An asynchronous filtering method for two-layer semi-markov jump systems

Abstract

This invention discloses an asynchronous filtering method for a two-layer semi-Markov skip system, belonging to the field of power system control technology. The method includes: constructing a two-layer semi-Markov model to capture the system's dynamics; saving network resources by introducing a dynamic memory event triggering protocol; establishing a filter model to obtain the system's filtering error model; obtaining sufficient conditions to ensure the system achieves the expected performance based on Lyapunov stability theory; proving the validity of the inequality conditions using Lyapunov functions and performance index functions; calculating the filter's gain matrix using simulation software; and constructing a filter design scheme based on asynchronous memory and ensuring key criteria for performance compliance. This invention saves network resources by introducing a dynamic memory event triggering protocol mechanism that includes historical transmission signals. Since system modes are difficult to obtain directly, this invention uses a class of non-homogeneous hidden semi-Markov models to solve this problem.

Description

Technical Field

[0001] This invention relates to the field of power system control technology, and specifically to an asynchronous filtering method for a two-layer semi-Markov skip system. Background Technology

[0002] In practical engineering, unavoidable disturbances often lead to random changes in system structure and parameters; such systems can be characterized as hybrid systems. Furthermore, Markov jump systems have been widely used in circuit systems, aircraft, vehicles, and communication networks. However, because Markov jump systems have fixed transition rates and residence times that follow an exponential distribution, they have limitations in constructing complex systems and cannot accurately reflect the dynamic changes of the system. To overcome these limitations, researchers have introduced semi-Markov jump systems. These models can more accurately describe the non-exponential distribution of residence time in real-world systems, as well as the complexity of transition rates that vary with time and environment. Researchers have used semi-Markov kernel techniques to evaluate the stability and performance of discrete-time semi-Markov jump systems, further relaxing the restrictions on residence time distribution. By approximating the original distribution associated with the semi-Markov process using a periodic distribution, it is shown that a more flexible method for handling semi-Markov jump systems is being explored. Although encouraging progress has been made in the research of semi-Markov jump systems, their control theory still needs further refinement. In particular, the jump times between states typically require additional calculation and derivation, making analysis and solutions more difficult. Therefore, researchers have simplified this problem by using average dwell time, reducing the complexity of random jumps by setting a minimum dwell time (i.e., the shortest time the system must remain in each mode). This method helps improve the accuracy of system stability analysis. However, for semi-Markov jump systems, the problem of accurately describing the switching process remains unsolved. To address these issues, this invention combines average dwell time theory with semi-Markov theory to establish a two-layer model.

[0003] In communication networks, data packet transmission is crucial for the stable operation of the network system. Specifically, data packet transmission refers to the process of efficiently transferring data from one node to another, ensuring the effective transmission of information between different devices or nodes within the network. Due to the limited bandwidth of communication networks, resource waste may occur. Therefore, adopting appropriate transmission protocols to improve network communication efficiency is particularly important. Early on, time-triggered protocols were widely used to improve network system communication efficiency. This protocol transmits data at fixed time intervals to ensure the regularity and predictability of communication. However, when data changes are infrequent or the system state is relatively stable, fixed-interval transmission leads to unnecessary resource consumption. To address this problem, researchers began exploring more flexible and efficient event-triggered protocols to optimize network resource utilization and improve communication efficiency. Event-triggered protocols mainly include the following types: static event-triggered protocols, dynamic event-triggered protocols, and memory event-triggered protocols. Static event-triggered protocols transmit data by setting predefined thresholds based on changes in system state, making them suitable for stable and predictable environments and effectively reducing redundant data transmission. To reduce unnecessary data transmission, static event-triggered protocols are commonly used to coordinate signal transmission. Compared to static event-triggered protocols, dynamic event-triggered protocols achieve higher resource utilization efficiency by dynamically adjusting triggering conditions. Internal dynamic variables are introduced into the traditional event-triggered mechanism to achieve higher resource utilization efficiency and more flexible system design requirements. Furthermore, compared to traditional memoryless event-triggered protocols, memory-triggered protocols incorporate historical transmission data. By effectively utilizing historical information, memory-triggered protocols can increase the triggering frequency when slow but significant changes occur, thereby improving system transient performance. Researchers have considered some useful historical transmission data packets in the triggering conditions, thus reducing conservatism. In summary, this invention's research on the filtering problem of a two-layer semi-Markov transition system under a dynamic memory-triggered protocol is of significant importance.

