A method for configuring zero-pole of high-order transmission function type wave trap

Abstract

This invention discloses a method for zero-pole configuration of a high-order transfer function notch filter, aiming to improve the dynamic response of a traditional second-order transfer function notch filter and achieve globally optimal configuration of the dynamic response of the zeros and poles of a high-order transfer function notch filter. Specifically, the method first elucidates the general form of a high-order transfer function notch filter and analyzes the high-dimensional multi-constraint problem involved in its zero-pole optimization configuration. Then, a multi-population genetic algorithm based on subspace prior partitioning is proposed. By introducing a low-computational-requirement pre-screening mechanism, low-confidence subspaces are pre-excluded, thereby significantly reducing the computational burden of the algorithm while maintaining excellent global optimization performance. The zero-pole optimization configuration process of this invention has been experimentally verified to be effective in improving the dynamic response of the notch filter. This method is applicable to motor control and signal processing applications.

Description

Technical Field

[0001] This invention relates to a method for zero-pole configuration of a high-order transfer function notch filter, belonging to the fields of electrical engineering and signal processing. Background Technology

[0002] With the rapid development and widespread application of modern motor control technology, high-performance motors such as permanent magnet synchronous motors have been widely used in high-end equipment fields such as industrial automation, new energy vehicles, precision CNC machine tools, and aerospace drives. However, during the motor control process, a series of harmonics are inevitably generated within the system. These harmonics not only reduce the accuracy of motor parameter identification but may also lead to misjudgment of faults and weaken the robustness of the control system.

[0003] Among existing harmonic filtering methods, notch filters are favored in cost-sensitive, mass-production motor drive systems due to their low computational cost and simple implementation. However, the second-order notch filters widely used in engineering still exhibit significant response delays when dealing with low-speed motor conditions, making it difficult to meet the requirements for fast dynamic response. More importantly, theoretically, increasing the order of the notch filter can introduce more zeros and poles, thereby gaining greater design freedom and achieving better dynamic performance. However, the zero-pole configuration problem of high-order notch filters then transforms into a complex optimization problem with high dimensions and multiple constraints. Using traditional exhaustive configuration methods to achieve the globally optimal configuration for dynamic response will face a huge computational burden. This has also resulted in the lack of research on the potential of high-order notch filters in improving dynamic response. Therefore, there is an urgent need to propose a low-computation-cost zero-pole configuration method to fully leverage the advantages of high-order transfer-function notch filters in improving dynamic response performance. Summary of the Invention

[0004] To address the limitations of traditional second-order notch filters in terms of insufficient dynamic response under low-speed conditions, and the optimization challenges faced by high-order notch filters in achieving optimal dynamic performance, such as high zero-pole configuration dimension and complex constraints, this invention proposes a computationally less computationally burdensome method for zero-pole configuration of high-order transfer function notch filters. First, this invention elucidates the general form of a high-order transfer function notch filter and analyzes the high-dimensional, multi-constraint problem involved in its zero-pole optimization configuration. Then, it proposes a multi-population genetic algorithm with priori partitioned subspaces (MPGA-PP), which introduces a low-computation-requirement pre-screening mechanism to pre-exclude low-confidence subspaces, thereby significantly reducing the computational burden of the algorithm while maintaining excellent global optimization performance. To solve the above problems, the technical solution adopted in this invention is as follows: A method for zero-pole configuration of a high-order transfer function notch filter includes the following steps: Step 1: Determine the order of the high-order transfer function notch filter and determine the zero-pole configuration constraints of the high-order transfer function notch filter. Step 2: Transform the zero-pole placement constraints of the high-order transfer function notch filter into a constrained optimization problem with the goal of fast dynamic response; Step 3: Within the preset range of zero and pole parameter values, a low-confidence subspace prior screening mechanism is used to initially screen out the range of zero and pole parameter values ​​that meet the requirements. The low-confidence subspace prior screening mechanism is as follows: First, the grid center sampling method is used to sample the zero and pole optimization space; then, the time-series scanning method is used to scan the obtained sampling points and screen out the sampling points that meet the response speed and amplitude-frequency gain constraints. Step 4: Use a combined binary search method k The mean clustering algorithm divides the range of zero and pole parameters that meet the requirements into several subspaces and executes a multi-population genetic algorithm to obtain the combination of zeros and poles with the global optimal dynamic response.

