Elastic control method of high-order nonlinear system under pulse false information injection attack

Abstract

The invention relates to an elastic control method of a high-order nonlinear system under pulse false information injection attack. The method comprises the following steps: acquiring a system output signal and a control input signal of a current control period, and a state estimation value vector and a fuzzy weight vector estimation value of a previous control period; inputting a system output signal and a control input signal of a current control period, and a state estimation value vector and a fuzzy weight vector estimation value of a previous control period into a nonlinear state observer based on a fuzzy logic system to obtain a state estimation value vector of the current control period; the controller based on the fuzzy logic system carries out backstepping recursion through the state estimation value vector of the current control period and the fuzzy weight vector estimation value of the previous control period to obtain an elastic control input signal; and obtaining a fuzzy weight vector estimator of the current control period through an adaptive law based on the state estimation value vector of the current control period. By adopting the method, state jump and performance damage of attacks can be resisted.

Description

Technical Field

[0001] This invention belongs to the field of intelligent control technology, and in particular relates to an elastic control method for high-order nonlinear systems under pulse false information injection attacks. Background Technology

[0002] With the development of Industry 4.0 and intelligent manufacturing technologies, Cyber-Physical Systems (CPS) technology has emerged. Through the deep integration of computing, communication and control technologies, it has the ability to monitor and precisely regulate physical processes in real time. It is widely used in key industrial fields such as power, transportation and manufacturing, and has become the core support for the efficient operation of modern industry.

[0003] Traditional technologies for addressing False Data Injection (FDI) attacks in CPS primarily focus on control methods for continuous FDI attacks. These methods optimize state estimation and feedback control logic to counteract the impact of continuous signal tampering on the system and ensure stable system operation.

[0004] However, actual attacks often exhibit pulse characteristics, meaning that attackers "hit and run" to evade detection, injecting false signals only momentarily at specific times, which can cause system state jumps. Traditional control methods based on continuous dynamics are difficult to directly handle such instantaneous changes. High-order nonlinear CPS itself has unknown dynamics, which further increases the difficulty of control design. There is a lack of systematic solutions that combine state estimation and flexible control capabilities, making it impossible to effectively cope with complex operating conditions under pulsed FDI attacks. Summary of the Invention

[0005] Therefore, it is necessary to provide an elastic control method for high-order nonlinear systems under pulse spurious information injection attacks, which can describe the dynamics of pulse attacks and system nonlinearity, has adaptive learning capabilities, and can guarantee system stability under attacks, in order to address the above-mentioned technical problems.

[0006] Firstly, this application provides an elastic control method for a high-order nonlinear system under a pulse false information injection attack, comprising:

[0007] Based on the current control cycle, obtain the system output signal of the current control cycle on the sensor side and the control input signal of the current control cycle on the controller side, and obtain the state estimation vector of the previous control cycle and the fuzzy weight vector estimation value of the previous control cycle on the nonlinear state observer side.

[0008] The system output signal of the current control cycle, the control input signal of the current control cycle, the state estimation vector of the previous control cycle, and the fuzzy weight vector estimation value of the previous control cycle are input into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation vector of the current control cycle. The nonlinear state observer based on the fuzzy logic system continuously updates the state estimation value through the instantaneous state jump caused by the pulse false information injection attack, and dynamically stabilizes the state estimation error based on the observer gain matrix.

[0009] Using a pre-defined controller based on a fuzzy logic system, the virtual control law and the actual control law are calculated by backstepping and recursively combining the state estimation vector of the current control cycle and the fuzzy weight vector estimation value of the previous control cycle, so as to obtain the elastic control input signal.

[0010] Based on the state estimate vector of the current control cycle, the fuzzy weight vector estimate of the current control cycle is obtained through an adaptive law.

[0011] In one embodiment, the system output signal of the current control cycle, the control input signal of the current control cycle, the state estimation vector of the previous control cycle, and the fuzzy weight vector estimation value of the previous control cycle are input into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation vector of the current control cycle, including:

[0012] Based on the preset attack model, determine whether the current control period is a non-attack time or an attack time;

[0013] If the current control period is a non-attack time, the state estimate vector for the current control period can be obtained using the following formula:

[0014]

[0015]

[0016]

[0017]

[0018] in, This is the vector of state estimates for the current control cycle; The state estimate vector of the current control cycle is the first... One state estimate; The state estimate vector of the current control cycle is the first... The instantaneous rate of change of each state estimate; The first state estimate vector of the previous control cycle One state estimate; The feedback gain parameter is the nonlinear state observer based on a fuzzy logic system. This is the system output signal for the current control cycle; This is the estimated value of the system output signal from the previous control cycle; This is the estimated value of the fuzzy weight vector from the previous control cycle; The first state estimate vector of the previous control cycle The fuzzy basis functions corresponding to each state estimate; This is the control input signal for the current control cycle;

[0019] If the current control period coincides with an attack, the state estimate vector for the current control period can be obtained using the following formula:

[0020]

[0021]

[0022] in, The state estimate vector of the current control cycle is the first... One state estimate; The first state estimate vector of the previous control cycle One state estimate; This is a false injection signal used in the current attack on the sensor side; This is a false injection signal for the current attack on the controller side; This is the moment of attack.

[0023] In one embodiment, a pre-defined controller based on a fuzzy logic system is used to perform backstepping recursive calculation of the virtual control law and the actual control law by combining the state estimation vector of the current control cycle and the fuzzy weight vector estimation value of the previous control cycle, thereby obtaining the elastic control input signal, including:

[0024] The error variable vector is obtained by performing inverse step coordinate transformation and first-order low-pass filtering on the state estimate vector of the current control cycle.

[0025] Based on the first virtual control law, the first virtual control quantity is calculated using the error variable vector, the state estimation error, and the estimated value of the fuzzy weight vector from the previous control cycle; the first virtual control law is... ,in, The first error variable in the error variable vector. The error is the estimation error for the first state. This is the estimated value of the first fuzzy weight vector from the previous control cycle. Let be the fuzzy basis function corresponding to the first state estimate of the state estimate vector in the current control cycle. The virtual control input feedback gain parameter; The feedback gain parameter is the nonlinear state observer based on a fuzzy logic system.

[0026] Based on the The virtual control law is calculated using the error variable vector, the state estimation error, and the estimated fuzzy weight vector from the previous control cycle. Virtual control quantity; the first Virtual control law is ,in, , The first error variable vector One error variable, For the first control cycle A fuzzy weight vector estimate The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate For the first The time derivative of the filtered signal;

[0027] Based on the actual control law, the elastic control input signal is calculated using the error variable vector, the state estimation error, and the estimated fuzzy weight vector from the previous control cycle; the actual control law is: ,in, The first error variable vector One error variable, For the first control cycle A fuzzy weight vector estimate The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate For the first The time derivative of the filtered signal.

[0028] In one embodiment, the state estimate vector of the current control cycle is subjected to backstepping coordinate transformation and first-order low-pass filtering to obtain an error variable vector, including:

[0029] The virtual control input from the previous control cycle is input into a first-order low-pass filter to obtain the filtered signal and its corresponding time derivative; the formula for the first-order low-pass filter is: ,in , The filter time constant; The time derivative of the filtered signal; For filtered signals; For the first control cycle One virtual control quantity;

[0030] If the current control cycle is a non-attack time, the error variable vector can be obtained using the following formula:

[0031]

[0032] in, This is the first error variable; This is the first state estimate in the state estimate vector for the current control cycle. The estimated value of the system output signal for the current control cycle; The first error variable vector One error variable; The state estimate vector of the current control cycle is the first... One state estimate; For filtered signals;

[0033] If the current control period is the attack time, the error variable vector can be obtained using the following formula:

[0034]

[0035] in, The first error variable vector One error variable; The first error variable vector before the attack One error variable; This is a false injection signal used in the current attack on the sensor side; This is a false injection signal for the current attack on the controller side; This is the moment of attack.

[0036] In one embodiment, based on the state estimate vector of the current control cycle, an adaptive law is used to obtain the fuzzy weight vector estimate of the current control cycle, including:

[0037] Calculate the corresponding fuzzy basis function based on each state estimate in the current control cycle's state estimate vector to obtain the fuzzy basis function vector;

[0038] Construct a diagonal fuzzy basis function matrix based on the fuzzy basis function vectors; the expression for the fuzzy basis function matrix is: ;in, The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate;

[0039] Based on the adaptive law, the time derivative of the fuzzy weight vector estimate for the current control cycle is calculated from the fuzzy basis function matrix and the state estimation error. Then, the fuzzy weight vector estimate for the current control cycle is obtained through numerical integration from the time derivative of the fuzzy weight vector estimate for the current control cycle. The adaptive law is: ,in, The time derivative of the fuzzy weight vector estimate for the current control cycle. The adaptive law gain matrix, For state estimation error, It is a positive definite matrix; The fuzzy basis function matrix; This is the weight attenuation coefficient; This is the estimated value of the fuzzy weight vector from the previous control cycle. For the control period; the expression for the fuzzy weight vector estimate of the current control period is: .

