Fractional-order ultra-precision platform identification method based on genetic optimization and data filtering

Abstract

The present invention discloses a fractional-order ultra-precision platform identification method based on genetic optimization and data filtering, which solves the technical problem of reduced control accuracy caused by the nonlinear characteristics of the system in high-precision motion control. The method comprises the following steps: first, an ultra-precision nonlinear platform system model including a fractional-order linear dynamic part and a static nonlinear part is established according to the GL definition. Then, data filtering is performed on the input and output of the system to convert complex colored noise into simple sliding average noise to improve the signal-to-noise ratio. Next, the crossover rule is improved, the crossover probability is dynamically adjusted, the global search capability and convergence speed of the algorithm are enhanced, and the accuracy and robustness of parameter estimation are improved. The present invention has high accuracy and good anti-interference ability in processing the parameter estimation problem of fractional-order nonlinear systems, and is suitable for modeling and control of complex industrial systems of ultra-precision platforms.

Description

Technical Field

[0001] The present invention belongs to the field of control science and engineering, and relates to a method for identifying and controlling a nonlinear system, and is particularly applicable to modeling and control of an ultra-precision motion platform. Background Art

[0002] In recent years, photolithography technology has been one of the technical difficulties in the manufacturing process of highly integrated high-performance chips. The entire photolithography machine can be roughly divided into: ultra-short wavelength photolithography light source, ultra-precision workpiece stage system, ultra-precision mask stage system, ultra-precision optical projection system, cooling system and lighting system. Among them, the ultra-precision platform completes the scanning exposure process through X-axis positioning movement and Y-axis scanning movement in cooperation with other subsystems of the photolithography machine. Therefore, this places extremely high demands on the platform's motion accuracy, stability and dynamic response. However, the nonlinear characteristics of the ultra-precision platform, such as friction, coupled vibration and thermal effects, bring huge challenges to system identification and control.

[0003] Traditional modeling and identification methods regard the entire ultra-precision platform controlled object as an integrated system, so that the linear and nonlinear links therein are identified as a whole, which greatly increases the complexity of the model and the error of system identification. The "linear system I-nonlinear system-linear system II" nonlinear system structure of the present invention is suitable for the model structure of most nonlinear systems. The linear and nonlinear subsystems are separated and analyzed separately, which is beneficial to improving the accuracy of system identification, so as to effectively compensate for the controller output and obtain better control effect. Therefore, it has great significance for practical engineering applications.

[0004] Fractional-order calculus has important applications in various practical industrial fields. The paper Identification of offractional order Hammerstein nonlinear ARMAX system with colored noise proposes a multi-innovation Levenberg-Marquardt (LM) method to identify fractional-order Hammerstein nonlinear ARMAX systems. The paper Identification of fractional-order transfer functions using exponentially modulated signals with arbitrary excitation waveforms performs parameter estimation based on the exponential modulation procedure to determine the coefficients in the fractional-order transfer function. The paper Iterative parameter and order identification for fractional-order nonlinear finite impulse response systems using the key term separation derives an auxiliary model gradient evolution identification algorithm for fractional-order finite impulse response systems. Compared with integer order, the fractional-order system of the present invention can better characterize long memory behavior and infinite-dimensional structure. Therefore, the study of fractional-order systems is of great significance. Compared with these methods, the present invention can be applied to various types of fractional-order systems, including linear, nonlinear, etc., and also has stronger anti-interference ability and robustness.

[0005] Nonlinear systems can be approximated by a block-structured nonlinear model to describe complex linear and nonlinear relationships. In recent years, scholars have conducted research on parameter estimation of nonlinear models. For example, the paper Identification of nonliear models based on variational bayesian approach in the presence of process noise uses Monte Carlo integration to perform a method of approximating the posterior distribution in the variational Bayesian framework of nonlinear systems. The paper Study of random forest to identify Wiener-Hammerstein system proposes a new evolutionary random forest for combined selection of dynamics in nonlinear models. Compared with the above literature, the method of the present invention proposes an efficient and accurate estimation method for fractional-order nonlinear systems, extending the integer-order research to the fractional-order field. Compared with other algorithms, the algorithm of the present invention can effectively model and identify parameters of ultra-precision motion platforms containing complex non-smooth nonlinear characteristics. By introducing data filtering technology, the parameter identification process can be further optimized and the anti-interference ability and robustness of the system can be improved.

[0006] Data filtering technology is an effective parameter estimation method, which is usually used in systems interfered by colored noise. It can achieve a more accurate identification model by extracting useful signals from noise information. At present, data filtering has received widespread attention. The paper The data filtering based multiple stage Levenberg-Marquardtalgorithm for Hammerstein nonlinear systems derives a data filtering multi-stage LM method to obtain interactive identification of each subsystem in the multi-input single-output Hammerstein model. The paper Data filtering-based recursive identification for an exponential autoregressive moving average model by using the multiinnovation theory applies recursive estimation based on data filtering to nonlinear exponential autoregressive models. The paper Data filtering based maximum likelihood gradient estimation algorithms for a multivariate equation-error system with ARMAnoise combines the maximum likelihood idea with data filtering to study multivariate equation error systems. By adopting data filtering technology, the convergence speed and computational efficiency are significantly improved. At present, for ultra-precision platform nonlinear systems containing typical single-valued non-smooth nonlinear links such as dead zone and saturation, some studies have used the key term separation principle to model and applied the identification method based on the general recursive algorithm for parameter identification. However, these methods have limitations when dealing with multi-valued nonlinear links such as gaps, and in practical applications, they may be affected by output noise and reduce the accuracy and stability of identification. Summary of the invention

[0007] The purpose of the present invention is to provide a fractional-order ultra-precision platform identification method based on genetic optimization and data filtering. First, a model of a nonlinear system of a fractional-order nonlinear ultra-precision platform is established, and an identification method based on an improved genetic algorithm with data filtering is proposed to estimate the unknown parameters of the nonlinear ultra-precision platform. Therefore, a genetic optimization and data filtering strategy proposed by the present invention is used to synchronously identify unknown parameters, which effectively improves the shortcomings of the traditional recursive method when facing complex noise.