[0004] In networked systems, latency and packet loss can impair information transmission, causing problems with data transmission from the system to the filter, potentially leading to asynchronous phenomena in semi-Markov transition systems. In early research on semi-Markov transition systems, researchers often assumed that system modes and filter modes were synchronized to simplify the model, neglecting the asynchronous problem. However, with further research, more and more scholars have recognized the limitations of this assumption and shifted their focus to modeling and handling asynchronous phenomena. Homogeneous hidden semi-Markov models are widely used, as they correlate filter modes with system modes through conditional probability matrices. For example, homogeneous hidden semi-Markov models are used to solve the asynchronous sliding mode problem in semi-Markov transition systems. However, because the model detection probability of homogeneous hidden semi-Markov models is time-invariant, it cannot better adapt to and predict the behavior of various complex systems. Therefore, to better address the challenges posed by time-varying state transition probabilities, this invention proposes a new hidden semi-Markov model. This is the main motivation behind this invention.

[0005] The above problems urgently need to be solved. To address this, an asynchronous filtering method for two-layer semi-Markov hopping systems is proposed. Summary of the Invention

[0006] The technical problem this invention aims to solve is how to effectively capture the dynamic characteristics of a system while improving its performance. It provides an asynchronous filtering method for two-layer semi-Markov skip systems. To effectively capture the system's dynamic characteristics, a two-layer semi-Markov model is constructed. In this model, the bottom layer employs a stochastic evolution mechanism based on a semi-Markov process, while the upper layer uses average dwell time theory to achieve process control dominated by deterministic switching signals. To improve system performance, a dynamic memory event triggering protocol mechanism incorporating historical transmission signals is specifically designed. Addressing the challenge of system pattern recognition, this invention innovatively employs a non-homogeneous hidden semi-Markov model. Subsequently, a memory-based filter design strategy is constructed, and sufficient conditions to ensure preset performance are obtained. Finally, simulation examples verify the practicality of the proposed method.

[0007] The present invention solves the above-mentioned technical problems through the following technical solution, and the present invention includes the following steps:

[0008] Step S1: A two-layer semi-Markov model dominated by deterministic switching signals was constructed to effectively capture the dynamic characteristics of the system;

[0009] Step S2: A dynamic memory event triggering protocol containing historical transmission signal triggering conditions is introduced;

[0010] Step S3: Construct the filter model to obtain the filter error system model;

[0011] Step S4: Based on Lyapunov stability theory, the mean square exponent of the filtering error system in step S3 is kept stable and satisfies... Performance indicators Inequality conditions;

[0012] Step S5: Use the Lyapunov function to prove that the inequality conditions in step S4 are valid;

[0013] Step S6: Calculate the gain matrix of the filter using simulation software;

[0014] Step S7: Achieve mean square exponential stability of the filtering error system based on the filter gain matrix and given system parameters from step S6, and satisfy... Performance indicators .

[0015] Furthermore, in step S1, the specific processing procedure is as follows:

[0016] S11: Consider a continuous-time semi-Markov jump system:

[0017] ;

[0018] in, and These represent the state, measured output, and signal to be estimated of the semi-Markov transition system, respectively. Indicates external disturbance; It is a non-homogeneous semi-Markov process in a finite set Take the value from; It is a known matrix;

[0019] S12: It is a deterministic switching signal, in a finite set Values, It is a nonhomogeneous Markov update process, whose transition probabilities are governed by the Markov process. and Domination, specifically in the following forms:

[0020] ;

[0021] S13: Order ,if This is called a semi-Markov process. Associated with the update process, when a system mode is activated, the probability distribution function is:

[0022] ;

[0023] S14: The switching behavior of a semi-Markov process can be determined in the following way:

[0024] .

