Parameter self-tuning method of feed servo system based on adaptive genetic algorithm

Abstract

The invention relates to a parameter self-tuning method of a feed servo system based on an adaptive genetic algorithm, and belongs to the technical field of dynamic performance optimization of numerical control machine tools. Parameter identification is performed on a mechanical transmission power system, and the feed servo system of the numerical control machine tool is established based on the identified parameters; a self-adaptive genetic algorithm is used for carrying out parameter self-tuning on the feeding servo system of the numerical control machine tool, the dynamic performance of the feeding servo system of the numerical control machine tool is optimized, and the precise adaptation characteristic of the feeding servo system of the numerical control machine tool on speed control is provided through self-tuning of speed ring parameters. The response speed, the stability and the error condition of the feeding servo system in the speed following process are revealed; position loop parameter self-tuning reveals the position positioning characteristics of the numerical control machine tool feeding servo system, helps to identify deviation accumulation and oscillation phenomena in position control and the overall positioning precision of the system, and optimizes the position dynamic response performance of machine tool moving parts.

Description

Technical Field

[0001] The present invention relates to the technical field of dynamic performance optimization of numerically controlled machine tools, and in particular to a parameter self-tuning method of a feed servo system based on an adaptive genetic algorithm. Background Art

[0002] In the existing technology, CNC machine tools are the core equipment of industrial mother machines, and the improvement of their processing accuracy has become a key breakthrough for industrial upgrading. However, in actual production, the unreasonable setting of machine tool motion control parameters has become a major bottleneck restricting the improvement of processing accuracy. Due to the lack of efficient parameter tuning methods, the current parameter tuning process relies heavily on the operator's experience, which is not only time-consuming and labor-intensive, but also difficult to achieve the best matching state, resulting in the machine tool being unable to fully exert its due performance. As the core functional component for CNC machine tools to achieve high-precision motion control, the feed servo system can effectively compensate for the dynamic errors of the mechanical transmission chain by self-tuning its controller parameters, significantly improving the positioning accuracy performance of the machine tool and providing reliable technical support for precision manufacturing.

[0003] Currently, most mechanical transmission identification work for feed servo systems is conducted under no-load conditions, lacking real-time identification of mechanical transmission parameters. Traditional parameter settings are inadequate. Intelligent algorithms can adjust parameters in real time, optimize system dynamic performance, and effectively suppress interference, necessitating in-depth research and application. Compared to traditional intelligent algorithms, adaptive genetic algorithms offer significant advantages. They utilize a roulette wheel selection mechanism and can adaptively adjust crossover and mutation probabilities, enabling more effective parameter tuning of feed servo systems to suppress interference generated by part processing.

[0004] Finally, the traditional parameter tuning method relies on manual experience and offline calculations, which makes it difficult to meet the real-time compensation requirements under complex working conditions. Summary of the Invention

[0005] In view of the above problems, the present invention provides a parameter self-tuning method for a feed servo system based on an adaptive genetic algorithm. The present invention performs parameter identification on a mechanical transmission power system, establishes a CNC machine tool feed servo system based on the identified parameters, and uses an adaptive genetic algorithm to perform parameter self-tuning on the CNC machine tool feed servo system, thereby optimizing the dynamic performance of the CNC machine tool feed servo system. The self-tuning of the speed loop parameters provides the CNC machine tool feed servo system with precise adaptation characteristics in speed control, and intuitively reveals the response speed, stability and error of the feed servo system in the speed following process; the self-tuning of the position loop parameters reveals the position positioning characteristics of the CNC machine tool feed servo system, helps identify deviation accumulation and oscillation phenomena in position control and the overall positioning accuracy of the system, thereby optimizing the position dynamic response performance of the machine tool moving parts, and realizing dynamic performance monitoring and real-time feedback of the parameter tuning process.

[0006] The present invention provides a parameter self-tuning method for a feed servo system based on an adaptive genetic algorithm, comprising:

[0007] Step S1: constructing an original CNC machine tool feed servo system and simplifying it to obtain a simplified CNC machine tool feed servo system;

[0008] Step S2: determining a mechanical transmission power system;

[0009] Perform parameter identification on the mechanical transmission power system based on the frequency domain least square method to obtain the identification parameters of the mechanical transmission power system;

[0010] Substituting the identified parameters into the simplified CNC machine tool feed servo system in step S1 to obtain the CNC machine tool feed servo system;

[0011] Step S3: Optimizing the controller parameters of the CNC machine tool feed servo system by using an adaptive genetic algorithm to complete the self-tuning of the CNC machine tool feed servo system parameters.

