Train multi-agent tracking control method under communication restriction

Abstract

The application discloses a communication-restricted multi-agent tracking control method for trains, which comprises the following steps: S1, for a high-speed train dynamics model, a train formation dynamics model is designed; S2, a non-singular sliding mode surface is designed, which is brought into a dynamics equation, an equivalent control law and a total control law are designed; S3, a fixed threshold strategy is designed based on a control input signal error, and an event trigger condition is constructed; S4, through information interaction of adjacent vehicles, unmeasured states are estimated by using local measurable states, full-state feedback is provided for a controller, and a distributed observer is designed; the application has the beneficial effects that an improved terminal sliding mode surface is designed by using a non-singular terminal sliding mode surface, wherein the non-singular terminal sliding mode surface is a coprime odd number, fraction exponent irreducibility is ensured, periodic oscillation is avoided, and the singularity problem of traditional sliding mode control in time is eliminated by the fraction exponent term.

Description

Technical Field

[0001] This invention belongs to the field of multi-agent cooperative formation control of high-speed trains, and specifically relates to a multi-agent tracking control method for trains with communication restrictions. Background Technology

[0002] With the rapid development of high-speed railway networks and the continuous increase in train operation density, multi-train cooperative control technology has become a core challenge for improving transportation efficiency and safety. Traditional independent control strategies are insufficient to meet the demands of highly dynamic and strongly coupled train group coordination, especially in real-world scenarios where communication delays, resource constraints, and complex disturbances coexist, posing severe challenges to the stability and tracking accuracy of the control system. While existing research has made progress in sliding mode control and time delay compensation, it largely focuses on single-vehicle or ideal communication assumptions, with few works systematically addressing the distributed cooperative control problem under time-varying time delays of multiple trains. Furthermore, it has limitations in balancing communication efficiency and robustness.

[0003] This paper conducts an in-depth study on the train tracking control problem under communication delay and resource constraints, and proposes a non-singular terminal sliding mode control method that integrates event triggering mechanism and delay prediction compensation. However, in more severe DoS attack scenarios, the communication network may face complete paralysis or intermittent interruption, making it difficult for existing communication-constrained solutions to meet the requirements of system stability and security. Summary of the Invention

[0004] This invention addresses train multi-agent formation control under conditions of communication delay and DoS attack, and conducts research on controller strategies based on stability analysis. The main research contents are as follows:

[0005] To address the communication delay and resource constraints in multi-train cooperative control, a cooperative strategy integrating non-singular sliding mode control and dynamic event triggering is proposed. By designing an improved terminal sliding surface structure and introducing a fractional exponential term to eliminate the singularity problem of traditional sliding mode control, the time delay compensation parameters are optimized using linear matrix inequalities (LMI) to ensure system stability under time-varying communication delays. To resolve the issues of unmeasurable states and communication resource conflicts, a distributed observer is constructed to achieve cooperative state estimation of neighboring trains. An event triggering mechanism dynamically adjusts the control update frequency, effectively balancing tracking accuracy and communication load.

[0006] To address the multi-agent cooperative control problem of trains under DoS attacks, a resilient tracking control strategy based on the Barrier Lyapunov Function (BLF) is proposed. A non-singular sliding mode surface is designed based on the formation tracking error, and neighbor vehicle information is used to compensate for communication interruptions caused by the attack. A distributed cooperative observer is constructed to achieve dynamic estimation of the leader vehicle's state. Displacement safety constraints are embedded into the control objective using the BLF to ensure that the formation tracking error remains within a preset safety boundary. The designed dynamic event triggering mechanism significantly reduces communication resource consumption, and the nonlinear sliding mode control law suppresses chattering during the attack recovery phase.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] The non-singular terminal sliding membrane formation control method based on linear matrix inequalities and the elastic tracking control strategy based on obstacle Lyapunov functions are described in the following steps:

[0009] S1: This paper focuses on the dynamics model of high-speed trains and designs a dynamics model for train formations;

[0010] S2: Design a non-singular sliding surface, substitute it with the dynamic equations, and design the equivalent control law and the overall control law;

[0011] S3: Design a fixed threshold strategy based on the error of the control input signal, and construct event triggering conditions;

[0012] S4: By exchanging information with neighboring vehicles, the unmeasurable state is estimated using the local measurable state, providing full-state feedback to the controller, and designing a distributed observer;

[0013] S5: Verify the effectiveness of the non-singular terminal sliding membrane formation control method based on linear matrix inequalities in this invention, and complete the coordinated operation of train formations;

[0014] S6: Design a DOS attack strategy based on BLF stability theory;

[0015] S7: To address communication disruptions caused by DoS attacks, a distributed cooperative observer and a state-constrained resilient controller based on BLF are designed.

