Denial of service attack nonlinear system communication scheduling method and system based on event triggering mechanism

Abstract

This invention discloses a communication scheduling method for a nonlinear system under a denial-of-service attack based on an event-triggered mechanism. Applied to a state-constrained nonlinear network physical system, it establishes a dynamic equation model for the nonlinear system under a denial-of-service attack; models the attack timing and provides constraints on attack frequency and duration; and generates a trigger time sequence based on an event-triggered mechanism to control the updating and maintenance of the actuator input signals. The event-triggered mechanism includes: real-time monitoring of whether the current time is during a communication interruption period; if the current time is during a communication interruption period, a time-triggered mode is used to determine the update time sequence of the control signals at a fixed time interval δ0; if the current time is during a normal communication period, an event-triggered mode is used to dynamically determine the trigger time based on the relationship between control error and the current system state. By monitoring the communication status in real time, it ensures that the update rhythm remains stable even under harsh attack conditions, balancing anti-interference capability and robustness.

Description

Technical Field

[0001] This invention belongs to the field of network security control technology for cyber-physical systems, and particularly relates to a communication scheduling method and system for a nonlinear system subjected to denial-of-service attacks based on an event-triggered mechanism. Background Technology

[0002] With the continuous advancement of Industry 4.0 and intelligent manufacturing, the widespread deployment of Cyber-Physical Systems (CPS) in critical infrastructures such as energy, power, and transportation has made the security and stability of their operation in complex network environments increasingly prominent. In particular, the frequent denial-of-service attacks targeting industrial control systems in recent years have exposed the vulnerability of control links to blockage, interruption, and even paralysis. For practical engineering systems with strong nonlinear characteristics and coupled behavior, the control objective not only includes achieving system state stability under disturbances and uncertainties, but also ensuring that the state strictly meets preset security constraints throughout the entire operation. Therefore, how to achieve network security stabilization control of state-constrained nonlinear systems under random DoS attacks and communication constraints has become a crucial problem that urgently needs to be solved.

[0003] In existing technologies, control strategies for nonlinear systems are mostly based on ideal communication settings, failing to adequately consider communication interruptions caused by DoS attacks. Furthermore, existing DoS attack-resistant fault-tolerant controls often employ a single time-triggered mechanism (i.e., fixed-period control). While this fixed-update strategy provides some robustness during attacks, during non-attack periods when communication is normal, fixed-period data transmission leads to unnecessary waste of communication resources and cannot adaptively adjust the communication frequency according to changes in the network environment. In addition, while traditional event-triggered control can effectively reduce communication load, existing research is mostly focused on linear systems or systems that do not consider state constraints. It lacks a hybrid scheduling mechanism that can automatically switch to fixed-interval updates when a DoS attack is detected and dynamically adjust the sampling frequency when communication recovers. This makes it difficult to ensure that the system strictly meets state constraints while simultaneously achieving efficient utilization of communication resources and robustness under attacks. Summary of the Invention

[0004] To address the aforementioned issues, the present invention aims to provide a communication scheduling method and system for nonlinear systems subjected to denial-of-service attacks based on an event-triggered mechanism. By monitoring the communication status in real time, the system switches to a time-triggered mode with a fixed interval δ0 when in the period of a DoS attack (communication interruption). This ensures that the control update rhythm remains stable even under severe attack conditions, thereby balancing the system's anti-interference capability and robustness in complex attack scenarios such as the superposition of high-frequency pulses and long-period blocking.

[0005] The technical solution provided by this invention is: a communication scheduling method for a nonlinear system subjected to denial-of-service attacks based on an event-triggered mechanism, applied to a state-constrained nonlinear network physical system, comprising the following steps: Establish a dynamic equation model for a nonlinear system subjected to a denial-of-service attack; The attack sequence is modeled and constraints on attack frequency and duration are given; The event-triggered mechanism generates a sequence of trigger times to control the updating and maintenance of the actuator input signals. This mechanism dynamically determines the trigger times based on the relationship between control error and the current system state, and the time interval between two consecutive triggers is not less than the minimum time interval τ. m .