[0025] Furthermore, in the semi-Markov transition system, the transition probability matrix is: ,in, , and .

[0026] Furthermore, in step S2, the specific processing procedure is as follows:

[0027] S21: The dynamic memory event triggering protocol is constructed as follows:

[0028] ;

[0029] ;

[0030] in, It is a given parameter. Represents internal dynamic variables. Indicates the sampling period; Indicates the latest release time; Indicates the sampling time, set The constraints are satisfied; Represents a weighted matrix. Indicates the threshold parameter; Indicates the number of data packets transmitted in the past; This indicates that the condition is met. Weighting factors;

[0031] S22: Order ,in , as well as ;

[0032] S23: Definition , It is a time delay function, and its expression is:

[0033] .

[0034] Furthermore, in step S3, the specific processing procedure is as follows:

[0035] S31: The filter model based on asynchronous memory is designed as follows:

[0036] ;

[0037] in, , ,matrix , and These are the filter parameters that need to be determined; For historically transmitted data packets, parameters Describe a discrete-time Markov process that belongs to the set ;in ;

[0038] S32: The asynchronous phenomenon is represented by a non-homogeneous hidden semi-Markov model, whose model detection probability matrix Given from the following:

[0039] ;

[0040] S33: Let , The filtering error system is represented as follows:

[0041] ;

[0042] in, , , , , , .

[0043] Furthermore, in step S4, the specific processing procedure is as follows:

[0044] S41: Given parameters , , , , , , , , , ;when , When setting the dimensions of a matrix, if for all , , , , Then, when the following inequality holds, the mean square exponential stability of the filtering error system can be achieved and the following conditions can be met: Performance indicators :

[0045] ;

[0046] in:

[0047] ;

[0048] ;

[0049] ;

[0050] ;

[0051] ;

[0052] ;

[0053] ;

[0054] ;

[0055] ;

[0056] .

[0057] Furthermore, in step S5, the specific processing procedure is as follows:

[0058] S51: Select the Lyapunov function as follows:

[0059] ;

[0060] ;

[0061] ;

[0062] ;

[0063] We obtain the following using weak infinitesimal operators:

[0064] ;

[0065] ;

[0066] ;

[0067] ;

[0068] S52: Based on Park's theorem, the following inequality is obtained:

[0069] ;

[0070] in:

[0071] ;

[0072] S53: For the proposed dynamic memory event triggering protocol, we get:

[0073] ;

[0074] S54: Introduce the free weight matrix as follows:

[0075] ;

[0076] S55: Combining steps S51-S54, the following is derived:

[0077] ;

[0078] in:

[0079] ;

[0080] S56: When According to the following expression:

[0081] ;

[0082] Derivation At that time, we obtained:

[0083] ;

[0084] S57: By a deterministic switching signal And any ,definition for In the interval Number of switches within, when At that time, if If it is established, then This is called the vibration range. This is called the average length of stay. Based on the definition of average length of stay, according to... get:

[0085] ;

[0086] S58: The following definition is established:

[0087] ;

[0088] This leads to the conclusion that:

[0089] ;

[0090] in:

[0091] ;

[0092] S59: Based on the following matrix:

[0093] ;

[0094] It can be seen that the filtering error system satisfies mean square exponential stability;

[0095] S510: Definition ,when At that time, the filtering error system will be verified. Performance indicators According to step S55, we can obtain:

[0096] ;

[0097] in:

[0098] ;

[0099] S511: For the matrix Applying Schur's complement lemma, we obtain:

[0100] ;

[0101] S512: Multiply both sides of step S511. Integrating, we get:

[0102] ;

[0103] S513: Under zero initial conditions and Therefore, the following inequality holds:

[0104] ;

[0105] in, ;

[0106] S514: According to and The following inequalities hold:

[0107] ;

[0108] S515: By substituting the expression from step S513 into step S514, we obtain:

[0109] ;

[0110] Therefore, the filtering error system is mean-square exponentially stable and satisfies Performance indicators .

