Recursive filtering method for state-saturated systems over high-speed networks under deception attacks

Abstract

The application provides a recursive filtering method for a state-saturated system based on a high-speed network under the influence of a deception attack. The method specifically comprises the following steps: constructing a dynamic model of a state-saturated system to be studied, wherein the dynamic model comprises a system state and a measurement output model, and conditions that functions in the dynamic model satisfy are set; constructing a new measurement model based on a high-speed network and a random deception attack according to the measurement output model, and setting constraints that the deception attack satisfies; designing a state-saturated recursive filter according to the dynamic model and the new measurement model; obtaining a filtering error dynamic equation according to the system dynamic model and the filter model, and calculating a filtering error covariance upper bound; obtaining an optimal filter gain by minimizing the filtering error covariance upper bound; and verifying the effectiveness of the recursive filtering method through a numerical simulation example. The simulation experiment verifies the effectiveness of the proposed recursive filtering method, and the method is suitable for online calculation and exhibits good real-time performance.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, and in particular relates to a recursive filtering method for state saturation systems based on high-speed networks under the influence of deception attacks. Background Technology

[0002] Filtering, a fundamental problem in signal processing and control, has long attracted widespread attention. Its core idea is to estimate the internal state of a system based on a system model and measured output. It is currently widely applied in numerous fields such as petrochemicals, smart grids, and process monitoring. As a relatively effective and commonly used filtering method, recursive filtering has attracted considerable attention. Typical recursive filtering methods include conventional Kalman filtering, extended Kalman filtering, and unscented Kalman filtering, all of which aim to minimize the covariance of the filtering error.

[0003] To date, there has been considerable research on recursive filtering; however, the vast majority of studies consider the scenario where the system state has no constraints. This seems impractical due to physical limitations. For example, a ship's attitude is constrained by the environment and its mechanical structure, leading to the so-called state saturation problem. For state-saturated systems, all system states are constrained within a defined range, which can result in system performance degradation or even instability. Furthermore, with the rapid development of network communication technology, networked systems have attracted considerable attention. However, the openness of shared networks makes data security difficult to guarantee. Major cybersecurity incidents have occurred frequently in recent years, with attackers often intercepting and tampering with transmitted data, severely impacting data security and system stability. Therefore, researching recursive filtering methods for saturated systems is crucial in the context of cyberattacks.

[0004] In addition, most of the systems currently studied are based on the assumption that the sampling rate of the system devices, the sampling rate of the measurement output, and the transmission rate of the communication network are all the same. However, in practical engineering, there are often multi-rate systems where the sampling periods of the state and measurement output can differ due to the different physical characteristics of the system devices and multiple sensors. For example, the sampling rate of the 13-b LM95172 temperature sensor is 0.37 kb / s, the maximum sampling rate of the 12-b accelerometer DT138 is 1.2 kb / s, while the data transmission rate of the process fieldbus process automation network is 31.25 kb / s, which will lead to oversampling. In this case, making full use of the transmitted information and optimizing the filter performance to the greatest extent is the ultimate goal of this invention, which has important theoretical and practical significance. Summary of the Invention

[0005] The purpose of this invention is to address the problems in existing technologies by proposing a recursive filtering method for state-saturated systems in high-speed networks under the influence of deception attacks. This method is applicable to filtering problems in ubiquitous networked systems and has general applicability.

[0006] This invention is achieved through the following technical solution: This invention proposes a recursive filtering method for a state-saturated system based on a high-speed network under the influence of deception attacks. Specifically, the method is as follows:

[0007] Step 1: Construct a dynamic model of the state-saturated system to be studied. The dynamic model includes the system state and measurement output model, and sets the conditions that the functions in the dynamic model must satisfy.

[0008] Step 2: Based on the measurement output model in Step 1, construct a new measurement model based on high-speed networks and random spoofing attacks, and set the constraints that the spoofing attacks must satisfy;

[0009] Step 3: Based on the dynamic model from Step 1 and the new measurement model from Step 2, design a state-saturated recursive filter;

[0010] Step 4: Based on the system dynamic model in Step 1 and the filter model in Step 3, obtain the dynamic equation of the filtering error and calculate the upper bound of the filtering error covariance.

[0011] Step 5: Obtain the optimal filter gain by minimizing the upper bound of the filter error covariance;

[0012] Step 6: Verify the effectiveness of the recursive filtering method through numerical simulation examples.

