A time-scheduled sliding mode formation control method for multi-agents under DoS attacks

Abstract

The present invention discloses a method for controlling a multi-agent scheduled time sliding mode formation under DoS attack, including establishing a non-periodic DoS attack model, considering the dynamic interaction, coordination relationship and external interference between agents, establishing a multi-agent dynamic model, describing the position change of the agents; designing a scheduled time sliding mode surface and its controller to achieve the adaptability of the nonlinear system of the multi-agent to the external environment and DoS attack; setting the Lyapunov sufficient condition of the scheduled time terminal sliding mode control; proving the stability of the system when there is no DoS attack through the unbounded Lyapunov function V2(x); setting a stability proof controller of the scheduled time stable system under DoS attack, and adjusting the initial parameters of the error system so that the nonlinear system completes the desired time-varying formation control under non-periodic DoS attack. The method of the present invention improves the stability of the nonlinear multi-agent system under non-periodic DoS attack and continuous external interference.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-agent collaborative control, and in particular to a method for controlling a multi-agent predetermined time sliding mode formation under DoS attacks. Background Art

[0002] Network control systems are widely used in many practical control systems such as intelligent transportation, smart grid, industrial process control, etc. due to their high transmission efficiency, flexible installation, and data sharing. However, due to the physical limitations of the network itself, the introduction of the network into the control system may cause communication delays, data loss, channel fading, network attacks, and other problems. Among them, network attacks can be divided into three main types: Denial of Service (DoS) attacks, spoofing attacks, and replay attacks. DoS attacks are one of the most common and easiest forms of attacks, and their purpose is to try to prevent communication between various physical components, thereby reducing system performance and even making the system unstable. Summary of the invention

[0003] In view of the shortcomings of the prior art described above, the present invention provides a method for controlling a multi-agent scheduled time sliding mode formation under DoS attacks. In order to enhance the security and robustness of the system and improve the collaboration efficiency of the multi-agent system, the present invention provides a method for controlling a multi-agent scheduled time sliding mode formation under DoS attacks, which is used for the application of multi-agent scheduled time formation under malicious network attacks.

[0004] To achieve the above effects, the technical solution of the present invention is as follows:

[0005] The present invention provides a method for controlling a multi-agent predetermined time sliding mode formation under a DoS attack, comprising the following steps:

[0006] Step 1: Establish a non-periodic DoS attack model:

[0007] For any time span, set The time interval that each agent interacts with its neighboring agents is called communication time. is called the k1th communication width, where and Respectively represent the start time and end time of the communication width k1th; δ k represents the kth communication width;

[0008] represents the DoS attack area, that is, the time when each pair of adjacent agents cannot communicate; It is called k2th attack width; Communication Width Attack Width sum; when and When , the non-periodic DoS attack becomes a periodic DoS attack;

[0009] Definition 1: For non-periodic DoS attacks, define the following formula:

[0010]

[0011] Where θ represents the attack width In time span The maximum proportion in , θ∈[0,1]; when θ=0, the non-periodic DoS attack becomes continuous communication, that is, there is no DoS attack;

[0012] Step 2: Consider the dynamic interactions, coordination relationships and external interference between agents, establish a multi-agent dynamic model to describe the position changes of agents; specifically:

[0013] Construct the dynamic equations of the nonlinear system of multiple agents under interference conditions:

[0014]

[0015] Among them, x i (t)∈R n and u i (t)∈R n are the state input and control input of agent i respectively; d i () is the external interference term of agent i;

[0016] Define the agent's error system:

[0017] e i () = x i ()-F i ()

[0018] In the formula, F i () represents the formation function of agent i;

[0019] Define the error relative status information of the error system:

[0020]

[0021] In the formula, A=[a ij ]∈R N×N is a weighted adjacency matrix; a ij represents the coefficient of the adjacency matrix; e i is the steady-state error of agent i, e j is the steady-state error of agent j;

[0022] Step 3: Design a predetermined time sliding surface and its controller to achieve the adaptability of the multi-agent nonlinear system to the external environment and DoS attacks;

[0023] Step 4: Set the Lyapunov sufficient conditions for the terminal sliding mode control at the predetermined time;

[0024] Definition 2: Set constant T c >0, for a predetermined time stable system There exists an unbounded Lyapunov function V2(x) that satisfies the following equation:

