledgment message; finally, in order to restore or reinitialize the communication topology of the intelligent agent system, a report on the detection of a DoS attack is sent to the intelligent communication management center, which is responsible for monitoring the network and making repair decisions;

[0015] The second step is to define the time series of the interaction topology of detecting the k-th attack and repairing the k-th attack, respectively, using t k s and Therefore, we use and Indicates the duration of no Dos attack and the duration of Dos attack; it should be emphasized that Including attack detection time, report transmission time and network repair time; Assume that the state of the interaction edge at node i is determined by ψ(t). If the attacker If we attack node i internally, then for any j, ψ(t)=0, otherwise ψ(t)=1; under DoS attack, the time-varying graph Describes the interaction topology between agents; the communication weight of this graph is composed of a and the Laplace matrix L ψ(t) Specified; Considering the dynamic equations of a high-order chain multi-agent system, a resilient distributed observer is constructed for each follower to address the impact of DoS attacks;

[0016] The third step is to construct the Lyapunov function and prove that the designed observer can estimate the leader's information in a finite time.

[0017] Step 3: By establishing a resilient distributed observer nonlinear protocol, the consistency tracking control problem of a group of linked list systems is transformed into the tracking control problem of a single linked list system;

[0018] The first step is to define the tracking error under DoS attack and without DoS attack respectively;

[0019] In the second step, the first-order subsystem and the higher-order subsystem are designed to stabilize the input signal of the tracking error, and the designed control protocol is given at the same time;

[0020] Step 4: Based on backstepping control, it is proved that for a given chain dynamics equation of a leader and followers, the designed observer and control protocol can achieve safe and consistent tracking in a finite time under DoS attack; finally, a group of wheeled mobile robots consisting of one leader and five followers is used as an example to verify the effectiveness of this method.

[0021] Compared with the prior art, the outstanding advantages and technical effects of the present invention are:

[0022] 1) The present invention addresses the problem that existing research on multi-agent systems under network attacks is mostly based on linear systems and can only reach consistency asymptotically in infinite time. The present invention studies the more general problem of finite-time secure consistency tracking of non-holonomic constrained chain systems.

[0023] 2) This paper develops a new resilient distributed observer to provide estimation of leader information in a finite time while resisting DoS attacks.

[0024] 3) The present invention adopts backstepping control technology to perform Lyapunov-based stability analysis on the finite-time stability of the system and conducts simulation studies on a group of wheeled robots. The results show that the proposed scheme has satisfactory performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 This is the control strategy block diagram under Dos attack.

[0026] Figure 2 Communication topology diagram of multi-agent system under DoS attack.

[0027] Figure 3 This is the timing diagram of the DoS attack signal (t=4s).

[0028] Figure 4 This is the timing diagram of the DoS attack signal (t=7s).

[0029] Figure 5 This is the simulation result of observation error under DoS attack (t=4s).

[0030] Figure 6 This is the simulation result of observation error under DoS attack (t=7s).

[0031] Figure 7 The simulation results of follower tracking under DoS attack (t=4s).

[0032] Figure 8 The simulation results of follower tracking under DoS attack (t=7s). DETAILED DESCRIPTION

[0033] The following embodiments will further illustrate the present invention with reference to the accompanying drawings.

[0034] This paper focuses on nonholonomically constrained high-order chained multi-agent systems, aiming to develop a secure control strategy that achieves finite-time convergence despite cyberattacks. By ingeniously integrating the essence of nonlinear control theory, adaptive control methods, and network communication technologies, it aims to construct an efficient and robust control framework, providing solid theoretical support and practical guidance for improving system security and responsiveness.

[0035] Lemma 1: For any vector x∈R m , if there exist scalar constants p, r such that p>r>0, then the following inequality holds:

[0036]

[0037] Lemma 2: For nonlinear systems α, β>0, x(0)=x0, ξ>1, if there exists a Lyapunov function V(x) such that

[0038]

[0039] Then V(x) will converge within a time T, satisfying Where T s is the total attack time.

[0040] prove:

[0041] When the attack does not occur, have Further available In the interval Integrating the two sides gives Sort out Further:

[0042]

[0043] When an attack occurs, Depend on have to Thus:

[0044]

[0045] Then V(t) can be Converges to 0. By have to make That is, x = q ξ , thus:

[0046]

[0047] Integrating both sides gives:

[0048]

[0049] It is concluded that:

[0050]

[0051] When x(t) can be derived = 0, so the equilibrium point is finite time stable. The upper bound of arctan(·) function is The convergence time is

[0052] Lemma three: take positive real numbers a, b, if c is in (0, 1] and is the ratio of two positive odd numbers, then |a c -b c |≤2 1-c |a-b| c Holds.