[0013] Furthermore, step one specifically includes:

[0014] Constructing a dynamic model of the system:

[0015]

[0016] In the formula, This indicates the k-th sampling time. Indicates the system status. This represents a zero-mean process with noise and a covariance of . For the system's measurement output, This represents measurement noise with zero mean and covariance of . The initial state z(ζ0) of the system is a random variable with a mean of 1 / 2. The covariance is Z(ζ0); ε(ζ k ), The system matrix is ​​of suitable dimension; σ(·) is the saturation function for a vector. The ιth element u ιThe saturation function satisfies the following condition:

[0017]

[0018] In the formula, sign(·) is the sign function, κ ι >0 indicates the saturation level of the ι-th element.

[0019] Furthermore, step two specifically involves:

[0020] Within a certain sampling period, the attack signal injected by the attacker at the j-th transmission time is as follows:

[0021]

[0022] in It is an unknown bounded signal and satisfies m > 0 is a given constant. And p≥1;

[0023] Considering the randomness of attack phenomena, the actual measurement signal received by the filter This can be represented as the following model:

[0024]

[0025] In the formula Let be a random variable that follows a Bernoulli distribution and satisfies:

[0026]

[0027] when When the time is specified, it indicates that the attack has not occurred, with a probability of 1. when At that time, the attack occurs with a probability of . is a given constant.

[0028] Furthermore, step three specifically includes:

[0029] First, in order to better estimate the system state, a dynamic model of the system within a sampling period is constructed:

[0030]

[0031] In the formula,

[0032]

[0033]

[0034]

[0035]

[0036] Secondly, construct a state-saturated recursive filter of the following form:

[0037]

[0038] in Indicates system state The estimate, For the designed filter gain, This represents the initial state of the filter.

[0039] Furthermore, step four specifically involves:

[0040] First of all, let The dynamic equation for the filtering error can be obtained as follows:

[0041]

[0042] Secondly, based on mathematical induction, matrix operations, the definition of covariance, and the properties of inequalities, the upper bound of the filter error covariance is obtained:

[0043]

[0044] In the formula Given a positive scalar, the other relevant parameters are:

[0045]

[0046]

[0047]

[0048] Furthermore, step five specifically includes:

[0049] Using the completing the square method, for Λ(ζ) in step four k+1 The formula was prepared, and the results are as follows:

[0050]

[0051] When the filter gain is When the upper bound of the filter error covariance is minimized, it is the optimal filter, and the minimum upper bound is:

[0052]

[0053] In the formula,

[0054]

[0055]

[0056] Furthermore, step six specifically includes:

[0057] Consider a time-varying linear system with state saturation levels κ1 = 1.2 and κ2 = 0.8, with the following parameters:

[0058]

[0059]

[0060] Setting process noise ρ(ζ) k The covariance of ) is Measurement noise v(ζ) k Covariance is Mean of the initial state of the system random variable The expectation is and constants

[0061] To verify the performance of the designed filter, a mean square error function is introduced:

[0062]

[0063] Based on the above parameters, and according to the optimal filter and the expression for the filter error covariance obtained in step five, the minimum upper bound of the error covariance and the filter gain can be calculated.

[0064] This invention also proposes a recursive filtering system for a state-saturated system based on a high-speed network under the influence of deception attacks. Specifically, the system is as follows:

[0065] Construction module: Constructs a dynamic model of the state-saturated system to be studied. The dynamic model includes the system state and measurement output model, and sets the conditions that the functions in the dynamic model must satisfy.

[0066] Setting Module: Based on the measurement output model in the construction module, construct a new measurement model based on high-speed networks and random spoofing attacks, and set the constraints that the spoofing attacks must satisfy;

[0067] Design module: Design a state-saturated recursive filter based on the dynamic model and the new measurement model;

[0068] Filtering module: Based on the system dynamic model and the filter model, the dynamic equation of the filtering error is obtained, and the upper bound of the filtering error covariance is calculated;

[0069] Minimization module: The optimal filter gain is obtained by minimizing the upper bound of the filter error covariance;

[0070] Verification module: Verify the effectiveness of the recursive filtering method through numerical simulation examples.

[0071] The present invention also proposes an electronic 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 recursive filtering method for a state saturated system based on a high-speed network under the influence of the deception attack.

[0072] The present invention also proposes a computer-readable storage medium for storing computer instructions, which, when executed by a processor, implement the steps of the recursive filtering method for a state-saturated system based on a high-speed network under the influence of the deception attack.