[0008] The idea of ​​the present invention is: taking the ultra-precision platform as the research object, a class of ultra-precision platform nonlinear systems with fractional nonlinearity is disclosed, and a new solution is provided for the identification of fractional nonlinear systems, data filtering is applied to the ultra-precision platform, and data filtering is combined with an improved genetic algorithm to achieve unbiased parameter estimation for the identification of ultra-precision platform nonlinear systems with fractional nonlinearity. Fractional-order control systems can more accurately describe the dynamic behavior of some physical systems than traditional integer-order control systems, but there are still few studies on fractional orders in nonlinear systems, and there are also many problems that need to be solved. The present invention studies the parameter estimation of ultra-precision platform nonlinear systems with fractional orders. The links with different characteristics of the ultra-precision platform are connected front and back in a modular form, and the system can be simplified into an input linear submodule, a memoryless nonlinear submodule and an output linear submodule. In order to improve the signal-to-noise ratio of the system, data filtering technology is used to simplify the noise interference of the system, and combined with genetic optimization (GA), an improved genetic optimization method based on data filtering is proposed. Finally, the simulation results show that the proposed method is suitable for fractional-order ultra-precision platform nonlinear systems, with high identification accuracy and accurate results.

[0009] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is specifically: a fractional-order ultra-precision platform identification method based on genetic optimization and data filtering, comprising the following steps:

[0010] Step 1) constructing a nonlinear system model of an ultra-precision platform with fractional order, and obtaining a fractional-order nonlinear system model including a fractional-order linear dynamic part and a static nonlinear part;

[0011] Step 2) constructing an identification method flow of an improved genetic algorithm based on data filtering;

[0012] Step 1) initializing the initial factors of the improved genetic algorithm based on data filtering;

[0013] Step 2) collecting the measurable input data of the ultra-precision platform, i.e., the thrust generated by the drive motor, and the output data, i.e., the moving position of the blade, as output, and constructing an output vector;

[0014] Step 3) Initialize the genetic algorithm and randomly generate L individuals as the initial population Θ 0 After k evolution times k = 1, 2, ..., construct the lth individual, which is related to the parameter vector θ f Estimated value of related;

[0015] Step 4) Calculate the evolution k-times population Θ k The fitness of each individual in

[0016] Step 5) Increase the number of evolutions k by one and calculate the estimated value of the intermediate variable in the input linear part Estimated values ​​of intermediate variables in the filtering part Output the estimated value of the intermediate variable in the linear part and the estimated value of the intermediate variable of the noise part

[0017] Step 6) Construct information vector Estimated value of Construct an estimate of the information matrix

[0018] Step 7) Select operator p l Acting on the group, after calculating the nonlinear crossover rule, the crossover operator Υ c Acting on the group, finally the mutation operator Υ m Act on groups;

[0019] Step 8) After the population is selected, crossed, and mutated, the next generation population is obtained, with the individual with the minimum fitness obtained in the evolution process. Output as the optimal solution;

[0020] Step 9) Set the fractional order α k Estimated value of and the filter parameter vector Estimated value of From the global best position Separate from

[0021] Step 10) Execute the evolution cycle until k reaches the total evolution times T, stop the evolution and output the final estimated value; otherwise, return to step 5);

[0022] The further optimization scheme of the fractional-order ultra-precision platform identification method based on genetic optimization and data filtering provided by the present invention is as follows: the step 1) comprises the following steps:

[0023] (1-1) The fractional order is described by the definition as follows:

[0024]

[0025] Among them, D α represents the fractional order with α as the order, f(·) is a nonlinear or linear function, h is the sampling time, g is the number of samples to calculate the derivative approximation at time t = gh, and the Newton binomial coefficient The calculation formula contains the Euler function, which is expressed as

[0026]

[0027] Where the gamma function Γ(·) is

[0028]

[0029] (1-2) Construct a structure of a nonlinear system model of an ultra-precision platform with fractional order. Based on this model, the expression of the nonlinear model of an ultra-precision platform with fractional order is constructed as follows:

[0030]

[0031] w(t)=B(s -α )r(t) (6)

[0032]

[0033] z(t)=w(t)+m(t) (8)

[0034] Among them, t is the time variable, and are the input and output signals of the system, is a function with zero mean and variance σ 2 Gaussian distributed white noise, input linear part intermediate variable The intermediate variables r(t) in the nonlinear part, w(t) in the output linear part, and m(t) in the noise part are all unknown, and H(·) is a nonlinear function. -α Represents the fractional post-shift operator: s -α μ(t)=D α μ(t-1), s -iα μ(t)=(s -i ) α μ(t)=D α μ(ti).

