CN111753681A - Fractional order system identification method based on Legendre wavelet multi-resolution analysis - Google Patents

Fractional order system identification method based on Legendre wavelet multi-resolution analysis Download PDF

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CN111753681A
CN111753681A CN202010526968.3A CN202010526968A CN111753681A CN 111753681 A CN111753681 A CN 111753681A CN 202010526968 A CN202010526968 A CN 202010526968A CN 111753681 A CN111753681 A CN 111753681A
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王春阳
杨晓策
牛启凤
王子硕
梁书宁
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Abstract

The invention discloses a fractional order system identification method based on Legendre wavelet multi-resolution analysis. The method comprises the following steps: first, Legendre wavelet order is selected over the [0,1] interval. Secondly, deducing an operation matrix of Legendre wavelet integral by using a block pulse function integral matrix, then expanding input and output signals of the system in the form of Legendre wavelets, continuously decomposing the output signals by using the multi-fraction analysis characteristic of the wavelets, abandoning high-frequency information, reducing the data length, and finally solving system parameters and orders by using a least square method and an IAE index as a target function in the form of cyclic fractional order integral. The method effectively solves the problems that the parameter estimation is slow and the noise pollution system cannot be identified due to single dimension, high order and multiple numbers of the traditional Legendre wavelet integral operation matrix identification method, accelerates the system identification speed and reduces the influence of noise on the identification precision.