[0040] Secondly, this application also provides an elastic control system for a high-order nonlinear system under pulse false information injection attacks, comprising:

[0041] The system monitoring module is used to acquire the system output signal of the current control cycle from the sensor side and the control input signal of the current control cycle from the controller side, and to acquire the state estimation vector of the previous control cycle and the fuzzy weight vector estimation value of the previous control cycle from the nonlinear state observer side.

[0042] The state estimation module is used to input the system output signal of the current control cycle, the control input signal of the current control cycle, the state estimation vector of the previous control cycle, and the fuzzy weight vector estimation value of the previous control cycle into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation vector of the current control cycle. The nonlinear state observer based on the fuzzy logic system continuously updates the state estimation value through instantaneous state jumps caused by pulse spurious information injection attacks, and dynamically stabilizes the state estimation error based on the observer gain matrix.

[0043] The elastic control module is used to utilize a preset controller based on a fuzzy logic system to perform backstep recursive calculation of the virtual control law and the actual control law by combining the state estimation value vector of the current control cycle and the fuzzy weight vector estimation value of the previous control cycle, so as to obtain the elastic control input signal.

[0044] The weighting module is used to obtain the fuzzy weighting vector estimate of the current control cycle based on the state estimation vector of the current control cycle through an adaptive law.

[0045] Thirdly, this application also provides a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the elastic control method for a high-order nonlinear system under any of the above-described pulse false information injection attacks.

[0046] Fourthly, this application also provides a computer-readable storage medium storing a computer program thereon, wherein the computer program, when executed by a processor, implements the steps of the elastic control method for a high-order nonlinear system under any of the above-described pulse false information injection attacks.

[0047] The aforementioned elastic control method for high-order nonlinear systems under pulsed false information injection attacks incorporates the instantaneous state jumps caused by the attack into the continuous dynamic updates of the nonlinear state observer. Based on a pulsed system framework, it models discrete, energy-limited attacks, accurately characterizing the attack signal's features and the resulting system state jumps. This accurately reflects the intermittent and sudden attack behaviors of attackers attempting to evade detection. By backstepping and recursively calculating the virtual and actual control laws, it effectively handles high-order nonlinear characteristics. Simultaneously, it integrates the average pulse interval method to solve the analytical challenges caused by attack jumps, deriving clear stability sufficient conditions and clarifying the quantitative relationship between the controller's convergence speed and the attack intensity correlation factor, as well as the lower bound of the attack frequency, providing clear guidance for engineering parameter tuning. Based on the current state estimate, it updates the fuzzy weight vector online using an adaptive law, thereby achieving real-time approximation and compensation for unknown nonlinear dynamics. Even when pulsed false information injection attacks continue, the accurate estimation of the high-order nonlinear system state, effective compensation for unknown dynamics, and elastic generation of control signals ultimately ensure that all signals in the closed-loop system remain bounded and stable, effectively resisting the state jumps and performance degradation caused by the attack. Attached Figure Description

[0048] To more clearly illustrate the technical solutions in the embodiments or related technologies of this application, the accompanying drawings used in the description of the embodiments or related technologies will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0049] Figure 1 This is a flowchart illustrating the elastic control method for high-order nonlinear systems under pulse false information injection attacks according to the present invention.

[0050] Figure 2 This is a flowchart illustrating the steps of step S103.

[0051] Figure 3The diagram shows the structure of the elastic control system for a high-order nonlinear system under the pulse false information injection attack of this invention. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0053] In one embodiment, such as Figure 1 As shown, a method for elastic control of a high-order nonlinear system under a pulse false information injection attack is provided. This embodiment illustrates the method by applying it to a terminal. It is understood that this method can also be applied to a server, or to a system including both a terminal and a server, and implemented through interaction between the terminal and the server. In this embodiment, the method includes the following steps:

[0054] S101. Based on the current control cycle, obtain the system output signal of the current control cycle on the sensor side and the control input signal of the current control cycle on the controller side, and obtain the state estimation vector of the previous control cycle and the fuzzy weight vector estimation value of the previous control cycle on the nonlinear state observer side.

[0055] Indicatively, the mathematical model of the controlled object is a high-order nonlinear system, and its continuous-time dynamics are described as follows: , ; , ,in, For the front A vector of states, For the entire system state, It is a control input. It is system output, function It is an unknown, smooth, nonlinear function, representing the uncertain dynamics within the system. The system output signal is acquired through sensors. However, due to the risk of pulsed FDI attacks on the communication network between the sensors and the controller, the acquired signal may actually be tampered with. It contains the actual output. With false injection signals The spurious injection signals exhibit pulse characteristics, only instantaneously superimposing at specific attack moments. The sequence of attack moments has a strictly monotonically increasing property, tending towards infinity. satisfy , For the Dirac function, This is a sequence of attack times. A spurious signal is injected into the sensor side and satisfies , It is an unknown positive scalar, conforming to Assumption 1. Control input signal. The signal originating from the controller output is also susceptible to pulse FDI attacks. The tampered control input signal contains both the real control input and spurious injected signals from the corresponding channels. The amplitudes of both types of spurious injected signals satisfy bounded constraints, meaning there exists an unknown positive scalar such that the absolute value of the spurious injected signal never exceeds this scalar. , , False signal injection to the controller side and , It is an unknown positive scalar, conforming to Assumption 1.

[0056] The state estimate vector of the previous control cycle With fuzzy weight vector estimate All are stored in the cache module of the system control unit. The preset control period must have a value that meets the following requirements. , This serves as a lower bound for the average pulse interval, conforming to Assumption 2, to ensure that the data update frequency matches the system's dynamics and attack response requirements. Furthermore, the state estimate vector is... A dimensional column vector, where the _i_i is a column vector of _i_i, and the ... The element corresponds to the system's first... The estimated value of each state, For higher-order nonlinear systems, the order is given, and the dimension of the state estimate vector is consistent with the system state dimension. The fuzzy weight vector estimate is the aggregated weight matrix, derived from... The weight vectors of each fuzzy logic system are composed of their respective weight vectors, and the dimension of each sub-weight vector corresponds to the number of fuzzy rules in the fuzzy logic system. Consistent, the selection of the number of fuzzy rules needs to be determined based on the complexity of the nonlinear dynamics of the system. It is usually fixed after verification through offline simulation in order to balance the approximation accuracy and computational complexity.

[0057] S102. Input the system output signal of the current control cycle, the control input signal of the current control cycle, the state estimation vector of the previous control cycle, and the fuzzy weight vector estimation value of the previous control cycle into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation vector of the current control cycle. The nonlinear state observer based on the fuzzy logic system continuously updates the state estimation value through the instantaneous state jump caused by the pulse false information injection attack, and dynamically stabilizes the state estimation error based on the observer gain matrix.

[0058] This paper illustrates how a nonlinear state observer, built upon a fuzzy logic system (FLS), can accurately estimate the state of a real system under conditions where a pulsed FDI attack causes a sudden jump in system state. The design of FLS is based on Lemma 1, which states that for any system defined on a compact set... Unknown smooth function on There exists an optimal weight vector. With fuzzy basis functions , making ,in To approximate the error and , It is a positive scalar. The observer's gain matrix. Designed as a Hurwitz matrix, in the form of... ,in To obtain the feedback gain parameter, the Lyapunov equation is solved. Sure, and The matrix is ​​positive definite, ensuring the observer itself has asymptotic stability and can dynamically suppress state estimation errors. State estimation error is the difference between the true state in a high-order nonlinear system and the state estimate output by the nonlinear state observer based on the fuzzy logic system. Specifically, it is a vector composed of the deviations between each true state variable and its corresponding estimate. In the scenario of impulsive FDI attacks, the unknown nonlinear dynamics inherent in the system itself, despite the approximation capability of the fuzzy logic system, will still leave a small approximation error. Furthermore, impulsive FDI attacks on the sensor and controller sides can tamper with the transmitted signals, causing distortion of the observer's input data and thus inducing estimation bias. Other factors include the small difference between the observer's initial state and the system's true initial state, and the subtle delays that may exist during data synchronization within the control cycle. The magnitude of the state estimation error directly reflects the observer's accuracy in tracking the true state of the system and is a crucial feedback information in the entire elastic control method. The nonlinear state observer suppresses error divergence through the Hurwitz property of the gain matrix, while the adaptive law dynamically adjusts the fuzzy weight vector estimate based on this error. The controller also incorporates an error correction term to optimize the control law design. Ultimately, through the synergistic effect of multiple components, the state estimation error converges to a bounded region, providing accurate state feedback support for the effective implementation of the elastic control strategy. For example, a single... The estimation error formula for each state is as follows: , The formula for the global state estimation error vector is: .in, Indicates the first The instantaneous estimation error of each state, For higher-order nonlinear systems One real state variable (over time) (Dynamic changes, cannot be directly measured) The output of the nonlinear state observer based on fuzzy logic system is the first Each state estimate Let be the system order. for The column vector aggregates the instantaneous estimation errors of all states, comprehensively reflecting the degree of deviation of the observer from the true state of the system.