[0035] Among them, A(s -α )、B(s -α )、C(s -α )、D(s -α ) means s -α The constant polynomials are defined as follows:

[0036]

[0037] In the formula, the polynomial factor a p 、b q 、c i ,d j is an unknown parameter.

[0038] (1-3) The nonlinear model of the ultra-precision platform nonlinear system with fractional order is constructed as follows:

[0039] The nonlinear submodule is

[0040]

[0041] in, are the polynomial coefficients, is a polynomial function, n β is the order of the nonlinear polynomial. In order to ensure the uniqueness of the parameters, let the nonlinear prime coefficient be β1=1 and the nonlinear prime function be So, there is

[0042]

[0043] A(s -α ) is substituted into formula (4), and we get

[0044]

[0045] D α μ(tp) is the fractional input variable, a p is the corresponding fractional-order coefficient, p = 1, 2, ..., n a Medium a is a fractional order. Then the nonlinear module output r(t) becomes

[0046]

[0047] (1-4) The linear model of the ultra-precision platform nonlinear system with fractional order is constructed as follows:

[0048] B(s -α ) is substituted into formula (6), and we get

[0049]

[0050] D α r(tq) is a fractional intermediate variable, b p is the corresponding fractional-order coefficient, q = 1, ..., n b Medium b is a fractional order. Multiply both sides of equation (7) by C(s -α ), and C(s -α ) and D(s -α ) into it, we get

[0051] C(s -α )m(t)=D(s -α )e(t)

[0052]

[0053] Extract the key term m(t) as

[0054]

[0055] D α m(ti) is the intermediate variable of fractional-order noise, c i is the corresponding fractional-order coefficient, i = 1, 2, ..., n c Medium c is the fractional order. α e(tj) is fractional colored noise, d j is the corresponding fractional-order coefficient, j = 1, 2, ..., n d Medium d is the fractional order.

[0056] (1-5) Construct an identification model of a fractional-order ultra-precision platform nonlinear system:

[0057] According to formula (8), the output of the system is

[0058]

[0059] In equation (1), the sampling time l in the GL definition is set to 1, and the above equation can be transformed into

[0060]

[0061] The information vector φ(t, α) and parameter vector θ of the fractional-order nonlinear system are defined as

[0062]

[0063] -D α m(t-1),…,-D α m(ti),…,-D α m(tn c ),

[0064]

[0065] n=n a +n b +n c +n d +n β -1

[0066] Where e(t) is the noise input of the nonlinear system of the ultra-precision platform. According to the GL definition,

[0067]

[0068] The final output z(t) is expressed as

[0069] z(t)=φ T(t, α)θ+e(t) (16)

[0070] The further design of the fractional-order ultra-precision platform identification method based on genetic optimization and data filtering is that the steps 1) to 10) specifically include the following steps:

[0071] (2-1) Initialize the initial factors of the improved genetic algorithm based on data filtering. Set the evolutionary generation counter k to 0; set the maximum evolutionary generation T, the crossover coefficients γ1, γ2, γ3, γ4 of the improved genetic algorithm, and the mutation rate γ5;

[0072] (2-2) Collecting measurable input data and output data of the ultra-precision platform and constructing an output vector;

[0073] Using data filtering, multiply both sides of the system equation by the filter to obtain a new system with simplified noise interference. The original system is described as

[0074]

[0075] Using the noise term C(s -α ) is filtered and we get

[0076] C(s -α )z(t)=B(s -α )C(s -α )r(t)+D(s -α )e(t) (18)

[0077] The filtered output z f (t), filtering intermediate term r f (t) and w f (t) is defined as

[0078]

[0079]

[0080]

[0081] Get the controlled moving average system

[0082]

[0083] The polynomial B(s -α )、D(s -α ) and r f Substituting the expanded form of (t) into the above formula, we get

[0084]

[0085] Substituting equation (11) into equation (23), we get

[0086]

[0087] If the sampling time is set to 1 in the GL definition, the above equation can be transformed into

[0088]

[0089] The filtered information vector φ(t, α) and parameter vector θ f Defined as

[0090]

[0091] -D α r(t-1),…,-D α r(ti),…,-D α r(tn c ),

[0092]

[0093] n=n a +n b +n c +n d +n β -1

[0094] in,

[0095]

[0096]

[0097] The filtered model becomes

[0098]

[0099] Among them, z f (t) is the system output after filtering, φ f (t, α) is the filtered information vector and θ f is the parameter vector after filtering.

[0100] M represents the data length, and the output vector Z M for

[0101]

[0102] (2-3) Randomly generate L individuals as the initial population Θ 0 After k evolution times k = 1, 2, ..., construct the lth individual, which is related to the parameter vector θ f Estimated value of related;

[0103] Since the fractional order α is unknown, the dimension of the search space is set to n+1. After k times of evolution k=1, 2, ..., the lth individual is estimated by and composition

[0104]

[0105] in, and is the parameter vector θ f and the estimated value of fractional order α, respectively defined as

[0106]

[0107] (2-4) Calculate the population Θ when the number of evolutions k k The fitness of each individual

[0108] (2-5) Let the number of evolutions k increase by one, and calculate the estimated value of the intermediate variable in the input linear part Estimated values ​​of intermediate variables in the filtering part Output the estimated value of the intermediate variable in the linear part and the estimated value of the intermediate variable of the noise part

[0109] Substituting these estimates for the original unknowns yields

[0110]

[0111] (2-6) Constructing information vector Estimated value of Construct an estimate of the information matrix

[0112] According to equations (28)-(35), we can get the estimated value of the information vector for

[0113]

[0114] Information Matrix for

[0115]

[0116] The corresponding fitness function is

[0117]

[0118] Here, ||·|| represents the bi-norm of the vector.