[0025]

[0026] but At scheduled time T c The interior is stable; T c >0, and α∈(0,1), we get T(x0)≤T and T=supT(x0); the concept of scheduled time synchronization is satisfied, so the expression of the unbounded Lyapunov function V2(x) is a scheduled time Lyapunov sufficient condition;

[0027] Step 5: Prove the stability of the system without DoS attack by using the unbounded Lyapunov function V2(x);

[0028] Step 6: Set the stability proof controller of the predetermined time stable system under DoS attack, and adjust the initial parameters of the error system so that the nonlinear system can complete the desired time-varying formation control under non-periodic DoS attack.

[0029] Furthermore, after step 1, the following steps are further included:

[0030] Assumption 1: For periodic DoS attacks, 0<δ<ω<+∞ is satisfied, so that k=0,1,2,…;

[0031]

[0032] Under assumption 1, the time span of each communication width is no less than δ, the sum of the communication width and the attack width is no greater than ω, and the span of the attack width is no greater than ω-δ;

[0033] Lemma 2: If Assumption 1 holds, then θ≤1-δ / ω, and for k=0,1,2,…, we obtain the following inequality:

[0034]

[0035] Further, step 3 includes: in order to stabilize the error system within a predetermined time, constructing a sliding surface equation for a predetermined time:

[0036]

[0037] in, 4>0, and α∈(0,1);

[0038] The controller input of the error system is designed as:

[0039] u i (t)= d (t)-(D i + i )signS i (t)+

[0040] -(1ξ i ()+C2sign(ξ i ())| i (t)| 1-α +3sign(ξ i (t))|ξ i (t)| 1+α +4sign(ξ i (t)))

[0041] -(5S i (t)+6sign(S i (t))|S i (t)| 1-α +7sign(S i (t))|S i (t)| 1+α +8sign(S i (t)))

[0042] Among them, F d Represents the preset formation function; C1-C4 are all parameters related to the preset time sliding surface, and C5-C8 are all parameters related to the controller; N represents the number of agents; and the error system will reach the predetermined time sliding surface within the predetermined time Tc2.

[0043] Furthermore, step 4 includes:

[0044] The classical terminal sliding mode control is improved to obtain the predetermined time terminal sliding mode control; the terminal sliding mode is described by the following first-order dynamic equation:

[0045]

[0046] Where x∈R 1 ; α>0; β>0 is a constant; q and p are both odd numbers, and q <p<2q;

[0047] The time it takes for the error system to converge from any initial state x(0)≠0 to the equilibrium state x=0 in the sliding mode is t s , t s Determined by the following formula:

[0048]

[0049] Assume that the origin of the error system is time-stable and the settling time t c For all e(0) T(e(0))≤t c , where e(0) represents the steady-state error at t=0, and the constant t c >0 is related to the parameters of the error system and is called fixed time;

[0050] Lemma 1: For T > 0, if there exists an unbounded Lyapunov function V1() that stabilizes the system for a predetermined time Then the following equation is satisfied:

[0051]

[0052] In the formula, is a predetermined time stable system; T is the predetermined time, and α∈(0,1).

[0053] Furthermore, there are the following constraints in step 5:

[0054] Assumption 2: Assume that the boundary of the internal uncertain nonlinear term is:

[0055] |(,t)|≤M i ,=1,2,……,

[0056] Assumption 3: Assume that the boundary of the external interference term is:

[0057] | i |≤D i ,=1,2,……,

[0058] Assumption 4: Assume that the formation function F i is a positive system with the following boundaries:

[0059]

[0060] Assumption 5: Setting parameter D i , i and M i The following conditions are met:

[0061] ρ i ≥D i + i ,=1,2,……,

[0062] Where Di represents the upper bound of the absolute value of the external interference term of agent i, ρ i represents the lower bound of the formation function of agent i.

[0063] Furthermore, after step 5, the following steps are included: constructing a Lyapunov energy function V(t):

[0064]

[0065] The derivative of the Lyapunov energy function V(t) along the time t direction is:

[0066]

[0067] Substituting the sliding surface equation of the predetermined time, the controller of the error system, and the dynamic equation of the nonlinear system, we get:

[0068]

[0069] Definition 2 is satisfied; therefore, the error system will reach the predetermined time sliding surface within the predetermined time Tc2.