[0111] Furthermore, in step S6, the specific processing procedure is as follows:

[0112] Step S61: Let , , ;

[0113] Step S62: Define , , Substituting this into step S58, we get:

[0114] ;

[0115] in:

[0116] ;

[0117] ;

[0118] ;

[0119] ;

[0120] ;

[0121] ;

[0122] Step S63: At this point, use simulation software to calculate the value of the filter gain using the given matrix parameters.

[0123] Compared with the prior art, the present invention has the following advantages: In order to effectively capture the dynamic characteristics of the system, the asynchronous filtering method for the two-layer semi-Markov hopping system constructs a two-layer semi-Markov model. By introducing a dynamic memory event triggering protocol triggering mechanism that includes historical transmission signals, network resources are saved. Since the system modes are difficult to obtain directly, the present invention adopts a class of non-homogeneous hidden semi-Markov models to solve this problem. Attached Figure Description

[0124] Figure 1 This is a flowchart illustrating the asynchronous filtering method for a two-layer semi-Markov jump system based on a dynamic memory event triggering protocol in an embodiment of the present invention.

[0125] Figure 2 These are modal diagrams of the system and its filter in the embodiments of the present invention;

[0126] Figure 3 This is a system dynamic trajectory diagram in an embodiment of the present invention;

[0127] Figure 4 This is a dynamic trajectory diagram of the filter in an embodiment of the present invention;

[0128] Figure 5 This is a filtering error curve diagram in an embodiment of the present invention;

[0129] Figure 6 This describes the changing trend of internal dynamic variables in the embodiments of the present invention;

[0130] Figure 7 This is a diagram showing the triggering times of the memory event triggering protocol;

[0131] Figure 8 This is a diagram showing the triggering times of the dynamic memory event triggering protocol in this embodiment of the invention. Detailed Implementation

[0132] The embodiments of the present invention are described in detail below. These embodiments are implemented based on the technical solution of the present invention, and provide detailed implementation methods and specific operation processes. However, the scope of protection of the present invention is not limited to the following embodiments.

[0133] like Figure 1 As shown, this embodiment provides a technical solution: an asynchronous filtering method for a two-layer semi-Markov hopping system based on a dynamic memory event triggering protocol, comprising the following steps:

[0134] S1: A two-level semi-Markov model dominated by deterministic switching signals was constructed to effectively capture the dynamic characteristics of the system.

[0135] In this embodiment, the specific processing procedure in step S1 is as follows:

[0136] S11: Consider a continuous-time semi-Markov jump system:

[0137]

[0138] in, and These represent the state, measured output, and signal to be estimated of the semi-Markov transition system, respectively. Indicates external disturbance; It is a non-homogeneous semi-Markov process that occurs on finite sets. Take the value from; It is a known matrix.

[0139] S12: It is a deterministic switching signal, which is in a finite set. Values, It is a nonhomogeneous Markov update process, whose transition probabilities are governed by the Markov process. and Domination, specifically in the following forms:

[0140]

[0141] S13: Order ,if This is called a semi-Markov process. Associated with the update process; based on the preceding description, when a system mode is activated, the probability distribution function is:

[0142]

[0143] S14: Therefore, the switching behavior of a semi-Markov process can be determined in the following way:

[0144]

[0145] S15: In a semi-Markov transition system, the transition probability matrix is... ,in, , and .