[0005] In the above technical solution, further, in step 1, the high-order transfer function notch filter is... n Notch filter of type transfer function. n The general form of a notch filter with a first-order transfer function can be expressed as: In the above equation, s It is the Laplace operator; G nth-o ( s )express n Input-output transmission characteristics of a notch filter with a first-order transfer function in the complex frequency domain; ω h The notch filter frequency; z i This is the first one that needs to be configured. i One zero point; p q This is the first one that needs to be configured. q One extreme point; The specific method for determining the zero-pole configuration constraints of a high-order transfer function notch filter is as follows: To avoid affecting the amplitude-frequency gain of the DC component, the amplitude-frequency gain at frequency 0 must be 1. Therefore, all zeros / poles must be configured under the following constraints: Furthermore, to avoid introducing unintended amplitude-frequency gain across the entire frequency domain, the zeros and poles of high-order transfer function notch filters should be configured based on the following conditions: in, s = jω ,in j Represents the imaginary unit. ω Angular frequency; ε It is the preset maximum allowable gain of the amplitude-frequency characteristic.

[0006] Furthermore, in step 2, the zero-pole configuration constraints need to be transformed into a constrained optimization problem with the goal of fast dynamic response. The specific method is as follows: in, ℂ 2n-2 Representative by 2 n A set consisting of -2 complex numbers, t s * ( x )yes x The corresponding per-unit settling time under a unit step response, x It is a matrix of zeros and poles that needs to be configured.

[0007] Furthermore, in step 3, to reduce the computational burden of the algorithm, a low-confidence subspace prior screening mechanism is used within the range of zero and pole parameters to initially screen out the range of zero and pole parameter values ​​that meet the requirements. Specifically: First, a grid center sampling method is used to sample the zero-pole optimization space. Specifically, the optimization space is defined as the range of zero-pole parameter values. Sampling is performed in the optimization space with a uniform step size (each sampling point corresponds to a smaller range of zero-pole parameter values). Each sampling point divides the optimization space into a uniform grid. Then, an additional sampling point is added at the geometric center of each grid. For any point in the optimization space, the distance to the nearest sampling point is defined as... r Take all of the optimization space. r The longest geometric distance is called the coverage radius. r cov ,satisfy: in, Ω Represents the entire optimization space; D For the set of sampling points; x It is any point in the optimization space; d It is any sampling point in the optimization space; Then, a time-series scanning method is used to scan the sampling points obtained by the grid center sampling method, and the sampling points that meet the response speed and amplitude-frequency gain constraints are selected (corresponding to the required zero-pole parameter value range). An early stopping mechanism is introduced into the time-series scanning method to quickly identify low-confidence subspaces that do not meet the response speed or amplitude-frequency gain constraints. The early stopping judgment formula is: In the above formula, t a * For the first time-series scan a Individual standardization moment; ω b For the first time-series scan b One frequency; γ This is the convergence error band for adjusting the time. y ( t a * )for t a * corresponding n Step response of a notch filter with a first transfer function; T s * It is the set threshold for step response adjustment time. ε It is the preset maximum allowable gain of the amplitude-frequency response; | G nth-o ( jω b )| is ω b corresponding n Amplitude-frequency gain of a notch filter with a first-order transfer function.