[0119] (2-7)

[0120] ① Select operator p l Acting on the population: The probability of an individual being selected for reproduction is proportional to its fitness. If p l is the probability of the lth individual being selected, then:

[0121]

[0122] is the fitness of the lth individual, and L is the total number of individuals in the population.

[0123] ②After calculating the nonlinear crossover rule, the crossover operator Υ c Acting on the population: The selected individuals generate new offspring through crossover operation. If the crossover rate is Υ c , and consider two individuals and Descendants It can be expressed as:

[0124]

[0125] Among them, tanh is the hyperbolic tangent function, γ1 and γ4 are used to change the crossover rate Υ c The maximum and minimum values ​​of γ2 are used to adjust the steepness of the transition area between the maximum and minimum values, γ3 is used to adjust the range of the diffusion area, and T is the maximum number of evolution times;

[0126] ③Finally, the mutation operator Υ m Acting on a group: with a small probability Υ m Modify some genes of newborn individuals to introduce mutations and increase the diversity of the population. nk , the mutation operation can be expressed as:

[0127] Θ′ nk =Θ nk +δ, with probability Υ m

[0128] δ is a small random variation, Υ m is the mutation rate.

[0129] (2-8) After the selection, crossover and mutation operations, the next generation of population is obtained, with the individual with the minimum fitness obtained in the evolution process. Output as the optimal solution;

[0130]

[0131] Among them, f(·) is the fitness function, argmin(·) represents the value of the independent variable when the function takes the minimum value;

[0132] (2-9) The fractional order α k Estimated value of and the filter parameter vector Estimated value of From the optimal solution Separate it out, that is

[0133]

[0134] (2-10) Execute the evolution cycle until k reaches the total evolution times T, stop the evolution and output the final estimated value; otherwise, return to step 2-5);

[0135] Compared with the prior art, the present invention has the following beneficial effects:

[0136] (1) The present invention establishes an identification model for a fractional-order ultra-precision platform. Ultra-precision platforms are widely used in the fields of microelectronics, precision instrument manufacturing, and ultra-precision equipment processing. In order to achieve the expected precise control effect, the structural model and parameters of the system must be identified relatively accurately. The key technology of the present invention is to derive the structure of the fractional-order ultra-precision platform system based on data filtering and improved genetic methods, and to solve the parameter identification problem in two cases where the fractional order is unknown. In order to solve the parameter identification problem, the concept of data filtering is used to convert the autoregressive moving average noise into a simple moving average noise, and a new system with simplified noise interference is used to estimate the unknown parameters. Since the research object is an autoregressive moving average noise system, a more accurate identification model can be obtained by using data filtering technology, which significantly improves the convergence speed of the algorithm. Therefore, the technology of the present invention obtains a more accurate model, reduces the computational pressure in the prior art, and greatly improves the computational efficiency of the identification method.

[0137] (2) The present invention identifies ultra-precision platforms with fractional orders based on an improved genetic optimization method based on data filtering. The present invention adopts a genetic algorithm. Since the genetic algorithm can solve the optimization problem of complex nonlinear parameter estimation, all parameters and fractional orders can be identified simultaneously by utilizing its fast search capability. Since the research object is a fractional-order nonlinear system under colored noise, the original system is filtered by a linear filter based on data filtering technology, which greatly enhances the anti-interference and robustness of the algorithm; in addition, a nonlinear crossover rule is introduced to solve the premature convergence problem of the genetic algorithm and improve the identification accuracy of the algorithm. The present invention deeply draws on and integrates genetic algorithms, data filtering technology and other optimization strategies, improves the shortcomings of genetic algorithms, and cleverly integrates these technologies together, which not only improves the identification accuracy of ultra-precision platform nonlinear systems with fractional orders, but also improves the convergence speed of the improved genetic optimization method and enhances the robustness of the algorithm. In short, the identification method of the present invention has accurate calculation, high calculation efficiency, and strong identification accuracy, and is fully suitable for parameter estimation of ultra-precision platform nonlinear systems with fractional orders. BRIEF DESCRIPTION OF THE DRAWINGS

[0138] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0139] Figure 1 It is a schematic diagram of the structure of the ultra-precision motion platform of the present invention;

[0140] Figure 2 is a structural diagram of the fractional-order nonlinear model of the present invention;

[0141] Figure 3 It is a schematic diagram of the flow of improved genetic optimization fractional-order nonlinear system identification based on data filtering of the present invention;

[0142] Figure 4 is the relationship between the estimation error Δ and the number of evolutions k of the improved genetic optimization method based on data filtering in Example 1 of the present invention;

[0143] Figure 5 is a fitting curve of the real output and the estimated output of the improved genetic optimization method based on data filtering for the fractional-order nonlinear system of embodiment 1 of the present invention;

[0144] Figure 6 is the relationship between the estimation error Δ and the number of evolutions k of the improved genetic optimization method based on data filtering in Example 2 of the present invention;

[0145] Figure 7 is the variation of each parameter estimation value of the unknown fractional-order nonlinear system with the evolution k in Embodiment 2 of the present invention;

[0146] Figure 8 This is how the fractional order estimate of the unknown fractional-order nonlinear system of embodiment 2 of the present invention changes with the evolution k. DETAILED DESCRIPTION

[0147] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described here are only used to explain the present invention and are not used to limit the present invention.