[0146] Step S2: In order to save network resources, a dynamic memory event triggering protocol containing historical transmission signal triggering conditions is specially introduced;

[0147] It should be noted that, based on dynamic event-triggered protocol methods, only the most recently released data packet is typically considered. Parameters. This method is highly conservative because it ignores certain useful historical transmission packets.

[0148] In this embodiment, the specific processing procedure in step S2 is as follows:

[0149] S21: The dynamic memory event triggering protocol is constructed as follows:

[0150]

[0151] as well as:

[0152]

[0153] in, It is a given parameter. Represents internal dynamic variables. Indicates the sampling period; Indicates the latest release time; This indicates the sampling time. Clearly, the set... The constraints are met. This represents a weighted matrix, while This refers to the threshold parameter. This indicates the number of data packets transmitted in the past. This indicates that the condition is met. Weighting factors.

[0154] S22: Order ,in , ,as well as .

[0155] S23: Definition , It is a time delay function, and its expression is:

[0156] .

[0157] Step S3: Construct the filter model to obtain the filter error system model;

[0158] In this embodiment, the specific processing procedure in step S3 is as follows:

[0159] S31: The filter model based on asynchronous memory is designed as follows:

[0160]

[0161] in, , ,matrix , and These are the filter parameters that need to be determined. Historically transmitted data packets. Incorporated into filter design. Parameters Describe a discrete-time Markov process that belongs to the set .

[0162] S32: The asynchronous phenomenon is represented by a non-homogeneous hidden semi-Markov model, whose model detection probability matrix is... Given from the following:

[0163]

[0164] S33: Let , The filtering error system can be represented as

[0165]

[0166] in , , , .

[0167] In this embodiment, the specific processing procedure in step S4 is as follows:

[0168] S41: Given parameters , , , , , , , , , ,when , When it is a matrix of appropriate dimensions, if for all , , , , Then, the mean square exponential stability of the filter error system and the following inequality must be satisfied. Performance indicators :

[0169]

[0170] in:

[0171]

[0172]

[0173] In this embodiment, the specific processing procedure in step S5 is as follows:

[0174] S51: Select the Lyapunov function as follows:

[0175]

[0176]

[0177]

[0178]

[0179] We obtain the following using weak infinitesimal operators:

[0180]

[0181]

[0182]

[0183]

[0184] S52: Based on Park's lemma, the following inequality can be obtained:

[0185]

[0186] in:

[0187]

[0188] S53: For the proposed dynamic memory event triggering protocol, we get:

[0189]

[0190] S54: Introduce the free weight matrix as follows:

[0191]

[0192] S55: Combining steps S51-S54, the following is derived:

[0193]

[0194] in:

[0195]

[0196] S56: When According to the following expression:

[0197]

[0198] Derivation At that time, it can be obtained

[0199]

[0200] S57: By a deterministic switching signal And any ,definition for In the interval Number of switches within, when At that time, if If it is established, then This is called the vibration range. This is called the average length of stay. Based on the definition of average length of stay, according to... get:

[0201]

[0202] S58: Defined as follows:

[0203]

[0204] roll out:

[0205]

[0206] in:

[0207]

[0208] S59: Based on the following matrix:

[0209]

[0210] It can be seen that the filtering error system satisfies mean square exponential stability.

[0211] S510: Definition when At that time, the filtering error system will be verified. Performance indicators From step S55, we can obtain:

[0212]

[0213] in:

[0214]

[0215] S511: For the following matrix:

[0216]

[0217] Applying Schur's complement lemma, we get:

[0218]

[0219] S512: Multiply both sides of step S511 by... Integrating, we get:

[0220]

[0221] S513: Under zero initial conditions and Therefore, the following inequality holds:

[0222]

[0223] in,

[0224] S514: According to and The following inequalities hold:

[0225]

[0226] S515: By substituting the expression from step S513 into step S514, we obtain:

[0227]

[0228] Therefore, the filtering error system is mean-square exponentially stable and satisfies Performance indicators .