[0016] S8: Verify the effectiveness of the elastic tracking control strategy based on the obstacle Lyapunov function of this invention and complete the coordinated operation of train formation.

[0017] Furthermore, the multi-train cooperative control dynamics model established in step S1 is specifically as follows:

[0018]

[0019] in Let i be the current displacement of train i. Speed ​​and velocity together constitute the system's state variables, fully characterizing the train's motion state. Nonlinear resistance term, constant resistance term This represents mechanical frictional resistance independent of speed, such as wheel-rail contact friction and bearing friction. Linear speed-related terms. It mainly consists of the viscous drag component of air resistance and is proportional to speed. (Secondary velocity related terms) The dynamic pressure component corresponding to air resistance is proportional to the square of the velocity and is the dominant source of resistance when the train is running at high speed. The mass normalization factor 1 / m converts the total resistance into units of acceleration, reflecting the dynamic response of the train's mass to resistance. This refers to time-varying communication delays caused by transmission delays and data processing times in wireless communication networks. ,in This represents the maximum permissible time delay. For bounded disturbances, it represents unmodeled dynamics and environmental disturbances, such as track gradient changes, crosswind disturbances, and sensor noise. ,in The upper bound of the known disturbance is a key parameter in controller design.

[0020] Define the tracking error between the lead vehicle and the follower vehicle:

[0021]

[0022] in For the desired spacing, the displacement error Directly reflects the deviation between the actual distance and the expected value, speed error The degree of speed synchronization is reflected, and its convergence directly affects the adjustment rate of displacement error.

[0023] Differentiating equation (2) and substituting it into the dynamic model equation (1), we obtain the error dynamic equation:

[0024]

[0025] Further, in step S2, a non-singular sliding surface is designed. Substituting the dynamic equations, the equivalent control law and the overall control law are designed. The non-singular sliding surface is designed as follows:

[0026]

[0027] in For coprime odd numbers, ensure the fractional exponent. It cannot be reduced to a minimum, thus avoiding periodic oscillations. At the same time... , making This ensures the exponent of the derivative term. . As an adjustable parameter, increasing β can accelerate the convergence of the sliding surface, but may exacerbate control chattering.

[0028] In the ideal sliding mode ( Under these conditions, ignoring disturbances Substituting into the error dynamics equation (3):

[0029]

[0030] After unfolding, you get

[0031]

[0032] The equivalent control law is obtained:

[0033]

[0034] Equivalent control is based on an ideal model, but disturbances exist in real systems. In response to modeling errors, a toggle option is added to suppress disturbances and modeling errors:

[0035]

[0036] in To ensure that the strength of the switching control is sufficient to overcome the upper bound of the disturbance, the linear damping term... Proportional feedback is used to suppress chattering near the sliding surface and accelerate the convergence process.

[0037] Combining the equivalent control law (7) and the switching control law (8), the overall control law is:

[0038]

[0039] Furthermore, step S3 designs a fixed threshold strategy based on the control input signal error, and constructs the event triggering conditions as follows:

[0040] Define the event triggering conditions as follows:

[0041]

[0042] Furthermore, in step S4, through information exchange with neighboring vehicles, the unmeasurable state is estimated using the locally measurable state, providing full-state feedback to the controller, and a distributed observer is designed:

[0043]

[0044] Among them, local correction terms , The observer gain can be obtained through the measurable displacement error. Driven observation convergence. Neighbor coupling strength. Multi-vehicle state collaborative estimation is achieved using communication topology information. Nonlinear compensation term. Resistance calculations based on estimated velocity enhance model matching accuracy.

[0045] Furthermore, step S5 verifies the effectiveness of the non-singular terminal sliding membrane formation control method based on linear matrix inequalities of the present invention, and completes the coordinated operation of train formations:

[0046] To analyze observation errors Based on convergence, Lyapunov candidate functions are selected:

[0047]

[0048] in , is a symmetric positive definite matrix, determined through LMI design.

[0049] Furthermore, step S6, based on the BLF stability theory, designs a DOS attack strategy:

[0050] For ease of analysis, it is assumed that an attack sequence exists. ,in , , These are two positive integers. Assume a DoS attack occurs at... It happens all the time, among which Furthermore, it is assumed that the DoS attack will last for [duration]. That is, the communication topology of the train multi-agent system in The train system suffered a DoS attack, paralyzing its communication topology. No attacks occurred within the time interval, and the system communication topology returned to normal. The intermittent nature of DoS attacks stems from the limited resources of attackers and the periodic balance of energy replenishment mechanisms. Constrained by parameters such as cloud computing resource scheduling capabilities and the need for attack concealment, DoS attack energy exhibits a decaying characteristic. When the attack traffic intensity drops to a preset threshold, the system will trigger a resource protection mechanism to enter a dormant state, reconstructing the attack resource pool through methods such as zombie node reorganization and virtual server leasing. This "attack-dormant-reconstruction" cycle results in a periodic and intermittent DoS attack pattern, i.e., the attack cycle: ,in Set the attack duration and data packet loss rate simultaneously: (Maximum attack strength is limited).