[0006] Preferably, the dynamic equation model of the nonlinear system is as follows: Where x is the system state, w is an uncertain parameter, u is the control input, and the input gain b(w) > 0 is within a known positive definite upper bound b. M and lower bound b m A scalar function that varies within an interval, g(·) is a continuously differentiable nonlinear function; The preset set of security constraints is as follows: , where k c >0 is a known constant.

[0007] Preferably, the modeling of the attack timing and the constraints on the attack frequency and duration include: The range of the p-th attack is denoted as... ,in, The moment the attack was triggered. Duration of the attack; In time interval Within this framework, we define a set of rejected times and a set of allowed times, respectively: Set of rejected times Allowed time sets and by the set of rejected times With allowed time set Describe the communication interruption and the available time period respectively.

[0008] Preferably, the modeling of the attack timing and the constraints on the attack frequency and duration are based on the following settings: There exists a constant ξ > 0. Such that, over any time interval [τ,t), the number of attack triggers n(τ,t) satisfies ; There exist constants κ ≥ 0 and T ≥ 1 such that, over any time interval [τ, t), the measure of the rejected time set satisfies .

[0009] Preferably, the updating and maintenance of the actuator input signals based on the event-triggered mechanism to generate a trigger time sequence includes: Define sampling time sequence k∈N in adjacent sampling intervals Within this range, the control input error is: ; The event triggering mechanism specifies that the sampling time should be updated as follows: in, δ0 is a fixed sampling interval when a DoS attack exists, satisfying δ k This refers to the event trigger interval; When a DoS attack is present, the actual input to the executor is: .

[0010] Preferably, the event trigger interval δ k for: Where σ > 0 is a design parameter. To avoid the Zeno phenomenon.

[0011] Preferably, the method further includes: Define the existence of constants , making ; Define the nominal control law , where m1>0 is a constant, P(x)≥1 is a smooth positive function, and its lower bound is related to γ(x) and the DoS trigger threshold.

[0012] Based on the same concept, the present invention also provides a communication scheduling system for a nonlinear system subjected to a denial-of-service attack based on an event-triggered mechanism, comprising: The system model building unit is used to establish a dynamic equation model of a nonlinear system under a denial-of-service attack. The attack timing modeling unit is used to model the attack timing and provide constraints on the attack frequency and duration. The scheduling unit is used to generate a sequence of trigger times based on an event-triggered mechanism to control the updating and maintenance of the input signals of the actuator. The event-triggered mechanism includes: Real-time monitoring to determine if the current moment is within a period of communication interruption due to a denial-of-service attack; If the current time is during a communication interruption period, a time-triggered mode is adopted, and the update time sequence of the control signal is determined at a fixed time interval δ0; If the current time is within a normal communication period, an event-triggered mode is adopted. The trigger time is dynamically determined based on the relationship between control error and the current system state, and the time interval between two consecutive triggers is not less than the minimum time interval τ. m .

[0013] Based on the same concept, the present invention also provides an electronic device, comprising: The memory is used to store the processing program; A processor, which, when executing the processing program, implements the communication scheduling method for a nonlinear system subjected to a denial-of-service attack based on an event-triggered mechanism as described above.

[0014] Based on the same concept, the present invention also provides a readable storage medium storing a processing program, which, when executed by a processor, implements the communication scheduling method for a denial-of-service attack nonlinear system based on an event-triggered mechanism as described above.

[0015] Because the present invention adopts the above technical solution, it has the following advantages and positive effects compared with the prior art: 1. By monitoring the communication status in real time, when the system is in a DoS attack (communication interruption) period, it switches to a time-triggered mode with a fixed interval δ0 to ensure that the control update rhythm remains stable even under the harsh environment of being attacked. This ensures that the system's anti-interference capability and robustness are both taken into account in complex attack scenarios such as high-frequency pulses and long-cycle blocking superposition. When the communication is normal, the system uses an event-triggered mode to dynamically determine the trigger time based on the control error and system status, realizing the adaptive adjustment of the sampling and update strategy according to the network status.