[0008] Furthermore, step 4 specifically involves: First, a combined binary search method is used. k The mean clustering algorithm divides all data points within the required range of zero and pole parameters into clusters. k Within each cluster, the specific steps include: (1) Set the initial range of the binary search k Low and k High ; (2) Determine the intermediate value of the binary search k Mid The specific formula for its calculation is as follows: Among them, floor( x The function is a floor function that returns a value less than or equal to the floor function.x The largest integer; (3) Based on step (2), randomly select k Mid An initial centroid; (4) Assign the nearest centroid to each data point and form a cluster, specifically using... k The mean clustering algorithm divides all data points within the required range of zero and pole parameters into clusters. k In each cluster, the partitioning criterion is to minimize the sum of squared errors from each data point to its corresponding cluster centroid: in, ℂ g Indicates inclusion m The first data point g The parameter set of each cluster; μ g It is the first g The centroid (i.e., arithmetic mean) of a cluster. x l It is the first g The first cluster l One data point; (5) For each cluster, calculate the arithmetic mean and update the centroid. The calculation formula is as follows: (6) If the error between two consecutive centroid calculations is less than δ ( δ =0.001), which means the centroid has converged, then jump to step (7); otherwise jump to step (4). (7) Solve for the maximum cluster radius R k_max The calculation formula is as follows: (8) Evaluate the clustering results, if R k_max ≤ R max (If the maximum cluster radius is preset), then proceed to step (9); otherwise, proceed to step (10). (9) Record k Mid A feasible solution, and try smaller ones. k The value is then jumped to step (11), and the expression is: (10) Try bigger k The value, whose expression is: (11) If kLow > k High If yes, then proceed to step (12); otherwise, proceed to step (2). (12) Obtain the minimum feasible solution k * And its clustering results. Among them, those satisfying the preset maximum cluster radius R max Minimum under the condition k The value is denoted as k * , k * It can be represented as: in, Represents positive integers; Then, based on the subspaces of zero and pole parameter ranges obtained by clustering, a multi-population genetic algorithm is executed independently in each subspace to finally obtain the combination of zeros and poles with the global optimal dynamic response, which is the optimal zero and pole configuration scheme.

[0009] The inventive principle of this invention is as follows: 1. To address the significant response delay of traditional second-order notch filters under low-speed motor conditions, this invention employs a high-order notch filter structure. By introducing additional freely configurable zeros and poles, the design freedom of the notch filter is expanded to improve dynamic response. 2. In the selection of parameters for high-order notch filters, the configuration of their zeros and poles faces complex optimization challenges due to high dimensions and multiple constraints. This invention proposes a Multi-Population Genetic Algorithms with Priori Partitioned Subspaces (MPGA-PP). By introducing a pre-screening mechanism, low-confidence subspaces are eliminated in advance, thereby avoiding the waste of computational power caused by optimizing in those subspaces. Ultimately, while maintaining excellent global optimization performance, the computational burden of the algorithm can be significantly reduced.

[0010] The beneficial effects of this invention are as follows: By introducing a high-order notch filter structure, the degree of freedom in system design parameters is increased. By constructing its parameterized mathematical model and establishing a complete constraint system for zero-pole configuration, this invention overcomes the inherent limitation of slow dynamic response of traditional second-order notch filters under low-speed motor conditions, and improves the overall response speed of the system.

[0011] By utilizing Multi-Population Genetic Algorithms with Priori Partitioned Subspaces (MPGA-PP), a low-confidence subspace prior screening mechanism is introduced to pre-exclude low-confidence subspaces. This reduces the complexity and computational cost of notch filter design (i.e., zero-point and pole configuration) while ensuring optimal dynamic performance of the notch filter.