[0051] Define the attack indication function:

[0052]

[0053] Furthermore, step S7 addresses communication disruptions caused by DoS attacks by designing a distributed cooperative observer and a BLF-based state-constrained resilient controller:

[0054]

[0055] in The estimated velocity is corrected by displacement measurement error correction, which drives the convergence of the observed values. Enhanced collaborative estimation by utilizing information from adjacent trains; Acceleration estimation is indirectly corrected by displacement error; It is a set of adjacency matrices; The strength of the cooperative coupling; This is the attack indication function.

[0056] Furthermore, step S8 verifies the effectiveness of the elastic tracking control strategy based on the obstacle Lyapunov function of the present invention, and completes the coordinated operation of train formations:

[0057] Construct the global Lyapunov-Krasovskii functional:

[0058]

[0059] Compared with the prior art, the beneficial effects of the present invention are:

[0060] 1. This invention proposes an improved terminal sliding surface structure using a non-singular terminal sliding surface design. ,in For coprime odd numbers, ensure the fractional exponent. To avoid irreducibility and periodic oscillations, traditional sliding mode control is eliminated through a fractional exponential term. This addresses the singularity problem at time. It avoids the divergence of control inputs when the error approaches zero, significantly reducing actuator chattering and improving the smoothness and engineering applicability of control inputs. A dynamic event-triggered mechanism is used to design triggering conditions. The system dynamically adjusts the control update frequency and optimizes the trigger threshold parameters using LMI. This reduces communication load under time-varying delays while ensuring stability through LMI, effectively balancing tracking accuracy and resource consumption. A distributed observer is designed, constructing an observer based on neighbor vehicle information coupling (Equation 3.23). It uses locally measurable states to estimate unmeasurable speeds and designs the gain matrix L through LMI to achieve high-precision state estimation under communication delays, providing reliable full-state feedback for the controller. For time delay compensation and LMI stability analysis, Lyapunov-Krasovskii functionals and LMI techniques are introduced to optimize the time delay compensation parameter τmax, maintaining stability even under time-varying delays (τmax). This resolves singularities and communication conflicts under time delays.

[0061] 2. This invention utilizes Lyapunov functions for designing barrier control based on BLF displacement constraint control, thereby incorporating safety constraints. An embedded sliding surface ensures that displacement errors remain within preset boundaries, avoiding collision risks and improving safety. A DoS attack elastic compensation strategy is adopted during the attack period ( Switching observer mode relies solely on neighbor vehicle information to compensate for communication interruptions, and uses nonlinear terms... Suppressing observations has the advantage that the system can still recover quickly even with an error of 30% attack intensity. A composite sliding surface design is employed, introducing an integral term. and time delay compensation item This eliminates steady-state errors, accelerates convergence during the attack recovery phase, and enhances robustness. It maintains security constraints under DoS attacks, reduces attack recovery time by 50%, improves control input smoothness, and provides resilient protection for high-density railway networks. Attached Figure Description

[0062] Figure 1 Design framework for multi-agent resilient tracking control strategy of vehicles in response to BLF-based DoS attacks;

[0063] Figure 2 A communication time-delay train non-singular sliding mode formation control framework based on LMI;

[0064] Figure 3 A diagram showing the DoS attack cycle; Detailed Implementation

[0065] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0066] This invention addresses train multi-agent formation control under conditions of communication delay and DoS attack, and conducts research on controller strategies based on stability analysis. The main research contents are as follows:

[0067] To address the communication delay and resource constraints in multi-train cooperative control, a cooperative strategy integrating non-singular sliding mode control and dynamic event triggering is proposed. By designing an improved terminal sliding surface structure and introducing a fractional exponential term to eliminate the singularity problem of traditional sliding mode control, the time delay compensation parameters are optimized using linear matrix inequalities (LMI) to ensure system stability under time-varying communication delays. To resolve the issues of unmeasurable states and communication resource conflicts, a distributed observer is constructed to achieve cooperative state estimation of neighboring trains. An event triggering mechanism dynamically adjusts the control update frequency, effectively balancing tracking accuracy and communication load.

