CN115453871A - Non-linear system modeling method based on IDE extended multidimensional Taylor network - Google Patents
Non-linear system modeling method based on IDE extended multidimensional Taylor network Download PDFInfo
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Abstract
The invention discloses a non-linear system modeling method and a system based on an IDE extended multidimensional Taylor network, wherein the method comprises the following steps: determining a system to be modeled and input variables of the system; performing fractional power expansion processing on input variables of the system through a Taylor expansion formula to construct an expanded multi-dimensional Taylor network model; and identifying the parameters of the expanded multi-dimensional Taylor network model by combining a differential evolution algorithm, and constructing an optimal expanded multi-dimensional Taylor network model. By using the method and the device, the input power of the multidimensional Taylor network can be expanded into a non-integer, and the error of the system is reduced, so that the modeling of a complex nonlinear body system with higher precision is realized. The invention is used as a nonlinear system modeling method and system based on the IDE extended multidimensional Taylor network, and can be widely applied to the technical field of trend prediction.
Description
Technical Field
The invention relates to the technical field of trend prediction, in particular to a nonlinear system modeling method based on an IDE extended multidimensional Taylor network.
Background
The system modeling is beneficial to the reasonable design of a control system and the adjustment of parameters of a regulator, and the system modeling method generally comprises three types: the first is mechanism modeling, and the modeling method establishes an analytic mathematical model of the system according to the transfer mechanism of the system; the second is data modeling, which is a system model obtained by analyzing statistical rules of data based on a large amount of experimental data; the third is composite modeling, namely a mechanism + data modeling method, wherein a model structure is determined through a mechanism, and model parameters are determined through data;
the modeling is a process from input to output of a simulation system, for example, a motor system is modeled, the input of the motor system is voltage, different voltages correspond to different motor rotating speeds, if the output rotating speed of the motor needs to be accurately controlled, the relation between the voltage and the rotating speed needs to be known, the relation between the voltage and the rotating speed can be obtained by establishing an accurate model, and the modeling is very important in some neighborhoods needing accurate control;
for example, the electric quantity of the lithium ion battery is estimated, the electric quantity of the lithium ion battery cannot be directly measured, in order to better use the battery and bring convenience to the use requirements of people, a model needs to be established to estimate the electric quantity of the battery, the residual electric quantity of the battery is estimated through some measurable quantities, and the model is required in many fields, such as new energy automobiles, mobile phone batteries and the like, and a more accurate model is established to be beneficial to reasonably planning and using the battery by people;
with the rapid development of production technology, industrial production systems are increasingly complex, and some systems have nonlinear, time-varying and fractional order characteristics. Therefore, it is difficult to establish an accurate mathematical model, and with the occurrence of an artificial neural network, the strong nonlinear approximation capability of the artificial neural network can effectively solve the problem, but the neural network is equivalent to a black box, the change process of the artificial neural network is difficult to explain, the initial weight is difficult to determine, the generalization capability is poor, the artificial neural network is easy to fall into local optimum, and the fractional order characteristics of different systems are different, but the existing multidimensional taylor network-based modeling method approximates the system by a polynomial of integer power without considering the fractional order characteristics of the system, so that a new error is generated, and when the system has the stronger fractional order characteristics, the modeling error of the nonlinear system is larger.
Disclosure of Invention
In order to solve the above technical problems, an object of the present invention is to provide a method and a system for modeling a nonlinear system based on IDE extended multidimensional taylor network, which can extend the power of the input of the multidimensional taylor network to a non-integer number and reduce the error of the system, thereby realizing a more accurate modeling of a complex nonlinear system.
The first technical scheme adopted by the invention is as follows: a nonlinear system modeling method based on IDE extended multidimensional Taylor network includes the following steps:
determining a system to be modeled and input variables of the system;
performing fractional power expansion processing on input variables of the system through a Taylor expansion formula to construct an expanded multi-dimensional Taylor network model;
and identifying the parameters of the expanded multi-dimensional Taylor network model by combining a differential evolution algorithm, and constructing an optimal expanded multi-dimensional Taylor network model.