[0059] Furthermore, the observer's working logic is divided into non-attack time and attack time. During non-attack time, the state estimate is updated through the continuous dynamic equation, and the state estimate of the previous cycle, the current attacked signal, the fuzzy weight estimate, and the fuzzy basis function output are integrated to track the normal evolution process of the system. During attack time, the observer directly responds to the state jump and corrects the state estimate based on the spurious injection signals from the sensor side and the controller side to avoid the accumulation of estimation bias caused by the attack.

[0060] S103. Using a pre-set controller based on a fuzzy logic system, the virtual control law and the actual control law are calculated by backstepping and recursively combining the state estimation vector of the current control cycle and the fuzzy weight vector estimation value of the previous control cycle, so as to obtain the elastic control input signal.

[0061] The schematic FLS-based controller employs a backstepping recursive strategy, suitable for constructing control laws for high-order nonlinear systems. By progressively introducing virtual control variables, it transforms the complex high-order system into a series of low-order subsystems, achieving precise control of the high-order nonlinear system and effective defense against pulse spurious information injection attacks. The core idea of ​​the backstepping recursive strategy is to decompose the high-order nonlinear system into n low-order subsystems. By progressively designing the virtual control variables for each subsystem, the actual control input of the system is finally derived. This strategy effectively handles the nonlinear coupling characteristics of the system while avoiding the "complexity explosion" problem encountered in traditional control methods. Specifically, the controller defines the error variable through coordinate transformation. An error feedback mechanism is constructed by comparing the state estimate with the filtered virtual control input. Virtual control law With actual control law The designs all incorporate state estimation errors. Fuzzy weight estimation With fuzzy basis functions Simultaneously, a feedback gain parameter is introduced. Adjust the system response speed and stability margin. Simultaneously introduce a first-order low-pass filter to smooth the virtual control input, obtaining the filtered signal. Filter parameters Selected based on system dynamic response requirements, ensuring the filtered signal accurately tracks the original virtual control quantity while reducing high-frequency jitter in the control input, and the actual control law. It also introduced This approach further enhances the system's resistance to attacks through an energy suppression mechanism, ensuring that the closed-loop system signal is bounded. By designing a three-level control law, combined with the unknown dynamic approximation capability of fuzzy logic systems and the decoupling characteristics of backstepping recursion, the resulting elastic control input signal can accurately compensate for the combined effects of nonlinear dynamics and pulse attacks, ensuring stable system operation.

[0062] S104. Based on the state estimate vector of the current control cycle, obtain the fuzzy weight vector estimate of the current control cycle through the adaptive law.

[0063] This is an illustrative example of dynamically updating the fuzzy weight vector estimate through an adaptive law, with the design goal of making... Able to approximate optimal weights This improves FLS's ability to handle unknown nonlinear dynamics. The approximation accuracy. The expression for the adaptive law is: ,in It is a positive definite adaptive gain matrix, and the values ​​of its diagonal elements determine the weight update speed, which is adjusted according to the system convergence requirements and stability constraints. This is the weight decay coefficient, used to prevent excessive growth of weight estimation and ensure system robustness; The fuzzy basis function matrix is ​​in diagonal form, consisting of the fuzzy basis functions corresponding to each state estimate. constitute; Let the state estimation error vector be... This represents the solution matrix of the Lyapunov equations. The fuzzy weight vector estimate for the current control cycle. It was calculated through numerical integration using the Euler integral method. This ensures that weight updates are synchronized with the control cycle, achieving real-time adaptive compensation for unknown dynamics and attack effects. The entire process, through the coordinated work of the observer, controller, and adaptive law, combined with the average pulse interval method, satisfies... , It ensures the bounded stability of the closed-loop system and effectively resists pulsed FDI attacks.

[0064] In the aforementioned elastic control method for high-order nonlinear systems under pulse false information injection attacks, the system output signal from the sensor side, the control input signal from the controller side, and the state estimation vector and fuzzy weight vector estimation value from the previous control cycle are acquired based on the current control cycle. This data is then input into a preset nonlinear state observer based on a fuzzy logic system. Leveraging the fuzzy logic system's ability to approximate unknown dynamics of high-order nonlinear systems, and combined with the stability design of the observer's gain matrix, the method accurately responds to instantaneous state jumps caused by pulse false information injection attacks, achieving continuous dynamic state estimation updates and stabilizing state estimation errors. The controller, based on the fuzzy logic system, combines the current state estimation vector with the fuzzy weight vector estimation value from the previous cycle, and calculates the virtual and actual control laws through a backstepping recursive strategy to generate elastic control input signals. Simultaneously, based on the current state estimation vector, the fuzzy weight vector estimation value is dynamically updated using an adaptive law, continuously optimizing the approximation accuracy of the fuzzy logic system. Ultimately, this effectively resists the impact of pulse false information injection attacks, overcomes the control challenges caused by unknown dynamics of high-order nonlinear systems, and ensures the stability and control performance of the closed-loop system.

[0065] In one embodiment, the system output signal of the current control cycle, the control input signal of the current control cycle, the state estimation vector of the previous control cycle, and the fuzzy weight vector estimation value of the previous control cycle are input into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation vector of the current control cycle, including:

[0066] S11. Based on the preset attack model, determine whether the current control period is a non-attack time or an attack time.

[0067] Indicatively, the attack model is a mathematical model constructed based on the dynamic characteristics of pulsed false information injection attacks. It includes three key elements: the attack time sequence, the constraints of the false injection signal, and the attack frequency constraints, providing a quantitative basis for determining the operating condition type of the current control cycle. The attack time sequence is a strictly monotonically increasing set of time points. Each time point in this set corresponds to one pulse attack, and the sequence tends towards infinity over time, consistent with the actual scenario of attackers continuously launching intermittent attacks. The spurious injection signal constraint clarifies the amplitude boundaries of the spurious injection signals on the sensor and controller sides; that is, there exists an unknown positive scalar such that the absolute values ​​of the two types of spurious injection signals never exceed their corresponding scalars, i.e., there exists an unknown positive constant. and This makes it possible for all ,have , This constraint stems from the objective limitations of attackers circumventing detection and limited energy, and is an important guarantee of the model's rationality. The attack frequency constraint limits the upper limit of the number of attacks per unit time, meaning there exists a normal number. (Average length of stay) and In any time interval Number of attacks within satisfy ,and There is a lower world This ensures that attacks do not occur at an unlimited frequency, allowing time for the control strategy to respond.

[0068] Specifically, the core logic for determining whether the current control period is a non-attack or attack period is to match the time interval of the current control period with the attack time sequence in the preset attack model. The time interval of the control period starts at the end of the previous control period and ends at the end of the current control period. If the interval does not contain any time point from the attack time sequence, it is determined to be a non-attack period; if the interval contains a time point from the attack time sequence, it is determined to be an attack period. Due to the instantaneous nature of pulse attacks, the attack time usually appears as a single discrete point within the time interval of the control period and does not span more than one control period. Therefore, there is no need to consider the case of multiple attack times overlapping. The judgment process needs to be verified in conjunction with the attack frequency constraint. If the number of judged attacks within a certain time period exceeds the upper limit of the frequency constraint, it is considered a misjudgment, and the matching relationship between the attack time sequence and the control period needs to be rechecked to ensure the accuracy of the judgment result and avoid deviations in the observer state estimation due to misjudgments.

[0069] S12. If the current control cycle is a non-attack time, the state estimate vector for the current control cycle is obtained using the following formula:

[0070]

[0071]

[0072]

[0073]

[0074] in, This is the vector of state estimates for the current control cycle; The state estimate vector of the current control cycle is the first... One state estimate; The state estimate vector of the current control cycle is the first... The instantaneous rate of change of each state estimate; The first state estimate vector of the previous control cycle One state estimate; The feedback gain parameter is the nonlinear state observer based on a fuzzy logic system. This is the system output signal for the current control cycle; This is the estimated value of the system output signal from the previous control cycle; This is the estimated value of the fuzzy weight vector from the previous control cycle; The first state estimate vector of the previous control cycle The fuzzy basis functions corresponding to each state estimate; This is the control input signal for the current control cycle.