[0068] To address the multi-agent cooperative control problem of trains under DoS attacks, a resilient tracking control strategy based on the Barrier Lyapunov Function (BLF) is proposed. A non-singular sliding mode surface is designed based on the formation tracking error. Neighboring vehicle information is used to compensate for communication interruptions caused by the attack, and a distributed cooperative observer is constructed to achieve dynamic estimation of the leader vehicle's state. Displacement safety constraints are embedded into the control objective using the BLF to ensure that the formation tracking error remains within a preset safety boundary. The designed dynamic event triggering mechanism significantly reduces communication resource consumption, and chattering during the attack recovery phase is suppressed through a nonlinear sliding mode control law. The train control structure is as follows: Figure 1-2 As shown.

[0069] To achieve the above objectives, the technical solution adopted in this application is a non-singular terminal sliding membrane formation control method based on linear matrix inequalities and an elastic tracking control strategy based on obstacle Lyapunov functions. The specific steps are as follows:

[0070] S1: This paper focuses on the dynamics model of high-speed trains and designs a dynamics model for train formations;

[0071] S2: Design a non-singular sliding surface, substitute it with the dynamic equations, and design the equivalent control law and the overall control law;

[0072] S3: Design a fixed threshold strategy based on the error of the control input signal, and construct event triggering conditions;

[0073] S4: By exchanging information with neighboring vehicles, the unmeasurable state is estimated using the local measurable state, providing full-state feedback to the controller, and designing a distributed observer;

[0074] S5: Verify the effectiveness of the non-singular terminal sliding membrane formation control method based on linear matrix inequalities in this invention, and complete the coordinated operation of train formations;

[0075] S6: Design a DOS attack strategy based on BLF stability theory;

[0076] S7: To address communication disruptions caused by DoS attacks, a distributed cooperative observer and a state-constrained resilient controller based on BLF are designed.

[0077] S8: Verify the effectiveness of the elastic tracking control strategy based on the obstacle Lyapunov function of this invention, and complete the coordinated operation of train formation.

[0078] The overall train control steps are as follows: Figure 1 As shown in the figure, the succession relationship between S1, S2, S3, S4, S5, S6, S7, and S8 is marked.

[0079] To facilitate subsequent analysis and proof, the relevant lemmas are explained here:

[0080] Lemma 1 (Young's Inequality)

[0081] set up And satisfy the conjugate condition Then the inequality holds. If and only if Take the equality sign. When in matrix form, the inequality is: This is an adjustable parameter used to optimize constraints and reduce the conservatism of inequalities.

[0082] Lemma 2 (Jensen Inequality)

[0083] set up It is a convex function. And weight satisfy ,have like If f is a concave function, the inequality is reversed. When it is in integral form, if f is a convex function, for integrable functions... and probability measure ,have .

[0084] The overall train control procedures are as follows:

[0085] S1: This paper focuses on the dynamics model of high-speed trains and designs a dynamics model for train formations;

[0086] This paper focuses on the dynamics model of high-speed trains. The running resistance experienced by a high-speed train during operation only includes basic resistance, and its dynamic behavior can be described by a system of second-order nonlinear differential equations:

[0087]

[0088] in Let i be the current displacement of train i. Speed ​​and velocity together constitute the system's state variables, fully characterizing the train's motion state. Nonlinear resistance term, constant resistance term This represents mechanical frictional resistance independent of speed, such as wheel-rail contact friction and bearing friction. Linear speed-related terms. It mainly consists of the viscous drag component of air resistance and is proportional to speed. (Secondary velocity related terms) The dynamic pressure component corresponding to air resistance is proportional to the square of the velocity and is the dominant source of resistance when the train is running at high speed. The mass normalization factor 1 / m converts the total resistance into units of acceleration, reflecting the dynamic response of the train's mass to resistance. This refers to time-varying communication delays caused by transmission delays and data processing times in wireless communication networks. ,in This represents the maximum permissible time delay. For bounded disturbances, it represents unmodeled dynamics and environmental disturbances, such as track gradient changes, crosswind disturbances, and sensor noise. ,in The upper bound of the known disturbance is a key parameter in controller design.

[0089] Define the tracking error between the lead vehicle and the follower vehicle:

[0090]

[0091] in For the desired spacing, the displacement error Directly reflects the deviation between the actual distance and the expected value, speed error The degree of speed synchronization is reflected, and its convergence directly affects the adjustment rate of displacement error.

[0092] Differentiating equation (2) and substituting it into the dynamic model equation (1), we obtain the error dynamic equation:

[0093]

[0094] S2: Non-singular sliding surface design is as follows:

[0095]

[0096] in For coprime odd numbers, ensure the fractional exponent. It cannot be reduced to a minimum, thus avoiding periodic oscillations. At the same time... , making This ensures the exponent of the derivative term. . As an adjustable parameter, increasing β can accelerate the convergence of the sliding surface, but may exacerbate control chattering.