[0012] Optionally, the identification parameters include: equivalent mass, viscous damping coefficient and Coulomb friction of the mechanical transmission power system.

[0013] Optionally, the expression of the mechanical transmission power system is:

[0014]

[0015] Where, I is the input current of the motor, K is the torque constant of the motor, is the motor acceleration of the mechanical transmission power system, is the motor speed of the mechanical transmission power system, m is the equivalent mass of the mechanical transmission power system, c is the viscous damping coefficient of the mechanical transmission power system, and f is the Coulomb friction of the mechanical transmission power system.

[0016] Optionally, the specific steps of performing parameter identification include:

[0017] After linearizing the mechanical transmission power system, discrete fast Fourier transform is performed to obtain the mechanical transmission frequency domain linear equation;

[0018] The mechanical transmission frequency domain linear equation is converted into matrix form, and then the frequency domain least square method is used to calculate and obtain the identification parameters.

[0019] Optionally, the specific steps for completing the complete CNC machine tool feed servo system parameter self-tuning include:

[0020] Preset the range of control parameter values and randomly select N groups of control parameters;

[0021] Construct fitness objective function;

[0022] Represent each set of control parameters as a corresponding population, and obtain N populations;

[0023] Determine multiple populations for the i-th iteration; i = 1, 2, 3…I, where I represents the total number of steps;

[0024] Based on the fitness objective function, obtaining the fitness of each population in the i-th iteration step;

[0025] Perform roulette wheel selection on multiple populations in the i-th iteration step to obtain the updated population of the i-th iteration step;

[0026] Perform crossover, mutation and sorting operations on the updated population of the i-th iteration step to obtain the final population of the i-th iteration step;

[0027] After traversing I iteration steps, I final populations are obtained and sorted, and the control parameters corresponding to the final population with the largest fitness are selected as the optimal control parameters to complete the parameter self-tuning.

[0028] Optionally, the fitness objective function is expressed as:

[0029]

[0030] BsJ n =∫0 ∞ (w1|e(t,n)|+w2u 2 (t,n)+w3|y(t,n)-y(t-1,n)|)dt+w4t u

[0031] Where, e(t,n) is the system error of the nth population at time t; u(t) is the controller output of the nth population at time t; y(t,n) is the actual output of the nth population at time t; t u is the adjustment time, which represents the time required for the CNC machine tool feed servo system to reach a stable state from the initial state; w1 is the weight coefficient of the error term, w2 is the weight coefficient of the control input term, w3 is the weight coefficient of the output smoothing term, w4 is the weight coefficient of the time penalty term, BsJ is the performance evaluation index, which measures the dynamic performance and control effect of the CNC machine tool feed servo system, and fn is the fitness value of the nth population.

[0032] Optionally, the specific steps of performing crossover, mutation, and sorting on the updated population of the i-th iteration step to obtain the final population of the i-th iteration step include:

[0033] Get the average fitness variance of the population, which is represented as population diversity;

[0034] Obtaining the adaptive crossover rate of each population based on population diversity;

[0035] Based on the adaptive crossover rate of each population, two populations are cross-operated to obtain multiple secondary updated populations, and the multiple secondary updated populations are sorted in ascending order according to the corresponding fitness to obtain multiple populations after one sorting;

[0036] Obtain the adaptive mutation rate of each population based on population diversity;

[0037] Based on the adaptive mutation rate of each population, mutation operations are performed on the sorted multiple populations to obtain multiple three-times updated populations. The multiple three-times updated populations are arranged in ascending order according to the corresponding fitness to obtain multiple populations after secondary sorting. The population with the largest fitness is selected as the final population in the i-th iteration step.