[0053] Lemma four: take positive real numbers c, d, f(x, y) is a positive value function, then |x| c |y| d ≤cf(x,y)|x| c+d / (c+d)+df -c / d (x,y)|y| c+d / (c+d) holds.

[0054] Lemma five: for p = 1, 2,..., k-1, there is a first-order derivable function H k,p So that

[0055] Lemma six: for the nonlinear system x(0) = x0, ξ>1, alpha, beta>0, then the equilibrium point is stable in .

[0056] The application first gives the communication topology of the multi-agent system according to graph theory, then establishes an attack model for Dos attack, considers the influence of the attack on the multi-agent system to cause the change of the communication topology, and thus develops a new distributed observer, adopts a backstepping control scheme, and proves that the chained multi-agent system can realize safe tracking control in finite time under the attack. Figure 1 The control strategy block diagram under the Dos attack of the application is given.

[0057] Step 1: without loss of generality, it is assumed that the multi-agent system is composed of n followers and a leader, numbered 0, 1,..., n.

[0058] Step 1.1: define the communication topology graph of the multi-agent system based on graph theory ​Represents the set of agents in the network, E represents the set of edges that interact between agents, and a directed edge (i, j)∈E starts from node i and ends at node j, indicating that agent j can receive information from agent i.

[0059] Step 1.2: Graph The adjacency matrix The definition is as follows:

[0060]

[0061] Its Laplace matrix Defined as:

[0062]

[0063] When the i-th follower can obtain the leader information, then g i is a normal number, otherwise it is 0. Let G=diag(g1,g2,…,g n ).

[0064] Step 2: Consider the dynamic equations of a high-order chain multi-agent system and develop an observer to estimate the leader's information in the scenario of DoS attack.

[0065] Step 2.1: Consider a nonholonomic constrained multi-agent system consisting of n followers and a leader. The dynamic equation of the i-th follower is:

[0066]

[0067] Among them, x i (t) = [x 1,i ,x 2,i ,…,x m,i ] T ∈R m is the state of the i-th agent, u i (t)=[u 1,i ,u 2,i ] T ∈R 2 represents the control protocol to be designed. The dynamic equation of the leader is:

[0068]

[0069] Where x0(t)=[x 1,0 ,x 2,0 ,…,x m,0 ] T ∈R m is the leader’s state, u0(t)=[u 1,0 ,u 2,0 ]T ∈R 2 Provides control input for the leader.

[0070] Step 2.2: To prevent the impact of DoS attacks, build the following elastic distributed observer for each follower:

[0071] 1) When an attack occurs, The observer is designed as:

[0072]

[0073] For s=1,2,i=1,2,...,n, we have:

[0074]

[0075] 2) When the attack does not occur, The observer is designed as:

[0076]

[0077] For s=1,2,i=1,2,...,n, we have:

[0078]

[0079] in, are the estimates of the leader state x0 and input u0 by each follower, α and β are positive even and positive odd numbers, ρ, ξ, a and b are the observer gains. At the same time, the error variable is defined

[0080] Step 2.3: Prove that for a given leader and follower chain dynamics equation, if assumptions 1-3 hold, there exist positive even and odd integers α and β, such that α < β, and there exist constants ρ0, ξ0, a0, and b0 such that:

[0081]

[0082] θ s ≥d s , then the elastic observer can accurately provide estimates of the leader's state and input under DoS attacks, and the observer error can converge to zero in time.