[0097] In the ideal sliding mode ( Under these conditions, ignoring disturbances Substituting into the error dynamics equation (3):

[0098]

[0099] After unfolding, you get

[0100]

[0101] The equivalent control law is obtained:

[0102]

[0103] Among them, the nonlinear compensation term To counteract the effect of the drag term and ensure system linearity. (Lead vehicle acceleration tracking term) Achieve dynamic synchronization with the leader's vehicle. Error feedback adjustment item. Through the coupled feedback of displacement and velocity errors, the drive system converges along the sliding surface.

[0104] Equivalent control is based on an ideal model, but disturbances exist in real systems. In response to modeling errors, a toggle option is added to suppress disturbances and modeling errors:

[0105]

[0106] in To ensure that the strength of the switching control is sufficient to overcome the upper bound of the disturbance, the linear damping term... Proportional feedback is used to suppress chattering near the sliding surface and accelerate the convergence process.

[0107] Combining the equivalent control law (7) and the switching control law (8), the overall control law is:

[0108]

[0109] Considering the physical limitations of the actuator, input saturation

[0110]

[0111] S3: Define the following event triggering conditions:

[0112]

[0113] Update the control variable when the conditions are met. Otherwise keep Among the parameters This represents the absolute error threshold, controls the triggering frequency, and determines the sensitivity of the triggering conditions. Increasing this threshold... This can reduce the number of triggers, but may increase tracking error; reducing This improves accuracy but increases the communication overhead. State-dependent threshold. With the leader's vehicle status norm Related to prevent the system from running at high speed ( (If the value is large), it triggers too frequently. Also, to avoid the Zeno phenomenon, implicit guarantees are needed.

[0114] During the event interval Internally, control the amount of water. The actual control error is:

[0115]

[0116] Under event-triggered conditions, the actual control input is updated discretely:

[0117]

[0118] Define control error:

[0119]

[0120] Introducing equation (14) into equation (3), the system dynamic equation is rewritten as:

[0121]

[0122] in This is the time-delay effect matrix.

[0123] S4: Design a distributed observer:

[0124]

[0125] Among them, local correction terms , The observer gain can be obtained through the measurable displacement error. Driven observation convergence. Neighbor coupling strength. Multi-vehicle state collaborative estimation is achieved using communication topology information. Nonlinear compensation term. Resistance calculations based on estimated velocity enhance model matching accuracy.

[0126] Consider the state observation error of the i-th following vehicle ,but

[0127]

[0128] in For the output matrix, Observer gain matrix, The deviation is nonlinear and satisfies the Lipschitz continuity condition. Designed using LMI. ,make It is a Hurwitz matrix.

[0129] Define the extended error vector:

[0130]

[0131] in The overall error dynamic equation is:

[0132]

[0133] in This is for controlling errors triggered by events.

[0134] S5: Verify the effectiveness of the non-singular terminal sliding membrane formation control method based on linear matrix inequalities in this invention, and complete the coordinated operation of train formations.

[0135] To analyze observation errors Based on convergence, Lyapunov candidate functions are selected:

[0136]

[0137] in , is a symmetric positive definite matrix, determined through LMI design.

[0138] right Differentiate and substitute into the dynamic equation of observation error:

[0139]

[0140] Assuming the nonlinear function satisfies the Lipschitz condition, there exists a constant. , so that:

[0141]

[0142] Applying Young's inequality to the cross terms Perform upper bound estimation:

[0143]

[0144] in To adjust the parameters. Due to the boundedness of the perturbation. We can obtain:

[0145]

[0146] in These are design parameters.

[0147] Based on the above analysis, the Lyapunov derivative satisfies:

[0148]

[0149] To ensure It is necessary to ensure that:

[0150]

[0151] make The inequality can be rewritten as:

[0152]

[0153] Solving symmetric matrices using the convex optimization tool (MATLAB LMI Toolbox) sum matrix This makes the above inequality hold. The observer gain matrix is... Furthermore, the observation error converges exponentially:

[0154]

[0155] in This is the convergence rate-related constant.

[0156] Considering time delay And event-triggered systems:

[0157]

[0158] in It is the control quantity at the moment the event is triggered. It is a bounded time-varying time-delay. Bounded perturbation, construct the LK functional:

[0159]

[0160] in:

[0161]

[0162]

[0163]

[0164] in It is a symmetric positive definite matrix. As event trigger weight, Used to quantify the control error energy introduced by event triggering.