[0016] 2. During normal communication periods outside of attacks, an event-triggered mechanism is employed, where the system only updates control signals when control errors or state changes exceed preset thresholds. Compared to traditional periodic triggering, this on-demand communication method effectively avoids unnecessary data transmission, significantly reduces communication load, and thus greatly improves the efficiency of communication resource utilization in the network physical system.

[0017] 3. By setting the time interval between two consecutive triggers to be no less than the minimum time interval τ m This effectively prevents the infinitely high-frequency triggering (i.e., the Zeno phenomenon) that may be caused by the event triggering mechanism. It ensures that the communication scheduling strategy is physically feasible, protects the actuators and communication channels from the influence of excessively high-frequency signals, and thus guarantees the reliability and safety of the system in practical engineering applications. Attached Figure Description

[0018] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings, wherein: Figure 1 This is a schematic diagram of the overall architecture of the nonlinear affine system with uncertain parameters and input gain according to the present invention; Figure 2 This is a curve showing the change of the system state under constraints in one embodiment; Figure 3 This is a curve showing the change of the system state norm trajectory under constraints in one embodiment. Figure 4 This is a curve showing the change in the sampling time interval of the system in one embodiment. Detailed Implementation

[0019] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. The advantages and features of the present invention will become clearer from the following description and claims. It should be noted that the drawings are all in a very simplified form and use non-precise ratios, and are only used to facilitate and clarify the illustration of the embodiments of the present invention.

[0020] It should be noted that all directional indications (such as up, down, left, right, front, back, etc.) in the embodiments of the present invention are only used to explain the relative positional relationship and movement of each component in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indication will also change accordingly.

[0021] First Embodiment This embodiment provides a nonlinear affine system with uncertain parameters and input gain, and its overall implementation architecture is as follows: Figure 1 As shown, to address potential random denial-of-service (DoS) attacks on the sensor-actuator communication link, this method first models the attack timing and provides constraints on attack frequency and duration. Based on this, an event-triggered communication scheduling strategy is introduced. By real-time monitoring of whether the current moment falls within a communication interruption period caused by a DoS attack, the control update rhythm can adaptively adjust according to the actual occurrence of the DoS attack. When the system detects that the current moment is a communication interruption period, it adopts a time-triggered mode to determine the update time sequence of the control signal at a fixed time interval δ0 in order to cope with the communication uncertainty in the attack environment. When the system detects that the current time is within a normal communication period, it adopts an event-triggered mode and dynamically determines the trigger time based on the relationship between control error and the current system state, thereby effectively reducing the communication load. At the same time, the time interval between two consecutive triggers is set to be no less than the minimum time interval τ. m This is to effectively avoid the Zeno phenomenon.

[0022] To strictly ensure that the system state does not violate physical constraints, this scheduling method combines obstacle function technology to explicitly embed state constraints into the control design, ensuring that the closed-loop trajectory always remains within a preset safety set, thereby effectively avoiding the risk of state exceeding limits. By constructing a suitable Lyapunov function and combining it with the average dwell time condition of a DoS attack, it is proven that the system can achieve semi-global asymptotic stability of the closed-loop system under finite-intensity DoS attacks, ensuring that the system state converges to the equilibrium point. Compared with existing technologies, this invention, in complex attack scenarios such as high-frequency pulses and long-period blocking, balances communication resource utilization efficiency and anti-interference capability, significantly reducing communication load while strictly maintaining state constraints, thereby effectively improving the security and operational reliability of nonlinear network physical systems in complex network threat environments.

[0023] This invention claims a communication scheduling method for a nonlinear system subjected to a denial-of-service attack based on an event-triggered mechanism, applicable to a state-constrained nonlinear network physical system, specifically including the following implementation steps: Step S1: Establish a dynamic equation model of the nonlinear system of a denial-of-service attack.