Further, the system to be modeled comprises a single-input single-output nonlinear discrete time-varying system and a multiple-input multiple-output nonlinear discrete time-varying system, and the expression thereof is as follows:
y(k)＝f[y(k-1),y(k-2),…,y(k-n 1 ),u(k-1),u(k-2),…,u(k-n 2 )]
in the above formula, f [. Cndot.)]Representing a non-linear function, u (k) representing the input of the system, y (k) representing the output of the system, n 1 Representing the maximum delay of the system output, n 2 Representing the maximum delay of the system input.
Further, the building of the extended multidimensional taylor network model includes a three-layer structure, which specifically includes:
the multi-dimensional Taylor network model adopts a forward single-middle-layer structure and comprises an input layer, a middle layer and an output layer, wherein the middle layer represents a processing layer of the multi-dimensional Taylor network and carries out Taylor expansion processing on input variables of the input layer.
Further, the step of performing fractional power expansion processing on input variables of the system through a taylor expansion formula to construct an extended multidimensional taylor network model specifically includes:
acquiring time series input and output data of a nonlinear discrete time-varying system and inputting the data to an input layer of the expanded multi-dimensional Taylor network model to obtain input variables of the expanded multi-dimensional Taylor network model;
carrying out integer order expansion processing on input variables of the expanded multi-dimensional Taylor network model according to a multivariate Taylor formula to obtain a primarily expanded polynomial form;
carrying out fractional order expansion processing on the preliminarily expanded polynomial form to obtain an expanded polynomial form;
abandoning high-order infinite small terms in a multi-order expanded polynomial form, and constructing an intermediate layer of the expanded multi-dimensional Taylor network model;
multiplying a polynomial of the middle layer of the expanded multi-dimensional Taylor network model by the corresponding product term weight and taking the product term as an output layer of the expanded multi-dimensional Taylor network model;
and constructing an expanded multi-dimensional Taylor network model by combining an input layer, an intermediate layer and an output layer of the expanded multi-dimensional Taylor network model.
Further, the expression of the multivariate Taylor expansion is as follows:
in the above formula, x i,o Represents the variable x i At the value of o, the number of bits in the bit is,
representing a high order infinite item, x n It is shown that there are n inputs,
it is shown that the h-order partial derivative is calculated,
the derivation of order m +1 is shown.
Further, the expression of the function after taylor series multi-order expansion of the expanded nonlinear function is as follows:
in the above formula, R m Representing a high order infinite element with respect to (x-a), (x-a) representing a Taylor expansion in the neighborhood of point a, g (n) The expression function g derives n times.
Further, the expression of the extended multidimensional taylor mesh model is as follows:
in the above formula, m represents the highest expansion order, n represents the input of the expanded multidimensional Taylor network model, W represents the weight, namely the parameter to be identified, X represents the product term,
represents the output of the expanded multidimensional taylor network, h represents the order of expansion, i represents the ith input variable, k represents the time, and T represents the transposed symbol.
Further, the step of identifying and processing the parameters of the extended multi-dimensional taylor network model by combining a differential evolution algorithm to construct an optimal extended multi-dimensional taylor network model specifically includes:
initializing parameters of an expanded multi-dimensional Taylor network model to generate a random population, wherein the parameters of the expanded multi-dimensional Taylor network model comprise a weight of a product term of the expanded multi-dimensional Taylor network model and a power of an input nonlinear function;
carrying out mutation treatment on the random population through a first mutation strategy and a second mutation strategy to obtain a mutated population individual;
performing cross treatment on the varied population individuals to obtain a new population;
evaluating and calculating new population individuals through a fitness function to obtain an error value, and selecting optimal population individuals according to the error value to construct an optimal population;
performing next iterative identification according to the processes of the optimal population cyclic variation treatment and the cross treatment until preset conditions are met, and outputting final population individuals;
and constructing an optimal extended multi-dimensional Taylor network model according to the final population individuals.