[0165] Differentiate the LK functional V:

[0166]

[0167] Derivative of the principal term:

[0168]

[0169] Derivative of the integral term:

[0170]

[0171] Derivative of double integral term, using Lemma 1:

[0172]

[0173] Estimating the upper bound of the integral term:

[0174]

[0175] but

[0176]

[0177] Derivative of the event-triggered error term:

[0178]

[0179] Substitute the derivatives of each component and simplify:

[0180]

[0181] Applying Lemma 1 and Lemma 2 to handle cross terms :

[0182]

[0183] Assumption (Given constants), define the augmented state vector:

[0184]

[0185] Then the derivative inequality (41) can be written as:

[0186]

[0187]

[0188] The sub-blocks are defined as follows:

[0189]

[0190]

[0191]

[0192] To meet It must meet the following requirements: and By adjusting This can ensure system stability, if LMI is feasible. Then the closed-loop equation satisfies: The exponential convergence.

[0193] According to the solution A matrix can be used to design the gain of a sliding mode controller.

[0194]

[0195] Known event triggering conditions:

[0196]

[0197] Assume that the event triggering conditions implicitly control the limits of error:

[0198]

[0199] Substituting equation (45) into the corresponding LMI constraint:

[0200]

[0201] Solving event trigger parameters using LMI At the same time, determine the trigger threshold for the event:

[0202]

[0203] S6: Based on BLF stability theory, design a DOS attack strategy:

[0204] For ease of analysis, it is assumed that an attack sequence exists. ,in , , These are two positive integers. Assume a DoS attack occurs at... It happens all the time, among which Furthermore, it is assumed that the DoS attack will last for [duration]. That is, the communication topology of the train multi-agent system in The train system suffered a DoS attack, paralyzing its communication topology. No attacks occurred within the time interval, and the system communication topology returned to normal. The intermittent nature of DoS attacks stems from the limited resources of attackers and the periodic balance of energy replenishment mechanisms. Constrained by parameters such as cloud computing resource scheduling capabilities and the need for attack concealment, DoS attack energy exhibits a decaying characteristic. When the attack traffic intensity drops to a preset threshold, the system will trigger a resource protection mechanism to enter a dormant state, reconstructing the attack resource pool through methods such as zombie node reorganization and virtual server leasing. This "attack-dormant-reconstruction" cycle results in a periodic and intermittent DoS attack pattern, i.e., the attack cycle: ,in Set the attack duration and data packet loss rate simultaneously: (Maximum attack strength is limited), DoS attack intervals are as follows: Figure 3 As shown.

[0205] Define the attack indication function:

[0206]

[0207] S7: To address communication disruptions caused by DoS attacks, design a distributed cooperative observer and a state-constrained resilient controller based on BLF:

[0208]

[0209] in The estimated velocity is corrected by displacement measurement error correction, which drives the convergence of the observed values. Enhanced collaborative estimation by utilizing information from adjacent trains; Acceleration estimation is indirectly corrected by displacement error; It is a set of adjacency matrices; The strength of the cooperative coupling; This is the attack indication function.

[0210] When no DoS attack occurs, the state observation error is defined for the following vehicle i: Then its dynamic equation is:

[0211]

[0212] in This is the local dynamics matrix; For observer gain; satisfy .

[0213] Global error of N following vehicles:

[0214]

[0215] Its dynamic equation is:

[0216]

[0217] in Let be the time delay error vector. It is a non-linear deviation. For disturbance, satisfy ,and For independent error dynamics of each vehicle, For adjacent vehicle information coupling items, This refers to nonlinear deviations and disturbances.

[0218] Under normal circumstances, the system utilizes local measurements Corrections are performed, but this is only possible when an attack is present. The observer degenerates into the following form:

[0219]

[0220] Remove all local measurements Related items and Only through the coupling of information from adjacent trains To maintain estimation capability, its error state equation becomes:

[0221]

[0222] The global error state equation becomes:

[0223]

[0224]

[0225] Differentiating equation (62) and applying Lemma 1, we get:

[0226]

[0227] By selecting a sufficiently large and satisfy connectivity ( ), we can get The error exponent converges to the residual set:

[0228]

[0229] Design of a state-constrained elastic controller based on BLF

[0230] Due to traditional sliding surfaces Unable to restrain A state constraint mechanism needs to be embedded in the controller, and a BLF design is required.

[0231]

[0232] Its derivative is:

[0233]

[0234] in ; ; For safety boundaries; For gain coefficient; when At that time, sliding surface item A virtual repulsive force is generated to prevent out-of-bounds movement.

[0235] Design the sliding surface using displacement error constraint formula (65):

[0236]

[0237] in The term is used to eliminate steady-state error; This term is used to compensate for phase lag caused by time delay. Since the observer configuration changes in attack mode, increasing observation error, an observation error compensation term is introduced into the sliding surface to enhance robustness:

[0238]

[0239] in To compensate for gain, used to suppress The effect on the sliding surface.