[0083] prove:

[0084] By adopting a bottom-up structure, it is proved that each follower can estimate the leader's information in a finite time under DoS attack. First, it is proved that each follower can accurately estimate the leader's input in a finite time. When s = 2, for ε u2,i Taking the derivative we get:

[0085]

[0086] in, Let U2=[ε u2,1 ,ε u2,2 ,…,ε u2,n ] T , then the above formula can be rewritten as a compact vector form:

[0087]

[0088] Construct a Lyapunov function Taking its derivative we can get the following result:

[0089] 1) When the attack does not occur, have:

[0090]

[0091] Among them,||·|| P is the p-norm, and Lemma 1 gives Then there is Similarly, Then there is Thus we can get Substituting into (20) we can get:

[0092]

[0093] 2) When the attack occurs, at time At least one agent is attacked, which results in an adjacency weight of ψ(t)a ij The Laplace matrix (L ψ(t) +G) has more than two eigenvalues ​​of 0, then λ min (L ψ(t) +G)=0. Therefore, similar to 1), it can be proved that

[0094] Therefore, from Lemma 2 and the comparison lemma, we can obtain that V1 can converge to 0 in a finite time, and for 1, 2, ..., n, there is ε u2,i = 0, the upper bound of the convergence time is in

[0095] According to similar steps, it can be proved that for any 1,2,…,n, ε can be reached within time T1. u1,i =0.

[0096] Next, we prove that each follower can estimate the leader's state in a limited time under Dos attack. We can get:

[0097] in From the above steps, we can see that when t>T1, Let Θ m =[ε m,1 ,ε m,2 ,…,ε m,n ] T , then the error vector ε m,i It can be written in compact vector form Construct a Lyapunov function Taking its derivative we can get the following result:

[0098] 1) When the attack does not occur, have:

[0099]

[0100] 2) When the attack occurs, at time There is a DoS attack in the memory, which paralyzes the connectivity of the communication graph, resulting in λ min (L ψ(t) + G)=0, and the same method as the previous case can be used to obtain

[0101] Therefore, from Lemma 2, we can get that V2=0 is stable in a finite time, that is, ε m,i = 0, the upper bound of the convergence time is in

[0102] The dynamic equation of each agent's m-1th state in the absence of DoS attacks and in the absence of DoS attacks is as follows:

[0103]

[0104] in, When t≥T1+T2, Let Θ m-1 =[ε m-1,1 ,ε m-1,2 ,…,ε m-1,n ] T , construct a new Lyapunov function A similar procedure is used to prove that in finite time there is ε m-1,i = 0. Using the recursive structure, we can prove that ε 2,i At time T obv =T1+(m-1)T2 converges to zero. In addition, ε 1,i Converges to zero in time T1+T2. Therefore, each follower can provide an estimate of the leader's information in a finite amount of time.

[0105] Note 1: It is worth noting that by T obv =T1+T2=T1 obv +(m-1)T2 obv +mT s It can be seen that the convergence time of the observer is T obv The total duration of the attack T s The longer the DoS attack lasts, the slower the convergence speed. If convergence is to be achieved within a predetermined time, the upper bound of the total duration of the attack needs to meet T s ≤(T obv -T1 obv -(m-1)T2 obv ) / m.

[0106] Step 3: By establishing a resilient distributed observer nonlinear protocol, the consistency tracking control problem of a group of linked list systems is transformed into the tracking control problem of a single linked list system.

[0107] Step 3.1: Define tracking error Substituting (8), (10) and (12) and taking the derivatives we get:

[0108]

[0109] in, From Theorem 1, we know that when t≥T obv When k=1,2,…,m, ε k,i = 0. Thus when t≥T obv The above formula can be rewritten as:

[0110]

[0111] Step 3.2: By designing the first-order subsystem (25) for stabilizing δ 1,i The input signal u 1,i , used in the high-order subsystem (26) to stabilize δ 2,i ,δ 2,i ,…,δ m,i The input signal u 2,i To achieve the desired control goal, that is, under DoS attacks, each follower can track the state of the estimated leader at a faster convergence speed within a limited time.

[0112] Considering formula (25), we first design a tracking protocol as follows:

[0113]

[0114] From Lemma 2, we can see that δ 1,i =0 at time T obv+T2 internal stability, when t≥T obv +T2 has Therefore, formula (26) can be rewritten as:

[0115]

[0116] Design tracking algorithm 2,i Accelerate the convergence of Equation (28) and make it stable in a limited time. To achieve this goal, a backstepping control method consisting of m-1 steps is adopted. In each step, a virtual control protocol is designed. Where k=3,4,…,m-1. Finally, design the control input u 2,i As shown below:

[0117]

[0118] in, μ and ν are positive even and odd numbers respectively, and for p=2,3,…,m, they all need to satisfy 0<χ p <1, then σ>0,γ=2-2χ2,η m-1,i It is a virtual protocol to be designed. m-2 >0 is a sufficiently large constant, φ m-2,i is a first-order differentiable function.

