Parameter identification method of fractional order system based on regeneration kernel space machine learning

Abstract

The invention discloses a parameter identification method of a fractional order system based on regenerative nuclear space machine learning. The method comprises the following steps: (1) determining a state equation of the fractional order system; (2) preparing a discrete data set of a sample orbit used by an identification equation through numerical simulation, and obtaining a continuous approximate sample orbit; (3) converting the fractional order system into an integral system; (4) utilizing the obtained approximate sample orbit information to construct functional information of the parameters to be identified; and (5) selecting a proper regenerative kernel Hilbert space kernel function, and giving out estimation of to-be-identified parameters of the fractional order system by utilizing a representation theorem in regenerative kernel space machine learning. The method has good robustness, when the number of obtained sample orbit information is very small, classical methods such as least squares cannot process the information, and the method provided by the invention can construct more functional sample information with the help of limited sample orbit information so as to further process the information.

Description

Technical Field

[0001] The present application relates to the technical field of machine learning and pattern recognition, and in particular to a parameter identification method for a fractional-order system based on reproducing kernel space machine learning. Background Art

[0002] Fractional calculus has memory and non-locality, and is very suitable for describing systems with memory and hereditary properties in the real world. Fractional-order systems can more accurately describe complex systems with historical dependence and global correlation, and have important applications in many fields such as earthquake analysis, robotics, medical imaging, signal analysis and processing, etc. Due to the non-locality of fractional-order derivatives, the research on fractional-order system identification methods is much more difficult and challenging than integer-order system identification.

[0003] One of the technical difficulties in fractional-order system identification is the processing of discrete sample information. Usually, only discrete information of the output sample trajectory can be obtained. This discrete information contains noise and requires data processing to obtain an ideal continuous approximate trajectory.

[0004] Another technical difficulty in fractional-order system identification is the estimation of orbital information derivatives. Conventional identification methods require sample orbital derivative information. Unfortunately, numerical derivatives are extremely sensitive to noise, which can produce artificial noise. The error between the derivative of the approximate sample orbit obtained through data processing and the true orbital derivative is generally large, or even seriously deviated, which leads to instability in system identification and difficulty in obtaining ideal identification parameters. Summary of the invention

[0005] In order to solve the above technical problems, an embodiment of the present application provides a parameter identification method for a fractional-order system based on reproducing kernel space machine learning, which takes the orbit (functional data) as the basic data unit, and embeds the orbital information of the system into the reproducing kernel space for research through a defined placeholder kernel, thereby avoiding the use of sample orbital derivative information, thereby improving the robustness of the identification method.

[0006] The present application embodiment provides a parameter identification method for a fractional-order system based on reproducing kernel space machine learning, including:

[0007] Step S1: construct an application model based on the application scenario, establish and construct the state equation of the fractional-order system, and the parameters to be identified;

[0008] Step S2: constructing a discrete data set of sample orbits required for identifying the state equation through numerical simulation sampling, and denoising the noise in the discrete data set through a regularized kernel function method to obtain a continuous approximate sample orbit;

[0009] Step S3: converting the fractional order system into an integral system;

[0010] Step S4: constructing functional information of the parameters to be identified using the approximate sample orbits obtained after step S3;

[0011] Step S5: Select a suitable reproducing kernel Hilbert space kernel function based on the functional information; use the representation theorem in reproducing kernel space machine learning to give an estimate of the parameters to be identified of the fractional-order system.

[0012] Furthermore, the application scenarios in step S1 include lithium-ion battery state of charge estimation or switched capacitor converter efficiency analysis or economic short-term forecasting.

[0013] Furthermore, the state equation of the fractional-order system in step S1 is where t∈[0,T], D q represents the q-order Caputo fractional derivative, 0<q<1, x:[0,T]→Ω∈R (Ω is a compact set), g:R→R, f(t) is a known continuous function, β(t) is the function to be identified, and T is the upper limit of the variable t.

[0014] Furthermore, the step S2 specifically includes:

[0015] Step S21: Prepare the data set required for training, sample track The discrete information of, N represents the number of tracks;

[0016] Step S22: Process the discrete data set using the regularized kernel regression function method to obtain a continuous approximate sample trajectory.

[0017] Furthermore, in step S3, the fractional order system is converted into an integral system.

