A dynamic event-triggered semi-markov system threshold-dependent filtering method

Abstract

The application discloses a dynamic event-triggered semi-Markov system threshold-dependent filtering method, and relates to the technical field of industrial process control and state estimation. The method aims to explicitly analyze the influence of a triggering threshold on filtering performance under a dynamic event-triggering mechanism, and design an adaptive filter gain which simultaneously depends on system modal, resident time and real-time triggering threshold. In combination with a redundant transmission channel strategy, the method can effectively resist data loss, ensure high precision and robustness of system state estimation, and significantly reduce network communication burden.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of industrial process control and state estimation, in particular to a dynamic event-triggered semi-Markov system threshold-dependent filtering method. BACKGROUND

[0002] State estimation is the core of modern control systems, and its accuracy is highly dependent on the integrity and real-time performance of measurement signals. SMJS (Semi-Markov Jump Systems) can more accurately model the random switching process in actual systems, such as power circuit switching, load changes, etc., compared with traditional Markov jump systems, because it can describe more general residence time distribution.

[0003] However, in actual networked control systems, the transmission of measurement signals often suffers from data blocking or loss due to network congestion, instability or malicious attacks, which seriously damages the filtering performance. To alleviate this problem, existing technologies mainly use event-triggered mechanisms (such as static or dynamic event triggering) to reduce unnecessary network transmission, and use redundant communication channels to improve transmission reliability.

[0004] However, the existing technology has the following defects:

[0005] 1. Threshold influence is ignored: Although the existing dynamic event-triggered mechanism can adjust the threshold, its design is often decoupled from the filter design, and the quantitative relationship between the triggering threshold, data transmission frequency and the final filtering performance (such as H performance) is not revealed in depth, and there is a lack of clear rules for optimizing the design of the threshold.

[0006] 2. Lack of filter adaptability: For SMJS, its system parameters will change dramatically during mode switching. However, the existing filter gain is usually fixed or only depends on the mode, which cannot adapt to the real-time changing triggering threshold, resulting in a decrease in estimation accuracy during mode switching.

[0007] 3. Lack of robustness: The reliability of a single transmission channel is insufficient, and once data loss occurs, the filter cannot update the state. Although existing research has paid attention to redundant channels, it has not been jointly optimized with the threshold-adaptive filter.

[0008] Therefore, there is an urgent need for a robust filtering scheme that can jointly optimize the triggering strategy and filtering performance, and can adapt to the communication state (such as threshold change, channel packet loss). SUMMARY

[0009] To solve the above technical problems, the present application provides a dynamic event-triggered semi-Markov system threshold-dependent filtering method, comprising the following steps:

[0010] S1, a state space model of a semi-Markov jump system with time delay is established, and the motor is dynamically switched in different operating modes;

[0011] S2, a dynamic event triggering mechanism is designed and deployed at the sensor end;

[0012] S3, a redundant transmission channel is configured, N independent parallel transmission channels are set, and N>=2, the trigger measurement signal generated by the sensor is transmitted to the filter end through the N channels at the same time; the filter end receiving logic is: as long as at least one channel transmits successfully, it is considered that the data at this time is valid; only when all N channels fail to transmit, it is determined that data loss occurs at this time;

[0013] S4, an adaptive filter depending on a trigger threshold is designed; a filter is constructed to obtain a filtering error system, and the model of the filter is:

[0014]

[0015] Among them, indicates the state estimation value, indicates the measurement signal actually received by the filter, indicates the estimation signal output by the filter, indicates the time that the system has stayed in the current mode, 、 、 indicates the filter gain matrix to be solved, the values of the three are jointly scheduled by the trigger threshold theta (k), the system mode iota (k) and the residence time tau (k);

[0016] S5, the filter design problem is converted into a convex optimization problem;

[0017] S6, a group of optimal filter gain matrices corresponding to different modes, residence time intervals and trigger threshold intervals are obtained by solving the convex optimization problem, and are stored in the filter for real-time calling;

[0018] S7, the filter gain matrix obtained is substituted into the system, and the filter dynamically adjusts the parameters according to the current system mode, residence time and trigger threshold, so as to estimate the state of the semi-Markov system under the trigger mechanism.

