A fractional order controller parameter tuning method based on rotating Hankel matrix and multi-objective genetic algorithm NSGA3

Abstract

The application is a fractional order controller parameter setting method based on a rotating Hankel matrix and a multi-objective genetic algorithm NSGA3. The application relates to the technical field of target constraint controller optimization design, and utilizes the multi-objective genetic algorithm NSGA3 to optimize fractional order controller parameters, so as to complete the design of a fractional order controller meeting the required indexes of a system; the setting of a reference point, the adaptive standardization of a population, the correlation operation and the individual reservation operation. The fractional order controller design method provided by the application has a significant improvement in overshoot and regulation time compared with existing methods. Under the condition of meeting the same required indexes of a system, the slope of a phase characteristic curve obtained by the fractional order controller designed by the method of the application is closer to 0, and the robustness of the system is improved.

Description

Technical Field

[0001] This invention relates to the field of target-constrained controller optimization design technology, and is a fractional-order controller parameter tuning method based on the rotated Hankel matrix and the multi-objective genetic algorithm NSGA3. Background Technology

[0002] Fractional controller parameter tuning methods are mainly divided into time-domain methods and frequency-domain methods. The time-domain method primarily uses the dominant pole method, which allocates dominant poles based on the system's desired time-domain indices to obtain the relationships between unknown parameters, and then calculates the corresponding controller parameters. This method has low tuning efficiency and high computational cost, and is not as widely used as the frequency-domain method. The frequency-domain method is mainly a parameter tuning method based on gain margin and phase margin. It uses the system robustness constraints as the objective function and the performance indices satisfied in the frequency domain as constraints, obtaining the constraint equations satisfied by the system. The Fmincon nonlinear function in the MATLAB optimization toolbox is used to solve these equations, completing the parameter tuning of the fractional controller. This method is relatively mature and widely used in fractional controller parameter tuning. However, due to the non-uniqueness of the solutions to the nonlinear equations, the efficiency and accuracy of parameter tuning are not high. Therefore, the above methods still have the following drawbacks:

[0003] (1) Traditional fractional-order controller parameter tuning methods involve large computational loads, repeated system simulation and parameter tuning, and low tuning efficiency.

[0004] (2) It cannot simultaneously take into account the time domain performance and frequency domain performance of the system, resulting in poor system robustness.

[0005] (3) It is difficult to solve the nonlinear equations satisfied by the system, and the accuracy of the obtained fractional-order controller parameters is low. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, this invention aims to provide a tuning method based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3, thereby improving the tuning efficiency of parameters and the robustness of the system.

[0007] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0008] This invention provides a method for tuning fractional-order controller parameters based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3. The invention provides the following technical solutions:

[0009] A method for tuning fractional-order controller parameters based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3, the method comprising the following steps:

[0010] Step 1: Approximate the fractional operator by rotating the Hankel matrix;

[0011] Step 2: Optimize the fractional-order controller parameters using the multi-objective genetic algorithm NSGA3 to complete the design of a fractional-order controller that meets the system's required performance indicators;

[0012] Step 3: Setting up reference points, adaptive standardization of the population, association operations, and individual retention operations.

[0013] Preferably, step 1 specifically comprises:

[0014] Set f(t) to be continuous in the interval [a, b], and generate N equally spaced nodes in the interval [0, t] with a step size of h, where t i =t0+ih (i=0,1,2,...,N), the first derivative of f(t) is expressed using backward difference as follows:

[0015]

[0016] The backward difference derivative expression for N nodes is determined as follows:

[0017]

[0018] The above system of equations can be expressed in vector matrix form as follows:

[0019]

[0020] Where F is the vector matrix of f′(t), It is a first-order backward difference vector matrix, F N It is a matrix of function value vectors obtained at equidistant nodes;

[0021] The nth derivative backward difference vector matrix of f(t) is obtained through the above transformation. It is expressed as follows:

[0022]

[0023] Among them, w j Given by equation (5)

[0024]

[0025] according to Letnikov's definition of the fractional derivative:

[0026]

[0027] Where k = 0, 1, 2, ..., N, the fractional derivative under the definition is actually calculated by... k Approximating the equation using the backward difference at position , the above equation can be written in the following matrix form:

[0028]

[0029] Backward difference coefficient matrix of fractional derivative It is expressed as follows:

[0030]

[0031] Difference coefficient w k The calculation is completed using equation (9).