[0024] Dynamics model of a nonlinear affine system: in, For system status, To control the input, For uncertain parameters, set W is a compact set containing the origin, and the function... Continuously differentiable and satisfying Input gain It is a scalar function.

[0025] The system state needs to satisfy a preset set of security constraints: in, Let these be known constants. Further define the safe subset. ,in , Given a constant, This is a subset of the safe operating parameters. The control objective is to design a feedback control law with uncertain parameters and input gain, such that the solution to the closed-loop system is globally existent and bounded, and holds for all t ≥ 0. And achieve state convergence to the equilibrium point.

[0026] Step S2: Model the attack sequence and provide constraints on the attack frequency and duration.

[0027] The sensor-actuator communication link may be vulnerable to non-periodic, random denial-of-service attacks. The p-th attack interval is denoted as... ,in, τ is the moment the attack is triggered. p ≥0 represents the attack duration; when τ p =0 indicates a pulse attack.

[0028] In time interval Within this framework, the set of rejected times and the set of allowed times are defined as follows: The set of rejected times and the set of allowed times describe the communication interruption and available time periods, respectively.

[0029] To describe the frequency and duration of DoS attacks, the following quantitative constraints are set: Setting 1: There exists a constant. , such that in any interval Above, number of attack triggers satisfy: Setting 2: There exists a constant. T≥1, such that on any interval [τ,t), the measure of the rejected time set satisfies: A barrier function is constructed for the preset set of safety constraints, and a coordinate transformation is performed to convert the original state-constrained system into an unconstrained system.

[0030] Define the barrier function μ(x) as: And order This gives the coordinate transformation .

[0031] when Bounded and When, it can be proven that the original state satisfies This achieves equivalent preservation of state constraints.

[0032] In the absence of DoS attacks, the transformed unconstrained system is written as: in, This is the sampling error term.

[0033] Based on the transformed unconstrained system, a robust state feedback control law is designed to generate control update times and actual actuator inputs.

[0034] First, by It can be seen that a smooth function exists. , , making .make ,definition ,but .

[0035] Simultaneously, the input gain is set to satisfy setting 3: a constant exists. , making , .

[0036] Based on this, design the nominal control law (robust state feedback control law): in, Let P(x) ≥ 1 be a constant, and let P(x) ≥ 1 be a smooth positive function with a lower bound of 1 and 2. It is related to the DoS trigger threshold.

[0037] Step S3: Generate a trigger time sequence based on an event-triggered mechanism to control the updating and maintenance of the actuator input signals. The event-triggered mechanism dynamically determines the trigger time based on the relationship between the control error and the current system state, and the time interval between two consecutive triggers is not less than the minimum time interval τ. m .

[0038] To generate control over update timing, an event-triggered mechanism is introduced: Define sampling time sequence , In adjacent sampling intervals Within, the control input error is .

[0039] The event triggering mechanism specifies that the sampling time should be updated as follows: in, δ0 is a fixed sampling interval when a DoS attack exists, satisfying .

[0040] Event trigger interval δ k Determined by the following conditions: in, α>0 is a design parameter. Used to avoid the Zeno phenomenon.

[0041] When a DoS attack is present, the actual input to the executor satisfies the following condition: the current moment belongs to the allowed time set. hour When it belongs to the set of rejected times hour This demonstrates the impact of communication interruptions on the control channel.

[0042] This invention constructs a barrier function and performs coordinate transformation to equivalently convert the original state-constrained system into an unconstrained system. It then designs a robust state feedback control law that explicitly depends on the barrier function. This ensures that when the system state approaches a preset safety constraint boundary, the barrier function exhibits infinite gain characteristics, automatically enhancing the control effect and forcibly pulling the state back to a safe region. The system can strictly maintain the state trajectory within a preset safe set, effectively avoiding the risk of state exceeding limits and meeting the hard safety requirements of the physical system.