[0075] Indicatively, during non-attack periods, the state estimation vector is obtained through a combination of continuous dynamic equations and integral operations. The core principle is to leverage the continuous evolution characteristics of the observer to track the normal operating state of the system, while simultaneously using a fuzzy approximation mechanism to compensate for unknown nonlinear dynamics. The state estimation vector for the current control cycle... for A dimensional vector, obtained through integration, with the integration interval from the end of the previous control cycle to the end of the current control cycle. Each element corresponds to a system... The estimated values ​​for each state are given, and the vector dimension is consistent with the system order to ensure a complete estimate of the system's entire state.

[0076] Instantaneous rate of change of state estimate It is the core of constructing the equations of continuous dynamics, for From 1 to The state, whose instantaneous rate of change changes from the first state of the previous control cycle. State estimate, observer feedback gain parameter The attacked system output signal during the current control cycle The estimated output signal value of the system in the previous control cycle The product of the differences, the estimated value of the i-th fuzzy weight vector in the previous control cycle The fuzzy basis function corresponding to the i-th state estimate of the previous control cycle The product of these three factors is the sum of their components. Among them, As the first in the previous control cycle Each state estimate provides a basic dynamic reference for the evolution of the current state; For feedback correction items, where Feedback gain parameters to the observer This is the system output signal after the current control cycle is attacked. The difference between the two values ​​represents the estimated output signal of the system in the previous control cycle and reflects the deviation at the output level. By adjusting the feedback gain parameter, the state estimate can be corrected in real time. For fuzzy approximation terms, It is the estimated value of the fuzzy weight vector from the previous control cycle. It is the first control cycle of the previous period The fuzzy basis functions corresponding to the nth state estimate, and their product constitute the fuzzy logic system's state for the nth state. An approximation output of an unknown nonlinear dynamic is used to compensate for the inherent nonlinear characteristics of the system. Fuzzy basis function. The membership function is obtained by normalization of the fuzzy membership function. The membership function adopts a Gaussian function, and its center and width parameters are determined by offline optimization to ensure effective approximation of nonlinear dynamics.

[0077] For the nth state, its instantaneous rate of change The expression and the previous The core difference between the states lies in the fact that... Replace with the control input signal of the current control cycle. Because the first order of a higher-order nonlinear system The evolution of each state is directly driven by the control input. By introducing the control input signal, the observer can track the dynamic changes of the system under the control action.

[0078] Integration This is a crucial step in converting the instantaneous rate of change into an estimate of the current control cycle state, with the integration interval set to the end time of the previous control cycle. Until the end of the current control cycle This ensures that the integral result fully reflects the continuous evolution of the state within the current control cycle. The state estimate from the previous control cycle... As the initial value for integration, it ensures the continuity of state estimation, avoids abrupt state changes caused by control cycle switching, and enables the observer output to smoothly track the true state of the system.

[0079] S13. If the current control period is the attack time, the state estimation vector for the current control period is obtained using the following formula:

[0080]

[0081]

[0082] in, The state estimate vector of the current control cycle is the first... One state estimate; The first state estimate vector of the previous control cycle One state estimate; This is a false injection signal used in the current attack on the sensor side; This is a false injection signal for the current attack on the controller side; This is the moment of attack.

[0083] As an illustration, at the moment of attack, a pulsed false information injection attack will cause an instantaneous change in the system state. Traditional continuous integration state update methods can no longer accurately capture this abrupt change. Therefore, an update logic that directly superimposes the false injection signal is adopted to ensure that the state estimate can respond promptly to the attack's impact. Attack moment It is a specific element in the attack time sequence in the preset attack model. Each attack time falls within a certain control period, and there is only one attack time within that control period, ensuring the uniqueness of the state update.

[0084] For the first to The state estimate for the current control cycle. The update formula reflects the impact of sensor-side attacks, and its expression is: .in For the first control cycle The state estimate is the base value for the state estimate before the attack occurs. This value is directly retrieved from the cache to ensure the accuracy of the state information before the attack. The signal is a false injection signal that is currently being attacked on the sensor side. Its amplitude satisfies the bounded constraints in the preset attack model. This signal is false data injected by the attacker into the communication link between the sensor and the controller, which will cause the system output signal to be tampered with. By directly superimposing this signal into the state estimate, the observer can synchronously perceive the state deviation caused by the attack.

[0085] For the The state estimate for the current control cycle. The update formula is , and the previous The key difference between these states lies in the addition of a spoofed injection signal from the controller side indicating the current attack. This is because the nth state is directly driven by the control input, and the communication link between the controller and the actuator is also susceptible to pulse spoofing attacks. This refers to the spurious injected signal in the link, whose amplitude also satisfies the bounded constraint. This can be achieved by simultaneously superimposing... and This enables the observer to fully compensate for attacks from both the sensor and controller sides on the first... The impact of each state is minimized, ensuring the completeness and accuracy of the state estimation. The entire update process requires no integration calculations; it directly completes the transition update based on the state estimation value before the attack and the spurious injection signal, quickly responding to the instantaneous characteristics of pulse attacks and avoiding the accumulation of state estimation bias caused by the attack.

[0086] In one embodiment, such as Figure 2 The process involves using a pre-defined controller based on a fuzzy logic system, combining the state estimation vector of the current control cycle and the fuzzy weight vector estimation value of the previous control cycle to perform backstepping recursive calculation of the virtual control law and the actual control law, thereby obtaining the elastic control input signal, including:

[0087] S201. Perform inverse step coordinate transformation and first-order low-pass filtering on the state estimate vector of the current control cycle to obtain the error variable vector.

[0088] This diagram illustrates how the state space of a high-order nonlinear system is transformed into an error space, facilitating control law design, through backstepping coordinate transformation. By defining error variables, a quantized relationship is established between the system state and the reference input, providing a foundation for the subsequent construction of virtual and actual control laws. The implementation logic of the coordinate transformation is closely related to the system order. The hierarchical system, by gradually introducing virtual control quantities as intermediate reference signals, will... The order system is decomposed into Each of the interconnected first-order subsystems corresponds to an error variable, enabling decoupled control of the complex system.

[0089] The introduction of a first-order low-pass filter aims to address the "complexity explosion" problem that may occur during the backstepping recursion process. It smooths the virtual control input, eliminating computational redundancy and control jitter caused by higher-order derivative terms. The filter's expression is: ,in The filter time constant is selected to balance filtering effect and response speed. It is usually a small positive number to ensure that the filtered signal can quickly track the original virtual control quantity, while suppressing high-frequency noise. It is the time derivative of the filtered signal, reflecting the dynamic rate of change of the filtered signal; For the current control cycle, the [number]th A filtered signal is used as the reference input for the next subsystem; For the first control cycle A virtual control variable is directly called from the cache to ensure the timeliness of the filtered input.

[0090] The construction of the error variable vector needs to distinguish between non-attack and attack scenarios. In the non-attack scenario, the first error variable... The system output signal estimate is directly equivalent to the current control cycle. ,and The first state estimate of the state estimate vector Consistent because the system output is directly represented by the first state, simplifying the logic of relating output error to state error; for From 2 to Error variables It is defined as the current control cycle number. State estimates With the i-th filtered signal The difference, which directly reflects the first The degree of deviation between the state of each subsystem and the reference input. At the attack time, the error variable needs to be directly superimposed with a spurious injection signal to respond to the state transition caused by the attack, from the 1st to the 2nd. Error variables Error variable value before the attack False signal injection on the sensor side The sum of the nth error variables Then, two types of spurious injected signals are superimposed: one from the sensor side and one from the controller side. and This ensures that error variables can fully capture the impact of attacks on system state deviations, providing an accurate basis for attack compensation in control laws.

[0091] S202. Based on the first virtual control law, the first virtual control quantity is calculated using the error variable vector, the state estimation error, and the estimated value of the fuzzy weight vector from the previous control cycle; the first virtual control law is... ,in, The first error variable in the error variable vector. The error is the estimation error for the first state. This is the estimated value of the first fuzzy weight vector from the previous control cycle. Let be the fuzzy basis function corresponding to the first state estimate of the state estimate vector in the current control cycle. The virtual control input feedback gain parameter; represents the feedback gain parameter of a nonlinear state observer based on a fuzzy logic system.