[0070] Furthermore, step 6 includes:

[0071] Considering the nonlinear system dynamic equation, error system, predetermined time sliding surface and controller expression, we get

[0072] when hour,

[0073]

[0074] Integrate both sides of the above equation:

[0075]

[0076] get:

[0077]

[0078] when hour,

[0079]

[0080] Integrate both sides of the above equation:

[0081]

[0082] in, hour,

[0083]

[0084] because (e(t)) is monotonically increasing, and we get

[0085]

[0086] when hour,

[0087]

[0088]

[0089] Finally, we get:

[0090]

[0091] when hour,

[0092]

[0093] in, hour,

[0094]

[0095] when When , we use the inductive method to obtain

[0096] when hour,

[0097] Finally get By adjusting the initial parameters of the error system, the nonlinear system can achieve the desired time-varying formation control under non-periodic DoS attacks; the initial parameters are X(0) and F(0), where X(0) and F(0) represent the initial position and initial formation function of the agent, respectively.

[0098] The present invention designs a predetermined time sliding mode formation control method for multiple agents under DoS attacks, which solves the impact of non-periodic DoS attacks and external interference on system stability; in particular, the system settling time range is directly related to the adjustable parameters, which facilitates the control of the design of the present invention to meet the expected convergence time requirements; the method not only has potential applications in robot collaboration, but also has important significance in multiple fields such as drone formation control, bringing technological innovation to the formation control field of multi-agent systems.

[0099] Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0100] Based on the classic scheduled time Lyapunov stability theory and the characteristics of terminal sliding mode, the present invention constructs a new scheduled time sliding mode surface. The method of the present invention improves the stability of nonlinear multi-agent systems against non-periodic DoS attacks and continuous external interference. It enables multiple agents to realize multi-agent time-varying formation within an adjustable predefined time under malicious network attacks. There are the following three beneficial effects:

[0101] 1. Enhanced system stability: By introducing the method of the present invention, the multi-agent system can remain stable in the face of uncertainty and interference, reducing the oscillation and unstable behavior of the multi-agent system, thereby improving the overall stability of the multi-agent system;

[0102] 2. Strong controllability: The present invention combines the predetermined time stability condition, so that the multi-agent system can converge at the predetermined time; the multi-agent system can flexibly adjust the convergence time under different scenarios and task requirements, ensuring that the multi-agent system can adapt to diverse environments;

[0103] 3. Practical application value: The method of the present invention is suitable for the application of multi-agent systems in the fields of robot collaboration, social networks, traffic flow control, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0104] Figure 1 The flowchart of the scheduled time sliding mode control scheme under non-periodic DoS attack;

[0105] Figure 2 It is a schematic diagram of non-periodic intermittent communication;

[0106] Figure 3 (a) and (b) are interference d i Multi-agent trajectories x with and without aperiodic DoS attacks i ()contrast;

[0107] Figure 4 (a) and (b) are interference d i The error of multi-agent with and without aperiodic DoS attack i ()contrast;

[0108] Figure 5 (a) and (b) are interference d i Multi-agent controller u with and without aperiodic DoS attack i () output value comparison;

[0109] Figure 6 (a) and (b) are interference d i() Comparison of the 2D formation coordinates of multi-agents at different times with and without aperiodic DoS attacks;

[0110] Figure 7 (a) and (b) are interference d i () Comparison of trajectories of different agents with and without aperiodic DoS attacks;

[0111] Figure 8 (a) and (b) are interference d i () Trajectories of different agents at 0-1s for multi-agents with () and without aperiodic DoS attacks;

[0112] Fig. 9 is the attack time T of the non-periodic DoS attack off and the communication time T on . DETAILED DESCRIPTION

[0113] The following will describe the embodiments of the present invention with reference to the accompanying drawings and preferred embodiments. Those skilled in the art will easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be understood that the preferred embodiments are only for illustrating the present invention, not for limiting the scope of protection of the present invention.

[0114] It should be noted that the illustrations provided in the following embodiments are only schematic illustrations of the basic concept of the present invention, and thus the drawings only show components related to the present invention rather than being drawn according to the number, shape and size of components in actual implementation. In actual implementation, the type, quantity and proportion of each component may be changed arbitrarily, and the component layout may also be more complicated.