[0038] Compared with the prior art, the present invention has at least the following beneficial effects:

[0039] (1) The CNC machine tool feed servo system established by the present invention combines mechanical transmission frequency domain characteristic identification with dynamic time domain modeling to achieve multi-physical field coupling;

[0040] (2) The present invention integrates frequency-domain least squares parameter identification and adaptive genetic algorithm to complete the coordinated tuning of parameters of the speed loop, current loop, and position loop within the control cycle;

[0041] (3) The present invention dynamically adjusts the adaptive crossover rate (0.6-0.9) and the adaptive mutation rate (0.01-0.1) through multiple iterations to optimize the dynamic performance of the CNC machine tool feed servo system and effectively suppress the influence of sudden load changes. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The drawings are only for purposes of illustrating particular embodiments and are not to be considered limiting of the invention.

[0043] Figure 1 Schematic diagram of the process of the parameter self-tuning method of the feed servo system of the present invention. DETAILED DESCRIPTION

[0044] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited by the specific embodiments disclosed below.

[0045] A specific embodiment of the present invention, as Figure 1, discloses a parameter self-tuning method for a feed servo system based on an adaptive genetic algorithm, which is used to suppress the load processing of CNC machine tools and improve the operating stability and processing accuracy of CNC machine tools under complex working conditions. The specific implementation steps are as follows:

[0046] Step S1: constructing an original CNC machine tool feed servo system and simplifying it to obtain a simplified CNC machine tool feed servo system;

[0047] It can be understood that the CNC machine tool feed servo system is a direct drive CNC machine tool feed servo system;

[0048] Optionally, the specific steps of simplification include:

[0049] When establishing the CNC machine tool feed servo system, factors that have little impact on the performance of the CNC machine tool feed servo system are ignored, and are used to highlight the main dynamic characteristics of the CNC machine tool feed servo system;

[0050] Optionally, the factors having little impact on the performance include: slow changes in temperature, slow changes in humidity, tooth side clearance, and torsional elastic deformation of the shaft.

[0051] The present invention focuses on the coupling effect between the mechanical structure and the electrical part of the CNC machine tool feed servo system, and constructs the original CNC machine tool feed servo system based on the coupling relationship.

[0052] It will be appreciated that the mechanical structure includes damping characteristics and load conditions;

[0053] The electrical part includes the electric drive winding characteristics and back electromotive force;

[0054] Furthermore, the CNC machine tool feed servo system in step S1 includes a current loop transfer function, a speed loop closed-loop transfer function, and a position loop transfer function.

[0055] Furthermore, the expression of the current loop transfer function is:

[0056]

[0057] Among them, G i is the response speed of the current loop of the CNC machine tool feed servo system, s represents the Laplace variable, which is used to describe the dynamic characteristics of the current loop, T f is the low-pass filter constant; T SPWM is the inverter constant, and the closed-loop current loop bandwidth w is obtained based on b , the expression is: Because (T SPWM +T f) value is extremely small, the closed-loop bandwidth of the current loop is extremely high, it can quickly track the command signal, and its dynamic response is much stronger than the speed loop with large mechanical inertia. Its response time is in the millisecond level, and the dynamic change of the back electromotive force can be regarded as quasi-static in the working frequency band. In the steady-state analysis of the feed servo system, the transient tracking and amplitude characteristics of the current loop can be equivalent to a proportional link with a gain of 1.

[0058] Furthermore, the expression of the closed-loop transfer function of the speed loop is:

[0059]

[0060] Among them, G v (s) is the response speed of the speed loop of the CNC machine tool feed servo system, R v (s) is the input speed, C v (s) is the actual output speed, m is the equivalent mass, c is the viscous friction coefficient, K m is the motor thrust coefficient, K vfr is the feedforward proportional controller, K vp is the speed proportional gain, K vi is the speed integral gain.

[0061] Furthermore, the expression of the position loop transfer function is:

[0062]

[0063] Where R x (s) is the position loop input transfer function, C x (s) is the position loop output transfer function, K pvfr is the position loop speed feedforward gain, K pafr is the acceleration feedforward gain of the position loop, K pp is the position loop proportional gain.

[0064] Step S2: determining a mechanical transmission power system;

[0065] Perform parameter identification on the mechanical transmission power system based on the frequency domain least square method to obtain the identification parameters of the mechanical transmission power system;

[0066] Substituting the identified parameters into the simplified CNC machine tool feed servo system in step S1 to obtain the CNC machine tool feed servo system;

[0067] Optionally, the identification parameters in step S2 include: equivalent mass, viscous damping coefficient and Coulomb friction of the mechanical transmission power system.