[0119] Step 4: Prove that for a given chain dynamics equation of a leader and a follower, if assumptions 1-3 hold, applying control schemes (27) and (29) and distributed observers (10)-(13) can achieve the goal of obv +Security consistency tracking within T2+T3, where

[0120] Step 4.1: Consider the first equation of the system (28) and construct a Lyapunov function Taking the derivative we get Design a virtual protocol as follows:

[0121]

[0122] Among them, η 1,i =δ 2,i ,φ 1,i >0 is a first-order differentiable function, then can be rewritten as:

[0123]

[0124] Step 4.2: Based on backstepping control, let Consider the second equation of system (28) and construct the Lyapunov function Taking the derivative we get:

[0125]

[0126] For the third term in the above formula, we have:

[0127]

[0128] From Lemma 3 we can get Further, from Lemma 4 we can get Where r1 is a positive constant, and the above formula can be obtained:

[0129]

[0130] Considering η 1,i =δ 2,i and From the designed virtual protocol We can get:

[0131]

[0132] The variables defined previously have Thus we can get:

[0133]

[0134] Combining (34) and (35) we can get:

[0135]

[0136] in is a first-order differentiable function. Substituting the above formula into the last term of (31) yields:

[0137]

[0138] The second inequality uses Lemma 3, is a first-order differentiable function. Substituting (33) and (37) into (31), we can obtain:

[0139]

[0140] Design a virtual protocol:

[0141]

[0142] Substituting (39) into (38) yields:

[0143]

[0144] Step 4.3: In step k-1 of the backstepping control design, let Constructing Lyapunov functions Taking the derivative we get:

[0145]

[0146] Next, we prove that the above formula also holds true at step k, and construct the Lyapunov function Combining (28) and (41) and taking the derivative, we can get:

[0147]

[0148] From formula (36), we can get:

[0149]

[0150] Among them, H 1,i is a first-order differentiable function. By the similar proof process of Lemma 5, for v = 2, 3, ..., k, we have:

[0151]

[0152] Established, where H k-1,i is a first-order differentiable function. For v = 2, 3, ..., k, we can know from the similar proof process of (35) that in is a first-order differentiable function, combined with (28), we have:

[0153]

[0154] in, is a first-order differentiable function. Arrangement yields:

[0155]

[0156] Then for the last term of (42) we have:

[0157]

[0158] According to the similar proof process of Lemma 3 and Lemma 4 in (33), we have:

[0159]

[0160] Substituting (47) and (48) into (42) yields:

[0161]

[0162] Design a virtual control protocol:

[0163]

[0164] Substituting equation (50) into (49) yields:

[0165]

[0166] Step 4.4: In step m-1 of the backstepping control design, select the Lyapunov function According to the recursive process at step k, we can get:

[0167]

[0168] Combining control protocols (29) and (52) we can obtain:

[0169]

[0170] For l = 3, 4, ..., m, Lemma 3 gives:

[0171]

[0172] Then there is And by 0<χ p <1, then From Lemma 1 we can get:

[0173]

[0174] Depend on From Lemma 1:

[0175]

[0176] Combining (53), (55) and (56), we can get the following inequality:

[0177]

[0178] Noting that γ = 2-2χ2, we can get:

[0179]

[0180] From Lemma 6, we can get W m,i Can be in time Inner convergence, high-order chain multi-agent systems can complete safe consistency tracking.

[0181] Step 4.5: Take a group of wheeled mobile robots consisting of a leader and five followers as an example to verify the effectiveness of this technique. The kinematic model of the robots is shown below:

[0182]

[0183] Among them, xi , y i is the position of the robot, θ i is the orientation of the robot, v i , ω i are the linear and angular velocities of the robot, respectively. Consider the following coordinate transformation:

[0184]

[0185] Convert the motion model to:

[0186]

[0187] Experimental verification:

[0188] The following evaluation verifies the ability and efficiency of the control scheme proposed in this invention in the face of different duration DoS (Denial of Service) attacks in a multi-mobile robot system.

[0189] 1. Experimental design and environment

[0190] Experimental object: 6 non-complete wheeled mobile robots.

[0191] Attack type: DoS attack, aimed at disrupting or delaying network communication to destroy system stability.

[0192] Communication topology: linear exchange topology, which is common in distributed systems, but it is more sensitive to single point failures. As shown in Figure 2 , consider the exchange of information between agents in the linear exchange topology under DoS attack. It is worth mentioning that considering the line communication topology will be a challenging case, as any node failure will interrupt network communication.