[0032]

[0033] Through the above transformation, the fractional operator is expressed by the approximate value of the vector. The matrix in equation (8) is rotated 90° counterclockwise, which is called the Hankel matrix. It is converted into matrix multiplication for calculation, making the calculation of fractional calculus operators simpler.

[0034] Preferably, the reference point is specifically:

[0035] A predefined set of reference points is used to ensure solution diversity. This set of reference points is defined in a structured manner or set by the user. Reference points are uniformly generated on the normalized edge plane according to the following formula. These reference points lie on a (m-1)-dimensional hyperplane, where m is the number of objective objectives. When each objective is divided into H parts, the number of reference points is...

[0036]

[0037] Preferably, the adaptive standardization of the population specifically involves: first solving for the minimum value of all objectives of this generation of the population, and then selecting the current population S. t The minimum value of each dimension of the objective of an individual constitutes the ideal point of the current population, and the population S is then... t Perform a translation operation to make the ideal point the origin. Then calculate the extreme points. Take the point corresponding to the minimum value of the above scalar function, where w i The unit direction vectors of the coordinate axes;

[0038] Construct a hyperplane, find the intercepts, and normalize the objective. The three lines formed by the extreme points and the origin and ideal points constitute a surface. The intersection of this surface with the three coordinate axes is the intercept a1, a2, and a3 to be solved. After finding the intercepts, normalize them according to the following equation.

[0039]

[0040] Preferably, the association operation specifically involves: associating each individual in the group with a corresponding reference point: the line connecting the origin and the reference point is used as the reference line, and S is calculated. t The distance from an individual to each reference line is used to determine the relationship between that individual and the corresponding reference line.

[0041] Preferably, the individual retention operation specifically involves: selecting individuals to enter the next generation; two situations may occur in the association operation: one is that the reference point is associated with one or more individuals; the other is that no individual is associated with it; retention principle: individuals corresponding to reference points with fewer connections should be retained to maintain diversity. In the selection process, NSGA3 emphasizes the dominance relationship and, in order to ensure individual diversity, also emphasizes the number of individuals associated with each parameter point.

[0042] Preferably, the method further includes a multi-objective optimization process, specifically:

[0043] Step S1: For a certain number of individuals in the population, perform random initialization;

[0044] Step S2: Calculate the objective function value for each individual in the population;

[0045] Step S3: For those that do not meet the performance indicators, the process will automatically proceed to the next generation of optimization.

[0046] Step S4: Assign fitness values ​​to each individual in the population, and scalarize the vector of objective function values ​​of each individual to convert it into a single fitness value;

[0047] Step S5: Select a certain number of individuals from the population based on the obtained fitness values, and perform genetic operations such as crossover and mutation on the individuals according to a certain probability to form a new generation of offspring individuals.

[0048] Step S6: Recalculate the objective function value for the new generation of child individuals, insert the child individuals into the parent individuals, and form a new generation of parent individuals;

[0049] The above process is repeated until the optimization objective is met, at which point it stops.

[0050] A fractional-order controller parameter tuning system based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3, the system comprising:

[0051] The preprocessing module approximates the fractional operator by rotating the Hankel matrix;

[0052] The design module utilizes the multi-objective genetic algorithm NSGA3 to optimize the parameters of the fractional-order controller, thereby completing the design of a fractional-order controller that meets the system's required indicators.

[0053] The multi-objective optimization module includes setting reference points, adaptive standardization of the population, association operations, and individual retention operations.

[0054] A computer-readable storage medium having a computer program stored thereon, which is executed by a processor to implement a fractional-order controller parameter tuning method based on a rotated Hankel matrix and a multi-objective genetic algorithm NSGA3.

[0055] A computer device includes a memory and a processor, the memory storing a computer program, and the processor executing the computer program to implement a fractional-order controller parameter tuning method based on a rotated Hankel matrix and a multi-objective genetic algorithm NSGA3.

[0056] The present invention has the following beneficial effects:

[0057] Compared with existing methods, the fractional-order controller design method proposed in this invention significantly improves overshoot and settling time. Under the same system requirements, the phase characteristic curve of the fractional-order controller designed using this invention has a slope closer to 0, thus improving the system's robustness. Attached Figure Description

[0058] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0059] Figure 1 This is a diagram of the fractional-order controller structure.

[0060] Figure 2 This is a diagram of the servo system architecture.