[0019] The further defined technical solutions of the application are:

[0020] Further, in step S1, the system mode is described by a semi-Markov process, and the residence time obeys an arbitrary distribution:

[0021]

[0022] Where x(k) represents the system state vector, y(k) represents the measured output, z(k) represents the signal to be estimated, k represents the discrete time step, which is the index of the system at the k-th sampling, and k=0 corresponds to the initial time; ω(k) and ν(k) represent external disturbances and belong to ι(k) represents the modal signal described by the semi-Markov process, taking values ​​in the finite set M={1,...,m}, where m represents the total number of system modes; d(k) represents the time-varying state delay, satisfying d1≤d(k)≤d2, where d1 and d2 are given positive integers; A represents the initial state of the system. ι(k) A d,ι(k) B ι(k) C ι(k) D ι(k) and E ι(k) Indicates system parameters.

[0023] As described above, in a threshold-dependent filtering method for a dynamic event-triggered semi-Markov system, step S1 involves the motor dynamically switching between three different operating modes, i.e., the system mode number m=3; a state vector is then established. w m Let i represent the load angular velocity and i represent the current. The system model switches with the semi-Markov chain ι(k), where ι(k)∈{1,2,3} is the system mode at time k, and its evolution follows a semi-Markov process.

[0024] As described above, in a threshold-dependent filtering method for a dynamic event-triggered semi-Markov system, in step S1, N... r K represents the mode of the system during the r-th switching. r T represents the time of the r-th switch. r Let T represent the modal dwell time of the system during the r-th transition, and T r =K r+1 -K r The probability that the system switches to mode b after a running time δ in mode a is:

[0025]

[0026] The mode transition probabilities of the system are:

[0027]

[0028] The probability density function for system mode transition is:

[0029]

[0030] System modes:

[0031] ;

[0032] The system operates for a finite amount of time in a certain mode. The upper bound of the residence time is indicated; the system parameter matrix changes with the different values ​​of mode ι(k), and the system parameter matrix under different modes is obtained. Thus, a state-space model of a three-mode semi-Markov jump system is established.

[0033] As described above, in a threshold-dependent filtering method for a dynamic event-triggered semi-Markov system, the triggering condition in step S2 is defined by the following inequality:

[0034]

[0035] Where k represents the discrete time step, which is the index of the system at the k-th sampling, and k=0 corresponds to the initial time; k m This indicates the time of the m-th trigger. This indicates the difference between the current measured value y(k) and the last triggered measured value y(k). m The difference between θ(k) and θ(k) represents the time-varying dynamic adjustment trigger threshold. This represents the weight parameters of the trigger to be designed.

[0036] As described above, in a threshold-dependent filtering method for a dynamic event-triggered semi-Markov system, step S2 sets the update law for the trigger threshold θ(k) as follows:

[0037]

[0038] in, This represents the upper and lower bounds of the trigger threshold θ(k). ε=8.

[0039] As described above, in a threshold-dependent filtering method for a dynamic event-triggered semi-Markov system, step S3 involves configuring two transmission channels, each with a set random packet loss rate; and then using... and This describes whether data transmission through the primary channel and the redundant channel was successful at time k. α(k) = 1 indicates successful transmission through the primary channel, and α(k) = 0 indicates failed transmission through the primary channel. β(k) = 1 indicates successful transmission through the redundant channel, and β(k) = 0 indicates failed transmission through the redundant channel. The probability distribution is as follows: , ,set up .

[0040] As described above, in a threshold-dependent filtering method for a dynamic event-triggered semi-Markov system, step S5 involves constructing and solving a set of linear matrix inequality constraints to obtain a set of filter gain matrices that meet the requirements. This process includes the following sub-steps:

[0041] S5.1 Definition and estimation error The resulting filtering error system is:

[0042]

[0043] in,

[0044]

[0045] Among them, A a A d,a B a C a D a and E a Let n represent the system parameters when ι(k) = a. x n y n ω n v Let x(k), y(k), ω(k), and v(k) represent the dimensions of the vectors x(k), y(k), ω(k), and v(k).