[0073] This invention provides a novel solution to the recursive filtering problem in state-saturated systems based on high-speed networks under the influence of deception attacks. Compared to existing inventions, its innovations are as follows: 1) It considers the problem of multiple transmissions of the same data during communication between the sensor and the filter, achieving full utilization of information; 2) It establishes a unified recursive filtering framework that can simultaneously tolerate the negative impacts of state saturation and deception attacks; 3) Simulation experiments verify the effectiveness of the proposed recursive filtering method, and the method is applicable to online computation, demonstrating good real-time performance. Attached Figure Description

[0074] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0075] Figure 1 This is a schematic diagram of the core steps of the method described in this invention;

[0076] Figure 2 This diagram illustrates the moment when the measurement signal of an embodiment of the present invention is attacked.

[0077] Figure 3 The filter states of an embodiment of the present invention are shown. Tracking the real state z1(ζ) k A schematic diagram of the curve;

[0078] Figure 4 The filter states of an embodiment of the present invention are shown. Tracking the real state z2(ζ) k A schematic diagram of the curve;

[0079] Figure 5 A schematic diagram of the mean square error and minimum upper bound trajectory of an embodiment of the present invention is shown. Detailed Implementation

[0080] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0081] Figure 1 The flowchart of the recursive filtering method for a state-saturated system based on a high-speed network under the influence of a spoofing attack, provided by the present invention, is shown. The recursive filtering method for a state-saturated system based on a high-speed network under the influence of a spoofing attack specifically includes the following steps:

[0082] Step 1: Construct a dynamic model of the state-saturated system to be studied. The dynamic model includes the system state and measurement output model, and sets the conditions that the functions in the dynamic model must satisfy.

[0083] Step 2: Based on the measurement output model in Step 1, construct a new measurement model based on high-speed networks and random spoofing attacks, and set the constraints that the spoofing attacks must satisfy;

[0084] Step 3: Based on the dynamic model from Step 1 and the new measurement model from Step 2, design a state-saturated recursive filter;

[0085] Step 4: Based on the system dynamic model in Step 1 and the filter model in Step 3, obtain the dynamic equation of the filtering error and calculate the upper bound of the filtering error covariance.

[0086] Step 5: Obtain the optimal filter gain by minimizing the upper bound of the filter error covariance;

[0087] Step 6: Verify the effectiveness of the proposed recursive filtering method through numerical simulation examples.

[0088] Explanation of symbols used in the embodiments

[0089] Describe an n-dimensional Euclidean space. Let m represent the set of all n×m real matrices. Let I represent the set of positive integers, where P > 0 indicates that P is a real symmetric positive definite matrix, and I represents the identity matrix of appropriate dimension. Let diag{A1, A2, ..., A...} n} represents a diagonal block consisting of matrices A1, A2, ..., A n A block diagonal matrix, where col{…} represents column vectors, [a ij ] l×l Let M represent an l×l dimensional matrix. T Let M denote the transpose of matrix M. -1Let M be the inverse matrix. Let X be the probability space to be established, where Prob is a probability measure that sums to 1. For real symmetric matrices X and Y, the symbol X≥Y (X>Y) indicates that XY is positive semi-definite (positive definite), and the symbol X≤Y (X<Y) indicates that XY is negative semi-definite (negative definite). ||x|| represents the Euclidean norm of vector x, and ||A||=(trace(A T A)) 1 / 2 Let be the norm of matrix A. Represents the expectation of x, symbol This is the Hadamard product, where min represents the minimum value operation and tr represents the trace operation. If the matrix dimension is not explicitly specified in the specification, it is assumed to be suitable for matrix algebra operations.

[0090] This embodiment studies the recursive filtering problem of networked systems with state saturation and random deception attacks based on high-speed communication networks.

[0091] Step one specifically involves: constructing a dynamic model of the state-saturated system under study. This dynamic model includes the system state and measurement output models, and defining the conditions that the functions in the dynamic model must satisfy. Specifically:

[0092]

[0093] In the formula, This indicates the k-th sampling time. Indicates the system status. This represents a zero-mean process with noise and a covariance of . For the system's measurement output, This represents measurement noise with zero mean and covariance of . The initial state z(ζ0) of the system is a random variable with a mean of 1 / 2. The covariance is Z(ζ0). The system matrix is ​​of suitable dimension. σ(·) is the saturation function for a vector. The ιth element u ι The saturation function satisfies the following condition:

[0094]

[0095] In the formula, sign(·) is the sign function, κ ι >0 indicates the saturation level of the ι-th element.