[0061] Figure 3 This is a flowchart of the fractional-order controller design method of the present invention;

[0062] Figure 4 A comparison diagram of the step response curves obtained by applying the multi-objective fractional-order controller design method of the present invention to the same system with the existing single-objective design method;

[0063] Figure 5 This is a comparison diagram of Bode plots obtained by applying the multi-objective fractional-order controller design method of the present invention to the same system as the existing single-objective method. Detailed Implementation

[0064] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0065] In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0066] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances. Specific Implementation Example 1:

[0068] according to Figures 1 to 5 As shown, the specific optimization technical solution adopted by the present invention to solve the above-mentioned technical problems is: The present invention relates to a method for tuning fractional-order controller parameters based on a rotated Hankel matrix and a multi-objective genetic algorithm NSGA3.

[0069] A method for tuning fractional-order controller parameters based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3, characterized by the following steps:

[0070] Step 1: Approximate the fractional operator by rotating the Hankel matrix;

[0071] Step 2: Optimize the fractional-order controller parameters using the multi-objective genetic algorithm NSGA3 to complete the design of a fractional-order controller that meets the system's required performance indicators;

[0072] Step 3: Setting up reference points, adaptive standardization of the population, association operations, and individual retention operations. Specific Implementation Example 2:

[0074] The only difference between Embodiment 2 and Embodiment 1 of this application is that:

[0075] Step 1 specifically involves:

[0076] Set f(t) to be continuous in the interval [a, b], and generate N equally spaced nodes in the interval [0, t] with a step size of h, where t i =t0+ih (i=0,1,2,…,N), the first derivative of f(t) is expressed using backward difference as follows:

[0077]

[0078] The backward difference derivative expression for N nodes is determined as follows:

[0079]

[0080] The above system of equations can be expressed in vector matrix form as follows:

[0081]

[0082] Where F is the vector matrix of f′(t), It is a first-order backward difference vector matrix, F N It is a matrix of function value vectors obtained at equidistant nodes;

[0083] The nth derivative backward difference vector matrix of f(t) is obtained through the above transformation. It is expressed as follows:

[0084]

[0085] Among them, w j Given by equation (5)

[0086]

[0087] according to Letnikov's definition of the fractional derivative:

[0088]

[0089] Where k = 0, 1, 2, ..., N, the fractional derivative under the definition is actually calculated by... k Approximating the equation using the backward difference at position , the above equation can be written in the following matrix form:

[0090]

[0091] Backward difference coefficient matrix of fractional derivative It is expressed as follows:

[0092]

[0093] Difference coefficient w k The calculation is completed using equation (9).

[0094]

[0095] Through the above transformation, the fractional operator is expressed by the approximate value of the vector. The matrix in equation (8) is rotated 90° counterclockwise, which is called the Hankel matrix. It is converted into matrix multiplication for calculation, making the calculation of fractional calculus operators simpler. Specific Implementation Example 3:

[0097] The only difference between Embodiment 3 and Embodiment 2 of this application is that:

[0098] The reference point is specifically:

[0099] A predefined set of reference points is used to ensure solution diversity. This set of reference points is defined in a structured manner or set by the user. Reference points are uniformly generated on the normalized edge plane according to the following formula. These reference points lie on a (m-1)-dimensional hyperplane, where m is the number of objective objectives. When each objective is divided into H parts, the number of reference points is...

[0100] Specific Implementation Example 4:

[0102] The only difference between Embodiment 4 and Embodiment 3 of this application is that:

[0103] The adaptive standardization of the population specifically involves: first, finding the minimum value of all objectives for this generation of the population, and then selecting the current population S. t The minimum value of each dimension of the objective of an individual constitutes the ideal point of the current population, and the population S is then... t Perform a translation operation to make the ideal point the origin. Then calculate the extreme points. Take the point corresponding to the minimum value of the above scalar function, where w i The unit direction vectors of the coordinate axes;

[0104] Construct a hyperplane, find the intercepts, and normalize the objective. The three lines formed by the extreme points and the origin and ideal points constitute a surface. The intersection of this surface with the three coordinate axes is the intercept a1, a2, and a3 to be solved. After finding the intercepts, normalize them according to the following equation.