[0046] Construct the Lyapunov function as follows:

[0047]

[0048] in,

[0049]

[0050] Among them, Q and Represents the Lyapunov matrix;

[0051] S5.2. The following sufficient conditions for satisfying the H∞ performance are derived mathematically, and the linear matrix inequalities that satisfy the conditions are:

[0052]

[0053] in, This represents a 7×7 symmetric matrix, where the non-zero elements are:

[0054]

[0055] in, This represents the probability that the system switches to mode b after a running time δ in mode a; Indicates robust performance metrics; The dimension is The identity matrix, The dimension is The identity matrix; The dimension is The zero matrix, The dimension is The zero matrix;

[0056] S5.3, Assuming a matrix variable dependent on the trigger threshold θ(k) and It consists of a constant and a trigger threshold θ(k), and its expression is: ;

[0057] The solvable constraints are further obtained as follows:

[0058]

[0059] in, This represents a 12×12 pairwise matrix, where the non-zero elements are:

[0060]

[0061]

[0062] in, and The subscripts s and t take values ​​of 1 or 2, representing the upper and lower bounds of the trigger threshold θ(k); and This represents the matrix variable to be solved, used to calculate the correlation matrix of the filter gain; , , X represents the intermediate matrix variable resulting from the coupling of the trigger threshold and the Lyapunov matrix; a Y a Z a This represents the introduced relaxation matrix, used to calculate the correlation matrix of the filter gain. The formula for calculating the correlation matrix of the filter gain is:

[0063] .

[0064] As described above, in a threshold-dependent filtering method for a dynamic event-triggered semi-Markov system, step S6 involves solving the constraint to obtain a set of matrix variables that meet the performance requirements. and ;

[0065] In step S7, the measurement signal y(k) acquired by the sensor passes through the event triggering mechanism. The triggering mechanism adjusts the trigger threshold θ(k) based on y(k), and then determines whether the data meets the triggering condition. If it does, the trigger updates the trigger time k. m =k, and measure the signal y(k) m After being sent out, the signal passes through the main channel and redundant channels, and the final signal received by the filter receiver is... The filter is based on the trigger threshold θ(k) at this time, and the matrix variables solved offline. and The filter gain at this point is obtained using the following formula:

[0066]

[0067] Ultimately, the estimation of the signal z(k) to be estimated in the system is achieved.

[0068] The beneficial effects of this invention are:

[0069] (1) In this invention, the filter gain is adaptive to the real-time changing trigger threshold, system mode and dwell time, which can respond more finely to the dynamic changes of the system and significantly improve the state estimation accuracy;

[0070] (2) In this invention, a quantitative relationship between the trigger threshold and the H∞ filtering performance was established through theoretical analysis, and the design rules for the threshold were provided, thus achieving the optimal balance between communication resources and filtering performance;

[0071] (3) In this invention, the redundant channel strategy greatly reduces the probability of complete data loss due to single channel failure. Combined with the adaptive filter, the system can still maintain reliable operation when the network is unstable or attacked, thus having stronger robustness.

[0072] (4) In this invention, the dynamic event triggering mechanism ensures that data transmission is only performed when necessary, and the redundant channel is only activated after triggering. The combination of the two ensures reliability while maximizing the saving of network resources and effectively reducing the communication burden. Attached Figure Description

[0073] Figure 1 This is a schematic diagram of the overall structure of the system in an embodiment of the present invention;

[0074] Figure 2 This is a diagram of redundant channel data transmission in an embodiment of the present invention;

[0075] Figure 3 The output response diagrams of the filter and system in this embodiment of the invention are shown.

[0076] Figure 4 This is a comparison chart of the estimation performance of the filter proposed in this embodiment of the invention and a general threshold-independent filter. Detailed Implementation

[0077] This embodiment provides a threshold-dependent filtering method for a dynamically event-triggered semi-Markov system, comprising the following steps:

[0078] S1. Establish a state-space model of a semi-Markov jump system with time delay, such as...Figure 1 As shown, the system modes are characterized by semi-Markov processes, and their residence times follow an arbitrary distribution:

[0079]

[0080] Where x(k) represents the system state vector, y(k) represents the measured output, z(k) represents the signal to be estimated, k represents the discrete time step, which is the index of the system at the k-th sampling, and k=0 corresponds to the initial time; ω(k) and ν(k) represent external disturbances and belong to ι(k) represents the modal signal described by the semi-Markov process, taking values ​​in the finite set M={1,...,m}, where m represents the total number of system modes; d(k) represents the time-varying state delay, satisfying d1≤d(k)≤d2, where d1 and d2 are given positive integers; due to the existence of the time delay d(k), the state-space equation may generate a non-existent state vector x(k) within the time step [-d2,0], such as x(-d2). However, in reality, within the time step [-d2,0], the system remains in its initial state, i.e., the expression... ; A represents the initial state of the system. ι(k) A d,ι(k) B ι(k) C ι(k) D ι(k) and E ι(k) Indicates system parameters.