The method of the invention has the beneficial effects that: the invention utilizes the multidimensional Taylor network to model the system without any prior knowledge of the system, only utilizes the observation data of the system time sequence to determine the input variable of the model, further performs Taylor expansion on the complex nonlinear system, expands a complex nonlinear function into a polynomial to approximate the complex nonlinear function by solving the partial derivative of a certain variable, expands the power of the input of the multidimensional Taylor network into a non-integer through an m-order polynomial form obtained by the Taylor expansion, and finally identifies the power of the polynomial of the complex nonlinear function according to the fractional order characteristic of the system through an improved differential evolution algorithm (IDE), thereby reducing the error of the modeling, realizing the modeling of the complex nonlinear system with higher precision, and improving the diversity of the population data and being beneficial to finding the global optimum.
Drawings
FIG. 1 is a flow chart of the steps of a method for modeling a non-linear system based on an IDE extended multidimensional Taylor network according to the present invention;
FIG. 2 is a schematic flow chart of the method for constructing an extended multi-dimensional Taylor network model according to the present invention;
FIG. 3 is a schematic structural diagram of an extended multidimensional Taylor network model constructed based on a single-input single-output nonlinear discrete time-varying system;
FIG. 4 is a schematic structural diagram of an extended multidimensional Taylor network model constructed based on a multi-input multi-output nonlinear discrete time-varying system according to the present invention;
FIG. 5 is a schematic diagram of a framework for improving the differential evolution algorithm of the present invention;
FIG. 6 is a schematic workflow diagram of the improved differential evolution algorithm of the present invention;
FIG. 7 is a comparison graph of simulation experimental data of a fitting output curve of the extended multi-dimensional Taylor network model and the traditional multi-dimensional Taylor network model in a training phase;
FIG. 8 is a comparison graph of simulation experimental data of the fitting output curve of the extended multi-dimensional Taylor network model and the traditional multi-dimensional Taylor network model in the testing stage.
Detailed Description
The invention is described in further detail below with reference to the figures and the specific embodiments. The step numbers in the following embodiments are provided only for convenience of illustration, the order between the steps is not limited at all, and the execution order of each step in the embodiments can be adapted according to the understanding of those skilled in the art.
The invention relates to a modeling method based on a multidimensional Taylor network, which belongs to one of data modeling, only needs observation data of a time sequence of a system, uses input and output data of a system historical moment according to a mathematical principle of a Taylor expansion formula, approaches the output of the system at the current moment in a polynomial form, thereby establishing a model of the system, the multidimensional Taylor network does not need any prior knowledge of the system, and can model the system only by using the observation data of the system time sequence, the actual electromagnetic coupling, friction force, viscous force and the like are proved to be fractional phenomena, namely, the system containing electromagnetic coupling, friction force, viscous force and the like is essentially a fractional order system, however, the existing modeling method based on the multidimensional Taylor network does not consider the fractional order characteristic of a complex system, the invention establishes an expanded multidimensional Taylor network model of non-integer power, can regard the traditional multidimensional Taylor network as a special case of the invention, the power of the polynomial can be improved according to the fractional order characteristic of the system, thereby reducing the error of the modeling, realizing the more-precise and the more-precise linear Taylor network model can be established only by using a closed-order model, and the extended parameter can be defined by the extended Taylor network algorithm, and the extended model of the continuous multi-dimensional Taylor network can be established.