[0043] By monitoring communication status in real time, the system switches to a fixed-interval δ0 time-triggered mode during DoS attack (communication interruption) periods. This ensures that the control update rhythm remains stable even under severe attack conditions, thus balancing the system's anti-interference capability and robustness in complex attack scenarios such as high-frequency pulses and long-period blocking. During normal communication periods, the system uses an event-triggered mode to dynamically determine the trigger time based on control errors and system status, enabling adaptive adjustment of the sampling and update strategy according to network conditions. The triggering conditions and control inputs explicitly depend on system status and attack timing information, automatically adjusting the sampling and update rhythm based on changes in network service quality and the evolution of DoS attack intensity. This overcomes the problem of slow response to changes in attack patterns in purely offline parameter tuning during actual operation, significantly improving the dynamic adaptability of the control strategy and the stability and reliability of the closed-loop system under random DoS attack environments.

[0044] During normal communication periods outside of attacks, an event-triggered mechanism is employed, where the system only updates control signals when control errors or state changes exceed preset thresholds. Compared to traditional periodic time-triggered mechanisms, this on-demand communication method effectively avoids unnecessary data transmission, significantly reduces communication load, and thus greatly improves the efficiency of communication resource utilization in the network physical system.

[0045] By setting the time interval between two consecutive triggers to be no less than the minimum time interval τ m This effectively prevents the infinitely high-frequency triggering (i.e., the Zeno phenomenon) that may be caused by the event triggering mechanism. It ensures that the communication scheduling strategy is physically feasible, protects the actuators and communication channels from the influence of excessively high-frequency signals, and thus guarantees the reliability and safety of the system in practical engineering applications.

[0046] Based on the above specific implementation, this invention constructs a Lyapunov function and derives the closed-loop error system inequality by combining the constraints of DoS attack frequency and duration.

[0047] Consider the Lyapunov function .

[0048] Combining the system dynamics model, feedback control law, and triggering mechanism, the upper bound of its derivative is derived as follows: in, and , And related to DoS parameters.

[0049] Using trigger conditions Furthermore, the DoS duration constraint is obtained as follows: And it satisfies exponential decay outside the DoS attack range.

[0050] To prove that the closed-loop system is semi-global asymptotically stable under finite-strength DoS attack conditions.

[0051] M N >0 represents the parameters related to control gain and DoS mean dwell time. The constant related to T Let represent the start and duration of the j-th attack.

[0052] By selecting control parameters, And order It can guarantee It holds true in the average sense.

[0053] Under the conditions of settings 1–3 and the parameters described above, any initial state belongs to the set. And satisfy In this case, using the comparison principle and the DoS average dwell time theory, we can obtain: Depend on As can be seen from the properties of the barrier function, z(t) is bounded, thus guaranteeing that for all t≥0, That is, the state constraints are always satisfied, and Follow Asymptotic convergence, thereby achieving semi-global robust stabilization.

[0054] The method of the present invention is then verified through simulation using a specific nonlinear system. The simulation results are as follows: Figures 2-4 As shown.

[0055] Establish the dynamic equations and state constraints of the nonlinear system.

[0056] Consider the following nonlinear system: in, For system status, For controlling input.

[0057] The initial state of the system is selected as follows: .

[0058] The known upper and lower bounds of the control gain are set to b. m =0.2,b M =2b.

[0059] State constraints use spherical sets The constraint parameter takes the value k. c =1.5, k c0 =0.01, Among them, k c For the state constraint boundary, k c0 and Used to define a safe working subset and an initial feasible region, thereby ensuring that the system's operating trajectory always remains within the preset safe range.

[0060] The nonlinear gain function is selected based on the control law structure.

[0061] For the constrained network security control law of a two-dimensional system, smooth positive functions are designed for the two state components respectively. , This selection not only satisfies the lower bound constraint on the function P(x) in theoretical analysis, but also provides a greater feedback gain when the state deviates from the equilibrium point, thereby enhancing the ability to suppress nonlinear terms and uncertain parameters.

[0062] In some embodiments, disturbance, uncertainty, and DoS attack and trigger parameters are set.

[0063] The disturbance parameter in the system's nonlinear term is taken as w=0.8.