[0068] Optionally, the expression of the mechanical transmission model is:

[0069]

[0070] Where, I is the input current of the motor, K is the torque constant of the motor, is the motor acceleration of the mechanical transmission power system, is the motor speed of the mechanical transmission power system, m is the equivalent mass of the mechanical transmission power system, c is the viscous damping coefficient of the mechanical transmission power system, and f is the Coulomb friction of the mechanical transmission power system.

[0071] Optionally, the specific steps of performing parameter identification include:

[0072] Linearize the mechanical transmission power system to obtain the mechanical transmission linear equation;

[0073] Perform discrete fast Fourier transform on both sides of the mechanical transmission linear equation to achieve frequency domain conversion and obtain the mechanical transmission frequency domain linear equation;

[0074] The mechanical transmission frequency domain linear equation is converted into a matrix form to obtain the mechanical transmission frequency domain matrix. The mechanical transmission frequency domain matrix is calculated based on the frequency domain least squares method to obtain the identification parameters.

[0075] The expression of the mechanical transmission linear equation is:

[0076]

[0077] in, represents the frequency domain derivative of the velocity signal at time k, j is the imaginary part; w k represents the angular frequency at time k; represents the actual speed at time k; represents the collected current at time k; I o represents the bias current; represents the thrust coefficient of the mechanical transmission model at time k; sign(.) represents the sign function; is the time domain complex variable matrix of the Fourier transform of the system state; is the Fourier transform of the observed variable in the frequency domain; θ is the parameter vector to be identified, m is the equivalent mass of the mechanical transmission power system, and c is the viscous damping coefficient of the mechanical transmission power system.

[0078] Alternatively, the expression of the mechanical transmission frequency domain matrix is:

[0079]

[0080] in, is the output matrix of mechanical transmission frequency domain, is the input matrix of mechanical transmission frequency domain.

[0081] Optionally, the identification parameter expression of the mechanical transmission power system is:

[0082]

[0083] in, represents the conjugate transpose of the input matrix in the mechanical transmission frequency domain, Re(.) represents the real part extraction function of the complex number, and θ represents the identification parameter.

[0084] Step S3: Optimizing the controller parameters of the CNC machine tool feed servo system by using an adaptive genetic algorithm to complete the self-tuning of the CNC machine tool feed servo system parameters.

[0085] Optionally, the specific steps for completing the complete CNC machine tool feed servo system parameter self-tuning include:

[0086] Preset control parameter value range; it can be understood that the control parameter is a controller parameter of the CNC machine tool feed servo system;

[0087] Based on the control parameter value range, randomly select N groups of control parameters;

[0088] Optionally, the controller parameters include position loop parameters, speed loop parameters and current loop parameters;

[0089] Exemplarily, the speed loop parameters include the proportional coefficient K pp , speed feedforward coefficient K pvfr , acceleration feedforward coefficient K pafr ;

[0090] Represent each set of control parameters as a corresponding population, and obtain N populations;

[0091] Determine multiple populations for the i-th iteration; i = 1, 2, 3…I, where I represents the total number of steps.

[0092] Constructing a fitness objective function, and obtaining the fitness of each population in the i-th iteration step based on the fitness objective function;

[0093] Perform roulette wheel selection on multiple populations in the i-th iteration step to obtain the updated population of the i-th iteration step;

[0094] Perform a crossover operation on the updated population of the i-th iteration step and sort them according to the fitness of each updated population to obtain multiple populations after one sorting;

[0095] Perform mutation operations on the multiple populations after the first sort and sort them according to the fitness of each population after the first sort to obtain multiple populations after the second sort. Select the population with the largest fitness as the final population in the i-th iteration step;

[0096] After traversing I iteration steps, I final populations are obtained. The I final populations are arranged in ascending order according to the fitness of each final population. The control parameters corresponding to the final population with the largest fitness are selected as the optimal control parameters, which are used as the optimal parameters for the CNC machine tool feed servo system to complete the parameter self-tuning.