[0092] Indicatively, the first virtual control law, applied to the first first-order subsystem after backstepping decomposition, uses error feedback, state estimation error correction, and nonlinear dynamic compensation to rapidly converge the first error variable to a bounded region near zero. The expression for the control law is: .in, The first error variable in the error variable vector is the core feedback signal of the control law, which directly reflects the state deviation of the first subsystem. The virtual control input feedback gain parameter is a positive scalar. Adjusting this parameter can change the convergence speed of the error variable. The larger the gain, the faster the error converges, but it is necessary to avoid excessive gain that could cause system oscillation. The feedback gain parameter of the nonlinear state observer is kept consistent with the gain parameter of the state estimation stage to ensure the coordinated stability of the control law and the observer. The first state estimation error is equal to the difference between the first true state of the system and the corresponding estimated value. This parameter is introduced to use the estimation bias information of the observer to correct the virtual control quantity and improve the control accuracy. The estimated value of the first fuzzy weight vector of the previous control cycle is stored in the cache, and its dimension is consistent with the number of fuzzy rules in the corresponding fuzzy logic system. The fuzzy basis function corresponding to the first state estimate of the current control cycle takes a value between 0 and 1, and the sum of all basis functions is 1. The product of the two constitutes the approximate output of the fuzzy logic system for the first unknown nonlinear dynamic. By introducing this negative term into the control law, active compensation for the unknown nonlinear characteristics of the system is achieved, and the influence of nonlinear dynamic on the system stability is weakened.

[0093] Specifically, when calculating the first virtual control quantity, the first fuzzy weight vector estimate from the previous cycle is first retrieved from the system cache. Combined with the first state estimate of the current cycle Calculate the fuzzy basis function Then, synchronously obtain the first error variable. Error with the first state estimation Substitute all parameters into the control law expression, and obtain the first virtual control quantity through linear operations and matrix multiplication. This control quantity will serve as the reference input basis for the second subsystem, providing support for subsequent recursive calculations.

[0094] S203, based on the first The virtual control law is calculated using the error variable vector, the state estimation error, and the estimated fuzzy weight vector from the previous control cycle. Virtual control quantity; the first Virtual control law is ,in, , The first error variable vector One error variable, For the first control cycle A fuzzy weight vector estimate The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate For the first The time derivative of the filtered signal.

[0095] Indicative, the first The virtual control law is designed for the intermediate first-order subsystem after backstepping decomposition. Its design logic, based on the first virtual control law, adds a time derivative term for the filtered signal to compensate for the dynamic delay introduced by the low-pass filter, ensuring accurate tracking of the system dynamics by the control law. The expression of the control law is as follows: .in, The first error variable vector The error variable reflects the first error variable. The deviation between the state of each subsystem and the reference input is the core feedback basis of the control law; The virtual control input feedback gain parameter for the corresponding subsystem is a positive scalar. The gain parameter of each subsystem can be adjusted independently according to the dynamic characteristics of the subsystem to optimize the response performance of each subsystem. The gain parameters are fed back to the observer and kept consistent with the corresponding parameters in the state estimation stage to ensure the coordination between control and observation. The first state estimation error is used to achieve global correction of the control laws of each subsystem by introducing global state estimation error information; For the first control cycle A fuzzy weight vector estimate For the current control cycle number The fuzzy basis functions corresponding to the nth state estimate, and their product constitutes the basis function of the nth state estimate. An approximation output of an unknown nonlinear dynamic is used to compensate for the nonlinear characteristics of the subsystem. For the first The time derivative of the filtered signal is calculated by numerical differentiation of the output signal of the first-order low-pass filter. Its function is to cancel the phase lag and amplitude attenuation generated during the filtering process, and to ensure that the dynamic characteristics of the reference input can be accurately transmitted to the current subsystem.

[0096] Calculate the first When using virtual control inputs, the output of a first-order low-pass filter must first be passed through the input. Solve for its time derivative Numerical differentiation methods, specifically the central difference method, can be used to balance computational accuracy and real-time performance; then, the previous cycle's... The estimated value of the fuzzy weight vector In combination with the current number State estimates Calculate fuzzy basis functions Synchronously collect the i-th error variable Error with the first state estimation Substitute all parameters into the control law expression to complete the calculation. The resulting i-th virtual control quantity is filtered and used as the reference input of the i+1-th subsystem, realizing the continuous advancement of the backstep recursion.

[0097] S204. Based on the actual control law, the elastic control input signal is calculated using the error variable vector, the state estimation error, and the estimated fuzzy weight vector from the previous control cycle; the actual control law is... ,in, The first error variable vector One error variable, For the first control cycle A fuzzy weight vector estimate The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate For the first The time derivative of the filtered signal.

[0098] The actual control law applies to the last first-order subsystem after backstepping decomposition. Its output is directly used as the elastic control input signal to drive the actuator to act on the controlled object. By introducing an energy suppression term, the system's resistance to pulse attacks is enhanced. The expression of the control law is as follows: .in, The first error variable vector Each error variable reflects the state deviation of the last subsystem and is directly related to the final control performance of the system. The feedback gain parameter of the actual control quantity is a positive scalar value and is a key parameter for adjusting the overall response speed and stability margin of the system. The gain parameters are fed back to the observer to maintain consistency with the state estimation stage, ensuring closed-loop coordination between control and observation; The first state estimation error provides global error correction information for the actual control quantity; As an energy suppression term, it limits the magnitude growth of the error variable by imposing a secondary penalty on the error variable. Especially when a pulse attack causes a sudden change in error, it can quickly consume the error energy, suppress error amplification, and improve the robustness of the system. For the first control cycle A fuzzy weight vector estimate The product of the two is used to compensate for the unknown nonlinear dynamics of the last subsystem. The time derivative of the nth filtered signal is obtained by differentiating the output of the low-pass filter. It is used to compensate for the filtering delay and ensure that the dynamic characteristics of the reference input are accurately transmitted to the actual control quantity.

[0099] Specifically, when calculating the elastic control input signal, the following steps must be performed sequentially: solving for the time derivative of the filtered signal, calculating the fuzzy basis function, acquiring error variables and state estimation errors, substituting all parameters into the actual control law expression, and completing the calculation through linear operations, matrix multiplication, and nonlinear term operations. The resulting elastic control input signal... The output constraints of the actuator must be met, meaning that the amplitude and rate of change of the control signal must not exceed the rated parameters of the actuator. If the constraints are exceeded, saturation processing is performed to ensure the physical feasibility of the control signal. Ultimately, the actuator acts on the high-order nonlinear system to achieve comprehensive suppression of pulse false information injection attacks and unknown nonlinear dynamics of the system, thus ensuring the stable operation of the system.

[0100] In one embodiment, the state estimate vector of the current control cycle is subjected to backstepping coordinate transformation and first-order low-pass filtering to obtain an error variable vector, including:

[0101] S21. Input the virtual control quantity from the previous control cycle into a first-order low-pass filter to obtain the filtered signal and its corresponding time derivative; the formula for the first-order low-pass filter is: ,in , The filter time constant; The time derivative of the filtered signal; For filtered signals; For the first control cycle A virtual control variable.

[0102] The core function of a first-order low-pass filter is to smooth the virtual control input, eliminating the computational complexity and control signal jitter caused by higher-order derivative terms during the backstep recursion, and providing a stable reference input for subsequent error variable construction. The filter's expression is: ,in The value range is 2 to This range corresponds to the reference input processing starting from the second subsystem in the backstepping method. Since the first subsystem has no pre-set virtual control quantity, no filtering operation is required. Among them, The filter time constant is a key parameter for adjusting the filtering effect and dynamic response. It is a positive scalar value, and usually a small value is selected according to the dynamic characteristics of the system. This ensures that the filtered signal can quickly track the original virtual control quantity, while effectively suppressing high-frequency noise and control jitter. The time derivative of the filtered signal reflects the rate of change of the filtered signal over time. It is solved by numerical differentiation of the filtered signal. The central difference method is often used to balance the calculation accuracy and real-time performance, and to avoid introducing additional lag in the derivative solution. For the current control cycle, the [number]th The filtered signal, as the first... The reference input of each subsystem has smoothing properties that can reduce the risk of subsystem state oscillations. For the first control cycle A virtual control quantity is directly called from the system cache to ensure that the filtered input is synchronized with the control cycle and to guarantee the timeliness of the reference input. The essence of the filtering process is to dynamically correct the virtual control quantity through a first-order inertial element, so that the output filtered signal retains the core control logic of the original virtual control quantity and has better dynamic smoothness, laying the foundation for the accurate construction of error variables.

[0103] S22. If the current control cycle is a non-attack time, the error variable vector is obtained using the following formula:

[0104]

[0105] in, This is the first error variable; This is the first state estimate in the state estimate vector for the current control cycle. The estimated value of the system output signal for the current control cycle; The first error variable vector One error variable; The state estimate vector of the current control cycle is the first... One state estimate; This is the filtered signal.