Referring to fig. 1 and 2, the present invention provides a method for modeling a non-linear system based on an IDE extended multidimensional taylor network, the method comprising the steps of:
s1, determining a system to be modeled and input variables of the system;
specifically, the system to be modeled comprises a single-input single-output nonlinear discrete time-varying system and a multi-input multi-output nonlinear discrete time-varying system, and the multi-dimensional taylor network model adopts a forward single-middle layer structure, and comprises an input layer, a middle layer and an output layer, wherein the middle layer represents a processing layer of the multi-dimensional taylor network, and taylor expansion processing is performed on input variables of the input layer, and with reference to fig. 3, an expression of the single-input single-output nonlinear discrete time-varying system is as follows:
y(k)＝f[y(k-1),y(k-2),…,y(k-n 1 ),u(k-1),u(k-2),…,u(k-n 2 )]
in the above formula, f [ ·]Representing a non-linear function, u (k) representing the input of the system, y (k) representing the output of the system, n 1 Representing the maximum delay of the system output, n 2 Represents the maximum delay of the system input;
referring to fig. 4, the expression of the multiple-input multiple-output nonlinear discrete time-varying system is as follows:
y(k)＝f[y(k-1),y(k-2),…,y(k-n 1 ),u(k-1),u(k-2),…,u(k-n 2 )]
in the above equation, f [ · ] represents a nonlinear function in vector form, u (k) represents a system input in vector form, and y (k) represents a system output in vector form;
wherein, f (·) = [ f ] 1 (·),f 2 (·),…,f s (·)] T ∈R s For an unknown non-linear vector function, y (k) = [ y 1 (k),y 2 (k),…,y s (k)] T ∈R s ，u(k)＝[u 1 (k),u 2 (k),…,u l (k)] T ∈R l Respectively the s-dimensional output and the l-dimensional input of the system, n 1,j (j＝1,2,…,s),n 2,i (i =1,2, …, l) respectively, the maximum delay of the jth output component and the ith input component, and n 1 ＝{n 1,1 ,n 1,2 ,…n 1,s },n 2 ＝{n 2,1 ,n 2,2 ,…n 2,l At time k, the jth output quantity of the system can be expressed as:
y i (k)＝f i (y 1 (k-1)，···，y 1 (k-n 1，1 )，···，y s (k-1)，···，y s (k-n 1，s )，u 1 (k-1)，···，u 1 (k-n 2，1 )，u l (k-1)，···，u l (k-n 2，1 ))
according to the principle of multivariate Taylor expansion, an n-element function f [. Cndot. ] is conductive in the m +1 order of a certain field, and then the function can be expanded into a polynomial of the order of m or less at the point.
S2, performing fractional power expansion processing on input variables of the system through a Taylor expansion formula to construct an expanded multi-dimensional Taylor network model;
s21, acquiring time series input and output data of the nonlinear discrete time-varying system and inputting the data to an input layer of the expanded multi-dimensional Taylor network model to obtain input variables of the expanded multi-dimensional Taylor network model;
specifically, input of a model is determined, input and output observation data of a time sequence of a nonlinear system are used as input of an expanded multi-dimensional Taylor network, and the model is designed to be expanded for m times according to precision requirements;
s22, carrying out integer order expansion processing on input variables of the expanded multi-dimensional Taylor network model according to a multivariate Taylor formula to obtain a primarily expanded polynomial form;
s23, carrying out fractional order expansion processing on the preliminarily expanded polynomial form to obtain an expanded polynomial form;
specifically, the N-ary function f [. Cndot. ] has m +1 continuous partial derivatives in a certain domain N (o), and the expansion of the function at that point is:
in the above formula, x i，o Represents the variable x i At the value of o, the number of bits in the bit is,
representing a high order infinite item, x n It is shown that there are n inputs,
it means that the h-order partial derivative is obtained,
the method comprises the following steps of (1) solving m +1 order partial derivatives;
let the function f (x) in a certain neighborhood I of point a be denoted as (x-a) v (v is more than or equal to 0 and less than or equal to 1) and a real analytic function value, the function can be developed into the following positive fraction power series in the neighborhood:
and S24, carrying out taylor series multi-order expansion on the expanded polynomial form, abandoning high-order infinite small terms of the multi-order expanded polynomial, constructing an intermediate layer of the expanded multi-dimensional taylor network model, multiplying the polynomial of the intermediate layer of the expanded multi-dimensional taylor network model by the corresponding product term weight and taking the product term as an output layer of the expanded multi-dimensional taylor network model.