[0115] Example

[0116] In recent years, the collaborative control of multi-agent systems has also received widespread attention and achieved many important results, such as consistency control, formation control, and inclusion control. As a classic problem of collaborative control, formation is a typical collective behavior, which requires all agents to form a preset formation through communication with adjacent agents. Multi-agent formation control under malicious network attacks is crucial to avoid threats to unmanned vehicles, drones and other agents during their missions.

[0117] This embodiment proposes a method for controlling a multi-agent sliding mode formation under DoS attack. Figure 1 , including the following steps:

[0118] Step 1: Establish a non-periodic DoS attack model

[0119] In the related control methods of DoS attacks, the prior art usually considers the formation control problem under periodic DoS attacks; but there are great restrictions on the periodicity requirements of DoS attacks; therefore, the present invention studies non-periodic DoS attacks, which are a more common DoS attack mode.

[0120] Non-periodic DoS attack model:

[0121] Periodic DoS attacks are only a small part of non-periodic DoS attacks, and they can also be included in non-periodic DoS attacks; Figure 2 , for any time span, set It represents the time interval for each agent to interact with its neighboring agents, i.e., the communication time. It is called the k1th (i.e., kth point) communication width, where and Respectively represent the start time and end time of the communication width k1th; δ k represents the kth communication width;

[0122] represents the DoS attack area, that is, the time when each pair of adjacent agents cannot communicate; It is called k2th attack width; Communication Width Attack Width sum; when and When , the non-periodic DoS attack becomes a periodic DoS attack;

[0123] Definition 1: For non-periodic DoS attacks, define the following formula:

[0124]

[0125] Where θ represents the attack width In time span The maximum proportion of θ∈[0,1]; when θ=0, the non-periodic DoS attack becomes continuous communication, that is, there is no DoS attack; without loss of generality, it is always assumed that θ∈(0,1);

[0126] Step 2: Model the nonlinear system of multi-agents, consider the dynamic interaction, coordination relationship and external interference between agents, establish a multi-agent dynamic model, and describe the position changes of agents; specifically:

[0127] Construct the dynamic equations of the nonlinear system of multiple agents under interference conditions:

[0128]

[0129] Among them, x i (t)∈R n and u i (t)∈R n are the state input and control input of agent i respectively; d i () is the external interference term of agent i;

[0130] Define the agent's error system:

[0131] e i () = x i ()-F i ()

[0132] In the formula, F i () represents the formation function of agent i;

[0133] Define the error relative status information of the error system:

[0134]

[0135] In the formula, A=[a ij ]∈R N×N is a weighted adjacency matrix; a ij represents the coefficient of the adjacency matrix; e i is the steady-state error of agent i, e j is the steady-state error of agent j;

[0136] Step 3: Design a predetermined time sliding surface and its controller to achieve the adaptability of the multi-agent nonlinear system to the external environment and DoS attacks;

[0137] Step 4: Set the Lyapunov sufficient conditions for the terminal sliding mode control at the predetermined time;

[0138] Definition 2: Set constant T c >0, for a predetermined time stable system There exists an unbounded Lyapunov function V2(x) that satisfies the following equation:

[0139]

[0140] but At scheduled time T c The interior is stable; T c>0, and α∈(0,1); the parameters are predetermined, and the convergence time of the scheduled time stable system can be flexibly adjusted according to actual needs; we get T(x0)≤T and T=supT(x0); the concept of scheduled time synchronization is satisfied, so the expression of the unbounded Lyapunov function V2(x) (i.e., the Lyapunov scheduled time stable function) is a scheduled time Lyapunov sufficient condition;

[0141] Prove that the settling time meets

[0142]

[0143] Step 5: Prove the stability of the system without DoS attack by using the unbounded Lyapunov function V2(x);

[0144] Step 6: Set the stability proof controller of the predetermined time stable system under DoS attack, and adjust the initial parameters of the error system so that the nonlinear system can complete the desired time-varying formation control under non-periodic DoS attack.