Description

Fractional order system identification method based on Legendre wavelet multi-resolution analysis
Technical Field
The invention belongs to the technical field of modeling and identification methods, and mainly relates to a fractional order system identification method based on Legendre wavelet multi-resolution analysis.
Background
The fractional calculus has been developed for over 300 years as the popularization of the classical integer calculus. Although the fractional calculus is an ancient subject, it attracts people's attention two hundred years after the development of peace and quiet, and permeates into various subjects. Before this, the description of the system is based on integral order calculus, however, in the actual system, the dynamic process of the system is often represented by nonlinearity, the description of the system by adopting integral order differential equation needs to construct nonlinear equation, and sometimes some artificial empirical parameters or assumed conditions need to be introduced, so that some characteristics of the system can be ignored. In addition, due to the diversity and complexity of things, the development process of things is not only related to the current state, but also related to the historical state, so that a system model established by using integral calculus cannot well describe a complex system.
Fractional calculus has global correlation and historical memory compared to integer calculus. A great deal of research data shows that the description of a complex system with history dependency or distributed parameters by utilizing the fractional calculus is more accurate, so the fractional calculus is widely applied to the aspect of system modeling in recent years. For example: thermal diffusion problems in boilers, elastoplastic behavior of metals, electrical performance problems in fuel cells, viscoelastic behavior of polymeric materials, and the like.
At present, fractional order models are widely used in many fields such as electromagnetism, biomedicine, physics, elastomechanics, and the like. The modeling method utilizing fractional calculus can more vividly describe the actual system. However, in the modeling process, the fractional order modeling process is more complicated than the integer order system due to the introduction of the fractional order calculus. In an actual system, the mechanism modeling method is difficult to accurately represent the dynamic characteristics of the system under the influence of internal factors and external disturbance, and the system identification method can utilize the mathematical relationship between system input and system output to build a system model under the condition that an internal operation mechanism is fuzzy or the external disturbance is unknown, so that the complexity of a modeling process is greatly reduced.
In recent years, an integral operation matrix method based on an orthogonal function is used for parameter identification of a fractional order system, the method utilizes an orthogonal basis function to expand the fractional order system in the form of an integral matrix, a complex fractional order model is further converted into an algebraic equation, and system parameters are determined through solving the algebraic equation. However, in the solution process of the method, when the parameter identification problem is converted into the optimization problem, the parameter optimal value needs to be searched through various optimization algorithms, the result obtained by the optimization method is a random value, a large amount of data is needed in the identification process, and a high-dimensional operation matrix is formed, so that the problem of slow optimization speed exists. Meanwhile, the optimization algorithm depends on the initial value to a great extent, and when noise exists in the system, the identification precision is greatly influenced, and the problem that engineering application is difficult exists.
Disclosure of Invention
The invention provides a fractional order system identification method based on Legendre wavelet multiresolution analysis, and aims to solve the problems that a system identification process is complex, the identification data volume is large, the identification speed is low, system parameters under the influence of noise cannot be accurately estimated, and engineering application is difficult.
In order to achieve the purpose of the invention, the invention provides a fractional order system identification method based on Legendre wavelet multi-resolution analysis, which comprises the following steps:
the method comprises the following steps: selecting Legendre wavelet order in [0,1] interval
1.1 constructing a Legendre polynomial into wavelet form;
1.2 selecting Legendre wavelets with lower orders and the size of an initial scale space;
step two: deducing an operation matrix of Legendre wavelet integral by using a block pulse function integral matrix;
step three: expanding input and output signals of the system in a Legendre wavelet form;
step four: output signals are continuously decomposed by utilizing wavelet multi-fraction analysis characteristics, high-frequency information is abandoned, and the data length is reduced;
4.1 output data 2nMultiple rejection, n is 1,2,3, …
Decomposing the scale space layer by layer to obtain a set of nested subspace sets
Figure BDA0002533927850000021
Wherein Wn-1Is VnIs a low resolution space VnMissing details, symbols of
Figure BDA0002533927850000022
Defined as the orthogonal sum.
4.2 reconstruction of the output Signal Using the residual data
Dividing two ends of the fractional order differential equation by the highest order of the differential to obtain an integral operator
Figure BDA0002533927850000023
Where f (t) and y (t) are the input and output of the system, β respectivelyn,βn-1,…,β0Is a fractional order of differential order, an,an-1,…,a0As a result of the parameters of the system,