[0229] Step S6: Calculate the gain matrix of the filter using simulation software;

[0230] In this embodiment, the specific processing procedure of step S6 is as follows:

[0231] make , , ;

[0232] definition , , Substituting this into step S58, we get:

[0233]

[0234] in:

[0235]

[0236]

[0237]

[0238]

[0239]

[0240]

[0241] At this point, simulation software can be used to calculate the filter gain value using given matrix parameters.

[0242] Step S7: Based on the filter gain matrix from Step S6 and the given system parameters (transfer rate matrix, detection matrix, residence time distribution parameters, and dynamic memory event triggering protocol parameters), achieve the mean square exponential stability of the filter error system and satisfy the following conditions: Performance indicators .

[0243] The feasibility of the safe and stable operation of this system will be verified through numerical examples below.

[0244] Consider a three-modal semi-Markov jump system with the following parameters:

[0245]

[0246]

[0247]

[0248] Consider a semi-Markov process where the dwell time follows a Weiber distribution probability distribution function.

[0249] in, Let Y represent the shape parameter and Y represent the scale parameter. Therefore, the transition rate is controlled by a deterministic switching signal, and the selected signal... for:

[0250] ,

[0251] Consider the transition matrix for:

[0252]

[0253] Taking the expectation of the transition rate matrix, we can obtain:

[0254]

[0255]

[0256] In addition, considering the asynchronous phenomenon, the time-varying detection matrix Modeled as a set of polyhedra with two vertices:

[0257] ,

[0258] Set other parameters as follows

[0259] .

[0260] Therefore, the filter gain matrix is ​​as follows:

[0261]

[0262]

[0263]

[0264] Given parameter value Choose an initial value. , and Based on the above parameter settings, the simulation results are as follows: Figures 2 to 8 As shown. Figure 2 Demonstrated the pattern and Waveform characteristics. State and The dynamic trajectories are respectively in Figure 3 and Figure 4 The results show that the filtering error system has always maintained stable operation. Figure 5 and Figure 6 Corresponding to Filtering error curve and internal dynamic variables The changing trend. Furthermore, the triggering times of the memory event triggering protocol and the dynamic memory event triggering protocol are shown through simulation results. Figure 7 and Figure 8 These charts clearly show that more valid data packets are transmitted in the initial stage, while the transmission volume decreases significantly as the system dynamics stabilize. These phenomena indicate that the dynamic memory event-triggered protocol can achieve faster convergence, thus significantly saving network resources.

[0265] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. An asynchronous filtering method for a two-layer semi-Markov skip system, characterized in that, Includes the following steps: Step S1: A two-layer semi-Markov model dominated by deterministic switching signals was constructed to effectively capture the dynamic characteristics of the system; Step S2: A dynamic memory event triggering protocol containing historical transmission signal triggering conditions is introduced; Step S3: Construct the filter model to obtain the filter error system model; Step S4: Based on Lyapunov stability theory, the mean square exponent of the filtering error system in step S3 is kept stable and satisfies... Performance indicators Inequality conditions; Step S5: Use the Lyapunov function to prove that the inequality conditions in step S4 are valid; Step S6: Calculate the gain matrix of the filter using simulation software; Step S7: Achieve mean square exponential stability of the filtering error system based on the filter gain matrix and given system parameters from step S6, and satisfy... Performance indicators .

2. The asynchronous filtering method for a two-layer semi-Markov skip system according to claim 1, characterized in that, In step S1, the specific processing procedure is as follows: S11: Consider a continuous-time semi-Markov jump system: ； in, and These represent the state, measured output, and signal to be estimated of the semi-Markov transition system, respectively. Indicates external disturbance; It is a non-homogeneous semi-Markov process in a finite set Take the value from; It is a known matrix; S12: It is a deterministic switching signal, in a finite set Values, It is a nonhomogeneous Markov update process, whose transition probabilities are governed by the Markov process. and Domination, specifically in the following forms: ； S13: Order ,if This is called a semi-Markov process. Associated with the update process, when a system mode is activated, the probability distribution function is: ； S14: The switching behavior of a semi-Markov process can be determined in the following way: 。 3. The asynchronous filtering method for a two-layer semi-Markov skip system according to claim 2, characterized in that, In the semi-Markov transition system, the transition probability matrix is: ,in, , and .