[0193] Simulate under two different duration Dos attacks as shown in Figure 3 and Figure 4 .

[0194] 2. Analysis of simulation results

[0195] Observation error: As can be seen from Figure 5 and 6 , even under DoS attack, the system can eventually reach a stable state, indicating that the proposed control scheme has good robustness. With the increase of attack duration, the convergence speed of the observer slows down. This shows that DoS attack has a direct impact on system performance, but the system can still recover to stability.

[0196] Follower tracking error: From Figure 7 and 8It can be seen that, under two different lengths of DoS attacks, the followers can successfully track the state of the leader, proving the effectiveness of the control scheme of the application. Similar to the observation error, the convergence speed of the tracking error also decreases with the increase of the attack length. However, the controller can still achieve convergence in a limited time, indicating that it has high regulation efficiency.

[0197] In summary, by Figure 5 、 Figure 6 The simulation results of the observation error and Figure 7 、 Figure 8 The simulation results of the follower tracking error, respectively, compare the simulation results under different lengths of Dos attack, and it can be seen that the multi-mobile robot system can finally realize the stability of the system, that is, each follower robot can successfully observe the state of the leader and successfully track. At the same time, by comparing the simulation results of different attack lengths, it can be seen that as the attack length increases, the convergence speed of the observer and the convergence speed of the follower tracking error also decrease, but the controller can still maintain high regulation efficiency and can converge in a limited time.

[0198] The system of the application can maintain stability when facing DoS attacks and successfully realize the observation and tracking tasks, which fully demonstrates the robustness of the control scheme. Although the increase of the attack length will affect the convergence speed, the controller can still complete the convergence in a limited time, indicating that it has high regulation efficiency. Considering the vulnerability of the linear exchange topology, the control scheme still performs well in such a challenging communication environment, indicating that it has strong applicability in practical applications.

[0199] The following gives the explanation of the symbols involved in the application:

[0200] a ij Indicates the weight edge of the adjacency matrix when not attacked;

[0201] L represents the Laplacian matrix when not attacked;

[0202] G=diag(g1,g2,…,g n ) represents gi indicates whether each follower can directly or indirectly obtain leader information;

[0203] x i (t)=[x 1,i ,x 2,i ,…,x m,i ] T ∈R m Indicates the state of the i-th agent;

[0204] u i (t)=[u 1,i ,u 2,i ] T∈R 2 represents the control input of the i-th agent;

[0205] x0(t)=[x 1,0 ,x 2,0 ,…,x m,0 ] T ∈R m Indicates the status of the leader;

[0206] u0(t)=[u 1,0 ,u 2,0 ] T ∈R 2 represents the leader’s control input;

[0207] d1,d2,d3,τ k ,k=2,3,…,m represents the upper bound of the leader’s input;

[0208] and Indicates the duration of no DoS attack and the duration of DoS attack;

[0209] Represents the time-varying graph under DoS attack;

[0210] represents the weight edge of the adjacency matrix when not attacked;

[0211] L ψ(t) represents the Laplace matrix under DoS attack;

[0212] ψ(t) represents the state of the interaction edge at node i;

[0213] ||·|| represents the Euclidean norm;

[0214] ∫ is the integral symbol;

[0215] represents the i-th follower’s estimate of the leader’s state x0;

[0216] represents the i-th follower’s estimate of the leader’s input u0;

[0217] α and β are positive even and positive odd numbers respectively;

[0218] ρ, ξ, a, b, θ1 and θ2 are observer gains;

[0219] ρ0,ξ0,a0,b0 represent the selected known constants;

[0220] sign is a mathematical symbol function;

[0221] represents the state observation error;

[0222] represents the input observation error;

[0223] λ min (·) represents the minimum eigenvalue of the matrix;

[0224] represents the tracking error;

[0225] It means that μ and ν are respectively a positive even number or a positive odd number;

[0226] γ, χ p ,p=2,3,…,m are the designed parameters;

[0227] η m-1,i A virtual protocol to be designed;

[0228] r m-2 ,σ is a constant greater than 0;

[0229] is a first-order differentiable function;

[0230] A virtual protocol to be designed;

[0231] H k-1,i is a first-order differentiable function.

[0232] The above embodiments are only preferred embodiments of the present invention and should not be considered to limit the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the patent coverage of the present invention.