[0106] Indicatively, during non-attack times, the construction of the error variable vector is based on the direct relationship between the system state and the reference input. An explicit coordinate transformation logic establishes the error quantification relationship, providing a clear feedback basis for control law design. The formula for constructing the error variable falls into two categories: when... When, the expression for the error variable is: .in As the first error variable, it directly reflects the deviation at the system output level; The estimated value of the system output signal for the current control cycle is the approximation result of the state estimation stage on the actual system output. The first state estimate is the state estimate vector for the current control cycle. Since the system output is directly represented by the first state variable, the three are equivalent. This design simplifies the mapping logic between output error and state error, and a feedback loop at the output level can be established without additional transformation.

[0107] when Values ​​range from 2 to When, the expression for the error variable is: .in The first error variable vector The error variable directly reflects the first error variable. The degree of deviation between the state of each subsystem and the reference input; The first value of the current control cycle state estimate vector The nth state estimate is the observer's estimate of the nth state. The tracking results of the system status; For the first The filtered signal, as the first... The smoothing characteristics of the reference inputs for each subsystem ensure more stable changes in error variables, avoiding misjudgments caused by reference input jitter. These two types of error variables together constitute a complete error vector, comprehensively covering the deviation information between the system output and the states of each subsystem, providing complete data support for subsequent error feedback correction of the control law.

[0108] S23. If the current control cycle is the attack time, the error variable vector is obtained using the following formula:

[0109]

[0110] in, The first error variable vector One error variable; The first error variable vector before the attack One error variable; This is a false injection signal used in the current attack on the sensor side; This is a false injection signal for the current attack on the controller side; This is the moment of attack.

[0111] As an illustration, during an attack, the injection of false pulse information causes a sudden change in the system state. The error variable needs to be quickly responded to by directly superimposing the false injection signal to ensure that the error feedback can accurately capture the deviation caused by the attack. The formula for constructing the error variable is divided into two cases based on the system state dimension. Values ​​range from 1 to When, the expression for the error variable is: .in The first error variable vector of the current control cycle One error variable, namely the error update value after the attack; The first error variable vector before the attack Each error variable, i.e., the output value of the error variable from the previous control cycle, is retrieved from the cache to ensure the continuity of the state before the attack. The spurious injection signal used in the current attack on the sensor side is false data injected into the sensor communication link by the attacker. Its amplitude satisfies bounded constraints. By directly superimposing this signal, the error variable synchronously reflects the impact of the sensor-side attack on the system state deviation. When, the expression for the error variable is: .in The first error variable vector One error variable; The first before the attack One error variable; Injecting false signals to the sensor side; The spurious injection signal used in the current attack on the controller side is fake data injected into the controller communication link by the attacker, and it also satisfies the amplitude bounded constraint. Because of the... Each state is directly driven by the control input, and the impact of attacks on both the sensor side and the controller side must be considered simultaneously. Therefore, two types of spurious injection signals are superimposed to ensure that the error variable can fully capture the state deviation caused by the dual attacks. The attack time is a discrete time point in the preset attack model, ensuring that the update of error variables is completely synchronized with the attack time, avoiding delayed feedback of the attack's impact.

[0112] In one embodiment, based on the state estimate vector of the current control cycle, an adaptive law is used to obtain the fuzzy weight vector estimate of the current control cycle, including:

[0113] S31. Calculate the corresponding fuzzy basis functions based on each state estimate in the current control cycle's state estimate vector to obtain the fuzzy basis function vector.

[0114] Indicatively, for each state estimate in the current control cycle state estimate vector, a corresponding fuzzy membership function is constructed, and then fuzzy basis functions are obtained through normalization. The fuzzy membership function adopts a Gaussian form, and its expression is as follows: ,in The index of the corresponding state estimate. The index of the fuzzy rule, with a value ranging from 1 to... , The number of fuzzy rules is given. The center and width parameters of the Gaussian membership function are determined through offline optimization. The optimization objective is to ensure that the membership function uniformly covers the range of state estimates, guaranteeing a comprehensive approximation of unknown nonlinear dynamics. Each state estimate corresponds to... After normalization, the fuzzy membership functions are obtained The normalization process involves dividing each membership function by the sum of all membership functions, ensuring that the values ​​of the fuzzy basis functions are between 0 and 1, and that the sum of all fuzzy basis functions is 1, thus satisfying the basic characteristics of a fuzzy logic system.

[0115] Specifically, the first value in the current control cycle state estimate vector is... State estimates structure A Gaussian membership function is used. Gaussian membership functions are characterized by good smoothness and high approximation accuracy, and their form can effectively capture the local characteristics of state variables. The center parameter and width parameter of the membership function are determined through offline optimization. The optimization objective is to make... The membership functions are evenly distributed across the possible range of the state estimates, ensuring that effective feature representations can be provided for state variables of different magnitudes. For each state estimate... The membership functions are normalized. The core of the normalization operation is to divide all membership function values ​​corresponding to a single state estimate by the sum of those membership function values, so that each normalized fuzzy basis function value falls between 0 and 1, and the sum of all fuzzy basis functions corresponding to the same state is 1. This process ensures the probabilistic meaning of the fuzzy basis functions, allowing the subsequent product of weights and basis functions to reasonably reflect the contribution of different fuzzy rules to nonlinear dynamics. Each state estimate is obtained through the above process. dimensional fuzzy basis function subvectors, all The fuzzy basis function subvectors corresponding to each state are combined sequentially to form a dimension of . The fuzzy basis function vector is generated. This vector aggregates the nonlinear characteristic information of all states, providing the basic data for the subsequent construction of the fuzzy basis function matrix, ensuring that the fuzzy logic system can comprehensively approximate the unknown dynamic by integrating the characteristics of all states.

[0116] S32. Construct a diagonal fuzzy basis function matrix based on the fuzzy basis function vectors; the expression for the fuzzy basis function matrix is: ;in, The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate.

[0117] Indicatively, the core of constructing the fuzzy basis function matrix is ​​to transform the dispersed fuzzy basis function vectors into a diagonal form that facilitates matrix operations, providing structural support for matrix multiplication operations in adaptive laws. The expression for the matrix is: .in, For the current control cycle number The fuzzy basis function subvectors corresponding to each state estimate have dimension . its transpose for The row vectors. During construction, the properties of diagonal matrices (diag) are used to represent the row vectors corresponding to each state. The row vectors are used as the diagonal elements of the matrix, and all elements in the off-diagonal positions are set to 0, forming a matrix with dimension 1. A square matrix. The dimension of this matrix is ​​determined by the system order. Dimension of fuzzy basis function subvectors corresponding to a single state Joint decision, The dimensionality of the square matrix ensures that it matches the dimension of the fuzzy weight vector estimate, satisfying the requirements of matrix multiplication. The purpose of constructing the diagonal matrix is ​​to... The outputs of independent fuzzy logic systems are aggregated, ensuring that the fuzzy basis function for each state interacts only with its own weight vector. This avoids feature interference between different states, simplifies the calculation process of the adaptive law, and improves computational efficiency within the control cycle. The diagonal structure of the matrix preserves the independence of the fuzzy features of each state while achieving unified management of feature information, providing an efficient computational platform for subsequent adaptive weight updates.

[0118] S33. Based on the adaptive law, the time derivative of the fuzzy weight vector estimate for the current control cycle is calculated from the fuzzy basis function matrix and the state estimation error. Then, the fuzzy weight vector estimate for the current control cycle is obtained through numerical integration from the time derivative of the fuzzy weight vector estimate for the current control cycle. The adaptive law is: ,in, The time derivative of the fuzzy weight vector estimate for the current control cycle. The adaptive law gain matrix, For state estimation error, It is a positive definite matrix; The fuzzy basis function matrix; This is the weight attenuation coefficient; This is the estimated value of the fuzzy weight vector from the previous control cycle. For the control period; the expression for the fuzzy weight vector estimate of the current control period is: .

[0119] The adaptive law is designed based on Lyapunov stability theory. By constructing a suitable adaptive update rule, the weight estimation error converges to a bounded region near zero, while ensuring the stability of the closed-loop system. The adaptive law prioritizes system stability, driving weight updates through state estimation error and introducing a decay term to prevent weight divergence. Finally, the fuzzy weight vector estimate for the current control cycle is obtained through numerical integration. The expression for the adaptive law is as follows: .in, The time derivative of the estimated fuzzy weight vector for the current control cycle directly determines the rate and direction of weight updates. The adaptive law gain matrix is ​​a positive definite symmetric matrix. The values ​​of its diagonal elements are determined according to the system's convergence speed requirements. The larger the value, the more sensitive the weight is to the error response. The optimal value needs to be determined by balancing the response speed and stability. The state estimation error vector consists of the difference between the true state of each system and the corresponding estimated value. It is the core driving signal for weight update and directly reflects the accuracy of fuzzy approximation and state estimation. It is a positive definite matrix, derived from the solution of the Lyapunov equation in the state observer design. Its function is to transform the state estimation error into a weight adjustment signal that meets the stability requirements, ensuring the cooperative stability of the adaptive law and the observer. It provides nonlinear dynamic feature information for the fuzzy basis function matrix of the current control cycle, enabling the weight update to specifically compensate for nonlinear effects; , is the weight decay coefficient, a positive scalar, used to suppress the excessive growth of the weight estimate, avoid system oscillation due to weight divergence, and improve system robustness; This is an estimate of the fuzzy weight vector from the previous control cycle, ensuring the continuity of weight updates and avoiding sudden changes in weights caused by cycle switching.