4. The asynchronous filtering method for a two-layer semi-Markov skip system according to claim 2, characterized in that, In step S2, the specific processing procedure is as follows: S21: The dynamic memory event triggering protocol is constructed as follows: ； ； in, It is a given parameter. Represents internal dynamic variables. Indicates the sampling period; Indicates the latest release time; Indicates the sampling time, set The constraints are satisfied; Represents a weighted matrix. Indicates the threshold parameter; Indicates the number of data packets transmitted in the past; This indicates that the condition is met. Weighting factors; S22: Order ,in , as well as ; S23: Definition , It is a time delay function, and its expression is: 。 5. The asynchronous filtering method for a two-layer semi-Markov skip system according to claim 4, characterized in that, In step S3, the specific processing procedure is as follows: S31: The filter model based on asynchronous memory is designed as follows: ； in, , ,matrix , and These are the filter parameters that need to be determined; For historically transmitted data packets, parameters Describe a discrete-time Markov process that belongs to the set ;in ; S32: The asynchronous phenomenon is represented by a non-homogeneous hidden semi-Markov model, whose model detection probability matrix Given from the following: ； S33: Let , The filtering error system is represented as follows: ； in, , , , , , .

6. The asynchronous filtering method for a two-layer semi-Markov skip system according to claim 5, characterized in that, In step S4, the specific processing procedure is as follows: S41: Given parameters , , , , , , , , , ;when , When setting the dimensions of a matrix, if for all , , , , Then, when the following inequality holds, the mean square exponential stability of the filter error system can be achieved and the following conditions can be met: Performance indicators : ； in: ； ； ； ； ； ； ； ； ； 。 7. The asynchronous filtering method for a two-layer semi-Markov skip system according to claim 6, characterized in that, In step S5, the specific processing procedure is as follows: S51: Select the Lyapunov function as follows: ； ； ； ； We obtain the following using weak infinitesimal operators: ； ； ； ； S52: Based on Park's theorem, the following inequality is obtained: ； in: ； S53: For the proposed dynamic memory event triggering protocol, we get: ； S54: Introduce the free weight matrix as follows: ； S55: Combining steps S51-S54, the following is derived: ； in: ； S56: When According to the following expression: ； Derivation At that time, we obtained: ； S57: By a deterministic switching signal And any ,definition for In the interval Number of switches within, when At that time, if If it is established, then This is called the vibration range. This is called the average length of stay. Based on the definition of average length of stay, according to... get: ； S58: The following definition is established: ； This leads to the conclusion that: ； in: ； S59: Based on the following matrix: ； It can be seen that the filtering error system satisfies mean square exponential stability; S510: Definition ,when At that time, the filtering error system will be verified. Performance indicators According to step S55, we can obtain: ； in: ； S511: For the matrix Applying Schur's complement lemma, we obtain: ； S512: Multiply both sides of step S511. Integrating, we get: ； S513: Under zero initial conditions and Therefore, the following inequality holds: ； in, ; S514: According to and The following inequalities hold: ； S515: By substituting the expression from step S513 into step S514, we obtain: ； Therefore, the filtering error system is mean-square exponentially stable and satisfies Performance indicators .

8. The asynchronous filtering method for a two-layer semi-Markov skip system according to claim 7, characterized in that, In step S6, the specific processing procedure is as follows: Step S61: Let , , ; Step S62: Define , , Substituting this into step S58, we get: ； in: ； ； ； ； ； ； Step S63: At this point, use simulation software to calculate the value of the filter gain using the given matrix parameters.

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