[0012] The proposed low-confidence subspace prior screening mechanism first creatively proposes a grid center sampling method to sample the zero-pole optimization space. Specifically, this method uses the range of zero-pole parameter values ​​as the optimization space, and first samples the optimization space with a uniform step size (each sampling point corresponds to a small range of zero-pole parameter values). Each sampling point divides the optimization space into a uniform grid, and then an additional sampling point is added at the geometric center of each grid. Compared with the traditional uniform grid sampling method, under the same coverage radius, the proposed method can reduce the number of sampling points by about 46%, thereby saving 53.8% of the computational cost. Secondly, a temporal scanning method is used to scan the sampling points obtained by the grid center sampling method to screen out the sampling points that meet the constraints of response speed and amplitude-frequency gain, thereby further reducing the number of sampling points that need to participate in the calculation. Attached Figure Description

[0013] Figure 1 A schematic diagram comparing the grid center sampling method used in this invention with the traditional uniform grid sampling method; Figure 2 A schematic diagram of the timing scan method in the present invention; Figure 3 The high-quality candidate regions and clustering subspace results of the third-order notch filter after pre-screening by the method of this invention are shown in the figure. Figure 4 A schematic diagram illustrating the working principle of the MPGA-PP proposed in this invention in a two-dimensional coefficient optimization space; Figure 5 Global optimal performance and convergence test diagram of zero-pole configuration of a third-order notch filter using the method of this invention. Detailed Implementation

[0014] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0015] Figure 2 This is a schematic diagram of the timing scan method in the present invention.

[0016] This invention will, in conjunction with an implementation example, specifically describe how to use the zero-pole configuration method of the aforementioned high-order transfer function notch filter. In this implementation example, a third-order filter is used as an example to perform optimal zero-pole configuration. The specific implementation steps of the proposed invention are as follows: Step 1: Determine the order of the high-order transfer function notch filter to be 3rd order, and determine the zero-pole configuration constraints of the notch filter; Step 2: Transform the zero-pole placement problem into a constrained optimization problem with the goal of achieving a fast dynamic response; Step 3: Use a low-confidence subspace prior screening mechanism within the preset range of zero and pole parameters to initially screen out the range of zero and pole parameter values ​​that meet the requirements. Step 4: Use a combined binary search method k The mean clustering algorithm divides the range of zero and pole parameters that meet the requirements into several subspaces and executes a multi-population genetic algorithm to obtain the combination of zeros and poles with the global optimal dynamic response.

[0017] First, in step 1, the third-order transfer function notch filter is expressed as follows: In the above equation, s It is the Laplace operator; G 3rd-o ( s () represents the input-output transmission characteristics of a third-order transfer function notch filter in the complex frequency domain; ω h The notch filter frequency; z 3 is the third zero point that needs to be configured; p q The first one that needs to be configured q One extreme point; Then, to avoid affecting the amplitude-frequency gain of the DC component, i.e., the amplitude-frequency gain at frequency 0 is 1, all zeros / poles must be configured under the following constraints: Furthermore, to avoid introducing unintended amplitude-frequency gain across the entire frequency domain, the zeros and poles of high-order transfer function notch filters should be configured based on the following conditions: In amplitude-frequency response analysis, s = jω ,in j Represents the imaginary unit. ω Angular frequency; ε It is the preset maximum allowable gain of the amplitude-frequency characteristic.

[0018] Secondly, in step 2, before applying the multi-population genetic algorithm based on subspace prior partitioning, it is necessary to transform the zero and pole placement constraints into a constrained optimization problem with the goal of fast dynamic response. The specific method is as follows: In the above equation, ℂ 2n-2 Representative by 2 n A set consisting of -2 complex numbers, t s * ( x )yes x The corresponding per-unit settling time under a unit step response, x It is a matrix of zeros and poles that needs to be configured.

[0019] Then, in step 3, in order to reduce the computational burden of the algorithm, a low-confidence subspace prior screening mechanism is adopted in the range of zero and pole parameters, specifically as follows: First, the grid center sampling method is used to sample and evaluate the zero-pole optimization space, as shown in the attached figure. Figure 1 As shown. Figure 1 This diagram illustrates a comparison between the grid center sampling method used in this invention and the traditional uniform grid sampling method. The black dots represent the notch filter coefficients to be sampled. For any point (red dot) in the optimization space, the distance to the nearest sampling point is defined as... r Take all of the optimization space. r The longest geometric distance is called the coverage radius. r cov It can be represented by the distance between the red and black dots in the diagram.