[0120] Furthermore, the fuzzy weight vector estimate for the current control cycle is solved using the Euler numerical integration method, with the integration formula being: .in, This is the final fuzzy weight vector estimate for the current period; To maintain consistency with the overall control cycle and ensure synchronization between the integral step size and data update frequency, this integration method multiplies the weight's time derivative with the control cycle to obtain the weight increment, which is then superimposed on the previous week's option value to achieve discretized weight updates. The entire process requires no complex calculations and can be completed rapidly within the control cycle. This ensures both the real-time nature of weight updates and the convergence and boundedness of weights through error-driven and decay constraints, providing reliable support for the continuous and accurate approximation of unknown nonlinear dynamics in fuzzy logic systems.

[0121] It should be understood that although the steps in the flowcharts of the embodiments described above are shown sequentially according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some steps in the flowcharts of the embodiments described above may include multiple steps or multiple stages. These steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential, but can be performed alternately or in turn with other steps or at least some of the steps or stages of other steps.

[0122] Based on the same inventive concept, this application also provides an elastic control system for a high-order nonlinear system under a pulse false information injection attack, used to implement the elastic control method for a high-order nonlinear system under a pulse false information injection attack as described above. The solution provided by this system is similar to the implementation scheme described in the above method. Therefore, the specific limitations of one or more embodiments of the elastic control system for a high-order nonlinear system under a pulse false information injection attack provided below can be found in the limitations of the elastic control method for a high-order nonlinear system under a pulse false information injection attack described above, and will not be repeated here.

[0123] In one exemplary embodiment, such as Figure 3 As shown, an elastic control system for a high-order nonlinear system under pulse false information injection attack is provided, comprising:

[0124] The system monitoring module 301 is used to acquire the system output signal of the current control cycle on the sensor side and the control input signal of the current control cycle on the controller side, and to acquire the state estimation vector of the previous control cycle and the fuzzy weight vector estimation value of the previous control cycle on the nonlinear state observer side, based on the current control cycle.

[0125] The state estimation module 302 is used to input the system output signal of the current control cycle, the control input signal of the current control cycle, the state estimation vector of the previous control cycle, and the fuzzy weight vector estimation value of the previous control cycle into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation vector of the current control cycle; the nonlinear state observer based on the fuzzy logic system continuously updates the state estimation value through the instantaneous state jump caused by the pulse false information injection attack, and dynamically stabilizes the state estimation error based on the observer gain matrix;

[0126] The elastic control module 303 is used to use a preset controller based on a fuzzy logic system to perform backstep recursive calculation of the virtual control law and the actual control law by combining the state estimation value vector of the current control cycle and the fuzzy weight vector estimation value of the previous control cycle, so as to obtain the elastic control input signal.

[0127] The weight module 304 is used to obtain the fuzzy weight vector estimate of the current control cycle based on the state estimate vector of the current control cycle through an adaptive law.

[0128] In one embodiment, the state estimation module 302 is further configured to:

[0129] Based on the preset attack model, determine whether the current control period is a non-attack time or an attack time;

[0130] If the current control period is a non-attack time, the state estimate vector for the current control period can be obtained using the following formula:

[0131]

[0132]

[0133]

[0134]

[0135] in, This is the vector of state estimates for the current control cycle; The state estimate vector of the current control cycle is the first... One state estimate; The state estimate vector of the current control cycle is the first... The instantaneous rate of change of each state estimate; The first state estimate vector of the previous control cycle One state estimate; The feedback gain parameter is the nonlinear state observer based on a fuzzy logic system. This is the system output signal for the current control cycle; This is the estimated value of the system output signal from the previous control cycle; This is the estimated value of the fuzzy weight vector from the previous control cycle; The first state estimate vector of the previous control cycle The fuzzy basis functions corresponding to each state estimate; This is the control input signal for the current control cycle;

[0136] If the current control period coincides with an attack, the state estimate vector for the current control period can be obtained using the following formula:

[0137]

[0138]

[0139] in, The state estimate vector of the current control cycle is the first... One state estimate; The first state estimate vector of the previous control cycle One state estimate; This is a false injection signal used in the current attack on the sensor side; This is a false injection signal for the current attack on the controller side; This is the moment of attack.

[0140] In one embodiment, the elastic control module 303 is further configured to:

[0141] The error variable vector is obtained by performing inverse step coordinate transformation and first-order low-pass filtering on the state estimate vector of the current control cycle.

[0142] Based on the first virtual control law, the first virtual control quantity is calculated using the error variable vector, the state estimation error, and the estimated value of the fuzzy weight vector from the previous control cycle; the first virtual control law is... ,in, The first error variable in the error variable vector. The error is the estimation error for the first state. This is the estimated value of the first fuzzy weight vector from the previous control cycle. Let be the fuzzy basis function corresponding to the first state estimate of the state estimate vector in the current control cycle. The virtual control input feedback gain parameter; The feedback gain parameter is the nonlinear state observer based on a fuzzy logic system.

[0143] Based on the The virtual control law is calculated using the error variable vector, the state estimation error, and the estimated fuzzy weight vector from the previous control cycle. Virtual control quantity; the first Virtual control law is ,in, , The first error variable vector One error variable, For the first control cycle A fuzzy weight vector estimate The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate For the first The time derivative of the filtered signal;

[0144] Based on the actual control law, the elastic control input signal is calculated using the error variable vector, the state estimation error, and the estimated fuzzy weight vector from the previous control cycle; the actual control law is: ,in, The first error variable vector One error variable, For the first control cycle A fuzzy weight vector estimate The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate For the first The time derivative of the filtered signal.

[0145] In one embodiment, an error variable module is also included, for:

[0146] The virtual control input from the previous control cycle is input into a first-order low-pass filter to obtain the filtered signal and its corresponding time derivative; the formula for the first-order low-pass filter is: ,in , The filter time constant; The time derivative of the filtered signal; For filtered signals; For the first control cycle One virtual control quantity;

[0147] If the current control cycle is a non-attack time, the error variable vector can be obtained using the following formula:

[0148]

[0149] in, This is the first error variable; This is the first state estimate in the state estimate vector for the current control cycle. The estimated value of the system output signal for the current control cycle; The first error variable vector One error variable; The state estimate vector of the current control cycle is the first... One state estimate; For filtered signals;

[0150] If the current control period is the attack time, the error variable vector can be obtained using the following formula:

[0151]

[0152] in, The first error variable vector One error variable; The first error variable vector before the attack One error variable; This is a false injection signal used in the current attack on the sensor side; This is a false injection signal for the current attack on the controller side; This is the moment of attack.

[0153] In one embodiment, the weighting module 304 is further configured to:

[0154] Calculate the corresponding fuzzy basis function based on each state estimate in the current control cycle's state estimate vector to obtain the fuzzy basis function vector;

[0155] Construct a diagonal fuzzy basis function matrix based on the fuzzy basis function vectors; the expression for the fuzzy basis function matrix is: ;in, The state estimate vector of the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate;

[0156] Based on the adaptive law, the time derivative of the fuzzy weight vector estimate for the current control cycle is calculated from the fuzzy basis function matrix and the state estimation error. Then, the fuzzy weight vector estimate for the current control cycle is obtained through numerical integration from the time derivative of the fuzzy weight vector estimate for the current control cycle. The adaptive law is: ,in, The time derivative of the fuzzy weight vector estimate for the current control cycle. The adaptive law gain matrix, For state estimation error, It is a positive definite matrix; The fuzzy basis function matrix; This is the weight attenuation coefficient; This is the estimated value of the fuzzy weight vector from the previous control cycle. For the control period; the expression for the fuzzy weight vector estimate of the current control period is: .

[0157] In one embodiment, a computer device is provided, including a memory and a processor, the memory storing a computer program, the processor executing the computer program to implement the steps in the above method embodiments.

[0158] In one embodiment, a computer-readable storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the steps in the above method embodiments.