[0064] The event trigger threshold is set to σ=0.2.

[0065] The minimum sampling interval and the time trigger interval during a DoS attack are respectively set to τ. m =0.001s, δ0 =0.002s.

[0066] DoS attacks satisfy the settings for average dwell time and duration, with parameters set as ξ=1 and τ. D =0.5, T=4.

[0067] In the simulation, the DoS attack is set to trigger at the following time points: 0.25s, 1.80s, 3.60s, 5.30s, 6.40s, 7.75s, 8.65s, 9.70s.

[0068] After each attack is triggered, the executor channel experiences a communication interruption within the corresponding time interval, and the control input switches between "keeping the most recently updated value" and "setting to zero" according to the segmented form given in the method.

[0069] Numerical simulation and state response analysis based on control law: Under the above parameter configuration, a numerical simulation was performed on a closed-loop system including an event triggering mechanism and obstacle function coordinate transformation. The state trajectory is as follows: Figure 2 As shown in the figure, the gray shaded areas represent the duration of each DoS attack, and their frequency and duration both satisfy the constraints in settings 1 and 2. It can be observed that in the initial stage of the attack, due to the initial state deviating from the equilibrium point, the state components x1 and x2 fluctuate to a certain extent, but under the action of the designed control law, they are always confined within the constraint set, and no out-of-bounds phenomenon occurs. When the attack ends and normal communication is restored, the state quickly returns to the convergence trend and eventually tends towards the origin. The red solid line and blue dashed line in the figure correspond to the evolution process of x1 and x2, respectively. Their peak values ​​are effectively suppressed in a short period of time, and no significant error accumulation occurs within the communication interruption interval.

[0070] The change of the system state norm |x(t)| is as follows Figure 3 As shown, after the initial brief transition phase, |x(t)| monotonically decays and converges to near the equilibrium point, while always remaining below the constraint boundary k. c (The red dashed line in the figure) shows that the combined effect of the obstacle function and the control law can effectively maintain state constraints. Even under the combined effects of random DoS attacks and system disturbances, the state convergence speed is still relatively fast, verifying the robustness of the controller to uncertainty and network attacks.

[0071] Analyze the sampling behavior and communication load under the event triggering mechanism.

[0072] Sampling time interval t k+1 -t k Changes over time, such as Figure 4 As shown, during normal communication without attacks, the trigger interval gradually increases as the system error decreases, indicating that the event-triggered mechanism can adaptively reduce the update frequency according to state changes, thereby reducing network load. When a DoS attack or communication interruption is detected, the sampling interval is compressed to a small fixed value, and the corresponding time-triggered mode is activated, enabling the controller to maintain a sufficient update frequency under limited bandwidth conditions. As the attack ends, the system automatically switches back from time-triggered to event-triggered mode, and the sampling interval returns to the normal range. Overall results show that the proposed event-triggered strategy significantly reduces communication overhead while ensuring system stability, and maintains good flexibility and fault tolerance during attacks.

[0073] The effectiveness of the method is verified by the comprehensive simulation results below.

[0074] pass Figures 2-4 The numerical results show that, under the selected DoS attack parameters and state constraint parameters, the state transition and event triggering control law based on the obstacle function adopted in this embodiment can guarantee that: for any condition satisfying the given conditions... In the initial state, the system trajectory always remains within the constraint set S, without exceeding the bounds. Under random and multiple DoS attacks, the closed-loop system still achieves asymptotic convergence to the equilibrium point. The sampling interval dynamically and adaptively changes according to the attack state and error, effectively balancing communication resource usage and control performance. In summary, this embodiment fully verifies the robustness and state constraint preservation capability of the proposed control method in low-dimensional nonlinear systems.

[0075] Based on the same concept, the present invention provides an electronic device, comprising: The memory is used to store the processing program; A processor, which, when executing the processing program, implements the communication scheduling method for a nonlinear system subjected to a denial-of-service attack based on an event-triggered mechanism as described above.

