Security control method of incomplete constraint chain type multi-agent system under Dos attack

Abstract

The invention discloses a security control method for an incomplete constraint chain type multi-agent system under a Dos attack, and relates to network security. The method comprises the following steps: 1) defining a communication topological graph of a multi-agent system based on a graph theory, giving an adjacent matrix and a Laplacian matrix of the communication topological graph, and considering the change of the communication topological graph of the system corresponding to the Laplacian matrix under a Dos attack and the input hypothesis of a leader; and 2) establishing a detection mechanism of a DoS attack model, considering a kinetic equation of a high-order chain type multi-agent system, and developing an observer to realize estimation of leader information in finite time. And 3) converting a consistency tracking control problem of a group of linked list systems into a tracking control problem of a single linked list system by establishing an elastic distributed observer nonlinear protocol. And 4) based on backstepping control, it is proved that the chain type kinetic equations of the given leader and follower achieve safe and consistent tracking within finite time under the Dos attack.

Description

Technical Field

[0001] The present invention belongs to the technical field of security control of non-complete constrained high-order chained multi-agent systems, and in particular to a security control method for non-complete constrained chained multi-agent systems under DoS attacks, in which the high-order non-complete chained systems can still maintain resilience when subjected to denial of service attacks and achieve consistency tracking within a limited time. Background Art

[0002] Network security is of indispensable importance for the distributed control of multi-agent networks that exchange information on interactive networks. However, in real-world environments, 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. It is particularly noteworthy that, unlike the common external interference (such as noise, faults, etc.) in typical control systems, cyber-physical attacks are intentional and unpredictable malicious behaviors of attackers. They are carefully designed and injected into the system to weaken or even destroy the performance stability of the entire system, and in severe cases even cause the system to lose control.

[0003] With the frequent occurrence of cyber attacks, the traditional safety control methods of linear multi-agent systems are unable to cope with non-holonomic constrained systems. Non-holonomic constrained multi-agent systems, such as mobile robots, have complex dynamic characteristics and are vulnerable to cyber attacks, especially in scenarios that require finite time convergence.

[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 consensus problem of multi-agent systems with impulsive control under deception attacks.

[0005] The current research focus is mostly on linear systems, while the exploration of nonlinear, nonholonomic constrained 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 deeply explore the safety issues in the consistency 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-complete constraint chain multi-agent systems under DoS attacks based on an observer to achieve consistency tracking within a limited time, in order to ensure that the system can converge rapidly in a short time, in view of the fact that it is difficult for multi-agent systems to adapt to the widely existing non-complete constraint scenarios through simple linearization processing, and the convergence speed problem of system tracking control has not been fully taken seriously when facing denial of service attacks.

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

[0008] The present invention develops a new type of elastic distributed observer, which can effectively estimate the information of the leader within a limited time, thereby effectively resisting the impact of denial of service attacks on the system; then, based on the precise information estimation provided by the observer, a set of finite-time control protocols are 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 limited time period, significantly improving the response speed and stability of the system.

[0009] The safety 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: Based on graph theory, define the communication topology of the multi-agent system, give the adjacency matrix and Laplace matrix of the communication topology, consider the changes of the Laplace matrix of the system communication topology under DoS attack, and make assumptions about the leader's input;

[0011] In the first step, without loss of generality, assume that the multi-agent system has n followers and one leader, numbered 0, 1, ..., n; define the system communication topology, adjacency matrix and Laplace matrix based on graph theory; 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] The second step is to assume that the communication topology of the agents is undirected and connected. The communication between agents may be cut off due to DoS attacks. 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, consider the dynamic equations of the high-order chain multi-agent system, and develop an observer in the DoS attack scenario to estimate the leader information in a limited time;

[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 neighbors of agent i receive the information, they send 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 max], 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 for detecting the kth attack and repairing the kth 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 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 weights of this graph are represented by a and the Laplace matrix L ψ(t) Specified; Considering the dynamic equations of a high-order chained 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 limited time.