[0240] Differentiate equation (68):

[0241]

[0242] Define the composite energy function:

[0243]

[0244] when When the log term is greater than zero, 0, therefore >0, hour This creates a natural constraint space.

[0245] Differentiate equation (70) and substitute it into equation (1):

[0246]

[0247] Design anti-attack control laws:

[0248]

[0249] in This is an estimated resistance value.

[0250] Substituting equation (72) into equation (71), equation (71) simplifies to:

[0251]

[0252] Using inequalities: And Young's inequality: After sorting, we get:

[0253]

[0254] Select make sure Negative definite, and Suppress error disturbances and ensure The system index is stable.

[0255] S8: Verify the effectiveness of the elastic tracking control strategy based on the obstacle Lyapunov function of this invention, and complete the coordinated operation of train formations:

[0256] Considering the combined effects of DoS attacks, event triggering, and latency, we define an extended error vector:

[0257]

[0258] Construct the global Lyapunov-Krasovskii functional:

[0259]

[0260] The main Lyapunov term (tracking error and observation error) is:

[0261]

[0262] Integral term of time delay:

[0263]

[0264] Double integral term (suppressing the effect of time delay derivative):

[0265]

[0266] Event triggering error term (quantized triggering error energy):

[0267]

[0268] in To control error in event triggering, These are the weight parameters.

[0269] right Differentiate each term separately:

[0270]

[0271] Using Leibniz's rule, for the equation Differentiate:

[0272]

[0273] Differentiating equation (79), we apply Lemma 2 to handle the double integral terms:

[0274]

[0275] Further utilize boundary conditions:

[0276]

[0277] Differentiating equation (80) yields:

[0278]

[0279] Combine all derivative terms:

[0280]

[0281] Using Lemma 1 to pair the perturbation terms and intersection Perform upper bound estimation:

[0282]

[0283]

[0284] Define the augmented state vector:

[0285]

[0286] Equation (86) can be rewritten as:

[0287]

[0288] Where the matrix The structure is as follows:

[0289]

[0290] The sub-block is defined as follows:

[0291]

[0292]

[0293]

[0294] To ensure It needs to meet the following requirements. and < Solving symmetric matrices using the MATLAB LMI toolbox and parameters

[0295] The event triggering condition is

[0296]

[0297] Among them, the basic threshold This can prevent triggering by noise or minor errors. For adaptive terms, close to the safety boundary At this time, the threshold should be increased to reduce the triggering frequency and prioritize security.

[0298] In LMI conditions, the event-triggered error term weight sum matrix Must meet By combining the triggering conditions, the relationship of the threshold parameters can be derived:

[0299]

[0300] in This is the displacement error weighting coefficient.