Specifically, let function f (x) = (x-a) v g (x), g ∈ C (I), where C represents a complex set, and since the function g (x) has various derivatives at point a, there are Taylor series and Taylor expansion as follows:
the binding function can be developed in the neighborhood as a positive fractional power series and the function g (x) has various derivatives available at point a as follows:
in the above formula, R m Representing a high order infinite element with respect to (x-a), (x-a) representing a Taylor expansion in the neighborhood of point a, g (n) Representing the function g to obtain n derivatives;
in summary, consider the constant term as x i The zero-order term of the system is removed, and the high-order infinite small term of the zero-order term is removed, so that the extended multi-dimensional Taylor network model of the single-input single-output system can be expressed as follows:
in the above formula, m represents the highest expansion order, n represents the input of the expanded multidimensional Taylor network model, W represents the weight, namely the parameter to be identified, X represents the product term,
representing the output of the expanded multidimensional Taylor network, h representing the expansion order, i representing the ith input variable, k representing the time, and T representing a transposed symbol;
wherein the content of the first and second substances,
the extended multidimensional taylor mesh model for a multiple-input multiple-output system is represented as follows:
the idea of expanding the multidimensional Taylor network is to approach a complex nonlinear function through the form of a polynomial, the expansion principle of the polynomial is that the function expanded according to the Taylor formula can be expanded into Taylor expansion of positive fractional power series in the neighborhood, and the form of an expanded multidimensional Taylor network model can be obtained by rounding off high-order infinite small terms, so that the obtained expanded multidimensional Taylor network mathematical expression is obtained through Taylor expansion instead of being set randomly, wherein the Taylor expansion principle is that a function is expanded into the polynomial to approach the function through solving the partial derivative of a certain variable, namely the expanded multidimensional Taylor network model can approach the complex nonlinear function;
the single-input single-output model and the multiple-input multiple-output model are mainly different in that a plurality of outputs are provided, and if 5 outputs are provided, that is, 5 outputs correspond to 5 nonlinear functions, the model for establishing the multiple-input multiple-output model needs to identify corresponding parameters of the 5 nonlinear functions, the single-input single-output model only has one nonlinear function, the multiple-input multiple-output model only has the influence of the multiple inputs on the multiple outputs, and the multiple-input multiple-output model is different from the nonlinear function corresponding to each output, namely, the model establishing mode of the single-input single-output model is also suitable for the multiple-input multiple-output model.
And S3, identifying the parameters of the expanded multi-dimensional Taylor network model by combining a differential evolution algorithm, and constructing an optimal expanded multi-dimensional Taylor network model.
Specifically, referring to fig. 5 and 6, compared with the conventional differential evolution, an Improved Differential Evolution (IDE) algorithm employs a method of a good point set to initialize a population, and under the condition of the same number of points taken, a good point sequence is more uniform than a point sequence selected by other methods, so as to ensure the diversity of population individuals, wherein the good point sequence means that each individual is arranged from good to bad in the whole population according to a fitness function, and a relatively good part of individuals is reserved; during cross operation, information among individuals is exchanged, so that population diversity is improved, during selection operation, the quality degree of the individuals is judged according to a fitness function, the selected individuals enter the next generation, an intelligent algorithm needs to be combined with a model, the error between the output of the model and the output of a system is calculated by combining parameters obtained by optimizing each generation with the model, and the quality of the parameters obtained by optimizing is evaluated and identified through the fitness function;
s31, selecting p individuals with the best fitness value in each generation of population, and performing mutation to form new individuals;
specifically, p individuals with the best fitness value are selected in each generation of population for mutation to form new individuals, and the expression of the first mutation strategy is as follows:
v i,g ＝x pbest +F i ×(x r1,g -x r2,g )
the first mutation strategy is to find the optimal individual;
s32, further selecting the optimal individual in the population for mutation;
specifically, the optimal individual in the population is selected for mutation, and the expression of the second mutation strategy is as follows:
stepsize＝R B ×(x best,g -R B ×x i,g )
v i,g ＝x i,g +0.5×F i ×stepsize
in the above formula, F represents a scaling factor, the value range of which is (0-1), x represents individual population, v represents individual obtained by variation, g represents evolution algebra of the population, i represents ith individual in the population, and R represents B Representing a random number that fits a positive distribution;
the second mutation strategy is to prevent local optimality;
s33, performing cross processing on the mutated individuals;