[0020] Based on the proposed grid center sampling method, each coefficient axis is divided into 646 intervals, and an additional sampling point is added within each grid cell, thus obtaining the coverage radius. r cov = 8.65e-4 pu. This is compared to the traditional uniform grid sampling method, at the same coverage radius... r cov In this case, the number of sampling points required for verification can be reduced by approximately 46%; Then, the timing scan method is used, as shown in the attached figure. Figure 2 As shown, an early stopping mechanism is introduced to quickly identify low-confidence subspaces that do not meet the response speed or amplitude-frequency gain constraints. This method progressively scans the time-domain step response. y ( t s * and amplitude-frequency characteristics | G 3rd-o ( jωIf the early stopping condition is triggered, the scanning process immediately terminates and the corresponding coefficient is classified as infeasible; conversely, if the complete scanning process does not violate any conditions, the coefficient is considered feasible. The specific formula for the early stopping condition is: In the above formula, γ This is the convergence error band for adjusting the time, which is determined based on the preset response error, usually selected as 2% or 5%, and 2% is used in this example; y for n Step response of a notch filter with a first transfer function; T s * This is the set threshold for the step response adjustment time; in this example, it is set to 0.595 pu. ε This is the preset maximum allowable gain of the amplitude-frequency characteristic, which is set to 1dB in this example.

[0021] Finally, step 4 specifically involves: First, a combined binary search method is used. k The mean clustering algorithm will cluster all data points (total) within the required range of zero and pole parameter values. n (each) is divided into k Within each cluster, the specific steps include: (1) Set the initial range of the binary search k Low and k High ; (2) Determine the intermediate value of the binary search k Mid The specific formula for its calculation is as follows: Among them, floor( x The function is a floor function that returns a value less than or equal to the floor function. x The largest integer; (3) Based on step (2), randomly select k Mid An initial centroid; (4) Assign the nearest centroid to each data point and form a cluster, specifically using... k Mean clustering algorithm will n Data points were divided into k In each cluster, the partitioning optimization criterion is to minimize the sum of squared errors from each data point to its corresponding cluster centroid: in, ℂ g Indicates inclusion m The first data pointg The parameter set of each cluster; μ g It is the first g The centroid (i.e., arithmetic mean) of a cluster. x l It is the first g The first cluster l One data point; (5) For each cluster, calculate the arithmetic mean and update the centroid. The calculation formula is as follows: (6) If the error between two consecutive centroid calculations is less than δ ( δ =0.001), which means the centroid has converged, then jump to step (7); otherwise jump to step (4). (7) Solve for the maximum cluster radius R k_max The calculation formula is as follows: (8) Evaluate the clustering results, if R k_max ≤ R max (If the maximum cluster radius is preset), then proceed to step (9); otherwise, proceed to step (10). (9) Record k Mid A feasible solution, and try smaller ones. k The value is then jumped to step (11), and the expression is: (10) Try bigger k The value, whose expression is: (11) If k Low > k High If yes, then proceed to step (12); otherwise, proceed to step (2). (12) Obtain the minimum feasible solution k * And its clustering results, as shown in the appendix. Figure 3 As shown. Among them, the preset maximum cluster radius is satisfied. R max Minimum under the condition k The value is denoted as k * , k * It can be represented as: in, Represents a positive integer.

[0022] To maximize the compactness of the subspace, this example sets the maximum allowed radius during clustering. R max The value is set to 0.013 pu. The proposed binary search-based method... k The mean clustering algorithm obtains the minimum number of clusters that satisfy the cluster radius constraint. k * = 12, at which point the maximum cluster radius is... R k_max = 0.0128 pu, the corresponding clustering results are attached. Figure 3 As shown. Figure 3 This image shows the high-quality candidate regions and clustering subspace results after pre-screening using the method of this invention for a third-order notch filter. To maximize the compactness of the subspace, the maximum allowable radius during clustering is... R max As small as possible.