Claims

1. A multi-agent tracking control method for vehicles with communication limitations, characterized in that: Includes the following steps: S1: Design a dynamic model for train formation based on the dynamic model of high-speed trains; S2: Design a non-singular sliding surface, substitute it with the dynamic equations, and design the equivalent control law and the overall control law; S3: Design a fixed threshold strategy based on the error of the control input signal, and construct event triggering conditions; S4: By exchanging information with neighboring vehicles, the unmeasurable state is estimated using the local measurable state, providing full-state feedback to the controller, and designing a distributed observer; S5: Verify the effectiveness of the non-singular terminal sliding membrane formation control method based on linear matrix inequalities in this invention, and complete the coordinated operation of train formations; S6: Design a DOS attack strategy based on BLF stability theory; S7: To address communication disruptions caused by DoS attacks, a distributed cooperative observer and a state-constrained resilient controller based on BLF are designed. S8: Verify the effectiveness of the elastic tracking control strategy based on the obstacle Lyapunov function of this invention, and complete the coordinated operation of train formation; Step S2: Design the non-singular sliding surface. Substitute the dynamic equations into the design, and design the equivalent control law and the overall control law. The non-singular sliding surface is designed as follows: ； in For coprime odd numbers, ensure the fractional exponent. Unreducible, to avoid periodic oscillations; at the same time , making This ensures the exponent of the derivative term. ; As an adjustable parameter, increasing β can accelerate the convergence of the sliding surface, but may exacerbate control chattering. In the ideal slip mode ( ), neglecting perturbation , substitute into error dynamics equation (3) : ； After unfolding, you get ； The equivalent control law is obtained: ； Equivalent control is based on an ideal model, but disturbances exist in real systems. In response to modeling errors, a toggle item is added to suppress disturbances and modeling errors: ； in To ensure that the strength of the switching control is sufficient to overcome the upper bound of the disturbance, the linear damping term... Proportional feedback is used to suppress chattering near the sliding surface and accelerate the convergence process. Combining the equivalent control law (7) and the switching control law (8), the overall control law is: ； Step S3 designs a fixed threshold strategy based on the control input signal error, and constructs the event triggering conditions as follows: Define the event triggering conditions as follows: 。 2. The communication-restricted multi-agent tracking control method for a train according to claim 1, characterized in that: The multi-train cooperative control dynamics model established in step S1 is as follows: ； in Let i be the current displacement of train i. Speed ​​constitutes the state variables of the system, fully characterizing the train's motion state; Nonlinear resistance term, constant resistance term Represents mechanical frictional resistance independent of speed; linear velocity-related terms. It mainly consists of the viscous drag component of air resistance and is proportional to speed; secondary velocity related terms. The dynamic pressure component, which corresponds to air resistance, is proportional to the square of the velocity and is the dominant source of resistance when a train is running at high speed. The mass normalization factor 1 / m converts the total resistance into units of acceleration, reflecting the dynamic response of the train's mass to resistance; This refers to the time-varying communication delay caused by the transmission delay and data processing time of the wireless communication network. ,in This is the maximum permissible time delay; For bounded perturbations, characterizing unmodeled dynamics and environmental disturbances; ,in The upper bound of the known disturbance is a key parameter for controller design; Define the tracking error between the lead vehicle and the follower vehicle: ； in For the desired spacing, the displacement error Directly reflects the deviation between the actual distance and the expected value, speed error It reflects the degree of speed synchronization, and its convergence directly affects the adjustment rate of displacement error; Differentiating equation (2) and substituting it into the dynamic model equation (1), we obtain the error dynamic equation: 。 3. The communication-restricted multi-agent tracking control method for a train according to claim 1, characterized in that: Step S4 involves exchanging information with neighboring vehicles, using locally measurable states to estimate unmeasurable states, providing full-state feedback to the controller, and designing a distributed observer: ； Among them, local correction terms , The observer gain can be obtained through the measurable displacement error. Driving observation convergence; neighbor coupling strength Multi-vehicle state collaborative estimation is achieved using communication topology information; nonlinear compensation term. Resistance calculations based on estimated velocity enhance model matching accuracy.

4. The communication-restricted multi-agent tracking control method for vehicles according to claim 1, characterized in that: Step S5 verifies the effectiveness of the non-singular terminal sliding membrane formation control method based on linear matrix inequalities of the present invention, and completes the coordinated operation of train formations: To analyze the convergence of the observation error , a Lyapunov candidate function is chosen: ； where is a symmetric positive definite matrix determined by LMI design.

5. The communication-restricted train multi-agent tracking control method of claim 1, wherein: Step S6, based on the BLF stability theory, designs a DOS attack strategy: For ease of analysis, it is assumed that an attack sequence exists. ,in , , Each is a positive constant; assuming the DoS attack occurs at... It happens all the time, among which Furthermore, it is assumed that the DoS attack lasts for a period of time. That is, the communication topology of the train multi-agent system in The train system suffered a DoS attack, paralyzing its communication topology. No attacks occurred within the time interval, and the system communication topology returned to normal. The intermittent nature of DoS attacks stems from the limited resources of attackers and the periodic balance of energy replenishment mechanisms. Constrained by cloud computing resource scheduling capabilities and attack concealment requirements, DoS attack energy exhibits a decaying characteristic. When the attack traffic intensity drops to a preset threshold, the system will trigger a resource protection mechanism to enter a dormant state, reconstructing the attack resource pool through zombie node reorganization and virtual server leasing. This "attack-dormant-reconstruction" cycle results in a periodic and intermittent DoS attack pattern, i.e., the attack cycle: ,in Set the attack duration and data packet loss rate simultaneously: Limit the maximum attack strength; Define the attack indication function: 。 6. The communication-restricted multi-agent tracking control method for vehicles according to claim 1, characterized in that: Step S7 addresses communication disruptions caused by DoS attacks by designing a distributed cooperative observer and a state-constrained resilient controller based on BLF: ； in The estimated velocity is corrected by displacement measurement error correction, which drives the convergence of the observed values. Enhanced collaborative estimation by utilizing information from adjacent trains; Acceleration estimation is indirectly corrected by displacement error; It is a set of adjacency matrices; The strength of the cooperative coupling; This is the attack indication function.

7. The communication-restricted multi-agent tracking control method for vehicles according to claim 1, characterized in that: Step S8 verifies the effectiveness of the elastic tracking control strategy based on the obstacle Lyapunov function of the present invention and completes the coordinated operation of train formation: Construct the global Lyapunov-Krasovskii functional: ； Among them, V5 is the main Lyapunov term, V6 is the time-delay integral term, V7 is the double integral term, and V8 is the event-triggered error term.

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