[0081] The motor dynamically switches between three different operating modes, i.e., the system mode number m=3; a state vector is established. w m Let i represent the load angular velocity and i represent the current. The system model switches with the semi-Markov chain ι(k), where ι(k)∈{1,2,3} is the system mode at time k, and its evolution follows a semi-Markov process.

[0082] Use N r K represents the mode of the system during the r-th switching. r T represents the time of the r-th switch. r Let T represent the modal dwell time of the system during the r-th transition, and T r =K r+1 -K r Therefore, the probability that the system switches to mode b after a running time δ in mode a is:

[0083]

[0084] The mode transition probabilities of the system are:

[0085]

[0086] The probability density function for system mode transition is:

[0087]

[0088] The core function of the Pr function is to calculate the probability of an event occurring. The difference between probability, transition probability, and probability density lies not in the Pr function itself, but in the event parameters it describes. Different definitions of events and conditions lead to different objects of Pr calculation and different physical meanings.

[0089] System modes:

[0090] .

[0091] Through mathematical statistics, the specific values ​​of the transition probability can be obtained as follows:

[0092] ,

[0093] The specific values ​​of the probability density function are:

[0094]

[0095] The system operates for a finite amount of time in a certain mode. Indicates the upper bound of the dwell time and sets it to... .

[0096] The system parameter matrix varies with the value of mode ι(k). The system parameter matrices for different modes are obtained as follows:

[0097]

[0098] Therefore, a state-space model of a three-modal semi-Markov transition system is established.

[0099] S2. Design a dynamic event triggering mechanism and deploy it at the sensor end; its triggering condition is defined by the following inequality:

[0100]

[0101] Where k represents the discrete time step, which is the index of the system at the k-th sampling, and k=0 corresponds to the initial time; k m This indicates the time of the m-th trigger. This indicates the difference between the current measured value y(k) and the last triggered measured value y(k). m The difference between θ(k) and θ(k) represents the time-varying, dynamically adjusted trigger threshold. This represents the weight parameters of the trigger to be designed.

[0102] The update law for setting the trigger threshold θ(k) is:

[0103]

[0104] The relevant parameters can be set by those skilled in the art based on experience. This represents the upper and lower bounds of the trigger threshold θ(k). ε=8.

[0105] S3. Configure redundant transmission channels: Set up N (N≥2) independent parallel transmission channels; the trigger measurement signal generated by the sensor is transmitted to the filter end simultaneously through these N channels; the receiving logic of the filter end is: as long as at least one channel is successfully transmitted, the data is considered valid at that moment; data loss is determined to have occurred at that moment only if all N channels fail to transmit.

[0106] In this embodiment, two transmission channels are configured, and each channel is assigned a random packet loss rate; specifically, they are respectively used... and This describes whether data transmission through the primary channel and the redundant channel was successful at time k. α(k) = 1 indicates successful transmission through the primary channel, and α(k) = 0 indicates failed transmission through the primary channel. β(k) = 1 indicates successful transmission through the redundant channel, and β(k) = 0 indicates failed transmission through the redundant channel. The probability distribution is as follows: , ,set up .

[0107] S4. Design a trigger threshold-dependent adaptive filter; construct the filter to obtain the filtering error system. The filter model is as follows:

[0108]

[0109] in, This represents the state estimate. This represents the actual measurement signal received by the filter. This represents the estimated signal output by the filter. This indicates the time the system has remained in the current mode. , , Let represent the filter gain matrix to be determined, whose values ​​are jointly determined by the trigger threshold θ(k), the system mode ι(k), and the dwell time τ(k).

[0110] The actual measurement signal received by the filter It is obtained by passing the system output signal y(k) through steps S2 and S3 respectively. Assuming N=2 in step S3, then The expression is .

[0111] S5. Transform the filter design problem into a convex optimization problem; construct and solve a set of linear matrix inequality constraints to obtain a set of filter gain matrices that meet the requirements. This includes the following steps:

[0112] S5.1 Define the filtering error system and, based on Lyapunov stability theory, analyze the mean square stability and H∞ performance index of the augmented system; define and estimation error The resulting filtering error system is:

[0113]

[0114] in,

[0115]

[0116] Among them, A a A d,a B a C a D a and E a Let n represent the system parameters when ι(k) = a. x n y n ω n v Let x(k), y(k), ω(k), and v(k) represent the dimensions of the vectors x(k), y(k), ω(k), and v(k).