[0148] like Figure 1 As shown, the functional module of the variable slit precision motion platform is mainly composed of an X-axis motion component, a Y-axis motion component, a driving component, an air-floating guide rail component, a measuring component, a compensation, a buffer and a safety protection device, etc. As moving parts with two degrees of freedom, the X-axis motion component and the Y-axis motion component have very similar structures and compositions, including blades, blade connectors, air-floating guide rails and a series of anti-collision and limiter components to ensure the flexibility and correlation of the mutual movement between the blade components. The blade holder is connected to the air-floating shaft through an air-floating shaft connector, and then connected to the lower guide rail through a slider, and four motors of the two degrees of freedom, XY, drive them respectively. Therefore, the present invention uses the thrust generated by the driving motor as input and the blade moving position as output to construct a nonlinear system structure of "linear system I-nonlinear system-linear system II".

[0149] See also Figures 1 to 8 , the technical solution of the embodiment is a fractional-order nonlinear ultra-precision platform parameter estimation technology. Compared with traditional integer-order control systems, fractional-order control systems have more advantages in accurately describing the dynamic characteristics of specific physical systems. However, in nonlinear systems, the discussion on fractional-order applications is still relatively rare, and there are many problems that need to be solved. In-depth parameter estimation research has been carried out on ultra-high precision nonlinear systems with fractional-order characteristics. The different characteristic parts of the ultra-precision platform are serially connected through a modular method, that is, the entire system is decomposed into an input linear sub-module, a memoryless nonlinear sub-module, and an output linear sub-module. In order to improve the signal-to-noise ratio of the system, data filtering technology is used to simplify noise interference, and an improved genetic optimization method based on data filtering is innovatively proposed. Simulation experiments show that the proposed method has good applicability to fractional-order ultra-high precision nonlinear systems, can achieve high-precision identification, and the results are accurate.

[0150] Example 1

[0151] Consider the following nonlinear model of an ultraprecision platform with unknown fractional order:

[0152]

[0153] w(t)=B(s -α )r(t)=(1+b1s -α )r(t)=(1+0.82884s -0.81246 )r(t)

[0154]

[0155] z(t)=w(t)+m(t)

[0156] Among them, the fractional order α is the unknown quantity to be identified, 0.81246 is its true value, and the parameter vector and position vector are expressed as:

[0157] θ f =[a1,α2,α3,b1,c1,d1] T

[0158] =[-0.86368, 0.06505, 0.80491, 0.82884, -0.93463, -0.72699] T

[0159]

[0160] In the simulation process, μ(t) uses an uncorrelated and measurable random sequence with zero mean and unit variance, and e(t) uses a white noise sequence σ with zero mean and unit variance. 2 =0.10 2 The accuracy of the identification result is measured by the parameter estimation error Δ, which is expressed as:

[0161]

[0162] Among them, ||·|| represents the bi-norm of the vector, Θ l is the true value of the parameter vector, is the parameter vector estimate.

[0163] Example 2

[0164] The nonlinear model of the ultra-precision platform with unknown fractional order is considered as follows:

[0165]

[0166] w(t)=B(s -α )r(t)=(1+b1s -α +b2s -2α )r(t)

[0167] =(1+0.61864s -0.95789 -0.67954s -1.91578 )r(t)

[0168]

[0169] z(t)=μ(t)+m(t)

[0170] The fractional order α is the unknown quantity to be identified, and 0.95789 is its true value. The parameter vector and position vector are expressed as:

[0171] θ f =[a1, a2, α2, α3, b1, b2, c1, d1] T

[0172] =[-0.87966, 0.92250, 0.21875, 0.85690, 0.61864, -0.67954, 0.79871, -0.41471] T

[0173]

[0174] In the identification process, MATLAB software is used to identify the parameters of the fractional-order nonlinear system under discussion by applying an improved genetic optimization method based on data filtering for the input μ(t), and finally a high-precision input-output model is established.

[0175] The fractional-order ultra-precision platform identification method based on genetic optimization and data filtering in Example 1 and Example 2 comprises the following steps:

[0176] (1) Construct a nonlinear system model of an ultra-precision platform with fractional order, and obtain a fractional-order nonlinear system model including a fractional-order linear dynamic part and a static nonlinear part. The specific steps are as follows:

[0177] Step 1: Construct a structure with a fractional-order ultra-precision platform nonlinear system model, see Figure 1 .

[0178] The fractional order is described by definition as follows:

[0179]

[0180] Among them, D α represents the fractional order with a as the order, f(·) is a nonlinear or linear function, h is the sampling time, g is the number of samples to calculate the derivative approximation at time t = gh, and the Newton binomial coefficient The calculation formula contains the Euler function, which is expressed as

[0181]

[0182] Where the gamma function Γ(·) is

[0183]

[0184] Step 2: Based on this model, the expression of the nonlinear model of the ultra-precision platform with multiple variables is constructed as follows:

[0185]

[0186] w(t)=B(s -α )r(t) (6)

[0187]

[0188] z(t)=w(t)+m(t) (8)

[0189] Among them, t is the time variable, and are the input and output signals of the system, is a function with zero mean and variance σ 2 Gaussian distributed white noise, input linear part intermediate variable The intermediate variables r(t) in the nonlinear part, w(t) in the output linear part, and m(t) in the noise part are all unknown, and H(·) is a nonlinear function. -α Represents the fractional post-shift operator: s -α μ(t)=D α μ(t-1), s -iα μ(t)=(s -i ) α μ(t)=D α μ(ti).