[0096] Step two specifically involves: based on the measurement output model from step one, constructing a new measurement model based on high-speed networks and random spoofing attacks, and setting constraints that the spoofing attacks must satisfy. Details are as follows:

[0097] Due to the introduction of high-speed networks, signals are transmitted multiple times in each sampling period, potentially leading to multiple network attacks. This invention aims to study common deception attacks in reality. Within a certain sampling period, the attack signal injected by the attacker at the j-th transmission moment is as follows:

[0098]

[0099] in It is an unknown bounded signal and satisfies m > 0 is a given constant. And p≥1.

[0100] Considering the randomness of attack phenomena, the actual measurement signal received by the filter This can be represented as the following model:

[0101]

[0102] In the formula Let be a random variable that follows a Bernoulli distribution and satisfies:

[0103]

[0104] when When the time is specified, it indicates that the attack has not occurred, with a probability of 1. when At that time, the attack occurs with a probability of . is a given constant.

[0105] Note: The methods used to obtain the above new measurement model include:

[0106] Assumption 1: The signal is transmitted periodically between the sensor and the filter, with a period h = h s / p.

[0107] This represents the s-th sampling period of the system.

[0108] Assumption 2: The first transmission time l0 is the same as the 0th sampling time ζ0, that is, l0 = ζ0.

[0109] Based on assumptions 1 and 2, we can obtain ζ k =ζ0+kh s =l0+kph=l kp That is, the p-th signal transmission occurs in the time interval (ζ). k-1 ,ζ k ].

[0110] Step three specifically involves designing a state-saturated recursive filter based on the state model from step one and the new measurement model from step two. The details are as follows:

[0111] First, in order to better estimate the system state, a dynamic model of the system within a sampling period is constructed:

[0112]

[0113] In the formula,

[0114]

[0115]

[0116]

[0117]

[0118] Secondly, construct a state-saturated recursive filter of the following form:

[0119]

[0120] in Indicates system state The estimate, For the designed filter gain, This represents the initial state of the filter.

[0121] Step four specifically involves: based on the system state model from step one and the filter model from step three, obtaining the dynamic equation for the filtering error, and calculating the upper bound of the filtering error covariance. Details are as follows:

[0122] First of all, let The dynamic equation for the filtering error can be obtained as follows:

[0123]

[0124] Secondly, based on mathematical induction, matrix operations, the definition of covariance, and the properties of inequalities, the upper bound of the filter error covariance is obtained:

[0125]

[0126] In the formula Given a positive scalar, the other relevant parameters are:

[0127]

[0128]

[0129]

[0130] Note: The methods used to obtain the upper bound of the above filtering error covariance include:

[0131] Lemma 1: For matrices function If satisfied So Solution With V k satisfy

[0132] Lemma 2: There must exist real numbers This makes the saturation function σ defined in formula (2) saturate. ι (·)satisfy

[0133] Lemma 3: For real matrices With diagonal random matrix The following results are true:

[0134]

[0135] Step five specifically involves minimizing the upper bound of the filter error covariance to obtain the optimal filter gain. The details are as follows:

[0136] Using the completing the square method, for Λ(ζ) in step four k+1 The formula was prepared, and the results are as follows:

[0137]

[0138] When the filter gain is When the upper bound of the filter error covariance is minimized, it is the optimal filter, and the minimum upper bound is:

[0139]

[0140] In the formula,

[0141]

[0142]

[0143] Step six specifically involves verifying the effectiveness of the proposed recursive filtering method through numerical simulation examples. The details are as follows:

[0144] By writing a Matlab program to plot simulation curves, the effectiveness of the recursive filtering method for state-saturated systems based on high-speed networks under the influence of deception attacks is demonstrated through simulation examples.

[0145] Consider a time-varying linear system with state saturation levels κ1 = 1.2 and κ2 = 0.8, with the following parameters:

[0146]

[0147]

[0148] Setting process noise ρ(ζ) k The covariance of ) is Measurement noise v(ζ) k Covariance is Mean of the initial state of the system random variable The expectation is and constants

[0149] To verify the performance of the designed filter, a mean square error function is introduced:

[0150]

[0151] Based on the above parameters, and according to the optimal filter and the expression for the filter error covariance obtained in step five, the minimum upper bound of the error covariance and the filter gain can be calculated. Figure 2 The moment when the measurement signal of an embodiment of the present invention was attacked is shown; Figure 3 The filter states of an embodiment of the present invention are shown. Tracking the real state z1(ζ) k The curve of ); Figure 4 The filter states of an embodiment of the present invention are shown. Tracking the real state z2(ζ) k The curve of ); Figure 5 The mean square error and minimum upper bound trajectory of an embodiment of the present invention are shown. Simulation results demonstrate that the filter design method of the present invention can effectively estimate the target state of the system.