[0017] Step 3: By establishing a resilient distributed observer nonlinear protocol, 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;

[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;

[0020] Step 4: Based on backstepping control, it is proved that for a given chain dynamics equation of a leader and followers, under DoS attack, the designed observer and control protocol can achieve safe and consistent tracking in a finite time; finally, a group of wheeled mobile robots consisting of a leader and five followers is taken 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 aims at 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 an estimate of the 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 research 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 This is the communication topology diagram of the multi-agent system under DoS attack.

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

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

[0029] Figure 5 Simulation results 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 in conjunction with the accompanying drawings.

[0034] This paper focuses on nonholonomically constrained high-order chained multi-agent systems and is committed to developing a security control strategy that can still achieve finite-time convergence under network attack environments. By cleverly integrating the essence of nonlinear control theory, adaptive control methods, and network communication technology, it aims to build an efficient and robust control framework to provide solid theoretical support and practical guidance for improving the security and response speed of the system.

[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 did not occur, have Further available In the interval Integrating both sides gives Arranged 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] Arranged:

[0050]

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

[0052] Lemma 3: Take positive real numbers a and b. If c∈(0,1] and is the ratio of two positive odd numbers, then |a c -b c |≤2 1-c |ab| c Established.

[0053] Lemma 4: Take positive real numbers c and d, f(x,y) is a positive 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 5: For p = 1, 2, ..., k-1, there exists a first-order differentiable function H k,p , so that

[0055] Lemma 6: For nonlinear systems x(0)=x0,ξ>1,α、β>0,then the equilibrium point is Internal stability.

[0056] The present invention firstly gives the communication topology of the multi-agent system according to graph theory, then establishes an attack model for DoS attacks, considers the impact of the attack on the multi-agent system leading to changes in the communication topology, and thus develops a new distributed observer. By adopting a backstepping control scheme, it is proved that the chained multi-agent system can achieve finite-time safe tracking control when attacked. Figure 1 A control strategy block diagram under Dos attack of the present invention is given.

[0057] Step 1: Without loss of generality, assume that the multi-agent system consists of n followers and one leader, numbered 0, 1, ..., n.

[0058] Step 1.1: Define the communication topology of the multi-agent system based on graph theory in 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 of 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 record 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 chained 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 dynamics 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 ith 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 leaders.

[0070] Step 2.2: To address 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 chain dynamics equation of a leader and a follower, if assumptions 1-3 hold, there exist positive even and odd numbers α and β, and the condition α<β is satisfied, and there exist constants ρ0,ξ0,a0,b0 that satisfy:

[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 The derivative is:

[0085]

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

[0087]

[0088] Constructing 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 get:

[0092]

[0093] 2) When the attack occurs, at the 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

[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, ε u2,i = 0, the upper limit of 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 Constructing 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 the 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, using the same method as the previous case, we can get

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

[0102] The dynamic equation of the m-1th state of each agent in the case of no DoS attack and DoS attack 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 It converges to zero in time T1+T2. Therefore, each follower can provide an estimate of the leader's information in a finite 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 is 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 satisfy 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 set 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. So 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 higher-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 attack, 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 know that δ 1,i = 0 at time T obv+T2 stable, when t≥T obv +T2 Yes Therefore, formula (26) can be rewritten as:

[0115]

[0116] Design tracking algorithm 2,i Accelerate the convergence speed 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 leader and follower chain dynamics equation, if assumptions 1-3 hold, under DoS attack, 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 The derivative is 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 Considering the second equation of system (28), construct the Lyapunov function The derivative is:

[0125]

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

[0127]

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

[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 obtain:

[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), we obtain:

[0143]

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

[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:

[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. After rearrangement, we can get:

[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), we obtain:

[0161]

[0162] Design a virtual control protocol:

[0163]

[0164] Substituting equation (50) into (49), we can obtain;

[0165]

[0166] Step 4.4: In step m-1 of the backstepping control design, select the Lyapunov function According to the recursive process in 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 obtain the following inequality:

[0177]

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

[0179]

[0180] From Lemma 6, we can get W m,i Can be in time Inner convergence, high-order chained 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 the 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 direction of the robot, v i ,ω i are the linear velocity and angular velocity 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 capability and efficiency of the control scheme proposed in the present invention when facing DoS (Denial of Service) attacks of different durations in a multi-mobile robot system.