[0190] Among them, A(s -α )、B(s -α )、C(s -α )、D(s -α ) means s -α The constant polynomials are defined as follows:

[0191]

[0192] In the formula, the polynomial factor a p 、b q 、c i ,d j is an unknown parameter.

[0193] Step 3: Construct a nonlinear model of the ultra-precision platform system with fractional order as follows:

[0194] The nonlinear submodule is

[0195]

[0196] in, are the polynomial coefficients, is a polynomial function, n β is the order of the nonlinear polynomial. In order to ensure the uniqueness of the parameters, let the nonlinear prime coefficient be β1=1 and the nonlinear prime function be So, there is

[0197]

[0198] A(s -α ) is substituted into formula (4), and we get

[0199]

[0200] D α μ(tp) is the fractional input variable, a p is the corresponding fractional-order coefficient, p = 1, 2, ..., n a Medium a is a fractional order. Then the nonlinear module output r(t) becomes

[0201]

[0202] Step 4: Construct a linear model of the ultra-precision platform nonlinear system with fractional order as follows:

[0203] B(s -α ) is substituted into formula (6), and we get

[0204]

[0205] D α R(tq) is a fractional intermediate variable, b p is the corresponding fractional-order coefficient, q = 1, ..., n b Medium b is a fractional order. Multiply both sides of equation (7) by C(s -α ), and C(s -α ) and D(s -α ) into it, we get

[0206] C(s -α )m(t)=D(s -α )e(t)

[0207]

[0208] Extract the key term m(t) as

[0209]

[0210] D α m(ti) is the intermediate variable of fractional-order noise, c i is the corresponding fractional-order coefficient, i = 1, 2, ..., n c Medium c is the fractional order. α e(tj) is fractional colored noise, d j is the corresponding fractional-order coefficient, j = 1, 2, ..., n d Medium d is the fractional order.

[0211] Step 5: Construct an identification model of the ultra-precision platform nonlinear system with fractional order:

[0212] According to formula (8), the output of the system is

[0213]

[0214] In equation (1), the sampling time l in the GL definition is set to 1, and the above equation can be transformed into

[0215]

[0216] The information vector φ(t, α) and parameter vector θ of the fractional-order nonlinear system are defined as

[0217]

[0218] -D α m(t-1),…,-D α m(ti),…,-D α m(tn c ),

[0219]

[0220]

[0221] n=n a +n b +n c +n d +n β -1

[0222] Where e(t) is the noise input of the nonlinear system of the ultra-precision platform. According to the GL definition,

[0223]

[0224]

[0225] The final output z(t) is expressed as

[0226] z(t)=φ T (t, α)θ+e(t) (16)

[0227] (2) Construct an identification method flow of an improved genetic algorithm based on data filtering, the specific steps are as follows;

[0228] Step 1: Initialize the initial factors of the improved genetic optimization algorithm based on data filtering;

[0229] Step 2: Collect the measurable input data of the ultra-precision platform, i.e., the thrust generated by the drive motor, and the output data, i.e., the moving position of the blade, to construct the output vector;

[0230] Step 3: Initialize the genetic optimization algorithm, set the maximum evolutionary generation, and randomly generate the initial population;

[0231] Step 4: Calculate the fitness of each individual in the group;

[0232] Step 5: Increase the number of evolutions by one, and calculate the estimated values ​​of the intermediate variables in the linear part of the input and output and the nonlinear part;

[0233] Step 6: Calculate the estimated value of the information vector and construct the estimated value of the information matrix;

[0234] Step 7: Apply the selection operator to the population, calculate the nonlinear crossover rule, apply the crossover operator to the population, and finally apply the mutation operator to the population;

[0235] Step 8: After the selection, crossover and mutation operations, the next generation of population is obtained, and the individual with the minimum fitness obtained in the evolution process is output as the optimal solution;

[0236] Step 9: Obtain the estimated value of the fractional order and the estimated value of the filter parameter vector from the optimal solution;

[0237] Step 10: Execute the evolution cycle until the number of evolutions reaches the total number of evolutions, stop the evolution and output the final estimated value; otherwise, return to step 5;

[0238] (3) According to the data filtering and genetic optimization process, the improved genetic algorithm based on data filtering is constructed as follows:

[0239]

[0240]

[0241] Φ M :=[φ f (M, α), …, φ f (t, α), …, φf (1, α)] T (26)

[0242]

[0243] Θ′ nk =Θ nk +δ, with probability Υ m (32)

[0244]

[0245] See also Figure 2 , the specific steps of the above method are:

[0246] (a) Initialize the initial factors of the improved genetic algorithm based on data filtering. Set the evolutionary generation counter k to 0; set the maximum evolutionary generation T, the crossover coefficients γ1, γ2, γ3, γ4 of the improved genetic algorithm, and the mutation rate γ5;