[0117] Construct the Lyapunov function as follows:

[0118]

[0119] in,

[0120]

[0121] Among them, Q and This represents the Lyapunov matrix.

[0122] S5.2 Derive the sufficient conditions for satisfying the performance index, which are expressed as a set of constraints with linear matrix inequalities as the core.

[0123] The designed threshold-dependent filter must possess stability and robustness. By applying mathematical methods such as Schur's complement lemma, the following sufficient conditions can be derived to guarantee the mean-square stability of the system and satisfy H∞ performance. The linear matrix inequality satisfying these conditions is:

[0124] .

[0125] in, This represents a 7×7 symmetric matrix, where the non-zero elements are:

[0126]

[0127] in, This represents the probability that the system switches to mode b after a running time δ in mode a; Indicates robust performance metrics; The dimension is The identity matrix, The dimension is The identity matrix; The dimension is The zero matrix, The dimension is The zero matrix.

[0128] S5.3 The key innovation lies in discretizing the interval to which the time-varying trigger threshold θ(t) belongs and transforming its dependency into a convex compact set or polyhedron representation, thereby transforming the infinite-dimensional optimization problem into a linear matrix inequality constraint problem on a finite number of vertices, thus avoiding computational complexity.

[0129] The linear matrix inequalities in step S5.2 contain the matrix variables that need to be solved. and Since the value of the trigger threshold θ(k) is uncertain, that is, the number of matrix variables that need to be solved is infinite, the linear matrix inequality constraints obtained in step S5.2 cannot be solved directly, thus obtaining the threshold-dependent filter parameters.

[0130] The key to resolving the above contradiction lies in: assuming a matrix variable that depends on the trigger threshold θ(k). and It consists of a constant and a trigger threshold θ(k), and its expression is: The remaining matrix variables depend on the trigger threshold. and Similar transformations are performed, but will not be elaborated upon here.

[0131] The solvable constraints are further obtained as follows:

[0132] .

[0133] in, This represents a 12×12 pairwise matrix, where the non-zero elements are:

[0134]

[0135]

[0136] in, and The subscripts s and t take values ​​of 1 or 2, representing the upper and lower bounds of the trigger threshold θ(k); and This represents the matrix variable to be solved, used to calculate the correlation matrix of the filter gain; , , X represents the intermediate matrix variable resulting from the coupling of the trigger threshold and the Lyapunov matrix; a Y a Z a This represents the introduced relaxation matrix, used to calculate the correlation matrix of the filter gain. The formula for calculating the correlation matrix of the filter gain is:

[0137] .

[0138] S6. By solving the convex optimization problem, a set of optimal filter gain matrices corresponding to different modes, residence time intervals, and trigger threshold intervals are obtained. , , The data is stored in the filter for real-time access; solving this constraint yields a set of matrix variables that meet the performance requirements. and .

[0139] S7. Solve the filter gain matrix , , Substituting these parameters into the system, the filter will dynamically adjust its parameters based on the current system mode ι(k), residence time τ(k), and trigger threshold θ(k), thereby achieving more accurate state estimation of the semi-Markov system under the triggering mechanism.

[0140] The measurement signal y(k) acquired by the system through the sensor passes through the event triggering mechanism. The triggering mechanism adjusts the trigger threshold θ(k) based on y(k), and then determines whether the data meets the triggering condition. If it does, the trigger updates the trigger time k. m =k, and measure the signal y(k) m After being sent out, the signal passes through the main channel and redundant channels, and the final signal received by the filter receiver is... The filter is based on the trigger threshold θ(k) at this time, and the matrix variables solved offline. and The filter gain at this point is obtained using the following formula:

[0141]

[0142] Ultimately, the estimation of the signal z(k) to be estimated in the system is achieved.

[0143] S8. Result Analysis: In the case of complete failure of the main channel, the transmission situation through the redundant channel is as follows: Figure 2 As shown; firstly, thanks to the event-triggered mechanism, the number of measurement signals that need to be transmitted is reduced; secondly, even if the main channel completely fails (due to network attacks, channel congestion, etc.), data transmission can still be completed through redundant channels, and the system output and filter output are as follows. Figure 3 As shown, the invented threshold-dependent filter can estimate the system output.