[0076] Based on the same concept, the present invention provides a readable storage medium storing a processing program, which, when executed by a processor, implements the communication scheduling method for a denial-of-service attack nonlinear system based on an event-triggered mechanism as described above.

[0077] The event-triggered denial-of-service attack nonlinear system communication scheduling method, if implemented as a program instruction and sold or used as an independent product, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this embodiment, essentially, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in software. This computer software is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this disclosure. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0078] Second Embodiment Based on the same concept, the present invention also provides a communication scheduling system for a nonlinear system subjected to a denial-of-service attack based on an event-triggered mechanism, comprising: The system model building unit is used to establish a dynamic equation model of a nonlinear system under a denial-of-service attack. The attack timing modeling unit is used to model the attack timing and provide constraints on the attack frequency and duration. The scheduling unit is used to generate a sequence of trigger times based on an event-triggered mechanism to control the updating and maintenance of the input signals of the actuator. The event-triggered mechanism includes: Real-time monitoring to determine if the current moment is within a period of communication interruption due to a denial-of-service attack; If the current time is during a communication interruption period, a time-triggered mode is adopted, and the update time sequence of the control signal is determined at a fixed time interval δ0; If the current time is within a normal communication period, an event-triggered mode is adopted. The trigger time is dynamically determined based on the relationship between control error and the current system state, and the time interval between two consecutive triggers is not less than the minimum time interval τ. m .

[0079] The communication scheduling system is applied to a nonlinear affine system with uncertain parameters and input gain. The nonlinear affine system has the following related characteristics: 1. Form of system dynamic equations: The system is described as a nonlinear affine system with the following dynamic equation: ẋ=g(x,w)+b(w)u, where x(t) is the system state vector, u(t) is the control input vector, w is the uncertain parameter of the system, which belongs to a compact set W containing the origin, g(x,w) is a nonlinear function of the state and the uncertain parameter, which is continuously differentiable and satisfies g(0,w)=0, and b(w) is the scalar input gain function.

[0080] 2. Uncertainty and Boundedness: Parameter uncertainty: The system parameter w is unknown, but varies within a known compact set W.

[0081] Input gain boundedness: The input gain b(w) is indeterminate, but its range is known. There exists a constant 0. m ≤b M Such that for all w∈W, b m ≤b(w)≤b M This boundedness is a key prior information for designing robust control laws.

[0082] 3. Nonlinear characteristics: The system has an explicit nonlinear term g(x,w), which makes the control design more complex than that for linear systems.

[0083] ​The assumption of nonlinear boundedness (i.e., the existence of a smooth function such that |g(x,w)|≤d(x)|x|) is used to handle nonlinear terms, providing a basis for control law design.

[0084] 4. State constraint requirements: The system state must strictly satisfy preset physical constraints. A spherical constraint set is typically defined as follows: S={x:|x| <k c}, where k c >0 is the constraint boundary. Simultaneously, define the safe working subset Ω={x:|x|≤k}. c -k{c0}},0 <k{c0}<k c This is to ensure that there is a safety margin between the initial state and the constraint boundary.

[0085] 5. Control objectives: Given uncertain parameters, input gain, and state constraints, design a feedback control law such that: For any initial state x(0)∈Ω, the solution of the closed-loop system exists globally and is bounded.

[0086] For all t≥0, the system state trajectory always remains within the constraint set S (i.e., |x(t)|). <k c ).

[0087] The origin of a closed-loop system is semi-global asymptotically stable, meaning the system state converges to the equilibrium point (origin).

[0088] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the specific identification content executed by the system and device described above can be referred to the corresponding process in the foregoing method embodiments.

[0089] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings, but the present invention is not limited to the above embodiments. Even if various changes are made to the present invention, if these changes fall within the scope of the claims of the present invention and their equivalents, they shall still fall within the protection scope of the present invention.