specifically, the cross operation is to improve the diversity of the population and realize the information exchange between two individuals by randomly pairing x i,g And v i,g The parameters in the method are exchanged to obtain a new individual which is an experimental individual u i,g The expression of the intersection is as follows:
in the above formula, rand (0,1) represents a random number, u represents a new individual obtained by crossing, x and v represent original individuals of a population, and j represents the jth parameter in the individual;
s34, selecting the optimal population individuals to perform next iteration optimization according to the processes of the optimal population cyclic variation processing and the cross processing until preset conditions are met, and outputting the final population individuals to construct an optimal expansion multi-dimensional Taylor network model;
specifically, the preset condition may be setting a threshold, and stopping the iterative identification process if the error value is smaller than the preset threshold, or setting a maximum evolution algebra, and stopping the iterative identification process no matter whether the number of population iterations reaches the maximum evolution algebra or not, regardless of whether the number of population iterations is smaller than the set threshold;
through the improvement, the improved differential evolution algorithm is simple in design, only a proper fitness function needs to be designed, and an identification algorithm suitable for a fractional order system can be designed no matter whether the system is conductive or linear, and compared with the traditional differential evolution algorithm, the improved differential evolution algorithm has the advantages of high solving precision and high convergence speed;
the finally constructed system model can know the output relation between the current output and the historical output of the system, so that the control and the prediction of the system are facilitated, the optimal parameter is found in a set range by optimizing the parameter, but for an intelligent algorithm, the found optimal parameter is possibly optimal in a certain range, namely local optimal, but not optimal in the set range, the diversity of the population is improved, the global optimal is facilitated to be found, and the constructed expanded multi-dimensional Taylor network model can be used for accurately predicting the output data of the next moment on the input data of the previous moment;
in the motor system, the input of the motor system is voltage, different voltages correspond to different motor rotating speeds, if the output rotating speed of the motor needs to be accurately controlled, the relation between the voltage and the rotating speed needs to be known, and the relation between the voltage and the rotating speed can be obtained by establishing an expanded multidimensional Taylor network model;
in the field of lithium battery estimation, the residual electric quantity of a lithium battery can be estimated through an established extended multi-dimensional Taylor network model;
the simulation experimental data of the invention are as follows:
the model expression is as follows:
in the above formula, u represents the input of the system, y represents the output of the system, and k represents the time;
the formula y (y) represents the output of a fractional linear system k and is used for verifying the correctness of the expanded multi-dimensional Taylor network;
referring to fig. 7 and 8, where ZMTN represents the fitting of an integer order MTN, that is, a conventional multidimensional taylor network model, and FMTN represents the fitting of a fractional order MTN, that is, an extended multidimensional taylor network model of the present invention, the simulation process is divided into two parts, namely, training and testing, where the training is a process of constructing the extended multidimensional taylor network model, and the testing is a process of verifying the reliability of the extended multidimensional taylor network model, and it can be obviously seen from the simulation diagram that the output curve of the extended multidimensional taylor network model proposed by the present invention is closer to the true output curve of the system, and the corresponding mean square error can be obtained by calculating data in the simulation diagram through the following evaluation index formula, which is shown as follows:
in the above formula, mse represents an evaluation criterion, and the smaller the calculated value of the evaluation criterion is, the better the proving effect is, y represents the output value of the system,
representing the output value of the model, i representing the time of day
The final calculated data is shown in the table below,
table 1 experimental simulation data table
| Error in training | Error of test | |
| Traditional multidimensional taylor net | 4.1146×10 -4 | 3.0404×10 -4 |
| Extended multidimensional taylor mesh | 1.0741×10 -4 | 3.1802×10 -5 |
| Lift (%) | 73.90 | 89.54 |
The present invention is not only applicable to the above-mentioned technical neighborhood, but also applicable to an environment where output data at the next time is accurately predicted based on input data at the previous time.
While the preferred embodiments of the present invention have been illustrated and described, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (8)
1. A non-linear system modeling method based on an IDE extended multidimensional Taylor network is characterized by comprising the following steps:
determining a system to be modeled and input variables of the system;
performing fractional power expansion processing on input variables of the system through a Taylor expansion formula to construct an expanded multi-dimensional Taylor network model;
and identifying the parameters of the expanded multi-dimensional Taylor network model by combining a differential evolution algorithm, and constructing an optimal expanded multi-dimensional Taylor network model.