[0145] As a preferred technical solution, in this embodiment, after step 1, the following steps are further included:

[0146] Assumption 1: For periodic DoS attacks, there exist positive scalars δ, ω, satisfying 0<δ<ω<+∞, such that k=0,1,2,…;

[0147]

[0148] Under assumption 1, the time span of each communication width is no less than δ, the sum of the communication width and the attack width is no greater than ω, and the span of the attack width is no greater than ω-δ;

[0149] Lemma 2: If Assumption 1 holds, then θ≤1-δ / ω, and for k=0,1,2,…, we obtain the following inequality:

[0150]

[0151] As a preferred technical solution, in this embodiment, step 3 includes: in order to stabilize the error system within a predetermined time, constructing a sliding surface equation for a predetermined time:

[0152]

[0153] in, 4>0, and α∈(0,1);

[0154] The controller input of the error system is designed as:

[0155] u i (t)= d (t)-(Di + i )signS i (t)+

[0156] -(1ξ i ()+C2sign(ξ i ())| i (t)| 1-α +3sign(ξ i (t))|ξ i (t)| 1+α +4sign(ξ i (t)))

[0157] -(5S i (t)+6sign(S i (t))|S i (t)| 1-α +7sign(S i (t))|S i (t)| 1+α +8sign(S i (t)))

[0158] Among them, F d Represents the preset formation function; C1-C4 are all parameters related to the preset time sliding surface, and C5-C8 are all parameters related to the controller; 8>0,i=1,2…N; N represents the number of agents;

[0159] And the error system will reach the predetermined time sliding surface within the predetermined time Tc2.

[0160] As a preferred technical solution, in this embodiment, step 4 includes:

[0161] The classic terminal sliding mode (TSM) control is improved to obtain the predetermined time terminal sliding mode control; the terminal sliding mode is described by the following first-order dynamic equation:

[0162]

[0163] Where x∈R 1 ; α>0; β>0 is a constant; q and p are both odd numbers, and q <p<2q;

[0164] The time it takes for the error system to converge from any initial state x(0)≠0 to the equilibrium state x=0 in the sliding mode is t s , t s Determined by the following formula:

[0165]

[0166] It can be seen that the settlement time t of the terminal sliding form s Adjustable, but related to the initial state x(0) of the system; Fixed-time terminal sliding mode control makes improvements on this basis: Assuming that the origin of the error system is time-stable, the settling time t c For all e(0) T(e(0))≤t c , where e(0) represents the steady-state error at t=0, and the constant t c >0 is related to the parameters of the error system, which is called fixed time; the scheduled time terminal sliding mode control used in the present invention is a special type of fixed time terminal sliding mode control, which can establish the relationship between system parameters and stable time, thereby avoiding overly conservative estimation of the settling time; Lemma 1 is a classic Lyapunov condition for scheduled time stability, and the present invention proposes a new Lyapunov sufficient condition for scheduled time stability (the condition and its proof are shown in Lemma 2);

[0167] Lemma 1: For T > 0, if there exists an unbounded Lyapunov function V1() that stabilizes the system for a predetermined time Then the following equation is satisfied:

[0168]

[0169] In the formula, is a predetermined time stable system; T is a predetermined time (constant), and α∈(0,1).

[0170] As a preferred technical solution, in this embodiment, step 5 has the following constraints:

[0171] Assumption 2: Assume that the boundary of the internal uncertain nonlinear term is:

[0172] |(,t)|≤M i ,=1,2,……,

[0173] Assumption 3: Assume that the boundary of the external interference term is:

[0174] | i |≤D i ,=1,2,……,

[0175] Assumption 4: Assume that the formation function F i is a positive system with the following boundaries:

[0176]

[0177] Assumption 5: Setting parameter D i , i and M i The following conditions are met:

[0178] ρ i ≥D i + i ,=1,2,……,

[0179] Where D i represents the upper bound of the absolute value of the external interference term of agent i, ρ i represents the lower bound of the formation function of agent i.

[0180] Prove that the multi-agent state vector form The vector form of control input and external disturbance is Vector form of formation error The vector form of the time-varying formation vector is

[0181]

[0182] As a preferred technical solution, in this embodiment, after step 5, the following steps are included: constructing a Lyapunov energy function V(t):

[0183]

[0184] The derivative of the Lyapunov energy function V(t) along the time t direction is:

[0185]

[0186] Substituting the sliding surface equation of the predetermined time, the controller of the error system, and the dynamic equation of the nonlinear system, we get:

[0187]

[0188] The specific process is:

[0189]

[0190] Definition 2 is satisfied; therefore, the error system will reach the predetermined time sliding surface within the predetermined time Tc2.