[0105] Specific Implementation Example 5:

[0107] The only difference between Embodiment 5 and Embodiment 4 of this application is that:

[0108] The association operation specifically involves: associating each individual in the group with a corresponding reference point; using the line connecting the origin and the reference point as the reference line, and calculating S... t The distance from an individual to each reference line is used to determine the relationship between that individual and the corresponding reference line. Specific Implementation Example Six:

[0110] The only difference between Embodiment Six and Embodiment Five of this application is that:

[0111] The individual retention operation specifically involves selecting individuals to enter the next generation. There are two scenarios for the association operation: one is that the reference point is associated with one or more individuals; the other is that no individual is associated with it. The retention principle is that individuals corresponding to reference points with fewer connections should be retained to maintain diversity. In the selection process, NSGA3 emphasizes the dominance relationship and, in order to ensure individual diversity, also emphasizes the number of individuals associated with each parameter point. Specific Implementation Example 7:

[0113] The only difference between Embodiment 7 and Embodiment 6 of this application is that:

[0114] The method also includes a multi-objective optimization process, specifically:

[0115] Step S1: For a certain number of individuals in the population, perform random initialization;

[0116] Step S2: Calculate the objective function value for each individual in the population;

[0117] Step S3: For those that do not meet the performance indicators, the process will automatically proceed to the next generation of optimization.

[0118] Step S4: Assign fitness values ​​to each individual in the population, and scalarize the vector of objective function values ​​of each individual to convert it into a single fitness value;

[0119] Step S5: Select a certain number of individuals from the population based on the obtained fitness values, and perform genetic operations such as crossover and mutation on the individuals according to a certain probability to form a new generation of offspring individuals.

[0120] Step S6: Recalculate the objective function value for the new generation of child individuals, insert the child individuals into the parent individuals, and form a new generation of parent individuals;

[0121] The above process is repeated until the optimization objective is met, at which point it stops. Specific Implementation Example 8:

[0123] The difference between Embodiment 8 and Embodiment 7 of this application lies only in:

[0124] The board provides a fractional-order controller parameter tuning system based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3, the system comprising:

[0125] The preprocessing module approximates the fractional operator by rotating the Hankel matrix;

[0126] The design module utilizes the multi-objective genetic algorithm NSGA3 to optimize the parameters of the fractional-order controller, thereby completing the design of a fractional-order controller that meets the system's required indicators.

[0127] The multi-objective optimization module includes setting reference points, adaptive standardization of the population, association operations, and individual retention operations. Specific Implementation Example Nine:

[0129] The difference between Embodiment Nine and Embodiment Eight in this application lies only in:

[0130] The present invention provides a computer-readable storage medium having a computer program stored thereon, which is executed by a processor to implement a fractional-order controller parameter tuning method based on a rotated Hankel matrix and a multi-objective genetic algorithm NSGA3. Specific Implementation Example 10:

[0132] The only difference between Embodiment 10 and Embodiment 9 of this application is that:

[0133] The present invention provides a computer device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it implements a fractional-order controller parameter tuning method based on a rotated Hankel matrix and a multi-objective genetic algorithm NSGA3. Specific Implementation Example Eleven:

[0135] The technical solution of the present invention will be further described in detail below.

[0136] Fractional order PI λ D μ The time-domain transfer function C(s) of the controller is expressed as follows:

[0137]

[0138] Fractional order PI λ D μ The controller structure diagram is as follows Figure 1 As shown.

[0139] Suppose that the fractional transfer function P(s) of a controlled system has the following form:

[0140]

[0141] The open-loop transfer function G(s) formed by the controlled object and the fractional-order controller is as follows:

[0142]

[0143] in

[0144]

[0145]

[0146] Taking the inverse Laplace transform of the above equations yields the fractional operators in the equations. By substituting the fractional differential operators in the above equations using equation (8), we obtain the following equations:

[0147]

[0148] in

[0149]

[0150]

[0151] Figure 2 This is a closed-loop system structure diagram consisting of a fractional-order controller C(s) and the controlled object P(s), where R(s), E(s), and Y(s) are the external input signal, error signal, and controlled object output, respectively.

[0152] The system's open-loop transfer function frequency response satisfies the following conditions:

[0153] (1) The open-loop transfer function of the control system at the cutoff frequency w cg The amplitude characteristics satisfy equation (20).

[0154] |C(jw cg )P(jw cg )|-1=0 (20)

[0155] (2) The open-loop transfer function of the control system at the cutoff frequency w cg The phase angle characteristics satisfy equation (21).