Claims

1. A security control method for a nonholonomic constraint chain multi-agent system under DoS attacks, characterized by The following steps are involved: Step 1: Define the communication topology of the multi-agent system based on graph theory. Give the adjacency matrix and Laplace matrix of the communication topology. Consider the changes in the Laplace matrix of the system communication topology under DoS attacks, and make assumptions about the leader's input. Step 2: Establish a detection mechanism for the DoS attack model. Considering the dynamics equations of the high-order chain multi-agent system, an observer is developed to estimate the leader information in a finite time under the DoS attack scenario. The detection mechanism of the DoS attack model is established, and the high-order chain multi-agent system dynamics equation is considered. In the DoS attack scenario, an observer is developed. The specific steps are as follows: In the first step, the detection mechanism under DoS attack can be summarized as follows: the information of agent i is transmitted to its neighbor j at time t; then, when the neighbor of agent i receives the information, it sends a confirmation report to agent i; Assume that there are multiple secure channels with different protocols from the communication channels between agents, which successfully transmit each confirmation message; if agent i max If this message is not received within ], it can be detected as a DoS attack, where Δ max It is defined as the maximum time to receive an acknowledgment message; finally, in order to restore or reinitialize the communication topology of the intelligent agent system, a report on the detection of a DoS attack is sent to the intelligent communication management center, which is responsible for monitoring the network and making repair decisions; The second step is to define the time series of the interaction topology of detecting the k-th attack and repairing the k-th attack respectively. and Therefore, we use and Indicates the duration of no DoS attack and the duration of DoS attack; Including attack detection time, report transmission time and network repair time; Assume that the state of the interaction edge at node i is determined by ψ(t). If the attacker If we attack node i internally, then for any j, ψ(t)=0, otherwise ψ(t)=1; under DoS attack, the time-varying graph Describes the interaction topology between agents; the communication weight of this graph is composed of a and the Laplace matrix L ψ(t) Specified, among which, represents the weight edge of the adjacency matrix when not attacked, a ij Represents the weighted edges of the adjacency matrix when not under attack; Considering the dynamic equations of a high-order chain multi-agent system, a resilient distributed observer is constructed for each follower to address the impact of DoS attacks; The third step is to construct the Lyapunov function and prove that the designed observer can estimate the leader's information in a finite time. Step 3: By establishing a resilient distributed observer nonlinear protocol, the consistency tracking control problem of a group of linked list systems is transformed into the tracking control problem of a single linked list system; Step 4: Based on backstepping control, it is proved that for a given chain dynamics equation of a leader and followers, the designed observer and control protocol can achieve safe and consistent tracking in a finite time under DoS attack; finally, a group of wheeled mobile robots consisting of one leader and five followers is used as an example to verify the effectiveness of the method.

2. The security control method of a non-holonomic constraint chain multi-agent system under DoS attack as claimed in claim 1 is characterized in that In step 1, the communication topology of the multi-agent system is defined based on graph theory, the adjacency matrix and Laplace matrix of the communication topology are given, and the changes of the Laplace matrix corresponding to the communication topology of the system under DoS attack are considered. At the same time, the specific steps of assuming the input of the leader are as follows: In the first step, without loss of generality, we assume that the multi-agent system has n followers and one leader, numbered 0, 1, ..., n; based on graph theory, we define the system communication topology, adjacency matrix and Laplace matrix; the leader's input satisfies d3≤|u 1,0 (t)|≤τ1, where d1, d2, d3, τ1 are positive constants, and u 2,0 (t) has a finite upper bound; the leader's state satisfies |x k,0 |≤τ k ,k=2,3,…,m, where τ k is a known positive constant; In the second step, it is assumed that the communication topology of the intelligent agents is undirected and connected; the communication between the intelligent agents is cut off due to a DoS attack. When no attack occurs, the communication topology of the system is an undirected graph, and at least one follower obtains the leader's information.

3. The security control method of a non-holonomic constraint chain multi-agent system under DoS attack as claimed in claim 1 is characterized in that In step 3, the consistency tracking control problem of a set of linked list systems is transformed into the tracking control problem of a single linked list system by establishing a resilient distributed observer nonlinear protocol. The specific steps include: The first step is to define the tracking error under DoS attack and without DoS attack respectively; In the second step, the first-order subsystem and the higher-order subsystem are designed to stabilize the input signal of the tracking error, and the designed control protocol is given.

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