[0191] As a preferred technical solution, in this embodiment, step 6 includes:

[0192] Considering the nonlinear system dynamic equation, error system, predetermined time sliding surface and controller expression, we get

[0193]

[0194] when hour,

[0195]

[0196] Integrate both sides of the above equation:

[0197]

[0198] get:

[0199]

[0200] when hour,

[0201]

[0202] Integrate both sides of the above equation:

[0203]

[0204] in, hour,

[0205]

[0206] because (e(t)) is monotonically increasing, and we get

[0207]

[0208] when hour,

[0209]

[0210] Finally, we get:

[0211]

[0212] when hour,

[0213]

[0214] in, hour,

[0215]

[0216] when When , we use the inductive method to obtain

[0217] when hour,

[0218] Finally get By adjusting the initial parameters of the error system, the nonlinear system can complete the desired time-varying formation control under non-periodic DoS attack, and the control objective of the present invention can be achieved; the initial parameters are X(0) and F(0), and X(0) and F(0) represent the initial position of the intelligent agent and the initial formation function, respectively.

[0219] Remark 2: Combined with Definition 2, using the sliding mode surface and the controller, the system error converges to 0 within a predetermined time, so that the nonlinear system with finite interference is formed according to the time-varying vector and can remain stable under non-periodic DoS attacks; the above is the predetermined time sliding mode control scheme proposed by the present invention.

[0220] As a preferred technical solution, in this embodiment, a simulation example will be provided to verify the theoretical results; the multi-agent system is composed of 5 agents described by formula (2), α=0.5,x1=5=40, and c4=c8=5;

[0221] The Laplace matrix L is designed as:

[0222]

[0223] The external disturbance settings are:

[0224] s i () = [0.1sin(t)0.1sin()] T ,i=1,2,…,5

[0225] Define the desired time-varying formation form of the multi-agent system as

[0226]

[0227] The initial state of the agent is chosen as:

[0228] x1(0)=[0.5-1] T ,2(0)=[-1 0.6] T ,3(0)=[0.3-0.2] T ,4(0)=[1 1] T ,5(0)=

[22] T

[0229] The sampling step of this numerical simulation is 0.0001, and the sampling time T is 20s. In the non-periodic DoS attack, the communication rate between dynamic agents is set to be no less than 70%, and ω can be randomly selected. k and δ k ;

[0230] Compare the data graph without DoS attack with the data graph with DoS attack; Figure 3-5 The multi-agent trajectory x in two cases are i (t), error e i () and controller u i () output values ​​are compared and local amplification is performed. Figure 6 The two-dimensional rendering of the relative positions of the five agents at t=2.5, t=5s, t=7.5, and t=10. The X-axis is the horizontal coordinate of each agent, and the Y-axis is the vertical coordinate of each agent. It can be seen that their positions remain relatively static.

[0231] Figure 7 is the trajectory diagram of the agent formation, where the X-axis is the horizontal coordinate of each agent, the Y-axis is the vertical coordinate of each agent, and the Z-axis is time; each agent updates its state according to time. After a period of time, according to the preset time-varying formation function, the formation remains relatively static, and the system can quickly recover stability during DoS attacks, which shows that the adopted formation control method has significant effectiveness and superior stability in this multi-agent system; to make the observation clearer, Figure 8 By zooming in on the two cases, we can see that at around t = 0.07, the system forms a preset formation, which is less than the predetermined stabilization time. It can be seen that despite the i () interference, but under the action of the sliding mode formation control law (5) at the predetermined time, the five agents can still form and maintain the desired formation at the predetermined time under the conditions of different initial positions, and the formation problem under DoS attack is indeed solved. The simulation results ( Figure 3-8 ) has well verified the theoretical analysis; Fig. 9 Shows the attack time T randomly generated by the system off and the communication time T on .

[0232] As a preferred technical solution, in this embodiment, a predetermined time sliding mode formation control method for multi-agents under DoS attacks is experimentally verified on a Matlab simulation test platform; the algorithm can enable the nonlinear multi-agent system to achieve predetermined time stability, and the system settlement time range is directly related to the adjustable parameters; and the impact of non-periodic DoS attacks and external interference on system stability is solved; the present invention demonstrates the stability of the present invention, and simulates it on a Matlab experimental platform to verify its superiority.