[0023] Then, based on the subspaces of zero and pole parameter ranges obtained by clustering, a multi-population genetic algorithm is executed independently in each subspace to finally obtain the combination of zeros and poles with the global optimal dynamic response, which is the optimal zero and pole configuration scheme.

[0024] Figure 4 This diagram illustrates the working principle of MPGA-PP proposed in this invention within a two-dimensional coefficient optimization space. As can be seen from the diagram, MPGA-PP can perform a preliminary fitness assessment across the entire optimization space, thereby identifying regions with a very low probability of containing the global optimum, thus avoiding the execution of genetic algorithms in these low-potential regions. This method effectively reduces the number of subspaces required for search.

[0025] Figure 5 The graph shows the global optimal performance and convergence test results for zero-pole configuration of a third-order notch filter using the method of this invention. The algorithm is run independently 10 times, with a population size of 300 in each subspace in each run, and 20 generations per iteration. Crossover probability... p c and mutation probability p m The values ​​were set to 0.9 and 0.1 respectively; and two optimization objectives were selected: f fit1 = t s * , f fit2 =| G nth-o ( jω )| maxFurthermore, to avoid the possibility that the optimal solution might lie on the boundary, the range of all subspaces was expanded by 20%. Simultaneously, to obtain the global optimal solution as a verification benchmark, an exhaustive scan search was performed, dividing each coefficient axis into 1000 intervals. The dynamic response of the obtained optimal zero-pole configuration notch filter was evaluated, and the evaluation results showed that the corresponding optimal settling time was... t s * = 0.59047 pu. The results show that the solution time of MPGA-PP (800 s) is much shorter than that of the exhaustive method (56 h), and the dynamic response settling time of the optimal zero-pole configuration notch filter obtained by MPGA-PP is close to the optimal settling time obtained by the exhaustive scan search. t s * The maximum difference between the two is less than 2×10 -4 Furthermore, the dynamic response adjustment time of MPGA-PP may be slightly faster than that of discrete exhaustive search (because the method of this invention can continuously search the region between discrete points).

[0026] The embodiments described above are merely some preferred embodiments of the present invention, and are not intended to limit the invention. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the invention. Therefore, all technical solutions obtained by equivalent substitution or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A method for configuring the zero-pole of a high-order message passing type wave trap, characterized in that, Includes the following steps: Step 1: Determine the order of the high-order transfer function notch filter and determine the zero-pole configuration constraints of the high-order transfer function notch filter. Step 2: Transform the zero-pole placement constraints of the high-order transfer function notch filter into a constrained optimization problem with the goal of fast dynamic response; Step 3: Within the preset range of zero and pole parameter values, a low-confidence subspace prior screening mechanism is used to initially screen out the range of zero and pole parameter values ​​that meet the requirements. The low-confidence subspace prior screening mechanism is as follows: First, the grid center sampling method is used to sample the zero and pole optimization space; then, the time-series scanning method is used to scan the obtained sampling points and screen out the sampling points that meet the response speed and amplitude-frequency gain constraints. Step 4: Use a combined binary search method k The mean clustering algorithm divides the range of zero and pole parameters that meet the requirements into several subspaces and executes a multi-population genetic algorithm to obtain the combination of zeros and poles with the global optimal dynamic response.

2. The method for zero-pole configuration of a high-order transfer function notch filter according to claim 1, characterized in that, In step 1, the higher-order transfer function notch filter is n A notch filter with a first-order transfer function is specifically represented as follows: In the above equation, s It is the Laplace operator; G nth-o ( s )express n Input-output transmission characteristics of a notch filter with a first-order transfer function in the complex frequency domain; ω h The notch filter frequency; z i This is the first one that needs to be configured. i One zero point; p q This is the first one that needs to be configured. q One extreme point; The specific zero-pole configuration constraints of the high-order transfer function notch filter are as follows: in, s = jω ,in j Represents the imaginary unit. ω Angular frequency; ε It is the preset maximum allowable gain of the amplitude-frequency characteristic.