[0097] Optionally, the fitness objective function is expressed as:

[0098]

[0099] BsJ n =∫0 ∞ (w1|e(t,n)|+w2u 2 (t,n)+w3|y(t,n)-y(t-1,n)|)dt+w4t u

[0100] Where, e(t,n) is the system error of the nth population at time t; u(t) is the controller output of the nth population at time t; y(t,n) is the actual output of the nth population at time t; t u is the adjustment time, which represents the time required for the CNC machine tool feed servo system to reach a stable state from the initial state; w1 is the weight coefficient of the error term, w2 is the weight coefficient of the control input term, w3 is the weight coefficient of the output smoothing term, w4 is the weight coefficient of the time penalty term, BsJ is the performance evaluation index, which measures the dynamic performance and control effect of the CNC machine tool feed servo system, and fn is the fitness value of the nth population.

[0101] Optionally, the specific steps of obtaining the updated population at the i-th iteration step include:

[0102] Get the total fitness A of multiple populations in the i-th iteration step i ;

[0103] Get the fitness of each population in the i-th iteration step as a percentage of the total fitness A i The probability of being selected is represented as the probability of being selected;

[0104] In the i-th iteration, after sorting each population by fitness, the corresponding cumulative probability is obtained based on the probability of selection of each population, which is represented by the area it occupies on the roulette wheel;

[0105] Based on the total fitness A i Set the random number selection range B i ; In the random number selection range B i Select the pth i random numbers;

[0106] Traverse the cumulative probability of each population in the i-th iteration step and find the first one that satisfies the cumulative probability greater than or equal to the p-th i A population of random numbers, as the pth i Screen the population and record it; the number of random numbers is the same as the population size; if it does not meet the requirement, let p i =p i +1;

[0107] Traverse P i Random numbers are screened to obtain the M i populations, among which M i Less than or equal to P i ;

[0108] Judge M i Is it equal to P i If so, the filtered M i The population is used as the updated population for the i-th iteration. If not, sort the multiple populations after screening in ascending order according to fitness, and select the top P populations with the largest fitness. i -M i The population pairs after screening M i The population is supplemented and P is obtained. i An updated population is used as the updated population for the i-th iteration step.

[0109] Optionally, the cumulative probability is expressed as:

[0110]

[0111] Where cum(n) is the cumulative probability of the nth population; N is the number of populations; fc(n) is the fitness value of the nth population, is the total fitness.

[0112] Optionally, the specific steps of obtaining multiple populations after a sorting process include:

[0113] Get the average fitness variance of the population, which is represented as population diversity;

[0114] Obtaining the adaptive crossover rate of each population based on population diversity;

[0115] Based on the adaptive crossover rate of each population, two populations are cross-operated to obtain multiple secondary updated populations, and the multiple secondary updated populations are sorted in ascending order according to the corresponding fitness to obtain multiple populations after one sorting;

[0116] Optionally, the specific steps of obtaining the multiple populations after secondary sorting include:

[0117] Obtain the adaptive mutation rate of each population based on population diversity;

[0118] Based on the adaptive mutation rate of each population, mutation operations are performed on the sorted populations to obtain multiple three-times updated populations. The multiple three-times updated populations are sorted in ascending order according to their corresponding fitness to obtain multiple populations after secondary sorting. The population with the largest fitness is selected as the final population in the i-th iteration step.

[0119] Optionally, the expression of the adaptive crossover rate is:

[0120]

[0121] P a =P a_min ,fc(n)>fc avg

[0122] Among them, P a is the adaptive crossover probability; P a_max is the maximum crossover probability, P a_min is the minimum crossover probability; fc avg is the average fitness value; fc max is the maximum fitness value;

[0123] Optionally, the expression of the adaptive mutation rate is:

[0124]

[0125] P b =P b_min ,fc>fc avg

[0126] Among them, fc avg is the average fitness value, fc max is the maximum fitness value, P b is the adaptive mutation probability; P b_max is the maximum mutation probability, P b_min is the minimum mutation probability.

[0127] Furthermore, after discretizing the transfer function of the velocity loop, the high-level language C++ is input into VisualStudio 2019, and the parameter self-tuning analysis of the velocity loop can be performed.