[0247] (b) Collect the measurable input data and output data of the ultra-precision platform and construct the output vector. Use data filtering to obtain a new system with simplified noise interference. Let M represent the data length and construct the output vector Z through equation (17): M ;

[0248] (c) Initialize the genetic algorithm and randomly generate L individuals as the initial population Θ 0 After evolving k times (k = 1, 2, ...), construct the lth individual, which is related to the parameter vector Estimated value of related;

[0249] (d) Calculate the k-th evolution population Θ k The fitness of each individual in

[0250] (e) Let the number of evolutions k increase by 1, and calculate the estimated value of the intermediate variable of the input linear part through equation (19): The estimated value of the intermediate variable of the filtering part is calculated by formula (23): The estimated value of the intermediate variable of the output linear part is calculated by formula (21) The estimated value of the intermediate variable of the noise part is calculated by formula (25):

[0251] (f) Construct the information vector through equations (18)-(27) Estimated value of The estimated value of the information matrix is ​​constructed by formula (28):

[0252] (g) By using equation (29), the selection operator p lActing on the group, after calculating the nonlinear crossover rule according to formula (30), the crossover operator Υ is converted into c Acting on the group, finally the mutation operator Υ is transformed into m Act on groups;

[0253] (h) After the selection, crossover and mutation operations, the next generation population is obtained. According to formula (33), the individual with the minimum fitness obtained in the evolution process is Output as the optimal solution;

[0254] (i) By using equations (34) and (35), the fractional order α k Estimated value of and the filter parameter vector Estimated value of From the global best position Separate from

[0255] (j) Execute the evolution cycle until k reaches the total evolution number T, stop the evolution and output the final estimated value; otherwise, return to step (e).

[0256] The variables are defined as follows:

[0257] Definition D α represents the fractional order with α as the order, f(·) is a nonlinear or linear function, h is the sampling time, g is the number of samples to calculate the derivative approximation at time t = gh, are Newton binomial coefficients, Γ(·) is the gamma function;

[0258] Define t as the time variable, and are the input and output signals of the system, is a function with zero mean and variance σ 2 Gaussian distributed white noise, is the intermediate variable of the input linear part, r(t) is the intermediate variable of the nonlinear part, w(t) is the intermediate variable of the output linear part, e(t) is the intermediate variable of the noise part, H(·) is the nonlinear function, s -α Represents the fractional post-shift operator: s -α μ(t)=D α μ(t-1), s -iα μ(t)=(s -i ) α μ(t)=D α μ(ti). Polynomial factor a p 、b q 、c i ,d j is an unknown parameter, in the nonlinear module, are the polynomial coefficients, is a polynomial function, n β is the order of the nonlinear polynomial, D α μ(tp) is the fractional-order input variable, ap is the corresponding fractional-order coefficient, p = 1, 2, ..., n a Medium a is the fractional order, D α r(tq) is a fractional intermediate variable, b q is the corresponding fractional-order coefficient, q = 1, 2, ..., n b Medium b is the fractional order, D α m(ti) is the intermediate variable of fractional-order noise, c i is the corresponding fractional-order coefficient, i = 1, 2, ..., n c Medium c is the fractional order. α e(tj) is fractional colored noise, d j is the corresponding fractional-order coefficient, j = 1, 2, ..., n d Medium d is the fractional order;

[0259] Define the information vector of the fractional-order nonlinear system as φ(t, α), and the parameter vector as θ; define e(t) as the noise input of the ultra-precision platform nonlinear system; define the filter output as z f (t), the filtering intermediate term is r f (t) and w f (t), the filtered information vector is φ f (t, α), the parameter vector after filtering is θ f ; Define the total number of individuals as L, k as the number of evolutions (initial value is 1), T as the total number of evolutions; define the estimated value of the intermediate variable in the linear part as The estimated value of the intermediate variable in the filtering part is The estimated value of the intermediate variable in the output linear part is The estimated value of the intermediate variable of the noise part is

[0260] Define M as the data length and the output vector as Z M ;definition is fitness; definition is the parameter vector θ f The estimated value of is the estimated value of fractional order α; define the information vector The estimated value of the information vector is The information matrix is ​​Φ L , the estimated value of the information matrix is The noise vector is vL ;p l is the probability that the lth individual is selected, is the fitness of the lth individual; c is the crossover operator, and For two random individuals, is a new individual; tanh is a hyperbolic tangent function, γ1 and γ4 are used to change the crossover rate Υ c The maximum and minimum values ​​of γ2 are used to adjust the steepness of the transition area between the maximum and minimum values, γ3 is used to adjust the range of the diffusion area, and T is the maximum number of evolution times; m is the mutation operator, Θ′ nk is a mutant individual, δ is a random small change; the individual with the minimum fitness is the optimal solution; is the fractional order α k The estimated value and is the filter parameter vector The estimated value of .

[0261] The identification effect of the designed identification method can be found in Figure 3-7 . According to the identification results, the method of the present invention has the following advantages in terms of identification error, estimation accuracy and convergence speed: for the two cases where the fractional order is unknown, as the evolution k increases, the identification error becomes smaller and smaller and eventually tends to zero, and the parameter estimation value and the fractional order estimation value also converge to the vicinity of the true value with the evolution. Compared with the data-free filtering algorithm, it has a faster convergence speed and a smaller parameter estimation error, while avoiding redundant calculations and improving computational efficiency. Therefore, the identification method can effectively identify fractional-order nonlinear systems, and the convergence speed is also very fast. The convergence curve eventually reaches near zero, the identification error of the model is very small, and the estimated output and the true output are very close, and the fit is very good. It is completely suitable for parameter estimation of fractional-order ultra-precision platform systems.