Figure BDA0002533927850000024
is a differential operator.
Substituting the Legendre wavelet integral operation matrix and the input and output operation matrices into a formula (2) to reconstruct a model output matrix
Figure BDA0002533927850000025
Figure BDA0002533927850000026
Step five: solving system parameters and orders by adopting a least square method and taking an IAE index as a target function in a cyclic fractional order integral mode
And 5.1, substituting the input and output signals and a Legendre wavelet integral operation matrix into the formula (2), and rewriting the input and output signals into a matrix form.
Figure BDA0002533927850000031
Wherein Y isTIn the form of a Legendre wavelet matrix for the output signal FTIn the form of a Legendre wavelet matrix of the input signal.
5.2 order
Figure BDA0002533927850000032
X＝[an；an-1；…；a0]，
Figure BDA0002533927850000033
Equation (4) can be rewritten as
AX＝B (5)
5.3 Using the least-squares method, the coefficient matrix X can be found, i.e.
X＝(ATA)-1ATB (6)
And 5.4, determining an IAE objective function, a fractional order integral coefficient range and a step length.
And 5.5, in the defined interval, circulating the fractional order integral order to find the corresponding system parameter when the IAE index is minimum, so that the model parameter can be accurately estimated.
Compared with the prior art, the invention can bring the following beneficial effects:
1. the method establishes the Legendre wavelet integral operation matrixes with different scales from a multi-scale space, and avoids the problem of low identification speed caused by multi-order or high-order Legendre wavelets.
2. The invention utilizes the wavelet multiresolution characteristic, outputs signals by layer decomposition, abandons high-frequency information, and achieves the purposes of reducing data length, reducing matrix dimension and improving identification speed, and simultaneously, the invention further accelerates the parameter estimation speed by combining a least square method.
3. The invention can effectively identify the system polluted by noise, has small calculated amount, is convenient for engineering application, and is suitable for various fields of electromagnetism, biomedicine, physics, elastomechanics and the like.
Drawings
FIG. 1 is a flow chart of a fractional order system identification method based on Legendre wavelet multiresolution analysis;
FIG. 2 illustrates the timing of system identification under different decomposition levels in example 1;
FIG. 3 shows the number of decomposition layers and performance index for different SNR in example 2.
Detailed Description
The present invention will be described in further detail with reference to the following embodiments and the accompanying drawings. The specific embodiments of the present invention and the description thereof are provided for the purpose of illustrating the invention and are not to be construed as limiting the invention.
Example 1:
the fractional order system can be described as
Figure BDA0002533927850000041
β therein2＝2.2，β1＝1.2，a2＝1，a1＝2，a0The system input f (t) is a sinusoidal signal, 3.
The method comprises the following steps: selecting Legendre wavelet order in [0,1] interval
1.1Legendre wavelet is defined over the [0,1] interval as:
Figure BDA0002533927850000042
wherein k is any positive integer, M is Legendre polynomial order, M is 0,1, …, M-1, n is 1,2,3, …,2k -1，
Figure BDA0002533927850000043
Figure BDA0002533927850000044
t is the normalized time for which the average value is,
Figure BDA0002533927850000045
for the orthogonal coefficient, the coefficient of expansion a is 2-kCoefficient of translation
Figure BDA0002533927850000046
1.2 selecting Legendre wavelet order of 2 and initial scale space of 29。
Step two: operation matrix for deducing Legendre wavelet integral by using block pulse function integral matrix
Block pulse function integral operation matrix
(IαBN)(t)≈FαBN(t) (9)
Wherein B isN(t)＝[φ1(t),φ2(t),φ3(t),…φN(t)]Is a block pulse basis vector, FαIs a block pulse function integral operation matrix.
FαIs shown as
Figure BDA0002533927850000047
Wherein f is1＝1，fp＝pα+1-2(p-1)α+1+(p-2)α+1,p＝2,3…N,Tf＝10,N＝512。
Legendre wavelet fractional order integral operation matrix PαIs expressed as
Pα＝ΨFαΨ-1(11)
Step three: expanding input and output signals of the system in the form of Legendre wavelets
The sinusoidal signal f (t) defined in the [0,1] interval can be expanded as:
Figure BDA0002533927850000051
wherein c isn,m＝<f(t),ψn,m(t)>For the coefficients of the Legendre wavelet,<,>representing the inner product, if equation (12) is truncated at infinity, equation (12) can be written as:
Figure BDA0002533927850000052
where superscript T is the transpose of the matrix and C and Ψ (T) are 2k-1M × 1 matrix
Figure BDA0002533927850000053
Figure BDA0002533927850000054
N ═ k × M is defined. In order to solve the Legendre wavelet coefficient C, nodes are set, and psi is enabledn,mTaking values at nodes such that Ψ (t) becomes a matrix of N × N, the nodes being defined as
Figure BDA0002533927850000055
The expansion of the input and output signals at the nodes using the Legendre wavelet results
Figure BDA0002533927850000056
Where f (t) is the system input, Y (t) is the system output, YTAnd FTAre Legendre wavelet matrix coefficients.
Step four: by utilizing wavelet multi-fraction analysis characteristics, output signals are continuously decomposed, high-frequency information is abandoned, and data length is reduced
4.1 output data 2nMultiple rejection, n is 1,2,3, …