[0144] Comparing the threshold-dependent adaptive filter proposed in this embodiment with a general threshold-independent filter, such as... Figure 4 As shown, the threshold-dependent filter in this embodiment can adjust system parameters based on the current trigger threshold, thus achieving a more accurate estimation than the threshold-independent filter.

[0145] The method in this embodiment aims to explicitly analyze the impact of the trigger threshold on the filtering performance under a dynamic event triggering mechanism, and to design an adaptive filter gain that depends on the system mode, dwell time, and real-time trigger threshold. Combined with a redundant transmission channel strategy, it can significantly reduce the network communication burden, effectively combat data loss, and ensure high accuracy and robustness of system state estimation.

[0146] In addition to the embodiments described above, the present invention may have other implementations. All technical solutions formed by equivalent substitution or equivalent transformation fall within the protection scope claimed by the present invention.

Claims

1. A threshold-dependent filtering method for a dynamically event-triggered semi-Markov system, characterized in that: Includes the following steps: S1. Establish a state-space model of a semi-Markov jump system with time delay, and dynamically switch the motor under different operating modes; The system's operating modes are represented by system modes. Different modes have different system parameters, and the system modes are characterized by semi-Markov processes, whose residence times follow arbitrary distributions. Where x(k) represents the system state vector, y(k) represents the measured output, z(k) represents the signal to be estimated, k represents the discrete time step, which is the index of the system at the k-th sampling, and k=0 corresponds to the initial time; ω(k) and ν(k) represent external disturbances and belong to ι(k) represents the modal signal described by the semi-Markov process, taking values ​​in the finite set M={1,...,m}, where m represents the total number of system modes; d(k) represents the time-varying state delay, satisfying d1≤d(k)≤d2, where d1 and d2 are given positive integers; A represents the initial state of the system. ι(k) A d,ι(k) B ι(k) C ι(k) D ι(k) and E ι(k) Indicates system parameters; S2. Design a dynamic event triggering mechanism and deploy it at the sensor end; the triggering condition of the triggering mechanism is defined by the following inequality: Where k represents the discrete time step, which is the index of the system at the k-th sampling, and k=0 corresponds to the initial time; k m This indicates the time of the m-th trigger. This indicates the difference between the current measured value y(k) and the last triggered measured value y(k). m The difference between θ(k) and θ(k) represents the time-varying dynamic adjustment trigger threshold. This represents the weight parameters of the trigger to be designed; The update law for setting the trigger threshold θ(k) is: in, This represents the upper and lower bounds of the trigger threshold θ(k). ε=8; S3. Configure redundant transmission channels, set up N independent parallel transmission channels, and N≥2. The trigger measurement signal generated by the sensor is transmitted to the filter end simultaneously through these N channels. The receiving logic of the filter end is: as long as at least one channel is successfully transmitted, the data is considered valid; data loss is determined to have occurred only when all N channels fail to transmit. S4. Design a trigger threshold-dependent adaptive filter; construct the filter to obtain the filtering error system. The filter model is as follows: in, This represents the state estimate. This represents the actual measurement signal received by the filter. This represents the estimated signal output by the filter. This indicates the time the system has remained in the current mode. , , Let represent the filter gain matrix to be determined. The values ​​of these three parameters are jointly determined by the trigger threshold θ(k), the system mode ι(k), and the dwell time τ(k). S5. Construct the Lyapunov function and derive a set of linear matrix inequality constraints. The matrix parameters include the filter gain, H∞ performance norm, and robustness index of the filter to be designed. The filter design problem is transformed into a convex optimization problem, with the objective function being to minimize the H∞ performance norm robustness index. A set of robust indicators The linear matrix inequality is transformed into a convex constraint, and the optimization variables are optimized. If the value is greater than 0, the filter gain matrix is ​​a real matrix, and the set of filter gain matrices that meet the requirements can be obtained by solving. S6. By solving the convex optimization problem, a set of optimal filter gain matrices corresponding to different modes, dwell time intervals and trigger threshold intervals are obtained and stored in the filter for real-time retrieval. S7. Substitute the solved filter gain matrix into the system. The filter dynamically adjusts its parameters based on the current system mode, dwell time, and trigger threshold, thereby estimating the state of the half-Markov system under the triggering mechanism.