3. The method for zero-pole configuration of a high-order transfer function notch filter according to claim 2, characterized in that, In step 2, the constrained optimization problem with the objective of achieving a fast dynamic response is specifically expressed as follows: in, ℂ 2n-2 Representative by 2 n A set consisting of -2 complex numbers, t s * ( x )yes x The corresponding per-unit settling time under a unit step response, x It is a matrix of zeros and poles that needs to be configured.

4. The method for zero-pole configuration of a high-order transfer function notch filter according to claim 3, characterized in that, In step 3: The method employs a grid center sampling approach to sample the zero-pole optimization space. Specifically, the optimization space is defined as the range of zero-pole parameter values. Sampling is first performed within this space with a uniform step size, dividing the optimization space into uniform grids at each sampling point. Then, an additional sampling point is added at the geometric center of each grid. For any point within the optimization space, the distance to the nearest sampling point is defined as... r ; Take all of the optimization space. r The longest geometric distance is called the coverage radius. r cov ,satisfy: in, Ω Represents the entire optimization space; D For the set of sampling points; x It is any point in the optimization space; d It is any sampling point in the optimization space; The timing scan method introduces an early stopping mechanism, and the early stopping judgment formula is as follows: In the above formula, t a * For the first time-series scan a Individual standardization moment; ω b For the first time-series scan b One frequency; γ This is the convergence error band for adjusting the time. y ( t a * )for t a * corresponding n Step response of a notch filter with a first transfer function; T s * It is the set threshold for step response adjustment time. ε It is the preset maximum allowable gain of the amplitude-frequency response; | G nth-o ( jω b )| is ω b corresponding n Amplitude-frequency gain of a notch filter with a first-order transfer function.

5. The method for zero-pole configuration of a high-order transfer function notch filter according to claim 4, characterized in that, Step 4 specifically involves: First, a combined binary search method is used. k The mean clustering algorithm divides the range of zero and pole parameters that meet the requirements into several subspaces. The specific method is as follows: (1) Set the initial range of the binary search k Low and k High , k Low and k High This represents the lower and upper bounds of the binary search; (2) Determine the intermediate value of the binary search k Mid The specific formula for its calculation is as follows: Among them, floor( x The function is a floor function that returns a value less than or equal to the floor function. x The largest integer; (3) Based on step (2), randomly select k Mid An initial centroid; (4) Assign the nearest centroid to each data point and form a cluster, specifically using... k The mean clustering algorithm divides all data points within the required range of zero and pole parameters into clusters. k In each cluster, the partitioning criterion is to minimize the sum of squared errors from each data point to its corresponding cluster centroid: in, ℂ g Indicates inclusion m The first data point g The parameter set of each cluster; μ g It is the first g The centroid of the cluster; x l It is the first g The first cluster l One data point; (5) For each cluster, update the centroid, and the calculation formula is as follows: (6) If the error between two consecutive centroid calculations is less than δ If the centroid converges, then proceed to step (7); otherwise, proceed to step (4). (7) Solve for the maximum cluster radius R k_max The calculation formula is as follows: (8) Evaluate the clustering results, if R k_max ≤ R max , R max If the maximum cluster radius is preset, then proceed to step (9); otherwise, proceed to step (10). (9) Record k Mid A feasible solution is to be considered, and smaller solutions are to be tried. k The value is then jumped to step (11), and the expression is: (10) Try bigger k The value, whose expression is: (11) If k Low > k High If yes, then proceed to step (12); otherwise, proceed to step (2). (12) Obtain the minimum feasible solution k * and its clustering results; among which, those satisfying the preset maximum cluster radius R max Minimum under the condition k The value is denoted as k * , k * Represented as: in, Represents positive integers; Then, based on the subspaces of the zero and pole parameter value ranges obtained by clustering, a multi-population genetic algorithm is executed independently in each subspace to finally obtain the zero and pole combination with the global optimal dynamic response, which is the optimal zero and pole configuration scheme.

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