[0128] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A parameter self-tuning method for a feed servo system based on an adaptive genetic algorithm, characterized in that: include: Step S1: constructing an original CNC machine tool feed servo system and simplifying it to obtain a simplified CNC machine tool feed servo system; Step S2, determining the mechanical transmission power system of the CNC machine tool feed servo system; Perform parameter identification on the mechanical transmission power system based on the frequency domain least square method to obtain the identification parameters of the mechanical transmission power system; Substituting the identified parameters into the simplified CNC machine tool feed servo system in step S1 to obtain the CNC machine tool feed servo system; Step S3: Optimizing the controller parameters of the CNC machine tool feed servo system by using an adaptive genetic algorithm to complete the self-tuning of the CNC machine tool feed servo system parameters.

2. The parameter self-tuning method of the feed servo system according to claim 1, characterized in that: The identification parameters include: equivalent mass, viscous damping coefficient and Coulomb friction of the mechanical transmission power system.

3. The parameter self-tuning method of the feed servo system according to claim 1, characterized in that: The expression of the mechanical transmission power system is: Where, I is the input current of the motor, K is the torque constant of the motor, is the motor acceleration of the mechanical transmission power system, is the motor speed of the mechanical transmission power system, m is the equivalent mass of the mechanical transmission power system, c is the viscous damping coefficient of the mechanical transmission power system, and f is the Coulomb friction of the mechanical transmission power system.

4. The parameter self-tuning method of the feed servo system according to claim 1, characterized in that: The specific steps for parameter identification include: After linearizing the mechanical transmission power system, discrete fast Fourier transform is performed to obtain the mechanical transmission frequency domain linear equation; The mechanical transmission frequency domain linear equation is converted into matrix form, and then the frequency domain least square method is used to calculate and obtain the identification parameters.

5. The parameter self-tuning method of the feed servo system according to claim 1, characterized in that: The specific steps to complete the complete CNC machine tool feed servo system parameter self-tuning include: Preset the range of control parameter values and randomly select N groups of control parameters; Construct fitness objective function; Represent each set of control parameters as a corresponding population, and obtain N populations; Determine multiple populations for the i-th iteration; i = 1, 2, 3…I, where I represents the total number of steps; Based on the fitness objective function, obtaining the fitness of each population in the i-th iteration step; Perform roulette wheel selection on multiple populations in the i-th iteration step to obtain the updated population of the i-th iteration step; Perform crossover, mutation and sorting operations on the updated population of the i-th iteration step to obtain the final population of the i-th iteration step; After traversing I iteration steps, I final populations are obtained and sorted, and the control parameters corresponding to the final population with the largest fitness are selected as the optimal control parameters to complete the parameter self-tuning.

6. The parameter self-tuning method of the feed servo system according to claim 1, characterized in that: The expression of the fitness objective function is: BsJ n =∫0 ∞ (w1|e(t,n)|+w2u 2 (t,n)+w3|y(t,n)-y(t-1,n)|)dt+w4t u Where, e(t,n) is the system error of the nth population at time t; u(t) is the controller output of the nth population at time t; y(t,n) is the actual output of the nth population at time t; t u is the adjustment time; w1 is the weight coefficient of the error term, w2 is the weight coefficient of the control input term, w3 is the weight coefficient of the output smoothing term, w4 is the weight coefficient of the time penalty term, BsJ is the performance evaluation index; fn is the fitness value of the nth population.

7. The parameter self-tuning method of the feed servo system according to claim 1, characterized in that: The specific steps of performing crossover, mutation and sorting on the updated population of the i-th iteration step to obtain the final population of the i-th iteration step include: Get the average fitness variance of the population, which is represented as population diversity; Obtaining the adaptive crossover rate of each population based on population diversity; Based on the adaptive crossover rate of each population, two populations are cross-operated to obtain multiple secondary updated populations, and the multiple secondary updated populations are sorted in ascending order according to the corresponding fitness to obtain multiple populations after one sorting; Obtain the adaptive mutation rate of each population based on population diversity; Based on the adaptive mutation rate of each population, mutation operations are performed on the sorted multiple populations to obtain multiple three-times updated populations. The multiple three-times updated populations are arranged in ascending order according to the corresponding fitness to obtain multiple populations after secondary sorting. The population with the largest fitness is selected as the final population in the i-th iteration step.

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