[0018]

[0019] Furthermore, in step S4, the output approximate sample trajectory information Choose a different represents the number of moments selected in the interval (0,T], constructing the functional Functional information available

[0020] Furthermore, in step S5, a suitable regeneration kernel space H is selected according to the smoothness of β(t), by To obtain an estimate of β(t),

[0021] Beneficial effects of the present invention:

[0022] 1. It overcomes the difficulty of processing discrete sample information in fractional-order system identification. The fractional-order derivative of the sample orbit contains the derivative of the sample orbit, and the derivative error of the approximate sample orbit is large. Converting the fractional-order system into an integral system can avoid using the information of the sample orbit derivative, thereby improving the stability of the identification method and having better robustness.

[0023] 2. Taking orbit (functional data) as the basic data unit, the orbit information of the system is embedded into the reproducing kernel space through the defined placeholder kernel for research;

[0024] 3. When the number of sample orbital information obtained is very small (far less than the number of parameters to be identified), or even only one sample orbital information, classical methods such as least squares will not be able to handle it, because the parameters that need to be determined by least squares estimation are often more than the number of sample information; this scheme can construct more functional sample information with the help of limited sample orbital information, and the estimated parameters to be determined are equal to the number of sample information, which is still effective when the number of sample orbital information is very small. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0026] Figure 1 It is a flowchart of this application;

[0027] Figure 2 is the fractional-order system parameter of this application Schematic diagram of comparison with the true parameter function β(t);

[0028] Figure 3 This application Plot of the absolute error versus β(t). DETAILED DESCRIPTION

[0029] In order to make the purpose, features, and advantages of the present application more obvious and easy to understand, the technical solutions in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the embodiments described below are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0030] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments.

[0031] In the description of the present application, it should be understood that the terms "upper", "lower", "top", "bottom", "inside", "outside", etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present application and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as a limitation on the present application.

[0032] like Figure 1 The figure shows a flow chart of a parameter identification method for a fractional-order system based on machine learning in a reproducing kernel space, which specifically includes the following steps:

[0033] Step S1: construct an application model based on the application scenario, and establish and construct the state equation of the fractional-order system.

[0034] In this application, application scenarios include lithium-ion battery state of charge estimation, switched capacitor converter efficiency analysis, and short-term forecasting of China's economy. This method can be applied to any application scenario that can generate a fractional-order system model.

[0035] As a preferred solution, the state equation of the constructed fractional-order system is:

[0036] where t∈[0,T], D q represents the q-order Caputo fractional derivative, 0<q<1, x:[0,T]→Ω∈R (Ω is a compact set), g:R→R, f(t) is a known continuous function, β(t) is the function to be identified, and T is the upper limit of the variable t.

[0037] Step S2: construct a discrete data set of sample orbits required for identifying the state equation through numerical simulation sampling, and denoise the noise in the discrete data set through a regularized kernel function method to obtain a continuous approximate sample orbit. Specifically, it can be divided into two steps:

[0038] Step S21: Prepare the data set required for training, sample track The discrete information of, N represents the number of tracks;

[0039] Step S22: Process the discrete data set using the regularized kernel regression function method to obtain a continuous approximate sample trajectory.

[0040] Step S3: Convert the fractional order system into an integral system. This step takes into account that the fractional order derivative of the sample orbit contains the derivative of the sample orbit, and the derivative error of the approximate sample orbit is large. In order to ensure the stability of the identification method, it is necessary to avoid using the information of the sample orbit derivative. Therefore, it is necessary to convert the fractional order system into an integral system.

[0041] As a preferred solution, in this step, the fractional order system is converted into an integral system.

[0042] Step S4: Using the obtained approximate sample orbit information Choose a different represents the number of moments selected in the interval (0,T];

[0043] Constructing functionals Functional information available

[0044] make

[0045] Step S5: Based on the above functional information, a suitable reproducing kernel Hilbert space kernel function is selected. Generally, a suitable reproducing kernel space H is selected based on the smoothness of β(t). To obtain an estimate of β(t),

[0046] By using the representation theorem in reproducing kernel space machine learning, numerical simulation is performed, the identified parameter results are compared with the actual parameters, and the predicted curve is plotted and compared with the actual curve to verify the accuracy and stability of the identification method.

[0047] The following is explained in conjunction with specific embodiments.

[0048] Step S1: When studying economic growth analysis, the following state equation of the fractional order system is used: where t∈[0,1], f(t) is the input of the economic system, also known as the driving function, x(t) is the GDP data function, and β(t) represents the GDP growth rate. In order to compare the accuracy of the estimated function, we take β(t)=sint is the function to be identified.