[0262] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A fractional-order ultra-precision platform identification method based on genetic optimization and data filtering, characterized in that: The steps include: Step 1) constructing a nonlinear system model of an ultra-precision platform with fractional order, and obtaining a fractional-order nonlinear system model including a fractional-order linear dynamic part and a static nonlinear part; Step 2) constructing an improved genetic optimization identification method based on data filtering; The step 2) specifically includes the following steps: Step 1) initializing the initial factors of the improved genetic optimization algorithm based on data filtering; Step 2) collecting the input data of the ultra-precision platform as the thrust generated by the drive motor and the output data as the moving position of the blade, and constructing an output vector; Step 3) Initialize the genetic optimization algorithm, set the maximum evolutionary generation, and randomly generate the initial population; Step 4) Calculate the fitness of each individual in the group; Step 5) Increase the number of evolutions by one, and calculate the estimated values ​​of the intermediate variables of the linear part of the input and output and the nonlinear part; Step 6) Calculate the estimated value of the information vector and construct the estimated value of the information matrix; Step 7) Apply the selection operator to the population, calculate the nonlinear crossover rule, apply the crossover operator to the population, and finally apply the mutation operator to the population; Step 8) After the population is selected, crossed over, and mutated, the next generation population is obtained, and the individual with the minimum fitness obtained in the evolution process is output as the optimal solution; Step nine) obtaining an estimate of the fractional order and an estimate of the filter parameter vector from the optimal solution; Step 10) executing the evolution cycle until the number of evolutions reaches the total number of evolutions, stopping the evolution and outputting the final estimated value; Otherwise, return to step 5); Steps 1) to 10) in step 2) specifically include the following steps: (2-1) Initialize the initial factors of the improved genetic algorithm based on data filtering, set the evolutionary generation counter k to 0; set the maximum evolutionary generation T, the crossover coefficients γ1, γ2, γ3, γ4 of the improved genetic algorithm, and the mutation rate γ5; (2-2) Collect the measurable input data and output data of the ultra-precision platform, use data filtering to obtain a new system with simplified noise interference, let M represent the data length, and construct the output vector Z M ; (2-3) Initialize the genetic algorithm and randomly generate L individuals as the initial population Θ 0 , after k evolution times k = 1, 2, ..., construct the lth individual, which is related to the parameter vector θ f Estimated value of related; (2-4) Calculate the evolution k-times population Θ k The fitness of each individual in (2-5) Let the number of evolutions k increase by 1, and calculate the estimated value of the intermediate variable in the input linear part Estimated values ​​of intermediate variables in the filtering part Output the estimated value of the intermediate variable in the linear part and the estimated value of the intermediate variable of the noise part (2-6) Constructing information vector Estimated value of Construct an estimate of the information matrix (2-7) The selection operator p l Acting on the group, after calculating the nonlinear crossover rule, the crossover operator Υ c Acting on the group, finally the mutation operator Υ m Act on groups; (2-8) After the selection, crossover and mutation operations, the next generation of population is obtained, with the individual with the minimum fitness obtained in the evolution process. Output as the optimal solution; (2-9) The fractional order α k Estimated value of and the filter parameter vector Estimated value of From the global best position Separate from (2-10) Execute the evolution cycle until k reaches the total evolution number T, stop the evolution and output the final estimated value; otherwise, return to step (2-5).

2. The fractional-order ultra-precision platform identification method based on genetic optimization and data filtering according to claim 1 is characterized in that: The step 1) comprises the following steps: (1-1) Using GL definition to describe fractional differentials is as follows: Among them, D α represents the fractional order with α as the order, f(·) is a nonlinear or linear function, h is the sampling time, g is the number of samples to calculate the derivative approximation at time t = gh, and the Newton binomial coefficient The calculation formula contains the Euler function, which is expressed as Where the gamma function Γ(·) is (1-2) A structure of a nonlinear system model of an ultra-precision platform with fractional order is constructed. The expression of the model is as follows: ω(t)=B(s -α )r(t) (6) z(t)=w(t)+m(t) (8) Among them, t is the time variable, and are the input and output signals of the system, is a Gaussian white noise with zero mean and variance σ, and the input linear intermediate variable The intermediate variables r(t) in the nonlinear part, w(t) in the output linear part, and m(t) in the noise part are all unknown. H(·) is a nonlinear function. -α Represents the fractional post-shift operator: s -α μ(t)=D α μ(t-1), s -iα μ(t)=(s -i ) α μ(t)=D α μ(ti); (1-3) The nonlinear model of the ultra-precision platform nonlinear system with fractional order is constructed as follows: In order to ensure the uniqueness of parameters, the nonlinear submodule is defined as in, are the polynomial coefficients, is a polynomial function, n β is the order of the nonlinear polynomial. By derivation and setting the sampling time in the GL definition to 1, the output of the system is The information vector φ(t, α) and parameter vector θ of the fractional-order nonlinear system are defined as n=n a +n b +n c +n d +n β -1 The output z(t) can be expressed as z(t)=φ T (t, α)θ+e(t) (11) Where e(t) is the noise input of the nonlinear system of the ultra-precision platform.

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