2. The modeling method of the nonlinear system based on the IDE extended multidimensional Taylor network as claimed in claim 1, wherein the system to be modeled comprises a single-input single-output nonlinear discrete time-varying system and a multiple-input multiple-output nonlinear discrete time-varying system, and the expression is as follows:
y(k)＝f[y(k-1),y(k-2),…,y(k-n 1 ),u(k-1),u(k-2),…,u(k-n 2 )]
in the above formula, f [. Cndot.)]Representing a non-linear function, u (k) representing the input of the system, y (k) representing the output of the system, n 1 Representing the maximum delay of the system output, n 2 Representing the maximum delay of the system input.
3. The method of claim 1, wherein the building of the extended multidimensional taylor network model comprises a three-layer structure, which specifically comprises:
the multi-dimensional Taylor network model adopts a forward single-interlayer structure and comprises an input layer, an interlayer and an output layer, wherein the interlayer represents a processing layer for expanding the multi-dimensional Taylor network and carries out Taylor expansion processing on input variables of the input layer.
4. The method as claimed in claim 3, wherein the step of constructing the extended multidimensional taylor network model by performing fractional power expansion on input variables of the system through taylor expansion formula specifically comprises:
acquiring time series input and output data of a nonlinear discrete time-varying system and inputting the data to an input layer of the expanded multi-dimensional Taylor network model to obtain input variables of the expanded multi-dimensional Taylor network model;
carrying out integer order expansion processing on input variables of the expanded multi-dimensional Taylor network model according to a multivariate Taylor formula to obtain a primarily expanded polynomial form;
carrying out fractional order expansion processing on the preliminarily expanded polynomial form to obtain an expanded polynomial form;
abandoning high-order infinitesimal terms in a polynomial form after multi-order expansion, and constructing an intermediate layer of the expanded multi-dimensional Taylor network model;
multiplying a polynomial of the middle layer of the expanded multi-dimensional Taylor network model by the corresponding product term weight and taking the product term as an output layer of the expanded multi-dimensional Taylor network model;
and constructing an expanded multi-dimensional Taylor network model by combining an input layer, an intermediate layer and an output layer of the expanded multi-dimensional Taylor network model.
5. The method according to claim 4, wherein the expression of the integer order expansion is as follows:
in the above formula, x i,o Represents the variable x i At the value of o, the number of bits in the bit is,
representing a high order infinite item, x n Indicates that there are n inputsIn the method, the raw materials are added,
it is shown that the h-order partial derivative is calculated,
which represents the derivation of order m + 1.
6. The method according to claim 4, wherein the functional expression of the fractional order extension process is as follows:
in the above formula, R m Representing a high order infinite element with respect to (x-a), (x-a) representing a Taylor expansion in the neighborhood of point a, g (n) The expression function g derives n times.
7. The method of claim 6, wherein the expression of the extended multidimensional taylor network model is as follows:
in the above formula, m represents the highest expansion order, n represents the input of the expanded multi-dimensional Taylor network model, W represents the weight, namely the parameter to be identified, and X tableThe term of the product is shown in the figure,
represents the output of the expanded multidimensional taylor network, h represents the order of expansion, i represents the ith input variable, k represents the time, and T represents the transposed symbol.
8. The method for modeling the nonlinear system based on the IDE extended multidimensional taylor network as recited in claim 4, wherein the step of constructing the optimal extended multidimensional taylor network model by identifying the parameters of the extended multidimensional taylor network model in combination with a differential evolution algorithm specifically comprises:
initializing parameters of an expanded multi-dimensional Taylor network model to generate a random population, wherein the parameters of the expanded multi-dimensional Taylor network model comprise a weight of a product term of the expanded multi-dimensional Taylor network model and a power of an input nonlinear function;
carrying out mutation treatment on the random population through a first mutation strategy and a second mutation strategy to obtain a mutated population individual;
performing cross treatment on the varied population individuals to obtain a new population;
evaluating and calculating new population individuals through a fitness function to obtain an error value, and selecting optimal population individuals according to the error value to construct an optimal population;
performing next iterative identification according to the processes of the optimal population cyclic variation processing and the cross processing until preset conditions are met, and outputting final optimal population individuals;
and constructing an optimal expansion multi-dimensional Taylor network model according to the final optimal population individuals.
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