Multi-resolution analysis refers to the analysis of functions from different temporal frequencies, representing the functions as a combination of components with temporal and frequency resolution. With multi-resolution analysis, a known function can be projected into a mutually orthogonal function space. The wavelet functions with the same size form the basis of one function space, and the size of the basis is different in different function spaces, so that different frequency characteristics are represented. By continuous projection, the components of the original function in any subspace can be obtained step by step. The function is divided into an approximation part and a detail part, namely a low-frequency component which can be projected into a large-scale space by the function; and the detail part refers to the high-frequency component of the projection of the function in a small scale space.
From multi-resolution analysis, the function spaces of wavelet function structures with different scales are nested, i.e.
Figure BDA0002533927850000057
The difference between two adjacent function spaces satisfies the following relationship
Figure BDA0002533927850000061
Decomposing the space layer by using a formula (19) to obtain a group of nested subspace sets
Figure BDA0002533927850000062
Wherein Wn-1Is VnIs a low resolution space VnMissing details, symbols of
Figure BDA0002533927850000063
Defined as the orthogonal sum.
Therefore, for the function f ∈ VnCan be represented by the following multiresolutions:
f＝f0+w0+w1+…+wn-1(20)
wherein f is0∈V0，wn-1∈Wn-1，n∈Z+。
By continuously decomposing the output signal, the output signal is analyzed,abandoning high frequency information to accelerate the system identification speed. Decomposing 4 layers, the output data length is 28,27,26,25。
Principle analysis: conventional integral operation matrices unfold signals from a single scale by fixing the wavelet order M and the scale coefficient k. The accuracy of parameter estimation depends on the values of M and k, that is, the larger the values of k and M are, the higher the system identification accuracy is, but the larger M is, the slower the matrix generation speed is, and further the system identification speed is reduced, and because the k value is fixed, all information of the signal is projected in the same scale space, so that effective signals and noise cannot be distinguished. The multi-resolution analysis method can select the wavelet with a lower order, reduce the space scale by continuously increasing the k value and improve the parameter estimation precision. According to the wavelet multi-resolution characteristic, high-frequency information can be projected in a small-scale space, noise signals usually have high-frequency characteristics, the high-frequency coefficients are abandoned and set to be 0, the effect of reducing noise pollution can be achieved, and the identification speed is further increased.
4.2 reconstruction of the output Signal Using the residual data
Dividing the two ends of the formula (1) by the highest order of the differential to obtain an integral operator
Figure BDA0002533927850000064
Substituting the Legendre wavelet integral operation matrix and the input and output operation matrices into a formula (2) to reconstruct a model output matrix
Figure BDA0002533927850000065
Figure BDA0002533927850000066
Step five: solving system parameters and orders by adopting a least square method and taking an IAE index as a target function in a cyclic fractional order integral mode
5.1 substituting the input and output signals and Legendre wavelet integral operation matrix into formula (2), and rewriting the matrix form
Figure BDA0002533927850000071
Wherein Y isTIn the form of a Legendre wavelet matrix for the output signal FTIn the form of a Legendre wavelet matrix of the input signal.
5.2 order
Figure BDA0002533927850000072
X＝[a2；a1；a0]，
Figure BDA0002533927850000073
Then formula (23) can be rewritten as
AX＝B (5)
5.3 Using the least-squares method, the coefficient matrix X can be found, i.e.
X＝(ATA)-1ATB (6)
5.4IAE objective function of
Figure BDA0002533927850000074
The variation range and the step length of the fractional order differential order are β2＝(1.5:0.1:2.5)，β1＝(0:0.1:1.5)。
And 5.5, in the defined interval, circulating the fractional order integral order to find the corresponding system parameter when the IAE index is minimum, so that the model parameter can be accurately estimated. Under different levels, the system identification results are as follows:
the time taken for identification was 47.741634 seconds when layer 1 was decomposed. IAE index 0.000173097918930, the identification result is
1.0000y(t)+2.0000D2.2y(t)+3.0001D1.2y(t)＝f(t) (22)
The time taken for identification was 38.251226 seconds when decomposing layer 2. IAE index 0.000758400602904, the identification result is
0.9998y(t)+2.0001D2.2y(t)+3.0002D1.2y(t)＝f(t) (23)
The time taken for identification was 31.845872 seconds when the 3 layers were decomposed. IAE index 0.003121937768676, the identification result is
0.9990y(t)+2.0019D2.2y(t)+3.0031D1.2y(t)＝f(t) (24)
The time taken for identification was 25.438197 seconds when the 4 layers were decomposed. IAE index 0.012638068112973, the identification result is
0.9958y(t)+2.0075D2.2y(t)+3.0124D1.2y(t)＝f(t) (25)
As can be clearly seen from fig. 2, the system identification speed can be effectively increased by using the method and the system identification device.
Example 2:
consider the following fractional order system
Figure BDA0002533927850000081
β therein2＝2.2，β1＝1.3，a2＝1.5，a1＝1.4，a01, the system input f (t) is a step signal,
the same method as in example 1 was used, except that the range and the step size of the fractional order differential order was β2＝(1.5:0.1:2.5)，β1White gaussian noise with signal to noise ratios of 20dB,30dB and 40dB, respectively, is added to the output signal (0:0.1: 1.5). The number of decomposition layers and performance indicators for different snr are shown in fig. 3. As can be seen from fig. 3, the IAE index is decreasing with the increasing number of decomposition layers, which indicates that the multiresolution analysis method has a good effect on suppressing noise.