[0156]

[0157] (3) To ensure the system's robustness to gain changes, this invention requires the system phase to satisfy the crossover frequency w. cg Based on the "flat" foundation, let the system be at w cg The surrounding area is largely flat, enhancing robustness. The phase derivative of the open-loop transfer function of the improved system satisfies the following relationship:

[0158]

[0159] (4) Define the integral of the absolute value of error (ITAE) index to constrain the steady-state error of the system:

[0160]

[0161] (5) To ensure the system has output interference immunity, in the low-frequency band w≤w s The system sensitivity function S(s) satisfies equation (24).

[0162]

[0163] (6) To achieve high-frequency noise reduction in the system, in the high-frequency band w≥w t The sensitivity function T(s) satisfies equation (25).

[0164]

[0165] By rotating the Hankel matrix, the series of fractional differential equations satisfied by the system are transformed into algebraic differential equations. The conditions satisfied by the system constitute a constrained nonlinear programming optimization problem. Solving this problem using NSGA3 yields the parameters (K) of the fractional controller. p ,K i ,K d ,λ,μ).

[0166] By solving equations (20-25) using the multi-objective genetic optimization algorithm NSGA3, the following optimization model for the multi-objective fractional problem is constructed:

[0167] min f={f1(x),f2(x),f3(x)…},X=[K p ,K i ,K d ,λ,μ]

[0168]

[0169] The flowchart of the NSGA3 algorithm for solving the above fractional multi-objective problem is as follows: Figure 3 As shown.

[0170] The fractional-order controller design method of this invention will be introduced below using a gyroscope-stabilized platform servo system as an example.

[0171] The transfer function P(s) of a certain type of gyroscope stabilization platform is as follows:

[0172]

[0173] The initial population size of the NSGA3 optimization algorithm is set to 200, the maximum number of iterations is set to 10, and the crossover percentage and mutation percentage are both 0.5. Here, the actual engineering requirements for the gyroscope stabilization platform system bandwidth frequency are greater than 10 rad / s, and the phase margin is... It should be greater than 70°; the sensitivity function should satisfy the condition |S(jw)|≤-20dB, where w≤w s =0.01 rad / s; the complementary sensitivity function satisfies the condition |T(jw)| ≤ -20 dB, where w ≥ w t =100 rad / s. Therefore, the preset system parameters are as follows: w cg =10 rad / s, A = B = -20 dB, w s =0.01rad / s, w t =100 rad / s.

[0174] After 10 iterations of calculation, the optimized fractional-order controller that satisfies the system performance indicators is as follows:

[0175]

[0176] In the technical background, the traditional single-objective design method uses equation (22) as the objective function and equations (20) and (21) as nonlinear constraints, and solves them using the Fmincon function in the MATLAB optimization toolbox. The multi-objective design method of this invention is compared with existing technologies below. The obtained closed-loop system step response curve is shown below. Figure 4 As shown, the Bode diagram of the open-loop system is as follows: Figure 5 As shown.

[0177] from Figure 4 It can be seen that the fractional-order controller design method proposed in this invention significantly improves overshoot and settling time compared with existing methods. Figure 5 It can be concluded that, under the same system requirements, the phase characteristic curve of the fractional-order controller designed by the method of this invention has a slope closer to 0, and the robustness of the system is improved.

[0178] In the description of this specification, references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or N embodiments or examples. Furthermore, those skilled in the art can combine and integrate the different embodiments or examples described in this specification and the features of different embodiments or examples without contradiction. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of the present invention, "N" means at least two, such as two, three, etc., unless otherwise explicitly specified. Any process or method described in the flowcharts or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or more N executable instructions for implementing custom logical functions or processes, and the scope of preferred embodiments of the invention includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order according to the functions involved, as will be understood by those skilled in the art to which embodiments of the invention pertain. The logic and / or steps represented in the flowcharts or otherwise described herein, for example, can be considered as a ordered list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-included system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device. More specific examples (a non-exhaustive list) of computer-readable media include the following: an electrical connection having one or N wires (electronic device), a portable computer disk drive (magnetic device), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic device, and portable optical disc read-only memory (CDROM).Furthermore, the computer-readable medium can even be paper or other suitable media on which the program can be printed, since the program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in a computer memory. It should be understood that various parts of the invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, the N steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0179] The above description is merely a preferred embodiment of a fractional-order controller parameter tuning method based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3. The scope of protection for this method is not limited to the above embodiments; all technical solutions falling within this conceptual framework are within the scope of protection of this invention. It should be noted that for those skilled in the art, any improvements and variations made without departing from the principles of this invention should also be considered within the scope of protection of this invention.