[0159] For the device embodiments, since they basically correspond to the method embodiments, the relevant parts can be referred to in the description of the method embodiments. The device embodiments described above are merely illustrative. The components described as separate parts may or may not be physically separate, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this disclosure according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0160] The above-described embodiments are merely illustrative of several implementation methods of the embodiments of this application, and their descriptions are relatively specific and detailed. However, they should not be construed as limiting the scope of the patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the embodiments of this application, and these modifications and improvements all fall within the protection scope of the embodiments of this application.

Claims

1. A method for resilient control of high-order nonlinear systems under pulse false information injection attack, characterized in that, The method comprises: Based on the current control period, the system output signal of the current control period on the sensor side, the control input signal of the current control period on the controller side, and the state estimation value vector of the last control period and the fuzzy weight vector estimation value of the last control period on the nonlinear state observer side are obtained; The system output signal of the current control period, the control input signal of the current control period, the state estimation value vector of the last control period, and the fuzzy weight vector estimation value of the last control period are input into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation value vector of the current control period; the nonlinear state observer based on the fuzzy logic system continuously updates the dynamic state estimation value through the state transient jump caused by the pulse false information injection attack, and dynamically stabilizes the state estimation error based on the observer gain matrix; A preset controller based on a fuzzy logic system is used to combine the state estimation value vector of the current control period and the fuzzy weight vector estimation value of the last control period to perform backstepping recursive calculation of the virtual control law and the actual control law to obtain the elastic control input signal; Based on the state estimation value vector of the current control period, the fuzzy weight vector estimation value of the current control period is obtained through an adaptive law.

2. The method of claim 1, wherein, The system output signal of the current control period, the control input signal of the current control period, the state estimation value vector of the last control period, and the fuzzy weight vector estimation value of the last control period are input into a preset nonlinear state observer based on a fuzzy logic system to obtain the state estimation value vector of the current control period, comprising: Based on the preset attack model, it is determined whether the current control period is a non-attack time or an attack time; If the current control period is a non-attack time, the state estimation value vector of the current control period is obtained through the following formula: ； in, This is the vector of state estimates for the current control cycle; The first state estimate vector of the current control cycle One state estimate; The first state estimate vector of the current control cycle The instantaneous rate of change of each state estimate; The first state estimate vector of the previous control cycle One state estimate; The feedback gain parameter is the nonlinear state observer based on a fuzzy logic system. This is the system output signal for the current control cycle; This is the estimated value of the system output signal from the previous control cycle; This is the estimated value of the fuzzy weight vector from the previous control cycle; The first state estimate vector of the previous control cycle The fuzzy basis functions corresponding to each state estimate; This is the control input signal for the current control cycle; If the current control period is an attack time, the state estimation value vector of the current control period is obtained through the following formula: ； wherein is the i-th state estimate of the state estimate vector of the current control period; is the i-th state estimate of the state estimate vector of the previous control period; is the i-th state estimate of the state estimate vector of the previous control period; is the i-th state estimate of the state estimate vector of the previous control period; is a false injection signal of a current attack on the sensor side; is a false injection signal of a current attack on the controller side; is an attack time instant.

3. The method of claim 2, wherein, The preset controller based on the fuzzy logic system is used to combine the state estimation value vector of the current control period and the fuzzy weight vector estimation value of the last control period to perform backstepping recursive calculation of the virtual control law and the actual control law to obtain the elastic control input signal, comprising: The state estimation value vector of the current control period is subjected to backstepping method coordinate transformation and first-order low-pass filtering to obtain an error variable vector; a first virtual control quantity is calculated based on the first virtual control law, the error variable vector, the state estimation error and the fuzzy weight vector estimation value of the last control cycle; the first virtual control law is wherein, is a first error variable of the error variable vector, is a first state estimation error, is a first fuzzy weight vector estimation value of the last control cycle, is a fuzzy base function corresponding to a first state estimation value of the state estimation value vector of the current control cycle, is a virtual control quantity feedback gain parameter; is a feedback gain parameter of the nonlinear state observer based on the fuzzy logic system. Based on the first virtual control law, the first virtual control variable is calculated by the error variable vector, the state estimation error and the fuzzy weight vector estimation value of the last control period The virtual control law is Wherein, , The first error variable of the error variable vector is The first fuzzy weight vector estimation value of the last control period is The fuzzy base function corresponding to the first state estimation value of the state estimation value vector of the current control period is The time derivative of the first filtering signal is Based on the actual control law, the elastic control input signal is calculated using the error variable vector, the state estimation error, and the estimated value of the fuzzy weight vector from the previous control cycle; the actual control law is... ,in, The first of the error variable vectors One error variable, For the first control cycle The estimated values ​​of the fuzzy weight vectors. The first state estimate vector of the current control cycle The fuzzy basis functions corresponding to each state estimate For the first The time derivative of the filtered signal.

4. The method of claim 3, wherein, The state estimation value vector of the current control period is subjected to backstepping method coordinate transformation and first-order low-pass filtering to obtain an error variable vector, comprising: The virtual control quantity of the last control period is input into a first-order low-pass filter to obtain a filtered signal and a corresponding time derivative; the first-order low-pass filter corresponds to a formula , wherein , is a filter time constant; is a time derivative of the filtered signal; is the filtered signal; is the jthvirtual control quantity of the last control period; and j is an integer. If the current control period is a non-attack time, the error variable vector is obtained through the following formula: ； wherein is a first error variable; is a first state estimate in the state estimate vector for the current control period; is a system output signal estimate for the current control period; is a first error variable in the error variable vector; is a first state estimate in the state estimate vector for the current control period; is a first state estimate in the state estimate vector for the current control period; is a first state estimate in the state estimate vector for the current control period; is a filter signal; If the current control period is an attack time, the error variable vector is obtained through the following formula: ； wherein is the i-th error variable of the error variable vector; is the i-th error variable of the error variable vector; is the i-th error variable of the error variable vector; is the i-th error variable of the error variable vector; is a false injection signal of a current attack on the sensor side; is a false injection signal of a current attack on the controller side; is the attack time instant.

5. The method of claim 1, wherein, The fuzzy weight vector estimation value of the current control period is obtained through an adaptive law based on the state estimation value vector of the current control period, comprising: A corresponding fuzzy base function is calculated respectively according to each state estimation value in the state estimation value vector of the current control period, and a fuzzy base function vector is obtained; Construct a diagonal fuzzy basis function matrix based on the fuzzy basis function vectors; the expression for the fuzzy basis function matrix is: ;in, The state estimate vector for the current control cycle is the first... The fuzzy basis functions corresponding to each state estimate; The time derivative of the fuzzy weight vector estimation value of the current control period is calculated according to the fuzzy base function matrix and the state estimation error based on an adaptive law, and the fuzzy weight vector estimation of the current control period is obtained by numerical integration according to the time derivative of the fuzzy weight vector estimation value of the current control period; the adaptive law is wherein, is the time derivative of the fuzzy weight vector estimation value of the current control period, is an adaptive law gain matrix, is a state estimation error, is a positive definite matrix; is a fuzzy base function matrix; is a weight decay coefficient; is the fuzzy weight vector estimation value of the previous control period, is a control period; and an expression of the fuzzy weight vector estimation of the current control period is .

6. A resilient control system for a high-order nonlinear system under pulse false information injection attack, characterized in that, The system comprises: The system monitoring module is configured to, based on a current control period, acquire a system output signal of a sensor side in the current control period, a control input signal of a controller side in the current control period, and a state estimation value vector of a previous control period and a fuzzy weight vector estimation value of the previous control period of a nonlinear state observer side; The state estimation module is configured to input the system output signal of the current control period, the control input signal of the current control period, the state estimation value vector of the previous control period, and the fuzzy weight vector estimation value of the previous control period into a preset nonlinear state observer based on a fuzzy logic system, to obtain the state estimation value vector of the current control period; the nonlinear state observer based on the fuzzy logic system continuously dynamically updates a state estimation value by a state transient jump caused by a pulse false information injection attack, and dynamically stabilizes an estimation error based on an observer gain matrix; The elastic control module is configured to utilize a preset controller based on a fuzzy logic system to combine the state estimation value vector of the current control period and the fuzzy weight vector estimation value of the previous control period to perform backstepping recursion to calculate a virtual control law and an actual control law, and to obtain an elastic control input signal; The weight module is configured to, based on the state estimation value vector of the current control period, obtain the fuzzy weight vector estimation value of the current control period by an adaptive law. 7.A computer device, comprising a memory and a processor, wherein the memory stores a computer program, and the computer device is configured to perform the method according to any one of claims 1-6 when the computer program is executed by the processor. The processor executes the computer program to implement the method in any one of claims 1 to 5.

8. A computer-readable storage medium having stored thereon a computer program, characterized in that, The computer program is executed by the processor to implement the method in any one of claims 1 to 5.

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