[0189] 1. Experimental Design and Environment

[0190] Experimental subjects: 6 non-holonomic wheeled mobile robots.

[0191] Type of attack: DoS attack, which aims to destabilize the system by interrupting or delaying network communications.

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

[0193] exist Figure 3 and Figure 4 The simulation is performed under two DoS attacks of different durations as shown.

[0194] 2. Analysis of simulation results

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

[0196] Follower tracking error: From Figure 7 and 8It can be seen that under two different DoS attack durations, the follower can successfully track the leader's state, proving the effectiveness of the control scheme of the present invention. Similar to the observation error, the convergence speed of the tracking error also decreases with the increase of the attack duration. However, the controller can still achieve convergence within a limited time, indicating that its regulation efficiency is high.

[0197] In summary, Figure 5 , Figure 6 Simulation results of observation error and Figure 7 , Figure 8 Simulation results of follower tracking error. Comparing the simulation results under Dos attacks of different durations, it can be seen that the multi-mobile robot system can eventually achieve system stability, that is, each follower robot can successfully observe the state of the leader and successfully track it. At the same time, comparing the simulation results of different attack durations, it can be seen that as the attack duration increases, the convergence speed of the observer and the convergence speed of the follower tracking error also decreases, but the controller can still maintain a high adjustment efficiency and can converge within a limited time.

[0198] The system of the present invention can maintain stability in the face of DoS attacks and successfully achieve observation and tracking tasks, which fully demonstrates the robustness of the control scheme. Although the increase in attack duration will affect the convergence speed, the controller can still complete convergence within a limited time, indicating that it has high regulation efficiency. Considering the fragility of the linear switching topology, the control scheme can still perform well in such a challenging communication environment, indicating that it has strong applicability in practical applications.

[0199] The following is an explanation of the symbols involved in the present invention:

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

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

[0202] G=diag(g1,g2,…,g n ) indicates whether each follower can directly obtain the leader's information;

[0203] x i (t) = [x 1,i ,x 2,i ,…,x m,i ] T ∈R m Represents 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 ith 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 Laplacian 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 smallest 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 cannot 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 present invention.

Claims

1. A safety control method for a non-holonomic constraint chain multi-agent system under DoS attack, characterized by The following steps are involved: Step 1: Based on graph theory, define the communication topology of the multi-agent system, give the adjacency matrix and Laplace matrix of the communication topology, consider the changes of the Laplace matrix of the system communication topology under DoS attack, and make assumptions about the leader's input; Step 2: Establish a detection mechanism for the DoS attack model, consider the dynamic equations of the high-order chain multi-agent system, and develop an observer in the DoS attack scenario to estimate the leader information in a limited time; Step 3: By establishing a resilient distributed observer nonlinear protocol, 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; Step 4: Based on backstepping control, it is proved that for a given chain dynamics equation of a leader and followers, under DoS attack, the designed observer and control protocol can be applied to achieve safe and consistent tracking in a finite time; finally, a group of wheeled mobile robots consisting of a leader and five followers is taken as an example to verify the effectiveness of the method.

2. The security control method of the 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, the change of the Laplace matrix corresponding to the communication topology of the system under DoS attack is considered, and the specific steps of assuming the input of the leader are as follows: In the first step, without loss of generality, assume that the multi-agent system has n followers and one leader, numbered 0, 1, ..., n; define the system communication topology, adjacency matrix and Laplace matrix based on graph theory; 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 agent is undirected and connected; the communication between the intelligent agents is cut off due to Dos attacks. 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 the non-holonomic constraint chain multi-agent system under DoS attack as claimed in claim 1 is characterized in that In step 2, the detection mechanism of the DoS attack model is established, and the high-order chain multi-agent system dynamics equation is considered. In the scenario of DoS attack, 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 ], 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 for detecting the kth attack and repairing the kth attack, respectively. 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 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 weights of this graph are represented by a and the Laplace matrix L ψ(t) Specified; Considering the dynamic equations of a high-order chained 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 within a finite time.

4. The method for controlling the safety of a non-holonomically constrained chain multi-agent system under DoS attack as claimed in claim 1, 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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