Claims

1. A denial-of-service attack nonlinear system communication scheduling method based on an event triggering mechanism, characterized in that, The application is applied to a state-constrained nonlinear cyber-physical system, and comprises the following steps: a denial-of-service attack nonlinear system dynamics equation model is established; an attack timing is modeled and constraint conditions of attack frequency and duration are given; The event trigger mechanism is based on the relationship between the control error and the current system state to dynamically determine the trigger time, and the time interval between two consecutive triggers is not less than the minimum time interval τ m .

2. The event-triggered denial of service attack nonlinear system communication scheduling method of claim 1, wherein, the nonlinear system dynamics equation model is: where x is the system state, w is an uncertain parameter, u is the control input, and the input gain b(w) > 0 varies within a known positive definite upper bound b M and a lower bound b m a scalar function that varies within a known interval, and g(·) is a continuously differentiable nonlinear function; The preset safety constraint set is wherein k c >0 is a known constant.

3. The denial of service attack nonlinear system communication scheduling method based on an event trigger mechanism according to claim 2, characterized in that, the attack timing is modeled and constraint conditions of attack frequency and duration are given, which are based on the following settings: The pth attack interval is denoted as wherein, is the attack trigger time, is the attack duration; Within the time interval , a rejected time set and an allowed time set are defined, respectively: rejected time set , allowed time set , and the communication interruption and available time period are described by the rejected time set and the allowed time set , respectively.

4. The denial of service attack nonlinear system communication scheduling method based on an event trigger mechanism according to claim 3, characterized in that, an event-triggered mechanism is used to generate a trigger time sequence to update and maintain the input signal of the controller, there exists a constant ξ > 0, such that for any time interval [τ, t) the number of attack triggers n(τ, t) satisfies ; There exists a constant K > 0, T > 1 such that for any time interval [τ, t) the measure of the rejected time set satisfies .

5. The event-trigger based denial of service attack nonlinear system communication scheduling method of claim 1, wherein, the event-triggered mechanism specifies that the sampling time is updated as: Definition of the sequence of sampling instants , k e N within the adjacent sampling interval The control input error is: ; when the DoS attack exists, the actual input of the controller is: wherein, , δ0is a fixed sampling interval when the DoS attack exists, satisfying , δ k is an event-triggered interval; the method further comprises: 。 6. The denial of service attack nonlinear system communication scheduling method based on an event trigger mechanism according to claim 5, characterized in that, Event-triggered interval δ k is: where σ > 0 is a design parameter, to avoid the Zeno phenomenon.

7. The denial of service attack nonlinear system communication scheduling method based on an event trigger mechanism according to claim 6, characterized in that, comprising: defining a constant presence such that ; Defining the nominal control law where m1> 0 is a constant and P(x) > 1 is a smooth positive function with lower bound related to the DoS trigger threshold and y(x).

8. An event-triggered mechanism based denial-of-service attack nonlinear system communication scheduling system, characterized in that, a system model construction unit, configured to establish a denial-of-service attack nonlinear system dynamics equation model; an attack timing modeling unit, configured to model an attack timing and give constraint conditions of attack frequency and duration; a scheduling unit, configured to generate a trigger time sequence to update and maintain the input signal of the controller based on an event-triggered mechanism, wherein the event-triggered mechanism comprises: real-time monitoring of whether the current time is in a communication interruption period caused by a denial-of-service attack; if the current time is in the communication interruption period, a time-triggered mode is used to determine the update time sequence of the control signal at a fixed time interval δ0; comprising: If the current time is in the normal communication period, an event-triggered mode is adopted, the triggering time is dynamically determined based on the relationship between the control error and the current system state, and the time interval between two consecutive triggers is not less than the minimum time interval τ m .

9. An electronic device, comprising: a memory, configured to store a processing program; a processor, configured to implement the denial-of-service attack nonlinear system communication scheduling method based on the event-triggered mechanism according to any one of claims 1 to 7 when the processing program is executed. The readable storage medium stores a processing program, and the processing program is executed by a processor to implement the denial-of-service attack nonlinear system communication scheduling method based on the event-triggered mechanism according to any one of claims 1 to 7.

10. A readable storage medium, characterized by, ​

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