[0152] This invention also proposes a recursive filtering system for a state-saturated system based on a high-speed network under the influence of deception attacks. Specifically, the system is as follows:

[0153] Construction module: Constructs a dynamic model of the state-saturated system to be studied. The dynamic model includes the system state and measurement output model, and sets the conditions that the functions in the dynamic model must satisfy.

[0154] Setting Module: Based on the measurement output model in the construction module, construct a new measurement model based on high-speed networks and random spoofing attacks, and set the constraints that the spoofing attacks must satisfy;

[0155] Design module: Design a state-saturated recursive filter based on the dynamic model and the new measurement model;

[0156] Filtering module: Based on the system dynamic model and the filter model, the dynamic equation of the filtering error is obtained, and the upper bound of the filtering error covariance is calculated;

[0157] Minimization module: The optimal filter gain is obtained by minimizing the upper bound of the filter error covariance;

[0158] Verification module: Verify the effectiveness of the recursive filtering method through numerical simulation examples.

[0159] The present invention also proposes an electronic 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 recursive filtering method for a state saturated system based on a high-speed network under the influence of the deception attack.

[0160] The present invention also proposes a computer-readable storage medium for storing computer instructions, which, when executed by a processor, implement the steps of the recursive filtering method for a state-saturated system based on a high-speed network under the influence of the deception attack.

[0161] The memory in this application embodiment can be volatile memory or non-volatile memory, or it can include both volatile and non-volatile memory. The non-volatile memory can be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or flash memory. The volatile memory can be random access memory (RAM), which is used as an external cache. By way of example, but not limitation, many forms of RAM are available, such as static random access memory (SRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous linked dynamic random access memory (SLDRAM), and direct rambus RAM (DRRAM). It should be noted that the memory used in the methods described in this invention is intended to include, but is not limited to, these and any other suitable types of memory.

[0162] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. The computer program product includes one or more computer instructions. When the computer instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium accessible to a computer or a data storage device such as a server or data center that integrates one or more available media. The available media may be magnetic media (e.g., floppy disks, hard disks, magnetic tapes), optical media (e.g., high-density digital video discs (DVDs)), or semiconductor media (e.g., solid-state disks (SSDs)).

[0163] In implementation, each step of the above method can be completed by integrated logic circuits in the processor's hardware or by instructions in software. The steps of the method disclosed in the embodiments of this application can be directly implemented by a hardware processor, or by a combination of hardware and software modules in the processor. The software modules can reside in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory, and the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above method. To avoid repetition, detailed descriptions are omitted here.

[0164] It should be noted that the processor in the embodiments of this application can be an integrated circuit chip with signal processing capabilities. During implementation, each step of the above method embodiments can be completed by the integrated logic circuitry in the processor's hardware or by instructions in software form. The processor can be a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components. It can implement or execute the methods, steps, and logic block diagrams disclosed in the embodiments of this application. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the methods disclosed in the embodiments of this application can be directly embodied as being executed by a hardware decoding processor, or executed by a combination of hardware and software modules in the decoding processor. The software modules can be located in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory, and the processor reads the information in the memory and, in conjunction with its hardware, completes the steps of the above methods.

[0165] The above provides a detailed description of the recursive filtering method for state saturated systems in high-speed networks under the influence of deception attacks proposed in this invention. Specific examples have been used to illustrate the principles and implementation methods of this invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