Claims

1. A method for setting fractional-order controller parameters based on a rotated Hankel matrix and the multi-objective genetic algorithm NSGA3, characterized by: The method includes the following steps: Step 1: Approximate the fractional operator by rotating the Hankel matrix; Step 2: Setting the reference point for the fractional-order controller, adaptive standardization of the population, association operations, and individual retention operations; Step 3: Optimize the fractional-order controller parameters using the multi-objective genetic algorithm NSGA3 to complete the design of a fractional-order controller that meets the system's required performance indicators; The reference point is specifically: A predefined set of reference points is used to ensure solution diversity. This set of reference points is defined in a structured manner or set by the user. Reference points are uniformly generated on the normalized edge plane according to the following formula. These reference points lie on a (m-1)-dimensional hyperplane, where m is the number of objective optimization targets. When each objective is divided into H parts, the number of reference points is... (10)； The adaptive standardization of the population specifically involves: first, finding the minimum value of all objectives for this generation of the population, and then selecting the current population... The minimum value of each dimension of the objective of an individual constitutes the ideal point of the current population, and the population is then... Perform a translation operation to make the ideal point the origin; then calculate the extreme points. Take the point corresponding to the minimum value of the scalar function, where... The unit direction vectors of the coordinate axes; Construct a hyperplane, find the intercepts, and normalize the objective function. The three lines formed by each extreme point and the origin / ideal point constitute a surface. The intersections of this surface with the three coordinate axes are the intercepts we need to solve for. , and After finding the intercept, normalize it using the intercept according to the following equation; (11) The association operation specifically involves: associating each individual in the group with a corresponding reference point; using the line connecting the origin and the reference point as the reference line, and calculating... The distance of an individual to each reference line is used to determine the relationship between that individual and the corresponding reference line. The individual retention operation specifically involves selecting individuals to enter the next generation. There are two situations in the association operation: one is that the reference point is associated with one or more individuals; the other is that no individual is associated with it. The retention principle is that individuals corresponding to reference points with fewer connections should be retained to maintain diversity. In the selection process, NSGA3 emphasizes the dominance relationship and, in order to ensure individual diversity, also emphasizes the number of individuals associated with each parameter point. The method also includes a multi-objective optimization process, specifically: Step S1: For a certain number of individuals in the population, perform random initialization; Step S2: Calculate the objective function value for each individual in the population; Step S3: For those that do not meet the performance indicators, the process will automatically proceed to the next generation of optimization. Step S4: Assign fitness values ​​to each individual in the population, and scalarize the vector of objective function values ​​of each individual to convert it into a single fitness value; Step S5: Select a certain number of individuals from the population based on the obtained fitness values, and perform genetic operations such as crossover and mutation on the individuals according to a certain probability to form a new generation of offspring individuals. Step S6: Recalculate the objective function value for the new generation of child individuals, insert the child individuals into the parent individuals, and form a new generation of parent individuals; The above process is repeated until the optimization objective is met, at which point it stops.

2. The method according to claim 1, characterized in that: Step 1 specifically involves: set up In the interval Continuous within, within the interval Endogenous generation Each equidistant node has a step size of ,in , The first derivative is expressed using backward difference as follows: (1) Determine The backward difference derivative expression with +1 nodes: (2) The above system of equations can be expressed in vector matrix form as follows: (3) in, yes vector matrix, It is a first-order backward difference vector matrix. It is a matrix of function value vectors obtained at equidistant nodes; The above transformation yields... of backward difference vector matrix of the first derivative It is expressed as follows: (4) in, Given by equation (5) (5) according to Definition of fractional derivative: (6) in The fractional derivative under the definition is actually calculated by... Approximating the equation using the backward difference at position , the above equation can be written in the following matrix form: (7) Backward difference coefficient matrix of fractional derivative It is expressed as follows: (8) Difference coefficient The calculation is completed using equation (9): (9) Through the above transformation, the fractional operator is expressed as an approximation of a vector, and the matrix in equation (8) is rotated counterclockwise. This is called the Hankel matrix; converting it into matrix multiplication makes the calculation of fractional calculus operators simpler.

3. A computer-readable storage medium having a computer program stored thereon, characterized in that, The program is executed by the processor to implement the method as claimed in claim 1.

4. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that: The processor implements the method of claim 1 when executing the computer program.

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