Claims (1)

1. A fractional order system identification method based on Legendre wavelet multi-resolution analysis is characterized by comprising the following steps:
the method comprises the following steps: selecting Legendre wavelet order in the interval of [0,1 ];
1.1 constructing a Legendre polynomial into wavelet form;
1.2 selecting Legendre wavelets with lower orders and the size of an initial scale space;
step two: deducing an operation matrix of Legendre wavelet integral by using a block pulse function integral matrix;
step three: expanding input and output signals of the system in a Legendre wavelet form;
step four: output signals are continuously decomposed by utilizing wavelet multi-fraction analysis characteristics, high-frequency information is abandoned, and the data length is reduced;
4.1 output data 2nMultiple rejection, n is 1,2,3, …
Decomposing the scale space layer by layer to obtain a set of nested subspace sets
Figure FDA0002533927840000011
Wherein Wn-1Is VnIs a low resolution space VnMissing details, symbols of
Figure FDA0002533927840000012
Defined as the orthogonal sum;
4.2 reconstruction of the output Signal Using the residual data
Dividing two ends of the fractional order differential equation by the highest order of the differential to obtain an integral operator
Figure FDA0002533927840000013
Where f (t) and y (t) are the input and output of the system, β respectivelyn,βn-1,…,β0Is a fractional order of differential order, an,an-1,…,a0As a result of the parameters of the system,
Figure FDA0002533927840000014
is a differential operator;
substituting Legendre wavelet integral operation matrix, input and output operation matrix intoIn equation (2), the model output matrix is reconstructed
Figure FDA0002533927840000015
Figure FDA0002533927840000016
Step five: solving system parameters and orders by adopting a least square method and taking an IAE index as a target function in a cyclic fractional order integral mode
And 5.1, substituting the input and output signals and a Legendre wavelet integral operation matrix into the formula (2), and rewriting the input and output signals into a matrix form.
Figure FDA0002533927840000017
Wherein Y isTIn the form of a Legendre wavelet matrix for the output signal FTIn the form of a Legendre wavelet matrix of the input signal;
5.2 order
Figure FDA0002533927840000021
X＝[an；an-1；…；a0]，
Figure FDA0002533927840000022
Equation (4) can be rewritten as
AX＝B (5)
5.3 Using the least-squares method, the coefficient matrix X can be found, i.e.
X＝(ATA)-1ATB (6)
5.4 determining an IAE objective function, a fractional order integral coefficient range and walking;
and 5.5, in the defined interval, circulating the fractional order integral order to find the corresponding system parameter when the IAE index is minimum, so that the model parameter can be accurately estimated.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113888721A (en) * 2021-10-08 2022-01-04 西北工业大学 Three-dimensional image feature design method based on integer-fractional order orthogonal polynomial

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113888721A (en) * 2021-10-08 2022-01-04 西北工业大学 Three-dimensional image feature design method based on integer-fractional order orthogonal polynomial
CN113888721B (en) * 2021-10-08 2022-09-16 西北工业大学 Three-dimensional image feature design method based on integer-fractional order orthogonal polynomial

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