[0233] The pseudo code of the present invention is as follows:

[0234]

[0235]

[0236] The experimental hardware and software configuration used in this embodiment can be seen in the following table:

[0237]

[0238] It should be noted that the control scheme flow chart is as follows Figure 1 As shown, it includes the following parts:

[0239] (1) System model, adding nonlinear terms and continuous disturbances;

[0240] (2) Preset time control system, where the preset time and other adjustable parameters can change the controller and the preset time sliding surface, thereby changing the time when the system reaches the preset time sliding surface and the final stable time;

[0241] (3) The desired formation system can adjust and change the formation at any time;

[0242] (4) Non-periodic DoS attack model, making the attack more universal;

[0243]

[0244]

[0245]

[0246] The implementation of the embodiments of the present invention has the following beneficial effects:

[0247] The present invention discloses a method for controlling a multi-agent scheduled time sliding mode formation under DoS attack; the method comprises an improved scheduled time terminal sliding mode algorithm, a scheduled time sliding mode surface and a system dynamic equation with external interference and nonlinear terms; the method enables a plurality of agents to realize a multi-agent time-varying formation within a predefined time under a non-periodic DoS attack and a continuous external interference environment.

[0248] Based on the classic scheduled time Lyapunov stability theory and the characteristics of the terminal sliding mode, the present invention constructs a new scheduled time sliding mode surface and designs an improved scheduled time terminal sliding mode formation control algorithm, which solves the impact of non-periodic DoS attacks and external interference on system stability; the present invention demonstrates the stability of the system and simulates it on the Matlab experimental platform to verify its effectiveness; the simulation results show that the estimated bound of the convergence time of the present invention to achieve a stable formation is less conservative than the existing terminal sliding mode control algorithm and has better robustness. It provides a more reliable and efficient solution for the anti-DoS attack mode for the collaboration and control of intelligent agents under malicious network attacks in practical applications, and brings practical value to the technical development and application promotion in related fields.

[0249] The above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. For those skilled in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the embodiments here. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the claims of the present invention.