[0049] The specific steps of the identification method of β(t) are as follows:

[0050] Step S21, prepare the data set required for training. In this embodiment, a sample track is taken as an example (the subsequent steps are all taken as an example of a sample track, so the i in the following is different from the previous i. The i in the following represents the number of sampling information in the sample track, ranging from 1 to 200). For α0=0, the output sample track γ(t) is obtained. The sampling information of discrete values ​​at γ i , γ i Contains white noise with mean 0, standard deviation 0.01, and Gaussian distribution.

[0051] Select the reproducing kernel space H 1 [0,1], its reproducing kernel function is

[0052] S22. Use sampling information γ i , the orbit γ(t) is estimated by using the regularized kernel regression method, specifically: Thus, we obtain an approximate orbit From the representation theorem we can get Let G be the matrix, G ij =K1(t i ,t j ), b=(γ1,γ2,...,γ 200 ) Τ ,ξ=(ξ1,ξ2,...,ξ 200 ) Τ , then ξ=(G+λI) -1 b, where λ is the regularization parameter.

[0053] Step S3: Convert the fractional order system into an integral system

[0054] Step S4: obtain the continuous approximate sample trajectory Choose a different Constructing functionals Functional information available make

[0055]

[0056] Step S5: Select Gaussian regeneration kernel function r = 0.5, and its corresponding reproducing kernel space is H. To obtain an estimate of β(t) is the regularization parameter, let remember is a matrix, Using the reproducing kernel theory, we can prove that

[0057] Identified parameters For comparison with the true parameter β(t), see Figure 2 , Figure 3 By comparison, it can be found that the error between the parameters identified by the present invention and the real parameters is very small, and the present invention is less affected by the discrete sampling data containing noise, and has good robustness.

[0058] The preferred embodiments of the present invention are described in detail above, but the present invention is not limited to the specific details in the above embodiments. Within the technical concept of the present invention, various equivalent transformations (such as quantity, shape, position, etc.) can be made to the technical scheme of the present invention, and these equivalent transformations are all protected by the present invention.

Claims

1. A parameter identification method for fractional-order systems based on reproducing kernel space machine learning, characterized in that: include: Step S1: construct an application model based on the application scenario, establish and construct the state equation of the fractional-order system, and the parameters to be identified; Step S2: constructing a discrete data set of sample orbits required for identifying the state equation through numerical simulation sampling, and denoising the noise in the discrete data set through a regularized kernel function method to obtain a continuous approximate sample orbit; Step S3: converting the fractional order system into an integral system; Step S4: constructing functional information of the parameters to be identified using the approximate sample orbits obtained after step S3; Step S5: Select a suitable reproducing kernel space kernel function based on the functional information; use the representation theorem in the reproducing kernel space machine learning to give an estimate of the parameters to be identified of the fractional-order system.

2. The parameter identification method of fractional order system based on reproducing kernel space machine learning according to claim 1 is characterized in that: The application scenarios in step S1 include lithium-ion battery state of charge estimation, switched capacitor converter efficiency analysis, or economic short-term forecasting.

3. The parameter identification method of fractional order system based on reproducing kernel space machine learning according to claim 1 is characterized in that: The state equation of the fractional order system in step S1 is: where t∈[0,T], D q represents the q-order Caputo fractional derivative, 0<q<1, x:[0,T]→Ω∈R, Ω is a compact set, g:R→R, f(t) is a known continuous function, β(t) is the function to be identified, and T is the upper limit of the variable t.

4. The parameter identification method of fractional order system based on reproducing kernel space machine learning according to claim 1 is characterized in that: The step S2 specifically includes: step S21, preparing the data set required for training and constructing a sample track The discrete information of, N represents the number of tracks; Step S22: Process the discrete data set using the regularized kernel function regression method to obtain a continuous approximate sample trajectory.

5. The parameter identification method of fractional order system based on reproducing kernel space machine learning according to claim 3 is characterized in that: In step S3, the fractional order system is converted into an integral system. Where Γ(q) is the gamma function.

6. The parameter identification method of fractional order system based on reproducing kernel space machine learning according to claim 4 is characterized in that: The output approximate sample trajectory information in step S4 Choose different 0<T j ≤T, represents the number of moments selected in the interval (0,T], constructing the functional Functional information available 7. The parameter identification method of fractional order system based on reproducing kernel space machine learning according to claim 1 is characterized in that: In step S5, a suitable regeneration kernel space H is selected according to the smoothness of β(t), by To obtain an estimate of β(t) in

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