2. The threshold-dependent filtering method for a dynamic event-triggered semi-Markov system according to claim 1, characterized in that: In step S1, the motor dynamically switches between three different operating modes, i.e., the system mode number m=3; a state vector is established. w m Let i represent the load angular velocity and i represent the current. The system model switches with the semi-Markov chain ι(k), where ι(k)∈{1,2,3} is the system mode at time k, and its evolution follows a semi-Markov process.

3. The threshold-dependent filtering method for a dynamic event-triggered semi-Markov system according to claim 2, characterized in that: In step S1, N is used r K represents the mode of the system during the r-th switching. r T represents the time of the r-th switch. r Let T represent the modal dwell time of the system during the r-th transition, and T r =K r+1 -K r The probability that the system switches to mode b after a running time δ in mode a is: The mode transition probabilities of the system are: The probability density function for system mode transition is: System modes: ； The system operates for a finite amount of time in a certain mode. The upper bound of the residence time is indicated; the system parameter matrix changes with the different values ​​of mode ι(k), and the system parameter matrix under different modes is obtained. Thus, a state-space model of a three-mode semi-Markov jump system is established.

4. The threshold-dependent filtering method for a dynamic event-triggered semi-Markov system according to claim 3, characterized in that: In step S3, two transmission channels are configured, and a random packet loss rate is set for each channel; respectively using and This describes whether data transmission through the primary channel and the redundant channel was successful at time k. α(k) = 1 indicates successful transmission through the primary channel, and α(k) = 0 indicates failed transmission through the primary channel. β(k) = 1 indicates successful transmission through the redundant channel, and β(k) = 0 indicates failed transmission through the redundant channel. The probability distribution is as follows: , ,set up .

5. The threshold-dependent filtering method for a dynamic event-triggered semi-Markov system according to claim 4, characterized in that: In step S5, a set of linear matrix inequality constraints is constructed and solved to obtain a set of filter gain matrices that meet the requirements. This specifically includes the following sub-steps: S5.1 Definition and estimation error The resulting filtering error system is: in, Among them, A a A d,a B a C a D a and E a Let n represent the system parameters when ι(k) = a. x n y n ω n v Let x(k), y(k), ω(k), and v(k) represent the dimensions of the vectors x(k), y(k), ω(k), and v(k). Construct the Lyapunov function as follows: in, Among them, Q and Represents the Lyapunov matrix; S5.

2. The following sufficient conditions for satisfying the H∞ performance are derived mathematically, and the linear matrix inequalities that satisfy the conditions are: in, This represents a 7×7 symmetric matrix, where the non-zero elements are: in, This represents the probability that the system switches to mode b after a running time δ in mode a; Indicates robust performance metrics; The dimension is The identity matrix, The dimension is The identity matrix; The dimension is The zero matrix, The dimension is The zero matrix; S5.3, Assuming a matrix variable dependent on the trigger threshold θ(k) and It consists of a constant and a trigger threshold θ(k), and its expression is: ; The solvable constraints are further obtained as follows: in, This represents a 12×12 pairwise matrix, where the non-zero elements are: in, and The subscripts s and t take values ​​of 1 or 2, representing the upper and lower bounds of the trigger threshold θ(k); and This represents the matrix variable to be solved, used to calculate the correlation matrix of the filter gain; , , X represents the intermediate matrix variable resulting from the coupling of the trigger threshold and the Lyapunov matrix; a Y a Z a This represents the introduced relaxation matrix, used to calculate the correlation matrix of the filter gain. The formula for calculating the correlation matrix of the filter gain is: 。 6. The threshold-dependent filtering method for a dynamic event-triggered semi-Markov system according to claim 5, characterized in that: In step S6, the constraint is solved to obtain a set of matrix variables that meet the performance requirements. and ; In step S7, the measurement signal y(k) acquired by the sensor passes through the event triggering mechanism. The triggering mechanism adjusts the trigger threshold θ(k) based on y(k), and then determines whether the data meets the triggering condition. If it does, the trigger updates the trigger time k. m =k, and measure the signal y(k) m After being sent out, the signal passes through the main channel and redundant channels, and the final signal received by the filter receiver is... The filter is based on the trigger threshold θ(k) at this time, and the matrix variables solved offline. and The filter gain at this point is obtained using the following formula: Ultimately, the estimation of the signal z(k) to be estimated in the system is achieved.

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