Claims

1. A multi-agent scheduled time sliding mode formation control method under DoS attack, characterized in that: The following steps are involved: Step 1: Establish a non-periodic DoS attack model: For any time span, set k=0,1,2,…, The time interval that each agent interacts with its neighboring agents is called communication time. is called the k1th communication width, where and Respectively represent the start time and end time of the communication width k1 th; δ k represents the kth communication width; represents the DoS attack area, that is, the time when each pair of adjacent agents cannot communicate; It is called k2 th attack width; Communication Width Attack Width sum; when and When , the non-periodic DoS attack becomes a periodic DoS attack; Definition 1: For non-periodic DoS attacks, define the following formula: Where θ represents the attack width In time span The maximum proportion in , θ∈[0,1]; when θ=0, the non-periodic DoS attack becomes continuous communication, that is, there is no DoS attack; Assumption 1: For periodic DoS attacks, 0<δ<ω<+∞ is satisfied, so that k=0,1,2,…; Under assumption 1, the time span of each communication width is no less than δ, the sum of the communication width and the attack width is no greater than ω, and the span of the attack width is no greater than ω-δ; Lemma 2: If Assumption 1 holds, then θ≤1-δ / ω, and for k=0,1,2,…, we obtain the following inequality: Step 2: Consider the dynamic interactions, coordination relationships and external interference between agents, establish a multi-agent dynamic model to describe the position changes of agents; specifically: Construct the dynamic equations of the nonlinear system of multiple agents under interference conditions: Among them, x i (t)∈R n and u i (t)∈R n are the state input and control input of agent i respectively; d i (t) is the external interference term of agent i; Define the agent's error system: e i (t)=x i (t)-F i (t) In the formula, F i (t) represents the formation function of agent i; Define the error relative status information of the error system: In the formula, A=[a ij ]∈R N×N is a weighted adjacency matrix; a ij represents the coefficient of the adjacency matrix; e i is the steady-state error of agent i, e j is the steady-state error of agent j; Step 3: Design a predetermined time sliding surface and its controller to achieve the adaptability of the multi-agent nonlinear system to the external environment and DoS attacks; In order to stabilize the error system within a predetermined time, the sliding surface equation for the predetermined time is constructed: in, C4>0, and α∈(0,1); The controller input of the error system is designed as: u i (t)=F d (t)-(D i +M i )signS i (t)+-(C1ξ i (t)+C2sign(ξ i (t))|ξ i (t)| 1- α+C3sign(ξ i (t))|ξ i (t)| 1+α +C4sign(ξ i (t)))-(C5S i (t)+C6sign(S i (t))|S i (t)| 1-α +C7sign(S i (t))|S i (t)| 1+α +C8sign(S i (t))) Among them, F d Represents the preset formation function; C1-C4 are all parameters related to the preset time sliding surface, and C5-C8 are all parameters related to the controller; C8>0, i=1,2…N; N represents the number of agents; and the error system will reach the predetermined time sliding surface within the predetermined time Tc2; Step 4: Set the Lyapunov sufficient conditions for the terminal sliding mode control at the predetermined time; Definition 2: Set constant T c >0, for a predetermined time stable system There exists an unbounded Lyapunov function V2(x) that satisfies the following equation: but At scheduled time T c The interior is stable; T c >0, and α∈(0,1), we get T(x0)≤T and T=supT(x0); the concept of scheduled time synchronization is satisfied, so the expression of the unbounded Lyapunov function V2(x) is a scheduled time Lyapunov sufficient condition; The classical terminal sliding mode control is improved to obtain the predetermined time terminal sliding mode control; the terminal sliding mode is described by the following first-order dynamic equation: Where x∈R 1 ; α>0; β>0 is a constant; q and p are both odd numbers, and q <p<2q; The time it takes for the error system to converge from any initial state x(0)≠0 to the equilibrium state x=0 in the sliding mode is t s , t s Determined by the following formula: Assume that the origin of the error system is time-stable and the settling time t c For all e(0) T(e(0))≤t c , where e(0) represents the steady-state error at t=0, and the constant t c >0 is related to the parameters of the error system and is called fixed time; Lemma 1: For T > 0, if there exists an unbounded Lyapunov function V1(x) that stabilizes the system for a predetermined time Then the following equation is satisfied: In the formula, is a predetermined time stable system; T is the predetermined time, and α∈(0,1); Step 5: Prove the stability of the system without DoS attack by using the unbounded Lyapunov function V2(x); Step 6: Set a stability proof controller for the predetermined time stable system under DoS attack, and adjust the initial parameters of the error system so that the nonlinear system can complete the desired time-varying formation control under non-periodic DoS attack; Considering the nonlinear system dynamic equation, error system, predetermined time sliding surface and controller expression, we get when hour, Integrate both sides of the above equation: get: when hour, Integrate both sides of the above equation: in, hour, because V(e(t)) increases monotonically, and we get when hour, Finally, we get: when hour, in, hour, when When , we use the inductive method to obtain when hour, Finally get By adjusting the initial parameters of the error system, the nonlinear system can achieve the desired time-varying formation control under non-periodic DoS attacks; the initial parameters are X(0) and F(0), where X(0) and F(0) represent the initial position and initial formation function of the agent, respectively.

2. The method according to claim 1, characterized in that The following constraints exist in step 5: Assumption 2: Assume that the boundary of the internal uncertain nonlinear term is: |f(x,t)|≤M i ,i=1,2,……,N Assumption 3: Assume that the boundary of the external interference term is: |d i |≤D i ,i=1,2,……,N Assumption 4: Assume that the formation function F i is a positive system with the following boundaries: Assumption 5: Setting parameter D i , i and M i The following conditions are met: r i ≥D i +M i ,i=1,2,……,V Where D i represents the upper bound of the absolute value of the external interference term of agent i, v i represents the lower bound of the formation function of agent i.

3. The method according to claim 2, characterized in that Step 5 and after that include: constructing the Lyapunov energy function V(t): The derivative of the Lyapunov energy function V(t) along the time t direction is: Substituting the sliding surface equation of the predetermined time, the controller of the error system, and the dynamic equation of the nonlinear system, we get: Definition 2 is satisfied; therefore, the error system will reach the predetermined time sliding surface within the predetermined time Tc2.

Images