Claims

1. A recursive filtering method for state-saturated systems in high-speed networks under the influence of deception attacks, characterized by: The method is specifically as follows: Step 1: Construct a dynamic model of the state-saturated system to be studied. The dynamic model includes the system state and measurement output model, and sets the conditions that the functions in the dynamic model must satisfy. Step one specifically involves: Constructing a dynamic model of the system: (1) wherein represents the th sampling instant represents the system state, represents a zero-mean process noise with covariance , is the measurement output of the system, represents a zero-mean measurement noise with covariance , the initial state of the system is a random variable with mean and covariance ; , , , is a system matrix of appropriate dimension; is a saturation function, for a vector , the th element of which satisfies the following condition: (2) In the formula, For symbolic functions, Indicates the first The saturation level of each element; Step 2: Based on the measurement output model in Step 1, construct a new measurement model based on high-speed networks and random spoofing attacks, and set the constraints that the spoofing attacks must satisfy; Step 3: Based on the dynamic model from Step 1 and the new measurement model from Step 2, design a state-saturated recursive filter; Step 4: Based on the system dynamic model in Step 1 and the filter model in Step 3, obtain the dynamic equation of the filtering error and calculate the upper bound of the filtering error covariance. Step 5: Obtain the optimal filter gain by minimizing the upper bound of the filter error covariance; Step Six: Verify the effectiveness of the recursive filtering method through numerical simulation examples; Step six specifically involves: Considering state saturation level and The time-varying linear system has the following parameters: Setting process noise The covariance is Measure noise covariance is The mean of the initial state of the system ,random variable The expectation is and constants ; To verify the performance of the designed filter, a mean square error function is introduced: Based on the above parameters, and according to the optimal filter and the expression for the filter error covariance obtained in step five, the minimum upper bound of the error covariance and the filter gain are calculated.

2. The method according to claim 1, characterized in that: Step two specifically involves: Within a certain sampling period, the attacker... j The attack signals injected at each transmission moment are as follows: in It is an unknown bounded signal and satisfies , Given a constant, and ; Considering the randomness of attack phenomena, the actual measurement signal received by the filter This can be represented as the following model: (3) In the formula Let be a random variable that follows a Bernoulli distribution and satisfies: when When the probability is 1, it means the attack did not occur. ;when At that time, the attack occurs with a probability of . ; is a given constant.

3. The method according to claim 2, characterized in that: Step three specifically involves: First, in order to better estimate the system state, a dynamic model of the system within a sampling period is constructed: (4) In the formula, , ， , , , , Secondly, construct a state-saturated recursive filter of the following form: (5) in , Indicates system state The estimate, For the designed filter gain, This represents the initial state of the filter.

4. The method according to claim 3, characterized in that: Step four specifically involves: First of all, let The dynamic equation for the filtering error can be obtained as follows: Secondly, the filter error covariance is obtained based on mathematical induction, matrix operations, the definition of covariance, and inequality properties. Upper bound of difference: (6) In the formula Given a positive scalar, the other relevant parameters are: , , j = 0, 1, 2, ..., p-1; 。 5. The method according to claim 4, characterized in that: Step five specifically involves: Using the completing the square method, in step four... The formulation was prepared, and the results are as follows: (7) When the filter gain is When the upper bound of the filter error covariance is minimized, it is the optimal filter, and the minimum upper bound is: (8) In the formula, 。 6. A recursive filtering system for a state-saturated system based on a high-speed network under the influence of deception attacks, characterized in that: The system is specifically as follows: Construction module: Constructs a dynamic model of the state-saturated system to be studied. The dynamic model includes the system state and measurement output model, and sets the conditions that the functions in the dynamic model must satisfy. Setting Module: Based on the measurement output model in the construction module, construct a new measurement model based on high-speed networks and random spoofing attacks, and set the constraints that the spoofing attacks must satisfy; Design module: Design a state-saturated recursive filter based on the dynamic model and the new measurement model; Filtering module: Based on the system dynamic model and the filter model, the dynamic equation of the filtering error is obtained, and the upper bound of the filtering error covariance is calculated; Minimization module: The optimal filter gain is obtained by minimizing the upper bound of the filter error covariance; Verification module: Verifies the effectiveness of the recursive filtering method through numerical simulation examples; Constructing a dynamic model of the system: (1) In the formula, Indicates the first Each sampling time Indicates the system status. This represents a zero-mean process with noise and a covariance of . , For the system's measurement output, This represents measurement noise with zero mean and covariance of . initial state of the system Let be a random variable and its mean be Covariance is ; , , , A system matrix of appropriate dimension; For a vector, the saturation function is... The element The saturation function satisfies the following condition: (2) In the formula, For symbolic functions, Indicates the first The saturation level of each element; The verification module operates as follows: Considering state saturation level and The time-varying linear system has the following parameters: Setting process noise The covariance is Measure noise covariance is ; Mean of the initial state of the system ,random variable The expectation is and constants ; To verify the performance of the designed filter, a mean square error function is introduced: Based on the above parameters, and according to the optimal filter and filter error covariance expression obtained in the minimization module, the minimum upper bound of the error covariance and the filter gain are calculated.

7. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1-5.

8. A computer-readable storage medium for storing computer instructions, characterized in that, When the computer instructions are executed by the processor, they implement the steps of the method according to any one of claims 1-5.

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